8 THE EMOD YN A M 1C S FOR ENGINEERS CAMBRIDGE UNIVERSITY PRESS C. F. CLAY, MANAGER LONDON : FETTEKLANE, E.G. 4 NEW YORK : G. P. PUTNAM'S SONS BOMBAY } CALCUTTA I MACMILLAN AND CO., LTD. MADRAS J TORONTO : J. M. DENT AND SONS, LTD. TOKYO : MARUZEN-KABUSHIKI-KAISHA ALL BIGHTS RESERVED THE RMODYN AMIC S FOR ENGINEERS BY J. A. EWING, K.C.B., M.A., LL.D., D.Sc., F.R.S., M.lNST.C.E., M.INST.MEOH.E. Principal and Vice- Chancellor of the University of Edinburgh ; Honorary Fellow of King's College, Cambridge ; Formerly Professor of Mechanism and Applied Mechanics in the University of Cambridge ; Sometime Director of Naval Education *** I ** CAMBRIDGE AT THE UNIVERSITY PRESS 1920 PREFACE ALTHOUGH written primarily for engineers, it is hoped that this book may be of service to students of physics and others who wish to acquire a working knowledge of elementary thermo- dynamics from the physical standpoint. In presenting the fundamental notions of thermodynamics, the writer has adopted a method which his experience as a teacher encourages him to think useful. The notions are first introduced in a non-mathematical form ; the reader is made familiar with them as physical realities, and learns to apply them to practical problems; then, and not till then, he studies the mathematical relations between them. This method appears to have two advantages: it prevents the non-mathematical student from becoming bewildered on the threshold, and it saves the mathematical student from any risk of failing to realize the meaning of the symbols with which he plays. When the non-mathematical student comes to face the mathematical relations, which he must do if he is to pass beyond the rudiments of the subject, he finds it comparatively easy to build on the foundation of physical concepts he has already laid: there is perhaps no better way to learn the meaning and use of partial differential coefficients than by applying them to thermo- dynamic ideas, once these ideas are clearly apprehended. Accordingly the plan of the book is to begin with the elementary notions and their interpretation in practice, and to defer the study of general thermodynamic relations till near the end. Finally these relations are illustrated by applying them to characteristic equations of fluids, and in particular to steam, following Callendar's method. The chapter on Internal Combustion Engines gives occasion for introducing some results of experiments on the internal energy and specific heats of gases, and this matter is dealt with further in an appendix which attempts an elementary account of the molecular theory. In any exposition of the first principles of thermodynamics it is important to choose a way of dealing with temperature such that students may be led by simple and logical steps to understand the thermodynamic scale. The course followed 434763 vi PREFACE here is first to imagine an ideal gas which serves as thermometric substance, and also as the working substance in a Carnot engine. This gives a perfect-gas scale by reference to which the efficiency of any Carnot cycle is provisionally expressed, and from that the step to the thermodynamic scale is easy. The writer is indebted to Professor Callendar and his publisher, Mr Edward Arnold, for permission to include a much abbreviated version of his Steam Tables. By the recent publication of complete Tables, Professor Callendar has added substantially to the many obligations under which he has put all students of thermodynamics. The writer would also thank Mr J. B. Peace, of the Cambridge University Press, for various suggestions and for the interest he has taken in bringing out the book; and also Dr E. M. Horsburgh, of the Mathematical Department of this University, for his great kindness in reading the proofs. THE UNIVERSITY, EDINBURGH. March 1920. CONTENTS CHAPTER I FIRST PRINCIPLES ART. PAGE 1. The Science of Thermodynamics ...... 1 2. Heat-Engine and Heat-Pump 1 3. Efficiency of a Heat-Engine . 2 4. Coefficient of Performance of a Refrigerating Machine . . 2 5. Working Substance 2 6. Operation of the Working Substance in a Heat-Engine . . 3 7. Cycle of Operations of the Working Substance ... 4 8. The First Law of Thermodynamics . . . . . 5 9. Internal Energy . 5 10. Work done in Changes of Volume of a Fluid .... 6 11. Indicator Diagrams 7 12. Units of Force, Pressure, and Work 8 13. Units of Heat 9 14. Mechanical Equivalent of Heat 10 15. Scales of Temperature 10 16. Reckoning of Temperature from the "Absolute Zero'* . . 12 17. Properties of Gases: Charles' Law and Boyle's Law ... 13 18. Notion of a "Perfect" Gas . . . . . , . 14 19. Internal Energy of a Gas: Joule's Law 15 20. Specific Heats of a Gas 17 21. Constancy of the Specific Heats in a Perfect Gas ... 19 22. Reversible Actions 20 23. Adiabatic Expansion . 21 24. Isothermal Expansion 22 25. Adiabatic Expansion of a Perfect Gas 22 26. Change of Temperature in the Adiabatic Expansion of a Perfect Gas 23 27. Work done in the Adiabatic Expansion of a Perfect Gas . . 24 28. Isothermal Expansion of a Perfect Gas 24 29. Summary of results for a Perfect Gas 25 30. Fundamental Questions of Heat-Engine Efficiency ... 26 31. The Second Law of Thermodynamics 26 32. Reversible Heat-Engine. Carnot's Cycle of Operations . . 27 33. Carnot's Principle 28 34. Reversibility the Criterion of Perfection in a Heat-Engine . 29 35. Efficiency of a Reversible Heat-Engine 29 36. Carnot's Cycle with a Perfect Gas for Working Substance . 30 37. Reversal of this Cycle 32 38. Efficiency of Any Reversible Engine 34 39. Summary of the Argument 34 viii CONTENTS ART. 40. Absolute Zero of Temperature 35 41. Conditions of Maximum Efficiency 36 42. Thermodynamic Scale of Temperature 39 43. Reversible Engine receiving Heat at Various Temperatures . 42 44. Entropy 44 45. Conservation of Entropy in Carnot's Cycle . .45 46. Entropy-Temperature Diagram for Carnot's Cycle ... 46 47. Entropy-Temperature Diagrams for a series of Reversible Engines 47 48. No change of Entropy in Adiabatic Processes .... 48 49. Change of Entropy in an Irreversible Operation . . 48 50. Sum of the Entropies in a System ... .49 51. Entropy-Temperature Diagrams .... .50 52. Perfect Engine using Regenerator 52 53. Stirling's Regenerative Air-Engine ...... 53 54. Joule's Air-Engine . . . 55 CHAPTER II PROPERTIES OF FLUIDS 55. States of Aggregation 59 56. Formation of Steam under Constant Pressure . . . 60 57. Saturated and Superheated Steam . . . . . . 61 58. Relation of Pressure to Temperature in Saturated Steam . 62 59. Tables of the Properties of Steam 62 60. Relation of Pressure to Volume in Saturated Steam . . 64 61. Boiling and Evaporation 65 62. Mixture of Vapour with other Gases: Dalton's Principle . . 65 63. Evaporation into a space containing Air: Saturation of the Atmosphere with Water- Vapour ..... 66 64. Heat required for the Formation of Steam under Constant Pressure: Heat of the Liquid and Latent Heat ... 67 65. Total External Work done . 69 66. Internal Energy of a Fluid 69 67. The "Total Heat" of a Fluid 70 68. Change of the Total Heat during Heating under Constant Pressure . . . . . . . . . . 71 69. Application to Steam formed under Constant Pressure, from Water at C 71 70. Total Heat of a mixture of Liquid and its Saturated Vapour . 73 71. Total Heat of Superheated Vapour 73 72. Constancy of the Total Heat in a Throttling Process . . 74 73. Entropy of a Fluid 75 74. Mixed Liquid and Vapour: Wet Steam 76 75. Specification of the State of any Fluid ..... 77 76. Isothermal Expansion of a Fluid: Isothermal Lines on the Pressure- Volume Diagram 78 77. The Critical Point: Critical Temperature and Critical Pressure . 80 CONTENTS ix ART. PAGE -78. Adiabatic Expansion of a Fluid 81 79. Supersaturation 84 80. Change of Internal Energy and of Total Heat in Adiabatic Expansion. "Heat-Drop" 86 CHAPTER III THEORY OF THE STEAM-ENGINE 81. Carnot's Cycle with Steam or other Vapour for Working Substance . 88 82. Efficiency of a Perfect Steam-Engine. Limits of Temperature 90 83. Entropy-Tmperature Diagram for a Perfect Steam-Engine . 91 84. Use of "Boundary Curves" in the Entropy-Temperature Diagram .......... 92 85. Modified Cycle omitting Adiabatic Compression . . . 94 86. Engine with Separate Organs . . . . . . 96 87. The Rankine Cycle 98 88. Efficiency of a Rankine Cycle . . . . . . . 99 89. Calculation of the Heat-Drop 100 90. The Function G 102 91. Extension of the Rankine Cycle to Steam supplied in any State 104 92. Rankine Cycle with Steam initially Wet .... 104 93. Rankine Cycle with Steam initially Superheated ... 106 94. Reversibility of the Rankine Cycle ..... 109 95. Conditions of High Efficiency 110 96. Effect of Incomplete Expansion 112 97. Ideal Engine working with No Expansion . . . .114 98. Clapeyron's Equation .115 99. Application of Clapeyron's Equation to other Changes in Physical State 116 100. Entropy-Temperature Chart of the Properties of Steam . . 118 101. Holder's Chart of Entropy and Total Heat . . . . 121 102. Other Forms of Chart . . 125 103. Effects of Throttling . . 126 104. The Heat- Account in a Real Process . . . . .129 CHAPTER IV THEORY OF REFRIGERATION 105. The Refrigeration Process 133 106. Reversible Refrigerating Machine 134 107. Conservation of Entropy in a Perfect Refrigerating Process . 135 108. Ideal Coefficients of Performance 136 109. The Working Fluid in a Refrigerating Process . . .137 110. The Actual Cycle of a Vapour-Compression Refrigerating Machine .......... 138 111. Entropy-Temperature Diagram for the Vapour-Compression Cycle 141 x CONTENTS AET. PAGE 112. Refrigerating Effect and Work of Compression expressed in Terms of the Total Heat 144 113. Charts of Total Heat and Entropy for Substances used in the Vapour- Compression Process 145 114. Applications of the /< Chart in studying the Vapour- Compres- sion Process ......... 149 115. Vapour- Compression by means of a Jet. Water- Vapour Machine 155 1 16. The Step-down in Temperature. Use of an Expansion Cylinder in Machines using Air 157 117. Air-Machines. Joule's Air-Engine reversed . . . .158 118. Direct Application of Heat to produce Cold. Absorption Machines 160 119. Limit of Efficiency in the Use of High-temperature Heat to Produce Cold 164 120. Expression in Terms of the Entropy 166 121. The Refrigerating Machine as a means of Warming . . 168 122. The Attainment of Very Low Temperature. Cascade Method 169 123. Regenerative Method . 171 124. First Stage ........... 172 125. Second Stage 175 126. Linde's Apparatus 176 127. Liquefaction of Air by Expansion in which Work is done. Claude's Apparatus . . . . . . .178 128. Separation of the Constituents of Air 180 129. Baly's Curves 185 130. Complete Rectification . . . . . . .188 CHAPTER V JETS AND TURBINES 131. Theory of Jets 191 132. Form of the Jet (De Laval's Nozzle) 193 133. Limitation of the Discharge through an Orifice of Given Size 197 134. Application to Air 198 135. Application to Steam 199 136. Comparison of Metastable Expansion with Equilibrium Ex- pansion 203 137. Measure of Supersaturation 206 138. Retarded Condensation 207 139. Action of Steam in a Nozzle, continued 208 140. Effects of Friction 209 141. Application to Turbines 214 142. Simple Turbines 215 143. Compound Turbines 216 144. Theoretical Efficiency-Ratio . . . . . . .216 145. Action in Successive Stages 217 146. Stage Efficiency and Reheat Factor 218 CONTENTS xi ABT. PAGE 147. Real Efficiency-Ratio 219 148. Types of Turbines . .220 149. Performance of a Steam Turbine . . . . . 222 150. Utilization of Low Pressure Steam 223 CHAPTER VI INTERNAL-COMBUSTION ENGINES 151. Internal Combustion 225 152. The Four-Stroke Cycle 226 153. The Clerk or Two-Stroke Cycle 226 154. Ideal Action 227 155. Air Standard 229 156. Constant-Pressure Type 232 157. Diesel Engine . 234 158. Combustion of Gases. Molecular Weights and Volumes . . 235 159. The Gramme-Molecule or Mol 237 160. The Universal Gas-Constant 238 161. Specific Heats of Gases in Relation to their Molecular Weights. Volumetric Specific Heats ...... 239 162. Summary of Methods of expressing the Specific Heats . . 241 163. Measured Values of Specific Heats 241 164. Variation of Specific Heat with Temperature .... 243 165. Internal Energy of a Gas 245 166. Adiabatic Expansion of a Gas with Variable Specific Heat . 247 167. Ideal Efficiency as affected by the Variation of the Specific Heat with Temperature 249 168. Curve of Internal Energy for Typical Gas-Engine Mixture . 251 169. Action in a Real Engine. Analysis of the Indicator Diagram . 254 170. Measurement of Suction Temperature 257 171. The Process of Explosion 257 172. Effect of Turbulence 259 173. Radiation in Explosions 260 174. Molecular Energy of a Gas 261 175. Dissociation 265 CHAPTER VII GENERAL THERMODYNAMIC RELATIONS .176. Introduction 266 177. Functions of the State of a Fluid 266 178. Relation of any one Function of the State to two others . . 267 179. Energy Equations and Relations deduced from them . . 270 180. Expressions for the Specific Heats K v and K p . . . . 272 181. Further deductions from the Equations for E and / . . 275 182. The Joule-Thomson Effect 276 183. Unresisted Expansion 279 xii CONTENTS AET. PAGE 184. Slopes of Lines in the /<, T0, and IP charts, for any Fluid . 280 185. Application to a Mixture of Liquid and Vapour in Equilibrium : Clapeyron's Equation. Change of Phase .... 283 186. Compressibility and Elasticity of a Fluid .... 286 187. Collected Results . 286 CHAPTER VIII APPLICATIONS TO PARTICULAR FLUIDS 188. Characteristic Equation 290 189. Characteristic Equation of a Perfect Gas .... 290 190. Isothermal and Adiabatic Expansion of Ideal Gas . . . 292 191. Entropy, Energy, and Total Heat of Ideal Gas . . .293 192. Ratio of Specific Heats. Method of inferring y in Gases from the Observed Velocity of Sound . . . . .293 193. Measurement of y by Adiabatic Expansion. Method of Cle- ment and Desormes 294 194. Effect of Imperfection of the Gas on the Ratio of Specific Heats 295 195. Relation of the Cooling Effects to the Coefficients of Expansion 296 196. Forms of Isothermals. Diagrams of P and V, and of PV and P 298 197. Imperfect Gases. Amagat's Isothermals of PV and P . . 299 198. Isothermals on the Pressure- Volume Diagram . . . 303 199. Continuity of Liquid and Gas 304 200. Van der Waals' Characteristic Equation 306 201. Critical Point according to Van der Waals' Equation . . 309 202. Corresponding States 311 203. Van der Waals' Equation only Approximate . . / . . 313 204. Other Characteristic Equations: Clausius, Dieterici . . 315 205. Calendar's Equation 318 206. Deductions from the Callendar Equation . . . .321 207. The Specific Heats in Calendar's Equation .... 324 208. The Entropy, Energy, and Total Heat, in Calendar's Equation 325 209. Application to Steam 327 210. Total Heat and Entropy of Water . . . . . .334 211. Relation of Pressure to Temperature in Saturated Steam . 336 212. Formulas for the Latent Heat of Steam, and for the Volume of a Wet Mixture 338 213. Collected Formulas for Steam 338 214. Tables of the Properties of Steam 340 APPENDIX I EFFECTS OF SURFACE TENSION ON CONDENSATION AND EBULLITION 215. Nature of Surface Tension . . . . . . . 342 216. Need of a Nucleus 344 217. Kelvin's Principle 345 218. Ebullition .... 349 CONTENTS xiii APPENDIX II MOLECULAR THEORY OF GASES ART. PAGE 219. Pressure due to Molecular Impacts . . . . .351 220. Boyle's, Avogadro's, and Dalton's Laws .... 355 221. Perfect and Imperfect Gases 356 222. Calculation of the Velocity of Mean Square .... 356 223. Internal Energy and Specific Heat 357 224. Energy of Vibration 361 225. Planck's Formula 362 226. Effect of Extreme Cold on the Diatomic Molecules of Hydrogen 366 APPENDIX III TABLES OF THE PROPERTIES OF STEAM TABLE A. Properties of Saturated Steam, in relation to the Temperature 368 A*. Properties of Water at Saturation Pressure .... 369 B. Properties of Saturated Steam, in relation to the Pressure . 370 C. Volume of Steam in any Dry State 372 D. Total Heat of Steam in any Dry State 374 E. Entropy of Steam in any Dry State 376 F. Specific Heat, at constant pressure, of Steam in any Dry State 378 INDEX 379 CHAPTER I FIRST PRINCIPLES i. The Science of Thermodynamics treats of the relation of heat to mechanical work. In its engineering aspect it is chiefly con- ERRATA Page.90, lines 5 and 7, for "L" read "Lj". Page 123, line I, for "but at" read "but near". Page 123, delete last two sentences of footnote and substitute: " Hence also, at ywyx j f~ ^WY' / X \ r / * chart the constant-pressure line through the critical point runs level and suffers inflection there. Thus on the I chart the constant-pressure line through the critical point has zero curvature there, though it does not suffer inflection." Page 140, line 5, delete "in the same figure." Page 149, line 1,/or "coincides with " read "is a little above ". Page 328, second last line, for "1-982" read "1-984". Page 363, line 6, for "ergs" read "c.g.s." some of the heat disappears in the process of being let down : it is converted into the work which the engine does. In a Refrigerating Machine work has to be spent upon the machine to enable it to take in heat at a low level of temperature, and discharge heat at a higher level of temperature, just as work would have to be spent upon a water-wheel if it were used as a means of raising water by reversing its action, in such a way that the buckets were filled at a low level and emptied at a higher level, so that it should serve as a pump. It would be quite correct to speak of a refrigerating machine as a heat-pump. But again there is an important difference between the refrigerating machine and E.T. 1 CHAPTER I FIRST PRINCIPLES 1. The Science of Thermodynamics treats of the relation of heat to mechanical work. In its engineering aspect it is chiefly con- cerned with the process of getting work done through the agency of heat. Any machine for doing this is called a Heat-Engine. It is also concerned with the process of removing heat from bodies that are already colder than their surroundings. Any machine for doing this is called a Refrigerating Machine. It is convenient to study the thermodynamic action of heat- engines and refrigerating machines together, because one is the reverse of the other, and by considering both we arrive more easily at an understanding of the whole subject. 2. Heat-Engine and Heat-Pump. In a Heat-Engine heat is supplied, generally by the combustion of fuel, at a high tempera- ture, and the engine discharges heat at a lower temperature. Thus in a steam-engine heat is taken in at the temperature of the boiler and discharged at the temperature of the condenser. In any kind of heat-engine the heat is let down, within the engine, from a high level of temperature to a lower level of temperature, and it is by so letting heat down that the engine is able to do work, as a water-wheel is able to do work by letting water down from a high level to a lower level. But there is this important difference, that some of the heat disappears in the process of being let down : it is converted into the work which the engine does. In a Refrigerating Machine work has to be spent upon the machine to enable it to take in heat at a low level of temperature, and discharge heat at a higher level of temperature, just as work would have to be spent upon a water-wheel if it were used as a means of raising water by reversing its action, in such a way that the buckets were filled at a low level and emptied at a higher level, so that it should serve as a pump. It would be quite correct to speak of a refrigerating machine as a heat-pump. But again there is an important difference between the refrigerating machine and E.T. l : : : OTERMOD YNAMICS [CH. the reversed water-wheel : the refrigerating machine is a heat-pump which discharges more heat than it takes in, for the work which is spent in driving the machine is converted into heat, which has to be discharged at the higher level of temperature in addition to the heat that is taken in at the low temperature. 3. Efficiency of a Heat-Engine. From the point of view of practical thermodynamics the object of a heat-engine is to get work done with the least possible expenditure of fuel. In other words the ratio of the work done to the heat taken in should be as large as is practicable. This ratio is called the Efficiency of the engine as a heat-engine. The theory of heat-engines deals with the conditions that affect efficiency, and with the limit of efficiency that can be reached when the conditions are most favourable. 4. Coefficient of Performance of a Refrigerating Machine. In a refrigerating machine the object is to get heat removed from the cold body and pumped up to a higher level of temperature at which it can be discharged, and what is wanted is that this should be done with the least possible expenditure of work. The ratio of the heat taken in by the machine from the cold body to the work that is spent in driving the machine is called the Coefficient of Perform- ance. The theory of refrigeration deals with the conditions that will allow this ratio to be as large as possible. 5. Working Substance. In the action of a heat-engine or of a refrigerating machine there is always a working substance which forms the vehicle by which heat passes through the machine. It is because the working substance has a capacity for taking in heat that it can act as a vehicle for conveying heat from one level of temperature to another. In this process its volume changes, and it is by means of changes of volume on the part of the working sub- stance that the machine does work, if it is a heat-engine, or has work spent upon it, if it is a refrigerating machine. Accordingly, an important part of the science of thermodynamics deals with the properties of substances in relation to heat, and the connection between such properties in any substance. The substances with which we are chiefly concerned are fluids in the gaseous or liquid states. They include air and other gases, water and water- vapour, and also some fluids more easily vaporized than water, such as ammonia and carbonic acid, which are used as the working sub- stance in certain refrigerating machines. Each fluid has of course i] FIRST PRINCIPLES 3 its own characteristics ; but many of the relations between its pro- perties are of a general kind and may be studied without limitation to individual fluids. It will be seen, as we go on, that much of what has to be said applies equally, whatever fluid serves for working substance, and that in any one fluid the various properties are connected with one another in a way that is true for all fluids. Th e study of the thermodynamic relationships between the various properties of a fluid is useful, not only because of the direct light it throws on the action of heat-engines, but also because it enables a practically complete knowledge of the properties of a fluid in detail to be inferred from a comparatively small number of experi- mental data. We shall see later, for example, how such relation- ships have been made use of in calculating modern tables of the properties of steam from the results of careful measurements, made in the laboratory, of a very few fundamental quantities. 6. Operation of the Working Substance in a Heat-Engine. In general the working substance is a fluid which operates by chang- ing its volume, exerting pressure as it does so. But it is easy to imagine a heat-engine having a solid body for working substance, say a long rod of metal arranged to act as the pawl of a ratchet- wheel with closely pitched teeth. Let the rod be heated so that it lengthens sufficiently to drive the wheel forward through the space of one tooth. Then let the rod be cooled, say by applying cold water, the ratchet-wheel being meanwhile held from returning by a separate click or detent. The rod on cooling will retract so as to engage itself with the next succeeding tooth, which may then be driven forward by heating the rod again, and so on. To make it evident that such an engine would do work we have only to suppose that the ratchet-wheel carries round with it a drum by which a weight is wound up. The device forms a complete heat-engine, in which the working substance is a solid rod, doing work in this case not through changes of volume but through changes of length. While its length is increasing it is exerting force in the direction of its length. It receives heat by being brought into contact with some source of heat at a comparatively high temperature; it trans- forms a small part of this heat into work; and it rejects the re- mainder to what we may call a receiver of heat, which is kept at a comparatively low temperature. The greater part of the heat may be said simply to pass through the engine, from the source to the receiver, becoming degraded as regards temperature in the process. 12 4 THERMODYNAMICS [CH. This is typical of the action of all heat-engines : they convert some heat into work only by letting down a much larger quantity of heat from a high temperature to a relatively low temperature. The engine we have just imagined would not be at all efficient; the fraction of the heat supplied to it which it could convert into work would be very small. Much greater efficiency can be obtained by using a fluid for working substance and by making it act so that its own expansion of volume not only does work but also causes it to fall in temperature before it begins to reject heat to the cold receiver. 7. Cycle of Operations of the Working Substance. Generally in the action of a heat-engine or of a refrigerating machine the working substance returns periodically to the same state of tem- perature, pressure, volume and physical condition in all respects. Each time this has occurred the substance is said to have passed through a complete cycle of operations. For example, in a con- densing steam-engine, water taken from the hot-well is pumped into the boiler; it then passes into the cylinder as steam, then from the cylinder into the condenser, and finally from the condenser back to the hot- well; it completes the cycle by returning to the same condition in all respects as at first, and is ready to go through the cycle again. In other less obvious cases a little consideration shows that the cycle is completed although the same portion of working substance does not go through it again: thus in a non- condensing steam-engine the steam which has passed through the engine is discharged into the atmosphere, where it cools to the tem- perature of the feed- water, while a fresh portion of feed-water is delivered to the engine to go through the cycle in its turn. In the theory of heat-engines it is of the first importance to con- sider as a whole the cycle of operations performed by the working substance. If we stop short of the completion of the cycle matters are complicated by the fact that the substance is in a state different from its initial state. On the other hand, if the cycle is complete we know that whatever heat or other energy the substance contained within itself to begin with is there still, for the state of the substance is the same in all respects, and consequently any work that it has done must have been done at the expense of heat which it has taken in during the cycle. The total amount of energy it has parted with must be equal to the amount it has received, during the cycle, for its stock of internal energy is the same at the end as at the i] FIRST PRINCIPLES 5 beginning. We can at once apply the principle of the Conservation of Energy and say that for the cyclic process as a whole this equa- tion must hold good, Heat taken in = Heat rejected + Work done by the substance. And similarly, when the working substance in a refrigerating machine has been carried through a complete cycle of operations, the equation holds for the cycle as a whole, Heat taken in = Heat rejected Work spent upon the substance. 8. The First Law of Thermodynamics. The principle of the Conservation of Energy in relation to heat and work may be ex- pressed in the following statement, which constitutes the First Law of Thermodynamics : When mechanical energy is produced from heat a definite quantity of heat goes out of existence for every unit of work done; and, conversely, when heat is produced by the expendi- ture of mechanical energy the same definite quantity of heat comes into existence for every unit of work spent. 9. Internal Energy. We have used in Art. 7 a phrase which requires some further explanation the internal energy of a sub- stance. No means exist by which the whole stock of energy that a substance contains can be measured. But we are concerned only with changes in that stock, changes which may arise from the sub- stance taking in or giving out heat, or doing work, or having work spent upon it. If a substance takes in heat without doing work its stock of internal energy increases by an amount equal to the heat taken in. If it does work without taking in heat, it does the work at the expense of its stock of internal energy, and the stock is diminished by an amount equal to the work done. In general, when heat is being taken in and the substance is at the same time doing work, we have Heat taken in = Work done + Increase of Internal Energy. For any infinitesimalJy small step in the process, we may write dQ = dW + dE, where dQ is the heat taken in during the step, dW is the work done, and dE the increase of internal energy. In a complete cycle there is, at the end, no change of the internal energy E, and consequently for the cycle as a whole, i - 2 - W, where Qj - Q 2 is the net amount of heat received, namely the THERMODYNAMICS [CH. difference between the heat taken in and the heat rejected in the complete cycle, and W is the work done in the complete cycle. In this notation we are supposing W to be expressed in units of heat, as well as Q and E. It would be more correct to speak of W as the thermal equivalent of the work done. 10. Work done in Changes of Volume of a Fluid. In an engine of the usual cylinder and piston type the working fluid does work by changes of volume. The amount of work done de- pends only on the relation of the pressure to the volume in these changes, and not on the form of the vessel or vessels in which the changes of volume take place. Let the intensity of pressure of the fluid (that is to say the pressure on unit of area) be P while the piston moves forward through a small distance 8/. If the area of the piston is S the total force on it is PS and the work done is PS$l. But S81 SF, the change of volume: hence the work done is PSF rV for the small change of volume SF, or PdV for a finite change Jr t of volume from a volume V l to a volume F 2 during which the pressure may vary. In any complete cycle of operations the volume at the finish is the same as at the start, and the work done is \PdV taken round the cycle as a whole. It is very useful to represent graphically the work which a fluid does in changing its volume. Let a diagram be drawn in which the relation of the pres- sure of any supposed working substance to its volume is shown by rectangular coordinates as in fig. 1. Beginning with the state represented by the point A, where the pressure is AM and volume OM, suppose the substance to expand to a state B, where the pressure is BN and the volume | _ |M ON, and let the curve AB repre- Volume sent the intermediate states of Fl S- l pressure and volume. Then the work done by the substance in this rON expansion, which is PdV, is represented by the area MABN . ] under the curve AB. OM I] FIRST PRINCIPLES Again, if the substance undergoes any complete cycle of change (fig. 2) by expanding from A through B to C and by being compressed back through D to A, work is done by it while it is expanding from A to C, equal to the area MABCN, and work is spent upon it while it is being compressed from Volume C through DtoA, equal Fi S- 2 to the area NCDAM. The net amount of work which the substance does during the cycle is equal to the algebraic sum of those areas: in other words it is equal to the area of the closed figure ABCDA representing the complete cyclic operation, which area is | PdV. If on the other hand the operation were such as to trace the figure in the opposite direction, the substance being expanded from A to C through D and compressed from C to A through J?, the enclosed area would be a measure of the work expended upon the substance in the cycle. ii. Indicator Diagrams. This pressure-volume diagram is an example, and a generalization, of the method of representing work which Watt introduced by his invention of the Indicator, an instrument for automatically drawing a diagram to represent the changes of pressure in relation to changes of volume in the action of an engine. The figure A BCD A may be called the Indicator Diagram of the supposed action. The indicator consists of a small cylinder containing a piston which can move in it without sensible friction but is controlled by a stiff spring. This is put in free communication with one end of the working cylinder of the engine, so that the working substance presses on the indicator piston and displaces it, against the spring, through distances that are proportional to the pressure at every instant. Connected with the indicator piston is a pencil which rises or falls with it, the connection being made, generally, through a lever that gives the movements of the indicator piston a convenient magnification. A sheet of paper on which the pencil marks its 8 THERMODYNAMICS [CH. movements is caused to move through distances proportional to the motion of the engine piston, and at right angles to the path of the pencil. Thus a diagram is drawn like that of fig. 2, exhibiting a closed curve for each double stroke of the engine piston, and with coordinates which represent the changes of pressure and changes of volume. The enclosed area, when interpreted by reference to the appropriate scales of pressure and volume, measures the net amount of work done in the engine cylinder during the double stroke, so far as one side of the piston is concerned. If the engine is double-acting that is to say, if the working substance acts successively on the two sides of the engine piston during successive strokes a similar indicator diagram is taken for the other end of the cylinder as well. 12. Units of Force, Pressure, and Work. For engineering purposes, in speaking of pressure and of work, the common unit of force in British and American usage is the weight of 1 Ib. and in continental usage the weight of 1 kilogramme*. By the word "weight" we mean here the force with which the earth attracts these masses. When it is necessary to give scientific precision to either of these units of force one must specify a locality, or rather a latitude, because gravity acts rather more strongly as we go from the equator towards the pole. The same piece of material is more strongly attracted by the earth in London than in Paris, to the extent of one part in 5000, and more strongly in London than in New York to the extent of one part in 1000. If the weight of 1 Ib. of matter in mean latitude (45) be taken as unity, its weight in any other latitude A is 0-99735(1 + 0-0053 sin 2 A). The differences due to latitude are so small that for nearly all purposes they may be ignored. The usual units of pressure are the pound per square inch and the kilogramme per square centimetref. Another unit often used is the "Atmosphere," which means the pressure of the atmosphere with the barometer standing at 760 mm. in latitude 45, or 759-6 mm. in London. This is equal to a pressure in London of 14-689 pounds per square inch or 1-03274 kilogrammes per square centimetre. For scientific purposes the absolute (c.g.s.) unit of pressure, the * One kilogramme is 2-20462 Ibs. f Since 1 centimetre is 0-393702 inch, 1 kilogramme per sq. cm. is 14-223 pounds per sq. in., when both are measured at the same place, so that gravity acts alike on the pound and the kilogramme. i] FIRST PRINCIPLES 9 dyne per sq. cm., has the advantage that it is independent of gravity. One "Atmosphere" is equal to T0133 x 10 6 dynes per sq. cm., at any place. Pressures are also sometimes given in inches, or in millimetres, of mercury. One inch of mercury (at C.) is equivalent to 0-4912 pounds per square inch; one millimetre of mercury to 1*3596 grammes per sq. cm. jg,> The usual engineering units of work are the foot-pound and the metre-kilogramme or kilogrammetre. One kilogrammetre is 7-233 foot-pounds. 13. Units of Heat. For the purpose of reckoning quantities of heat we compare them with the quantity that is required to warm a unit mass of water from the temperature of melting ice to the temperature at which water boils under a pressure of one atmo- sphere. These two points serve to determine two fixed states of temperature that are quite definite and are independent of the particular way in which temperature may be measured. The unit of heat which is obtained by taking a certain fraction of this quantity of heat is described as the m,ean thermal unit. The mean thermal unit which will be used here is one-hundredth part of the heat required to warm one pound of water from the melting point to the boiling point at a pressure of one atmosphere. This unit is called the Pound-Calory. The reason why one-hundredth part is taken is that on the Centigrade scale of temperature the interval between these fixed points is divided into 100 degrees : consequently the pound-calory is the average amount of heat required to warm a pound of water through one degree Centigrade, between the melting point and the boiling point as limits. The actual amount required per degree need not be the same for each degree of the scale, and in fact is not the same, for the specific heat of water is not quite constant. Similarly, what is commonly called the British Thermal Unit (when the Fahrenheit scale is employed) would be defined as 1/180 of the quantity of heat required to warm 1 Ib. of water from the melting point to the boiling point, because on the Fahrenheit scale there are 180 degrees between the two fixed points. Again, the " Kilo-Calory" is one-hundredth of the amount of heat required to w r arm 1 kilogramme of water from the melting point to the boiling point, and the "gramme-calory" is one-thousandth of a kilo-calory 10 THERMODYNAMICS [CH. 14. Mechanical Equivalent of Heat. The experiments of Joule, begun in 1843 and continued for several years, demonstrated that when work is expended in producing heat a definite relation holds between the amount of heat produced and the amount of work spent. Causing the potential energy of a raised weight to be used up in turning a paddle which generated heat by stirring water in a vessel, and observing the rise of temperature so produced, Joule made the first determination of the number of units of work that are spent in producing a unit of heat. This number is called the mechanical equivalent of heat. Joule found that 772 foot-pounds were required to raise the temperature of one pound of water through one degree (Fahrenheit) on the thermometer he employed, at a particular part of the scale. Many later and more exact determinations were made by Joule himself and by other observers, using various methods of experi- ment. They agree in showing that Joule's original figure was rather low. The general result is to fix 1400 as the number of foot- pounds (in the latitude of London) that are equivalent to one Pound-Calory as defined in Art. 13. The corresponding value of the mechanical equivalent of the "British Thermal Unit" is 777-8 foot-pounds, and that of the Kilo-Calory is 426-7 kilogram- metres*. The mechanical equivalent of heat enters into many of the for- mulas of thermodynamics. It is often called Joule's Equivalent, and is generally represented by the symbol J. The symbol A is used for the reciprocal of Joule's equivalent, or 1/J. 15. Scales of Temperature. In the construction of an ordin- ary thermometer a fine tube of uniform bore is chosen, and a bulb is formed on it to contain the mercury or other liquid whose expan- sion is to be used as an indication of temperature. When it is filled the two fixed points are determined by placing the instrument (a) in melting ice, and (b) in the steam coming from water boiling under a pressure of one atmosphere. The position taken by the end of the column of liquid in the tube is marked for each of these two points. The distance between them is then divided into equal parts which are called degrees, 100 parts for the Centigrade scale and 180 for the Fahrenheit scale. By this construction equal steps in temperature are defined by equal amounts of expansion on the part of the * In absolute. (c.g.s.) units the gramme-calory will be taken in this book as equivalent to 4*1868 x 10 7 ergs, or cm-dynes. i] FIRST PRINCIPLES 11 selected liquid, or rather by equal amounts of difference between the expansion of the liquid itself and that of the glass in which it is contained, for it is the difference of expansion that determines the rise of the column in the tube. This common method of measuring temperature gives results that vary for different liquids and for different sorts of glass. Each of two mercury thermometers, for example, may have the fixed points correctly marked, and be of uniform bore, and yet if they are made of different sorts of glass they may give readings that differ by as much as half a degree (Centigrade) at the middle of the range between the fixed points, and may show very serious discrepancies sometimes amounting to as much as five degrees or more when they are applied to measure higher temperatures such as that of steam on its way to an engine. This illustrates the fact that the measurement of temperature by an ordinary thermometer gives an arbitrary scale, which cannot even be relied on to be the same in different instru- ments. Measurements of temperature are much less capricious if we select for the expanding substance any one of the so-called perman- ent gases, such as air, or nitrogen, or hydrogen, taking care of course to keep the pressure of the gas constant while it is employed to measure temperature by its changes of volume. Such an instru- ment is called a constant-pressure gas thermometer. It would be inconvenient for ordinary use ; but it serves to supply a scale with which the readings of an ordinary thermometer can be compared. Thus the readings of any mercury thermometer can be corrected to bring them into agreement with the scale of a gas thermometer if that scale be adopted as the standard scale in stating temperatures. Experiments on the expansion of various gases by heat have shown that all gases which are far from the conditions that would cause liquefaction expand very nearly alike. Thus if we compare an air thermometer with a nitrogen or a hydrogen thermometer we get practically the same scale except at extremely low temperatures such as those at which the gas is approaching the liquid state. Gases expand by almost exactly the same amount between the two fixed points, namely by 100/273 of the volume they have at the temperature of melting ice ; and at intermediate points, or at points beyond the range, their agreement with one another is almost perfect. Hence the scale of the gas thermometer is much to be preferred to that of any mercury thermometer as a means of stat- ing temperature. But there is another and even stronger reason for 12 THERMODYNAMICS [CH. this preference. We shall see later that it is possible to imagine a scale of temperature, based on general thermodynamic principles, which does not depend on the properties of any particular sub- stance : that scale is called the thermodynamic scale of temperature, and much use is made of it in thermodynamic reasoning. The scale of a gas thermometer is practically identical with the thermo- dynamic scale. Taking the hydrogen thermometer, in which the agreement is closest, Callendar has shown* that mid way between the fixed points the scale correction (that is, the difference between the numbers which state the same temperature on the hydrogen scale and the thermodynamic scale) is only 0-0013 of a degree, and that the temperature has to go up to about 1000 or down below 150 before the correction becomes as much as 0-1 of a degree. These figures are for hydrogen expanding under a constant pressure of one atmosphere. The differences between the scale of the gas thermometer and the thermodynamic scale are even less if a con- stant-volume type of gas thermometer be used, in which increments of temperature are measured by the increments of pressure that are required to keep the volume of the gas constant while it is heated. 1 6. Reckoning of Temperature from the "Absolute Zero." Experiment shows that the amount by which air or hydrogen or any other so-called " permanent " gas expands between the two fixed points that is to say in passing from the temperature of melting ice to that of boiling water (at a pressure of one atmo- sphere) is about 100/273 of the volume at the lower fixed point, care being taken that the pressure does not change. Hence if we adopt the scale of the gas thermometer as our scale of tempera- ture, and use Centigrade divisions, this result may be expressed by saying that when 273 cubic inches of gas at C. are heated under constant pressure to 1 the volume alters to 274 cubic inches. When the gas is heated to 2C. its volume becomes 275 cubic inches, and so on. Similarly if the gas be cooled from C. to 1 C. its volume changes from the original 273 cubic inches to 272, and so on. Putting this in a tabular form, let the volume be 273 at C. It will become 272 at - 1 C. and finally would be at - 273 C., * H. L. Callendar, " On the thermodynamical correction of the Gas Thermometer," Proc. Phys. Soc. vol. xvin, or Phil. Mag. January, 1903. i] FIRST PRINCIPLES 13 if the same law could be held to apply down to the lowest tempera- tures. Any actual gas would change its physical state before so low a temperature were reached, becoming first liquid and then solid, and the volume to which it would contract would consequently be not zero but the volume of the substance in the solid state. The above result may be concisely expressed by saying that if temperature be reckoned not from the ordinary zero but from a zero which is about 273 Centigrade degrees below it (more exactly 273-1), the volume of a gas, heated under constant pressure, is proportional to the temperature reckoned from that zero. The zero in question is spoken of as the Absolute Zero of temperature. Denoting any temperature on the ordinary scale by i and the corresponding temperature reckoned from the absolute zero by T, we have (using Centigrade degrees) T = / + 273-1. The absolute zero has been defined here by reference to the ex- pansion of a gas. But it will be seen later that the thermodynamic scale of temperature starts from a zero which is absolute in the sense that no lower temperature can possibly exist, and that the zero of the thermodynamic scale coincides with the zero of the gas scale as defined above*. 17. Properties of Gases : Charles' Law and Boyle's Law. The experimental fact that all "permanent" gases expand by very nearly the same fraction of their volume for a given increase of temperature, the pressure being kept constant, is known as Charles' Law. Another fundamental property of gases, discovered by the experiments of Boyle, is that when the volume of a gas is altered by altering the pressure, the temperature being kept constant, the volume varies inversely as the pressure. Thus if V be the volume of a given quantity of any gas, and P the pressure, then so long as the temperature remains unchanged, V varies inversely as P, or PV = constant. This is Boyle's Law. It is very nearly though not exactly true in gases such as air or oxygen or nitrogen or hydrogen: the deviations from it are very slight in any gas that is in conditions far removed from those which produce liquefaction. * The exact position of the absolute zero is uncertain to the extent of about one-tenth of a degree. Callendar places it at - 273-1 C. That figure is used in his determinations of the properties of steam, and is adopted in this book. 14 THERMODYNAMICS [CH. 18. Notion of a " Perfect'* Gas. In dealing with the pro- perties of gases and with the thermodynamics of heat-engines it is convenient to imagine a gas which exactly conforms to laws that are only very nearly true of real gases. Such a gas is called a "perfect" gas. The properties of real gases are most easily treated as small deviations from those of imaginary "perfect" gases obey- ing simple laws. Among real gases hydrogen probably comes nearest to the ideal of a perfect gas, but no real gas is in this sense strictly perfect. In a gas which is perfect in the sense of conforming exactly to Boyle's Law we should find PV strictly constant, so long as the temperature is constant. If we define the temperature scale by reference to the expansion of the gas we should also have V varying as the temperature T (reckoned from the absolute zero) under any constant pressure. Combining these two statements we should have PV = RT ........................... (1), where R is a constant. We may write, for any gas assumed to be perfect, where P and F are the pressure and volume at C. When the volume is reckoned per unit quantity of the gas we have a definite constant value of R for each gas, depending on the units employed and on the specific density of the gas in question. It should be noticed that when a gas satisfying this equation is heated under constant pressure and consequently expands, R is a measure of the amount of work done by the gas in this expansion for each degree through which the temperature rises. Let the original temperature of the gas be T l and its volume V l and let it be heated under constant pressure P till the temperature is T 2 and the volume F 2 . Then we have RT = PV^ and RT 2 = PF 2 , from which R (T 2 -T 1 )=P (F 2 - F^, which is the work done by the gas in expanding from V l to F 2 . Let the interval of temperature be 1, then R is equal to the work done. Thus R is numerically expressed in units of work per unit of mass and per degree: in foot-pounds per Ib. or in kilogrammetres per kilogramme. If we use the Centigrade degree in both cases the ratio of the number which expresses R in foot-pounds per Ib. to the number which expresses it in kilogrammetres per kilogramme is 3-28085, namely the number of feet in a metre. According to measurements by Regnault a cubic metre of dry air, i] FIRST PRINCIPLES 15 at a temperature of C. and pressure of 1 atmosphere as denned in Art. 12. contains 1*2928 kilogrammes. We should accordingly have for dry air, if it were "perfect," R = 1-03274 x 100 2 /l-2928 x 273'1 = 29-25, in kilogrammetres per kilogramme, at the latitude of London. The factor 100 2 is required to convert the pressure into kilogrammes per square metre. The corresponding value of R in foot-pounds per Ib. is 96-0. In this calculation air is treated as if it conformed exactly to Boyle's Law For the present it is to be understood that the symbol T stands for temperature measured on the scale of a gas ther- mometer, from a zero which is 273'1 below the melting point of ice. 19. Internal Energy of a Gas: Joule's Law. The Internal Energy of a given quantity of a gas depends only on the temperature. This is an inference from the fact established by experiments of Joule, that when a gas expands without doing external work and without taking in or giving out heat, and therefore without changing its stock of internal energy, its temperature does not change. Joule's Law is to be regarded as strictly true only of imaginary perfect gases: in any actual gas there is a slight departure from it, which is very small indeed in a nearly perfect gas such as hydrogen. The law was originally established by means of the following experiment. Joule connected a vessel containing compressed gas with another vessel which was empty, by means of a pipe with a closed stop-cock. Both vessels were immersed in a bath of water and were allowed to assume a uniform temperature. Then the stop-cock was opened, and the gas distributed itself between the two vessels, expanding without doing external work. After this the temperature of the water in the bath was found to have undergone no appreciable change. The temperature of the gas appeared unaltered, and no heat had been taken in or given out by it, and no work had been done by it. Since the gas had neither gained nor lost heat, and had done no work, its internal energy was the same at the end as at the begin- ning of the experiment. The pressure and volume had changed, but the temperature had not. The conclusion follows that the internal energy of a given quantity of gas depends only on its temperature, and not upon its pressure or volume; in other words, a change of pressure and volume not associated with a change of temperature 16 THERMODYNAMICS CH. does not alter the internal energy. Hence in any change of tempera- ture the change of internal energy is independent of the relation of pressure to volume during the operation: it depends only on the amount by which the temperature has been changed. The apparatus used by Joule in this experiment is shown in fig. 3. The vessel A was filled with air compressed to more than 20 atmospheres, and B was exhausted. Both vessels were immersed in a bath of water. The water in the bath was stirred and the temperature noted before the stop-cock C was opened. After the gas had come to rest in the two vessels the water was again stirred, and was found to have the same temperature as before, so far as tests made by a very sensitive ther- mometer could detect. In another form of the apparatus Joule separated the bath into three portions, one portion round each of the vessels and one round the con- necting pipe. When the stop-cock was opened the water surrounding A was cooled, but this was compensated by a rise of temperature in the water Fig. 3 surrounding B and C. The gas in A became colder in the act of expanding, but heat was given up in B and C as its eddying motion settled down, and when all was still there was neither gain nor loss of heat on the whole, so far as could be detected in this form of experiment. It is now, however, known that a very slight change of tempera- ture does in fact take place when a real gas expands without doing work. In later experiments by Joule and Thomson (Lord Kelvin) a more delicate method was adopted of detecting whether there is any change of internal energy when the pressure and volume change under conditions such that external work is not done. The gas was forced to pass through a porous plug by maintaining a constant high pressure on one side of the plug and a constant low pressure on the other. Care was taken to prevent any heat being gained or lost by conduction from outside. In this operation work was done upon the gas in forcing it up to the plug, and work was done by it when it passed the plug, by displacing gas under the lower pressure on the side beyond the plug. If no change of temperature took place, and if the gas conformed to Boyle's Law, these two quantities of work would be exactly equal, and consequently no external work i] FIRST PRINCIPLES 17 would be done on the whole. For let P l be the pressure and V the volume before passing the plug, and P 2 the pressure and V 2 the volume after passing the plug, the volumes being in both cases stated per Ib. of the gas. Then the work done upon the gas (per Ib.) as it approaches the plug is P^i , and the work done by it as it leaves the plug is P 2 F 2 . If the temperature is the same on both sides these quantities are. equal in a gas for which PV is con- stant at any one temperature. Thus a gas which is " perfect " in the sense that it conforms strictly both to Boyle's Law and to Joule's would in its passage of the plug have expanded without (on the whole) doing any work, and therefore without changing its internal energy, no heat being gained or lost. In such a gas no change of temperature should accordingly be found, as it passes the plug, and if a change of temperature is observed in any real gas it is due to the fact that real gases are not strictly "perfect." In the experiments of Joule and Thomson* small changes of tem- perature were in fact detected and measured in air and other real gases, on passing the porous plug. This Joule-Thomson effect, as it is called, is in general a cooling. Observations of the Joule-Thomson effect are of great value in determining exactly the properties of gases and vapours which are not perfect; and (as we shall see later) certain practical methods of liquefying gases under extreme cold depend upon the existence of this effect. In the imaginary perfect gas, however, the Joule-Thomson effect is entirely absent. There is no change of temperature in passing the plug, and there is also no change of internal energy, for no work is done and (by assumption) no heat is taken in or given out. It is important to notice that we assume the imaginary perfect gas to satisfy two conditions: it obeys Boyle's Law exactly and also Joule's Law exactly. These characteristics are independent of one another : it would be possible to have a gas satisfy one and not the other, but a gas is said to be perfect in the thermodynamic sense only when it satisfies both, and in that case certain other properties follow which will now be pointed out. 20. Specific Heats of a Gas. The Specific Heat of any sub- stance means the amount of heat required per degree to raise the temperature of unit quantity of the substance, under any assumed mode of heating. Thus when a substance is heated through a small interval of temperature dT the heat taken in (per Ib.) is * See Kelvin's Mathematical and Physical Papers, vol. i, p. 333. E. T. 2 18 THERMODYNAMICS [CH. KdT, where K is the specific heat for the particular conditions and mode of heating. In dealing with gases or other fluids two important modes of heating must be distinguished : we may heat them under conditions of constant pressure or of constant volume. We shall use the symbol K p to represent specific heat at constant pressure, and K v to represent specific heat at constant volume. Consider first the operation of heating unit quantity of a perfect gas at constant volume, from temperature T t up to temperature To. The heat taken in is ^ (rr \ K v (^2 - *!) No external work is done, for the volume (by assumption) does not change, and consequently all this heat goes to increase the stock of internal energy contained in the gas. But by Joule's Law the internal energy depends only on the temperature. Therefore if we heat the same quantity of the same gas in any other manner from T x to T 2 , the same change of internal energy must take place. Suppose then another manner of heating, namely at constant pressure. In that case the heat taken in is K, (T t - 2\). During this process external work is done, because the gas ex- pands, and its amount is P (r z ~~ PI) where Fj and F 2 represent the volumes at the beginning and end of the operation respectively, and P is the pressure, which by assump- tion is constant. Since PF 2 = RT 2 and PF 1 = RT l , we may write the expression for the external work in the form R (T, - Z\). This is in work units : in heat units it is AH (T t - TJ, where A is the reciprocal of Joule's equivalent (Art. 14). The difference between the heat taken in and the work done, namdy (K v - AR) (T 2 - TJ, is simply an addition to the stock of internal energy. But as was pointed out above, the change of internal energy must be the same in both modes of heating, and therefore K V = K P -AR (2). This important relation between the two specific heats in a perfect gas follows from the Laws of Boyle and of Joule. We have here taken K v and K v as applying throughout a finite i] FIRST PRINCIPLES 19 range of temperature from T 1 to T 2 . But this range may be made infinitesimally small without affecting the argument* and in that case K v and K v become the specific heats at a definite temperature. The conclusion holds that for any condition of the gas K V -K V = AH, and this is true whether the specific heats are or are not inde- pendent of the temperature. 21. Constancy of the Specific Heats in a Perfect Gas. From the above result it follows that if either of the two specific heats is constant the other must also be constant. To be constant the specific heat has to be independent both of the pressure and of the temperature. First as to independence of pressure: we have seen (Art. 19) that the internal energy of a perfect gas depends only on the temperature and is independent of the pressure. If we heat a perfect gas through 1 at any one temperature the change of internal energy is measured (Art. 20) by K V) no matter what is the pressure. Hence K v is independent of the pressure ; and since, by equation (2), K v is equal to K v + ^41?, it follows that K 9 also must be independent of the pressure. But a gas may conform to the Laws of Boyle and Joule without having K v and K v independent of the temperature, and if we are to treat them as constant we must make a further assumption regarding the properties of that convenient imaginary substance a perfect gas. Regnault's experiments showed that in some gases K 9 is nearly constant through a moderate range of temperature. But it is now known that in most gases the specific heat becomes distinctly greater at high temperatures. This variation will be discussed in Chapter VI ; for our present purpose it will simplify matters to think of an ideal gas in which the specific heat is constant. Accordingly, in dealing with a perfect gas, it is assumed that K p in such a gas is strictly independent of the temperature. This is a third assumed * Suppose the heating to be through a very small interval of temperature dT. In heating at constant volume, the heat taken in is K v dT, and all of it goes to increase the internal energy by an amount dE. Hence K v dT=dE. In heating at constant pressure through the same interval of temperature the heat taken in (dQ) does work dW and also adds to the internal energy by the amount dE. dQ is K v dT; and dW is PdV, which is equal to RdT. Hence From which K P -K V = AR. 22 20 THERMODYNAMICS [CH. characteristic of a perfect gas, additional to the two already de- scribed in Arts. 18 and 19. It does not in any way conflict with them : each of the three characteristics is independent of the others. With this further assumption we have, for any perfect gas, K p constant under all conditions, and consequently K v also constant under all conditions, since the difference between them is constant. 22. Reversible Actions. We have now to consider particular modes in which a working substance may be expanded or com- pressed and may take in or give out heat, and at the outset it is important to distinguish between actions that are reversible and those that are irreversible. in expansion or compression is reversible if it is carried out in such a manner that the operation can be reversed, with the result that the substance will pass back through all the stages through which it has passed during the expansion or compression and be in the same condition in all respects at each corresponding stage in This implies that the substance must expand smoothly, without setting up any motions within itself of a kind such that their kinetic energy is frittered down into heat through internal friction. The whirls and eddies which occur as a fluid enters or expands in the cylinder of an engine are irreversible, and in ideal reversible expan- sion we must suppose them absent. Reversible expansion implies that there are no losses of mechanical effect from any sort of inter- nal friction. It excludes throttling, such as occurs when a sub- stance expands through a valve or other constricted opening into a region of lower pressure where the kinetic energy of the stream and eddies is dissipated. In such cases the motion of the stream and eddies cannot be reversed. To get the substance back to the region of higher pressure would require an expenditure of more work than was done upon it during its expansion, and if we were to force it back we should find it had gained heat through the subsidence of the internal eddying motions, though no heat had come in from outside. The kind of expansion which takes place in Joule's experiment (Art. 19) is an extreme instance of irreversible expansion. A transfer of heat to or from any substance is reversible only if the substance is at the same temperature as the body from which it is taking heat or to which it is giving heat. Suppose, for instance, that a substance is taking in heat from a hot source and is expanding as i] FIRST PRINCIPLES 21 it does so. The expansion may be reversible in itself, that is to say it may involve no internal friction, but unless the temperature of the substance be the same as that of the source, the operation as a whole considered in its relation to the source cannot be reversed. So considered it is reversible only when the further condition is fulfilled that compression of the substance will reverse the transfer of heat, giving back to the source the heat that was taken from it. Any thermal contact between bodies at different temperatures involves an irreversible transfer of heat. Neither the expansions and compressions nor the transfers of heat that occur in a real engine are ever strictly reversible, some of them indeed are far from being reversible. But the study of an ideal engine, in which all the operations are reversible, is of fundamental importance in the science of thermodynamics, and it furnishes a basis for the critical analysis of actions in a real engine. 23. Adiabatic Expansion. There are two specially important kinds of reversible expansion, (1) Adiabatic and (2) Isothermal. Adiabatic expansion or compression means expansion or com- pression, carried out reversibly, in which no heat is allowed to enter or leave the substance. A curve drawn to show the relation of pressure to volume during the process is called an adiabatic line. Adiabatic action would be realized if we had a substance expanding, or being compressed, without change of chemical state, and without any eddying motions, in a cylinder which (along with the piston) was totally impervious to heat. From this definition it follows that the work which a substance does while it is expanding adiabatically is all done at the expense of its stock of internal energy; and the work which is spent upon a substance when it is being compressed adiabatically all goes to increase its stock of internal energy. In actual heat-engines the action is never strictly adiabatic, for there are always some exchanges of heat between the working sub- stance and the surface of the cylinder and piston. Very rapid com- pression or expansion may come near to being adiabatic by giving little time for any transfer of heat to occur. After what has been said already about reversibility, it is scarcely necessary to add that expansion through a throttle-valve is not adiabatic, though it may be (and generally is) done without letting heat enter or leave the substance. . In the adiabatic expansion of any substance work is done, and 22 THERMODYNAMICS [CH. since no heat is taken in or given out, there must be a decrease of internal energy equivalent to the amount of the work done by the substance. Taking the general equation (Art. 9) dQ = AdW + dE, which applies to any small change of state on the part of any sub- stance, we have dQ = when the action is adiabatic, and hence for an adiabatic expansion AdW = - dE. Here dW is the work done, A is the factor required to convert an expression for work into heat units (Art. 14), and dE is the change of internal energy. 24. Isothermal Expansion. Isothermal expansion or com- pression means expansion or compression carried out reversibly (as regards internal action) and without change of temperature. A curve drawn to show the relation of pressure to volume during isothermal expansion or compression is called an isothermal line. When a substance is expanding isothermally it takes in heat to maintain its temperature constant; it therefore must be in contact with a source of heat. When it is being compressed iso- thermally it gives out heat, and must be in contact with a receiver which can take heat from it. 25. Adiabatic Expansion of a Perfect Gas. Consider next the behaviour of a perfect gas during adiabatic expansion or com- pression. We have seen that in a small adiabatic expansion of any substance (Art. 23) dE = _ In a perfect gas dE = K v dT (Art. 20). Hence in the adiabatic expansion of a perfect gas APdV = - K v dT. But P = ETJV (Art. 18). Hence ARTdVjV + K v dT = 0, or, dividing by T, ARdV/V + K v dT/T = 0, which gives on integration AR log e V + K v log e T = constant ............ (3). i] FIRST PRINCIPLES 23 Writing K 9 - K v for AR (Art. 20), and dividing by K v , which is constant (Art. 21), (K V /K V - 1) log e V 4- log e T = constant. We shall write y for the ratio of the two specific heats, namely K,/K 9 . Thus we have y loge V log e V + log e T = constant ............ (4). Further, since PV '/T is constant, log e P + log e V log e T = constant. Adding these two equations loge-P + y log e V = constant ............... (5), which gives PV y = constant ........................ (6) as the equation of any adiabatic line in the pressure-volume diagram, for the adiabatic expansion of a perfect gas*. 26. Change of Temperature in the Adiabatic Expansion of a Perfect Gas. When a gas is expanding adiabatically its stock of internal energy is, as we have seen, being reduced, and hence its temperature falls, the change of internal energy being propor- tional to the change of temperature (Art. 20). Conversely, in adiabatic compression the temperature rises. The amount by which the temperature is changed (in a perfect gas) may be found by combining the equations PJTf = P 2 F 2 v and PjrjPjr, = T./T,. Multiplying them together we have r, JW T iV \y-i whence - , or T 2 = T, This result of course applies to compression as well as to expansion along an adiabatic line. It may be got directly from equation (4), which can be written log e T + (y - 1) log e F = constant; whence TF?- 1 = constant ..................... (7). Combining equations (6) and (7) and eliminating F, we obtain nl the further relation T/P y = constant. * It is to be remembered that loge , the * ' hyperbolic " or " Napierian " or " natural logarithm of any number, is 2-302585 times the common logarithm of the number. 24 THERMODYNAMICS [CH. 27. Work done in the Adiabatic Expansion of a Perfect Gas. In any kind of expansion of any fluid the work done in expanding from volume F-, to volume F 2 is W= { Vi PdV. If the nature of the expansion be such that PV n is constant, n being any index, then P at any point when the volume is V is P 1 F 1 n /F n , P! and F x being the pressure and volume in the initial state. In that case, for expansion from F x to F 2 , W = which gives on integration W = P,VS (F, 1 - - F 1 1 -)/(l - n), or ................................. > rl/ x So far we have made no assumption as to the nature of the working substance. Apply this result to a gas expanding adiabatically, for which the index n is equal to y (by Eq. 6, Art. 25). We then have since P 1 F I = RT t and P 2 F 2 - RT 2 . Further, it follows from Art. 23 that this expression (mul- tiplied by A) is the decrease of internal energy produced by the 28. Isothermal Expansion of a Perfect Gas. In a gas which satisfies the equation PF == RT, PV is constant during isothermal expansion or compression, and any isothermal line on the pressure- volume diagram is a rectangular hyperbola, the pressure varying inversely as the volume. To find the work done in the isothermal expansion of a gas from Fj to F 2 we have .^ W = 2 PdV .'F, and P = PjFj/F, from which W = P 3 F 3 f ^ ^ . J F, V i] FIRST PRINCIPLES 25 Integrating, W = P 1 V 1 (log, F 2 - log, V& FF-P^lo&p. *i Instead of writing P 1 V 1 we may write PF, since the product of P and F is constant throughout the process, and again, since PF - RT, v ? ........................ (10), where T is the temperature at which the process takes place. This expression serves to give either the work that is done by a gas in isothermal expansion, or the work that is spent upon it in isothermal compression; During the isothermal expansion or compression of a perfect gas there is no change of internal energy, since there is no change of temperature and the internal energy depends only on the temperature (Art. 19). Hence during isothermal expansion a perfect gas must take in an amount of heat equivalent to the work it does, namely ART log e F 2 /F 1} and during isothermal com- pression from F 2 to F x it must give out that amount of heat. 29. Summary of results for a Perfect Gas. It may be con- venient at this point to collect the results that have been found for actions occurring in perfect gases. It is assumed that the gas satisfies Boyle's Law (Art. 17) and Joule's Law (Art. 19) and that the specific heat (at constant pressure) is independent of the temperature. Further, the tem- perature is measured on the scale furnished by the expansion of the gas itself. Under these conditions we have the following results : PF TIT rid. , where R is a constant depending on the specific density of the gas ; where K v is the specific heat at constant pressure, K v the specific heat at constant volume and A is the reciprocal of Joule's equiva- lent. K v and K v are both constant. In adiabatic expansion: PF* = constant, or P ly /P 2 = where y is K V JK V . TVy- 1 = constant, or Ty/T, = 7-1 T/P y = constant, or 26 THERMODYNAMICS [CH. Heat taken in = 0. R(Ti-TJ PiFi-PjFa Work done = ^ *- y = -- * 2 . y-1 y-1 Decrease of Internal Energy = AR ( T ^~ T *> . In isothermal expansion : PV = constant, since T = constant. Heat taken in = ART iog e ^- 2 , "l Work done = RTlog e ^. "i Change of Internal Energy = 0. 30. Fundamental Questions of Heat-Engine Efficiency. We are now in a position to deal with the most fundamental questions of heat-engine efficiency, which may be stated in the following terms: (1) Having given a source from which heat may be taken in at a high temperature, and a sink or receiver to which heat may be rejected at a lower temperature, how may heat taken from the source be utilized to the best advantage for the purpose of producing mechanical effect ? In other words, how may the greatest amount of work be done by each unit of heat taken from the hqt source? (2) What fraction of the heat taken from the hot source is it theoretically possible to convert into work? In other words, what is the limiting efficiency of conversion? 31. The Second Law of Thermodynamics. So far as the P^irst Law of Thermodynamics (Art. 8) goes, it is not obvious that there is anything to prevent all the heat which the source can supply from being converted into work. But it will presently be seen that a limit is imposed as a consequence of the following principle, which is known as the Second Law of Thermodynamics : It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature. The Second Law says, in effect, that heat will not pass up automatically from a colder to a hotter body. We can force it to pass up, as in the action of a refrigerating machine, but only by applying an "external agency" to drive the machine. A heat- engine acts by letting heat pass down from a hotter to a colder body, not of course by direct conduction from one to the other, for that i] FIRST PRINCIPLES 27 is a mode of transfer in which the heat would do no work, but by making the working substance alternately take in heat from the hot body and reject heat to the cold body, and thereby undergo expansions and contractions in which its pressure is on the whole greater during expansion than during contraction, with the result that a part of the heat that is passing down through the engine is converted into work. In consequence of the Second Law it is only a certain fraction of the whole heat supplied by the hot body that can be converted into work by any such process. 32. Reversible Heat-Engine. Carnot's Cycle of Opera- tions. To the first of the above two questions (Art. 30) a correct answer was given by Sadi Carnot in a remarkable essay, published in 1824, entitled Reflexions sur la puissance motrice du feu el sur les machines propres a developper cette puissance. In this essay Carnot maybe said to have laid the foundation of thermodynamics. He pointed out that the greatest possible amount of work was to be obtained by letting the heat pass from the source to the receiver through an engine working in a strictly reversible manner not only as regards its own internal actions but also as regards the transfer of heat to it from the source and from it to the receiver. The engine conceived by Carnot is an engine every one of whose operations is reversible in the sense explained in Art. 22. He further showed how an engine might (theoretically) work in such a way as to satisfy this condition, its cycle consisting of these four reversible operations : (1) Isothermal expansion at the .temperature of the hot source (7\). During this operation heat is taken in reversibly from the hot source. (2) Adiabatic expansion, during which the temperature of the working substance falls from T l to T 2 (the temperature of the receiver). (3) Isothermal compression at the temperature of the receiver. During this operation heat is rejected reversibly to the receiver. (4) Adiabatic compression, by which the temperature of the working substance is raised from T. to Z\ . This completes the cycle by bringing the substance back to the condition in which it was assumed to be at the beginning of the first operation. In the cycle as a whole work is done by the substance: the average pressure in (1) and (2) being greater than in (3) and (4). 28 THERMODYNAMICS [CH. This cycle of operations, which is known as Carnot's Cycle, is entirely reversible. The working substance might be forced to go through it in the reversed direction, taking in heat from the cold body and giving out heat to the hot body. The transfers of heat would be exactly reversed, and at every stage the pressure and volume and temperature of the substance would be the same as when working direct. The work spent upon it would be equal to the work got from it in the direct action. Carnot's ideal engine accordingly affords a strictly reversible means of letting heat down from the hot source to the cold receiver. The argument by which Carnot proved that no heat-engine can utilize heat more completely than a reversible heat-engine utilizes it, in letting heat down from a given source to a given receiver, is substantially as follows. 33. Carnot's Principle. To prove that no other heat-engine, working between the same source and receiver of heat, can do the same amount of mechanical work as a reversible engine by taking in a smaller quantity of heat. Suppose there are two heat-engines R and S, one of which (R) is reversible, working between the same hot body or source of heat and cold body or receiver of heat, and each producing the same amount of mechanical work. Let Q be the quantity of heat which R takes in from the hot body. Now if R be reversed it will by the expenditure on it of the same amount of work give to the hot body the 'amount of heat it formerly took from it, namely Q. For this purpose set the engine S to drive R reversed. The work which S produces is just sufficient to drive R, and the two machines (S driving R) form together a self-acting machine unaided by any external agency. One of the two, namely S, takes heat from the hot body and the other, R, which is reversible, gives back to the hot body the amount of heat Q. Now if S could do its work by taking less heat than Q from the hot body the hot body would on the whole gain heat. No work is being done on the system from out- side, nor is any heat supplied from other sources, so whatever heat the hot body gains must come from the cold body. Therefore if S could do as much work as the reversible engine jR, with a smaller supply of heat, we should be able to arrange a purely self-acting machine through which heat would continuously pass up from a cold body to a hot body. This would be a violation of the Second Law of Thermodynamics. i] FIRST PRINCIPLES 29 The conclusion is that S cannot do the same amount of work with a smaller supply of heat than a reversible engine ; or, in modern language, that no other engine can be more efficient than a reversible engine, when they both work between the same two temperatures in source and receiver. Further, let both engines be reversible. Then the same argu- ment shows that each cannot be more efficient than the other. Hence all reversible engines taking in and rejecting heat at the same two temperatures are equally efficient. 34. Reversibility the Criterion of Perfection in a Heat- Engine. These results imply that, in the thermodynamic sense, reversibility is the criterion of what may be called perfection in a heat-engine. A reversible heat-engine is perfect in the sense that it cannot be improved on as regards efficiency: no other engine taking in and rejecting heat at the same two temperatures can obtain from the heat taken in a greater proportion of mechanical effect. Moreover, if this criterion be satisfied, it is, as regards effi- ciency, a matter of complete indifference what is the nature of the working substance or what, in other respects, is the mode of the engine's action. It is, therefore, a complete answer to the first question in Art. 30 to say that the greatest amount of work that is theoretically possible will be done by each unit of heat if the heat is supplied to an engine which works in such a way that every one of its operations is reversible. 35. Efficiency of a Reversible Heat-Engine. The second question in Art. 30 could not be answered by Carnot because in his time the doctrine of the Conservation of Energy was unknown, and it was not recognized that part of the heat disappears, as heat, in passing through the engine. Carnot realized that work is done by an engine through the agency of heat, but he did not know that it is done by the conversion of heat. It is remarkable that he was nevertheless able to conceive the idea of a reversible engine and to see that it is the most effective possible means 0*1" getting work done through the agency of heat. His argument as to this is per- fectly valid though it makes no use of the First Law of Thermo- dynamics. It is moreover extraordinarily general. There is no assumption in it as to the properties of any substance, nor as to the nature of heat, nor as to the way in which temperature is to be measured. All that he assumes about the temperatures of the 30 THERMODYNAMICS [CH. source and the receiver is that one is hotter than the other. The argument stands by itself, and the whole passage in which it is reproduced here (Art. 33) does not involve a reference to any of the results stated in earlier Articles. But for the purpose of answering the second question of Art. 30 we shall in the first place deal with one particular reversible heat-engine, namely a reversible engine which has a perfect gas for working substance, and shall calculate its efficiency with the help of the results previously obtained for perfect gases. It will be easy to go on from that to find a general answer to the question, What is the limiting efficiency of any heat-engine? Fig. 4. Carnot's Cycle, with a gas for working substance. 36. Carnot's Cycle with a Perfect Gas for Working Sub- stance. Consider then an ideal engine in which a substance may go through the operations of Carnot's Cycle (fig. 4). Imagine a cylinder arid piston composed of perfectly non-conducting material, except as regards the bottom of the cylinder, which is a conductor. Imagine also a hot body or indefinitely capacious source of heat A, kept always at a temperature T 19 also a perfectly non-conducting cover B, and a cold body or indefinitely capacious receiver of heat C, kept always at some temperature T 2 which is lower than T 1 . It is supposed that A or B or C can be applied at i] FIRST PRINCIPLES 31 will to the bottom of the cylinder. Let the cylinder contain 1 Ib. of a perfect gas, at temperature T l9 volume F , and pressure P a to begin with. The suffixes refer to the points on the indicator diagram, fig. 4. There are four successive operations : (1) Apply A, and allow the piston to advance slowly through any convenient distance. The gas expands isothermal ly at T t , tak- ing in heat from the hot source A and doing work. The pressure changes to P b and the volume to F 6 . The line ab on the indicator diagram represents this operation. (2) Removed and apply B. Allow the piston to go on advancing. The gas expands adiabatically, doing work at the expense of its internal energy, and the temperature falls. Let this go on until the temperature is T 2 . The pressure is then P c , and the volume F c . This operation is represented by the line be. (3) Remove B and apply C. Force the piston back slowly. The gas is compressed isothermally at T z , since the smallest in- crease of temperature above T 2 causes heat to pass into C. Work is spent upon the gas, and heat is rejected to the cold receiver C. Let this be continued until a certain point d is reached, such that the fourth operation will complete the cycle. (4) Remove C and apply B. Continue the compression, which is now adiabatic. The pressure and temperature rise, and, if the point d has been properly chosen, when the pressure is restored to its original value P a , the temperature will also have risen to its original value T . [In other words, the third operation cd must be stopped when a point d is reached such that an adiabatic line drawn through d will pass through a.] This completes the cycle. To find the proper place at which to stop the third operation, we have (by Art. 26), for the cooling during the adiabatic ex- pansion in stage (2), (F^r^-iv/r.-zyr,, and also, for the heating during the adiabatic compression in stage (4), (F./FJV- 1 = T a jT d - T /T 2 . Hence F c /F 6 =F d /F a , and therefore also F c /F d = F & /F a . That is to say, the ratio of isothermal compression in the third stage of the cycle is to be made equal to the ratio of isothermal expansion in the first stage, in order that an adiabatic line through 32 THERMODYNAMICS [CH. d shall complete the cycle. For brevity we shall denote either of these last ratios (of isothermal expansion and compression) by r. The following are the transfers of heat to and from the working gas, in the four successive stages of the cycle; quantities of heat are here expressed in work units : (1) Heat taken in from A = RT log e r (by Art. 28). (2) No heat taken in or rejected. (3) Heat rejected to C = RT 2 log. r (by Art. 28). (4) .No heat taken in or rejected. Hence, the net amount of external work done by the gas, being the excess of the heat taken in above the heat rejected in a com- plete cycle, is R ^ _ T j logg r . this is the area enclosed by the four curves in the figure. The Efficiency in this cycle, namely the fraction Heat converted into work Heat taken in is accordingly Another way of stating the result is to say that if we write Q a for the heat taken in from the hot source, and Q. 2 for the heat rejected to the cold receiver, then In these expressions the temperatures T and T 2 are understood to be measured on the scale of a perfect gas thermometer, and from the absolute zero. 37. Reversal of this Cycle. This cycle, being a Carnot cycle, is reversible. To realize the fact more fully we may consider in detail what will happen if we make the imaginary engine work backwards, forcing it to trace out the same indicator diagram in the opposite order. For this purpose we must expend work upon it from some other source of work. Starting as before from the point a (fig. 4) and with the gas at T 19 we shall require the following four operations : (1) Apply B and allow the piston to advance. The gas expands adiabatically, the curve traced is ad, and when d is reached the temperature has fallen to T 2 . i] FIRST PRINCIPLES 33 (2) Remove B and apply C. Allow the piston to go on advanc- ing. The gas expands isothermally at T 2 , taking in heat from C, and the curve dc is traced. (3) Remove C and apply B. Compress the gas. The process is adiabatic. The curve traced is cb, and when b is reached the temperature has risen to T lt (4) Remove B and apply A. Continue the compression, which is now isothermal at T l . Heat is now rejected to A, and the cycle is completed by the curve ba. In this process the engine is not on the whole doing work; on the contrary, a quantity of work is spent upon it equal to the area of the diagram, or R (T^ T 2 ) log g r, and this work is converted into heat. Heat is taken in from C in the first operation, to the amount RT 2 log e r. Heat is rejected to A in the fourth operation, to the amount RT^ log e r. In the first and third operations there is no transfer of heat. The machine js acting as a heat-pump. The action is now in every respect the reverse of what it was before. The substance is in the same condition at corresponding stages in the two processes. The same work is now spent upon the engine as was formerly done by it. The same amount of heat is now given to the hot body A as was formerly taken from it. The same amount of heat is now taken from the cold body C as was formerly given to it. This will be seen by the following scheme: Carnot's Cycle with a perfect gas. Direct. Work done by the gas = R(T 1 - T 2 ) log e r; Heat taken from A = RT l log e r; Heat rejected to C = RT 2 log e r. Carnofs Cycle with a perfect gas. Reversed. Work spent upon the gas = R (T l T 2 ) log e r; Heat rejected to A = RT 1 log e r; Heat taken from C = RT 2 log e r. In the second case the heat rejected to the hot body is equal to the sum of the heat taken in from the cold body and the work spent on the substance. This of course follows from the principle of the Conservation of Energy. E. T. 3 34 THERMODYNAMICS [CH. 38. Efficiency of Any Reversible Engine. The imaginary engine, then, of Art. 36 is reversible. Its efficiency, as we have seen, IS rr\ rp where 2\ is the temperature of the source from which it takes heat and T 2 is the temperature of the receiver to which it rejects heat. But we saw, by Art. 33, that all reversible heat-engines taking in and rejecting heat at the same two temperatures are equally efficient. Hence the expression measures the efficiency of any reversible heat-engine, and therefore (by Art. 33) also expresses the largest fraction of the heat supplied that can possibly be converted into work by any engine whatever operating between these limits. In other words, if we have a supply of heat at a temperature Tj , and a means of getting rid of heat at a temperature T 2 , then there is no possibility of converting more than that fraction of the heat into work. This is the measure of perfect efficiency. it is the theoretical limit beyond which the efficiency of a heat- engine cannot go. No engine can conceivably surpass this stan- dard, and as a matter of fact any real engine falls short of it, because no real engine is strictly reversible. 39. Summary of the Argument. Briefly recapitulated the steps of the argument by which we have reached this immensely important result are as follows. Following Carnot, we considered how any heat-engine works by taking in heat from a hot source and rejecting heat to a cold receiver, and established (by means of the reductio ad absurdum of a hypothesis which would conflict with the Second Law of Thermodynamics) the conclusion that no engine could do this more efficiently than a reversible engine does, that is to say, an engine which goes through a reversible cycle of operations. This led to the inference that all reversible engines working between the same temperatures of source and receiver were equally efficient, and consequently that an expression for the efficiency of any one of them would apply to all, and would mean the highest efficiency that is theoretically possible. Still following Carnot, we imagined a cycle which would be reversible, consisting of four stages, namely (1) isothermal expansion during which heat is taken in from the source, (2) adiabatic expansion during which ij FIRST PRINCIPLES 35 the temperature of the substance falls from the temperature of the source to that of the receiver, (3) isothermal compression during which heat is rejected to the receiver, (4) adiabatic com- pression during which the temperature of the substance rises again to that of the source. Up to this point there had been no assump- tion as to the use of any particular working substance. We next enquired what would happen in this cycle if a perfect gas were used as working substance. Taking for the scale of temperature a scale based on the expansion of a perfect gas *, and expressing on this scale the temperatures of source and receiver as T x and T 2 respec- tively, we found that a reversible engine, using a perfect gas for working substance, has an efficiency of Hence it was concluded that this expression measures the effi- ciency of any reversible engine working between these limits, and that this is the highest efficiency theoretically obtainable in any heat-engine. This general conclusion may also be stated, with equal gener- ality (for any reversible engine), in the form QJT, - Q 2 /2V where Q l is the heat taken in by the engine from the source at T l , and Q 2 * s the heat rejected by it to the receiver at T 2 . The efficiency of any heat-engine may be written whether the engine be reversible or not. In a reversible engine, or, as we may now call it, a thermodynamically perfect engine, this becomes , //r - J-2/^l' In an engine which falls short of reversibility a smaller fraction of the heat supply is converted into work and the heat rejected is relatively larger; Q 2 /T 2 is greater than d/j^. 40. Absolute Zero of Temperature. The zero from which 2\ and T 2 are measured is the zero of the gas thermometer, which was denned (Art. 16) as the temperature at which the volume of the gas would vanish if the same law of expansion continued to apply. * That is to say, a scale in which the temperature is proportional to the volume of the gas, when the pressure is kept constant. 32 36 THERMODYNAMICS [CH. But we can now give it another meaning. Taking the expression for the efficiency of a reversible heat-engine we see that if the cold receiver were at the temperature of the absolute zero (so that T 2 = 0) the efficiency would be equal to 1 : in other words, all the heat supplied to the engine would be converted into work. It is clearly impossible to imagine a receiver colder than that, for it would make the efficiency greater than 1 and thereby violate the First Law of Thermodynamics by making the amount of work done greater than the heat supplied. Hence the zero which we found on the gas scale is also an absolute thermodynamic zero, a temperature so low that it is inconceivable on thermodynamic grounds that there can be any lower temperature. The term "absolute zero" has consequently acquired a new meaning: with- out reference to the properties of any substance we see that it represents a limit below which temperature cannot go. This justifies the use of the word "absolute" as applied to a zero of temperature. 41. Conditions of Maximum Efficiency. From the above result it will be obvious that the availability of heat for trans- formation into work depends essentially on the range of tempera- ture through which the heat is let down, from that of the hot source to that of the cold body into which heat is rejected; it is only in virtue of a difference of temperature between bodies that conversion of any part of their heat into work becomes possible. No mechanical effect could be produced from heat, however great the amount of heat present, if all bodies were at a dead level of temperature. Again, it is impossible t<5 convert the whole of any supply of heat into work, because it is impossible to have a body at the absolute zero of temperature as the sink into which heat is rejected. If T l and T 2 are given as the highest and lowest temperatures of the range through which a heat-engine is to work, it is clear that the maximum of efficiency can be reached only when the engine takes in all its heat at T 1 and rejects at T 2 all that is re- jected. With respect to every portion of heat supplied to the engine the greatest ideal efficiency is Temperature of reception temperature of rejection Temperature of reception i] FIRST PRINCIPLES 37 Any heat taken in at a temperature below T 1 , or rejected at a temperature above T 2 , will be less capable of conversion into work than if it had been taken in at T 1 and rejected at T 2 , and hence, with a given pair of limiting temperatures, it is essential to maximum efficiency that no heat be taken in by the engine except at the top of the range, and no heat rejected except at the bottom of the range. Further, as we have seen in Art. 33, when the tem- peratures at which heat is received and rejected are assigned, an engine attains the maximum of efficiency if it be reversible. It may be useful to repeat here that in the transformation into work of heat supplied from a given source, the condition of reversi- bility is satisfied in the whole operation from source to receiver if (1) no part of the working substance is brought into contact during the operation with any body at a sensibly different tempera- ture, and (2) there is no dissipation of energy through internal friction. The first condition excludes any unutilized drop in tem- perature; the second excludes eddying motions and such like sources of waste, which arise in consequence of expansion through throttle-valves or constricted orifices, or in consequence of any cause that sets up dissipative motions within the substance. In a piston and cylinder engine we have to think of the substance as expanding by the gradual displacement of the piston, doing work upon it, and not wasting energy to any sensible extent by setting portions of itself into motion. There are to be no local variations of pressure within the cylinder, such as might occur in a fast-running engine through the inertia of the expanding fluid. When we proceed to deal in a later chapter with steam jets in relation to steam turbines, we shall see that it is possible to have (in theory) a reversible action, though the work done by the sub- stance in expanding is employed to give kinetic energy to the substance itself as a whole by forming a jet, because in that case the energy of the jet is recoverable when proper care is taken to control the formation of the jet. But the eddying motions spoken of here are of a different class: their energy is irrecoverable and for that reason they violate the condition of reversibility. It may also be worth while to repeat here that no real heat- engine can work between the source and the receiver in a strictly reversible manner. It cannot wholly escape eddying motions: it cannot wholly escape transfers of heat between the working sub- stance and bodies at other temperatures. In particular, since the working substance must in practice take in heat at a reasonable 38 THERMODYNAMICS [CH. rate from the hot source, the source is usually much hotter than the substance while heat is being taken in. This is, in practice, the most serious breach of reversibility in the transformation of heat by a steam-engine. It means that between the temperature of the source and the highest temperature reached by the working substance in its cycle of operations, there is a wasteful drop, a drop that is not utilized thermodynamically. If it were practicable in the steam-engine to avoid the drop between the temperature of the furnace gases and that of the water in the boiler a greatly increased efficiency of conversion would be attainable. If we leave this drop out of account, and take for the upper limit Tj , not the temperature of the furnace gases but the tempera- ture in the boiler, and if we also take for T 2 the temperature in the condenser, the fraction will measure the greatest fraction of the heat supplied to the boiler that can be converted into work, under ideally favourable (in other words, strictly reversible) conditions between the boiler and the condenser. The performance of any real engine falls short of this because it includes irreversible features, the chief of which are throttling actions in the steam -passages and exchanges of heat between the steam and the metal of the cylinder and piston. But although this limit of efficiency cannot be actually reached, it affords a valuable criterion with which to compare the per- formance of any real engine, and establishes an ideal for engine designers to aim at. It is important to realize that a substance may expand re- versibly although it is taking in heat from a source hotter than itself: in other words, there may be an irreversible drop of heat between the source and the substance, but no irreversible action within the substance. Thus the fluid in a boiler is at a definite temperature lower than that of the furnace while it is taking in heat from the furnace; there is accordingly an irreversible drop in this transfer of heat: but the formation and expansion of the steam may go on in a reversible manner. We can imagine all the internal actions of the working substance to be reversible, although as regards transfers of heat from the source or to the receiver. there is not reversibility. In that event the engine will still work as efficiently as possible between its own limits of temperature, namely the limits at which the substance takes in and rejects i] FIRST PRINCIPLES 39 heat, though it is no longer the most efficient possible con- trivance for utilizing the full range of temperature from source to receiver. Thus if we interpret T l and T 2 as the limits of temperature of the working substance itself without any reference to a source or a receiver T 1 being the temperature of the substance while it is taking in heat, and T 2 the temperature of the substance while it is rejecting heat, and if the internal actions of the substance are reversible, then (T^ - T 2 )/2\ still measures the efficiency of the engine. This fraction still expresses the greatest efficiency that is theoretically possible in any heat-engine working between the limits 2\ and T 2 . When we speak of a substance as taking, in heat at a stated temperature, or rejecting heat at a stated temperature, it is to be understood that the temperature of the substance itself is meant, though that may not be the temperature of the source or receiver; and when we speak of a substance as expanding or being com- pressed in a reversible manner we do not imply that it may not be taking in heat from a source hotter than itself or rejecting heat to a receiver colder than itself. A cycle of operations may be internally reversible, that is to say, reversible so far as actions within the working substance are concerned, although it happens to be associated with an irreversible transfer of heat to the working substance from the source or from the working substance to the receiver*. 42. Thermodynamic Scale of Temperature. Reference was made in Art. 15 to the fact (first pointed out by Lord Kelvin f) that thermodynamic principles allow a scale of temperature to be defined which is independent of the properties of any particular substance, real or imaginary. Up to the present we have based the scale on the properties of a perfect gas, taking a scale in which the degrees (or equal intervals of temperature) correspond to equal amounts of expansion on the part of a perfect gas kept at constant pressure. Using this scale we have seen that a rever- sible engine which works between the limits T 1 and T 2 , and takes in any quantity of heat Q x at T a , rejects at T 2 a quantity Q 2 equal to Q! T 2 /T l , and has an efficiency equal to (T 1 - T 2 )/T 1 . * We may imagine a source at T^ and receiver at T z to be substituted for the actual source and receiver, if these have a wider range of temperature, without affecting the action of the working substance. t Mathematical and Physical Papers, vol. I, p. 100; also pp. 233236. 40 THERMODYNAMICS [CH. Now imagine that the heat Q 2 , which is rejected by this engine, forms the supply of a second reversible engine taking in heat at T 2 and rejecting heat at a lower temperature T 3 , such that the interval of temperature through which it works (T 2 T 3 ) is the same as the interval through which the first engine works (T x T 2 ). Call each of these intervals AT. Let the heat Q 3 rejected by this second engine pass on to form the supply of a third reversible engine, work- ing through an equal interval AT and rejecting heat Q^ to a fourth reversible engine, and so on. We imagine a series of engines, every one of which is reversible, each passing on its rejected heat to form the supply of the next engine in the series, and each working through the same number of degrees on the perfect gas thermo- meter, AT. The efficiencies of the successive engines are AT/T 1; AT/T 2 , AT/T 3 , etc. The amounts of heat supplied to them are Qt , Q 2 = Qi r a / Z\ , Q 3 = Q 2 T 3 /T 2 = ^ TJ/T! , etc. Multiply in each case the heat taken in by the efficiency to find the amount of work done by each engine in the series, and we find that the amount of work done is the same for all the engines, namely Accordingly, we might define the interval of temperature for each engine, without reference to a perfect gas or to any other thermometric substance, as that interval which makes every engine in the series do the same amount of work ; and if we did so we should get a scale of temperature which is identical with the scale of the perfect gas thermometer. The above method of obtaining a thermodynamic scale of temperature may be put thus: Starting from any arbitrary con- dition of temperature at which we may imagine heat to be supplied, let a series of intervals be taken such that equal amounts of work will be done by every one of a series of reversible engines, each working with one of these intervals for its range, and each handing on to the engine below it the heat which it rejects, so that the heat rejected by the first forms the supply of the second, and so on. Then call these intervals of temperature equal. What the above proof shows is that the intervals thus defined to be equal are also equal when measured on the scale of the perfect gas thermometer: in other words, the thermodynamic scale and the i] FIRST PRINCIPLES 41 perfect gas scale coincide at every point. Any temperature T reckoned from zero on the scale of a perfect gas thermometer is also an absolute temperature on the thermodynamic scale. The conception, then, of a chain ot reversible heat-engines, each working through a small definite range, furnishes for the statement of temperature a scale which is really absolute in the sense of being independent of all assumptions about expansion or other behaviour of any substance. As the heat goes down from engine to engine in the chain, part of it is converted into work at each step, and the remainder passes on to form the heat-supply of the next engine. We have only to think of the steps as being such that the amount of heat converted into work is the same for each step, and that the remainder passes from engine to engine till all is converted. Thus if we have n engines in the chain, and if the whole quantity of heat supplied to the first engine is Qj , then the steps are such that each engine converts the quantity Q 1 /n of heat into work. When n steps are completed there is no heat left: all is converted into work. This means that the absolute zero of temperature has been reached : we may in fact define the absolute zero as the temperature which is reached in this manner. It is imagined to be reached by coming down through a finite number of steps of temperature, each step representing a finite fall in temperature. We define the absolute or thermodynamic scale by saying that these steps are to be taken as equal to one another. From this it will be seen that the conception of an absolute zero, and of an absolute thermodynamic scale with uniform intervals, does not depend an any notion about perfect gases or about the properties of any particular substance. We reach the absolute zero when, on going down through the chain of perfect engines, we come to a point at which the last fraction of the heat has been converted into work. That fixes the absolute zero. And we call the steps by which we have come equal steps of temperature, the steps being determined by the consideration that each engine in succession is to do the same amount of work out of the residue of heat received from the engine immediately before it in the series. That fixes the scale. Moreover the steps can be so taken, that the scale they give will agree at two fixed points with the ordinary thermometric scale, and will contain between those fixed points the same number of steps as the ordinary scale contains degrees. Thus suppose the initial temperature, at the top of the chain, is that of the boiling point of water, and that we have 373 engines in the chain. Then we find that it takes 100 steps to come 42 THERMODYNAMICS [CH. down to the temperature of melting ice, and 273 more steps* to complete the conversion of the remaining heat into work. This means that the uniform step of temperature on the thermodynamic scale is equal to the average of the intervals called degrees on any centigrade thermometer, when that average is taken between the freezing point and the boiling point (0 and 100), and that the absolute zero is at a point 273 of such steps* below the freezing point. But the thermodynamic scale would agree from point to point with the indications of the thermometer throughout the whole of the scale only if the thermometer could use a perfect gas as its expanding substance. Even with hydrogen, which is very nearly a perfect gas, there are slight divergences which were mentioned in Art. 15. 43. Reversible Engine receiving Heat at Various Tempera- tures. In Carnot's cycle it was assumed that there was only one source and one receiver of heat. All the heat that was taken in was taken in atT^; all the heat that was rejected was rejected atjT 2 . But an engine may take in heat in stages, at more temperatures than one, and may also reject heat in stages. With regard to every quantity of heat so taken in, the result still applies that the greatest fraction of it that can be converted into work is repre- sented by the difference between its temperatures of reception and rejection, divided by the absolute temperature of reception. And this is the fraction that will be converted into work provided the processes within the engine are reversible. Thus if Qj represents that part of the whole supply of heat which is taken in at T and Q 2 represents what is taken in at some other temperature T 2 , Q 3 at T 3 , and so on, and if T be the temperature at which the engine rejects heat, the whole work done, if the engine is reversible, is Q. (2\ - TQ) Q 2 (T, - T) Qi (f, - r ) 7ft m T BSC. 2 1 *2 2 3 We here take, for simplicity of statement, a single temperature of rejection T . A mechanically analogous machine would be a great water- wheel, working by gravity, and receiving water into its buckets from reservoirs at various levels, some of which are lower than the top of the wheel. Let M 1 , M 2 and so on be the weights of water * More exactly 273 and a fraction (Art. 16). i] FIRST PRINCIPLES 43 received at heights l^ , 1 2 etc. above any datum level, and let Z be the height above the same datum level at which the water leaves the wheel. If the wheel is perfectly efficient (and here again the test of perfect efficiency is reversibility) the work done is ^i (*i - W + ^2 (I* ~ W + ^3 (*s ~ W + etc. Comparing the two cases we see that the quantity Q 1 /T 1 is the analogue in the heat-engine of M 1 in the water-wheel, Q 2 /^2 is the analogue of M 2 , and so on. The amount of work which can be got out of a given quantity of heat by letting it down to an assigned level of temperature is not simply proportional to the product of the quantity of heat by the fall of temperature, but to the product of Q/T by the fall of temperature. On the strength of this analogy Zeuner has called the quantity Q/T the "heat weight" of a quantity of heat Q obtainable at a temperature T. Another way of putting the matter has a wider application. Let the engine as before take in quantities of heat represented by Qi , Q 2 , Q 3 etc. at 7\ , T 2 , T 3 , and let it reject heat at T', T", T'" etc., the quantities rejected being respectively Q', Q", Q'" etc. Then by the principle that in a reversible cycle the heat rejected is to the heat taken in as the absolute temperature of rejection is to the absolute temperature of reception, we have Q' Q" Q"' Qi Q 2 Qs jL i :: __ i :* __ i _ _r 4_ _r? 4_ _5? J_ rr\i i rpff i rpttt i rri i rjj t rri i ) from which S^=0, when the summation is effected all round the reversible cycle. In this summation heat taken in is reckoned as positive and heat rejected as negative. If the cycle is not reversible, the heat re- jected will be relatively greater, and therefore, for a non-reversible cycle, 2 (Q/T) will be a negative quantity. Some of the processes may be such that changes of temperature are going on continuously while heat is being taken in or given out, and if so we cannot divide the reception or rejection of heat into a limited number of steps, as has been done above. But the equa- tion may be adapted to the most general case by writing it integration being performed round the whole cycle. This holds for any internally reversible cycle. It means that when 44 THERMODYNAMICS [CH. a substance has passed through any series of reversible changes which cause it to return to its initial state, the quantities of heat which it has taken in and given out are so related to the tem- perature of the substance at each stage as to make this integral vanish for the cycle as a whole. If the cycle is not reversible SdQ/T is a negative quantity, because the amount of heat rejected is relatively larger than when the cycle is reversible. 44. Entropy. We have now to introduce an important thermo- dynamic quantity which serves many useful purposes. The Entropy of a substance is a function of its state which is most conveniently denned by reference to the heat taken in or given out when the state of the substance undergoes change in a reversible manner. In any such change, the heat taken in or given out, divided by the absolute temperature of the substance, measures the change of entropy. Thus if a substance which is either expanding reversibly or not expanding at all takes in heat SQ when its temperature is T, its entropy increases by the amount SQ/T. We shall see that the entropy of any substance in a definite state is a definite quantity, which has the same value when the substance comes back again to the same state after undergoing any changes. To give the entropy a numerical value we must start from some arbitrary point where, for convenience of reckoning, the entropy is taken as zero. We are concerned only with changes of entropy, and consequently it does not matter, except for convenience, what zero state is chosen for the purpose of calculating the entropy. \ Starting then from any suitable zero, each element SQ of the heat taken in has to be divided by T, which is the absolute tempera- ture of the substance when SQ is being taken in. The sum measures the entropy of the substance, on the assumption that no irreversible change of state has occurred during the process. We shall denote the entropy of any substance by . If the tem- perature is changing continuously while heat is being taken in, the change of entropy from any state a to any other state b is , [ dQ. -*.=] -Y provided there is no irreversible action within the substance during its change of state. ij FIRST PRINCIPLES 45 This definition of the entropy of a substance as a quantity which is to be measured by reckoning | -~ while the substance a * passes by a reversible process from any state a to any other state b, is consistent with the fact that the entropy is a definite function of the state of the substance, which means that it has only one possible value so long as the substance is in the same state. To prove this we must show that the same value is obtained for the entropy no matter what reversible operation be followed in passing from one state to the other: in other words, that is the same a -* for all reversible operations by which a substance might pass from state a to state b. Consider any two reversible ways of passing from state a to state b. If we suppose one of them to be reversed the two together will form a complete cycle for which (by Art. 43) = 0. Hence | ~ for one of them must be the same as J J> a *- for the other. It is therefore a matter of indifference, in the reckon- ing of entropy, by what " path " or sequence of changes the substance passes from a to b provided it be a reversible path: starting from any zero state the reckoning of the entropy in a given state will always give the same value, which shows that the entropy is simply a function of the actual state and does not depend on previous conditions. It is chiefly because the entropy of a substance is a definite function of the state, like the temperature, or the pressure, or the volume, or the internal energy, that the notion of entropy is im- portant in engineering theory. The entropy of a substance is usually reckoned per unit of mass, and numerical values of it reckoned in this manner are given in tables of the properties of steam and of the other substances which are used in heat-engines and refrigerating machines. But we may also reckon the entropy of a body as a whole when the state of the body is fully known, or the change of entropy which a body undergoes as a whole when it takes in or gives out heat. And we may also reckon the total entropy of a system of bodies by adding together the entropies of the several bodies that make up the system. 45. Conservation of Entropy in Carnot's Cycle. As a simple illustration of the uses to which the idea of entropy may be put, 46 THERMODYNAMICS [en. consider the changes of entropy which a substance undergoes when it is taken through Carnot's cycle (Art. 32). All four opera- tions are reversible. In the first, which is isothermal expansion at Tj , the entropy of the substance increases by the amount Q i /T 1 where Q 1 is the amount of heat taken in from the hot source. In the second operation no heat is taken in or given out and there is no change of entropy. In the third operation a quantity of heat Q 2 is rejected at T 2 : the entropy of the substance accordingly falls by the amount Q 2 /^2 ^ n the fourth operation there is again no transfer of heat and no change of entropy. It is only in the first and third operations that changes of entropy occur. Moreover they are equal, for Q.ilT l = Q 2 /T 2 , which shows that the substance has the same entropy as at first, when it has returned to the original state. During the first operation, while it was taking in heat, its entropy rose from the initial value, which we may call b such that , , n /rr 9b = 9a + ^i/^l- During the third operation, while the substance was rejecting heat, its entropy fell again from cf> b to a , and ,. = >- Q 2 /2V Taking the cycle as a whole, the thermal equivalent of the work done by the substance is Q Q 2 , and is accordingly equal to Further, the source of heat has lost an amount of entropy equal to Qi/T lt and the receiver has gained an equal amount of entropy, namely Q 2 /T 2 We may therefore regard the reversible engine of Carnot as a device which transfers entropy from the hot source to the cold receiver without altering the amount of the entropy so transferred. The amount of heat alters in the process of transfer, for an amount of heat Q 3 Q 2 disappears, which is the thermal equivalent of the work done; but the amount of entropy in the system as a whole does not change. If, on the other hand, we had to do with an engine which is not reversible, working between the same source and receiver, Q 2 would be relatively larger, since less of the heat taken in is con- verted into work. Hence Q 2 /T 2 would be greater than QijT^ and the amount of entropy would therefore increase in the transfer. 46. Entropy-Temperature Diagram for Carnot's Cycle. It is instructive to represent the changes of entropy in a Carnot I] FIRST PRINCIPLES 47 cycle by means of a diagram the two coordinates of which are the entropy of the working substance and its temperature (fig. 5). The first operation (isothermal expansion) is represented by ab, a straight line drawn at the level of temperature 2\: during this operation the entropy of the substance rises from a to < 6 . This is followed by adiabatic expansion be during which the tem- perature falls but the entropy does not change. Then isother- mal compression cd at tempera- ture jT 2 , during which the entropy returns to the initial value. Finally adiabatic compression da com- pletes the cycle. The area of the closed figure abed measures (in heat units) the work done during the cycle. The area mabn measured to the base line, which is the absolute zero of temperature, is the heat taken in from the source. The area mdcn is the heat rejected to the receiver. These figures are rectangles. m Entropy All this is true whatever be the working substance. Neither in Art. 45 nor here is any assumption made as to that. The diagram (fig. 5) applies to any engine going through the reversible cycle of Carnot whether it use a gas (as in Art. 36) or any other substance. 47. Entropy-Temperature Diagrams for a series of Rever- sible Engines. We may apply this method of representation to exhibit the action of the imaginary chain of reversible engines which was used in Art. 42 to establish a thermody- namic scale of temperature. Starting from any temperature T 1 let a reversible engine take in heat at that temperature, and go through the Carnot cycle of opera- tions represented by the rectangle abed. For this purpose it takes in heat equivalent to mabn and rejects heat equivalent to mdcn. Let its rejected heat pass on to the next Temperature a d f h u 9 e n m Entropy Fig. 6 48 THERMODYNAMICS [CH. engine of the series, which goes through the Carnot cycle dcef, and let the interval of temperature df be so chosen as to make the work done by the second engine equal to the work done by the first. From the geometry of the figure it is obvious that this requires df to be equal to ad, so that the area abed may be equal to the area dcef. Similarly in order that the work done by the third engine should be the same, we must have//* = df= ad, and so on. Thus these intervals constitute equal steps in a scale of temperature which is* based entirely on thermodynamic considera- tions, the condition determining the steps being simply this, that the same amount of work shall be done by the heat as it passes down through each step. 48. No change of Entropy in Adiabatic Processes. It fol- lows from the definition of entropy given in Art. 44 that when a substance is expanded or compressed in an adiabatic manner (Art. 23) its entropy does not change. An adiabatic line is con- sequently a line of constant entropy, or, as it is sometimes called, an isentropic line. Just as isothermal lines can be distinguished by numbers T lf T 2 etc. denoting the particular temperature for which each is drawn, so adiabatic lines can be distinguished by numbers x , 2 etc. denoting the particular value of the entropy for each. We might accordingly define the entropy of a substance as that characteristic of the substance which does not change in adiabatic expansion or compression, and this definition would be consistent with the method of reckoning entropy described in Art. 44. It is only in a reversible process that the change of entropy of a sub- stance is to be determined by reference to the heat it takes in or gives out. The definition of an adiabatic process (Art. 23) excludes any process that is not reversible. 49. Change of Entropy in an Irreversible Operation. It is important in this connection to realize that a substance may in- crease its entropy without having any heat communicated to it from outside. When a substance expands in an irreversible manner, as by passing through a throttle-valve from a region of high pres- sure to a region of lower pressure, it gains entropy. Work is then done by the substance on itself, in giving energy of motion to each portion as it passes through the valve, and this energy of motion is frittered down into heat as the motion subsides through internal friction. The effect is like that produced by the communication i] FIRST PRINCIPLES 49 of some heat, though none is taken in from outside the substance. Expansion through a throttle- valve may be regarded as consisting of two stages. The first stage is a more or less adiabatic expansion during which the substance does work in setting itself in motion: the second stage is the loss of this motion and the consequent generation within the substance itself of an equivalent amount of heat. There is accordingly a gain of entropy, which occurs because the process as a whole is not reversible. We cannot directly apply the definition of entropy given in Art. 44 to determine the amount by which the entropy of a sub- stance changes in an irreversible operation such as throttling. But when the final state is known it is in general easy to calculate the entropy corresponding to that state, by considering the amount by which the entropy would have changed if the substance had come to that state by a reversible operation for which $dQ/T measures the change. When a substance has passed through any complete cycle of operations its entropy is the same at the end as at the beginning, for the original state has been restored in all respects. This is true of an irreversible cycle as well as of a reversible cycle. But for an irreversible cycle $dQ/T does not vanish. It has a negative value (Art. 43) and it does not measure change of entropy, for it is only in an internally reversible action 'that the change of entropy is dQ/T. 50. Sum of the Entropies in a System. It is instructive to enquire how the sum of the entropies of all parts of a thermo- dynamic system is affected when we include not only the working substance but also the source of heat and the sink or receiver to which heat is rejected. Consider a cyclic action in which the working substance takes in a quantity of heat Q 1 from a source at T 1 and rejects a quantity Q 2 to a sink at T 2 . W T hen the cycle is completed the source has lost entropy to the amount Q i /T l : the working substance has returned to the initial state, and therefore has neither gained nor lost entropy : the sink has gained entropy to the amount Q 2 /T 2 . If the cycle is a reversible one Q^IT^ = Q 2 /^*2 > and therefore the system taken as a whole, consisting of source, substance and sink, has suffered no change in the sum of the entropies of its parts. But if the cycle is not reversible the action is less efficient, Q 2 bears a larger proportion to Q and Q 2 /T 2 is greater than Qi/T i . Hence in an irreversible action the sum of the entropies of the system as a whole becomes increased. This E. T. 4 50 THERMODYNAMICS [CH. conclusion has a very wide application: it is true of any system of bodies in which thermal actions may occur. It may be expressed in general terms by saying that when a system under- goes any change, the sum of the entropies of the bodies which take part in the action remains unaltered if the action is reversible, but becomes increased if the action is not reversible. No real action is strictly reversible, and hence any real action occur- ring within a system of bodies has th'e effect of increasing the sum of the entropies of the bodies which make up the system. This is a statement, in terms of entropy, of the principle that in all actual transformations of energy there is what Lord Kelvin called a universal tendency towards the dissipation of energy*. Any system, left to itself, tends to change in such a manner as to increase the aggregate entropy, which is calculated by summing up the entropies of all the parts. The sum of the entropies in any system, considered as a whole, tends towards a maximum, which would be reached if all the energy of the system were to take the form of uniformly diffused heat; and if this state were reached no further transformations would be possible. Any action within the system, by increasing the aggregate entropy, brings the system a step nearer to this state, and to that extent diminishes the availability of the energy in the system for further transformations. This is true of any limited system. Applied to the universe as a whole, the doctrine suggests that it is in the condition of a clock once wound up and now running down. As Clausius, to whom the name entropy is due, has remarked, "the energy of the universe is constant: the entropy of the universe tends towards a maximum." An extreme case of therm odynamic waste occurs in the direct conduction of a quantity of heat Q from a hot part of the system, at T lt to a colder part at T 2 , no work being done in the pro- cess. The hot part loses entropy by the amount Q/2\: the cold part gains entropy by the amount Q/T 2 , and as the latter is greater there is an increase in the aggregate quantity of entropy in the system as a whole. 51. Entropy-Temperature Diagrams. We shall now con- sider, in a more general manner, diagrams in which the action of a substance is exhibited by showing the changes of its entropy in relation to its temperature. Such a diagram forms an interesting * Mathematical and Physical Papers, vol. I, p. 511. I] FIRST PRINCIPLES 51 and often useful alternative to the pressure-volume or indicator diagram. One example, namely the entropy- temperature diagram for a Carnot cycle, has already been sketched in fig. 5. Let ckf> be the small change of entropy which a substance under- goes when it takes in the small quantity of heat dQ at any tem- perature T, it being assumed that in the process the substance undergoes only a reversible change of state. Then, by the definition of entropy (Art. 44), d whence Td = and JTd0= the integration being performed between any assigned limits. Now if a curve be drawn with T and c/> for coordinates, fTcty is the area under the curve. This by the above equation is equal to fdQ, which is the whole amount of heat taken in while the substance passes through the states which that portion of the curve repre- sents. Let ab, fig. 7, be any portion of the curve of (/> and T. The area of the cross-hatched strip, whose breadth is 80 and height 2\ is T8, which is equal to 8Q, the heat taken in during the small change 80. The whole area mabn or JTd0 between the limits a and b is the whole heat taken in while the sub- stance changes in a reversible manner from the state represented by a to the state represented by b. Simi- larly, in changing reversibly from state b to state a by the line ba the substance rejects an amount of heat which is measured by the area nbam. The base line ox corresponds to the absolute zero of temperature. When an entropy-temperature curve is drawn for any complete cycle of changes it forms a closed figure, since the substance returns to its initial state. To find the area of the figure we have to inte- grate throughout the complete cycle, and provided there has been no irreversible action within the substance, Entropy X Fig. 7. Entropy -Temperature Curve. Q! being the heat taken in and Q 2 tne neat rejected. But the difference between these is the heat converted into work, hence = W, 42 52 THERMODYNAMICS [CH. when the integration extends round a complete cycle and W is expressed in thermal units. Thus an entropy-temperature diagram, so long as it represents changes of state all of which are reversible, but not otherwise, has the important property in common with a pressure-volume diagram that the enclosed area measures the work done in a complete cycle. But the entropy-temperature diagram has an advantage not possessed by the pressure-volume diagram, in that it exhibits not only the work done, but also the heat taken in and the heat rejected, by means of areas under the curves. An illustration of this has already been given in speaking of the Carnot cycle (Art. 46), and others will be found in Chapter III. 52. Perfect Engine using Regenerator. Besides the cycle of Carnot there is (theoretically) one other way in which an engine can work between a source and receiver so as to make the whole action reversible, and thereby transform into work the greatest possible proportion of the heat that is supplied. Suppose there is, as part of the engine, a body (called a "regenerator") into which the working substance can temporarily deposit heat,, while the substance falls in temperature from the upper limit T to the lower limit T 2 , and suppose further that this is done in such a manner that the transfer of heat from the substance to the regenerator is reversible. This condition implies that there is to be no sensible difference in temperature between the working substance and the material of the regenerator at any place where they are in thermal contact. Then when we wish the substance to pass back from T 2 to T we may reverse this transfer, and so recover the heat which was deposited in the regenerator. This alternate storing and restoring of heat serves instead of adiabatic expansion and com- pression to make the temperature of the working substance pass from T 1 to T 2 and from T 2 to T x respectively. It enables the tem- perature of the substance to fall to T 2 before heat is rejected to the receiver, and to rise to T x before heat is taken in from the source. This idea is due to Robert Stirling, who in 1827 designed an engine to give it effect. For the present purpose it will suffice to describe the regenerator as a passage (such as a group of tubes) through which the working fluid can travel in either direction, whose walls have a very large capacity for heat, so that the amount alternately given to or taken from them by the working fluid causes no more than an insensible rise or fall in their temperature. i] FIRST PRINCIPLES 53 The temperature of the walls at one end of the passage is T 1 , and this falls continuously down to T 2 at the other end. When the working fluid at temperature T 1 enters the hot end and passes through, it comes out at the cold end at temperature T 2 , having stored in the walls. of the regenerator a quantity of heat which it will pick up again when passing through in the opposite direction. During the return journey of the working fluid through the re- generator from the cold to the hot end its temperature rises from T 2 to T! by picking up the heat which was deposited when the working fluid passed through from the hot end to the cold. The process is strictly reversible, or rather would be so if the regenerator had an unlimited capacity for heat, if no conduction of heat took place along its walls from the hotter parts towards the cold end, and if there were no loss by conduction or radiation from its ex- ternal surface. A regenerator satisfying these conditions is of course an ideal impossible to realize in practice. 53. Stirling's Regenerative Air-Engine. Using air as the working substance, and employing his regenerator, Stirling made an engine which, allowing for practical imperfections, is the earliest example of a reversible engine. The cycle of operations in Stirling's engine was substantially this (in describing it we treat air as a perfect gas): (1) Air, which had been heated to T 1 by passing through the regenerator, was allowed to expand isothermally through a ratio r, taking in heat from a furnace and raising a piston. Heat taken in (per Ib. of air) = RT t log e r (by Art. 28). (2) The air was caused to pass through the regenerator from the hot to the cold end, depositing heat and having its tempera- ture lowered to T 2 , without change of volume. Heat stored in regenerator = K v (T x T 2 ). The pressure of course fell in propor- tion to the fall in temperature. (3) The air was then compressed isothermally at T 2 , through the same ratio r to its original volume, in contact with a receiver of heat. Heat rejected = RT 2 log e r. (4) The air was again passed through the regenerator from the cold to the hot end, taking up heat and having its temperature raised to T 1 . Heat restored by the regenerator = K v (T l T z ). This completed the cycle. 54 THERMODYNAMICS [CH. The efficiency is RT l log e r - RT 2 log e r _ T t - T 2 ~RT 1 log.r T! The indicator diagram of this action is shown in fig. 8. Stirling's engine is important, not as a present- day heat-engine (though it has been revived in small forms after a long interval of disuse), but because it is typical of the only mode, other than Carnot's plan of adiabatic expansion and adiabatic compression, by which the action of a heat-engine can be made reversible. A modified form of regenerative en- gine was devised later by Ericsson, who kept the pressure instead of the volume constant while the working substance passed through the regenerator, and so got an indicator diagram made up of Volume Fig. 8. Ideal Indicator diagram of Air-Engine with Regener- ator (Stirling). jj two isothermal lines and two lines of constant pressure. 1 The entropy-temperature diagram of a regenerative engine is of the type shown in fig. 9. r l The isothermal operation of taking in heatfat T is represented by db ; be is the cooling of the substance from T l to T 2 in its passage through the re- generator, where it deposits heat : cd is the isothermal rejection of heat at T 2 ; and da is the restoration of heat by the regenerator while the substance passes through it in the opposite direction, by which the tem- perature of the substance is raised from T 2 to Tj. Assuming the action of the re- generator to be ideally perfect, be and ad are precisely similar curves whatever be their form. The area of the figure is then equal to the area of the rectangle which would represent the ordinary Carnot cycle (fig. 5). The equal areas pbcq and ndam measure the heat stored and restored by the regenerator. When the working substance is air and the regenerative changes take place either under constant volume, as in Stirling's engine, Fig. 9. Entropy-tempera- ture diagram of perfect engine using a Regene- rator. FIRST PRINCIPLES 55 or under constant pressure, as in Ericsson's, the specific heat K being treated as constant, ad and be are logarithmic curves with the equation , g^ (f> = | Y~ = ^ lge T + constant, K being K v in Stirling's process and K v in Ericsson's. 54. Joule's Air-Engine. A type of air-engine was proposed by Joule which, for several reasons, possesses much theoretical interest. Imagine a chamber C (fig. 10) full of air (temperature T 2 ), which is kept cold by circulating water or otherwise; another chamber A heated by a furnace and full of hot air in a state of Fig. 10. Joule's proposed Air-Engine. compression (temperature 2\) ; a compressing cylinder M by which air may be pumped from C into A, and a working cylinder N in which air from A may be allowed to expand before passing back into the cold chamber C. We shall suppose the chambers A and C to be large, in comparison with the volume of air that passes in each stroke, so that the pressure in each of them may be taken as sensibly constant. The pump M takes in air from C, compresses it adiabatically until its pressure becomes equal to the pressure in A> and then, the valve v being opened, delivers it into A. The indicator diagram for this action on the part of the pump is the diagram fdae in fig. 11. While this is going on, the same quantity of hot air from A is admitted to the cylinder N, the valve u is Ihen closed, and the air is allowed to expand adiabatically in N until its pressure falls to the pressure in the cold chamber C. During 56 THERMODYNAMICS [CH. the back stroke of N this air is discharged into C. The operation of N is shown by the indicator diagram ebcf in fig. 11. The area fdae measures the work spent in driving the pump; the area ebcf is the work done by the air in the working cylinder N. The difference, namely, the area abed, is the net amount of work obtained by carrying the given quantity of air through a complete cycle. Heat is taken in when the air has its temperature raised Fig. 11. Indicator diagram in Joule's Air-Engine. on entering the hot chamber A. Since this happens at a pressure which is sensibly constant, the heat taken in QA = KV (^6 ~- ^V) where T b = T lt the temperature of A, and T a is the temperature reached by adiabatic compression in the pump. Similarly, the heat rejected Q - K (T T \ where T d = T 2 , the temperature of C. and T c is the temperature reached by adiabatic expansion in N. Since the expansion and compression both take place between the same terminal pressures, the ratio of expansion and compression is the same. Calling it r, we have y T (Art. 26), and hence also T/TT 7i Jt"m T~ ZV _ T T T -*- a -*- c * i Hence 'a T, Qc T t T c ' and the efficiency FIRST PRINCIPLES 57 This is less than the efnciencyof a perfect engine working between the same limits of temperature ( - 1 -J because the heat is not taken in and rejected at the extreme temperatures. The atmosphere may take the place of the chamber C: that is to say, instead of having a cold chamber, with circulating water to absorb the rejected heat, the engine may draw a fresh supply at each stroke from the atmosphere, and discharge into the atmosphere the air which has been expanded adiabatically in N. The entropy-temperature diagram for this cycle is drawn in fig. 12, where the letters refer to the same stages as in fig. 11. After adiabatic compression da, the air is heated in the hot chamber A, and the curve ab for this process has the equation = K,(] g.T-log.T^. Fig. 12. Entropy- temperature dia- gram in Joule's Air- Engine. Then adiabatic expansion gives the line be, and cd is another logarithmic curve for the rejection of heat to C by cooling under constant pressure. The ratio -^ , which is represented by -=- in J- 5 CO fig. 11 and by -j- in fig. 12, shows the proportion which the volume of the pump M must bear to the volume of the working cylinder N. The need of a large pump would be a serious draw- back in practice, for it would not only make the engine bulky but would cause a relatively large part of the net indicated work to be expended in overcoming friction within the engine itself. In the original conception of this engine by Joule it was in- tended that the heat should reach the working air through the walls of the hot chamber, from an external source. But instead of this we may have combustion of fuel going on within the hot chamber itself, the combustion being kept up by the supply of fresh air which comes in through the compressing pump, and, of course, by supplying fuel either in a solid form from time to time through a hopper, or n a gaseous or liquid form. In other words, the engine may operate as an internal- combustion engine. Internal-combustion engines, essentially of the Joule type, em- ploying solid fuel have been used on a small scale, but by far the 58 THERMODYNAMICS [CH. i most important development of the type is to be found in engines which work by the explosion or burning of a mixture of air with combustible gas or the vapour of a combustible liquid. The thermodynamics of internal-combustion engines will be con- sidered in a later chapter. We shall also see later (Chapter IV) that a practicable re- frigerating machine, using air for working substance, is obtained by making Joule's Air-Engine work as a heat-pump. CHAPTER II PROPERTIES OF FLUIDS 55. States of Aggregation. In the previous chapter the only substances whose properties were discussed were imaginary ones, namely perfect gases. We have now to treat of real substances, such as steam, carbonic acid, or ammonia, which serve as work- ing substances in heat-engines or refrigerating machines, and to examine their action and properties in the light of thermodynamic principles. Any such substance may exist in three states of aggregation, solid, liquid and gaseous. We are mainly concerned with the liquid and gaseous states, in either of which the substance is spoken of as a fluid. The working fluid in an engine is often a mixture of the same substance in the two states of liquid and vapour; but in some stages of the action it may consist entirely of liquid, in others entirely of vapour. The vapour of a substance may be either saturated or superheated. A vapour mixed with its liquid, and in equilibrium with it, must be saturated. Any attempt to heat the mixture would result in more of the liquid turning into saturated vapour. But when a vapour has been removed from its liquid it may be heated to any extent, thereby becoming superheated. Thus when steam is formed in a boiler it is necessarily saturated when the bubbles leave the water, but it may be superheated on its way to the engine by passing through hot pipes which cause its temperature to rise above that of the boiler. Any of the so-called permanent gases, such as hydrogen, or oxygen or nitrogen, is a superheated vapour which can be reduced to the saturated condition by greatly lowering its temperature. At any one pressure the saturated vapour of a substance can have but one temperature: the superheated vapour at the same pressure may have any temperature higher than that. In the change of state from solid to liquid, and again in the 60 THERMODYNAMICS [CH. change from liquid to vapour, heat is taken in, though the substance does not rise in temperature while the change is going on. The heat so taken in was said in the phraseology of old writers to be- come latent, and the name Latent Heat is still applied to it. Thus the heat taken in by unit mass of a substance in passing, without change of pressure, from the solid to the liquid state is called the latent heat of the liquid, and the heat taken in by unit mass in passing, without change of pressure, from the state of liquid to 'that of vapour is called the latent heat of the vapour. The latent heat of water is 80 thermal units, which means that unit mass of ice takes in 80 thermal units while it melts, the thermal unit being one-hundredth part of the quantity of heat required to warm a unit mass of water from to 100 centigrade. The temperature at which ice melts is only very slightly affected by the pressure (see Art. 99), and the latent heat of water is practically the same at all pressures ordinarily met with. If we assume the pressure to be one atmosphere, ice melts at the tem- perature which is taken for the lower fixed point (0 C.) in gradua- ting a thermometer (Art. 15). At a pressure of one atmosphere water boils at the temperature which is taken for the upper fixed point of the thermometer (namely 100 C.), and the latent heat of the vapour is 539-3 thermal units. We shall see immediately that the temperature at which the change from liquid to vapour occurs, and also the amount of heat taken in during the change, depends greatly on the pressure. At higher pressures the temperature of boiling is higher and the amount of latent heat is less. In describing the properties of fluids it will save circumlocution to speak usually of water, taking it as typical of the rest. It is itself of special interest to the engineer, being the working substance of the steam-engine, and the numerical values by which its pro- perties are expressed are. better known than those that relate to other fluids. But the definitions and thermodynamical principles which will be stated must be understood as applying to fluids in general. We have now to consider in more detail some of the points that have been briefly summarized in this Article. 56. Formation of Steam under Constant Pressure. The properties of steam, or of any other vapour, are most conveniently stated by referring in the first instance to what happens when it is nj PROPERTIES OF FLUIDS 61 formed under constant pressure. This is substantially the process which occurs in the boiler of a steam-engine when the engine is at work. To fix the ideas we may suppose that the vessel in which steam is to be formed is a long upright cylinder fitted with a frictionless piston which may be loaded so that it exerts a constant pressure on the fluid below. Let there be, to begin with, at the foot of the cylinder a quantity of water (which for convenience of state- ment we shall take as one unit of mass, 1 Ib. say), and let the piston rest on the surface of the water with a pressure P. Let heat now be applied to the bottom of the cylinder. As heat enters the water it produces the following effects in three stages : (1) The temperature of the water rises until a certain tem- perature T s is reached, at which steam begins to be formed. The value of T s depends on the particular pressure P which the piston exerts. Until the temperature T s is reached there is nothing but water below the piston. (2) Steam is formed, more heat being taken in. The piston, which is supposed to continue to exert the same constant pressure, rises. No further increase of temperature occurs during this stage, which continues until all the water is converted into steam. During this stage the steam which is formed is saturated. The volume which the piston encloses at the end of this stage the volume, namely, of unit mass of saturated steam at pressure P and con- sequently at temperature T 3 will be denoted by V s . (3) If more heat be allowed to enter after all the water has been converted into steam, the volume will increase and the tem- perature will rise. The steam is then superheated: its temperature is above the temperature of saturation. 57. Saturated and Superheated Steam. The difference between saturated and superheated steam may be expressed by saying that if water (at the temperature of the steam) be mixed with steam, some of the water will be evaporated if the steam is superheated, but none if the steam is saturated. Steam in contact with water, and in thermal equilibrium with it, is necessarily saturated. When saturated its properties differ considerably, as a rule, from those of a perfect gas, but when superheated they approach "those of a perfect gas more and more closely the farther the process of superheating is carried, that is to say, the more the temperature is raised aboveT s ,the temperature of saturation corre- sponding to the given pressure P. 62 THERMODYNAMICS [CH. 58. Relation of Pressure to Temperature in Saturated Steam. The temperature T s at which steam is formed under the conditions described in Art. 56, which is called the temperature of saturation, depends on the value of P. The relation of pressure to the temperature of saturation was determined with great care by Regnault, in a series of classical experiments to which much of our knowledge of the properties of steam is due*. Regnault's obser- vations extended from temperatures below the zero of the centi- grade scale, where the vapour whose pressure was measured was that given off by ice, up to 220 C. The pressures found by him, expressed in millimetres of mercury, were as follows, omitting those below C. as not relevant to steam-engine calculations : Pressure of saturated steam Temperature C. in mm. of Mercury 4-60 25 23-55 40 54-91 50 91-98 75 288-50 100 760-00 130 2030-0 160 4651-6 190 9426 220 17390 It will be seen from these figures that the pressure of saturated steam rises with the temperature at a rate which increases rapidly in the upper regions of the scale. Various empirical formulas have been devised to express the relation of pressure to temperature in saturated steam and to allow tables to be calculated in which inter- mediate values are shown. When a table is available, however, it is more convenient to find the pressure corresponding to a given temperature, or the temperature corresponding to a given pressure directly from it, either interpolating or drawing a portion of the curve connecting pressure with temperature when the values con- cerned lie between those that are stated in the table. 59. Tables of the Properties of Steam. At the end of this book a number of Tables will be found showing not only the re- lation of the pressure to the temperature of saturation, but also various other properties of steam which are of use in engineering * M4m. Inst. France, 1847, vol. xxi. An account of Regnault's methods of experiment and a statement of his results expressed in British measures will be found in Dixon's Treatise on Heat (Dublin, 1849). n] PROPERTIES OF FLUIDS 63 calculations. Tables of the properties of steam have been calculated by Professor Callendar, by methods which will be explained later, and have been published under the title of The Callendar Steam Tables*. From Callendar's tables, which give the most authori- tative results now available, a selection has been made, with his permission, for the purposes of this book. The figures which are given for the pressure of saturated steam at various temperatures are not taken directly from the measure- ments of Regnault, but are inferred from a characteristic equation which Callendar has devised to express the relation between pres- sure, volume and temperature within the working range. The validity of that equation (within the range to which the tables apply) is demonstrated by the general agreement of the quantities calculated from it with the best experimental results, in measure- ments not only of the pressure at saturation but of other properties of steam. The pressures, however, which are stated in these tables do agree very closely with the results of Regnault's observations quoted above. It is only at the highest pressures that an appreci- able difference will be found, and even there it is not material. In other respects the Callendar tables will be found to differ somewhat widely from the earlier tables of such authorities as Rankinef or ZeunerJ, which have been accepted as standards and copied into many text-books. When these were calculated the only available data of value were those furnished by the experiments of Regnault. But more recent researches have supplied additional data which in some particulars modify his, and it is now clear that Regnault's figures require revision and in some cases considerable amendment. The various properties of steam, or of any other vapour, are linked together in such a manner that the relations between them must satisfy certain thermodynamic equations. This affords a test of consistency, and in the light of such investigations the figures given in the old tables are now known to be not even mutually consistent. Callendar's tables give a set of values that are* consistent amongst themselves and are also in good agree- ment with the most trustworthy experimental results. Further re- searches may in time lead to a still closer adjustment of the figures to the results of observation, but Callendar's values for the various * London, Edward Arnold, 1915. Students should obtain a copy of these Tables, which contain fuller particulars than are quoted here. f Rankine, A Manual of the Steam Engine and other Prime Movers. J Zeuner, Technische Thermodynamik, vol. n. (Trans, by J. F. Klein, 1907-). 64 THERMODYNAMICS [CH. quantities may be accepted not only as mutually consistent, from the thermodynamic point of view, but as certainly correct enough for the purposes of the engineer. 60. Relation of Pressure to Volume in Saturated Steam. Among the quantities shown in the tables is the volume F fi , in cubic feet per lb., of saturated steam at various temperatures and at various pressures. The volume of a given quantity of saturated steam at any assigned temperature or pressure is a quantity difficult to measure by direct experiment, and the volumes which are given in steam tables are generally inferred from the results of experiments on other properties which can be more easily measured. Successful measurements of volume have however been carried out* and the results are in general agreement with the figures stated in these tables. The relation of P to F 8 in saturated steam is approximately expressed by an empirical formula PF/* = constant. With P in pounds per sq. inch and V s in cubic feet per lb. this gives PF S H = 490. WithP in kilogrammes per square centimetre and V s in cubic metres per kilogramme, it becomes PF S H - 1-786. This formula applies well from a pressure of say 1 pound per square inch up to 300 pounds per square inch. Within these limits it gives values which agree to one part in a thousand with those in the tables. The student will find it useful to draw curves, with the data of the tables, showing the relation between the pressure and the temperature of saturated steam, and also the relation of pressure to volume, especially within the range usual in steam-engine practice. He will observe that the rate of change of pressure with respect to change of temperature increases rapidly as the temperature rises, and hence that in the upper part of the range a very small elevation of temperature in a boiler is necessarily associated with a large increment of pressure. The pressure shown by a pressure-gauge on a boiler is the excess of pressure in the boiler above the pressure of the atmosphere. * See especially 0. Knoblauch, R. Linde and H. Klebe, Mitteilungen tiber For- schungsarbeiten herausgegeben vom Verein deutscher Ingenieure, Heft 21, 1905. n] PROPERTIES OF FLUIDS 65 Consequently the true or "absolute" pressure in the boiler is to be found by adding, to the reading of a correct gauge, the pressure which corresponds to the height of the barometer at the time ; this is generally about 14- 7 pounds per square inch or 1*033 kilogrammes per square centimetre. 61. Boiling and Evaporation. The familiar case of water boiling in a kettle or other open vessel is only a special example of the formation of steam under constant pressure. There the constant pressure is that of the atmosphere, and consequently the tempera- ture at which the water boils is about 100 C.* Water in the open evaporates slowly at any temperature lower than that at which it boils. Though the pressure of the vapour so formed is lower than that of the atmosphere and may be very much lower the vapour is able to escape from the surface by diffusion : the atmosphere is not displaced and the pressure on the surface of the water is still that of the air. As the temperature of water in the open is raised this slow evaporation from the surface becomes more rapid, but it is only when the temperature reaches the value which corresponds (for saturated steam) to the given atmo- spheric pressure that the water boils : the vapour is then formed in bubbles at the pressure of the atmosphere, and it escapes not by diffusion but by displacing the superincumbent air. 62. Mixture of Vapour with other Gases: Dalton's Prin- ciple. In what has been said about the relation of pressure and volume to temperature in the saturated state, it has been assumed that in the process of formation there is simply a mixture of the liquid with its vapour, no other substance being present. This is substantially true in a steam boiler or in the evaporator of a refrigerating machine. But the case is different when the vapour has to mix with another gas or gases. A principle discovered by Dalton then applies, that the pressure in any closed space con- taining a mixture of two or more gases at any given temperature is very approximately equal to the sum of the pressures which each of the gases would exert separately if the others were absent, that is to say if each of the gases (at the same temperature) alone occupied the whole space. These pressures, which are added together to make up the actual pressure, are called "partial * Water in the open boils at 100 C. when the atmospheric pressure has its standard value, which corresponds to a barometer reading (corrected to 0C.) of 760 mm. at sea level in latitude 45, or 759-6 mm. in London (see Art. 12). E.T. 5 C6 THERMODYNAMICS [CH. pressures." An important instance of the application of Dalton's principle is considered in the next article. 63. Evaporation into a space containing Air: Saturation of the Atmosphere with Water-Vapour. When water evapor- ates in a closed space containing air, the process goes on until a definite amount of it has become mixed, as vapour, with the air already there. When this has happened, and a state of equilibrium is reached, the air is said to be saturated with water-vapour. The amount of water- vapour that a given volume of air will take up in this way depends upon the temperature : it is very nearly the same amount as would be required to fill the same space with saturated steam at that temperature if the air were not present. By Dalton's principle the pressure of the mixed gases, namely the air and the water-vapour mixed with the air, is very nearly the same as the sum of the pressures which each would exert separately: that is to say the pressure in the given space after the water-vapour has been formed is greater than the pressure which the air would exert in that space, if the water-vapour were not there, by an amount which is nearly equal to the pressure of saturated steam at the temperature of the mixture. It is approximately true to say that each of the constituents of the mixed atmosphere in the closed space behaves as if it occupied the whole volume, and contributes to the pressure just as if the other constituent were absent. This is very nearly accurate at ordinary pressures. It becomes less accurate when the pressure is high : the amount of water-vapour required to saturate the atmosphere is then somewhat less than the rule would require. As an example, suppose air at 25 C. (77 Fah.) to be saturated with water- vapour. At that temperature one Ib. of saturated steam would (by the Tables) occupy 692*4 cubic feet, and therefore one cubic foot weighs 0-00144 Ib. Consequently each cubic foot of the air takes up 0-00144 Ib. of water-vapour in reaching the state of saturation at that temperature. And since the corresponding pressure of water-vapour is 0-46 pound per sq. inch, the pressure in an enclosed space containing this moist air is greater by 0-46 pound per sq. inch than it would be if the water-vapour were removed and the dry air alone were left to fill the same space at the same temperature. In other words 0-46 pound per sq. inch is the "partial pressure" of the water- vapour present in the air under the assumed conditions. ii] PROPERTIES OF FLUIDS 67 When the amount of water-vapour present in air is less than enough to cause saturation the water-vapour is held in a super- heated state. If the temperature of the mixture be lowered, a point is reached at which the air becomes saturated, and any further lowering of the temperature causes some of the vapour to be de- posited as liquid on the walls of the containing vessel, or on any particles of dust that may be present. Any solid particles will serve as nuclei for condensation. The water condensed on such nuclei forms a mist of minute drops which fall so slowly that they seem to be held in suspension. The temperature at which water begins to be deposited from moist air is called the dew-point. Condensation of some of the water contained in air will also occur on any cold surface (colder than the dew-point) with which the air comes in contact : this results from local cooling of the air close to the surface in question. Thus in a refrigerating plant with pipes that convey a liquid colder than the freezing point through the warm atmosphere of the engine-room, a coating of ice forms round the pipes. For the same reason an effective way to dry air is to make it cold and drain away the water condensed in the process : at the lowest temperature the air remains saturated, but the amount of water required to saturate it at a low temperature is very small, and when it is allowed to become warm again without taking up more water it will be far from saturation. 64. Heat required for the Formation of Steam under Constant Pressure: Heat of the Liquid and Latent Heat. Return now to the imaginary experiment of Art. 56, where steam is formed under the constant pressure of a loaded piston, nothing but water or water-vapour being present and enquire what amount of heat has to be supplied in each stage of the operation. In the first stage the substance is wholly in the condition of water which is being heated from the initial temperature to T s , the temperature at which the second stage begins. During this first stage the heat taken in (per Ib. of the water) is approximately equal to one thermal unit for each degree by which the temperature of the water rises. It would be exactly equal to that if the specific heat of water were constant and equal to unity, but this is not the case. At about 30 C. the specific heat of water is less than unity; it passes a minimum value thereabouts of 0-9967, and then increases, becoming appreciably greater than unity at such temperatures as are found in steam boilers. Thus for instance to heat 1 Ib. of water 68 THERMODYNAMICS [CH. from C. to 80 C. requires 79-9-thermal units instead of 80. On the other hand, to heat it from C. to 200 C., under a pressure sufficient to prevent steam from forming, requires nearly 203-2 thermal units instead of 200. These figures will indicate how far it is legitimate to estimate the heat taken in during the first stage as one unit per degree. More accurate values of the heat of the liquid, that is to say the heat taken in during the first stage, can be found by means of the Steam Tables (see Art. 69). During this first stage, while the substance is still liquid, nearly all the heat that is taken in goes to increase the stock of internal energy. There is scarcely any external work done, for the volume is only slightly increased. Thus in heating water from C. to 200 C. (under a pressure of 225 24 pounds per sq. inch) the volume of the water changes from 0-0160 cubic ft. per Ib. to 0-0185. The external work done during this heating is therefore 225-24 x 144 x 0-0025 or 81 foot-pounds. This is equivalent to barely 0-06 thermal unit, and is negligible in comparison with the 203-2 units of heat that are taken in. In the second stage, the liquid changes into saturated steam with- out change of temperature. The heat that is taken in during this stage constitutes what is called the Latent Heat of the vapour. We shall denote it by L. Values of the latent heat of saturated steam are given in the tables. For steam formed under a pressure of one atmosphere (saturation temperature 100 C.) the latent heat is 539-3: with lower pressures of formation it is greater, and with higher pressures it is less. At the end of the second stage the sub- stance contains no liquid; it is spoken of as dry saturated steam: at any earlier point, when the substance consists partly of saturated steam and partly of water, it may be spoken of as wet steam. The latent heat of a vapour may be defined as the amount of heat which is taken in by unit mass of the liquid while it all changes into saturated vapour under constant pressure, the liquid having been previously heated up to the temperature at which the vapour is formed. A considerable part of the heat taken in during this process is spent in doing external work, since the substance expands against the constant pressure P. It is only the remainder of the so-called latent heat L that can be said to remain in the fluid and to con- stitute an addition to its stock of internal energy. The amount spent in doing external work during the second stage is AP (V, - V w ), n] PROPERTIES OF FLUIDS 69 where V s is the volume of the saturated vapour and V w is the volume of the liquid at the same temperature and pressure, A being the factor for converting units of work into thermal units. The excess of L above this quantity measures the amount by which the internal energy increases during the second stage. Thus for instance when water at 200 C. and a pressure of 225-24 pounds per sq. inch is converted into steam, of the 467-41 thermal units taken in, 47-61 units are spent in doing external wprk* and 419-8 units go to increase the stock of internal energy. 65. Total External Work done. In the two stages together the whole amount of external work done is to be found by taking the whole increase of volume and multiplying it by the pressure. If we assume that the water is originally at C. its volume may be taken as 0-0160. In converting water from C. to saturated steam at 200 C. under constant pressure the external work done is found thus to be equivalent to 47-67 thermal units: this is 0-06 units more than the external work of the second stage, for it in- cludes the small amount already referred to as having been done during the first stage. The whole increase of internal energy, from water at C. to saturated steam at any temperature, is equal to the whole amount of heat taken in, less the equivalent of the external work done. This in fact is only a particular example of the general principle stated in Art. 9, that when any substance ex- pands in any manner, taking in heat and doing work, the heat taken in is equal to the work done plus the increase of internal energy. In the case here considered the action is going on under constant pressure, but the statement applies to any change of state whatever. 66. Internal Energy of a Fluid. No matter what changes a substance may undergo, its internal energy will return to the same value when the substance returns to the same condition in all respects. In other words the internal energy is a function of the actual state of the substance and is independent of the way in which that state has been reached. Thus the internal energy of 1 Ib. of saturated steam at a particular pressure is a definite quantity which is the same whether the steam has been formed by boiling under constant pressure or in any other manner. Steam formed in a closed vessel of constant volume, for example, would have the same internal energy as steam at the same pressure but formed under conditions of constant pressure, though the amount of * The volume of the water is 0-0185 cubic ft. and of the steam 2-0738 cubic ft. The value of AP(V S - V w ) is therefore 47-61. 70 THERMODYNAMICS [CH. heat taken in during its formation would be different, for no external work is done in the process of formation in a closed vessel of con- stant volume. In that case the heat taken in would be equal to the increase of internal energy. We have no means of measuring the total stock of internal energy in a substance, and can deal only with changes in the stock. But by taking some arbitrary starting point as a zero from which the internal energy E is reckoned we can give E a numerical value for any other state of the substance. That value really expresses the difference from the internal energy in the zero state. The usual convention is to write E = when the substance is in the liquid condition at a temperature of C., and at a pressure equal to the vapour-pressure corresponding to that temperature. We may call this, for brevity, the zero state of the substance. Following this convention we take E = for water at C. The value of E for saturated water-vapour at C. will then be 564-21 thermal units (see Tables in Appendix). That this agrees with other figures in the tables will be seen by considering the conversion of water at C. to steam at C. under constant pressure. The only heat taken in is L, which is 594-27 units, and of this the external work AP (V s V w ) represents 30-06 units: the difference measures E. Values of E for saturated steam at various temperatures are given in the tables. It will be seen that they increase slowly with the temperature. 67. The "Total Heat " of a Fluid. We come now to another function of the state of any substance, a function which is of very great use in thermodynamic calculations. It is generally called the "Total Heat" and is represented* by the letter /. The "total heat" / is defined for any state of the substance by the equation I = E + APF. That is to say / is equal to the sum of the internal energy and the external work which would be done if the substance could be imagined to start from no volume at all and to expand to its actual volume, under a constant pressure equal to its actual pressure. Since the pressure, volume, and internal energy are all functions of the actual state, I is also a function of the actual state : its value is independent of how the state has been reached. In steam, for example, the heat taken in during formation depends on how the * Callendar in his Tables uses H to represent this function. In view of the fact that Rankine and other writers have used H in another sense the author prefers to use a different symbol. n] PROPERTIES OF FLUIDS 71 steam is formed, but the "total heat" / depends only on the final condition. The total heat can be calculated for any condition of a substance, whether in the state of liquid or of saturated or super- heated vapour. It is measured in thermal units per Ib. Values of the total heat of saturated steam and also of water under satura- tion pressure at various temperatures are given in the tables. The total heat of steam increases progressively with the temperature, rather more rapidly than does the internal energy. It follows from the definition of / that in the zero state of any substance, at which E is reckoned to be zero, / is not equal to zero but to a small positive quantity depending on the volume of the liquid and its pressure at that state. Since E is then zero / is equal to AP V Q , where P is the pressure at the zero state, namely the vapour- pressure at C., and V is the volume of the liquid at C. and pressure P . For water this quantity AP Q V Q is quite negligible, amounting as it does to 0-000146 thermal unit. For carbonic acid it is about 1 thermal unit, for ammonia and sulphurous acid it is much less. 68. Change of the Total Heat during Heating under Con- stant Pressure. An important property of the function / is that when any substance is heated under constant pressure the change of / is equal to the amount of heat taken in. To prove this, let Q be the amount of heat taken in while the substance expands under constant pressure P from a state in which the volume is V^ and the internal energy is E 1 to another state in which the volume is V 2 and the internal energy is E 2 . Then the amount of external work done is P (V 2 V-^) and, by the conservation of energy, Q-Ez-Et + AP (F 2 - F x ), which may be written Q = E 2 + APV< 2 - (E l + APV^ or Q = / 2 - I 19 where I I is the total heat in the first state and / 2 is the total heat in the second state. 69. Application to Steam formed under Constant Pressure, from Water at o C. The above proposition applies to every stage of the imaginary experiment of Art. 56. Referring to that experi- ment, assume that to begin with there is under the piston 1 Ib. of water at C. and at the pressure P at which steam is to be formed. By definition of the total heat, / = E + APV, 72 THERMODYNAMICS [CH. E at the beginning may be taken as zero *. Hence the value of / for the water at C. may be taken as APV Q , where F is the volume of 1 Ib. of water at C. and P is the pressure at which steam is to be formed. At the end of the first stage where I w represents the value of I for water at the temperature at which steam is about to form f. When values of /, are known this allows Q 1? the heat taken in during the first stage, to be more accurately calculated than by the rough method of Art 64. Values of I w are included in the steam tables. During the second stage an amount of heat equal to L is taken in at constant pressure, and the total heat changes from I w to I s , where I is the total heat of saturated steam. Hence The sum L + Q l is the whole heat of formation, in the experi- ment of Art. 56. Thus the "total heat" of steam is equal to the heat of formation under constant pressure, plus a small quantity which is the thermal equivalent of the work that would be done in lifting the piston far enough to admit the original volume of the water. The quantity APV forms a very small part of the "total heat": it is only 0-37 thermal unit when the temperature of formation is 200 C. and it is much less at lower temperatures. These remarks and the following tabular scheme will serve to show how the total heat of saturated steam (or other vapour) is related to the heat of formation under constant pressure. But the student should accustom himself to think of the total heat without reference to any process of formation, as a property which a substance possesses in its actual state a property which is just as simply a function of the state as is the temperature, or the pressure, or the volume, or the internal energy, or the entropy, which we shall have to consider presently J. * The convention of Art. 66 makes E = f or water at C. and pressure P . Here we have water at C. and pressure P, which is higher than P : but the higher pressure does not cause the internal energy of water at C. to differ appreciably from zero. f In Calendar's Tables this quantity I w is written h. J The function here called Jbhe total heat /, namely E + APV, was introduced by Willard Gibbs (Trans, of the Connecticut Academy, vol. m; Collected Scientific Papers, vol. I, page 92), and was first called the " Total Heat " by Callendar (Phil. Mag, 1903, vol. v, p. 50). Its great importance in technical thermodynamics was emphasized by Mollier, who employed it in charts for exhibiting the properties of steam and other substances. The use of such charts will be described later. II] PROPERTIES OF FLUIDS 73 j Total Heat, I \ 1 APV Internal energy acquired by the water in being heated from 0C., Internal energy External work acquired during done during change change of state from of state from water to steam, water to steam, L-AP(V B -V W ) AP(V.-V U ) External work AP done while the water is being heated, AP(V W -V ) \ Heat taken in during second stage, L Heat taken in during first stage, Q 1 70. Total Heat of a mixture of Liquid and its Saturated Vapour. It follows from Art. 68 that while a liquid is being con- verted into vapour, under constant pressure, the total heat / increases in proportion to the amount of vapour that is formed. At any intermediate stage in the process, if we call q the fraction that is vaporized and 1 q the fraction that is still liquid, the total heat of the mixture is 7 , /-, \ r ql s + (1 - q)I w , which may be written 7 4- L Similarly, while a vapour is being condensed under constant pressure, / becomes less by an amount measured by the heat given out, which is proportional, at any intermediate stage, to the fraction then condensed. 71. Total Heat of Superheated Vapour. When steam, or any other vapour, becomes superheated (as in the third stage of the experiment of Art. 56) by continuing the heating process under constant pressure after the saturated condition has been reached, the value of / becomes increased above the value I 8 , by an amount equal to the heat so taken in. We might find the total heat of superheated steam by calculating the supplementary amount taken in during the process of superheating, provided we knew the specific heat of the vapour during the process of heating it, under constant pressure, from its temperature of saturation to its actual tempera- ture. But this specific heat is not a constant: it diminishes slowly as the temperature rises, and it is greater at high pressures than at low pressures. A better way of finding the total heat in superheated steam is to use an equation, devised by Callendar, which gives the 74 THERMODYNAMICS L CH - total heat of the superheated vapour directly for any condition of temperature and pressure, without reference to the mode of forma- tion. This will be described in a later chapter, and a selection of numerical values will be found in the tables. From them the heat taken in during superheating at constant pressure may be found as /' I s , where /' is the total heat in the superheated state and I s the total heat in the saturated state at the same pressure. In engineering practice, the superheating of steam is generally carried out at constant pressure: the steam on leaving the boiler passes through a group of tubes forming a superheater, kept hot by the furnace gases, and while taking up heat from these tubes its pressure remains equal (or nearly equal) to that in the boiler. Super- heating is rarely carried further than 400 C. and not often so far. 72. Constancy of the Total Heat in a Throttling Process. An important property of the function /, in any substance, is that it does not change when the substance passes through a valve or other constricted opening, such as the porous plug of the Joule- Thomson experiment mentioned in Art. 19, by which it becomes throttled or "wire-drawn" so that its pressure drops. A practical instance of this kind of action occurs when steam passes through a partially closed orifice or "reducing valve." Eddies are formed in the fluid as it rushes through the constricted opening, and the energy expended in forming them is frittered down into heat as they subside. Fig. 13 To prove that / is constant in such an operation we shall consider what happens while a unit quantity of the substance passes through a constricted opening (as in fig. 13), and, to make the matter clear, imagine this unit quantity to be separated from the rest of the substance by two frictionless pistons, one of which (A) slides in the pipe that leads to the constriction and the other (B) slides in the pipe that leads away from it. On one side, as the substance comes up, let its pressure be P 2 , volume F x and internal energy E 1 . On the other side, after passing the constriction, let its pressure be P 2 , volume F 2 and internal energy E 2 . As each portion approaches the constriction, work is done upon it by the substance behind pushing n] PROPERTIES OF FLUIDS 75 in the imaginary piston A, and the amount of that work done while unit quantity is passing is P^Vi. After each portion has passed the constriction it does work upon the substance in front by pushing out the imaginary piston B, and the amount of that work is P 2 V 2 for the whole unit quantity. Any excess of the work done by the substance on piston B over the work done upon it by piston A must be supplied by a reduction in its stock of internal energy. Hence from which E 2 + AP 2 V 2 = E l + AP 1 V 1 , or /2 = Ji- Thus the total heat does not change in consequence of the thrott- ling. The imaginary pistons were introduced only to make the reasoning more intelligible ; the argument holds good whether they are there or not. It applies to any fluid, and to any action in which there is a frictional fall of pressure. We might accordingly describe the quantity / as that property of a substance which does not change in a throttling process*. 73. Entropy of a Fluid. In reckoning the entropy of a fluid we follow the same convention as in reckoning internal energy : the entropy of the liquid at C. is taken as zero. Consider, as before, a process in which the liquid is first heated under constant pressure and then vaporized at that pressure. During the heating of the liquid from an initial temperature T to any temperature T (on the absolute scale) the entropy increases by the amount f T dQ _ T ~ To where a is the specific heat at constant pressure. If cr could be treated as constant this would give on integration a(log e T-log 6 T ). In the case of water a is not far from constant and equal to unity. Hence a rough value of the entropy of water w at any temperature T is given by the expression log e T - log e 273. * It is assumed that no heat is taken in or given out, and "also that the velocity in the pipes is so small that no account need be taken of any difference in the kinetic energy of the stream in the pipes before and after passing the constriction, once the eddies have subsided. If the stream has acquired an appreciable amount of kinetic energy after the process, there will be a corresponding reduction in /. (See Art. 104.) 76 THERMODYNAMICS [CH. More accurate values of are obtained by using a formula devised by Callendar which will be given when the derivation of his tables is described (Chap. VIII). In the tables there is a column for the entropy of water at various temperatures, the pressure in each case being the saturation pressure at that temperature. It is the amount of entropy which the water has at the end of the first stage in Art. 56, when steam is just about to be formed. During the second stage an additional amount of heat L is taken in at constant temperature T s , namely the temperature at which steam is formed under the given pressure. Hence the entropy increases by the amount ^- , and we have, for the entropy of * s saturated steam, L s = w + 7p * s Values of cf> s are given in the tables. During superheating there is a further increase of entropy as the substance takes in more heat. The entropy of superheated steam at various pressures and temperatures will be found in one of the tables. It can be calculated by means of a formula which will be given later. 74. Mixed Liquid and Vapour : Wet Steam. In many of the actions that occur in steam-engines and refrigerating machines we have to do not with dry saturated vapour but with a mixture of saturated vapour and liquid. In the cylinder of a steam-engine, for example, the steam is generally wet; it contains a proportion of water which varies as the stroke proceeds. When any such mixture is in a state of thermal equilibrium the liquid and vapour have the same temperature, and the vapour is saturated. What is called the dryness of wet steam is measured by the fraction q of vapour which is present in unit mass of the mixture. When the dryness is known it is easy to determine other quantities. Thus, reckoning in every case per unit mass of the mixture, we have : Latent Heat of wet steam = qL = q (I s I w ) (1), Total Heat of wet steam, I Q = I w + qL = I s - (1 - q) L ...(2), Volume of wet sleam, V q = qV s + (1 - q) V w (3), which is very nearly equal to qV s unless the mixture is so wet as to consist mainly of water; Entropy of wet steam, Q = w + ^ = cf> s - (1 " g) L ...(4). * * n] PROPERTIES OF FLUIDS 77 From (2) it follows that when the total heat I q of wet steam is known, the dry ness may be found by the equation Combining (2) and (4), and eliminating q t we have I Q = I W + T s (fa - fa) (6), which is a convenient expression for finding the total heat of wet steam when the data are the temperature and the entropy. An alternative form is In these expressions I w is the total heat of water, and I s that of dry saturated steam, at the temperature of the wet mixture. All these formulas apply to a mixture of any liquid with its vapour. 75. Specification of the State of any Fluid. We have now spoken of the following quantities, which are functions of the state of the substance. They all depend on the actual state, not on how that state has been reached: The temperature, T. The pressure, P. The volume, V. The Internal Energy, E. The Total Heat, /. These four are reckoned per unit quantity of the substance. The Entropy, fa A substance may change its state in many different ways : it may for instance take in heat at constant volume or while expanding; it may expand or be compressed with or without taking in heat; expansion may take place through a throttle-valve or under a piston. But in any change of state whatever, the amount by which each of these quantities is altered depends only on what the initial and final states are, and not at all on the particular process by which the change of state has been effected. There are other quantities, such as the heat taken in, or the work done, which depend on how the change of state has taken place. In dealing with them we have to distinguish between one process of change and another, even when both processes bring the sub- stance from the same initial to the same final condition. 78 THERMODYNAMICS [CH. The working substance may be a liquid, or a mixture of liquid and vapour, or a dry-saturated or superheated vapour. The condition of a dry-saturated vapour is only a boundary condition between that of wetness and that of superheat. To specify completely the state at any instant it is enough to give either the pressure or the temperature and one of the other four quantities named in this list. Thus if P and V are given the state is fully denned : all the other quantities can then be determined, provided, of course, we have sufficient experimental knowledge of the characteristics of the substance. Or we may specify the state by giving another pair of quantities, such as T and (/>. or P and 7, or < and /. More generally, any two of these six quantities will serve as data in specifying the state, so long as the substance is homogeneous ; but when the substance is a mixture of liquid and vapour the pressure and temperature do not suffice without some other par- ticular such as the dryness q. With regard to these functions it may be useful to repeat here that T is constant in isothermal expansion; is constant in adiabatic expansion; / is constant in expansion through a throttle- valve or porous plug. 76. Isothermal Expansion of a Fluid: Isothermal Lines on the Pressure-Volume Diagram. A saturated vapour can expand isothermally only when it is wet: the process corresponds to the second stage in the experiment of Art. 56 ; it goes on at constant pressure and involves change of part of the liquid in the wet mixture into vapour. Similarly, isothermal compression of a wet vapour involves condensation of part of it. Isothermal lines on the pressure-volume diagram for a mixture of vapour and liquid are straight lines of uniform pressure. It is instructive to consider the general form of the isothermal lines as the fluid passes successively through the stages of being (1) entirely liquid, (2) a mixture of vapour and liquid, (3) entirely vaporous, by having its pressure gradually reduced under con- ditions such that the temperature remains constant throughout the process. Imagine for instance a cylinder to contain a quantity of the liquid under pressure applied by a loaded piston, and let the cylinder stand on a body at a definite constant temperature, which will supply enough heat to it to maintain the temperature un- changed when the pressure of the piston is gradually relaxed and n] PROPERTIES OF FLUIDS 79 the volume consequently increases. Starting from a condition of very high pressure, say at A (fig. 14), when the contents of the cylinder are wholly liquid, let the load on the piston be slowly reduced so that the pressure gradually falls. The contents at first remain liquid, until the pressure falls to the saturation value for the given temperature, namely the pressure at which vapour begins to form. Thus we have in the pressure-volume diagram a line A t B t to represent what happens while the pressure is falling during this first stage; the contents are then still liquid. The volume of the liquid increases, but only very slightly, in consequence of the pressure being relaxed, and hence A 1 B l in the diagram is nearly but not quite vertical. At B vapour begins to form, and continues forming until all the liquid becomes vapour. This is represented by B l C l , a stage during which there is no change of pressure. At Cj there is nothing but saturated vapour. Then, if the fall of pressure continues, a line C 1 D 1 is traced, the progressive fall of pressure being associated with a progressive increase of volume. The temperature, by assumption, is kept constant throughout. At D 19 or at any point beyond C 19 the vapour has become superheated, be- cause its pressure is lower than the pressure corresponding to saturation, and hence its tem- perature is higher than the temperature corresponding to saturation at the actual pres- sure. Any such line ABCD is an isothermal for the substance in the successive states of liquid (A to B), liquid and vapour mixed (B to C), saturated vapour (at C), superheated vapour (C to D). Now take a much higher temperature. We get a similar isothermal A 2 B 2 C 2 D 2 ; and at a still higher temperature another isothermal A 3 B 3 C 3 D 3 , and so on. The higher the temperature the nearer do B and C approach each other, and if the temperature be made high enough the horizontal portion of the isothermal line vanishes. VOLUME Fig. 14. Isothermal Lines. 80 THERMODYNAMICS [CH. 77. The Critical Point: Critical Temperature and Critical Pressure. A curve (shown by the broken line) drawn through B^BJEtz , etc. is continuous with one passing through C^CgCg , and it is only within the region of which this curve is the upper boundary that any change from liquid to vapour takes place. The branch B 1 B 2 B 3 , which shows the volume of the liquid, meets the branch CjCgCg, which shows the volume of the saturated vapour, in a rounded top. The summit of this curve represents a state which is called the Critical Point. The temperature for an isothermal line E that would just touch the top of this curve is called the Critical Temperature. We might define the critical temperature in another way by saying that if the temperature of a vapour is above the critical temperature no pressure, however great, will cause it to Jiquefy. The pressure at the critical point is called the Critical Pressure; at any higher pressure the substance cannot exist as a non-homogeneous mixture, partly liquid and partly vapour. Starting from D and increasing the pressure, the temperature being kept constant, we may trace any of the isothermals back- wards. The initial state is then that of a gas (a superheated vapour) . If the temperature is low enough we have a discontinuous process DCBA : as the pressure increases C is reached when the vapour is saturated and condensation begins : at B condensation is complete, and from B upwards towards A we are compressing liquid. At any point between C and B the substance exists in two states of aggregation; part is liquid and part is vapour. But if the tem- perature is above the critical temperature the isothermal is one that lies altogether outside of the boundary curve, shown by the broken line; in that case the substance does not suffer any sharp change of state as the pressure rises. It passes from the state of a gas to that of a liquid in a continuous manner, following a course such as is indicated by the lines F or G, and at no stage in the process is it other than homogeneous. The critical temperature for steam is about 365 C., and the corresponding pressure is about 2950 pounds per square inch. In the action of an ordinary steam-engine the critical point is never approached. But with carbonic acid, whose critical temperature is only about 31 C., the behaviour in the neighbourhood of the critical point, and above it, is of great practical importance in connection with refrigerating machines which employ carbonic acid as working substance. Gases such as air, hydrogen, oxygen and so forth, are vapours n] PROPERTIES OF FLUIDS 81 which under ordinary conditions are very highly superheated. Their critical temperatures are so low that it is only by extreme cooling that they can be brought into a condition which makes liquefaction possible. The critical temperature of hydrogen is 241 C. or 32 absolute. Even helium, the most refractory of the gases, has been liquefied, but only by cooling it to a temperature within about 5 degrees of the absolute zero. 78. Adiabatic Expansion of a Fluid. When a saturated vapour expands adiabatically it becomes wet; and if it is initially wet (unless very wet*) it becomes wetter. Its temperature, pressure, and total heat fall. The fact that its entropy remains unaltered allows the change of condition to be investigated as follows, if we assume that the liquid and vapour in the mixture are in thermal equilibrium throughout the process. For greater generality we shall suppose the vapour to be wet to begin with. Let the initial temperature be T and the initial dry- ness q l . In this state the entropy is L x being the latent heat of the vapour and Wi the entropy of the liquid, both at the temperature 2\. These quantities are found in the tables. Let the substance expand adiabatically to any lower temperature T 2 , at which the latent heat is L 2 and the entropy of the liquid is (f> W2 : we have to find the resulting value of the dryness, q 2 . The entropy may now be expressed as and since there has been no change of entropy this is equal to the initial value . Hence This equation serves to determine the dryness after expansion, and once it is known the volume V q is readily found as in Art. 74. Its exact value is q 2 V 8t + (1 # 2 ) PW,, which is practically equal in ordinary cases to q 2 V Sz , V ' s is equal to Si and the expression for the wetness after expansion to any temperature T 2 becomes = T2 As an example of the calculation, let steam initially dry and saturated at a temperature of 190 C. (P x = 182-1 pounds per sq. inch) expand adiabatically to a pressure of one atmosphere (tem- perature 100 C.). The entropy, which remains constant during expansion, is 1-5613, Wa is 0-3119, and L 2 is 539-3. With these data q 2 is 0-864, 13-6 per cent, of the steam has become liquefied, and the volume which was originally 2-534 cub. ft. per Ib. is 23-157 cub. ft. after expansion. Similarly, if the substance is entirely liquid in the initial state, the pressure being sufficient to prevent vapour from forming, adiabatic expansion will cause some of it to vaporize. Its initial entropy is w ^ , and since this does not change, ?--(, -&,), L 2 after expansion to a temperature T 2 . Thus, when water initially at 190 C., and at the corresponding saturation pressure of 182-1 pounds per sq. inch, expands adia- batically to a pressure of one atmosphere, q 2 becomes 0-154: in other words 15 per cent, of the water vaporizes in consequence of the expansion. The resulting volume is 4-127 cub. ft. per Ib. Conversely, if the wet mixture in this condition were compressed adiabatically it would become wetter during compression, and would be wholly condensed by compression when the pressure reached 182-1 pounds per square inch. An approximation to the form of the pressure- volume curve for the adiabatic expansion of wet steam is sometimes obtained by using an equation of the type PV m = constant, and selecting a value of the index m appropriate to the initial state. Zeuner gives for the index m the formula m = 1-035 + 0-1 and I b = E b + AP b V b ; from which I a -I b = E a - E b + A (P a V a - P b V b ) = A (area mabn + area eamo area/Zwo) = A (area eabf). This is true whatever be the condition of the fluid before expansion : it applies for example to superheated as well as to saturated or wet steam, or to any gas. Volume Fig. 15 n] , PROPERTIES OF FLUIDS 87 It may be instructive to the student to have the same proof put in a somewhat different form. From the equation which defines the total heat / in any state, namely, / = E + APV, we have by differentiation dl = dE + Ad (PV) = dE + APdV + AVdP. But in any small change of state it follows from the conservation of energy that the increase of internal energy plus the work done by the fluid is equal to the heat taken in, or dE + APdV = dQ, where dQ is the heat taken in during the change. Hence in any small change of state dl = dQ + AVdP. In an adiabatic operation dQ = 0, and hence in that case dl = AVdP. ' Therefore if the fluid expands adiabatically from state a to state /; the resulting decrease in its total heat, namely - / = A I" VAP. . h This integral is the area eabf of the pressure-volume diagram (fig. 15). It is the whole work done in a cylinder when the fluid is admitted at the pressure corresponding to state a, then expanded adiabatically to state b, and then discharged at the pressure corre- sponding to state b. The decrease of total heat in expansion, I a I b , is called the "Heat-drop." It is a quantity of much importance in the theory of heat-engines. The above equation shows that under adiabatic conditions the whole work done in the cylinder, when expressed in heat units, is measured by the heat-drop. In the next chapter this principle will be applied to infer from the heat- drop the work that can be done in steam-engines under various assumed conditions, and it will be shown how to calculate the heat- drop which occurs in adiabatic expansion from any initial state. CHAPTER III THEORY OF THE STEAM-ENGINE 81. Carnot's Cycle with Steam or other Vapour for Work- ing Substance. We are now in a position to study the action of a heat-engine employing water and steam, or any other liquid and its vapour, as the working substance. To simplify the first con- sideration of the subject as far as possible, let it be supposed that we have, as before, a long cylinder, composed of non-conducting material except at the base, and fitted with a non-conducting piston ; also a source of heat A at some temperature 2\ ; a receiver of heat, or as we may now call it, a condenser, C, at some lower temperature T 2 ; and also a non-conducting cover B (as in Art. 36). Then Carnot's cycle of operations can be performed as follows. To fix the ideas, suppose that there is unit mass of water in the cylinder to begin with, at the temperature T x . (1) Apply A, and allow the piston to rise against the constant pressure P 1 which is the saturation pressure corresponding to the temperature 2\. The water will take in heat and be converted into steam, expanding isothermally at the temperature 2\. This part of the operation is shown by the line ab in fig. 16. (2) Remove A and apply B. Allow the expansion to continue adiabatically (be), with falling pressure, until the temperature falls to T 2 . The pressure will then be P 2 , namely, the pressure which corresponds in the steam table to T 2 , which is the temperature of the cold body C. (3) Remove B, apply C, and compress. Steam is condensed by rejecting heat to C. The action is isothermal, and the pressure remains P 2 . Let this be continued until a certain point d is reached, which is to be chosen so that adiabatic compression will complete the cycle. (4) Remove C and apply B. Continue the compression, which is now adiabatic. If the point d has been rightly chosen, this will complete the cycle by restoring the working fluid to the state of water at temperature T t . CH. m] THEORY OF THE STEAM-ENGINE 89 The indicator diagram for the cycle is drawn in fig. 16, the lines be and da having been calculated by the method of Art. 78, for a particular example in which the initial pressure is 90 pounds per square inch (T x = 433), and the expansion is continued down to the pressure of the atmosphere, 14-7 pounds per square inch (T 2 = 373). a Isothermal ft sm i Fig. 16. Garnet's Cycle with water and steam for working substance. Since the process is reversible, and since heat is taken in only at T l and rejected only at T 2 , the efficiency (by Art. 38) is The heat taken in per unit mass of the liquid is L l9 and therefore the work done is L (T T ) a result which may be used to check the calculation of the lines in the diagram by comparing it with the area which they enclose. It will be seen that the whole operation is strictly reversible in the thermodynamic sense. 90 THERMODYNAMICS [CH. Instead of supposing the working substance to consist wholly of water at a and wholly of steam at b, the operation ab might be taken to represent the partial evaporation of what was originally a mixture of steam and water. The heat taken in would then be (q b <7 )L,/and as the cycle would still be reversible the area of the diagram would be 82. Efficiency of a Perfect Steam-Engine. Limits of Tem- perature. If the action here described could be realized in practice, we should have a thermodynamically perfect steam- engine using saturated steam. Like any other perfect heat-engine, an ideal engine of this kind has an efficiency which depends upon the temperatures between which it works, and upon nothing else. The fraction of the heat supplied to it which such an engine would convert into work would depend simply on the two temperatures, and therefore on the pressures, at which the steam was produced and condensed respectively. It is interesting therefore to consider what are the limits of temperature between which steam-engines may be made to work. The temperature of condensation is limited by the consideration that there must be an abundant supply of some substance to absorb the rejected heat; water is actually used for this purpose, so that T 2 has for its lower limit the temperature of the available water-supply. To the higher temperature T I and pressure P x a practical limit is set by the mechanical difficulties, with regard to strength and to lubrication, which attend the use of high-pressure steam. In steam motor-cars pressures of 1000 pounds per sq. inch have been used, but with engines and boilers of the ordinary construction the pressure ranges from about 300 pounds per sq. inch downwards. This means that the upper limit of temperature, so far as satur- ated steam is concerned, is about 215 C. A steam-engine, there- fore, under the most favourable conditions, comes very far short of taking full advantage of the high temperature at which heat is produced in the combustion of coal. From the thermodynamic point of view the worst thing about a steam-engine is the irre- versible drop of temperature between the combustion-chamber of the furnace and the boiler. The combustion of the fuel supplies heat at a high temperature : but a great part of the convertibility m] THEORY OF THE STEAM-ENGINE 91 of that heat into work is at once sacrificed by the fall in temperature which is allowed to take place before the conversion into work begins. If the temperature of condensation be taken as 20 C., as a lower limit, the efficiency of a perfect steam-engine, using saturated steam and following the Carnot cycle, would depend on the value of P l , the absolute pressure of production of the steam, as follows : Perfect steam-engine, with condensation at 20 C., P! in pounds per sq. inch being 50 100 150 200 250 300 Highest ideal efficiency = -288 -330 -355 -373 -384 -399 These numbers express what fraction of the heat taken in by the working substance would be convertible into work under the ideally favourable conditions of the Carnot cycle. But it must not be supposed that these values of the efficiency are actually attained, or are even attainable. Many causes con- spire to prevent steam-engines from being thermodynamically perfect, and some of the causes of imperfection cannot be removed. These numbers will serve, however, as one standard of comparison in judging of the performance of actual engines, and as illustrating the advantage of high-pressure steam from the thermodynamic point of view. We shall see in Art. 87 that there is another standard with which the performance of a real steam-engine may more appropriately be compared. 83. Entropy-Temperature Diagram for a Perfect Steam- Engine. The imaginary steam-engine of Art. 81 has the same very simple entropy-temperature diagram as any other engine which follows Carnot's cycle. The four operations are represented by the four sides of a rectangle (fig. 17). The first operation changes water (at the upper limit of temperature) into saturated steam at the same temperature; the entropy accordingly changes from (f> w to s . This is shown by the constant-temperature line ab in fig. 17. In the second operation which is adiabatic ex- pansion the entropy does not change, and the temperature falls to the lower limit, at which heat is to be rejected: this is repre- sented by the line of constant entropy be. In the third operation, cd, the temperature remains constant and the entropy is restored to its original value, heat being rejected to the cold body. In the fourth operation which is adiabatic compression the entropy does not change, and the temperature rises to the upper limit; 92 THERMODYNAMICS [CH. m Entropy Fig. 17 the substance has returned to its initial state in all respects. In order to be comparable with other diagrams which will follow, fig. 17 is sketched for a particular example in which P 1 i s 1 80 pounds per sq. inch, and P 2 is 1 pound per sq. inch: consequently ^ is 189-5 C. and / 2 is 38-7 C. Expressed in terms of entropy, the heat taken in (during ab) is T 1 (c/) s w ). This is represented by the area under ab measured down to the absolute zero of temperature, namely the area mabn. The heat rejected (during cd) is T 2 ((f> s cf) w ) and is repre- sented by the area ncdm. The thermal equivalent of the work done in the cycle is accordingly (T 1 - T 2 ) (cf) s - w ), and is represented by the area abed, enclosed by the lines which represent the four reversible operations. The efficiency is (y, - j.) 0. - &.) = 2\ - T. _ In the example for which the diagram is drawn, with the data stated above, the numerical value of this is 0-326. 84. Use of "Boundary Curves" in the Entropy-Tempera- ture Diagram. In fig. 18 the diagram of fig. 17 is drawn over again, with the addition of a curve through a which represents the values at various temperatures of (f> w , the entropy of water when steam is just about to form, and a curve through b which repre- sents at various temperatures the value of cf) s9 the entropy of dry saturated steam. These curves are called Boundary Curves. They are readily drawn from the data in the steam tables. Any point on the boundary curve through a would relate to the entropy of water; between the two curves any point in the diagram relates to a mixture of water and steam ; to the right of the boundary curve through b any point would relate to steam in the superheated state. We are not at present concerned with the outlying regions but only with the space between the two curves, within which the points c and d fall. Let the line cd be produced both ways to meet the Ill THEORY OF THE STEAM-ENGINE 93 boundary curves in e and s. Then the ratio of cs to es represents the fraction of the steam which becomes condensed during the adiabatic expansion be from the condition of saturation at b. To prove this we may first consider the meaning of any hori- zontal (isothermal) line such as se on the entropy-temperature diagram between the two boundary curves. It re- a L \b presents complete con- densation of 1 Ib. of dry saturated steam, under constant temperature and pressure. During its conversion from the con- dition of dry saturated steam (at s) to water (at e) the steam gives out a quantity of heat which is measured by the area under the line, namely the area osel. Any inter- mediate point in the line represents a mixture of water and steam; thus c m Entropy Fig. 18 represents a mixture which, though it has actually been produced by adiabatic expansion from 6, might have been produced by partial condensation from s under constant pressure, a process which would be represented by sc, or by partial evaporation under the same constant pressure from e, a process which would be represented by ec. Now if the mixture at c were completely condensed under constant pressure to e, the heat given out would be measured by the area ncel. This heat is given out by the condensation of that part of the mixture which consisted of steam. Hence the fraction which existed at c as steam, or in other words the dryness of the mixture at c, is measured by the ratio of the areas ncel to osel, which is equal to the ratio of the lengths ec to es. Hence also the ratio cs to es measures the wetness of the mixture at c. An entropy-temperature diagram on which the boundary curves are drawn therefore gives a convenient means of determining the wetness of steam at any stage in the process of adiabatic expansion. It is only necessary to draw a vertical line through the point repre- senting the initial condition. That line represents the adiabatic THERMODYNAMICS [CH. process, and the segments into which it divides a horizontal line drawn from one boundary curve to the other at any level of temperature represent the proportions of water and steam in the resulting mixture. This is true not only of the final stage, when adiabatic expansion is complete, but of any intermediate stage; for the argument given above obviously applies to a horizontal line drawn at any temperature between the two boundary curves. Similarly the point d which represents the wet mixture at the beginning of adiabatic compression da, shows by the ratio of segments ds to de what is the proportion of water to steam at which the third stage of the cycle has to be arrested, in order that adiabatic compression may bring the mixture wholly to the state of water when the cycle is completed by the operation da. The student should compare this graphic method of studying the wetness resulting from adiabatic expansion with the calcula- tions given in Art. 78. He will observe that both have the same basis. At any temperature T the length es of the isothermal line drawn from the water boundary curve at e to the steam boundary curve at s is L/T, and the intercept ec up to any intermediate point c on that line is qL/T 9 where q is the dryness of the mixture at the point c. The same principle of course holds for the entropy-tem- perature diagram of any other fluid. 85. Modified Cycle omitting Adiabatic Compression. Con- sider next a modification of the Carnot cycle of Art. 81. Let the first and second operations occur as they do there, but let the third operation be continued until the steam is wholly condensed. The substance then consists of water at T 2 , and the cycle / a is completed by heating it, in the condition of water, from T 2 to 2\. In the simple engine of Art. 81, where all the operations occur in a single ^ vessel, this could be done by increasing the pressure exert- Fig 19 ed by the piston from P 2 to P 15 after condensation is complete, then removing the cold body C and applying the hot body A. The water is therefore heated at P! and no steam is formed till the temperature reaches 2\. The pressure-volume diagram (or indicator diagram) of a cycle Ill THEORY OF THE STEAM-ENGINE 95 modified in this manner is shown by abce in fig. 19. The sketch is not drawn to scale. As before, ab is the operation of forming- steam, from water, at T l and P x ; be is adiabatic expansion from T l and Pj to T 2 and P 2 . Then ce is complete condensation at T 2 and P 2 . The fourth operation ea now involves two stages, first raising the pressure of the condensed water from P 2 to Pj and then heating it from T 2 to 2\. During both of these stages the changes of volume are negligible in comparison with those that take place in the other operations. The entropy-temperature diagram for this modified cycle is shown by abce in fig. 20, where the same letters as in fig. 19 are used for corresponding operations. As in the Carnot cycle, ab represents the conver- sion of a pound of water at T into dry saturated steam at T 13 and be re- presents its adiabatic ex- ^ 7 pansion to T 2 , resulting in a wet mixture at c, the dryness of which is mea- sured by the ratio ec/es. Then ce represents the complete condensation at T 2 of the steam in this wet mixture, and ea, which practically coincides with m the boundary curve, repre- sents the re-heating of the \ fntropy Fig. 20 condensed water from T 2 to T l , after its pressure has been raised to P l so that no steam is formed during this operation*. The working substance behaves reversibly throughout all these operations, and therefore the work done in the cycle is represented by the area abce in the entropy-temperature diagram of fig. 20. The diagram further exhibits the heat taken in and the heat re- jected. The whole heat taken in is measured by the area leabn, and of this the area learn measures the heat taken in during the last * The line ea in both diagrams, figs. 19 and 20, really stands for a broken line ea'a, where ea' represents the raising of pressure from P 2 to P x at constant tempera- ture T 2 , and a! a represents the heating from T 2 to T^ at constant pressure PI. In fig. 19 a' practically coincides with a; in fig. 20 a' practically coincides with e. 96 THERMODYNAMICS [CH. operation, while the water is being re-heated, and the area mabn measures the heat taken during the first operation, while the water is turning into steam. The area ncel measures the heat re- jected, namely during the condensing process ce. To express algebraically the work done in the cycle, refer to the indicator diagram, fig. 19, and let the lines ba and ce be produced to meet the line of no volume in j and k. Then, by Art. 80, the areajbck is an amount of work equivalent to the difference of total heats /-/, namely the "heat-drop" of a pound of steam in expanding adia- batically from the condition at b to the condition at c. The small area jaek is (P x P 2 ) V Wt where F Wg is the volume of a pound of water at T 2 , which we may take to be practically constant for the purposes of this calculation. Hence the expression I t -I e -A (P 1 - P 2 ) V Wl is the thermal equivalent of the work done in the cycle. If figs. 19 and 20 were both carefully drawn to scale for any particular example, a measurement of the enclosed area abce in either figure would give a result in agreement with this calculation. 86. Engine with Separate Organs. The importance of the modified cycle described in Art. 85 lies in'the fact of its being the Fig. 21 nearest approach to the Carnot cycle that can be aimed at when the operations of boiling, expanding and condensing are conducted in separate vessels. The imaginary engine of fig. 16 had one organ Illj THEORY OF THE STEAM-ENGINE 97 only a cylinder which also served as boiler and as condenser. We come nearer to the conditions that hold in practice if we think of an engine with separate organs, shown diagrammatically in fig. 21, namely a boiler A kept at T l9 a non-conducting cylinder and piston By and a surface condenser C kept at T 2 . To these must be added a feed-pump D which returns the condensed water to the boiler. Provision is made by which the cylinder can be put into connection with the boiler or condenser at will. With this engine the cycle of fig. 19 can be performed. An in- dicator diagram for the cylinder B is sketched in fig. 22. Steam is admitted from the boiler, giving the line jb. At b " cut-off " occurs, that is to say the valve which admits steam from the boiler to the cylinder is closed. The steam in the cylinder is then expanded adiabatically to the pressure of the condenser, giving the line be. At c the "exhaust" valve is opened which connects the cylinder with the condenser. The piston then returns, discharging the steam to the condenser and giving the line ck. The area jbck Fig. 22 Fig. 22 a represents the work done in the cylinder B. The condensed water is then returned to the boiler by the feed-pump, and the indicator diagram showing the work expended upon the pump during this operation is sketched in fig. 22 a. It is the rectangle keaj; where ke represents the up-stroke in which the pump fills with water at the pressure P 2 , and aj represents the down-stroke in which it discharges water to the boiler against the pressure P x . If we superpose the diagram of the pump on that of the cylinder we get their difference, namely abce (fig. 19), to represent the net amount of work done by the fluid in the cycle. It is the excess of the work done by the fluid in the cylinder over that spent upon it in the pump. Taking the two parts separately, the adiabatic heat-drop, B. T. 98 THERMODYNAMICS [CH. is the thermal Tequivalent of the work done by the fluid in the cylinder, and /p j> \v 1*1 ~~ r 1) y w a is the work spent upon the fluid in the feed-pump. Accordingly the difference, namely is, as before, the thermal equivalent of the work obtained in the cycle as a whole. 87. The Rankine Cycle. This cycle is commonly called the Rankine Cycle. Like the Carnot cycle it represents an ideal that is not practically attainable, for it postulates a complete absence of any loss through transfer of heat between the steam and the surfaces of the cylinder and piston. But it affords a very valuable criterion of performance by furnishing a standard with which the efficiency of any real engine may be compared, a standard which is less exacting than the cycle of Carnot, but fairer for comparison, inasmuch as the fourth stage of the Carnot cycle is necessarily omitted when the steam is removed from the cylinder before con- densation. A separate condenser is indispensable, in any real engine that pretends to efficiency. The use of a separate condenser was in fact one of the great improvements which distinguished the steam-engine of Watt from the earlier engine of Newcomen, where the steam was condensed in the working cylinder itself. The introduction of a separate con- denser enabled the cylinder to be kept comparatively hot, and thereby reduced immensely the loss that had occurred in earlier engines through the action of chilled cylinder surfaces upon the entering steam. But a separate condenser, greatly though it adds to efficiency in practice, excludes the compression stage of the Carnot cycle, and consequently makes the Rankine cycle the proper theoretical ideal with which the performance of a real engine should be compared. The efficiency of the Rankine cycle is less than that of a Carnot cycle with the same limits of temperature. This is because, in the Rankine cycle, the heat is not all taken in at the top of the range. In the Rankine cycle, as in Carnot's, all the internal actions of the working substance are, by assumption, reversible, and consequently each element of the whole heat- supply produces the greatest possible mechanical effect when regard is had to the temperature at which that element is taken in. in] THEORY OF THE STEAM-ENGINE 99 But part of the heat is taken in at temperatures lower than T lt namely while the working substance is having its temperature raised from T 2 to T in the fourth operation. Hence the average efficiency is lower than if all had been taken in at T lt as it would be in the cycle of Carnot. Each pound of steam does a larger amount of work in the Rankine cycle than it does in the Carnot cycle. This will be apparent when the areas are compared which represent the work in the corresponding diagrams : the area abce with the area abed in fig. 20. But the quantity of heat that has to be supplied for each pound in the Rankine cycle is also greater, and in a greater ratio: it is measured by the area leabn, as against mabn. Hence the efficiency is less in the Rankine cycle. One may put the same thing in a different way by saying that, in the Rankine cycle, of the whole heat-supply the part learn does only the comparatively small amount of work ead, and the remainder of the heat-supply, namely mabn, does the same amount of work as it would do in a Carnot cycle. 88. Efficiency of a Rankine Cycle. Taking in the first instance a Rankine cycle in which the steam supplied to the cylinder is dry and saturated, the whole amount of heat taken in is the quantity required to convert water at P l and T 2 into saturated steam at P x . This quantity is I S1 - {I w? + A(P 1 - P 2 ) F^}, for the total heat of the water at P and T 2 is greater than I Wt by the quantity A(P 1 -P 2 )V Wz . The work done is (by Art. 85) equal to the heat-drop minus the work spent in the feed-pump, or I S} I c A (P l P 2 ) V w ^ where I c is the total heat of the wet mixture after adiabatic expansion. The efficiency in the cycle as a whole is therefore The feed-pump term A (P x P 2 ) V w ^ is relatively so small that it is often omitted in calculations relating to ideal efficiency, just as it is omitted in stating the results of tests of the performance of real engines. In such tests it is customary to speak of the work done per Ib. of steam, without making any deduction for the work that has to be spent per Ib. in returning the feed- water to the boiler. But in the complete analysis of a Rankine cycle the feed- pump term has to be taken into account, and it is only then that the area of the entropy-temperature diagram gives a true measure of the work done. It should be clearly understood that the heat-drop, 72 100 THERMODYNAMICS [CH. by itself, is not an accurate measure of the work done in the Rankine cycle as a whole, nor is the heat-drop equal to the enclosed area of the entropy-temperature diagram, until the thermal equiva- lent of the work spent in the feed-pump has been deducted from it. If however we are concerned only with the work done in the cylinder of the ideal engine, then the heat-drop alone has to be reckoned. It is the exact measure of that work. The ratio of the heat-drop to the heat supplied shows what proportion of the supply is converted into work in the cylinder, under the ideal conditions of adiabatic action : it is a ratio nearly identical with the efficiency of the Rankine cycle, and even more useful as a standard with which to compare the performance of a real engine. In the actual per- formance of any real engine the amount of work done in the cylinder necessarily falls short of the adiabatic heat-drop because the working substance loses some heat to the cylinder walls. The extent to which it falls short is a matter for trial, and once that has been ascertained by trials of engines of given types, estimates may be made of the performance of an engine under design, using the adiabatic heat-drop as the basis of the calculation, with a suit- able allowance for probable waste. 89. Calculation of the Heat-Drop. It is therefore essential to be able to calculate the heat-drop in ideal engines under any assigned initial and final con- ditions. For this purpose we / \ have to find / c , the total e/ c \s heat of wet steam after adia- / \^ batic expansion. One way of doing so would be first to calculate the dryness q and then apply equation (2) of Art. 74, I q = I w + qL. But equations (6) and (7) of that Article give a more con- venient method, which is available here because we know the entropy of the mixture. These expressions may be directly obtained by considering what amount of heat the wet mixture would have to part with if it were to be wholly condensed, and what amount of heat it would have to take up if it were to be wholly evaporated, under the constant Fig. 23 m] THEORY OF THE STEAM-ENGINE pressure corresponding to the temperature of saturation T in either case. To bring a mixture at c (fig. 23) into the condition of water at e would require the removal of a quantity of heat equal to the area under ec, namely T ((/> w ), where < is the entropy at c and cf) w is the entropy of water (at e). On the other hand, to bring it to the condition of saturated steam would require the addition of a quantity of heat equal to the area under cs, namely T (< s ). Hence the total heat of the mixture at c is Of these two expressions the second is the more convenient because steam tables generally give more complete sets of values of (f> s than of c/> w . The entropy Si . Under these conditions the total heat after adiabatic expansion is I.~I H -T^ H -4^' and the heat-drop is i*-i.-i-i*+T t (t H -j. To take a numerical example, let the steam be supplied in a dry saturated state at a pressure P 1 of 180 pounds per square inch, and let it expand adiabatically to a pressure P 2 of 1 pound per square inch, at which it is condensed. With these data we find from the tables Z\ = 462-58, T 2 = 311-84, ^ = 1-5620, ^ = 1-9724, I Sl = 668-53, /, f = 612-46. Hence the total heat after adiabatic expansion to the assumed pressure of condensation is I c = 612-5 - 311-8 (1-9724 - 1-5620) = 484-5. >; C ^ 5. /^THERMODYNAMICS [CH. And the heat-drop, I Si - I c = 668-5 - 484-5 = 184-0. If we consider the Rankine cycle as a whole, the feed-pump term A (P 1 - P 2 ) V w is (180 - 1) 144 X 0-0161 1400 Deducting this from the heat-drop we have 183-7 pound-calories as the thermal equivalent of the net amount of work done in the Rankine cycle. The heat supplied is I 8i - I w ^ - A (Pi - P 2 ) V W2 = 668-5 - 38-6 - 0-30 - 629-6. Hence the efficiency of this Rankine cycle is i^l 7 = 0-2918. 629-6 This example will serve, incidentally, to show how unimportant is the feed-pump term. It reduces the amount of work done by less than one part in six hundred. If we had left it out of account, and taken the heat-drop in full as the numerator in reckoning the efficiency, the figure obtained would have been 0-2923: the difference is insignificant*. A Carnot cycle with the same limits of temperature would (Art. 81) have the efficiency 0-326. The difference between this and 0-292 shows the loss which results in the Rankine cycle from not supplying all the heat to the best possible thermodynamic advantage, namely at the top of the temperature range. It amounts in this instance to not quite 3| per cent, of the whole heat-supply. 90. The Function G. In his steam tables Callendar gives numerical values of a function G, defined by the equation G = Tcf> - I, which applies to steam in any state, wet, dry-saturated, super- heated, or supercooled. By the help of this function the process of calculating the heat-drop may be slightly shortened. G has the important property that it is constant during a process of evapora- tion or condensation at constant pressure. For in any step of such a * Accordingly a good approximation to the efficiency of the Rankine cycle is obtained by leaving out the term A (P x - P 2 ) ^w 2 in both numerator and denomi- nator of the complete expression in Art. 88, and writing it simply irA /.. -i,.' in] THEORY OF THE STEAM-ENGINE 103 process 87 = T8 and T is constant; consequently SG = 0. Hence the value of G for a wet mixture at temperature T and entropy <, such as the mixture at c (fig. 20) resulting from adiabatic expansion, is the same as G s , the (tabulated) value of G for dry-saturated steam at the same pressure. Therefore to find I c , the total heat of the wet mixture, we have The heat-drop is then determined as before, by subtracting I c from the total heat before expansion. Taking the same numerical example as in Art. 89, T is 311-84, c/) is 1-5620 and G s (for saturated steam at a pressure of 1 pound per square inch) is 2-61 by the tables. This gives I c = 311-84 x 1-5620 - 2-61 = 484-5, and the heat-drop from the dry-saturated state before expansion is 668-5 - 484-5 or 184 as before. Or we may obtain the heat-drop even more directly thus, when tabulated values of G are available. The relation / = Tcj> - G holds for any state of the substance. Hence between any two points (b) and (c) on the same adiabatic line the heat-drop /,-/.= (T, -T t )i- (G, - G.). In the present example G b is the value of G for saturated steam at P = 180, which (by the tables) is 54-10. G c is equal to the value for saturated steam at P = 1, which is 2-61. The difference of temperature T b T c is 150-74 degrees. Hence the heat-drop is 150-74 x 1-5620 - (54-10 - 2-61) = 184-0, which agrees with the result found above by less direct methods. The use of the function G in this connection is only a matter of convenience. The procedure in Art. 89 gives the heat-drop readily enough, though not quite so shortly, without the help of G*. * (with its sign reversed) is one of three functions to which Willard Gibbs gave the name of " Thermodynamic Potentials ": see his Scientific Papers, vol. I. He represented them by the symbols \f/ t %, and f. Of these, ^ is E - T. This function was called by Helmholtz the "Free Energy"; it is used in the theory of solution and other applications of thermodynamics to chemistry, a subject outside the scope of this book. The function x is the total heat /, namely E + APV, and is, as we have seen, of particular importance in the thermodynamics of engineering. The function f is E-T+APV or / -T; hence 0= -f. This function is useful in treating of the equilibrium of different states or "phases" of the same substance. One example of such equilibrium occurs in wet steam, which 104 THERMODYNAMICS [CH. 91. Extension of the Rankine Cycle to Steam supplied in any State. In the Rankine cycle described in Arts. 86-87 the steam was supplied to the cylinder in the dry-saturated state. But the term Rankine cycle is equally applicable whatever be the con- dition of the working substance on admission, whether wet, dry- saturated, or superheated. As regards the action in the cylinder, all that is assumed is that the substance is admitted at a constant pressure P 19 is expanded adiabatically to a pressure P 2 and is discharged at that pressure, and that in the process there is no transfer of heat to or from the metal, nor any other irreversible action. In these conditions the heat-drop in adiabatic expansion from P x to P 2 is the thermal equivalent of the area jbck in fig. 22 (compare also Art. 80) and therefore measures the work done in the cylinder, no matter what the condition of the substance on admission may be. This applies to wet steam or superheated steam just as much as to dry-saturated steam. 92. Rankine Cycle with Steam initially Wet. A complete Rankine cycle for steam that is wet on admission to the cylinder is shown on the entropy-temperature diagram by the figure ab'c'e (fig. 24). The point b' is placed so that the ratio ab' to ab is a mixture of two "phases," liquid and vapour. The functions \f/ and f or - will be referred to again in Chap. VII. From the engineering point of view it may be useful to point out that these functions have the following property. Referring to Art. 80, fig. 15, we have seen that when any fluid expands adiabatically from any state a to any other state b, the thermal equivalent of the area eabf, or A j VdP, is the heat-drop, /-/&; and that the area mabn or A\PdV is the loss of internal energy, E a -E b . Similarly, if ab in that diagram represent an isothermal process, we have two corresponding propositions, with regard to the functions G and \f/: When any fluid expands isothermally from any state a to any state b, the thermal equivalent of the area eabf, or AjVdP, is O a - G b ; and that of the area mabn, or AjPdV, is ^ a ~^b- To prove this, we have by definition \f/=E- T$. Hence in an isothermal process, But Td<(> is the heat taken in, which is equal to the gain of internal energy plus the work done, or Td(f) =dE + APd V. ri Therefore d^= -APdV, and ^ a -^ b =A I PdV. J a Again, we have by definition G T(f> - 1. Hence in an isothermal process, dG = Td-dL But (by definition) I=E + APV, from which = Td+AVdP. Therefore dG = - A VdP, and G a -G b =Al l> VdP. Ill] THEORY OF THE STEAM-ENGINE 105 is equal to q t the assumed dryness on admission. The line b'c' represents adiabatic expansion from P x to P 2 , c'e represents condensation at P 2 , and ea. re- presents as before the heating of the condensed water. The total heat before adia- batic expansion is I Wi + q-J^i or I 8l (1 qj L 1 and the heat supplied is the excess of this e quantity above Fig. 24 The entropy during adiabatic expansion is Wi + q.L./T, or < Si - (1 - The total heat after adiabatic expansion is The heat-drop is got by subtracting this from the total heat before adiabatic expansion. Or the heat-drop may be found, as soon as is calculated, by using the expression The efficiency which, as before, is practically equal to the heat- drop divided by the heat supplied, is slightly less than when the steam is saturated before expansion; the reason being that the proportion of heat supplied at the upper limit of temperature is now rather less, because part of the water remains unconverted into steam. As a numerical example let q 1 be 0-9, and let the limits of pressure be the same as in the example of Art. 89. Then the total heat per Ib. of the mixture before expansion, which is I Sj 0-lL l5 is 668-53 - 0-1 X 476-2 = 620-9. The heat supplied is 620-9 - 38-9 = 582-0. The entropy is i = 1-5620 - ~ 1 ' 4591 ' The total heat after expansion / Sj5 - T 2 (< S2 - ), or - G 2 , is 452-4; the heat-drop is therefore 168-5, and the same figure is obtained for it by the direct formula (T 1 - T 2 ) - (G l - G 2 ). Allowing for the feed-pump term, the efficiency in the complete Rankine cycle is 0-289, as against 0-292 when there was no initial wetness. 106 THERMODYNAMICS [CH. In practice the steam supplied to an engine would be wet only if there were condensation in the steam-pipe, such as would occur if it were long or insufficiently covered with non-conducting material, or if the boiler "primed." Priming is a defective boiler action which causes unevaporated water to pass into the steam- pipe along with the vapour. The above example will show that a moderate amount of wetness reduces the ideal efficiency only very slightly; it has no more than a small effect on the figure for the Rankine cycle. But its practical effect in reducing the efficiency of an actual engine is much greater, because the presence of w r ater in steam increases the exchanges of heat between it and the metal of the cylinder, and consequently makes the real action depart more widely from the adiabatic conditions which are assumed in the ideal operations of the Rankine cycle. 93. Rankine Cycle with Steam initially Superheated. On the other hand if the steam be superheated before it enters the engine, the exchanges of heat between it and the metal are reduced ; the action becomes more nearly adiabatic, and the performance of the real engine approaches more closely the ideal of the Rankine cycle. This is the chief reason why superheating improves the efficiency of a real engine of the cylinder and piston type. In steam turbines it is beneficial partly for the same reason and partly because it reduces internal friction in the working fluid by keeping it drier than it would otherwise be during its expansion through the successive rings of blades. Superheating is now very generally em- ployed in steam engineering. It is therefore important to consider in some detail the Rankine cycle for steam that is initially superheated. In the entropy-temperature diagram (fig. 25) the line bb r repre- sents the process of superheating steam that was dry-saturated at b. During this process its entropy and its temperature both in- crease, and when the pressure and temperature at any stage in the superheating are known the corresponding entropy is found from the tables relating to superheated steam. If we assume that the pressure during superheating is constant, and equal to the boiler pressure, the line bb' is an extension, into the region of superheat, of the constant-pressure line ab. During the process of superheating the steam takes in a supplementary quantity of heat equal to the area under the curve bb', measured down to the base line, namely nbb'n'. This quantity of heat may also be found from the tables, being equal to the excess of the total heat / b ', Ill THEORY OF THE STEAM-ENGINE 107 over that of saturated steam of the same pressure. Callendar's tables give values of the total heat of superheated Steam, as well as its en- tropy, for a wide range of pressures and temperatures. During the subsequent pro- cess of adiabatic expansion b'c' the steam loses super- heat, and if the process is carried so far that the adia- batic line through b' crosses the boundary curve, it be- comes saturated and then wet, and the final condition is that of a wet mixture at c'. The total heat of this wet mixture is found by the method already described. The work done in the Rankine cycle as a whole is the area eabb'c', and the heat taken in is the area L ntropy leabb'ri. Both these quan- Fi 25 tities are readily calculated without the help of the diagram. To find the work done in the cycle we have only to calculate the heat-drop during adiabatic expansion, namely, I b > / c ', and subtract from that the small term which is the thermal equivalent of the work done in the feed- pump, nam'ely, A (P^ P 2 ) V w ^. The heat supplied is As a numerical example we may again take P = 180 and P 2 = 1, and assume that superheating is carried so far as to raise the tem- perature of the steam to 400 C., which is a limit very rarely ex- ceeded in practice. As a rule the temperature after superheating is considerably lower than this. With these data the steam tables show that the total heat of the superheated steam is 780-8 and its entropy is 1-7633. The heat-supply is 780-8 - 38-9 = 741-9. After adiabatic expansion the steam is wet, and its total heat, which is I 8f - T 2 ( S2 - <) or T 2 c/> - G 2 , is 547-2. The adiabatic heat-drop is therefore 233-6. 108 THERMODYNAMICS [CH. Or we can find the heat-drop very directly by help of Callendar's values of G. By Art. 90 it is (T/ - T 2 ) s is L/T and height is ST. Its area is the thermal equivalent of the work done; hence SP (V s - V u .) = JLST/T, as before. In the vaporization of a liquid V s is greater than V w , and heat (JfT1\ -jp J is positive, which means that increasing the pressure raises the boiling point. When the change of volume V s V w is known for any substance, the equation may evidently be used to find the amount by which the boiling point is raised per unit increase of pressure. 99. Application of Clapeyron's Equation to other Changes in Physical State. The reasoning by which this equation was arrived at was general: it applies to any reversible change in the state of aggregation of any substance, to the change from solid to liquid as well as to the change from liquid to vapour. The engine whose indicator diagram was sketched in fig. 29 may have anything for working substance, and the isothermal line in the in] THEORY OF THE STEAM-ENGINE 117 first operation, during which heat is taken in, may be drawn to represent the change of volume corresponding to any change of state. In the example already dealt with, the change was from liquid to vapour. But we might begin with a solid substance pre- viously raised to the temperature at which it begins to melt (under a given pressure), and draw the line to represent the change of volume that occurs in melting, while the pressure remains con- stant. The substance does external, work against that pressure if it expands in melting; or has work spent upon it if (like ice) it con- tracts in melting. The steps of the argument are not affected, and hence the equation may be written thus, with reference to any such transformation of state, V" V = ^ T dP' where V is the volume of the substance (per unit of mass) in the first state, V" its volume in the second state, A is the heat absorbed in the transformation, and dT/dP is the rate at which the tempera- ture of the transformation (say the melting point or boiling point) is altered by altering the pressure. If a solid body expands in melting V" is greater than V and (since the latent heat A is positive) it follows that dT/dP is positive: in other words the melting point is raised by applying pressure. On the other hand if the body contracts in melting V" V is negative and dP/dT is negative: in other words the melting point is lowered by applying pressure. This is the case with ice. From the known amount by which ice contracts when it melts, James Thomson (in 1849) applied this method of reasoning to predict that the melting point of ice would be lowered by about 0-0074 C. for each atmosphere of pressure, and the result was afterwards verified experimentally by his brother, Lord Kelvin*. The lower of the two fixed points used in graduating a thermo- meter (Art. 15) is the temperature at which ice melts under a pressure of one atmosphere. If this pressure were removed, as it might be by putting the ice in a jar exhausted of air by means of * See Kelvin's Mathematical and Physical Papers, vol. I, p. 156 and p. 165. The numerical result stated in the text is obtained as follows: A pound of water changes its volume in freezing from 0-0160 to 0-0174 cub. ft., and gives out 80 calories. HenCG dT _ 0-0014 x 273 _ dp' 80x1400 - and if dP be one atmosphere or 2160 pounds per sq. ft., 8T is 2160 x 0-00000341 or 0-0074 C. 118 THERMODYNAMICS [CH. an air-pump, the temperature of melting would be raised. The water-vapour given off at the melting point has a pressure of only O09 pounds per square inch, and consequently if no air were present, and if the only pressure were that of its own vapour, ice would melt at approximately 0-0074 C., for the pressure would be reduced by nearly one atmosphere. The temperature at which ice melts under these conditions is called the Triple Point, because (in the absence of air) water-stuff can exist at that par- ticular temperature in three states, as ice, as water, and as vapour, in contact with one another and in equilibrium. 100. Entropy-Temperature Chart of the Properties of Steam. Besides serving to illustrate the operations of ideal engines, a diagram in which the coordinates are the entropy and the temperature may be used as a general chart for exhibiting graphically the properties of steam or of any other fluid. The student will find it instructive to draw for himself a chart for steam, on section paper, to a scale large enough for reasonably accurate measurement. The general character of such a chart will be apparent from fig. 30. It includes the boundary curves already described, which represent the relation of entropy to temperature in saturated steam and in water at the same temperature and pressure. Between these is the wet region, where the substance can exist in equilibrium only as a mixture of liquid and vapour. Beyond the steam boundary, to the right, is the region of superheated vapour. Now let a system of Lines of Constant Pressure be drawn. Each of these shows the relation of to T while the substance changes its state in the manner of the imaginary experiment of Art. 56. Starting from the extreme left, a line of constant pressure for water is practically indistinguishable from the boundary curve; strictly it lies a little to the left of that curve, reaching it only when the temperature is such that steam begins to form. Then it crosses the wet region as a horizontal straight line, T being constant during the conversion of the substance from liquid into vapour. After reaching the steam boundary the line of constant pressure rises rapidly during the process of superheating. In the figure, five representative lines of constant pressure are drawn, namely those for P = 2, 20, 80, 200 and 500 pounds per square inch. When a sufficient number of such lines are drawn it is easy, by graphic interpolation, to mark on the chart the position of a point Ill] THEORY OF THE STEAM-ENGINE 119 corresponding to any assigned condition of the substance as to pressure and temperature, and to trace, by measurement instead of by calculation, the changes which ensue during adiabatic expansion. The convenience of the chart for such purposes is increased by including a system of Lines of Constant Total Heat. Examples of these lines are shown in fig. 30, for each interval of 25 calories from ;oc. 0-25 0-75 1-50 175 1-0 1-25 Entropy Entropy -temperature Chart for Water and Steam. 2-0 2-25 Fig. 30. / = 600 to / = 800 calories. They are specially useful in the region of superheat ; they may however be drawn in the wet region also. As an example the line for / = 650 is continued into the wet region; it undergoes a sharp change of direction in crossing the steam boundary. Each of these lines represents what occurs in a throttling process. The lines of constant total heat tend, at the extreme right, to become nearly straight lines of constant tempera- ture : this is because the vapour behaves more nearly like a perfect 120 THERMODYNAMICS [en. gas the more the pressure is reduced. In a perfect gas, as we saw in Art. 19, T is constant when the expansion is of such a nature as to keep / constant. Another useful addition is a set of Lines of Constant Dryness in the wet region. These are drawn in the figure for q ='0-1, 0-2, 0-3, 0-4, 0-5, 0-6, 0-7, 0-8, and 0-9. They divide each horizontal width between the two boundary curves into equal parts (see Art. 84). Lines of Constant Volume may also be drawn in the manner already described (Art. 96). With the aid of such a chart one may find, for example, by drawing the appropriate adiabatic (vertical) line, that steam with an initial pressure of 200 pounds per square inch, superheated to 400 C., becomes saturated when its pressure falls, by adiabatic expansion, to 16 pounds. Continued into the wet region the adia- batic cuts the constant dryness line q = 0-9 at 50 C., showing that there is 10 per cent, of water present when the pressure has fallen to 1*8 pounds. The heat-drop may be inferred, but for its measure- ment a better form of chart is one which will be described in the next Article. By drawing a vertical line to represent the adiabatic expansion of a mixture of steam and water, it is easy to trace the changes that occur in the proportion of water to steam. In the region of ordinary working pressures the line for q = 0*5 is nearly vertical. Hence if there is about 50 per cent, of water present at the begin- ning of adiabatic expansion, nearly the same percentage will be found as the expansion goes on. When the steam is much wetter than this to begin with, adiabatic expansion makes it drier. In fig. 30 a conjectural curve has been added (shown by a broken line) connecting the water and steam boundary curves in the region of high pressure, where, at present, there are no data for a precise determination of the entropy. This broken line is simply a smooth curve forming a continuation of each boundary curve, and drawn so that it touches the isothermal for 365 C., that being the critical temperature for water. It is at the critical tem- perature that the distinction between (f> s and curve. At sufficiently high pressures the lines of constant pressure would pass, in the form of continuous curves, clear of the rounded top, from the region of water to that of superheated steam. m] THEORY OF THE STEAM-ENGINE 121 The water boundary curve is concave on the left for tempera- tures below 250 C., because the rise of entropy per degree, which is o-/T, where o- is the specific heat of water, becomes less as T in- creases, o- being nearly constant at low temperatures. But at higher temperatures the specific heat of water increases so fast as to make cr/T increase with rising temperature: the curve accord- ingly bends over to the right as it approaches the critical point. In the next chapter we shall have occasion to refer to examples of entropy-temperature charts for other fluids. In one of these carbonic acid the region which is practically important, in con- nection with refrigerating processes, includes the rounded top whose summit is the critical point. In that diagram the lines of constant pressure in the liquid are clearly distinguishable from the boundary curve of the liquid state. 101. Mollier 's Chart of Entropy and Total Heat. While the entropy-temperature diagram is invaluable as a means of exhibiting thermodynamic cycles and as a help towards under- standing them, another diagram, introduced in 1904 by Dr R. Mollier *, is of greater service in the solution of practical problems. By taking for coordinates the entropy and the total heat, Mollier constructs a chart which from this point of view has advantages that entitle it to the first place among devices for representing graphically the thermodynamic action of steam in steam-engines, or of the working fluids in refrigerating machines. Its applica- tions in refrigeration will be dealt with in the next chapter. As regards steam it furnishes the most convenient way to measure the heat-drop in adiabatic expansion, whatever be the initial state as to superheat, and consequently to find the greatest theoretical output that is attainable when the initial pressure and temperature, and the final pressure, are assigned. We have seen that this can be calculated when tables as complete as Callendar's are available; and also that it can be found by the aid of an entropy-temperature chart on which lines of constant total heat have been drawn. But the Mollier chart allows a graphic solution to be obtained with great directness and ease. For practical purposes the Mollier I(f> chart is drawn so as to show only the steam boundary curve and the region immediately above and below it, but it is instructive to consider the complete * R. Mollier, "Neue Diagramme zur technischen Warmelehre," Zeitschrifl des Vereines deutscher Ingenieure, 1904, p. 271. See also his Neue Tabdlen und Diagramme fur W asserdampf, Berlin (Julius Springer), 1906, 122 THERMODYNAMICS [CH. chart for water and steam, which is sketched in skeleton form and to a very small scale in fig. 31. There ea is the water boundary curve and bs is the steam boundary curve. The straight lines between them, such as db and es, are constant-pressure lines : one of these (for P = 200 pounds per sq. inch) is continued across the boundary into the region of superheat; the curve bb r represents the process of superheating at that pressure. The slope of any line of constant pressure is a measure of the temperature, for at con- 800 .,/ 700 600 500 400 9=0-7 200 100 Entropy

Chart for Water and Steam. stant pressure dl = dQ = Td = T. In the wet region the temperature along any line of constant pressure is constant, being the temperature of saturation for that pressure, and therefore any constant-pressure line in that region is straight. It crosses the steam boundary without change of slope, but gradually bends upwards in the region of superheat as the temperature rises, for its slope continues to be a measure of the temperature. The water and steam boundaries are connected, as in fig. 30, by a conjectural line through the critical point. The critical point m] THEORY OF THE STEAM-ENGINE 123 fl6ar is not at the summit of this line, but at its point of inflection, which is also its point of maximum slope. At the critical point the continuous boundary curve, shown by the broken line, would touch a curve of constant pressure, and consequently its slope there, dl/d(f>, is equal to the critical temperature, the absolute value of which is 638. The broken line is accordingly drawn to have a slope of 638 units of / for 1 unit of < at its steepest part where, for some distance, it is very nearly straight*. Each constant-pressure line in the wet region may have its length between the two boundary curves divided into parts which express the dryness q at successive stages in the process of vapori- zation, just as in the T(f> chart. Since the heat taken in up to any stage of that process is proportional to q (Art. 70), equal distances along the line, corresponding as they do to equal increments of total heat, correspond to equal changes of dryness. In this way lines of constant dryness are determined, some of which are shown in the sketch. It is useful to have a system of lines of constant temperature drawn in the region of superheat : two such lines are shown in fig. 31. When they and the constant-pressure lines in that region have been drawn it is easy to mark the point which corresponds to any assigned condition of the steam as to temperature and pressure. Thus b' is the point corresponding to steam with a pressure of 200 pounds, superheated to 400 C. Then by drawing a vertical straight line through the point so found, we exhibit the process of adiabatic expansion. The length of that line, down to the final pressure, measures the adiabatic heat-drop, and therefore gives a very simple and direct means of finding the greatest amount of work ideally obtainable from a pound of the working substance. Thus the heat-drop in adiabatic expansion down to a pressure of one pound per square inch is determined by measuring b'c on the scale of /. The position of c among the lines of constant dryness * The slope of the boundary curve, which is ( -=- ) , is equal, at the critical point, to the slope of the constant-pressure line which touches it there, namely ( -=- } . But i-\ = T, since in any constant-pressure change dl = Td. Hence at the critical point ~) =T. Hence also, at that point, ^ 2 = (~ . But " , which \d(f> / \atpr/ t \<*0/8 \* ( / > /8 is the slope of the boundary curve in the entropy -temperature chart (fig. 30), is zero at the critical point. Hence at the critical point ( ^ , 2 j = ; that is to say the boundary curve in the /0 chart there undergoes inflection. 124 THERMODYNAMICS [CH. shows how much of the steam is condensed by this adiabatic expansion. The advantage of a high vacuum, to which attention was drawn in Art. 95, will be obvious from the effect of the final pressure on the length of b'c. A throttling process is represented by a horizontal straight line, since / is constant. Lines of constant temperature in the super- heated region become nearly straight and horizontal at very low pressures, for the behaviour of the vapour then approximates to that of a perfect gas. 800 m 3 1-4 1-5 1-6 1-7 1-8 1-9 2 Fig. 32. Mollier's Chart of Total Heat and Entropy. 2-2 A complete Rankine cycle is shown by the closed figure eaWce, where ea is the heating of the feed- water, ab the formation of steam in the boiler, W its subsequent superheating, b'c its adiabatic expansion to the pressure of the condenser, and ce its condensation at that pressure. For the practical use of the diagram, however, there is no need to include the whole cycle. What is wanted is the region to the right, where the quality of the steam before and after expansion is exhibited, especially the from Ill] THEORY OF THE STEAM-ENGINE 125 = 1-5 to 2 and from / = 450 to 800; and by restricting the chart to this region open scales may be used without making it unduly large. Fig. 32 gives, in miniature form, a Mollier Chart for the useful region, showing a few lines of constant pressure, also lines of constant temperature in the region of superheat, and lines of constant dryness in the wet region*. 1 02. Other Forms of Chart. Besides the foregoing diagram Mollier introduced another in which the coordinates are the 800 T 7 / / . _*s / / ___ c / / ^T l~ /" [/~ V V 750 N 7 7 y~ 700 / / /?^/0/? /O/ 1 / Superheat/ 1 , I / / ~~~^-- jL 7 ^, 5: /Boundary Curve 650 >^^ / / Wei Region /I i \ /-/ tf-0-9 i 600 LX^I ,*> rt rj Pressure Pounds per sq inch 100 200 300 400 Fig. 33. Mollier's Chart of Total Heat and Pressure. 500 pressure and the total heat. A skeleton PI chart for steam is shown in fig. 33 for the region useful in practice. It has the property that lines of constant temperature and lines of constant volume are straight. It includes the steam boundary curve and a part of the wet region below it, which is mapped out by lines of constant dryness as in the other charts. Through the wet region and the region of * A chart of this kind, exhibiting Calendar's figures on a scale large enough for practical use, has been drawn by Prof. Dalby for his book on Steam Power (E. Arnold) and may be purchased separately. 126 THERMODYNAMICS [CH. superheat above, lines of constant volume are drawn. They are straight in the region of superheat, and sensibly straight in the wet region, but they undergo an abrupt change of direction on crossing the boundary. (See Arts. 208 and 209.) Various other charts may be devised by selecting for the two coordinates other pairs of properties from the list given in Art. 75. In any such chart the characteristics of the fluid are exhibited by drawing systems of curves, each of which represents the relation that holds between the two properties chosen for coordinates when the state alters in such a manner that some third property is kept constant. By drawing several such systems of curves a compre- hensive graphical substitute for numerical tables may be con- structed. The particular properties selected for the coordinates, and for the curves, may depend on the type of problem or problems for which the chart is wanted. Callendar gives, as an adjunct to his steam tables, a chart in which the coordinates are the total heat and the logarithm of the pressure. With respect to all such devices it may be said that, so far as steam is concerned, the publication of full tables, which include the region of superheat, render graphic tabulation less necessary. It is now comparatively easy to find any required quantities directly from the tables, or by interpolation from them, with greater accuracy than is reached in measuring from a chart. But for certain purposes the graphic process is sufficiently exact and more convenient. All students should in any case make themselves acquainted with the entropy-temperature chart, and also with the Mollier chart of entropy and total heat : the former because it will help them to understand cyclic processes ; the latter as an instru- ment for dealing with practical problems in steam engineering and mechanical refrigeration. 103. Effects of Throttling. We have already seen (Art. 72) that when a throttling process is carried out under conditions that prevent heat from entering or leaving the substance the total heat / does not change. Lines of constant total heat on any of the diagrams accordingly show the changes in other quantities which are brought about by throttling. It is the process that occurs when a .fluid passes through a "reducing valve" or other constricted orifice such as the porous plug of the Joule -Thomson experiment (Art. 19). It is not what occurs when a jet is formed, as in the nozzle of steam turbines. In that process, which will be m] THEORY OF THE STEAM-ENGINE 127 dealt with later, the stream of vapour acquires kinetic energy that may be turned to useful account; whereas in throttling, any kinetic energy acquired in passing through the constriction is immediately dissipated by internal friction. In a perfect gas throttling produces no change of temperature (Art. 19), but in steam and other vapours it produces a cooling effect which is measured as the fall of temperature per unit fall of pressure under the condition that / is constant, or Cooling effect = ( j= } \a"/ Values of this quantity for steam under various conditions can be deduced from Callendar's tables. In steam that is highly superheated, especially at low pressure, it is small, for the con- dition of the steam then approaches that of a perfect gas, but if the steam is saturated or only slightly superheated the cooling effect of throttling is much greater. Thus with steam at a pressure of 20 pounds per square inch, the cooling effect is only 0-0513 at 400 C. but is 0-338 at the temperature of saturation. These are the falls of temperature, due to throttling, for a drop in pressure of one pound per square inch. The cooling effect plays an im- portant part in determining the values of the total heat and other properties of the vapour, in the method used by Callendar*. Using the values given in his tables for the total heat of * Callendar tabulates for steam a quantity (called by him SC) which is the product of the cooling effect and the specific heat at constant pressure. It is a quantity of heat, namely the number of calories which would have to be given to each Ib. of the throttled steam to restore it, at constant pressure, to the tempera- ture it had before throttling, when the amount of throttling is such that the pressure ((J.T\ -jpj : it is equal to ~ ( TIP ) aa( * ' IB m dep en d enfc f tne pressure (as will be shown later). The values of "$(7" or p which are given in the table for saturated steam therefore apply also to superheated steam at the same temperature. The cooling effect C may be found by dividing the tabulated values of "/SC"' by the specific heat. The specific heat, which is ( -y^ ) , changes only slowly with the \al )p temperature. It may therefore be found from the tables, for any given pressure and temperature, by noting the difference between values of / at that pressure and at temperatures above and below the given temperature, and taking the amount by which / changes per degree. Thus, for example, at a constant pressure of 20 Ibs. the rate at which / changes with the temperature is 0-509 calory per degree in the neighbourhood of saturation. For saturated steam of that pressure "SO" is given as 0-172; hence the cooling effect of throttling, per pound drop of pressure, is 0-172/0-509 or 0-338, as has been stated in the text. 128 THERMODYNAMICS [CH. superheated steam, it is easy to calculate how much the steam is cooled by any given drop of pressure in throttling. Let saturated steam, for example, at 200 pounds per square inch be throttled down to a pressure of 20 pounds. The value of /, which remains constant, is 669-7. At 20 pounds the table shows that this value corresponds to a temperature of 163-8. The saturation tempera- ture for 20 pounds is 108-9. The original temperature was 194-3. Throttling has therefore cooled the steam by 30-5; but at the same time it has caused it to become superheated to the extent of 54-9. The apparent paradox, that throttling both cools a vapour and superheats it, is due to the fact that when the pressure is reduced by throttling the saturation temperature has fallen more than the actual temperature has fallen. Hence saturated steam is super- heated by throttling, and steam that is initially superheated becomes more superheated. Similarly, a mixture of vapour and liquid is partially dried by throttling; it may be completely dried and even superheated if there is not much initial wetness and if there is a sufficient pressure-drop. This is illustrated in fig. 30 by the line of constant total heat for / = 650, which is drawn partly in the wet region and partly beyond it. It shows the effect of throttling on a wet mixture that contains 6-8 per cent, of water at a pressure of 500 pounds ; the steam becomes dry when the pressure is reduced to 37 pounds, corresponding to the temperature of 128 at which the line crosses the saturation curve or steam boundary. The process of throttling is still more simply shown by horizontal lines (/ = constant) in the Mollier diagram (fig. 31). By drawing such lines through the points on the saturation curve for P = 1 and P = 15 it will be seen that 12 per cent, of water can be re- moved from steam at 200 pounds pressure by throttling it down to 1 pound, or fully 6 per cent, by throttling it to atmospheric pressure. Similarly, it is easy to trace the extent to which liquid will evaporate in escaping through a throttle- valve from a region of high pressure to a region of lower pressure. The method of drying by throttling has been applied as a means of determining the percentage of water present in steam. For this purpose a device is used that is called, rather inappropriately, a " throttling calorimeter." Its essential feature is a pipe through which a sample of the steam to be tested can be passed, containing within it a diaphragm with a pin-hole orifice, or a throttle-valve or porous plug, through which the steam has to pass. There are pressure-gauges on both sides, and a thermometer to read the in] THEORY OF THE STEAM-ENGINE 129 temperature of the steam immediately after passing the obstruc- tion. Both parts of the pipe must be thermally insulated, so that no heat is lost, nor conveyed by conduction from one part to the other. The amount of steam passing, which may be regulated by means of another valve beyond the obstruction, should be such that the steam after throttling is appreciably superheated, in order that no wetness may be left in it ; complete dryness is ensured by seeing that the temperature after throttling is somewhat higher than the saturation temperature. Then from the tables we find /' the total heat which corresponds to the temperature and pressure as observed after throttling. Since there has been no change in the total heat, this must be equal to I w + qL, where these quan- tities refer to the state before throttling. Hence the initial dryness is found, namely I' I I' I 9= ~TT = 7^V In practical applications of this method a porous plug is to be preferred to a throttle-valve because the thermometer can be placed close to it and the temperature measured after the throttled stream has lost its kinetic energy and before it has suffered loss of heat. It is difficult in any case to secure that the sample tested by any such apparatus shall be properly representa- tive, in respect of the moisture it carries ; and consequently little reliance can be placed on tests that are carried out by diverting a portion of a steam supply into a throttling calorimeter, as a means of determining the general wetness of the supply. 104. The Heat- Account in a Real Process. The processes which have been considered in this chapter as going on in a steam- engine are ideal in the sense that they have been assumed to be adiathermal: that is to say, there is no transmission of heat to or from the working substance except what is originally taken from the source or finally rejected to the receiver; in all the intermediate operations the working substance has been enclosed in vessels that are assumed to transmit no heat. The assumed processes are also ideal in the sense that they are internally reversible. The process of throttling, which is a typically irreversible process, did not occur in the ideal engine cycles. In dealing with it also, however, we postulated adiathermal conditions; it was assumed in the argument of Art. 72 that no heat passed by conduction through the con- taining walls to or from the space outside. E. T. 9 130 THERMODYNAMICS [CH. Discarding these limitations we may now draw up, in general terms, a balance-sheet or heat-account for any real process, which will include thermal loss to the space outside and also irreversible actions within the engine or other apparatus. Whether the apparatus considered be an engine cylinder, or the series of cylinders of a compound engine, or a turbine, or a thrott- ling device, we may in all cases compare the state of the fluid at entry and at exit, as for example in the admission pipe of an engine and in the exhaust pipe. We imagine a steady flow of the working fluid through the apparatus. At entry let its pressure be P 1 , its volume (per Ib.) V lt and its internal energy E 1 . At exit let its pressure be P 2 , its volume V 2 and its internal energy E 2 . To make the comparison complete we may write K 1 for the kinetic energy (also per Ib.) of the stream as it enters, and K 2 for its kinetic energy as it leaves. In passing through the apparatus the fluid will, in general, do external work, and also lose by conduction some heat to external space. Let W represent, in thermal units, this output of work, and let Q t represent the heat lost by conduc- tion to external space, both of these quantities (like the others) being reckoned per pound of the fluid that passes through. Each pound that enters the apparatus represents a supply of energy equal to Kj_ + E 1 + APJf^ , for E is the internal energy it carries, and P^Vi is the work done by the fluid behind in pushing it in. But E + AP 1 V 1 is equal to / x , the total heat per pound of the fluid in its actual state at entry. Similarly, each pound that leaves the apparatus represents a rejection of energy amounting to K 2 + E 2 + AP 2 V 2 , for E 2 is the internal energy which the fluid carries out, and P 2 V 2 is the work spent upon it by the fluid behind in pushing it out. E 2 + AP 2 V 2 is equal to 7 2 , the total heat per pound of the fluid in its actual state at exit. Hence, by the con- servation of energy, for the apparatus as a whole, K 1 + I 1 = K 2 + I 2 + W +Q Z . The terms on the left of this equation represent the energy that enters the apparatus ; the terms on the right show how it is disposed of in the issuing stream, in output of useful work, and in leakage of heat. The terms K and K 2 are usually very small, except when the apparatus is one for forming a steam jet, in which case K 2 is the useful term: this will be considered in a later chapter. When the change of kinetic energy in the stream is practically negligible, m] THEORY OF THE STEAM-ENGINE 131 as it is between the admission pipe and exhaust pipe of an engine, we have /, = /, + FT + Q,. And when, in addition, the apparatus does not allow any appreci- able amount of heat to escape to the outside (Q t = 0), we have A - / 2 = w. This means that when there is a steady flow of a working sub- stance through any thermodynamic apparatus, the output of work is measured by the actual Heat-Drop, whether the internal action is or is not reversible, provided there is no loss of heat to the outside by conduction through the walls. The actual heat-drop must not be confused with the adiabatic heat-drop, which is the difference between I 1 and that value which the total heat would reach if there were adiabatic expansion to the exit pressure P 2 . The actual heat-drop /j 7 2 is identical with the adiabatic heat-drop only when there is no loss of heat to the outside and when, in addition, the internal action is wholly reversible. Any irreversible feature in the internal action will increase / 2 above the value which would be reached by adiabatic expansion, and will consequently diminish the output of work. In the extreme case of a throttling process there is no output of work, and therefore I 2 = I 19 provided there is no loss of heat to the outside. Any loss of heat to the outside in a throttling process will make 7 2 correspondingly less, for we then have / 2 = I I Q t . The losses of thermodynamic effect in a real engine, which make W less than the ideal output, namely the value corresponding to the adiabatic heat-drop, arise partly from loss of heat to the outside and partly from two kinds of irreversible internal action. One of these two kinds is mechanical; the other is thermal. In the mechanical kind, the action involves fluid friction within the working substance. It is of the same nature as that which occurs in throttling: there is irreversible passage of the working substance from one part of the engine to another where the pressure is lower, as for instance the passage of steam through somewhat constricted openings into the cylinder, or its passage, on release after incom- plete expansion, into the exhaust pipe, with a sudden drop of pres- sure: or again, there is the same kind of irreversibility in a turbine in the frictional losses that attend the formation of steam jets or in the friction of the jets on the turbine blades. These are all instances of mechanical irreversibility. In the second kind of 92 132 THERMODYNAMICS [CH. m irreversible action there is exchange of heat between the working substance and the internal surface of the engine walls. The hot steam, on admission to a cylinder which has just been vacated by a less hot mixture of steam and water, finds the surfaces colder than itself. A part of it is accordingly condensed on them, which re- evaporates after the pressure has fallen through expansion. This alternate condensation and re-evaporation involves a considerable deposit and recovery of heat in a manner that is not reversible, for it takes place by contact between fluid and metal at different temperatures. The action may occur without loss of heat to the outside : it would occur, for instance, in an engine with a conducting cylinder covered externally with a "lagging" of non-conducting material. Its effect, like that of throttling or fluid friction gener- ally, is to reduce the output of work below the limit that is attainable only in a reversible process, and it does this by making the actual heat-drop I 1 / 2 less than the adiabatic heat-drop. The equation W = I t I 2 takes account of both kinds of irre- versibility of the effect of thermal exchanges within the apparatus, as well as of any throttling or frictional effects in the action of the working substance. But it does not take account of heat lost to the outside, and for that the term Q z has to be deducted, making W = I 1 -I 2 -Q l . The full statement of the heat-account in a real process may be expressed as follows: When there is a steady flow of a working substance through any thermodynamic apparatus, the output of work is measured by the actual heat-drop from entrance to exit, less any heat that escapes by conduction to the outside, and less any gain of kinetic energy of the issuing stream over the entering stream; or, in symbols W = /! - 7 2 - Q z - (K 2 - KJ, all these quantities being expressed in thermal units, and reckoned per unit quantity of the working substance. This equation also applies to reversed heat-engines, or heat- pumps, which will be considered in the next chapter, but in them the quantity W is negative: work is expended on the machine instead of being produced by the machine. In such machines Q l is also generally negative, for as a rule the apparatus is colder than its surroundings and the leakage of heat is inwards. CHAPTER IV THEORY OF REFRIGERATION 105. The Refrigeration Process. Refrigeration is the re- moval of heat from a body that is colder than its surroundings. In cold storage, for example, the contents of a chamber are kept at a temperature lower than that of the air outside, by extracting the heat which continuously leaks in through the imperfectly in- sulating walls. To maintain a refrigerating process requires ex- penditure of energy. It is generally done by means of a mechani- cally driven heat-pump, working on what is essentially a reversed heat-engine cycle. It may also be done by the direct use of high- temperature heat without intermediate conversion of that heat into work. We shall consider later the direct application of heat to effect refrigeration, but shall in the first instance treat of re- frigerating machines driven by the expenditure of mechanical power. Any process of refrigeration involves the use of a working substance which can be made to take in heat at a low temperature and discharge heat at a higher temperature. The heat is discharged by being given up to the air outside or to any water that is available to receive it. The process is a pumping-up of heat from the level of temperature of the cold body, at which it must be taken in, to the level at which it may be discharged. These levels should be as near together as is practicable, in order that no unnecessary work may be done: in other words the action of the working substance should be confined to the narrowest possible range of temperature. The temperature of discharge should be no higher than is necessary to get rid of the heat, and the lower limit should be 110 lower than will ensure transfer of heat into the refrigerating substance from the cold body whose heat is to be extracted. Let T l be the temperature at which heat is discharged and T 2 the temperature at which it is taken in from the cold body. Con- sider a complete cycle in the action of the working substance. Let Q l be the quantity of heat which is discharged and Q 2 the quantity which is taken in from the cold body; and let W be the thermal 134 THERMODYNAMICS [CH. equivalent of the work spent in driving the refrigerating machine. Then, by the conservation of energy, Q! = Q 2 + W. The useful refrigerating effect is measured by Q 2 > an d the tw co- efficient of performance," which is the ratio of that effect to the work spent in accomplishing it (Art. 4) is =* . 1 06. Reversible Refrigerating Machine. We have first to en- quire what is the highest possible coefficient of performance when the limits of temperature 2\ and T 2 are assigned. We know by the principle of Carnot (Arts. 33, 39) that when heat passes down from Tj to T 2 through a heat-engine, the ideally greatest efficiency in the conversion of heat into work is obtained when the engine is thermodynamically reversible. In that case Qj = 3 *i 2V The output of work W is Q x Q 2 . Hence the ideally greatest output of work is related to Q 2 , the heat rejected at the lower limit of temperature, by the equation A corresponding proposition in the theory of refrigeration is that the ideally greatest coefficient of performance of a refrigerating machine, working to pump up heat from T 2 to T 19 is obtained when the machine is thermodynamically reversible. In that case the same relation holds, namely Qi = Q 2 T, zy and the amount of work W which is spent in driving the machine (and is equal to Q 1 Q 2 ) is related to Q 2 by the equation w ^ 2 \ i ~ 2) T 2 z In other words, the greatest amount of work that is theoretically obtainable in letting heat pass down through a given range of temperature is the least amount of work that will suffice to pump up the same quantity of heat through the same range. To show that no refrigerating machine can be more efficient than one that is reversible, we shall use an argument like that of Art. 33. Let E, fig. 34, be a reversible refrigerating machine, reversed and IV] THEORY OF REFRIGERATION 135 Cold Body therefore serving as a heat-engine. It takes a quantity of heat, sa y Qi> from the hot body and delivers a quantity Q 2 to the cold body, converting the difference into work. Let all the work W which it develops be employed to drive a refrigerating machine R; and assume that there is no loss of power in the connecting mech- anism. Accordingly the two machines, thus coupled, form a self-acting combina- tion. If it were conceivable that the machine R could have a greater coefficient of performance than the reversible machine J5, that Fig. 34 would mean that the ratio of Q 2 to W would be greater in R than in E. Hence (W being the same for both) R would take more heat from the cold body than E gives to it, and R would also^give more heat to the hot body than E takes from it. The result would be a continuous transfer of heat from the cold body to the hot body by means of a purely self-acting agency. This would be contrary to the Second Law of Thermodynamics : we conclude therefore that no re- frigerating machine can have a higher coefficient of performance than a reversible machine working between the same limits of temperature. It follows that all reversible refrigerating machines, working between the same limits of temperature, have the same coefficient of performance. It also follows that the value of this coefficient is that which would be found in a reversed Carnot cycle, namely W T! - T 2 ' This is the ideally highest coefficient it measures the performance of what may be called a perfect refrigerating machine. The coefficient of performance in any real machine is necessarily less, for the cycle of a real machine falls short of reversibility. 107. Conservation of Entropy in a Perfect Refrigerating Pro- cess. We saw in Art. 45 that a perfect, or reversible, heat-engine, 136 THERMODYNAMICS [CH. such as Carnot's, may be regarded as a device which transfers entropy from a hot body to a cold body without altering the amount of the entropy so transferred, although the amount of heat which enters the engine is greater than the amount of heat which leaves the engine. The entropy taken from the hot body, namely Q 1 /T 1 , is equal to the entropy given to the cold body, namely Q 2 /T 2 ; it may be said to pass through the engine without change, though the heat that passes through is reduced in the process by the amount which is converted into work, namely, by the amount Q 1 Q^ Similarly a perfect, or reversible, refrigerating machine or heat- pump may be regarded as a device which transfers entropy from a cold body to a hot body without altering the amount of the entropy so transferred, although the amount of heat which enters the machine is less than the amount which leaves the machine. The action is in every particular a reversal of that of the perfect heat-engine. Entropy to the amount Q 2 /T 2 is taken from the cold body, and entropy to the equal amount Q^T^ is given to the warmer body to which heat is discharged. The amount of heat which is pumped up increases from Q 2 to Qj in the process, because an amount of work equivalent to Q Q 2 is expended in driving the machine and is converted into heat within the machine. 1 08. Ideal Coefficients of Performance. The following table shows the values of the coefficient of performance in a perfect or reversible refrigerating process, for various ranges of temperature. These are ideal figures, representing a theoretical limit which cannot be reached in practice. Though they relate to conditions of reversibility which are not fully attainable in a real machine, they illustrate clearly the practical importance of making the range of temperature as small as possible, by taking in the heat at a tem- perature no lower than can be helped and by discharging it after the least practicable rise. Coefficients of Performance of a Perfect Refrigerating Machine. Lower limit Upper limit of temperature of temperature (Centigrade) (Centigrade) 10 20 30 40 50 - 20 8-4 6-3 5-1 4-2 3-6 - 15 10-3 7-4 5-7 4-7 4-0' - 10 13-1 8-8 6-6 5-3 4-4 - 5 17-9 10-7 7-7 6-0 4-9 27-3 13-6 9-1 6-8 5-5 5 55-6 18-5 11-1 7-9 6-2 10 28-3 14-1 9-4 7-1 IV] THEORY OF REFRIGERATION 137 B The importance of a narrow range of temperature in refrigeration is further illustrated by fig. 35. It gives the en- tropy-temperature dia- grams of three reversible refrigerating processes, in ^ all of which the upper limit of temperature (TJ I- is the same, and the same amount of work is . spent. Each of the three supposed processes is Entropy Fig. 35 ideally efficient: it is a reversed Carnot cycle, and its entropy- temperature diagram is a rectangle. The area of the rectangle represents the work spent, and the area under it, down to the absolute zero of temperature, represents the amount of heat that is taken from the cold body, and therefore measures the refrigerat- ing effect. The three processes for which the diagram is sketched differ only in the temperature T 2 of the cold body from which heat is extracted. That temperature is relatively high in the first case (a), lower in case (b) and lower still in case (c). The refrigerating effect is measured by the area AD in the first case, by BD in the second, and by CD in the third. The result of lowering T 2 is very apparent, in reducing the amount of refrigeration that is ideally capable of being done by a given expenditure of work. 109. The Working Fluid in a Refrigerating Process. The working substance in a refrigerating cycle may be a gas which remains gaseous throughout, such as air. More commonly it is a fluid which is alternately condensed and evaporated. During evaporation at a low pressure the fluid takes in heat from the cold body: it is then compressed and gives out heat in becoming con- densed at a relatively high pressure. The selection of the fluid is governed by practical considerations. Water is used in some cases, but a serious drawback to its use is the very large volume and low pressure of the vapour at low temperatures. There are obvious advantages in using a fluid whose vapour-pressure is neither incon- veniently small at the lower limit of temperature nor inconveniently large at the upper limit. The fluids most commonly used are ammonia and carbonic acid. Ammonia has a very convenient range of vapour-pressure throughout the range of temperature with which we are concerned in practical refrigeration. With carbonic 138 THERMODYNAMICS [CH. acid the vapour-pressure is considerably higher, the critical point is reached at a temperature that may come within the range of operation, and the thermodynamic efficiency is somewhat less. Notwithstanding these objections carbonic acid is frequently pre- ferred, especially on board ship, where it is more harmless should any of the fluid escape by leakage into the room containing the machine. For use on land, especially where the highest thermo- dynamic efficiency is aimed at, ammonia is usually chosen. Other fluids with lower vapour-pressures are occasionally used, such as sulphurous acid, ethyl chloride, and methyl chloride. no. The Actual Cycle of a Vapour-Compression Refri- gerating Machine. If the reversed Carnot cycle were actually followed, the choice of working fluid would make no difference to the efficiency: the coefficient of performance for any fluid would have the value shown in Art. 106, namely T 2 /(T 1 - T 2 ). But a part of the reversed Carnot cycle is omitted in practice, with the result that the coefficient is reduced, and the extent of the reduction depends on the nature of the fluid; it is greater in carbonic acid than in ammonia. To carry out a reversed Carnot cycle, with separate organs for the successive events which make up the cycle, would require : (1) A compression cylinder in which the vapour is compressed from the pressure corresponding to T 2 to the pressure corresponding to Tj. (2) A condenser in which it is condensed at T x . A typical form of this organ would be a surface condenser in which the working fluid gives up its heat to circulating water. (3) An expansion cylinder in which it expands from T l to T 2 . (4) An evaporator in which it takes up heat at T 2 from the cold body from which heat is to be extracted. This vessel is some- times called the "refrigerator." In nearly all refrigerating machines the expansion cylinder is omitted for reasons of practical convenience, and the fluid streams from (2) to (4) through a throttle-valve with an adjustable opening, called the "regulator" or "expansion- valve." In passing the ex- pansion-valve the pressure of the working fluid falls to that of the evaporator: its temperature falls to T 2 and part of it becomes evaporated before it begins to take in heat from the cold body. The omission of an expansion cylinder, with the substitution for it of an expansion-valve, reduces the coefficient of performance for IV] THEORY OF REFRIGERATION 139 two reasons. The work which would be recovered in the expansion cylinder is lost; and also the refrigerating effect in the evaporator is reduced, for more of the liquid is vaporized in the act of streaming through the expansion-valve than would be vaporized in adiabatic expansion, consequently less is left to be evaporated by subse- quently taking in heat from the cold body. The loss of efficiency from these two causes is not, however, very important under ordinary conditions. To omit the expansion cylinder is a consider- able simplification of the machine, all the more as the effective volume of such a cylinder would need adjustment relatively to that of the compression cylinder in order to secure the best effect under varying conditions as to the limits of temperature. Rather than Fig. 36. Organs of a Vapour-Compression Refrigerating Machine. introduce this complication it is worth while to make a slight sacrifice of thermodynamic efficiency. In the usual type of vapour- compression refrigerating machine, accordingly, the expansion cylinder is omitted, and the organs are those shown diagrammatically in fig. 36. They are, (1) the com- pression cylinder B; (2) a condenser A such as a coil of pipe, cooled by circulating water, in which the working substance is condensed under a relatively high pressure and at the upper limit of tem- perature T x ; (3) an expansion- valve or regulator R through which it streams from AtoC; (4) the evaporator C, in which it is vaporized at a low pressure by taking in heat from the cold body at the lower limit of temperature. The evaporator may for instance be a coil of pipe taking in heat from the surrounding atmosphere of a cold chamber; often it is a coil surrounded by cold circulating brine 140 THERMODYNAMICS [CH. which serves as a vehicle for conveying heat to the working sub- stance from a cold chamber or from cans for ice-making or other objects that are to be refrigerated. The action of the compression cylinder is shown by the indicator diagram, fig. 37, in the same figure. During the forward stroke of the compressor the valve leading to A is shut and that leading from C is open. A volume V 1 of the working vapour is taken in from C at a uniform pressure corresponding to the lower limit T 2 . In most actual cases what is taken in is not dry-saturated vapour but a wet mixture, the wetness of which is regulated by adjusting the expansion valve R. This is in order that the subsequent compression may not produce much (if any) superheating. It is possible to make the compression wholly "wet" by taking in a sufficiently wet mixture: more generally the expansion- valve is adjusted so that the vapour is moderately wet to begin with, and becomes slightly Fig. 37. Indicator Diagram of Compression Cylinder. superheated by compression. At the end of the forward stroke the valve leading from C closes and the piston is forced to move back compressing the vapour or wet mixture in the cylinder until its pressure becomes equal to that in A. This compression reduces the volume of the fluid in the cylinder to V 2 . The valve leading to A then opens, and the back-stroke is completed under a uniform pressure while the working substance is discharged into A and condensed there. The valves of the compressor are spring valves which open and close automatically in consequence of the changes in pressure, and are situated in the cover of the cylinder in such a manner as to make the clearance negligibly small. To complete the cycle, the same quantity of working substance is allowed to pass directly from A to C through the expansion- valve R. This step is not reversible (Art. 22). The temperature T x at which condensation takes place, is in practice necessarily a good deal higher than that of the circulating water by which the condenser is kept cool, for a large amount of IV THEORY OF REFRIGERATION 141 heat has to be discharged from the condensing vapour in a limited time. But it is important that the condensed liquid should be no warmer than is unavoidable before it passes the expansion- valve. Accordingly the condenser is arranged (sometimes by the addition of a separate vessel called a "cooler") so that the condensed liquid is brought as nearly as possible to the lowest temperature of the available water-supply before it passes the valve, though it may have been condensed at a considerably higher temperature. The d ff f ah Fig. 38. The Vapour-Compression Cycle, using Ammonia. b '9 f ~^ U Fig. 39. The Vapour-Compression Cycle, using Carbonic Acid. advantage of this will be obvious when we consider, in the next article, the thermal effects of each step in the cycle. in. Entropy-Temperature Diagram for the Vapour-Com- pression Cycle. The complete cycle is exhibited in the entropy- temperature diagram of fig. 38, which is drawn for ammonia as working substance, and fig. 39, which is drawn for carbonic acid. There dg and ch are portions of the boundary curves. The point a represents the condition of the mixture which is drawn into the compression cylinder, when compression is about to begin; its 142 THERMODYNAMICS [CH. d wetness is measured by the ratio ah/gh. The line ab represents adiabatic compression to the pressure of the condenser. The next process consists of cooling and condensation at this constant pres- sure: it is made up of three stages, be, cd and de. In the first stage, be, the superheated vapour is cooled to the temperature at which condensation begins; in the next stage, cd, the vapour is completely condensed ; in the third stage, de, the condensed liquid is cooled to the lowest available temperature before it passes the expansion- valve. The lines be, cd, and de form parts of one line of constant pressure. In fig. 38 de is practically indistinguishable from the boundary line, but in fig. 39 the ^ distinction is very apparent be- cause we are there dealing with a liquid that is highly com- pressible in consequence of its nearness to the critical state. The line ef represents the pro- cess of passing through the expansion valve, in which the pressure falls to that of the evaporator. This is a throttling process, for which / is constant (Art. 72) : ef is therefore a line of constant total heat; its direction changes in fig. 39 in crossing the boundary curve. By passing the expansion- valve the working substance comes into the condition shown by the point /. The proportion which is converted into vapour by the mere act of passing the valve is shown by the ratio gf/gh. Lastly we have the process of effective evaporation when the substance is usefully extracting heat from the brine or other cold body by evapo- rating in the refrigerator. This is represented by the line fa, during which the dryness changes from gf/gh to ga/gh. om Fig. 40 IV] THEORY OF REFRIGERATION 143 The refrigerating effect, that is to say, the amount of heat taken in from the cold body, is represented by the area under the line fa, measured down to a base-line corresponding to the absolute zero of temperature, namely the area mfan (fig. 40). The amount of heat rejected during cooling and condensation of the vapour and subsequent cooling of the condensed liquid, is the area under the lines be, cd and de, namely the area nbcdeo. The thermal equivalent of the work spent in carrying the working substance through the complete cycle which is simply the work spent on it in the compressor is the difference between those two quantities, namely the area nbcdeo minus the area mfan. It should be noted that the work spent is not measured by the area abcdefa, /' \ a Fig. 41. Cycle for Carbonic Acid, with compression above the Critical Pressure. enclosed by the lines which represent the complete cycle, because the cycle includes an irreversible step ef (see Art. 51). In consequence of that the work spent is greater than the enclosed area by the amount oefm. As a further example we may take a compression process (fig. 41), with carbonic acid for working substance, in which the temperature of the cooling water is so high that the pressure during cooling is above the critical pressure. The line be is accordingly a continuous curve lying entirely outside of the boundary curve. The working substance passes from the state of a superheated vapour at b to the state at e without any stage corresponding to cd in fig. 39, in which it is a mixture of liquid and vapour. As before, the refri- gerating effect is measured by the area under fa: the heat rejected to the cooling water is measured by the area under be : the difference 144 THERMODYNAMICS [CH. between these two quantities measures the work spent, and is greater than the area of the closed figure abefa by the area under the irreversible step ef. 112. Refrigerating Effect and Work of Compression ex- pressed in Terms of the Total Heat. While it is instructive to state, as in the preceding article, the refrigerating effect, the work of compression, and the heat rejected, in terms of areas on the entropy-temperature diagram, it is much more useful, for purposes of practical calculation, to express these as follows in terms of the total heat of the substance at the various stages of the operation. The refrigerating effect, that is to say the amount of heat taken in from the cold body, is I a I ft where I a is the total heat at a and If is the total heat at/. This is because the (reversible) opera- tion fa is effected at constant pressure (Art. 68). For the same reason the amount of heat rejected to the condenser and cooler is I b I e , where those quantities designate the total heat at b and at e respectively. Further, in the process ef of passing the expansion-valve there is no change of total heat, by the principle proved in Art. 72. Consequently, I f = I e . We may therefore state the amount of heat rejected as I b I f . Again, the work spent in the compressor is (in thermal units) I b I a . It is the thermal equivalent of the area of the indicator f* diagram in fig. 37, namely A J VdP, which is equal to 7 & I a by a the general principle proved in Art. 80. We are dealing here with the increase of total heat in adiabatic compression instead of its decrease in adiabatic expansion. That these results are in agreement with one another is seen by considering the heat-account of the cycle as a whole : Work spent = Heat rejected Heat taken in. It ~Ia = (A-//) - (/.-/,). The coefficient of performance, which is the ratio of the heat taken in from the cold body to the work spent in the compressor, is It will be obvious that the numerical value of this coefficient would be reduced if we were to omit the cooling after condensation, which is represented by the line de. For in that case f would be iv] THEORY OF REFRIGERATION 145 shifted to the right, to a point on a line of constant total heat through d, and I f would be increased. The refrigerating effect would be lessened; but the work spent in producing it would be the same as before, for the indicator diagram of the- compression process, which is measured by I b I a , is not affected. The values of I a and I b depend only on the state of the substance at a and at b respectively, and are the same as before. 113. Charts of Total Heat and Entropy for Substances used in the Vapour-Compression Process. The above results will show that calculations of performance, as regards refrigerating effect, heat rejected, and work expended, become very easy when we can find the total heat of the liquid just before the expansion valve and that of the vapour before and after compression. This is readily done if data are available for drawing a Mollier chart of entropy and total heat for the working substance. Fairly complete data are available for ammonia, carbonic acid, and sulphurous acid. I(f> charts for these substances will be found in a Report of the Refrigeration Research Committee of the Institution of Mechanical Engineers*. In drawing these charts a geometrical device is resorted to for the purpose of making the diagrams at once open and compact, with the effect that measurements may be made with sufficient accuracy on a chart of reasonable size. This device, which Mollier originally adopted in drawing his / chart for carbonic acid, is to use oblique coordinates, as illustrated in fig. 42. The lines of con- stant / are horizontal: the lines of constant charts; as applied to them, the device gives a better separation of lines that run more or less diagonally across the sheet, like the lines in fig. 31 (Art. 101). There is consequently a great gain in clearness and in the power of accurately measuring those changes of I that take place in refrigerating processes. The in- clination selected for the oblique axis will depend on the degree of opening out that is convenient in any particular chart. In the case of fig. 42 it is 5 along the slope to 1 vertically, and hence a measurement of / if made along a line of constant (/> would have to Fig. 42. Use of oblique coordinates in the 70 chart. be interpreted on a scale five times as coarse as the normal scale for/. An I(f> chart for ammonia drawn with oblique coordinates is shown (on a small scale*) in fig. 43. In this case the amount of shearing is moderate, for the slope of the lines of constant entropy is only two to one. The diagram, for the useful region, consists of a fan-like group of lines of constant pressure extending as straight lines through the region of wetness from the liquid boundary to the vapour boundary or saturation curve, and then as curves into * For similar charts drawn in fuller detail and on a scale large enough for use in solving problems, reference should be made to the publications cited above. IV] THEORY OF REFRIGERATION 147 the region of superheat. Lines of constant temperature are also drawn in the region of superheat, and lines of constant dryness (shown as broken lines in the chart) are drawn by dividing the 160- 140- 100 tool ^=200 1=150 1-100 70C- 30- 2tf- fo-2 Fig. 43., 70 chart for Ammonia. straight portion of each line of constant pressure into a number of equal parts. This chart should be compared with that shown for water and steam in fig. 31 (Art. 101) in which, however, there was no shearing, for rectangular coordinates were employed. Allowing 102 148 THERMODYNAMICS [CH. for that difference the remarks made in Art. 101 apply here. The slope of any constant-pressure line, when properly interpreted with reference to the coordinates used in the drawing, measures the temperature, for T = dl/d(f>. Hence there is no abrupt change of direction between the straight part of any such line and its 1-70 Fig. 44. 70 chart for Carbonic Acid. curved continuations into the liquid region at one end and into the region of superheat at the other. This of course applies to any substance. The / chart for sulphurous acid is generally similar to the chart for ammonia. The I(f> chart for carbonic acid is shown on a small scale in fig. 44. It shows the region round about the critical point. That point IV] THEORY OF REFRIGERATION 149 coincides with the point of inflection of the continuous boundary curve (Art. 101 ). Constant-pressure lines are drawn for pressures that are higher than the critical pressure as well as for the wet region. The principle already stated applies to these lines, that the slope at any point (due regard being had to shearing) measures the tem- perature. In passing up along any line of constant pressure above the critical pressure, the slope, which measures the tem- perature, increases continu- ously*. The straight portions of the constant-pressure lines, within the boundary curve, are divided by broken lines which are lines of constant dryness. Lines of constant temperature are also drawn in the region outside the boundary curve. In the region within the boundary, where the state is that of a mixture of .saturated vapour and liquid, these lines would of course be straight, and would coincide with lines of constant pressure. To avoid confusion the straight portions of the constant-temperature lines are omitted in the figure. 114. Applications of the ! Chart in studying the Va- pour-Compression Processf. We are now in a position to represent the vapour-compres- * As Messrs Jenkin and Pye have pointed out (loc. cit., p. 365) in correcting the earlier chart of MolJier, there is no point of inflection in any of these lines. For, since j~ } =T, --S = -- Fig. 45. Refrigeration cycle Uac-ed on the 70 chart for Carbonic Acid. , which is a positive quantity throughout Uie whole course of any line above the critical pressure, as will be seen by reference to the entropy-temperature diagram. A point of inflection would require ( -y-~ ) to be zero. \a0Vp Some of the constant-pressure lines were erroneously drawn with inflections in Mollier's original /0 chart for carbonic acid, which was reproduced in the author's book on The Mechanical Production of Cold. t Parts of this article are taken from an appendix (by the present writer), to the Report of the Refrigeration Research Committee of the Institution of Mechanical Engineers, 1914. 150 THERMODYNAMICS [CH. sion refrigerating process by diagrams which exhibit the changes of total heat in relation to entropy. With the help of / charts numerical values of the total heat are readily found by measure- ment at each stage in the assumed cycle. To trace a refrigerating cycle on the appropriate chart, begin as before at a point a (fig. 45) which represents the state of the sub- stance when it is about to enter the compressor. This point is on the constant-pressure line corresponding to the process of evapora- tion in the cold body or evaporator (fig. 36), and its distance from the two boundary curves corresponds to the proportion of vapour to liquid in the mixture. If the compression is to be completely "dry," a will be on the boundary curve (at a^) : more generally the substance is slightly wet when compression begins. The straight line ab, drawn parallel to the lines of constant entropy on the chart, is the process of adiabatic compression. The position of b is deter- mined by the intersection of this line with a line of constant pressure corresponding to the known upper limit of pressure at which con- densation is to occur. The temperature reached in the process of compression is seen by the position of b among the lines of constant temperature. In general there will be some superheating. But if the mixture is so wet to begin with that the adiabatic line through a does not cross the boundary curve during compression before the upper limit of pressure is reached there is none, and in that case the process is spoken of as " wet" compression. This would be the case if compression had begun at a c instead of a. By beginning at a it carries the substance into the region of superheat before compression is completed at b. Next we have the constant-pressure process of cooling and condensation and further cooling, repre- sented in its three stages by the lines be, cd, and de, the position of e being fixed by the temperature to which the liquid is known to be cooled before it reaches the expansion-valve. Then a hori- zontal straight line through e (a line of constant total heat) represents the process of passing through the expansion -valve, and determines a point f, on the evaporation line, which exhibits the condition in which the substance enters the evaporator. The process of evaporation fa, which is the effective refrigerating process, completes the cycle. The values of / , 7 & , I e and I f (which is the same as I e ) are read directly by measurement from the chart. As has been already pointed out, the work spent in compressing the substance is I b / a , and the refrigerating effect is I a I f . We may illustrate the use of the chart by some examples. Take first a case in which the working substance is carbonic acid, with iv] THEORY OF REFRIGERATION 151 10 C. as the temperature of evaporation, 25 C. as the tem- perature of condensation, and 1 5 C. as the temperature to which the substance is cooled before passing the expansion-valve. The diagram for the performance of an ideal machine under these con- ditions is sketched in fig. 45, assuming various degrees of dryness at the beginning of the compression. If the substance is then entirely dry the operation starts at a lt namely, the end of the evaporation line for 10 C., and compression brings it to fe x which is on a line of constant pressure equal to the pressure of saturated vapour at 25 C., namely, 930 pounds per sq. inch. But the vapour is considerably superheated at b l9 its temperature there (as the lines of constant temperature show) being 58 C. The work spent in compression, which is most accurately found by reading off the length of the line aj)-^ on a scale which makes that length a direct measure of the change of /, is 8*7. We next trace the process of condensing and cooling, under the constant pressure of the condenser. From bj_ to c the gas is losing its superheat; from c to d it is being condensed; and from d to e it is being cooled as a liquid. The point e is found by the intersection of the line of constant pressure under which the process is carried out with the line of constant temperature for 15 C. Next draw ^/parallel to the lines of constant total heat to meet the evaporation line for - 10 C. The refrigerating effect / flj - I f is 47-9. The coefficient of performance is therefore -5-5. This cycle corresponds to completely dry compression. Suppose on the other hand that the compression is just wet enough to avoid any superheating. In that case it must commence at a c in order that the adiabatic line representing the compression may pass through c on the boundary curve. Then the work done in compression is smaller than before, for a c c is smaller than a-Jb^ The refrigerating effect is also smaller, for fa c is smaller than /tip The coefficient of performance is now found to be 5-54. Between these two there is a certain degree of dryness which gives a slightly higher coefficient of performance than either. This may be shown by taking a succession of points for various states of dryness between a c and a^ as the starting point of the cycle, and working out the coefficient of performance for each. But we may reach the same conclusion more directly as follows, by a general method which is applicable to any Iff) chart: The refrigerating effect for any state of initial dryness, a, is proportional (on some scale) to the length fa. The work done is 152 THERMODYNAMICS [CH. proportional (on another scale) to the length ab. Hence the position of b which will give the highest coefficient of performance is that which gives the smallest ratio of ab to fa. This is found by drawing a tangent from /to the line of constant pressure on which b lies. By applying this method the point b has been determined in the figure, and hence the point a is found at which compression should begin if the coefficient of performance is to have its maximum value. In the example that value is 5-72, and is obtained when the initial dry ness is about 0-87. As another example, still with carbonic acid, take the same con- ditions as before, except that the condensed liquid, instead of being cooled at 15 C. before expansion, reaches the valve at the tem- perature of condensation, namely 25 C. In that case the process of expansion corresponds to the line df d in fig. 45, the rest of the cycle remaining as before. For maximum coefficient of performance, under these conditions, compression should no longer start from a but from a point so chosen that the adiabatic line through it reaches the constant-pressure curve b^c at the point where the tangent from f d meets that curve. This corresponds to an initial dryness of about 0-95, and the maximum coefficient so obtained is 4'39. When this value is compared with that found in the previous example, namely 5-72, it will be obvious that a serious loss of efficiency is caused by omitting to cool the condensed liquid before it reaches the expansion- valve*. A further example will serve to illustrate the application of the Ij" the temperatures being taken as TI throughout. Q may also be found experimentally, by observing the drop of temperature TI - T' which takes place when the gas expands from P A to PB through a Joule- Thomson orifice without any interchange of heat. iv] THEORY OF REFRIGERATION 175 drop. There is also a continuous gradient along the return pipe from T 2 , on the low-pressure side of the valve, to T 1 at the exit. The flow and return streams are in close thermal contact, and at each point there is an excess of temperature in the flow which allows heat to pass by conduction into the return, except at the entrance where, under the ideal condition which we have postulated of perfect interchange, the temperature of both flow and return is 2V This state of things is diagrammatically represented in fig. 56. There the flow and return are represented as taking place in straight pipes, one inside the other to provide for interchange of heat. Entering along the inner pipe A the compressed gas expands through a constricted orifice E (equivalent to an expansion- valve) into a vessel from which it returns by the outer pipe B. The vessel is provided with a stop-cock C by which that part of the fluid which is liquefied can be drawn off when the second stage of the operation has been reached. In the temperature diagram (fig. 56 a) MN represents the length of the interchange^ DM is the initial (and final) temperature T l9 GN is T 2 , and FG is the Joule-Thomson drop. DF is the gradient for the flow-pipe, and GD for the return. 125. Second Stage. When this gradient has become established the gas begins to liquefy, the apparatus does not become any colder, and the action enters on the second stage, which is one of thermal equilibrium. A certain small fraction of the gas is continuously liquefied and may be drained off as a liquid through the stop-cock C. The larger fraction, which is not liquefied, continues to escape through the interchanger and to leave the apparatus at the same temperature as before, namely the temperature T 1 equal to that of the entering gas. Call this unliquefied fraction q; then 1 q represents the fraction that is drawn off as a liquid at the tempera- ture T 2 . Since the apparatus is now neither gaining nor losing heat on the whole, its heat-account must balance; from which where I A is the total heat per Ib. of the gas entering at A, IB is the total heat per Ib. of the gas leaving at /?, and Ic is the total heat per Ib. of the liquid leaving at C. In this steady working the aggregate total heat of the fluid passing out is equal to that of the fluid passing in. The fluid, as a whole, takes up no heat in passing through the apparatus. 176 THERMODYNAMICS [CH. Suppose now that the liquid leaving at C were evaporated at its boiling point T 2 , and then heated at the same pressure from T 2 to T lt The heat required to perform that operation would be (1 - q) (L + K v (2\ - T,)]. But that hypothetical operation would result in this, that the whole of the fluid then leaving the apparatus would b restored to the temperature of entry, namely T 19 since the part which escapes at B is already at that temperature. Hence the heat re- quired for it is equal to the quantity Q as defined in Art. 123. We therefore have from which 1 - q = r . L+ K, 9 (l l -r- 1 2 ) This equation allows the fraction that is liquefied to be calculated when Q is known*. The fraction so found is the ideal output of liquid, for we have assumed that there is no leakage of heat from without, and that the action of the interchanger is perfect in the sense that the outgoing gas is raised by it to the temperature of entry. Under real conditions there will be some thermal leakage, and the gas will escape at a temperature somewhat lower than T l : the effect is to diminish the fraction actually liquefied. The fraction 1 q is increased by using a larger pressure-drop. It is also increased by reducing the initial temperature T ; thus the output of a given apparatus can be raised by using a separate refrigerating device to pre-cool the gas. Pre-cooling is indispensable if the method is to be applied to a gas in which, like hydrogen, the Joule-Thomson effect is a heating effect at ordinary temperatures, but becomes a cooling effect when the initial temperature is suffi- ciently low. 126. Linde's Apparatus. The principle of regenerative cooling described in the preceding article was first successfully applied by Linde in 1895 for the production of extremely low temperatures, and for the liquefaction of air, by means of an apparatus shown diagrammatically in fig. 57. It consists of an interchanger CDE formed of two spiral coils of pipes, one inside the other, enclosed in a thermally insulating case. A compressing pump P delivers air under high pressure through the valve H into a cooler J where * The specific heat of the vapour is here treated as constant from T 2 to T lf which is very nearly true at low pressures. IV] THEORY OF REFRIGERATION 177 the heat developed by compression is removed by water circulating in the ordinary way from an inlet at K to an outlet at L. The highly compressed air then passes on through the pipe BC to the inner worm and after traversing the worm it expands through the throttle-valve R into the vessel T, thereby suffering a drop in temperature. Then it returns through the outer worm F and, being in close contact with the inner worm, gives up its cold to the gas that is still on its way to expand. Finally it reaches the com- pression cylinder P through the suction- valve G, and is compressed to go again through the cycle. During the first stage it simply goes round and round in this way; but when the second stage is reached and condensation begins, the part that is liquefied is drawn off at V and the loss is made good by pumping in more air through the stop- valve at A by means of an auxiliary low-pressure pump, not shown Fig. 57. Linde's Regenerative Apparatus. in the sketch, which delivers air from the atmosphere to the low- pressure side of the circulating system. Linde showed that by keeping this lower pressure fairly high, it is practicable to reduce the amount of work that has to be spent in liquefying a given quantity of air. He pointed out that while the cooling effect of expansion depends upon the difference of pressures P A and P B on the two sides of the expansion- valve, the work done in compressing the air in the circulating system depends on the ratio of P A to P R . It is roughly proportional to the logarithm of that ratio, for it approximates to the work spent in the isothermal compression of a perfect gas, which (by Art. 28) is RT log e r, where r is the ratio of the volumes or of the pressures. If, for example, P A is 200 atmospheres and P B is one atmosphere, the cooling effect is proportional to 199 and the work of the compressing pump is roughly proportional tojog 200. If on the other hand the E. T. 12 178 THERMODYNAMICS [CH. back pressure PR is 50 atmospheres, the cooling effect is propor- tional to 150 and the work of the main compressing-pump to log 4. The cooling effect is reduced by only about one-fourth, while the work is reduced by nearly three-fourths. After allowing for the extra amount of work that has, in the second case, to be spent on the auxiliary pump in supplying air at 50 atmospheres to replace the fraction which is liquefied, there is still a marked advantage, in point of thermodynamic efficiency, in using a closed cycle with a moderately high back pressure. The Linde process is employed on a commercial scale to liquefy air as a first step in the separation of its constituents. A Linde plant at Odda, in Norway, liquefies about one hundred tons of air daily for the purpose of supplying nitrogen for use in the manu- facture of cyanamide, an artificial nitrogenous fertilizer which is formed by passing gaseous nitrogen over hot calcium carbide. The method by which the constituent gases are separated will be presently described. 127. Liquefaction of Air by Expansion in which Work is done. Claude's Apparatus. The drop in temperature which a gas undergoes in passing from a region of high pressure to a region of low pressure would be greater if the process were conducted reversibly, as by expansion in a cylinder in which the gas does mechanical work. We should still have the small Joule-Thomson cooling effect, but in addition there would be the (generally much larger) cooling effect that is due to the energy which the gas loses in doing work. Early attempts made by Siemens, Solvay , and others to reach very low temperatures by applying a thermal interch anger to an expansion cylinder, failed mainly because the cylinder soon reached a temperature at which the lubricant froze. This difficulty was successfully overcome in 1902 by Claude, who found that the difficulties attendant on expansion in a working cylinder down to a temperature below the critical point of air could be overcome by using certain hydrocarbons as lubricants. A hydrocarbon such as petroleum- ether does not solidify but remains viscous at a tem- perature as low as 160 C. Using a lubricant of this kind Claude succeeded, as an experimental tour de force, in liquefying air in an expansion cylinder furnished with a regenerative counter- current thermal interchanger : the expansion cylinder simply taking the place of the expansion-valve in an apparatus such as that of Art. 126. He also found that the liquid, once it begins to form, IV] THEORY OF REFRIGERATION 179 serves itself as a lubricant, and no other need then be supplied. Under these conditions, however, there is little if any advantage in using an expansion cylinder, for the volume of the fluid at the lowest extreme of temperature is so small as to make the work of expansion insignificant. There is not much additional cooling: at the same time it is far less practicable to secure thermal insulation with an expansion cylinder than with a Joule-Thomson orifice. Claude subsequently obtained a more economical result by giving the apparatus the modified form shown in fig: 58. In that arrange- ment part of the compressed air expands in a working cylinder to a temperature which may be just below the critical temperature, and the air which is cooled (but not liquefied) by that expansion is used as a cooling agent on the remainder of the air, with the Fig. 58. Claude's method. result that some of the latter is liquefied under the higher pressure at which it is supplied. The supply comes in, at a pressure of 40 atmospheres or so, through the central pipe of the counter-current interchanger M. Part of it passes into the expansion cylinder D where it expands doing work, and is then discharged through the condensing vessel L, where it serves as the cooling agent to maintain a temperature somewhat lower than 140 C., the critical tem- perature of air. The remainder of the compressed air enters the tubes of L and is condensed there, under pressure, dropping as a liquid into the chamber below from which it can be drawn off. In a further development of this invention Claude made the expansion compound, and caused the expanded gas to act as a cooling agent after each stage, becoming itself warmed up in the process. The expanded gas is thereby prepared to suffer further 122 180 THERMODYNAMICS [CH. expansion without an excessive fall of temperature. During its expansion the gas in the cylinder is not so near the liquid state as to make expansion in a working cylinder of little use. The arrangement with compound expansion is illustrated in fig. 59. Air under pressure enters, as before, through the central pipe of M. Part of it goes to the first expansion cylinder A, does work there, and proceeds at reduced pressure, and at a temperature below the critical point, through the outer vessel of the condenser L 19 in the inner tube of which some of the compressed air is being condensed. This warms up the expanded air- to some extent, and it then passes on to complete its expansion in B, which again brings its temperature down sufficiently to allow it to act as condensing agent for the Eig. 59. Claude's later method with compound expansion. remaining portion of the air under pressure, in the second con- denser L 2 . This division of the expansion into two (or it may be more than two) stages is equivalent to making the process as a whole more nearly isothermal, so that the air need not at any stage deviate very widely from a temperature which is just sufficiently below the critical point to allow liquefaction to go on under the pressure at which the air is supplied*. 128. Separation of the Constituents of Air. The lique- faction of air enables the constituent gases to be separated because * G. Claude, Comptes Rendus, 11 June 1906, and 22 Oct. 1906. See also his book on Liquid Air, Trs. Cottrell, 1913. An article by Professor E. Mathias in Revue generate, des Sciences, 15 Sept. 1907, contains an interesting account of the whole subject of the industrial liquefaction of air. iv] THEORY OF REFRIGERATION 181 in re-evaporation they have different boiling points. The boiling point of nitrogen, under atmospheric pressure, is about 195 C. or 13 lower than that of oxygen, which is 182 C. When a quan- tity of liquefied air evaporates freely both gases pass off, but not in the original proportion in which they are mixed in the liquid. The nitrogen evaporates more readily, and the liquid that is left becomes richer in oxygen as the evaporation proceeds. This difference in volatility between oxygen and nitrogen makes it possible to carry out a process of rectification analogous to the process which is used by distillers for extracting spirit from the "wash" or fermented wort, which is a weak mixture of a ohol and water, by means of a device known as the Coffey Still. In the still patented by Aeneas Coffey in 1880 there is a rectifying column consisting of a tall chamber containing many zig-zag shelves or baffle plates. The wash enters at the top of the column and trickles slowly down, meeting a current of steam which is admitted at the bottom and rises up through the shelves. The down-coming wash and the up-going steam are thereby brought into close contact and an exchange of fluid takes place. At each stage some of the alcohol is evaporated from the wash and some of the steam is condensed, the heat supplied by the condensation of the steam serving to evaporate the alcohol. The condensed steam becomes part of the down-coming stream of liquid : the evaporated alcohol becomes part of the up-going stream of vapour. Finally at the top a vapour comparatively rich in alcohol passes off: at the bottom a liquid accumulates which is water with little or no alcohol in it. A temperature gradient is established in the column : at the bottom the temperature is that of steam, and at the top there is a lower temperature approximating to the boiling point of alcohol. The wash enters at this comparatively low temperature, and takes up heat from the steam as it trickles down. Linde applied the same general idea in a device for separating the less volatile oxygen from the more volatile nitrogen of liquid air. In this device, the primary purpose of which was to obtain oxygen, there is a rectifying column down* which liquid air trickles, starting at the top at a temperature a little under 194 C. or 79 absolute, which is the boiling point of liquid air under at- mospheric pressure. As the liquid trickles down it meets an up-going stream of gas which consists (at the bottom) of nearly pure oxygen, initially at a temperature of about 91 absolute, that being the boiling point of oxygen under atmospheric pressure. As the gas 182 THERMODYNAMICS [CH. rises and comes into close contact with the down-coming liquid, there is a give and take of substance: at each stage some of the rising oxygen is condensed and some of the nitrogen in the down-coming liquid is evaporated; the liquid also be- comes rather warmer. By the time it reaches the bottom it consists of nearly pure oxygen: the nitrogen has almost completely passed off as gas, and the gas whr t passes off at the top con- sk s very largely of nitrogen. More precisely it consists of nitrogen mixed with about 7 per cent, of oxygen : in other words, out of the whole original oxygen content of the air (say 21 per cent.) two- thirds are brought down as liquid oxygen to the bottom of the column, while one-third passes off unseparated along with all the nitrogen. The oxygen that gathers at the bottom is with- drawn for use, and is evaporated in serving to liquefy fresh com- pressed air, which is pumped into the apparatus to undergo the pro- cess of separation. The cold gases that ^ire leaving the ap- paratus, namely the oxygen which is the useful product, and the nitrogen which passes off as waste gas at the top of the column, are made to traverse counter- current interchangers on their way out, so as to give up their cold to the incoming compressed air Fig 60 Linde > s apparatus of 190 2 for that is on its way to be liquefied. extracting oxygen by rectification. In the diagram, fig. 60, these counter-current interchangers are omitted for the sake of clearness, but the essential features of the condensing and rectifying apparatus are shown. The figure is iv] THEORY OF REFRIGERATION 183 based on one given in Linde's patent of June 1902, which describes the invention by which a process of rectification has been success- fully applied in the extraction of oxygen from air. There A is the rectifying column, consisting in this instance of a vertical chamber stacked with glass balls, through the interstices of which the liquid trickles down. The lower part B contains an accumulation of fluid which, when the apparatus has been at work long enough to establish a uniform regime, consists of nearly pure liquid oxygen. Compressed air, which has been cooled by passing through a counter-current interchanger, enters at C, becomes liquefied in the vertical condenser pipes D, which are closed at the top, and drops down into the vessel E. It gives up its latent heat to the oxygen in B, thereby evaporating a part of that, and so supplying a stream of gaseous oxygen which begins to pass up the rectifying column. On its way up, this stream of gas effects an exchange of material with the liquid air which is trickling down: gaseous oxygen is condensed and returns with the stream to the vessel B, while nitrogen is evaporated and passes off at the top of the column, at N 9 mixed with some oxygen. The escaping gas goes through an interchanger, taking up heat from the in- coming compressed air. The accumulation of nearly pure liquid oxygen in B overflows into the lower vessel F, where a supplementary supply of compressed air entering at G is employed to evaporate it by means of a similar arrangement of condenser tubes open at the bottom and closed at the top, this air becoming itself condensed in the process, and falling as a liquid into the vessel H. The liquefied air from E and from H is still under pressure: it passes up through expansion- valves K to the top of the rectifying column, where it is discharged over the glass balls at a pressure not materially above that of the atmosphere. This secures the necessary difference in tempera- ture between the bottom and top of the column. The com- pressed air plays the part of heater and evaporator of the liquid oxygen at the bottom, at the comparatively high temperature of about 91 absolute, before it undergoes rectification. In other words, it not only corresponds to the "wash" of the Coffey still, but it also serves as the equivalent of the heater by which the liquid at the bottom of the still gives off an upward current of steam. Gaseous oxygen, the desired product in this case, passes off at 0, and like the waste gas, consisting mainly of nitrogen, which escapes at A 7 , it goes through a counter-current interchanger, taking up heat 184 THERMODYNAMICS [CH. from the compressed air which enters partly at C and partly at G. It is the waste gas in this process that forms the analogue of the rectified spirit which is the useful product of the Coffey still. At first, when the machine begins working, the air is highly com- pressed, but after the operation has gone on for some time, and a steady state is approached, a much lower pressure is sufficient. It must be high enough to make the air liquefy at the temperature of the liquid oxygen bath, say 91 absolute, and in practice it is kept PERCENTAGE OF OXYGEN IN VAPOUR S8SS.S8SS88 i i i 7 / / / / / y ^ / ^ ^ ^ 10 20 30 40 50 60 70 80 90 10 PERCENTAGE OF OXYGEN IN LIQUID. Fig. 61 higher than this to ensure that the drop in temperature at the expansion- valve may be sufficient to make good any losses due to leakage of heat from outside, and to imperfect interchange in the counter-current apparatus. For some time after the apparatus is first started the rectifying action is imperfect, but as the process goes on the liquid contents of the vessel B become richer and richer in oxygen, the rectification becomes more complete, and the pressure may be reduced. Under practical conditions it is easy to secure that the gaseous product shall be pure to the extent of containing 98 per cent, of oxygen. iv] THEORY OF REFRIGERATION 185 129. Baly's Curves. The action of the rectifying column will be made more intelligible if we refer to the results of experiments published in 1900 by Baly*, which deal with the nature of the evaporation in mixtures of liquid oxygen and nitrogen. Given a mixture of these liquids in any assigned proportion, equilibrium between liquid and vapour is possible only when the vapour contains a definite proportion of the two constituents, but this proportion is not the same as that Jn the liquid mixture. Say for example that the liquid mixture is half oxygen and half nitrogen, then according to Baly's experiments the vapour proceeding from such a mixture will consist of about 22 per cent, of oxygen and 78 per cent, of nitrogen. With these proportions there will be equilibrium. If however a vapour richer than this in oxygen be brought into contact with the half-and-half liquid, part of the gaseous oxygen will condense and part of the liquid nitrogen will be evaporated, until the proportion giving equilibrium is reached. The curve, fig. 61 , shows, for each proportion in the mixed liquid, what is the corresponding proportion in the vapour necessary for equilibrium: in other words what is the proportion which the constituents have in the vapour, when that is being formed by evaporation of the mixed liquid, in the first stages of such an eva- poration, before the proportion in the liquid changes. In this curve the base-line specifies the proportion of oxygen in the liquid mixture, from to 100 per cent., and the ordinates give the proportion of oxygen in the corresponding vapour, when the vapour is formed under a pressure equal to that of the atmosphere. Much the same general relation will hold at other pressures. It will be seen from the curve that when the evaporating liquid mixture is liquid air (oxygen 21 per cent., nitrogen 79 per cent.), the proportion of oxygen present in the vapour that is coming off is about 7 per cent, or a little less. This is what occurs at the top of the rectifying column in the apparatus of fig. 60. The liquid that is evaporating there is freshly formed liquid air, and hence the waste gases carry off about 7 per cent, of oxygen. Coming down the column the liquid finds itself in contact with gas containing more oxygen than corresponds to equilibrium. Accordingly oxygen is condensed and nitrogen is evaporated at each stage in the descent, in the effort at each level to reach a condition of equilibrium between the liquid and the vapour with which it is there in contact. * Baly, Phil. Mag., vol. XLIX, p. 517, 1900. 186 THERMODYNAMICS [CH. Fig. 62 is another form of Baly's curve, the form, namely, in which the results of the experiments were originally shown. There the ordinates represent the absolute temperature (in centigrade 88 87 86 ^ Q: I- S kj 83 80 79 30 40 50 60 70 80 PERCENTAGE OF OXYGEN. Fig. 62 degrees) at which, under atmospheric pressure, the mixed liquid Kr ils, and two curves are drawn which show by means of the scale v e base-line the percentage constitution of (1) the liquid, vapour, when the condition of equilibrium between liquid iv] THEORY OF REFRIGERATION 187 and vapour is attained*. A horizontal line drawn across the curves at any assigned level of temperature shows the composition of vapour and liquid respectively for that temperature, when the two are in equilibrium. Taking an intermediate point between the top and bottom of the rectifying column, and drawing the line for the corresponding temperature, we should find the respective compositions of liquid and vapour there to approximate to the values found from the two curves, this approximation being closer the more slowly the liquid trickles down, and the more intimate the contact between liquid and gas. If a similar condition of equilibrium holds at each stage in the process of liquefying a mixture of the gases, these curves may also be taken as showing what is the proportion of the constituents in the mixed liquid at each stage while condensation of the mixed gas proceeds. Thus when air containing 21 per cent, of oxygen begins to liquefy, the liquid initially formed should, under equilibrium conditions, be much richer in oxygen: the proportion of oxygen in it, according to the curve, is 48 per cent. These conditions are approximately realized when the process known as "scrubbing" is resorted to in the liquefaction of air. By this process, which will be presently described in the form in which it has been practically carried out by Claude, a partial separation between the two constituents is effected during the act of liquefaction. * The following figures are given by Baly: Absolute Temperature 77-54 Percentage In Vapour of Oxygen In Liquid 78 2-18 8-10 79 6-80 21-60 80 12-00 33-35 81 17-66 43-38 82 23-60 52-17 83 29-95 59-55 84 36-86 66-20 85 44-25 72-27 86 52-19 77-80 87 60-53 82-95 88 69-58 87-60 89 79-45 91-98 90 89-80 96-15 90-96 100 100 188 THERMODYNAMICS [CH. 130. Complete Rectification. In Linde's invention of 1902 the rectifying process is incomplete, for although the process yields nearly pure oxygen it leaves a part of the oxygen to escape in the waste gas and it does not yield pure nitrogen. In a commer- cial process for the manufacture of oxygen this is of no consequence : for the raw material costs nothing, and the nitrogen is not wanted. But a modification of the process enables the separation to be made substantially complete, should it be desired to complete it, and allows approximately pure nitrogen to be obtained, as well as pure oxygen. The modification consists in extending the rectifying column upwards and in supplying it at the top with a liquid rich in nitrogen. A fractional method of liquefaction is adopted, which separates the condensed material at once into two liquids, one containing much oxygen and the other little except nitrogen. The latter is sent to the top of the rectifying column, while the former enters the column at a lower point, appropriate to the proportion it contains of the two constituents. Practically pure nitrogen passes off as gas at the top, and practically pure oxygen from the bottom. Fig. 63 is a diagram showing this modified process in a form given to it by Claude. The counter- current interchangers which are of course part of the actual apparatus are omitted from the diagram. Compressed air, cooled by the interchanger on its way, enters the condenser at A. The condenser consists of two sets of vertical pipes, communicating at the top, where they all open into the vessel J5, but separated at the bottom. The central pipes, which open from the vessel A, are one set : the other set form a ring round them and drain into the vessel C. Both sets are immersed in a bath, S, of liquid which, when the machine is in full operation, consists of nearly pure oxygen. The condensation of the compressed air causes this oxygen to be evaporated. Part of it streams up the rectifying column Z>, to be condensed there in carrying out the work of rectification and consequently to return to the vessel below. The rest of the evaporated oxygen, forming one of the useful products, goes off by the pipe E at the side. In these features the apparatus is substantially the same as Linde's, but there is a difference in the mode of condensation of the compressed air. Entering at A it first passes up the central group of condenser pipes, and the liquid which is formed in them contains a relatively large proportion of oxygen. This liquid drains back into the vessel A, where it collects, and the gas which has survived condensation in these pipes goes IV] THEORY OF REFRIGERATION 189 N COMPRESSED AIR Fig. 63. Claude's apparatus for the complete separation of oxygen and nitrogen. 190 THERMODYNAMICS [CH. iv on through B to the outer set of pipes, is condensed in them, and drains into the other collecting vessel C. It consists almost wholly of nitrogen. Then the liquid contents of C are taken (through an expansion-valve) to the top of the rectifying column, while those of A enter the column lower down, at a level L, chosen to correspond with the proportion of the constituents. The result is to secure practically complete rectification, and the second product of the machine commercially pure nitrogen passes off at the top through the pipe N and may be collected for use. The action in the central pipes of the condenser is to be inter- preted in the light of Baly's curves. The first portions of the air to be condensed trickle down the sides of these pipes and are " scrubbed" by the air as it ascends : that is to say they are brought into such intimate contact with the ascending air that a condition of equilibrium between liquid and vapour is at least closely ap- proximated to. The condition of equilibrium when gases of the composition of air are being condensed requires, as we have seen, that about 48 per cent, of the liquid should consist of oxygen*. Accordingly the liquid which collects in the vessel A is of this degree of richness, or near it. And by making the condenser pipes long enough it is clear that little or no oxygen will be left to pass over through B into the other pipes. It is true of course that in the upper parts of the central pipes the liquid that is formed con- sists largely of nitrogen, but as this trickles down the pipe in which it has been condensed there is a give and take between it and the ascending gas, precisely analogous to that which occurs in a rectify- ing column, and when the liquid reaches the bottom it has been so much enriched in oxygen as to be nearly or completely in equili- brium with the gaseous air, and therefore contains about 48 per cent. When the 48 per cent, liquid from A is discharged through an expansion- valve into the rectifying column at L, it produces an atmosphere which has the composition of air (21 per cent, of oxygen). Hence the part of the column which extends above this point has for its function to reduce the percentage of oxygen in the ascending gas from 21 per cent, to nil, and this is done in the second stage of rectification, by means of the liquid from C which consists almost wholly of nitrogen f. * That proportion, as has been pointed out in speaking of Baly's curves, relates to experiments made at atmospheric pressure. At the higher pressure under which condensation takes place in Claude's apparatus it may not be exactly the same. f For further particulars of some of the subjects treated in this Chapter reference should be made to the author's book on The Mechanical Production of Cold. CHAPTER V JETS AND TURBINES 131. Theory of Jets. We have now to consider the manner in which a jet is formed in the discharge, through an orifice, of steam or any other gas under pressure. To simplify matters it will be assumed that the fluid takes in no heat and gives out no heat to other bodies during the operation; in other words that the jet is formed under adiathermal conditions. Suppose a gas to be flowing through a nozzle or channel of any form, from a region where the pressure is relatively high to one where it is lower. Each element of the stream expands, and the work which it does in expanding gives energy of motion to the element in front of it. The whole stream therefore acquires velocity in the process and also increases in volume. Let A and B (fig. 64) be imaginary partitions, across which it flows, taken at right angles to the direction of the stream lines, A being in the region of higher F - 64 pressure. Let P a be the pressure at A, v a the velocity there, and V a the volume which unit quantity of the gas has as it passes the imaginary partition at A. Similarly let P b , v b and V b be the pressure, velocity, and volume of unit quantity at B. Let E a and E b be the internal energy of the gas at A and B respectively. In flowing from A to B the velocity changes from v a to v b and there is consequently a gain of kinetic energy amounting, per unit of mass, to - . Each unit quantity of gas that enters the space between A and B has work done upon it by the gas behind, amounting to P a V a . In passing out of this space at B it does work on the gas in front amounting to P b V b . In flowing from A to B it loses internal energy amounting to E a - E b . Hence by the principle of the 192 THERMODYNAMICS [CH. conservation of energy, since by assumption no heat is taken in or given out, 2 _ v 2 6 ^r = E a -E b + P a V a -P b V b ............ (1). But E a + P a V a is I a> the total heat at A, and E b + P b F 6 is I b> the total heat at B, and the equation may consequently be written 2g "> ..................... ' The gain in kinetic energy is therefore equal to the loss of total heat, or what is commonly called the "heat-drop." We are treating E and / as if they were expressed in work units : when ex- pressed in heat units they have to be multiplied by the mechanical equivalent J. The equation applies as between any two places in the flow, and taking the process as a whole, from the initial condition in which the velocity is v 1 and total heat I 1 to the final condition in which the velocity is v 2 and total heat / 2 we have r -A- In many practical cases the initial velocity is zero or negligibly small, and then v z where v is the velocity acquired in consequence of the heat-drop. This is the fundamental equation from which to calculate the velocity which an expanding fluid acquires in a jet, starting from rest. So far there has been no assumption as to absence of losses through friction or eddy currents. If we assume, as an ideal case, that in the formation of the jet the fluid is expanding under such conditions that there is no conduction of heat to or from or within the fluid and also no dissipation of energy through friction or eddies, the heat-drop in the equation is that which occurs in expansion with constant entropy. W r e have already seen (Art. 80) that this heat-drop is equal to the area ABCD of the ideal indicator diagram (fig. 65) for adiabatic ex- eP\ pansion from the initial to the final state, or I VdP. J p. 2 Hence ft - ^ = ^^ AECT> = f 'VdP (5). v] JETS AND TURBINES 193 This result might also be inferred from the fact that, under the assumed conditions, the gas is doing all the work of which it is A_ ideally capable, as it expands from the first to the second state, in giving kinetic energy to its own stream. The gain of kinetic energy is, there- fore, equal to the area of the ideal Fig 65 indicator diagram. Assume that we may, with sufficient accuracy, express the ex- pansion in the ideal indicator diagram by a formula of the type py m = constant. Then the area of the diagram, namely VdP = ??-- (R V, - P. Fo) m 1 A & ' Hence when the expanding fluid starts from rest, at pressure P l , to form a jet, we have v 2 m = =^^L ] -w 'J PiFi - as an equation from which to find the velocity v when the pressure has fallen to any lower pressure P, under the assumed conditions of flow without friction or eddies and with no conduction of heat. Equation (6) is a particular case of Equation (4), namely the case where the expansion is isentropic and where the relation of pressure to volume in isentropic expansion admits of being expressed by the formula PV m = constant. 132. Form of the Jet (De Laval's Nozzle). As expansion of the fluid in a jet proceeds, the volume and velocity both increase. It is easy in frictionless adiabatic flow to calculate both, and in that way to determine the proper form to give to the nozzle or channel, to make provision for the increased volume, having regard to the increased velocity. At any stage the area of cross-section of the channel required for each Ib. of fluid discharged is equal to the volume per Ib. divided by the velocity. It is convenient to reckon the area of section per unit of mass in the discharge, and afterwards multiply by the number of Ibs. or kilogrammes. E.T. 13 194 THERMODYNAMICS [CH. Let M represent the discharge, namely the mass which passes through the nozzle per second, X the area of cross-section of the stream at any part of the nozzle, v the velocity there, and V the volume of the fluid there (per unit of mass); then vX X V M = and = . V M v On making the calculation for a gaseous fluid starting from rest and discharged into a region of much lower pressure, it will be found that in the earliest stages the gain of velocity is relatively great, but as expansion proceeds the increase of volume outstrips the increase of velocity. The result is that the ratio of volume to velocity at first diminishes, passes a minimum value, and then increases; and hence the channel to be provided for the discharge, after passing a minimum of cross-section, expands in the later stages. The proper form for the nozzle, to allow the heat-drop corresponding to a large drop in pressure to be utilized as fully as possible in giving kinetic energy to the stream, is therefore one in which the area of section at first contracts to a narrow neck or "throat" and afterwards becomes enlarged to an extent that is determined by the available fall of pressure. It is on this principle that De Laval's "convergent-divergent" nozzle /fig. 66) is designed. The throat, or smallest section, is ap- proached through a more or less rounded entrance which allows the stream lines to con- verge, and from the throat out- wards to the discharge end the nozzle expands in any gradual manner, generally in fact as a simple cone, until an area of section is reached which will correspond to the proper area of discharge for the final volume and velocity, the values of which depend upon the final pressure.. The divergent taper from the throat onwards is made sufficiently gradual to preserve stream-line motion as completely as is practic- able, and so avoid the formation of eddies which would dissipate the kinetic energy of the stream. A very short rounded entrance to the throat is sufficient to guard against eddies in the convergent portion of the stream, but in the divergent portion a much more gradual change of section is required. The nozzle shown in the figure was designed for an initial pressure of 250 pounds per sq. inch and a back pressure of about Ij pounds. By the back v] JETS AND TURBINES 195 pressure is meant the pressure in the space into which the fluid is discharged. In the design of such a nozzle the purpose is (1) to make the discharge have a given value, and (2) to give the stream as high a final velocity as possible by utilizing completely the energy of the fluid in expanding down to the back pressure. The data for the design are the initial pressure, the back pressure, and the intended amount of the discharge. It will be shown as we proceed that the area of section at the throat depends only on the initial pressure and the intended discharge; and that the enlargement from the throat to the final section depends further on the back pressure against which the stream is to escape. At any place in the nozzle the discharge per unit area of cross- section is At the throat, where the cross-section is least, this is a maximum. Consider now the ideal case of isentropic expansion in a nozzle when the fluid is one for which PV m is constant during such ex- pansion. Equation (6) is then applicable. The velocity at any point, the pressure there having fallen to P, is / and the volume is V = V l ( Hence for the discharge per unit area of section at the place where the pressure is P, we have m - 1 This may be applied to calculate the proper section X for a given discharge M when the pressure has fallen from the initial pressure PI to any assigned lower pressure P. For the purpose of designing a nozzle there are only two places where this calculation has to be made, namely at the throat, and at the end where the fluid escapes against the assigned back pressure. When the throat-section X t and the final section X f have been calculated, a suitable form for the nozzle is readily drawn; any smooth curve will serve for the con- vergent entrance, and any conical taper may be selected for the divergent extension from the throat to the end, provided it is neither so abrupt as to interfere with stream-line flow, nor so 132 196 THERMODYNAMICS [CH. gradual as to make the nozzle unduly long and thereby introduce unnecessary friction. To calculate the final section X f which will allow the energy of the fluid to be fully utilized by expansion down to the assigned back pressure, that pressure is to be taken for the value of P in Eq. (7). To calculate the section at the throat the pressure there has first to be found. The pressure at the throat is determined by the consideration that the discharge per unit of section (M/X) is there a maximum. If the expression for M/X in Eq. (7) is differ- entiated with respect to P/Pj and the differential written equal to zero, the resulting value of P/P t w r ill be that for which M/X is a maximum ; in other words it will be the value of P t jP l , where P t is the pressure at the throat. Eq. (7) may be written M 2gm Pl /PN /P ar ' V.irn PW \ft/ uv The condition for a maximum is found by differentiating the quantity under the second root : m 2 (~ p from which Further, by substituting this in Eq. (6a), we have for the velocity " aiethro " The volume (per Ib.) of the fluid at the throat is By combining these an equation is obtained for the discharge per unit of cross-section at the throat, X t V t m+l (m+\)V l ' From this equation the cross-section at the throat is found which will give an assigned discharge when the initial pressure is known. The ratio of the cross-section at any place, where the v] JETS AND TURBINES 197 pressure is P, to the cross-section at the throat, is readily found from Eq. (7 a) : _ m+l ni + l .(12). This expression is convenient in determining the proper amount of enlargement of the nozzle from the throat to the end when the back pressure is assigned. 133. Limitation of the Discharge through an Orifice of Given Size. It follows from these equations that the discharge through a given orifice under a given initial pressure P 1 depends only on the cross-section at the narrowest part of the orifice, and is indepen- dent of the back pressure, provided the back pressure is not greater than P t as calculated by Eq. (8). By continuing the expansion in a divergent nozzle after the throat is passed, the amount of the discharge is not increased, but the fluid acquires a greater velocity before it leaves the nozzle, because the range of pressure which is effective for producing velocity is increased. To put it in another way, we may say that the heat-drop down to the pressure at the throat determines the amount of the discharge, and the remainder of the heat-drop, which would be wasted if there were no divergent extension of the nozzle, is utilized in the divergent portion to give additional velocity to the escaping stream. This velocity is given in a definite and useful direction, whereas if there were no divergent extension of the nozzle the fluid, after leaving the nozzle, would expand laterally, and its parts would acquire velocity in directions such that no use could be made of the kinetic energy so acquired. Consider what happens with a nozzle such as that of fig. 67, which has no divergent extension. Fluid is expanding from a chamber where the pressure is P 1 into a space where the pressure is P 2 . Assume the back pressure P 2 to be less than P t as calculated by Eq. (8). In that case the pressure in Fi S- 67 the jet, where it leaves the nozzle, will be P t , and the further drop of pressure to P 2 will occur through scattering of the stream. The discharge is determined by Eq. (11). It is not in- creased by any lowering of the back pressure P 2 , because any 198 THERMODYNAMICS [CH. lowering of P 2 does not affect the final pressure in the nozzle, which remains equal to P t . Osborne Reynolds* explained the apparent anomaly by pointing out that the stream is then leaving the nozzle with a velocity equal to that with which sound (or any wave of extension and compression) is propagated in the fluid, and consequently any reduction of the pressure P 2 cannot be com- municated back against the stream : its effects are not felt at any point within the nozzle. The pressure in the stream at the orifice therefore cannot become less, however low the back pressure P 2 may be. But if P 2 is increased so as to exceed P t , the lateral scattering close to the orifice ceases, the velocity is reduced, the pressure at the orifice then becomes equal to P 2 , the discharge is reduced, and its amount is to be calculated by writing P 2 for P inEq. (7) or (Id). In applying these results to a nozzle of any form, the least section is to be regarded as the throat: if there is a divergent extension beyond the least section the amount of the discharge is not affected, though the final velocity of the stream is increased. Taking a nozzle of any form, and a constant initial pressure P l , if we reduce the back pressure P 2 from a value which, to begin with, is just less than P l9 the discharge increases until P 2 reaches the m value P! f - - J . After that, any further reduction of P 2 does not increase the discharge. But the velocity which the fluid acquires before it leaves the nozzle may then be augmented by lowering- Pa and adding to the divergent portion of the nozzle. The nozzle will be rightly designed when it provides for just enough expansion to make the final pressure equal to the back pressure; the jet then escapes as a smooth stream, and the energy of expansion is utilized to the full. If the nozzle does not carry expansion far enough; if in other words the final pressure exceeds the back pressure, energy will be wasted by scattering. If on the other hand the back pressure is too high for the nozzle, so that the nozzle provides for more expansion than can properly take place, vibra- tions are set up which cause some wastej". We shall now consider the application of these general results to air and to steam. 134. Application to Air. In applying the above formulas to * Phil Mag. March, 1886; Collected Papers, vol. n, p. 311. f For experiments on the effects of nozzles which carry expansion too far, or not far enough, see Stodola's book on the Steam Turbine. v] JETS AND TURBINES 199 any permanent gas, such as air, the index m is y, the ratio of the two specific heats (Art. 25). Its value for air may be taken as 1-40. Substituting this number in Eq. (8) we have, for a jet of air expanding under isentropic conditions, Hence if the jet is being delivered against a back pressure less than 0-528Pj a divergent extension of the nozzle is required to give the greatest possible velocity to the issuing stream, though the quantity delivered will be the same as that which would be delivered against a back pressure of 0-528P 1 . If the back pressure be increased it must exceed 0-528P 1 before there is any diminution in the discharge. As a numerical example, suppose that air, with an initial pressure of 300 pounds per sq. inch, is discharged through a convergent- diver- gent nozzle into the atmosphere, or against a back pressure of say 15 pounds per sq. inch. The pressure at the throat is 158-4 and, since the final ratio of pressures is one to twenty, the ratio of the final cross-section to the cross-section of the throat should, by Eq. (12), be Xf _ A/0-4019 - 0-3349 _ X* ~ VO'01385 - 0-00588 This is for the ideal case of isentropic expansion. Effects of friction are disregarded; they will be considered in Art. 140. 135. Application to Steam. In applying the general equations for isentropic expansion to steam, we have to distinguish between the type of expansion which occurs in a jet and the type of expan- sion which was treated of in Art. 78. In that article the expansion was assumed to be isentropic (adiabatic); 3 = const.t In both cases the expansion is isentropic and therefore * Observations of the appearance of escaping jets support this conclusion. They show that when steam initially dry (but not necessarily superheated) escapes from a divergent nozzle in which it has expanded through a considerable ratio, no particles of water become visible until the steam has travelled some distance from the orifice. See Stodola, Zeitschrift des Vereines deutscher Ingenieure, 1913. f Here, and on p. 204, the 6 of Calendar's equation is written /3, to avoid con- fusion with the 6 of the diagram. 204 THERMODYNAMICS [CH. But though the entropy and the pressure are the same at b as at c, the fluid is in two very different states. At b it is a homogeneous gas ; at c it is a wet mixture. At c its temperature is the temperature of saturation corresponding to its pressure there ; at b its temperature is much lower, being determined by the equation P (V ]8) = RT, 3^ which makes T/Pw = const. The volume is of course less at b. The heat-drop from a to c is the thermal equivalent of the work represented by the area eacf, and the heat-drop from a to & is the thermal equivalent of the work represented by eabf, since both types of expansion are adiabatic (see Art. 80). Hence the heat- drop is less in the metastable expansion, by an amount that is equivalent to the area abc, and the total heat at b is therefore greater than the total heat at c by that amount. The total heat I c of the mixture at c, after equilibrium expansion, may be found by the method described in Art. 89 or Art. 90. The total heat / & in the metastable fluid at b may be found by reckoning the heat-drop from the initial value I a . Since the volume at any stage in the metastable expansion is 10 Then, since I a -I b = A (Area eabf) = A VdP, I a -I b = APp (V a - 0) dPIPu + Al dP j p b q Suppose now that after sudden expansion to b, along the curve ab, the metastable fluid at b is allowed to become stable by partially condensing under constant pressure, without any gain or loss of heat. Its temperature will rise to the saturation value for that pressure; it will, therefore, come to have the same temperature as the mixture at c, but it will be somewhat drier, because its total heat remains equal to 7 & which, as we have seen, is greater than the total heat I c of the mixture at c. Its volume will, therefore, increase to a point d, which is beyond c. If we write q c for the dryness at c of steam that has expanded in a stable state, or state of equilibrium as a whole, from a to c, and v] JETS AND TURBINES 205 q d for the dryiiess at d of steam that has expanded in a metastable state to b and has subsequently attained equilibrium, by water separating out at constant pressure, without loss or gain of heat, the difference of total heats is I d - I c - L (q d - q c ). But I d = I b and I b = I c + A (Area abc). Hence L (q d q c ) = A (Area abc). In attaining equilibrium the fluid as a whole has gained entropy, for a , or < c in the equilibrium state, by the amount that would convert the equilibrium mixture at c into the equilibrium mixture at d. Thus L(q d -q c ) A (Area, abc) d < s/ ki ^ c5 CO IT \ > CN o \ g Uj CO 3 ^ i^ Q) c II <* W \ ^P "S 0> QD 15 u, o X. CD 0= fill . c e> il C i, but some is in two bands whose wave-lengths are about 2*7//, and between 14 and 15/>t. In one case the radiation comes from vibrating molecules of H 2 O, in the other from vibrating molecules of CO 2 . It is also found that cold CO 2 absorbs strongly the radiation from a CO flame, and water- vapour absorbs strongly the radiation from a hydrogen flame. It may be concluded that the modes of free vibration of a molecule of cold CO 2 or water- vapour have periods corresponding to the chief wave-lengths which the gas gives out when it is so violently agitated as to become a source of radiation. This happens when the molecules are formed by the coming together of their constituent atoms. It is further found | that a mixed or compound gas burning to form CO 2 and H 2 O gives out both wave-lengths (4'4/x, and 2'8/x), and that the whole energy it radiates is equal to the sum of the energies separately computed for the molecules of H 2 O and CO 2 that are formed by its combustion. For equal volumes of H 2 O and CO 2 , at the same flame temperature, the radiation from CO 2 appears to be about 2J times that from H 2 O. These results point to the conclusion that when -a gas-engine mixture is fired, the energy that is radiated comes almost entirely from molecules of CO 2 and H 2 O in the burnt gases : very little of it comes from the nitrogen or the surplus oxygen. 174. Molecular Energy of a Gas. According to the kinetic theory of gases, the internal energy E of a gas is made up of the communicable energies of its molecules, and each molecule may, in general, have communicable energy of these three kinds: (1) Energy of translation of the molecule as a whole, (2) Energy of rotation of the molecule about an axis through its centre of mass, (3) Energy of vibration. * /m. stands for millionths of a metre. The wave-lengths in the visible spectrum range from about 0'39 to 0'77/u. f R. von Helmholtz : see the Third Report of the British Association Committee. 262 THERMODYNAMICS [CH. It is to energy of the first kind that the pressure of the gas is due. The kinetic theory shows* that, in a gas for which PV = RT, the energy of translation is ^RT. The pressure, in kinetic units, is numerically equal to two-thirds of the energy of translation of the molecules in unit volume of the gas. When the gas is heated, this energy increases in direct proportion to T. Hence if all the internal energy of the gas were in this form we should have (reckoning E from the absolute zero of temperature), and the specific heat would be constant. K v would then be equal to f R, which would make K v = %R, and y = f or T667. This is nearly true of actual monatomic gases : in such gases E consists entirely, or almost entirely, of energy of translation of the molecules. The second kind, energy of rotation, becomes an important part of the whole when the molecule comprises two or more atoms. We may conceive the molecule of a diatomic gas such as O 2 or N 2 to consist of paired atoms held at a definite distance apart like the heavy ends of a dumb-bell. Such a structure may, in the course of its encounters, acquire energy of rotation about any axis per- pendicular to the line joining the two atoms, but riot about that line. In addition to its three degrees of freedom of translation it consequently has two effective degrees of freedom of rotation ; hence five in all are effective out of the six degrees of freedom which it possesses as a rigid body. According to the kinetic theory the encounters between the mole- cules, when the gas is in a steady state as to pressure and tempera- ture, cause the energy of translation and rotation, (1) and (2) together, to become equally divided among as many of these six degrees of freedom as are effective. Hence in a perfect diatomic gas, besides the energy of translation, which is tj>RT, there is an amount of energy of rotation equal to RT due to the two freedoms of rotation, making f RT in all for the five effective degrees of freedom. Consequently in such a gas, if there were no energy except what is comprised in (1) and (2), we should find E = %RT; the specific heat would be constant, K v would be f R, K v would be \ R, and y would be \ or 1-4. These values agree well with those found in actual diatomic gases such as nitrogen or air, so long as the gases are cold. But, as * See Appendix II. vi] INTERNAL-COMBUSTION ENGINES 263 we have already seen, the specific heats become distinctly higher at high temperatures and y becomes less. This means that, in addition to items (1) and (2), there is in these gases some energy of vibration (3), the amount of which is insignificant at low tem- peratures, but becomes comparatively important when the gas is highly heated. It does not increase proportionally to T but in a more rapid ratio. In triatornic gases such as H 2 O or CO 2 , and in gases of a more complex constitution, there are three effective freedoms of rotation as well as three freedoms of translation, making six in all, between which the energy comprised under items (1) and (2) is equally shared. Thus items (1) and (2) account for an amount of energy equal to f RT . If there were no more, namely no energy of vibra- tion, the specific heat would be constant, K v would be 3/2, K p would be 47?, and y would be f or 1-333. In water-vapour and carbonic acid the value of y even at low temperatures is less than 1-333: in water-vapour it is about 1-3 and in carbonic acid it is a little lower. (Art. 163.) From this, and also from the fact that moderate heating considerably raises the specific heat, it may be inferred that even at low temperatures the molecules of these gases have some energy of vibration. Its pro- portion to the whole energy is increased by heating the gas. The amount by which the energy of vibration augments the specific heat in any gas may be inferred from the value of y if we assume the gas-law PV = RT to apply. Take for instance a tri- atomic gas. K v , if there were no vibration, would be 37?; let nR be the amount by which vibrational energy increases it. Then from which y = (4 + n)/(3 + n). Suppose that y has the value 1-30 instead of 1-333: this makes n = , and the specific heat K v is therefore 10 per cent, greater because of vibration. The value of n increases with the tempera- ture. At the temperature reached in a gas-engine explosion y for CO 2 is probably not much more than 1-14, which would correspond to a specific heat approaching 67?. (See Art. 224.) The phrase "energy of vibration" is to be understood as including all the kinds of energy which the molecule may acquire in the course of its encounters with other molecules, except energy of rotation as a whole and energy of translation as a whole. All such forms of energy are internal to the molecule itself: they may be due to 264 THERMODYNAMICS [CH. relative motions of its parts or to electrical disturbances within it, or within its atoms. It is to energy of vibration that the radiation given out by a heated gas is attributed. When a gas-engine mixture is fired the energy generated by the explosion is at first concentrated in the newly-formed molecules of CO 2 and H 2 O and spreads to the other molecules as a result of encounters. We may conjecture that it is at first mainly vibratiorial, and the encounters transform part of it into energy of translation. It is clear that the newly-formed molecules possess much more than their normal proportion of energy of vibration; much more, that is to say, than they would possess if the burnt mixture were kept without loss of heat long enough to let equilibrium be attained between the different kinds of energy, or were re-heated to the same temperature after being cooled. Some time, perhaps only a very short time, must elapse before a condition of equilibrium is reached. If the gas were enclosed, after combustion, in a vessel impervious to heat, while this process is going on, the energy of translation would increase at the expense of the energy of vibration, and the temperature would therefore rise though the total energy undergoes no change. So far as it goes, this process of attaining equilibrium has an effect like continued combustion or "after-burning." The time taken to reach equili- brium is not known. If the process is not very soon completed it may account for the fact that measurements of specific heat made by means of an explosion in a closed vessel give values somewhat greater than those that are got when the gas is heated in other ways. It has been suggested that the molecules of a hot gas emit radiation mainly when they undergo structural change. If this view be correct we should expect a gas mixture to radiate more energy immediately after explosion than when it is maintained at the same temperature, or re-heated to the same temperature after cooling. Hopkinson's and David's experiments show that in an explosion the gas continues to radiate for a second or so after maximum pressure. This may only mean that the special vibrations (special in violence or in kind) that are set up during the act of formation, to which radiation is ascribed, subside rather slowly. There is in any case an action going on. in all hot gases, that tends to maintain such vibrations, namely the breaking up of some mole- cules by exceptionally violent encounters, which is called dissocia- tion, and their subsequent re-formation. vi] INTERNAL-COMBUSTION ENGINES 265 175. ''Dissociation. In any gas, however homogeneous, and at any temperature, the molecules at a given instant have widely various speeds. Some of the encounters may be so violent as to break up compound molecules, separating them into parts which after a time meet fresh partners and re-combine. The probability of such disruptive encounters is obviously greater the hotter the gas is. In a hot gas in equilibrium, a process of dissociation and re-combination goes on continually, to an extent depending on the temperature, with the result that at any instant a, certain proportion of the gas is in the dissociated state. The proportion dissociated depends also on the pressure: at high pres- sure it is less than at low pressure, for the same temperature. According to measurements by Nernst and others the amount of H 2 O dissociated, under a pressure of one atmosphere, is barely 2 per cent, at a temperature of 2000 C., barely 1 per cent, at 1800, and 0-02 per cent, at 1227 C. At a pressure of ten atmospheres these numbers are about halved. In CO 2 , at one atmosphere, the proportion dissociated at 1650 C. is about 1 per cent, and at 1200 about 0-03 per cent. At such temperatures there is probably no sen- sible dissociation in nitrogen. These figures are open to some doubt*, but if they can be accepted as applying to the conditions of a gas- engine mixture after explosion (conditions which are not those of equilibrium) it appears that dissociation plays no considerable part in that action. So far as it has any effect it reduces, very slightly, the chemical contraction, by substituting some molecules of H 2 and O 2 for molecules of H 2 O, and some molecules of CO and O 2 for mole- cules of CO 2 ; for the same reason it reduces slightly the immediate development of thermal energy, leaving a small proportion of the available chemical energy of the gaseous fuel to be developed later, as the proportion of dissociated molecules diminishes with falling temperature. The effect is therefore equivalent to a continued combustion or " after-burning." Or, if we regard the whole thermal energy as being developed at once, and then a small portion of it as being absorbed by the breaking up of some of the molecules in consequence of their encounters, the effect of dissociation is indistinguishable from that of increased specific heat. * See the Second Report of the British Association Committee on Gaseous Ex- plosions, 1909. CHAPTER VII GENERAL THERMODYNAMIC RELATIONS 176. Introduction. In the earlier chapters but little use was made of formal mathematics in introducing the reader to the fundamental ideas of thermodynamics. To most students there is an advantage in having these ideas so presented : their physical significance is more likely to be appreciated. Once that is grasped, the student may proceed to a more mathematical treatment with less risk that the real meaning of the symbols will be obscured in the analysis. But a mathematical treatment must be resorted to if we wish to express with anything like completeness the relations that hold between the various properties of a fluid. One of the uses to which these relations can be put is in framing tables or charts of the properties of the fluid. By their aid such tables can be compiled from a small number of experimental data, and the experimental data themselves, as well as the numbers com- puted from them, can be tested for thermodynamic consistency. The purpose of this chapter is to show how the methods of the differential calculus may be applied to obtain, by inference from the First and Second Laws of Thermodynamics, certain general relations between the properties of any fluid. With some of these results the reader of the earlier chapters is already acquainted. In the next chapter some applications of these general relations to particular substances will be considered, including imperfect gases, or real fluids in the state of vapour. In particular it will be explained how Callendar has employed them in calculating his tables of the properties of steam. 177. Functions of the State of a Fluid. Assume that we are dealing with unit mass of a homogeneous fluid. As was pointed out in Art. 75, the six quantities named there, P, V, T, E, /, and (/>, are all functions of the state of the fluid, that is to say their value depends only on the actual state. When the fluid passes in any manner from one state to another, each of these quantities changes CH. vn] GENERAL THERMODYNAMIC RELATIONS 267 by a definite amount which does not depend on the nature of the operation by which the change is effected, but only on what the state was before and what it is after the operation has taken place. This fact is expressed in mathematical language by saying that the differential of any of these quantities is a "perfect" differential. Other quantities might be added to the list, which are also functions of the state of the fluid, such as the quantities G (or , which is G) and i/r mentioned in Art. 90. In what follows it is to be understood that T means (as usual) the absolute temperature on the thermodynamic scale (Art. 42). We defined the entropy cf> in Art. 44 by the equation d(f> in a reversible operation ; and the fact that ^ is a function of the state was proved there as a consequence of the result that I = for a reversible cycle, a result which follows from the Second Law of Thermodynamics. The Second Law is therefore involved in treating as a function of the state. Hence the fact that d is a perfect differential is sometimes spoken of as a mathematical expression of the Second Law. It is important to notice that while -^ , which is d are also functions of the state, it follows that this is also true of , which is / Tcf>, and of i/r, which is E T where M and N are quantities depending on the relations of the functions to one another, and are therefore also functions of the state. This expression applies whether both functions X and F vary, or only one of them. If X varies but not F, then dY = and dZ = MdX : similarly if F varies but not X, dX = and dZ = NdY.- Hence , 7 T /dZ In this notation, [3^] means the rate of variation of Z with \dXJjr respect to X when F is constant. In the language of the calculus, r!7 \ [STR] is the partial differential coefficient of Z with respect to VaA/ Y X when F is constant, and (-7^1 is the partial differential co- \dY j x efficient of Z with respect to F when X is constant. We might regard the change of Z as occurring in two steps. In the first step suppose X to change and F to keep constant. The corresponding part of the change of Z is MdX, and M is the slope of the thermodynamic surface in a section-plane ZX. In the second step X is constant and F changes. The corresponding part of the change of Z is NdY, and N is the slope of the thermodynamic vn] GENERAL THERMODYNAM1C RELATIONS 269 surface in a section-plane ZY. The whole change of Z is the sum of these two parts, as expressed in equation (1). The slopes along the two section -planes are expressed in equation (2). Combining these equations we have These equations apply when X, Y, and Z are interpreted as any three functions of the state of a fluid. Thus, for instance, if we think of a small change of state in which the temperature changes from T to T + dT, and the pressure from P to P + dP, the consequent change of volume will be Similarly, if the volume and pressure change, the consequent change of temperature is Or again, the change of entropy consequent on a change of tem- perature and pressure is and so on. It will be obvious that a very large number of similar equations might be written out, each using one pair of functions of the state as independent variables, and expressing in terms of their variation the variation of some third function of the state. These are merely forms of the general equation (3). Returning now to the general form in X, F, and Z, suppose a small change of state to occur of such a character that the function Z undergoes no change. In that special case dZ = 0; the steps MdX and NdY cancel one another. Consequently g) dX - - (g) dY, \dXJ Y \dY/x when dX and dY are so related that there is no variation of Z. Hence the general conclusion follows that U) dXJr~ dY x dx z - This relation between the three partial differential coefficients 270 THERMODYNAMICS [CH. holds, in all circumstances, for any three functions of the state of any fluid. It may be expressed in these alternative forms : 'dX\ fdY^ 'dX\ ' Returning now to equation (1), dZ = MdX + NdY, the principles of the calculus show that when dZ is a perfect differ- ential, but not otherwise, (dM\ fdN\ = In dealing with functions which depend only on the actual state of the fluid the condition that dZ is a perfect differential is satisfied, and consequently equation (5) applies. We shall see immediately some of the results of its application. 179. Energy Equations and Relations deduced from them. Consider now the heat taken in when a small change of state occurs in any fluid. Calling the heat dQ we have, by the First Law, dQ = dE+dW ........................ (6), where dE is the gain of internal energy and dW is the work which the fluid does through increase of its volume. Since dW = PdV the equation may be written dE= dQ-PdV ........................ (7). Here and in what follows we shall assume that quantities of heat are expressed in work units. This simplifies the equations by allow- ing the factor J or A to be omitted. We are concerned for the present only with reversible operations. In any such operation dQ = Tdcj) ; hence dE = Tdcf>-PdV .............................. (8). Again, I = E + PV, by definition of /. Hence dl = dE + d (PV) = Td - PdV + PdV + VdP VdP ...... . ..... ..................... (9). vn] GENERAL THERMODYNAMIC RELATIONS 271 Again, = / - T, by definition of *. Hence dT s = dl - d (T) = VdP - dT (10). Again, iff = E T, by definition of i//. Hence ) = Td - PdV - (Tdcf) + dT) = - PdV - dT (11). But dE, dl, d, and dtfj are all perfect differentials. Hence, applying Eq. (5) in turn to Eqs. (8), (9), (10), and (11) we obtain at once the following four relations between partial differential coefficients : From (8), (%} - - (%\ ...(12). These are known as Maxwell's four thermodynamic relations. Expressed in words, the first one means that when any fluid ex- pands adiabatically (cf> = const.) the rate at which the temperature falls per unit increase of volume is equal to the rate at which the pressure would rise, per unit increase of entropy, if the fluid were heated at constant volume. The second means that when a fluid is compressed adiabatically the rate at which its temperature rises, per unit increase of pressure, is equal to the rate at which the vol- ume would increase per unit increase of entropy if the fluid were heated at constant pressure. The third means that when a fluid is heated at constant pressure, the rate at which the volume in- creases with the temperature is equal to the rate at which the entropy would be reduced per unit increase of pressure if the fluid were compressed isothermally. The fourth means that when a fluid is heated at constant volume the rate at which the pressure rises with the temperature is equal to the rate at which the entropy * For the sake of symmetry f, which is - G, is used here rather than G. 272 THERMODYNAMICS [CH. would increase with increase of volume if the fluid were expanded isothermally. The following further relations are immediately deducible from Eqs. (8) to (11). Taking Eq. (8), imagine the fluid to be heated at constant volume. Then dV = and dE = Tdcf> ; hence \ - T Again, imagine the fluid to expand adiabatically. Then d$ = and dE = - PdV; hence dE \ - P ,__ I "~" J. Similarly from Eq. (9) we obtain from Eq. (10) , F. and - * fr o mEq .(n) Collecting these results, 1 80. Expressions for the Specific Heats K v and K v . In general the specific heats of a fluid are not constant; they are functions of the state of the fluid. We shall proceed to find differential expressions connecting them with the temperature, volume and pressure. Such expressions enable other properties to be calculated when the relation between T, F, and P is known. Consider, as before, a small change of state during which the fluid takes in an amount of heat dQ while it expands in a reversible manner. Its entropy accordingly increases by an amount dc/> such that Tdcf> = dQ. Its temperature changes from T to T + dT and its vii] GENERAL THERMODYNAMIC RELATIONS 278 volume from V to V + dV '. Take, in the first place, the temperature and volume as the two independent variables by means of which the state of the fluid is specified. The change in any third quantity may be stated with reference to the changes in T and in V. Thus the heat taken in may be written dQ = K v dT + IdV (20). Here K v , which is the specific heat at constant volume, is ( ~u \al ) v and / is a symbol for ( -^ 7 ) . \(Lv / p Since dQ = Td, I = T (^} . b ^'< W,-Wr' Hence I = T (^} , (21), \ai / y and dQ = K v dT + T f-j=) dV. \dl I v Dividing both sides by T, we have r7P\ dV T This is a perfect differential, and therefore, by Eq. (5), - KV = (^\ (dP< Hence or dV) T T \dTJ v \dT)r 1 fdKA /d*P\ This is an important property of K v . To obtain a corresponding property of K P , take the temperature and pressure as the two independent variables and express the heat taken in with reference to them. The heat taken in, dQ, is the same as before, being still equal to Td. We may write dQ = K^dX + I'dP (24). Here K^, which is the specific heat at constant pressure, is dQ\ , ., . . (dQ , - and / is a symbol tor -== CLJ. ]p \ClJ. J y Since dQ = Tdfi. I' = T (^ . E. T. 18 274 THERMODYNAMICS fen. , **,.!, <$),-- <). (251 ' and dQ = K v dT-T dP. Dividing both sides by T, we have And by Eq. (5), since this is a perfect differential, (A\ *. (A\ ( d K\ \dP) T T '\dTJp\dTjf- (),"* (5), ............ ..... <> , , which is the property of K v corresponding to that of K v in Eq. (23). Further, from Eqs. (20) and (24), K p dT + I'dP = K v dT + IdF, or (K 9 -K v )dT = By writing dP = it follows that K K dV - i' ' dp Or by writing dV = 0, V By Eq. (21) or (25), either of these gives this important expres- sion for the difference between the two specific heats, .-.-',(), And since by Eq. (4) dP T \dT v > this result may be written From Eq. (28 a) it will be seen that K v can never be less than K v , for [5=] is essentially negative, increase of pressure causing vn] GENERAL THERMODYNAMIC RELATIONS 275 decrease of volume in any fluid, and therefore the whole expression on the right is positive. Accordingly K v is always greater than K v , except in the special case when one of the factors on the right- hand side is equal to zero, in which case K v is equal to K v . This is possible in a fluid which has a temperature of maximum density (as water has at about 4 C.). At the temperature of maximum density [-3=] =0, and consequently at that point K v K v = 0. \al J p Return now to Eqs. (22) and (26). In heating at constant volume dV = 0; hence by Eq. (22) In heating at constant pressure dP = 0; hence by Eq. (26) In an adiabatic operation ckf> = 0; hence by Eq. (22) K *( dT \ ( dp \ " and by Eq. (26) ^(^) = (~) (32). Further, by Eq. (4fo) (dT)p __ JT (dP\ KV fdT\ \dT) v T (dr)t or ^=(^} (^} (33). (dV\ K,(dT\ fdV\ \dTjp T \dPJ \dP) T This is the ratio usually called y. Thus in the adiabatic expansion of any fluid the slope of the PV line is y times its slope in isothermal expansion, 181. Further deductions from the Equations for E and /. ByEq.(7) dE = dQ-PdV. Hence by Eq. (20) dE = K v dT + Zr/F - P^F = K v dT + (l-P) dV. In heating at constant volume dV = ; hence dE\ v 182 (SI-*- 276 THERMODYNAMICS [en. In isothermal expansion dT = 0; hence, using Eq. (21), We may therefore write V ......... (36). Again, by Eq. (9) dl = dQ + VdP. Hence by Eq. (24) dl -- K v dT + I'dP + VdP = K v dT + (l f + V) dP. In heating at constant pressure dP = 0; hence In isothermal compression dT = 0; hence, using Eq. (25), (),-' *'- F - r (5), We may therefore write di = ff,dr + [V - T (^)J dP (39). 182. The Joule-Thomson Effect. In a throttling process dl = (Art. 72); hence, from Eq. (39), This is the "cooling effect" in the Joule-Thomson porous plug experiment of Art. 19; the cooling effect which the working fluid of a refrigerating machine undergoes in passing the expansion-valve (Art. 110); the cooling effect used cumulatively by Linde for the liquefaction of gases (Art. 123). It expresses the fall of temperature per unit fall of pressure when any fluid suffers a throttling operation, during which it receives no heat from outside. From Eq. (40) it follows that the cooling effect vanishes when \dT)p T' This occurs in any ideal "perfect" gas under all conditions, that is to say in a" gas which exactly satisfies the equation PV = RT. But it also occurs in real gases under particular conditions of temperature and pressure. A gas tested for the Joule-Thomson effect at moderate pressure, and at various temperatures, will be found to become warmer instead of colder on passing the plug if vii] GENERAL THERMODYNAMIC RELATIONS 277 the temperature exceeds a certain value. At that temperature, which is called the temperature of inversion of the Joule-Thomson effect, throttling produces no change of temperature. Above the temperature of inversion the effect of passing the plug is to heat the fdV\ V gas ; f j is then less than and the expression for the " cooling effect" is negative. Below the temperature of inversion the cooling effect is positive. The temperature of inversion depends to some extent on the pressure, in any one gas. It differs widely in different gases. In air, oxygen, carbonic acid, steam and most other gases it is so high that the normal effect of throttling is to make the gas colder ; in hydrogen, on the other hand, the normal effect of throttling is to make the gas warmer, for the temperature of inversion is exceptionally low, about 80 C.* In the Linde process it is essential that the gas to be liquefied should enter the apparatus at a temperature below its temperature of inversion: the process can be applied to hydrogen only by cooling the gas beforehand to a suitably low temperature. Taking Eqs. (38) and (40) together we have *.,-'(,---(), ......... <"> This product, K v I J , is the quantity of heat that would just suffice to neutralize the Joule-Thomson cooling effect per unit drop in pressure, if it were supplied to the fluid in the process of throttling. It may conveniently be represented by the single symbol p. It measures the cooling effect, per unit drop in pressure by throttling, as a quantity of heat (expressed in work units), while dT ( jrl measures that effect as a change in temperature^. \ (Mr i It follows that if the range through which the pressure falls in a throttling process is from P A to P B , the whole quantity of heat that would have to be supplied to neutralize the cooling effect is as was stated in a footnote to Art. 124 J. * This was found by Olszewski for a pressure-drop from 117 atmospheres to 1 atmosphere. t In Calendar's Steam Tables the quantity here called p is tabulated for steam under the heading "SC" (See Art. 103.) J Cf. E. Buckingham, Bulletin of the Bureau of Standards (Washington), vol. 6, 1909, p. 125. 278 THERMODYNAMICS [CH. Since / = E + PV we may write Eq. (41) in the form ............... <-> This is instructive as showing the analysis of the Joule-Thom- son effect into two parts. When an imperfect gas or vapour is throttled, that part of the effect which is measured by the first term arises from the fact that the internal energy is not constant at any one temperature but depends to some extent on the pressure. In other words, the first term is due to departure from Joule's Law. There is in general an additional part of the effect, measured by the second term. It is due to departure from Boyle's Law, according to which PV should be constant for constant tempera- ture. A gas may conform to Boyle's Law at a particular tempera- ture and still be imperfect : in that case it will show a cooling effect due to the first term alone. It is only when both terms vanish that the gas is perfect. Experiments which will be mentioned in the next chapter show that in an imperfect gas the term ( ) may be either V dr / T negative or positive according to the conditions of pressure and temperature (Art. 197). Hence that part of the Joule-Thomson effect which is due to deviation from Boyle's Law will under some conditions assist, and under other conditions oppose, that part of the effect which is due to deviation from Joule's Law. The latter part is always a cooling effect ; the former may be either a cooling or a heating effect. At the temperature of inversion the two parts cancel one another. It may help the student to understand Eq. (41 a) if we put the physical interpretation of that equation in another way. Suppose unit quantity of any fluid to undergo unit drop of pressure in passing a porous plug or other throttling device. We may then putdP = 1. Suppose also a quantity of heat p to be supplied to it from outside which just prevents any change of temperature. Then Eq. (41 a) takes the form p = dE + d (PV), which is equivalent to saying that in the complete process, "Heat supplied = Increase of internal Energy -f W T ork done by the fluid. Here d (PV) is the net amount of work done by the fluid, because it is the excess of P 2 F 2 , which is the work done by the fluid as it vir] GENERAL THERMODYNAMIC RELATIONS 279 leaves the apparatus, over P^V^ , which is the work spent upon the fluid as it enters the apparatus. 183. Unresisted Expansion. In the Joule-Thomson porous plug experiment the fluid, in expanding from a region of constant high pressure to a region of constant lower pressure, does some work on things external to itself, the net amount of which is P 2 F 2 - P.V,. This quantity is not zero except in special cases, But in the original Joule experiment with two closed vessels (Art. 19) the fluid did no work on anything external to itself. The expansion there may therefore be described as strictly un- resisted. This distinction between it and the Joule-Thomson mode of expansion is important. Imagine the two closed vessels of the Joule experiment to be completely impervious to heat, so that no heat passes out of, or into, the fluid as a whole during the process. Imagine also that heat may pass freely from the fluid in one vessel to the fluid in the other through the opening between them, so that after expansion T becomes the same in both as well as P. Under these conditions the internal energy E of the fluid as a whole is not altered by the expansion; for no heat is taken in or given out, and no work is done. This is true of any fluid. The characteristic, therefore, of such expansion is that E is unchanged, just as the characteristic of the Joule-Thomson expansion is that / is unchanged. In the unresisted Joule expansion each vessel may of course be of any size. Think of the second vessel, into which the fluid ex- pands, as consisting of a group of very small chambers which are successively opened, so that the volume of the fluid increases by steps, each dV. We still suppose the temperature of the fluid to attain equilibrium at each step, and no heat to come in from out- side. Then for each step dE = 0. With infinitesimal steps the process becomes continuous. The cooling effect in this imaginary process is not identical with the cooling effect in the Joule-Thomson experiment. In this process it is ( -y^ ) , namely the rate at \dv J ^ which the temperature falls with increase of volume, under the condition that E is constant. By Eq. (36), writing dE = 0, 280 THERMODYNAMICS [CH. and this, along with Eq. (35). gives dT\ _ fdP\ = fdE\ Eq. (42) expresses the cooling effect in this imaginary process as a fall of temperature, per unit increase of volume; Eq. (43) expresses it as a quantity of heat, per unit increase of volume, namely the quantity that would have to be supplied from outside to neutralize the change of temperature caused by the expansion. We may call this quantity of heat a. Hence in unresisted expansion from any volume V ' A to any volume V B , under adiathermal conditions (Joule's expansion with vessels made perfectly impervious to heat), the whole quantity of heat that would have to be supplied to neutralize the cooling effect is, for any fluid, A further interesting relation follows. By Eq. (28), we had K K K *- K = But by Eq. (35), A,so, byEq .( 41 ), On substituting these values, Eq. (28) takes the new form This, like all the relations given in the present chapter, is true of any fluid. We shall return to it later in connection with imperfect gases (Art. 194). 184. Slopes of Lines in the /<, T(f>, and IP charts, for any Fluid. The slope of any constant-pressure line in the I chart is equal to the absolute temperature, for, by Eq. (16), It follows that all constant-pressure lines in that chart have the same slope at points where they cross any one line of constant temperature, vn] GENERAL THERMODYNAMIC RELATIONS 281 To find an expression for f . J , which is the slope of a constant- temperature line in the / chart, we shall proceed by a process of substitution which may be followed in finding other partial differential coefficients. It will serve as an example of a general method. Starting with Eq. (9) dl = Td + VdP, we shall eliminate dP by substituting for it an expression in terms of dcf) and dT, got by applying the general relation of Eq. (4), namely, This substitution gives -h Hence, writing dT = 0, (U - T + V (45) P ............... (45a) ' rince.Eq.d4), Similarly, to find an expression for (-=7 ] , which is the slope of ' 09/17 a constant-volume line in the Ifi chart, we start from the same equation for dl, but eliminate dP by substituting an expression for it in terms of d and dV, namely This substitution gives di -\ T Hence, writing dV = 0, ............... (46a) ' since, by Eq. (12), 282 THERMODYNAMICS [en. Turning next to the T chart, the slope of a constant- volume line is given by Eq. (29), (dT\ T " arid the slope of a constant-pressure line by Eq. (30), (-)=:r- To find the slope of a line of constant total heat (rr) we mav again apply the method of substitution. Starting with the equation Td = dl - VdP, substitute for dP an expression in dT and dl (Eq. (4)), dp = AT \dl J i This gives Td* = [l-r (g) J a - F QdT, from which, writing dl = 0, dT\ Tdf But by Eq. (40), dT\ _ T(d\ ( V\dPh K dTp -rl dT - j- , (dT\ T r T (dV\ 1 Hence =-^-1 - T/ ^ ............ (47 a). \dJ I K v |_ V \dljp] Also, since dV\ dV\ dl\ . dl we may put this result in the form \ T In the IP chart the slope of an adiabatic, or line of constant entropy, is given by Eq. (17), V-F dPV from which it follows that all adiabatics have the same slope at points where they cross any one line of constant volume. The slope of a line of constant temperature is given by Eq. (38), a -V-T (*? vn] GENERAL THERMODYNAMIC RELATIONS 283 To find expressions for the slope of a line of constant volume, ( 7P ) ? we ma y P rocee d thus : dl = dE + d (PV) = dE + VdP + PdV. (dE (dp it follows that (g) F = V + K v (g)^ .................. (48). By Eq. (31) this may be written, . Two other expressions which are sometimes useful may conveni- ently be given here, one for ( -r= ) and one for ( - ) : \dP ) T \av I p Hence (S) T = - T( ^ ] - p( ^ (49) - *di\ m & di '} -ffi) f*)-^*) ,byEq.(16). vdF/p \dcf>'p\dVj P \dV p Hence, by Eq. (13) .(50). 185. Application to a Mixture of Liquid and Vapour in Equilibrium: Clapeyron's Equation. Change of Phase. Equation (50) is applicable not only to homogeneous fluids, but to a mixture of two phases of the same substance, in equilibrium with each other and therefore both at the same pressure and the same temperature. / and V are then to be reckoned for the mixture as a whole. Say for instance that the substance is a mixture, part liquid and part saturated vapour. Suppose the proportion of liquid to vapour to be changed by vaporizing some of the liquid part at constant pressure, and therefore also at constant temperature. During that process f-j^J is constant, for the volume of the \dv ) 284 THERMODYNAMICS [CH. mixture as a whole increases in proportion to the heat taken in. Instead of ( -^, ) in equation (50) we may therefore write \Q>V IP or V V V V * s y w v s y w where the suffixes s and w relate to the two states, when all is vapour and all is liquid respectively. Further, the condition that is con- stant may be dropped in writing the coefficient - , which is Ci-L no longer a partial differential coefficient. Since the vapour present in the mixture is always saturated, P is a function of T only ; -==, is simply the rate at which the pressure of saturation rises with the temperature. While the mixture is vaporizing or condensing under variable pressure it makes no difference in the relation of P to T whether the process is conducted with = constant, or with V = constant, or in any other way: during that process dP\ dP is the same as -r- . Hence when applied to an dl/v dl equilibrium mixture of liquid and vapour, or of any two phases, Eq. (50) may be written in the form fdP\ ( I 3= ] or \dl J $ \ V,-V W ~ AT' This is Clapeyron's Equation, which was arrived at in Art. 98 in another way. The same result may be got from Eq. (21): dV T dT v ' During vaporization at constant temperature [^21] is constant \dV J T and its value is - . Hence, dropping the suffix V for the ' s ~ * w reason just given, we have as before V V - Ld ^ w TdP' This result may be extended to any reversible change of phase which a substance undergoes at constant pressure. During any such change the two phases of the substance are in equilibrium with one another, and the temperature is constant. Writing A for vn] GENERAL THERMODYNAMIC RELATIONS 285 the heat taken in during the change of phase, and F'and V" for the volumes of the first and second phases respectively (as in Art. 99), we have \ rfT Similarly, the expression for (--- ) in Eq. (47 b), namely \d chart. A still more direct means of getting Clapeyron's Equation is to use the function G, which is Tcf) I or . By Eq. (10) , a : In any change of phase which occurs at constant temperature and constant pressure, such as the conversion of water into steam at constant pressure, dT and dP are both zero. Hence in such a change G is constant, as was pointed out in Art. 90, where this property of G was turned to account. Compare now the state of any substance at the beginning and end of a change of phase, during which G is constant. Use the suffix w for the first state (say water), and the suffix s for the second state (say steam): = $ s dT - V 8 dP = w dT - V w dP. * Used by Jenkin and Pye (Phil. Trans. A, vol. 534, p. 366) in correcting the T chart for carbonic acid. .286 THERMODYNAMICS [CH. Therefore F s -^ = (^-^)g. But cf> s cf> w = . Hence this again gives Clapeyron's Equation, V V = TdP' 1 86. Compressibility and Elasticity of a Fluid. Let a fluid be subjected to an increase of pressure dP, with the result that the dV volume is reduced from V to V dV . Then - -=- measures the volume strain, and the ratio of this strain to dP measures the com- pressibility. The reciprocal of the compressibility or ^(-TT?) measures what is called the elasticity of the fluid. Its value will obviously depend on the circumstances under which the compression takes place. We may for instance keep the temperature constant during the compression. In that case the expression for the elasticity becomes - V (jy\ This is called the isothermal elasticity of a fluid, and will be denoted here by e t . Or we may prevent any heat from leaving or entering the fluid during the compression. In that case the expression becomes - V [-~J . This, which is called the \&r/4 adiabatic elasticity of a fluid, will be denoted here by e^ . We have accordingly the two elasticities by Eq. (33). That is to say, the ratio of the adiabatic to the isother- mal elasticity is equal to y, the ratio of the specific heats. Since K p is greater than K v (Art. 180) ^ is greater than e t . 187. Collected Results. All the foregoing relations are true of any fluid. Before proceeding to apply them (in the next chapter) vn] GENERAL THERMODYNAMIC RELATIONS 287 to particular fluids, it will be useful to collect them here for con- venience of reference. dE = Td - PdV (8), dl = Td + VdP (9), d =VdP-dT (10), d*/j = - PdV - cf)dT (11), - .- , ..................... " (15) dV) T ..................... (15)) dl\ fdE dp T ),--*-(*) .................. (19) ' dv '' (27) - K -K -T( dP \ ( dV \ (28) A, K v - 1 ............... (28), 2 8 8 THERMODYNAMICS [en. I (30), P , <> *,- (5), <* r' ....... : " ........ (35) ' dE = K v dT + IT (^r) v - p ] dv '-- - (86) ' - 18 "- dl = KdT + v - T dP ......... (39), ...(40), ?) p (45), S=r + F()=r-F(^) (46), dcj)J v \d(f>J v \dVJt ?(S)rf.[-?(a)J.-f,-T(S^ a,-'* '-(a,-'-'. <' r(g)-p (). fUr/f \Cl-L Ip \Cir } f ( d -L\ = r ^ UF'P (50). In. a reversible change of phase at constant pressure V S -V W = ^ (51), .(52), >// V L and G s = G w , or T^ s -I s = T w - I w (53). The isothermal and adiabatic elasticities : j -^vfGS), (54)> <* -r( d ] -.(55), 9^or^ = y^ .'(56). K v E. T. 19 CHAPTER VIII APPLICATIONS TO PARTICULAR FLUIDS 1 88. Characteristic Equation. The general thermodynamic relations considered in Chapter VII can be applied to determine the properties of a particular fluid when an equation connecting one of its properties with two others is known. An equation of this kind is called the "Characteristic Equation" or "Equation of State" for the given fluid. It is based upon experimental know- ledge of how the numerical values of some one property, such as the volume, depend upon those of two other properties, such as the pressure and the temperature, these two being used as inde- pendent variables for specifying the state. The most usual form of characteristic equation is one connecting V with P and T. Such an equation, when it can be established, is of fundamental impor- tance in the calculation of other properties. But taken by itself it does not allow all the thermodynamic quantities to be determined : for that purpose it must be supplemented by data regarding the specific heat, or (what comes to the same thing) by data as to the relation of the internal energy to the temperature. 189. Characteristic Equation of a Perfect Gas. The simplest case to consider is that of an ideal gas conforming exactly to the equation py __ j^j 1 where R is a constant and T is the absolute temperature. on the thermodynamic scale. We discussed some of the properties of such a gas in Chapter I, but it will be instructive now, as a first example of the method, to show how certain results which were obtained there follow directly when this characteristic equation is inter- preted by applying to it some of the general relations of Art. 187, which hold for all fluids. By differentiating the characteristic equation of the ideal gas, we have PdV + VdP - RdT. CH. vm] APPLICATIONS TO PARTICULAR FLUIDS 291 Hence in such a gas, (dP\ R P (dV\ = R == V (dP\ P \dTJ v ~V~~T' \dTJp~ P~T ; \dV) T ~ ~ V ; By Eqs. (23) and (27) of Chap. VII, in any fluid, \w) T ~ Hence in the ideal gas, Thus it follows from the characteristic equation that both K v and K v are constant at any one temperature ; in other words they are independent of the pressure. They may however vary with tempera- ture: the characteristic equation gives no information on that point. By Eq. (28) of Chap. VII, in any fluid, K K = T (\ (} Hence in the ideal gas, K V T R R 7? Aj, - K v = 1 . y. p = K. This agrees with Art. 20. The factor A is omitted because quan- tities of heat are here expressed in work units (Art. 179). By Eq. (40), Chap. VII, in any fluid the cooling effect in the Joule-Thomson porous plug experiment is _L|W^ -v \ fdV\ V In the ideal gas ( -.-- } = ; hence the quantity in square brackets \al ) p 1 vanishes and there is no cooling effect. By Eq. (36), Chap. VII, in any fluid, In the ideal gas T ( T = ) = P, hence \dl )v dE = K v dT, 192 292 THERMODYNAMICS [CH. and since K v is independent of the pressure it follows that the internal energy of the ideal gas depends upon the temperature alone. By Eq. (39), Chap. VII, in any fluid, In the ideal gas T ( -j^ } = F, hence \dTJp dl = KdT, and since K v is independent of the pressure it follows that the total heat of the ideal gas also depends upon the temperature alone. These results show that a gas which conforms exactly to the characteristic equation PV = RT (T being the temperature on the thermodynamic scale) conforms exactly both to Boyle's Law (PV constant for any one temperature) and to Joule's Law (E a function of the temperature alone). It is therefore "perfect" in the sense of Art, 19. When the equation PV = RT was introduced in Art. 18 the symbol T denoted temperature on the scale of the gas thermometer, that is to say a scale denned by the expansion of the gas itself, and the gas was assumed to conform exactly to Boyle's Law. But if it also conforms exactly to Joule's Law, the scale of the gas ther- mometer coincides with the thermodynamic scale (Art. 42). 190. Isothermal and Adiabatic Expansion of Ideal Gas. In the ideal gas, since E depends upon the temperature alone, it is constant during isothermal expansion, and therefore the work done by the gas is equal to the heat it takes in. The pressure varies in- versely as the volume. By Eq. (33 a), Chap. VII, for the adiabatic expansion of any fluid, dP\ dP Hence in the ideal gas dP So that in the adiabatic expansion of an ideal gas, dP dV ^ + r-F = - If now we make the further assumption that y is constant, which VTII] APPLICATIONS TO PARTICULAR FLUIDS 293 is equivalent to assuming that the specific heat does not vary with temperature, this gives on integration log e P + y loge V = constant, or PV y = constant, which is the adiabatic equation of a perfect gas with constant specific heat, arrived at otherwise in Art. 25. 191. Entropy, Energy, and Total Heat of Ideal Gas. By Eqs. (8) and (9), Chap. VII, in any fluid, dE + PdV dl - VdP ~~T~ ~T~ In the ideal gas dE = K v dT; dI = K v dT, P R and since ^^* Hence if we again assume that the specific heat does not vary with the temperature, E = K V T + constant, / - K P T + constant, (f> = K v loge T + R loge V + constant = K p loge T - R loge P + constant. The values of the constants depend on what initial state is chosen as the starting-point of the reckoning. It is only changes in E, /, and c/> that can be determined by these formulas. 192. Ratio of Specific Heats. Method of inferring y in Gases from the Observed Velocity pf Sound. We saw (Art. 186) that in any fluid the ratio y of the two specific heats, K V /K V , is equal to the ratio of the adiabatic elasticity e^ to the isothermal elasticity e t . Also that /dP\ Hence in a gas for which PV RT, c i= V (^] = P > and ^ - yP. 294 THERMODYNAMICS [en. This relation has been used as a means of finding y experi- mentally in air and other gases which at ordinary temperatures and pressures very nearly conform to the equation PV = RT. The method is based on Newton's theory of the transmission of waves of sound. Newton showed that waves of compression and dilatation, such as those of sound, travel through any homogeneous fluid with a velocity which may be expressed as VeV, where V is as usual the volume of the fluid per unit mass (the reciprocal of the average density) and e is the elasticity, in kinetic units. It was afterwards pointed out by Laplace that in applying this result to the passage of sound through air or other gases e should be taken as the adiabatic elasticity e^, for the compressions and dilatations follow one another so fast as to leave no time for any substantial transfer of heat from the portions that are momentarily heated by compression to those that are momentarily cooled by expansion. Hence in air under atmospheric conditions, or in any other nearly perfect gas, sound travels at a rate equal to A/yPF. This fact is used as a means of determining y by measuring the velocity of sound or (what comes to the same thing) by measuring the wave- length in sound of a known pitch. In air at C. and a pressure of one atmosphere the values given by various observers for the velocity of sound range from 33,060 to 33,240 centimetres per second *. Under these conditions the volume of one gramme of air is 773-1 cubic cms., arid P is 1-0133 x 10 6 dynes per sq. cm. (Art. 12). Hence, taking an average of 33,150 for the velocity, 33,150 = V-y x 1-0133 x 10 6 x 773-1, which gives y = 1-403. 193. Measurement of y by Adiabatic Expansion. Method of Clement and Desormes. Another method of determining the value of y in a. gas is by an experiment due originally to Clement and Desormes and improved on by Gay-Lussac and others. A quantity of the gas is contained in a large vessel at a pressure some- what higher than that of the atmosphere, and at atmospheric temperature. There is a pressure-gauge attached, and a tap which may be opened to allow some of the gas to escape quickly. On opening the tap, the pressure falls suddenly to that of the atmo- sphere: when this happens the tap is at once closed. Then the * See Rayleigh's Theory of Sound, vol. II. vin] APPLICATIONS TO PARTICULAR FLUIDS 295 pressure of the gas that remains in the vessel slowly rises, because the temperature, which had been reduced by the sudden expansion of the gas in the vessel while the tap was open, rises gradually to the value which it had at first, namely the temperature of the surrounding atmosphere. When this process is complete the final pressure is noted. Let the original pressure be P lt the pressure of the atmosphere P 2 and the final pressure P 3 . The change from P! to P 2 is approximately adiabatic on account of its suddenness : the change from P 2 to P 3 occurs at constant volume. Let V , V z and V be the volumes of the gas per unit mass, at the three corre- sponding stages. Then F 2 = F 3 . We have, in the adiabatic ex- pansion, p y y _ p 77 y and since the initial and final temperatures are the same, P77 ._ p 77 ._ p 77 i*l -* 3" 3 ~~ r Z y 2* Hence ^ = (Wl = \^\ , teg P. -log P. y-iogp^iogp,- Values of y are accordingly found by observing these three pressures. Experiments by Lummer and Pringsheim, using this method in an improved form, give 1-4025 as the value of y for normal air. An earlier application of the method by Rontgen gave 1-405*. 194. Effect of Imperfection of the Gas on the Ratio of Specific Heats. It has been already mentioned that in a perfect diatomic gas the ratio y, as deduced from the molecular theory (see Appendix II), should not exceed 1-4. In air the ratio, according to all the evidence, is, at ordinary temperatures and pressures, slightly greater. This is due partly to the presence of about one per cent, of (monatomic) argon, but mainly to the fact that air is an imperfect gas, deviating to a small extent both from Boyle's Law and from Joule's Law. By Eq. (44), Chap. VII, in any fluid, (P + a) (V + p) ^p &v= y > where p is the cooling effect in the Joule-Thomson porous plug experiment (Art. 182), and a is the cooling effect that would be * See Preston's Theory of Heat, Chap. IV. 296 THERMODYNAMICS [CH. found in unresisted expansion (Art. 183), without gain or loss of heat in either case. In a perfect gas p and a are both nil, and the expression on the right becomes PF/T, as it should. With air (under usual conditions) both p and or are small positive quantities : p was measured in the Joule-Thomson experiments, and cr, though it has not been directly measured, can be inferred from known experi- mental data. Hence K v K v is a little greater than PF/T, which is the value it would have in a perfect gas. The ratio y is also a little greater in normal air than it would be in a perfect gas. In any fluid (P + q)(F + g) V K V T In air at ordinary temperatures the imperfection increases (P + a) (V + p) more than it increases K v . and consequently makes y slightly exceed the ideal value 1*4. But at high tempera- tures K v is much increased (because the molecules then acquire energy of vibration) and y is substantially reduced. 195. Relation of the Cooling Effects to the Coefficients of Expansion. The expressions for p and a given in Eqs. (41) and (43) of Chap. VII may be put in another form which is convenient in dealing with imperfect gases. By these equations, in any fluid, ' Here ( -==. ) may be written F, where a is the fractional increase \aljp of volume per degree, on the thermodynamic scale, when the fluid is heated at constant pressure. Measured at C. a is the co- efficient of expansion at constant pressure, or what is sometimes called the "volume-coefficient." Similarly f J may be written f$P, where /3 is the fractional increase of pressure per degree, on the thermodynamic scale, when the fluid is heated at constant volume. Measured at C. ft is w r hat is called the "pressure-coefficient." Hence at C. F + PQ = 273-la F , and P + a = 273-l&>P OJ the suffix being introduced to show that the quantities concerned are all to be taken as at C. The results of the Joule-Thomson porous plug experiments may vin] APPLICATIONS TO PARTICULAR FLUIDS 297 be used to calculate p . They showed that with air the cooling effect of passing the plug was nearly proportional to the drop in pressure. It was different for different initial temperatures, be- coming less when the initial temperature was raised. With air at C. the cooling effect (according to the formula in Art. 123) was 0-275 for a pressure-drop of one atmosphere in passing the plug. Hence, using C.G.S. units, for air at C. we should have dT\ 0-275 = ~ 10*' We may take K v as 0-241 calory (Art. 161) equivalent in C.G.S. units of work to 0-241 x 4-1868 x 10 7 . Multiplying the values of K v and ( -^= ) we obtain \ dP/I p, = 2-74. This is in cubic centimetres per gramme, the dimensions of p being the same as those of F, namely work volume ^ _ _ ______ pressure x mass mass We may apply this result of the porous plug experiment to calculate the coefficient of expansion when air, at C., is heated under a constant pressure of one atmosphere through one degree of the thermodynamic scale. We had _ ^o + />o 273 -1F * In air at C. and a pressure of one atmosphere, the volume of one gramme is 773-5 cub. cms. Hence under these conditions we should a , : . = 0-003675. 273-1 x 773-5 This is slightly larger than the mean coefficient that is found when the expansion of air at a constant pressure of one atmosphere is measured over a range of temperature from C. to 100 C. Again, taking the relation a value of cr can be inferred when the pressure-coefficient is known. If for air be taken as about 0-003674, P + a becomes 1-0034P , making a l Adding the two terms, the whole cooling effect in a fluid which obeys Van der Waals' equation would be RTb By making T sufficiently large the second term within the square brackets exceeds the first, which means an inversion of the effect. When the fluid is a gas at low pressure, and V is consequently very large compared with 6, the condition for inversion is that RTb =2a: in other words the inversion temperature in a gas at very low pressure would be 2a/Rb. vm] APPLICATIONS TO PARTICULAR FLUIDS 315 and to that extent the equation is satisfactory. But the amount of the cooling effect in such a gas as carbonic acid, when calculated from the Van der Waals equation (with constants which suit the form of the isothermal curves) falls much short of the cooling effect that is actually observed ; and if the constants of the equation are adjusted to make the observed and calculated cooling effects agree, then the equation does not accord with the observed figures for compressibility*. 204. Other Characteristic Equations: Clausius, Dieterici. Enough has been said to show that Van der Waals' equation cannot be brought into exact agreement with the deviations from Boyle's Law and Joule's Law which are found in an actual gas. The reason has already been indicated that the "constants" of the equation are not strictly constant. In particular the attraction between the molecules, on which a depends, is probably a function of the tem- perature, although it is treated in the equation as independent of the temperature. Various attempts have been made to modify the equation so as to bring it into closer accord with the known pro- perties of gases. None of these have been completely successful in giving a formula which will stand all tests throughout a very wide range of states, though in some respects the modified equations approximate better to the observed facts. Clausius | gives a characteristic equation which we may write in the form p = where a' and b' 9 as well as b and /?, are constants. On com- paring this with Eq. (1 a), it will be seen to differ from Van der Waals mainly by the presence of T in the denominator of the last term, which expresses the addition to P that is due to inter- molecular attractions. Clausius assumes that these attractions become reduced when the temperature rises; he thereby gets an equation which, while it gives to the isothermals the same general form as is given by the equation of Van der Waals, agrees better with the Joule-Thomson cooling effect. When the same method of finding the critical quantities is applied to it, by writing dP\ /d*P\ - , -== =0 and ( -= ) =0, ay J \dr V * See Callendar, Phil. Mag., Jan. 1903, pp. 58-60. f Phil. Mag., June, 1880. 316 THERMODYNAMICS [CH. one finds that V c = 3b + 26', V-y j; 27R (b + b') P - ~ 216 (b For carbonic acid Clausius gives his constants the following values : R = 0-003688, b = 0-000843, a' = 2-0935, b' 0-000977, the unit of pressure being again one atmosphere, and the quantity of gas considered being that which occupies unit volume at one atmo- sphere and C. With these constants the calculated critical temperature is 31 C. and the calculated critical pressure is 77 atmospheres. Clausius draws a theoretical isothermal curve of pressure and volume for carbonic acid at 13-1 C. calculated from his formula. This curve, which is reproduced in fig. 94, shows the form assumed by the James Thomson wave in the Clausius type of characteristic equation. The horizontal straight line BC, which exhibits the process of liquefaction, is so drawn that the crest and hollow of the wave shall have equal areas (Art. 199): this consideration deter- mines its height and therefore fixes the saturation pressure. The dotted portions of the curve exhibit imaginary states, comprised within the characteristic equation, which serve to establish theoretical continuity between the real state of homogeneous liquid AB and the real state of homogeneous vapour CD. A modified and more general type of Clausius equation is ob- tained by writing RT a'f(T) - v -b (v + vy ' where /(T) is a function of T such as to diminish with rising tem- perature. In the original equation of Clausius, f(T) = . Van !..!' der Waals has suggested* that f(T) may be e T C, where e is 2-7183, the base of the Napierian logarithms, and T c is the critical tem- perature. In that case, at the critical temperature f(T) would become equal to 1. This form of characteristic equation was adopted by Mollier in calculating his tables of the properties of carbonic acid*. * Mollier, Zeitechri/l fiir die gesammte Kalte-Industrie, vol. n, 1895 and vol. HI, 1896. viii] APPLICATIONS TO PARTICULAR FLUIDS 317 Still another characteristic equation of the same comprehensive kind is that of Dieterici *, who writes P(V -b) = UT V ..................... (18), where is again the number 2*7183, and a, b arid R are constants. Like the others, this formula is founded on the kinetic theory, h o 3( 20 10 Volume i i Fig. 94. Theoretical isothermal of C0 2 at 13-1 C. according to the equation of Clausius. and like them it reproduces the general features of isothermal curves under all conditions and gives a critical point. Since it has only two constants besides R, the principle of corresponding states holds good for the relation it establishes between P, V and T. It makes the critical temperature T c = -=-= , the critical * Annalen der Physik, vol. v, p. 51, 1901. 318 THERMODYNAMICS [CH. volume F c = 26, and the critical pressure P c = rp-^ Hence at the critical point the ratio RTJPV becomes equal to Je 2 or 3-695, a value which is in much better agreement with observed results than was the value 2-667 calculated from Van der Waals' equation (Art. 201). In respect also of the Joule-Thomson cooling effect and its inversion* Dieterici's equation gives a better agreement with experiment than does Van der Waals'. A more general form of the Dieterici equation is obtained by writing T n instead of T in the index term, thus introducing one more adjustable constant : - a P (V - b) = RT RTnV .................. (19). The critical temperature then becomes */ ^ . The principle of corresponding states would still apply to any group of substances for which n had the same value, since each substance in the group would still have only two constants individual to itself. 205. Callendar 's Equation. None of these equations is com- pletely successful in representing the behaviour of a fluid in all possible states. But for the practical purpose of enabling tables to be calculated which will show the properties of a fluid throughout a limited range of variation of state, it is not impossible to frame a characteristic equation which, by empirical adjustment of the con- stants, can be made to apply with sufficient accuracy and even with great accuracy within that range, though it may fail entirely when carried beyond the range. A conspicuous example of this less ambitious type of characteristic equation is one which Callendar has devised and applied to calculate his tables of the properties of steamf . It serves to express very exactly the observed properties of steam within the limits of pressure and temperature that are usual in steam-engine practice, but it has no application to higher pressures, and it makes no attempt to represent the continuity of the gaseous and liquid states. This equation, which Callendar takes as characteristic of any vapour, saturated or superheated, provided the pressure is much lower than the critical pressure, is (20), * See Porter, Phil. Mag., April 1906 and June 1910. t Callendar, Proc. Hoy. Soc. vol. 67, p. 266, 1900; Phil M ag., Jan. 1903; Encyc. Brit.. Articles "Thermodynamics" and "Vaporization." vm] APPLICATIONS TO PARTICULAR FLUIDS 319 T>nn where =- is, as usual, the ideal volume of a perfect gas : b is a constant representing the co- volume, as in other characteristic equations ; and c is a term which is not constant but is a function of the temperature. Callendar takes c = where C is a constant and n is a number depending on the nature of the gas. The term c represents the effect of inter-molecular forces, but instead of re- garding these forces as augmenting the influence of the external pressure (which Van der Waals did by adding the term -^ 2 to P) 9 Callendar represents by c their effect in reducing the volume below its ideal value, in consequence of the " co- aggregation " or tem- porary interlinking of some of the molecules during their en- counters. He calls c the "co-aggregation volume" and treats it, at the moderate pressures within which he applies the equation, as a function of the temperature only. This assumption would not be true under conditions of high density, but for a gas or saturated vapour at moderate pressures it gives results which agree remark- ably well with those of experiment. Before proceeding to apply Callendar 's equation, it may be useful to point out its relation to that of Clausius. We may write the equation of Clausius (Eq. (16) of Art. 204) in the form RT a'(V-b) p Now at low or moderate pressures the volume will be large, and the modifying terms on the right will be comparatively small. When V is large the effect of the second term will not be much altered if we take 7, as approximately equal to = , and also take P in that term as approximately equal to RTJV. When these sub- stitutions are made the equation becomes RT a' j/ _ _ _ _ I /} ' P which we may write in Callendar's form V RT C _u h -p~T^ + b > where the more general index n is substituted for 2 as the index of T, and C is written for a' JR. Callendar finds that the best agreement with observed results, 320 THERMODYNAMICS [CH. especially with observations of the Joule-Thomson cooling effect, is got by giving to n a value which is not necessarily 2 but may be greater or less than 2 according to the nature of the gas*. For oxygen or nitrogen or hydrogen he takes n to be 1-5: for carbonic acid at low pressure good results are got by taking it as 2 ; and for steam, in the calculation of his tables, he has taken it as -V . It must be emphasized that Callendar's equation applies only to gases and vapours at low and moderate pressures. That this is so will be obvious when one considers the form of the isothermal lines which it gives on a diagram of PV and P. We may write it PF = RT - cP + bP .................. (20). Since c is a function of T only, and is therefore constant along any one isothermal, this gives Hence in a gas which obeys Callendar's equation the isothermal lines would be straight, inclined downwards, with increasing P, if c is greater than b, and inclined upwards if b is greater than c. There would be no minimum of PV nor change of inclination along any isothermal line. The equation therefore can apply only under con- ditions such that the lines are substantially straight, namely at low or moderate pressures. Starting from P = the Jines are in fact nearly straight for some distance; and. as we saw in Art. 197, they slope down when the temperature is low and slope up when it is high. In any gas, at a sufficiently low temperature c is greater than b, and an isothermal line there will slope down. As the tem- perature increases for which the isothermal is drawn, c becomes less, C since c = =-^ , and a temperature is reached at which the line runs level (c = b). For any higher temperature than this the line slopes up, like the lines for hydrogen in fig. 89. The temperature at which the sign of the slope changes will be relatively low in a gas which, like hydrogen, has a very low critical temperature, and will be relatively high in a gas like carbonic acid, as might be inferred from the principle of corresponding states. In dealing with steam, the limits within which Callendar has applied his equation are from zero pressure to 500 pounds per square inch or 34 atmospheresf. Within this range it is not probable that * Phil. Mag., Jan. 1903, p. 95. f The critical pressure in water- vapour is about 200 atmospheres, or say six times as high as the pressure up to which Callendar's equation is held to apply. vm] APPLICATIONS TO PARTICULAR FLUIDS 321 any important error is introduced by treating the isothermals as straight lines on the diagram of PV and P. Besides representing accurately the behaviour of the substance within this range, when suitable values are chosen for the con- stants, Callendar's equation has the very convenient property that differential expressions deduced from it for the various quantities E, /, cf>, K p , K v and so forth, by applying the general thermo- dynamic relations of Chap. VII, take forms s.uch as may be readily integrated. Hence it enables numerical values of these quantities to be calculated, to any desired number of figures, which will be thermodynamically consistent with one another. It would be possible to fix the constants by reference only to experiments on the compressibility of the gas at various tempera- tures, if sufficiently accurate data of that kind were available. But Callendar prefers to fix them by reference mainly to observed values of the Joule-Thomson cooling effect. Their relation to the cooling effect will be apparent from what follows. 206. Deductions from the Callendar Equation. Taking the Callendar equation, ^y V = -- - c + b, since c = 2^' we have dc nC _nc ............... (26) , where K p f is the constant of integration. It is the limiting value of the specific heat K v when P = 0, at the temperature T. But since any gas in that infinitely rarefied condition may be treated as perfect, Callendar assumes that K v ' may be taken as having the same value at all temperatures to which the equation is applied. It should be noticed that this integration is performed along an isothermal line, and that the constant of integration is not necessarily the same for other temperatures. To treat K p ' as con- stant when the temperature is varied therefore involves an assump- tion which is independent of anything in the equation itself. Again, by Eq. (41) of Chap. VII we had for the measure of the Joule-Thomson cooling effect in any fluid .,-' (),-' Hence for a gas to which the Callendar equation applies, the cooling effect is RT + nc (n + l)c -b ........................... (27). As was explained in Art. 182, -gp is the fall of temperature per unit fall of pressure when the gas passes through a porous plug or any other throttling device, and p is the quantity of heat that would serve to maintain the original temperature, if it were supplied from outside during the process. From the above result it follows that Callendar's formula provides for the inversion of the cooling effect which is known to occur in real gases. When (n + 1) c is greater than b the expression for p is positive ; the gas is then cooled by throttling. This is the usual case. But when (n + I) c is less than b, p is negative; the gas is then warmed by throttling, as vm| APPLICATIONS TO PARTICULAR FLUIDS 323 hydrogen is at ordinary temperatures, and as any gas will be if the initial temperature is sufficiently high. By raising the initial tem- perature the quantity (n + 1) c is reduced } since c = -^ n . Inversion of the Joule-Thomson effect occurs when (n + 1) c = 6, or when nc = c + b. But, as we saw above, c + b is the slope of any isothermal line on the diagram of PV and P, namely f ,p j . Hence if the isothermal slopes up with a gradient steeper than nc the Joule-Thomson effect will be a heating; if it slopes up less steeply than this, or runs level, or slopes down, the effect will be a cooling. It will be apparent from these considerations that measure- ments of the cooling effect furnish an important means of settling the values of the constants in Callendar's equation, apart from direct determi- nation of the isothermal lines. Cal- ^ lendar in fact assumes that the co- ^ volume b is equal to the volume is which would be occupied if the gas g were all condensed to a liquid, and Q - then calculates the values of n and c from observations of the cooling effect*. An illustration may help to make some of the above points clear. In fig. 95, which is a diagram of the Amagat type, with PV and P for co- ordinates, isothermals are sketched Fig. 95. Amagat isothermals ac- (not to scale) for a gas obeying cording to Callendar's charac- Callendar's equation. They are, as we teristic e 1 uation - saw, straight lines within the range to which the equation is applied. AS is an isothermal drawn for a temperature such that the vapour becomes saturated at a moderate pressure, which is assumed to be within the range of pressure for which the equation holds good. Accordingly the line AS is straight, up to the saturation point S. The curved line through S is a portion of the boundary curve, below which lies the "wet" region, where the beginning of w Pressure Phil. Mag., Jan. 1903, p. 87. 212 324 THERMODYNAMICS [CH. condensation would be represented by a vertical straight line, SW, P as well as T being then constant. AS slopes downwards, and the effect of throttling, at that temperature, is to cool the gas. A'S' is an- other isothermal, drawn for a lower temperature, to which the same remarks apply. The effect of throttling is still to cool the gas at the higher temperature for which the horizontal isothermal BB is drawn (c = b), and at any temperature up to that of CC, which is the isothermal corresponding to the inversion of the Joule-Thomson effect, namely that for which (n + 1) c = b, the upward gradient of CC being equal to nc. At any higher temperature, such as that for which DD is drawn, the upward gradient is steeper and the effect of throttling is to heat the gas. 207. The Specific Heats in Callendar's Equation. The ex- pression given in Eq. (26) for K p , in a gas that conforms to Callendar's equation, enables the specific heat at constant pressure to be calculated for any temperature and any pressure within the range to which the equation applies, when the value of K p r (as- sumed constant) for the given gas is known, as well as the constants of the characteristic equation. In order to obtain a corresponding expression for the specific heat at constant volume it is most con- venient to write the characteristic equation in the form PU = RT, where U stands for V b + c. U is a function of V and T only. Differentiating with respect to T, keeping V constant, dU\ dc nc . But = = - ' and " Substituting these values, and remembering that = , we obtain from Eq. (28), Then from Eq. (29), /d?P\ _ Pnc r 2nc\ _ Rnc \dT*) v ~ 7i ^ r ^ --^r\ vm] APPLICATIONS TO PARTICULAR FLUIDS 325 Now by Eq. (23) of Chap. VII, in any fluid, = T dV T Hence in a gas or vapour to which Callendar's equation applies (dK v \ Rncf^ 2nc\ ~=- l - n + - In integrating we have to remember that at constant temperature dU = dV. Accordingly, r;r Rnc ( nc\ K v = -jj- f n - 1 ] + constant. Writing this in the form Pmr V I 1 ""^ \ i 17" f /OT \ K v = ~7fr( n - l ~77 } + K* (31), we see that the constant of integration K v ' is the limiting value of K v when P = 0, which (like K P ' 9 Art. 206) is taken as having the same value at all temperatures to which the equation is applied. Next, to find an expression for K v - K v . By Eq. (28) of Chap. VII, in any fluid, Hence in a gas to which Callendar's equation applies we obtain the relation (32), by substituting the values already found in Eqs. (28 a) and (22) for these two differential coefficients. In the limit when P = the volume becomes indefinitely great, TIC the term jj vanishes, and K 9 ' -K v ' = R ........................ (33), as we should expect from the fact that this gas is then to be regarded as perfect (compare Art. 189). It should be noted that the assump- tion that K p f is constant requires that K v ' should also be constant. 208. The Entropy, Energy, and Total Heat, in Callendar's Equation. To find an expression for the entropy we shall apply Eq. (26) of Chap. VII. which is true of any fluid, 326 THERMODYNAMICS [CH. Here, and in what follows, the factor A. which is 1/J (Art. 14), is introduced in order that heat quantities (including R) may be numerically stated in thermal units. By Eqs. (26) and (22), Art. 206, we have, using thermal units, An (n + 1 ) cP , j . (dV\ R Anc K = - f- - + K and A \dT) P = p + ^r in any gas that obeys CalJendar's equation. Hence in any such gas , - IT + il^ a - * a- _ tr, Integrating, this gives AncP f - K,' log. T- Slog. P--T- +B ...... (35), or < f > = K p 'l g,T-Rlog i P-- + B ...... (35 a), where B is the constant of integration. To find an expression for the internal energy E we may most conveniently use the general equation (8) of Chap. VII, dE - Tdt/> - APdV. In a gas that satisfies Callendar's equation . By substituting this and the value of dc/> in Eq. (34) we have dE = (K p f -R)dT + An (^dT - cdP\ = K v 'dT - And(cP) ........................... (36), from which E = K v f T - AncP + B r .................. (37), where B' is the constant of integration. Note that the internal energy falls short of the value it would have in a perfect gas by the amount AncP. To find the total heat we have, by definition / = E + APV . Hence, from Eqs. (37) and (20 a), in a Callendar gas / = (K 9 ' + R)T-A(n + I)cP + AbP + B' = K 9 'T - A [(n + 1) c - b] P + B' ............... (38). Further, since A[(n + l)c b] is, by Eq. (27), equal to the Joule- vm] APPLICATIONS TO PARTICULAR FLUIDS 327 Thomson cooling effect p expressed in thermal units per unit drop of pressure, we may write this expression for the total heat in the form / = K V 'T -pP+B' (38 a). On differentiating Eqs. (37) and (38) with respect to P, keeping T constant, we have T I 1 AVI t\ LT \df} T \dPjT~ These results agree with the expressions already given for the cooling effect. The whole cooling effect is ( -r= ) ; by Art. 182 \dJr J y it is made up of fd (PV)\ or Anc, and A I -^= J or ^4 (c b). Further, since [ 35 ] is constant for any one temperature, the slope \drjT of any constant-temperature line is constant, on a chart of / and P for steam (compare Art. 102). The lines slope downwards, with increasing P, and the slope is less at high temperatures, since c is then less. To complete the list, expressions may be added for the function (which is G) and the function 0, in a Callendar gas. These are found at once from the above results : = K 9 ' T(l - log e T) + RTlog e P- A(c-b)P-BT + 5'.. .(89), = E - Tcf> = K V 'T - K v 'Tlog e T + RTlog e P - BT + B' ...(40). All the foregoing deductions from Callendar's equation hold good for any gas or vapour to which the equation applies, whatever be the values of the constants, provided the specific heat at zero pres- sure may be taken as independent of the temperature within the range of application. 209. Application to Steam. In applying his equation to steam, Callendar assigns to the constant n a value such that nR = K v '. This relation, which is not true for all gases, gives a value of n for steam that is consistent with the observed effects of throttling. It has the practical advantage of allowing expressions for the be- haviour of the gas during adiabatic changes to take a very simple form. 328 THERMODYNAMICS [CH. When K v ' = nR it follows from Eq. (33) that K v ' - (n + 1) R, and in that case the expression for , Eq. (35 a), becomes , 4> = (n+I)R log e T - R log, P - AnC ^+B B ...(41). Now in adiabatic expansion (/> remains constant, and that can p happen in this expression only if _-^ is constant. Hence in the adiabatic expansion of a Callendar gas in which the relation nR = K v ' holds, the pressure and temperature are connected by the equation p constant ........................ (42). Further, it follows that in all such cases ^-= - - is constant during adiabatic expansion, because by the characteristic equation we have p (V _ b) x^ J- L yn+l and under the condition stated both terms on the right-hand side are constant. Again, under the same condition that nR = K v ', T n+l P(P-b) -p- - = constant, i whence T (V - b) H = constant ............... (43), and, multiplying by - -- , n+l P (V - b) n = constant .................. (44). All these results for adiabatic expansion are true of steam, within the limited range through which Callendar's equation is applicable. They hold good so long as the substance remains in the homo- geneous state of a gas, whether superheated, saturated, or super- cooled (Art. 79), and they cease to apply when part of it liquefies. In the calculation of his steam tables Callendar takes for the numerical value of the co- volume b the volume of unit mass of water at C., namely 0-01602 cubic feet per Ib. For R he takes 0-11012 in mean calories, corresponding to 1-982 per mol, and equivalent to 154-17 foot-pounds per Ib. For n he takes -g -. This vin] APPLICATIONS TO PARTICULAR FLUIDS 329 figure is based mainly on throttling experiments by Grindley*, Peakef, and Callendar himself J. He takes for C a value such that c is 0-4213 cubic foot at 100 C. This makes 157-52 x 10 6 C= 0-4213 (373-1) *, or 157-52 x 10 6 ; and c = ^ . Hence when V is the volume of 1 Ib. in cubic feet, P is the pressure in pounds per square foot, and T is the absolute temperature in centigrade degrees, the Callendar equation jgy V = -c + b becomes, for dry steam in any state, F= 154_^_l^|_xlO + . 01602 (45) . As a numerical example, let it be required to find the volume of 1 Ib. of steam at a pressure of 400 pounds per square inch and a temperature of 240 C. Here P = 400 x 144 and T = 513-1, making V = 1-3733 - 0-1456 + 0-0160 = 1-2437 cub. ft. This will be found to agree with the value in Table C (Appendix III) where the volume is tabulated for various pressures and for tempera- tures ranging from 400 C. down to the temperature of saturation and below it. The volumes below the temperature of saturation refer to water-vapour in a supercooled (metastable) state, such as that which is set up by adiabatic' expansion in the absence of nuclei on which condensation may occur. In this example the steam is slightly superheated, the saturation temperature for a pressure of 400 pounds being just under 230 C. Callendar also tabulates separately the "co-aggregation volume" c for various temperatures. Some of the values are given below. Co-aggregation volume c for Dry Steam in any state, in cubic feet per Ib. Temp. c Temp. c Temp. c Temp. c 1-192 70 0-5570 140 0-3000 210 0-1780 10 1-057 80 0-5061 150 0-2771 220 0-1663 20 0-9417 90 0-4611 160 0-2562 230 0-1555 30 0-8420 100 0-4213 170 0-2375 240 0-1456 40 0-7557 110 0-3857 180 0-2204 250 0-1366 50 0-6804 120 0-3540 190 0-2050 260 0-1282 60 0-6147 130 0-3255 200 0-1909 270 0-1205 Phil Trans. A, vol. 194, p. 1, 1900. f Proc. Roy. Soc A, vol. 76, p. 185, 1905. See Brinkworth, Phil. Trans. A, vol. 215, p. 383, 1915. 330 THERMODYNAMICS [CH. In further illustration of Eq. (45) various isothermal lines for steam are drawn to scale in fig. 96, showing PV in relation to P as 800 Pressure, Pounds per Sy. Inch . 100 100 200 300 400 500 Fig. 96. Isothermals for steam, from Callendar's equation, calculated from that equation. Here the pressures are expressed in pounds per sq. inch ; consequently the numerical values of PV vm] APPLICATIONS TO PARTICULAR FLUIDS 331 given in the figure must be multiplied by 144 if it is desired to have them in foot-pounds. The dotted continuations of the isothermal lines for 200 and 100 below the saturation curve represent values for supercooled or, as it is sometimes called, supersaturated vapour. The full lines drawn at constant pressure represent the first stages in the condensation of a wet mixture. It will be observed that at the highest temperature at which Callendar applies his formula to steam, namely 500 C., the isothermal still slopes down with increasing P. Throughout the whole working range the throttling of steam produces a cooling effect. Since the value assigned to R is 0-11012 calory, and that of n is V -, the relation K v ' = nR requires that K v ' shall be 0-36707 and K v f , which is K v ' + R, shall be 0-47719. We have next to show how the tabulated values of the total heat are calculated. The formula for /, Eq. (38), becomes, for steam in any homogeneous condition, whether superheated, saturated, or supercooled, Z = K P 'T - A (*fc - b) P + ', giving, in calories per lb., I = 0-47719T - - =- + B' ............ (46), 1400 where c and b are expressed in cubic feet per Jb., and P in pounds per sq. foot. To obtain a numerical value for #', which comes in as a constant of integration, we must fix some zero state from which the total heat of the substance is to be reckoned, or, what conies to the same thing, we must assign a numerical value to the total heat in some known state. In the calculation of his tables Callendar assumes that the total heat of water is zero at C. and is 100 at 100 C., under saturation pressure in each case*. * This assumption not only fixes the zero from which the total heat is to be reckoned, but also gives to the thermal unit a value very slightly greater than the mean calory as defined in Art. 13. The thermal unit of Callendar' s tables and formulas is one-hundredth of the change in total heat which water undergoes when it is heated from to 100 under the (varying) pressure of saturation, whereas the ordinary mean calory is one -hundredth of the change in total heat when water is heated through the same interval of temperature under a constant pressure of one atmosphere. Callendar' s unit is the larger of the two by about one part in four thousand. This difference is of no practical importance : it is so small as to be well within the limits of error of experiment. The figures for the mechanical equivalent of heat given in Art. 14 relate, strictly, to the larger unit, which is the one used in the tables. Callendar takes his calory to be equivalent to 4-1868 x 10 7 ergs; the corresponding value of the constant-pressure mean calory would be 4-1858 x 10 7 ergs. The relation between the two units will be made clear if we write out an energy 332 THERMODYNAMICS [CH. In passing from the state of water at 100 C. to that of dry saturated steam at 100, under a constant pressure of one atmo- sphere, the fluid takes in 539-30 calories, that being the latent heat as determined by experiment. Hence / for steam when T is 373-1 and P is 14-689 x 144 is 639-30. The value of c at that temperature is 0-4213; and b is 0-01602. Substituting these figures in the ex- pression for /, Eq. (46), we have 639-3 0-47719 x 373-1 - X 144 + B', from which B' = 463-995. For most purposes B' is taken as 464. The formula for the total heat of steam, in any condition within the Callendar range, accordingly becomes / = 0-47719T - + 464 ..... (46 a). 1400 Values of / calculated in this way are given in the tables. As an example, take the same case as before, namely steam at a pressure of 400 pounds per sq. inch and a temperature of 240 C. With these data the numbers are / = 244-84 - 25-29 + 464 = 683-55. The tabulated value (Table D) is 683-54. account for the process of warming water. Let E be the internal energy of unit mass of water at C. and the corresponding saturation pressure P , which is 0-892 x 144 pounds per sq. ft., and let 7 be the total heat in that state. I [) =E +AP V . L et E IOQ be the internal energy of water at 100 C. and the corresponding saturation pressure P 10 o which is one atmosphere or 14-689 x 144 pounds per sq. ft., and let / 100 be its total heat in that state. I 10 Q=E 100 +AP 100 V W Q. V is the volume at C., namely 0-016 cub. ft., and F 1CO is the volume at 100. Imagine the water, initially at and P to be under a piston. Increase the load on the piston till the pressure is P JOO . Since water is almost incompressible this does not sensibly change the volume, or the temperature, or the internal energy, which may be taken as still equal to E Q . Then heat the water under constant pressure to 100 : this requires the addition of 100 constant-pressure calories. In being heated the water expands from F to F 100 and therefore does work on the piston equal to AP 1CO (F 100 - F ). Hence the net gain of internal energy in the whole operation, expressed in constant- pressure calories, is F v , ftn . p . v v . ^loo-^o^W-^/iootKioo- K ), from which or 7 100 - 7 = 100 + A (P 100 - P ) F = 100-023. Thus the same change of total heat which is measured by 100 of Callendar' s units is measured by 100-023 constant-pressure units. vin] APPLICATIONS TO PARTICULAR FLUIDS 333 It follows from Eq. (37) that in steam E = 0-36709T - + 464 ............... (47). The above expressions are in terms of P and T. We may also express both E and / for steam in terms of P and F, eliminating T. Since K v ' nR, Eq. (37) may be written E = nRT - AncP + B'. But by the characteristic equation (20 #), when R is expressed in thermal units, RT - AcP = AP (V - b). Hence, in steam, E = AnP (V - b) + B' .................. (48), which gives, in thermal units, Again /, being equal to E + APV, becomes I = A(n + I)Pr- AnbP + B' ............ (49), giving, in thermal units, - 0-534P 1400 This relation may be written in the form / - B' nb A(n + 1)P n+ 1 10 13 -(50), Tr 3 x 1400 (J - 464) 10 F. - -J + -X0.016 ^ > + 0-0123 ;..(). Hence if we use p to denote the pressure in pounds per square inch, the volume, in cubic feet, of 1 Ib. of steam in any dry state, super- heated, saturated, or supercooled, is given by the formula 2-2436(7-464) P This affords a convenient means of calculating the volume when the total heat is known. Take again the same example as before : steam at 400 pounds per square inch and 240 C. The tabulated value of / is 683-54. Substituting it in the formula we find V to be 1-2314 -f 0-0123 = 1-2437 cubic feet, in agreement with the figure got from Eq. (45) and with the tabulated value of V. 334 THERMODYNAMICS [CH. By differentiating Eq. (49) with respect to P, keeping V constant, we obtain, in steam, ).,- A(n + l)V- Anb ............... (51), which is constant for any chosen value of F. It follows that lines of constant volume on a steam chart with P and / for coordinates are straight in the region of superheat*, as in fig. 33 (Art. 102). We shall next obtain a working formula for the entropy of steam in any dry state, by using Eq. (35) and finding the value of the constant B. The constant is found by working out, from indepen- dent data, the entropy of saturated steam at 100 C. Following the usual convention, the entropy of water at C. is taken as zero. It follows from what is known about the specific heat of water, as will be shown in the next article, that the entropy of water at 100 C. and a pressure of one atmosphere is 0-31186. In passing at that constant pressure from the state of water at 100 to the state of saturated steam, the substance takes in 539-3 units of heat at the absolute temperature 373-1 : its entropy therefore increases to 0-31186 + -=^- or 1-75732. At that temperature c is 373*1 0-4213, and P is 14-689 x 144. Hence by Eq. (35), 1-75732 = 0-47719 log e 373-1 - 0-11012 log e (14-689 x 144) 10 0-4213 x 14-689 x 144 3 * 373-1 x 1400 From which B = _ . 21964> Substituting this, and introducing the factor 2-302585 to convert common to Napierian logarithms, the formula for the entropy of dry steam in any state becomes (/) = 1-09876 Iog 10 T - 0-25356 Iog 10 P - 0-002381 ^ - 0-21964 ...... (52). Values of cj> are given in the tables (Table E) for the same range of pressure and temperature as was used in tabulating V and /. 21 o. Total Heat and Entropy of Water. It is known from the researches of Regnault and others that the specific heat of water is not constant, but increases with rising temperature. Callendar * In the wet region the constant- volume lines remain very nearly straight, for the above relation still holds with regard to that part of the steam which is un con- densed, and its volume constitutes nearly the whole volume of the wet mixture. viii] APPLICATIONS TO PARTICULAR FLUIDS 335 suggests * that this increase may be due to the presence of water- vapour dissolved in the water. He supposes that when water and its vapour are in equilibrium at any temperature a volume of the vapour equal to the volume of the water is contained within the water. Consequently when water is heated its total heat increases more rapidly than it would do if the specific heat were constant, for the heat that is required to form the dissolved vapour becomes a larger proportion of the whole heat. This theory gives results which are consistent with the experimental data, and Callendar adopts it in calculating, for his tables, the total heat and the entropy of water. It has the advantage of allowing each of these two quantities to be expressed in a simple manner. Let V 9 be the volume of 1 Ib. of saturated steam at any assigned temperature T, and let V w be the volume of 1 Ib. of water at the same temperature and pressure. Then, according to the theory, 1 Ib. of "water" in that state is really 1 Ib. of a solution, containing dissolved vapour; the water conceals within it a volume of saturated steam equal to V w . If the remainder were also turned inCc vapour, under constant pressure, we should have a total volumr of vapour equal to V w + (V 8 V w ) or V 8J and the heat taken ii- during the process would be the latent heat L. Hence -=="=. represents the heat that is required to produce the r* I .,. vapour initially ^resent in the water before the formation of any separate stearr hr-?ins. This heat had to be supplied while the water was being Banned up to the temperature of saturation ; it constitutes a part of the total heat of water /, . The other (and chief) part of the total he'at of water is supposed to increase at a uniform rate as the temperature rises : it may there- fore be represented by K (T T ), where AC is a constant and T T is the excess of temperature above C., which is taken as the starting-point in reckoning the total heat. Hence, adding the two parts, the total heat of water under saturation pressure at any temperature T is /irn rri \ , *** W "()" tt?u /KQ\ w = K (T- T ) + = JT~V -- v~ ...... ( ) y ~ * ' ~ ^ Here L , V Wo and V, o refer to the state at C. At that temperature the latent heat is 594-27, the volume of water is 0-016, and that of saturated steam is 3726 cub. ft. Hence p ' gf = 0-0029 calory. *So~ Y WQ * Phil Trans. A, vol. 199, p. 147, 1902. 336 THERMODYNAMICS [CH. This is the constant which has to be subtracted to make I w = when the temperature is C. To calculate K, we* have I w = 100 when T - T = 100, by Art. 209. L is then 539-3, V w is 0-01671 and F, is 26-789. Hence " from which K = 0-99666. The formula for I thus becomes I w = 0-99666 (T - 273-1) + = ^- - 0-003. ..(53 a). ' s * w Values of /, calculated by this formula are given in the tables. y Throughout the working range the ratio ~ ^TT~ is very nearly the same as V W JV S9 and its numerical value is approximately equal to 0-00004^?, where p is the saturation pressure in pounds per sq. inch. To find the entropy of water under saturation pressure at any temperature T, we may think of the water as being brought to its actual condition by two stages. Imagine it to be heated to that temperature in an "ideal" manner, namely without the formation of any dissolved steam, and then the dissolved steam to be formed at that temperature. The entropy depends only on the actual con- dition (Art. 44). Taking, as before, the entropy of water at C. to be zero, we therefore have zr. L O F_. T TV = 0-99666 lbg e + " - 0-00001 (54), ^0 * \ y 8 ~ y V)) which is the formula used by Callendar for the entropy of water under saturation pressure*. It follows that the value of G, or T /, for water under satura- tion pressure is G w = K T log e - - 0-00001T - K (T - T ) + 0-003... (55), ^o the term ^ ^- cancelling out. * 3 ~ ' W 2ii. Relation of Pressure to Temperature in Saturated Steam. A formula connecting the pressure with the temperature of steam in the saturated state is most easily obtained by making * Steam Tables, p. 7. vni] APPLICATIONS TO PARTICULAR FLUIDS 837 use of the fact that G, or T(f> /, has the same value tor the saturated vapour, at any temperature, as for the liquid at the same temperature and pressure (Arts. 90 and 185). By Eq. (39) of Art. 208 the value of G for steam in any state is K v ' T log e T - K v ' T - RT log e P + A (c - b) P + BT - B' . Hence for dry steam at saturation pressure P s G s = K v ' T log e T -K V 'T- RT log e P s + A (c - b) P 3 + BT - B' (56). Since G s = G w we get, by equating (55) with (56), R loge P s - ^ (C ~ 6)Ps = B + K - K ' + AC log e T + 0-00001 B' + K T + 0-003 -( K -K 9 ')log.T ...(57). This expression allows the saturation pressure P s to be found for any temperature. On giving the various constants the values a] ready stated, it becomes 0-11012 log e P s - ~- 8 = 5-89094 - - 0-51947 log, T ......... (57 a). Callendar* puts this in a form more suitable for calculation, by substituting 2-302585 log ]0 for log e , and 144^> s for P 8 , p s being the saturation pressure in pounds per square inch : - 2 1. 07 449 - - 4. 7 1734.o glo T ......... (576). The saturation pressure corresponding to any given temperature is found by working out the right-hand side of the equation and then adjusting the value of log p s until the two sides become equal. The pressures of saturated steam, thus deduced from Callendar's characteristic equation, agree very closely, throughout the range to which the equation applies, with those measured by Regnault, and the agreement between the calculated and measured figures is evidence of the soundness of Callendar's method. Further con- firmation is obtained when the volumes, as calculated by his equation, are compared with experimentally measured volumes both of saturated and of superheated steam. * Steam Tables, p. 27. E. T. 22 THERMODYNAMICS [en. 212. Formulas for the Latent Heat of Steam, and for the Volume of a Wet Mixture. From the equations LV - V ~ I S -I W and I w = id + = - ~ * 003 > where t is the temperature on the centigrade scale, we have 1(1 += ^=- ) = /.- irf + 0-008, \ ' $ ~ * w' or L = (I s - kT + 0-003) (l - j&\ (58), which Callendar* writes L = (/, - K T) (l - j*} (58 a), dropping the 0-003 as negligible in this calculation. For the volume of a wet mixture, V q (Art. 74), he gives the . formula* y J K t Y = f^t (59) ' To obtain this we have /= L+I - l + Kt+ LVw y s ' w on again dropping the 0-003 ; also I Q - Kt qL(V e - V, Hence W I.- id L(V S -V W ) r, _ V V * * s y s 213. Collected Formulas for Steam. For convenience of reference and use the formulas are collected here by means of which the quantities in the Steam Tables may be calculated. In these formuJas V is the volume of 1 Ib. in cubic feet ; P is the pressure in pounds per square foot, and p in pounds per square inch. Centigrade degrees are used in the reckoning of temperature and quantities of heat. T is the absolute temperature and t the temperature from C. The following expressions for V, /, E and apply to dry steam * Steam Tables, p. 10. vin] APPLICATIONS TO PARTICULAR FLUIDS 339 in any state, that is to say, superheated, saturated, or supercooled, but not to a mixture of steam and water. The volume: 154 . 17T 167 . 52 x 10 . p -^lo- - + 0-01602 (45). The total heat : where c is the "co-aggregation volume" iu cubic feet, namely = 157-52 x 10^ = . 4213 /373.1NV- yV- \ T / Equation connecting the volume with the pressure and total heat : V = The internal energy : 2-2436 (/ - 464) V = - } - + 0-0123 (506). 10 P(V- 0-016) +464 The entropy : = 1 -09876 Iog 10 T - 0-25356 Iog 10 P - 0-002381 ^ - 0-21964. . .(52). The following equation, which applies only when the steam is at saturation pressure, determines the relation of pressure to temperature in saturated steam : 0-4057 (c - 0-016) p s Iog 10 /? s - '-jr- 2903*39 = 21-07449- ^ - 4-71734 log lo r ...(576). When the saturation pressure for any given temperature T has been determined by means of this equation, the volume, total heat, energy and entropy of saturated steam at that temperature (V s , I s , E s and c/>J are found by the above formulas. The latent heat : / y \ L = (I s - 0-99666* + 0-003) 1 - =2-1 ......... (58), \ ' s / where V w is the volume of 1 Ib. of water at saturation pressure. Within the range usual in practice, the ratio V w jV s is very nearly equal to 0-00004^ s , where p g is the saturation pressure in pounds per square inch, and the working formula is L = (I 8 - 0-99666*) (] - 0-00004j9 s ). 22-2 340 THERMODYNAMICS ICH. The total heat of water at saturation pressure : TV I w = 0-99666* + TF-^ - 0-003 ......... (58 a), ' s ' w or, very nearly, within the working range, I w = 0-99666* + 0-00004p s L. The entropy of water at saturation pressure : T IV w = 0-99666 log e - + r(Ff _Vj ~ O'OOOOl ...(54). The function G, which is T - /: For dry steam in any state, which gives G = 1-09877T Iog 10 T - 0-69683T - 0-25356T Iog 10 P P(c- 0-016) -1400- For saturated steam, or water at saturation pressure, or a mixture of water and steam in equilibrium, G S = G W = K T log e Jr - K (T - T ) - 0-OOOOlT + 0-003 . . .(55), ^o which gives G s = G w = 2-2949T Iog 10 - 0-99666* 214. Tables of the Properties of Steam. The Steam Tables in the Appendix contain some representative numbers, but refer- ence should be made to Callendar's Tables for a more complete set. Tables A and B relate to the special case of saturated steam. When steam is saturated a single property, such as either the temperature or the pressure, fixes its state. In Table A the property which is assumed to be known is the temperature, and the table gives corre- sponding values of the pressure, volume, total heat, and entropy- all for the saturated state. It also gives the latent heat. Similarly Table A* gives the volume, total heat and entropy of water at saturation pressure. It also gives the function G, which is the same for water and for saturated steam. In Table B the property which is assumed to be known is the pressure, and corresponding values are given of the other properties in the saturated state, namely the temperature, volume, total heat, entropy, latent heat, and the function G. vni] APPLICATIONS TO PARTICULAR FLUIDS 341 Tables C, D and E relate to the general case of steam in any dry state, whether superheated or supercooled. A knowledge of two properties is then required to specify the state ; the two that are selected as independent variables in the tables are the temperature and the pressure. Table C gives the volume, Table D the total heat, and Table E the entropy, in relation to these two variables. In each table a zig-zag line indicates the boundary between the superheated state (above) and the supercooled state (below). In crossing this line the substance passes through the state of satura- tion. From Table D it is easy to find the heat of formation, under constant pressure, of steam in any condition of superheat. The total heat, at the given pressure and temperature, is obtained from the table, and the heat of formation is found by subtracting from that the total heat of water, at the same pressure and at the temperature of the feed. Again, from Table D, values may be inferred of the specific heat (K v ) of steam at temperatures and pressures within the range of the table. K P for any condition of the steam is equal to the amount by which / increases per degree of rise in temperature, at constant pressure. The change of / per degree is found by noting, in the appropriate pressure column, the amount by which / changes for an interval of -10, and dividing that by 10; this gives the mean value of K v over the interval, which is practically the same thing as K p at the middle temperature. Values of the specific heat at various constant pressures and for various temperatures have been deduced in this way from the tabulated values of the total heat, and are given separately in Table F. The zig-zag line has the same meaning as in the other tables ; the figures above it relate to super- heated steam. They show a decrease of K v with rising temperature, but at higher temperatures (beyond the range to which Calendar's equation applies) there is a marked increase, as was pointed out in Chap. VI. APPENDIX I EFFECTS OF SURFACE TENSION ON CONDENSATION AND EBULLITION 215. Nature of Surface Tension. In Arts. 135-138 it was pointed out that when water-vapour is suddenly expanded it assumes a metastable state, becoming supersaturated owing to what was there called a static retardation in the formation of drops. Wilson's experiments were cited to show that, in the absence of foreign nuclei, a vapour will become much supersaturated before drops will form, and it was mentioned that this is an effect of surface tension in the liquid. In this note some account will be given of what is meant by surface tension, and how it retards the formation of drops in a cooled vapour; also how it retards the formation of bubbles within a liquid when the liquid is boiled. The cohesive forces which the molecules of any liquid exert upon one another make the free surface of the liquid behave as if it were a stretched elastic skin. It is to this that the phenomena of capillarity are due the rise of a liquid column in a tube when the liquid is one that wets it, and the depression of the column when the liquid does not wet the tube. To this also are due the forms assumed by liquid films and by drops. It is the tension of the surface layer that makes a drop take a spherical shape when there are no disturbing forces: the drops of molten metal in a shot- tower, for example, become spheres as they fall freely, and solidify into spherical shot before they reach the bottom. A drop of mer- cury on a plate, or of dew on a leaf w r hich it does not wet, would be spherical were it not for the upward pressure of the support on which the drop rests ; the smaller the drop is the nearer does it come to being a sphere, for the disturbing force due to the weight is relatively unimportant in a small drop. As a result of surface tension, the energy contained in a drop of liquid is greater than the energy contained in an equal quantity when that forms part of a big mass of the same liquid at the same temperature, for energy is stored in the surface layer in much the same way as it would be stored by the stretching of an elastic skin. APP. l] EFFECTS OF SURFACE TENSION 343 We are concerned here only with thermodynamic aspects of sur- face tension, and especially with its influence on the formation of drops in an expanding vapour. We shall see that, as a consequence of surface tension, a small drop will evaporate into an atmosphere of supersaturated vapour, because the vapour pressure which is required to prevent evaporation from the curved surface of a drop is greater than the vapour pressure which is sufficient to prevent evaporation from a flat surface of the same liquid at the same temperature; in other words, that at any given temperature the saturation pressure for a small drop is greater than the normal saturation pressure. The film that is formed when a soap-bubble is blown, or when a soapy liquid is smeared over a ring or hoop of wire, consists of two surface layers, back to back, with some of the liquid between. When the film is very thin, as, for instance, when it looks black in reflected light just before it breaks, it may be said to consist of two surface layers only; but it can be made a hundred or more times thicker than that and have just the same tension, for the state of tension exists in the surface layers only. The tension of such a film, whether thick or thin, is the tension of two surface layers; in other words, it is twice the surface tension. The tension in a liquid film differs from that of a stretched sheet of india-rubber or other elastic membrane in these important respects : it does not change when the film contracts or is stretched, and it has necessarily the same value in all directions along the surface. Let a liquid film be formed on a U-shaped frame (fig. 97) by wetting a wire AB with the liquid, placing it over C, and then drawing it away in the direction of the arrow. The force that will have to be applied to draw it away or to Q hold it from coming back is 2SI where / is the length AB and S is the tension of the surface layer on each side of the film per unit of length. The quantity S so defined B Fig. 97 measures the surface tension of the liquid. In drawing the rod away through any distance x in the direction of the arrow the work done is 2Slx. Hence S also measures the work done in 344 THERMODYNAMICS [APP. forming a single surface layer, per unit of area of the layer; in other words, S measures the potential energy that is stored in each unit of area of the free surface of a liquid in consequence of its surface tension. It follows that the surface energy of a spherical drop (that is to say the potential energy which is due to its surface tension) is 4f7TT 2 S where r is the radius of the drop. The spherical form which a free drop assumes is the form which will make the surface energy (for a given volume) a minimum. A drop resting on a support takes such a form as will make its total potential energy a minimum, namely the sum of the energy of surface tension and the energy of position which the drop has in consequence of the height of its centre of gravity above the level of the support. 216. Need of a Nucleus. Imagine a drop to be evaporating under conditions that keep its temperature constant. Energy has to be supplied in proportion to its loss of mass to provide for the latent heat of the vapour that is formed. But the drop is losing surface energy in consequence of its diminution of surface, and to some extent this reduction of surface energy supplies the latent heat that is required; only the remainder has to be supplied from out- side the drop. Consequently a drop is more ready to evaporate than the same liquid in bulk, at the same temperature, and it will continue to evaporate into an atmosphere which would be saturated with respect to the same liquid in bulk. Moreover, as the drop gets smaller and smaller (if we assume that the reduction of size may go on without altering the nature of surface tension), a stage would be reached when the loss of potential energy due to con- traction of the surface would become sufficient to supply all the latent heat of the vapour that is passing off. In that event, no heat would have to be supplied from outside to complete the evaporation of the drop : it would become inherently unstable and would flash into vapour. For the same reason a drop cannot form except around a nucleus, and the larger the nucleus the more readily it forms. When particles of dust are present in expanding vapour, the first drops to be formed use them for nuclei, as was shown by Aitken (Art. 79), and only a small amount of supersaturation occurs before such drops begin to form. The cloud of particles observed by Wilson when dust-free air containing water-vapour is expanded enough to i] EFFECTS OF SURFACE TENSION 345 cause an eight-fold supers aturation are formed around much smaller nuclei which consist, probably, of accidental co-aggrega- tions of the molecules of the gas itself, or of electrically charged molecules, such as are always present in small numbers. It should be added that the presence of an electric charge greatly favours condensation of the vapour upon any nucleus. As an electrified drop evaporates, the charge remains behind ; the potential energy due to electrification therefore increases as the drop becomes smaller, for the energy due to a constant charge varies inversely as the radius of the sphere that carries it. In this respect the effect of an electric charge is opposite to that of surface tension. Hence when a drop is charged more energy has to be supplied from outside to make it evaporate than would be required if it were uncharged. An electri- cally charged drop will therefore evaporate less readily than an uncharged drop of the same size, and may grow larger in an atmo- sphere that is but little supersaturated or even not supersaturated at all. In vapour which is slightly supersaturated it is found that any "ionizing" action, such as that of an electric spark, or of Rontgen rays or of ultraviolet light, which sets free the ions or particles conveying unbound electric charges, brings about a cloud of condensation, by creating fresh nuclei, or by stimulating the powers of existing nuclei through causing them to acquire an electric charge*. 217. Kelvin's Principle. Confining our attention, however, to drops which are not electrically charged, we shall now consider how, as a consequence of surface tension, the equilibrium of a drop of given size depends on the state of supersaturation of the vapour around it. Assume the liquid and the vapour to be at the same temperature. Liquid with a flat surface is in equilibrium with the vapour above it when the vapour is at the pressure of saturation : there is then no tendency on the whole for the liquid to evaporate or for the vapour to condense, any evaporation that occurs being exactly balanced by an equal amount of condensation. Liquid in the form of a small drop is, owing to its curved surface, in equi- librium with the surrounding vapour only when the pressure of the vapour surrounding it exceeds the normal saturation pressure by a definite amount; in other words, only when the vapour is super- saturated. The degree of supersaturation necessary for equi- librium depends on the curvature of the surface, and therefore on * See Sir J. J. Thomson, On the Conduction of Electricity through Oases, Chap. VII; C. T. R. Wilson, Phil. Trans. A, vol. 192, 1889. 346 THERMODYNAMICS APP. B the smallness of the drop. This principle was first established by Lord Kelvin *. It is of fundamental importance in explaining the retarded condensation of expanding steam. We may apply Kelvin's general method as follows to find a relation between P s the normal pressure of saturation (which is the equilibrium vapour-pressure over the flat surface of a liquid f) and P' the equilibrium vapour-pressure over a curved surface, such as the surface of a small drop. Take for this purpose the curved surface at A, fig. 98, which is formed by __^ holding in the liquid a capillary tube of a material such that the liquid does not wet it. The column of liquid in the tube is accordingly depressed through some distance h, and if the bore is small enough the free surface at A is sensibly part of a sphere. Imagine the liquid to be contained in a closed vessel, and that the space C above it contains nothing but the vapour of the liquid. Let all be at one temperature T. The whole system is in equilibrium. Over the flat surface at B there is vapour whose pressure is P s : over the curved surface at A there is vapour of a higher pressure P' '. The difference P' - P s is equal to the weight of the column of vapour in the tube (per unit area of cross section) from the level of A to the level of B. Let a be the weight of unit volume of the vapour. If this were constant, the weight of the column of vapour in the tube (per unit area of section) would be simply ah. But a depends upon the pressure P; it is equal to 1/F and may therefore be written p The difference in the integrated between the level at B and the level at A. Compare next the hydrostatic pressures within the liquid just * Proc. Roy. Soc. Edin. vol. vn, 1870; Popular Lectures and Addresses, vol. I, p. 64. f Namely, the saturation pressure for any assigned temperature as given in tables of the properties of saturated steam. Fig. 98 if we take the equation PV= RT to apply. two vapour pressures is p f _ p i] EFFECTS OF SURFACE TENSION 347 under the surface at B and just under the surface at A. Just under the flat surface at B the hydrostatic pressure in the liquid is equal to the pressure of the vapour over the surface ;*t is therefore equal to P 8 . As we go down through the liquid to the level of A, the hydrostatic pressure increases by the amount ph, where p is the weight of unit volume of the liquid. Therefore just under the curved surface at A its value is P s + ph. But we may also calculate the hydrostatic pressure under the curved surface at A in another way. The top of the liquid column at A, which has a surface layer in tension, may be treated as a segment of a sphere of radius r. Its surface layer forms a cap whose surface tension S causes it to press down upon the liquid below with a pressure p such that rrr^p = 27rrS. That this is so will be seen at once by considering the equilibrium of a complete hemisphere of the same curvature and with the same surface tension. Round the circumference (27rr) of the horizontal plane forming the base of such a hemisphere there would be a vertical force 2nrS balancing the resultant force due to the pressure p acting on the area of the base, 77T 2 . Hence 2S and the hydrostatic pressure just under the curved surface is therefore equal to ; 2S L ~\~ r Equating the two expressions for this hydrostatic pressure, we have , or ~ = ph~ (P' ~ PS). Hence, since P' P s = fadh, 2S - = ph fadh = /(/> cr) dh. And since dP = adh, 2S p_-Gr dp = p dp n IPs* because or is small compared with p. On substituting P/RT for a this approximation gives . P' 2S or log e ^ = * 348 THERMODYNAMICS [APP. This applies to any liquid surface whose radius of curvature is r. It therefore expresses the relation of the pressure P' in the super- saturated vapour around a spherical drop of radius r to the normal pressure of saturation P s (over a flat surface) for the same tempera- ture, when the drop is in equilibrium, in the sense that it is neither growing by condensation nor shrinking by evaporation. The expression shows how the degree of supersaturation P'/P 8 necessary for the equilibrium of the drop increases when the size of the drop is reduced. For a drop of given radius any increase of P above the value so calculated would cause the drop to grow. The expression also shows what is the least size of drop that can exist in an atmosphere with a given degree of supersaturation: any drop for which r is smaller would disappear by evaporation. It is only when the drop is very small that the excess of P' over P s is at all considerable. This is best shown by numerical examples. If we take water- vapour at 10 C. or 283 absolute, and use c.g.s. units, RT (which is treated as equal to PV) is 1-30 x 10 9 . The surface tension of water at that temperature is about 76 dynes per linear centimetre, and p is 1 gramme per cubic centimetre. Hence P' 2 x 76 . 1-01 gl P, = F-30 x 10 9 x r x 2-303* ~ ~D ' where D is the diameter of the drop in millionths of a millimetre. The formula accordingly gives these results : Ratio of vapour-pressure Diameter of drop in equilibrium with the in millionths of drop to normal satura- a millimetre tion pressure for the same temperature (P'/P S ) 100 1-02 50 1-05 10 1-26 5 1-59 2 3-2 1 10-2 This means, for instance, that a drop of water two millionths of a millimetre in diameter will grow if the ratio of supersaturation in the vapour around it is greater than 3-2, but will evaporate if that ratio is less. Hence when the ratio of supersaturation is 3-2, drops will not form unless there are nuclei present which are at least big enough to be equivalent to spheres with a diameter of two millionths of a millimetre. In Wilson's experiments a cloud of mist was produced when the supersaturation was 8, which corresponds, by the formula, to a * To convert from Napierian to common logarithms. i] EFFECTS OF SURFACE TENSION 349 diameter of about 1-1 millionths of a millimetre. On the assumption that the formula may still be applied to such small nuclei, it might be inferred, if there were no ionization, that water-vapour contains many nuclei of that order of magnitude, which may be pairs or small groups of molecules co-aggregated by chance encounters. It will be obvious from Kelvin's principle that a drop of water cannot continue to exist in an atmosphere of saturated vapour. When the drop and the atmosphere are at the same temperature, the drop can exist only if the atmosphere around it is super- saturated. For any given degree of supersaturation there is a value of r (determined by the formula) such that a drop of smaller radius will evaporate and a drop of larger radius will grow. Thus the bigger drops in a cloud will tend to grow at the expense of the smaller drops. 218. Ebullition. Similar considerations govern the formation of bubbles in a boiling liquid. We may treat any small bubble as a spherical space of radius r, containing gas, bounded by a spherical envelope in which there is surface tension. Outside of that is the liquid, at a pressure P. In consequence of the surface tension in the envelope, the pressure inside the bubble P z must exceed P by the amount 2S/r, where S is the surface tension in the boundary surface of the bubble, making 2$ p i - p - -j- When r is very small this implies a great excess of pressure within the bubble. If no particles of air or other nuclei were present to start the formation of bubbles, boiling would not begin until the temperature were raised much above the point corresponding to the external pressure, and would occur with almost explosive violence. Once formed a bubble would be highly unstable, for as the radius increases the tension of the envelope becomes less and less able to balance the excess of pressure within it. This happens, to some extent, when water is boiled after being freed of air in solution: it is then said to boil with bumping. It follows that a pure liquid may be superheated, that is to say, raised above the temperature of saturation corresponding to the actual pressure. This is an example of a metastable state like the state that is produced when a vapour is supercooled without condensing, or when a liquid is supercooled without solidifying. Water at atmospheric pressure may be heated to 180 C. or more when it has been freed of air and when it is kept from contact with 350 THERMODYNAMICS [APP. i the sides of the vessel by supporting it in oil of its own density, so that the water takes the form of a large globule immersed in oil. In the ordinary process of boiling, a bubble contains in general some air or other gas besides the vapour of the liquid itself. With- out gas in it, the bubble could not exist in stable equilibrium. With gas in it, the bubble will be in stable equilibrium when the partial pressure due to the gas provides the necessary excess of the whole internal pressure P 4 over the external pressure P. Any reduction of the bubble's size would then raise the pressure of the gas more than enough to balance the increase of 2S/r. Let P v be the vapour-pressure inside the bubble. If we assume that the external pressure and temperature remain constant, the partial pressure due to the gas may be expressed as a/r s where a is a con- stant. Then P z = P v + a/r 3 , and the equation a 2S 2S a P * + ^ = P +^> OT P v- P =~-- 3 > determines the value of r at which the bubble is in equilibrium. The quantity P v P is the excess of the vapour-pressure in the bubble over the pressure in the liquid. Differentiating this with respect to r, to find the limiting condition for stability, we have d(P v -P) a 1 2S and therefore when P v P = - . 3r Hence for stability P v - P must be less than 4S/3r. This means that when a liquid containing bubbles of radius r is heated, the temperature will rise until the vapour-pressure within the bubbles exceeds the pressure in the liquid by the amount - , but when that point is reached the bubbles will become unstable and ebullition will begin. Callendar* calculates on this basis that with bubbles one millimetre in diameter water (under one atmosphere) will boil at a temperature of 100-05 C., and that to produce 10 of superheat the diameter of the bubbles must not i "2017 Enc. Brit., Article "Vaporization.' APPENDIX II MOLECULAR THEORY OF GASES 219. Pressure due to Molecular Impacts. According to the molecular theory, a gas consists of a very large number of particles called molecules moving with great velocity. Each molecule moves freely, with uniform velocity in a straight line, except when it encounters another molecule or the wall of the containing vessel. In an encounter the velocity changes in direction, and generally in amount, but there is no dissipation of energy; the mole- cules behave like perfectly elastic bodies. As a result of many encounters, a stable distribution of speed among the mole- cules is established, but the speed of any one molecule is being constantly changed, by its encounters, within very wide limits. The length of the free path, which it traverses between one en- counter and the next, is also quite irregular. The average of that length, or what is called the "mean free path," is very long com- pared with the dimensions of the molecule itself. This characteristic distinguishes a gas from a liquid. In a gas the average time during which a molecule is moving in its free path is very large compared with the time of an encounter. By the time of an encounter is meant the lime during which the molecule is either in contact with another, or so near it that there is a sensible force acting between them. When a gas is compressed, the mean free path is reduced, and the encounters become more frequent between one molecule and another and also between the molecules and the walls of the vessel. When a gas is heated the speed with which the molecules move is increased; we shall see immediately that their average kinetic energy is proportional to the temperature. The molecular theory is now well established: there is conclusive evidence that actual gases do consist of particles moving in the manner which the theory prescribes. The pressure of the gas, that is to say, the pressure which the gas exerts on every unit of surface of the containing vessel, is due entirely to the blows of the molecules upon the surface : the mo- mentum given to the surface by their blows, per unit of area and per unit of time, measures the pressure in kinetic units. 352 THERMODYNAMICS [APP. In any gas that is chemically homogeneous, all the molecules have the same mass. Call that mass m. Let N be the number of molecules present in unit of volume of the gas in any actual state as to pressure and temperature. Then mN represents the density, namely, the whole mass per unit of volume, and F, the volume per unit of mass, is equal to 1/mN. Before proceeding to consider the pressure caused by molecular blows, we shall make the following postulates: (1) That the molecules are perfectly free except during en- counters, -and therefore move in straight lines with uniform velocity, from one encounter to the next; (2) That the time^during which an encounter lasts is negligibly small in comparison with the time during which the molecule is free; (3) That the dimensions of a molecule are negligibly small in comparison with the free path. These three postulates are equivalent to assuming that the gas is perfect in the sense of Art. 18. They are not strictly true of any real gas ; but we shall assume them to be true in what immediately follows, and shall thereby deduce from the molecular theory a result which corresponds to the ideal formula PV= RT. Suppose the gas to be in equilibrium in a vessel at rest, and let the velocity v of any molecule be resolved into rectangular com- ponents v x , v y and v z , along three fixed axes. Consider the pressure due to molecular blows upon a containing wall, of area S, forming a plane surface at right angles to the direction of x. The contribution which any molecule makes to the pressure on that wall is due entirely to the component velocity v x : nothing is contributed by the components v y or v z . Any molecule which strikes the wall has the normal component of its velocity reversed by the collision. Hence the momentum due to the blow is 2mv x where v x is the normal component of the velocity and m is the mass of the molecule. Consider next how to express the sum of the effects of such blows in a given time. For this purpose we may think of the mole- cules as divided into groups according to their velocities at any instant. Let n be the number, in unit volume of the gas, whose ^-component of velocity, v x , has the same numerical value. Since the number of molecules is very great, we may take the number to be the same in one cubic inch (say) as in another. There will of course be very many such groups, each with a different value of n] MOLECULAR THEORY OF GASES 353 v x . Think, in the first place, only of those in the group n. Half of the whole number of molecules in the group are moving towards S ; the other half are moving away from it. At any instant of time there will therefore be within a small distance S# of the surface S, and moving towards it with component velocity v x , a number of molecules of that group equal to %nS8x. A molecule distant 8a? from S, and having a component velocity v x towards S, would reach S in a time $t = - , provided it did not encounter any other molecule on its way. Hence the number of blows delivered to S by molecules of that group, in the time 8t, would (on the same proviso) be equal to the number of such molecules as originally lay within a distance 8x, namely the number ^nSSx. Hence also the momentum due to the blows on the area S in the time 8t would be equal to ^nSSx x 2mv x , which becomes, per unit of area and per unit of time, Sx nmv x ~- = nmv x 2 , ot Sx since v x = -K- . This is the momentum contributed by one group only. The pressure P is made up of the sum of the quantities of momentum contributed by all the groups ; hence P = ZnmVy? = mLnv x 2 , or P = mNv^, where N is as before the whole number of molecules per unit of volume, and v x 2 is the average of v x 2 for all the molecules. Now the velocity v of any molecule is related to its components by the equation v z = v j + Vy 2 + ^ 2> Hence, if we write v 2 for the average value of v 2 for all the molecules, il* = vJ + vS + vf = 3v x 2 , since the motions take place equally in all directions. The square root of v' 2 is called the " velocity of mean square." It is not the same thing as the average velocity, but is the velocity a molecule would have whose kinetic energy is equal to the average kinetic energy of all the molecules. The expression for P may therefore be written P = $mNv*. 23 354 THERMODYNAMICS [APP. Further, since mN is the quantity of gas in unit volume, or 1/F, where V is (as usual) the volume of unit mass, this gives PF = J^. In obtaining this result we made (in order to simplify the argu- ment) a proviso that each molecule of a particular group, lying initially within the distance $x of the wall, struck the wall without encountering other molecules on the way. This is not true, but any encounter on the way does not affect the final result in a gas to which the three postulates apply. For in any encounter, some momentum, perpendicular to the wall, is simply transferred to another molecule, and reaches the wall without loss. The molecule which takes it up has to travel the full remainder of the distance in the direction of #, neither more nor less, since the dimensions of the molecules are negligibly small (Postulate 3), and no time is lost in the encounter (Postulate 2). Hence the general result of the encounters is not to alter the amount of momentum which reaches the wall in any given time, and the conclusion remains valid that py _ j~ 3 * Comparing this with the perfect-gas equation we see that v 2 is proportional to the absolute temperature; and consequently the average kinetic energy which the molecules possess in virtue of their velocity of translation is proportional to the absolute temperature. We shall call their energy of translation E' ; they may, in addition, have energy of other kinds. The energy of translation of the molecules E' is equal to |r; 2 per unit mass of the gas. Hence by the molecular theory and the pressure is equal to two-thirds of the energy of translation, per unit volume of the gas. It may be noted in passing that the molecular theory explains why a gas is heated by compression. Think of the gas as contained in a cylinder, and being compressed by the pushing in of a piston. Then any molecule which strikes the piston recoils with an increased velocity because it has struck a body that is advancing towards it. The component velocity v x normal to the piston is not simply reversed by the blow, but is increased by an amount 2z/, where v' is the velocity with which the piston is moving when the molecule n] MOLECULAR THEORY OF GASES 355 strikes it, for the quantity which is reversed is the relative velocity v x + v'. The result is that the motion of the piston in compressing the gas augments the average velocity of the molecules, and consequently increases v 2 , on which the temperature depends. 220. Boyle's, Avogadro's, and Dalton's Laws. These laws follow from the molecular theory, for gases that obey the three postulates. Keeping v 2 constant, we have the law of Boyle, PV = constant, since PV = ffi. If there are two gases at the same pressure, since P = in each ' Maxwell has shown that if two gases are at the same tem- perature, the average kinetic energy of a molecule is the same in both, or Hence if they are at the same pressure and the same temperature N = N 2 , that is to say, the number of molecules in unit volume is the same for both, which is Avogadro's Law. It follows that the density, or mass of unit volume, differs in the two gases in the ratio of the masses of their molecules ; or, in other words, the density is proportional to the molecular weight (Art. 158)*. Again, the molecular theory shows that in a mixture of two or more gases, each of which obeys the three postulates, P = m 2 + mNV 2 + etc. In other words, the partial pressure due to each constituent of the mixture is the same as it would be if the other constituents were not there. This is in agreement with Dalton's Law (Art. 62). * The number N of molecules per cubic centimetre, which is the same for different gases at the same temperature and pressure, is about 27-5 x 10 18 for any gas at C. and a pressure of one atmosphere (see Jeans' Dynamical Theory of Gases, p. 8). Their average distance apart, which is JT~ , is therefore about one V-tv three -millionths of a centimetre. The number of molecules per mol is 22400 N or 6-16 x 10 33 . The mass m of a molecule in any gas may be found by dividing the density mN by N. Since the density of oxygen is 0-001429 gramme per c.c. the mass of an oxygen molecule is about 52 x 10^ 24 grammes. The mass of a hydrogen molecule is one-sixteenth of that, or about 3-3 x 10~ 24 grammes, the ratio of the molecular weights being 2 to 32. 232 356 THERMODYNAMICS [APP, 221. Perfect and Imperfect Gases. Thus the molecular theory, for gases which satisfy the three postulates, gives results identical with those we already know as laws of ideal perfect gases. In a real gas the postulates do not strictly hold. The size of the molecules is not negligible, and in any encounter there is an appre- ciable time during which the molecules concerned exert forces on one another. There may even be temporary pairing or co-aggrega- tion on the part of some molecules. It is interesting to enquire, in a general way, how these departures from the ideal conditions affect the calculation of the pressure. For this purpose, consider the simple case in which one of a group of molecules, advancing towards the wall, meets a molecule, initially at rest, to which it passes on the whole of it's momentum, and the other molecule then completes the journey and delivers the blow. If there were no loss of time in the encounter, and if the second molecule could be regarded as travelling over exactly the remainder of the distance, the rate at which the wall receives momentum would be exactly the same as if the encounter had not taken place. But if there were loss of time in any encounter, such, for example, as would occur if the two colliding molecules nloved together for any appreciable time, with their velocity reduced below that of the molecule which was originally moving, then the rate at which the wall receives momentum would be reduced, with the result of reducing P. On the other hand, if the molecules have a finite size, so that the one which was initially at rest had less distance to travel in completing the journey, the rate at which the impacts succeed one another on the wall would be increased. with the result of increasing P. This indicates that the pressure in a real gas will differ from the ideal pressure, which is given by the equation PV = Jz; 2 , by two small terms, one positive, depending on the size of the molecules, and one negative, depending on their cohesion. Such, in effect, is the kind of modification which finds expression in characteristic equations like those of Van der Waals, Clausius, or Callendar. 222. Calculation of the Velocity of Mean Square. Taking, for any gas that may be treated as sensibly perfect, the equation P = it is easy to calculate the value of the velocity of mean square v when we know the density of the gas at a given pressure. The n] MOLECULAR THEORY OF GASES 357 product mN is the density, and we do not need to know m or N separately to find v. In oxygen, for example, at C., the density is 0-001429 gramme per c.c., when the pressure is one atmosphere, or 1-0133 x 10 6 dynes per sq. cm. (Art. 12). Hence in oxygen at /a x^roiss x~To 6 standard temperature and pressure, VISA/ -- ^.rirnToo^ ' C( l ua ' V \J vlV/J.TP^Jt7 to 461 metres per second. Similarly in nitrogen it is 4&3 metres per second, and in hydrogen 1839 metres per second. 223. Internal Energy and Specific Heat. Consider next the bearing of the molecular theory on the internal energy and specific heats of a gas. We have seen that, in an ideal gas, where E' is the energy of translation of the molecules, or This may be written RT = \E' or E' = *RT. E' is therefore proportional to the temperature. Now E' may or may not be the whole internal energy, E, which the gas acquires when it is heated. It will be the whole if, when the gas is heated, the molecules can only take up energy of translation, and cannot take up energy of rotation or energy of vibration (Art. 174). Suppose, for instance, that each molecule behaved like a perfectly smooth rigid billiard ball, or like a massive point with no appreci- able moment of inertia about any line passing through it. In that case, it could not have any energy of vibration, nor acquire any energy of rotation in the course of its encounters with other mole- cules, and the only kind of communicable kinetic energy would be energy of translation. We should then find E = E', and con- sequently E = \UT. When a gas of this kind is heated, we should therefore have dE = RdT. But in any gas (regarded as perfect) dE = K v dT and K P =K V + R. Hence for a gas whose molecules have energy of translation only K v = fl?, K p = |J?, K ^ and y or -=2 = - or 1-667. A v 6 This value of y would not apply if E were only a part of E. But it is found that in a monatomic gas, such as argon, or helium, or the 358 THERMODYNAMICS [APP. vapour of mercury, the value of y is in fact equal to 1-667 or very near it. The inference is that in a monatomic gas, the structure of the molecule is such that substantially all its communicable energy consists of energy of translation. In any gas each molecule possesses three degrees of freedom of translation, namely, freedom to move along each of three inde- pendent axes. Since E' = ^RT, each degree of freedom of trans- lation accounts for a quantity of kinetic energy equal to \ET. This is .true whatever be the number of atoms in the molecule, and whether or no the molecules have other energy besides energy of translation. Consider next a diatomic gas, each molecule of which consists of two atoms. According to modern views* an atom is a complex system made up of a minute positively charged central nucleus in which the mass of the atom is almost all concentrated, with electrically negative particles called electrons distributed around it, at distances which are large compared with the dimensions of the nucleus f. The structure of the atom and the nature of the forces between one atom and another in the molecule are still uncertain, but for our present purpose it will suffice to picture an atom as a massive point, surrounded by a massless quasi-elastic fender due to forces which keep other atoms at a distance. Under normal conditions a diatomic molecule is equivalent, as regards inertia, to two masses held some distance apart: dynamically it may be compared to a dumb-bell ; a more exact comparison would be to a light stick capable of some elastic extension and carrying a heavy ball at each end. Considered as a rigid body it has five effective degrees of freedom effective as regards the storing and communication of kinetic energy namely, three of translation and two of rotationf . The two effective degrees of freedom of rotation are about axes in a plane perpendicular to the line joining the two atoms : about that line itself, the system has no effective moment of * See Rutherford, Phil May., May, 1911; Bohr, Phil. Mag., July, Sept. and Nov. 1913; J. J. Thomson, Phil. Mag., April, 1919. f In an electrically neutral atom the positive electricity in the nucleus is equal to the negative electricity in the electrons. Removal of one or more of the electrons would therefore leave the atom as a whole positively charged : this happens when a gas is "ionized." J A free rigid body has six degrees of freedom : it can move parallel to itself along three independent axes, and it can rotate about these axes. Any possible movement is made up of these six components. In a diatomic molecule one of the degrees of freedom of rotation is ineffective as regards the communication of energy from one molecule to another in an encounter. n] MOLECULAR THEORY OF GASES 359 inertia. Under these conditions it can be shown that the ultimate result of collisions is that the kinetic energy becomes equally shared by each of the five degrees of freedom. The energy of translation E f is equal to ^RT, and each degree of freedom of translation accounts for an amount of energy equal to %RT. It follows that each of the two degrees of freedom of rotation accounts in addition for %RT, and that the energy of translation and rotation together amounts to ^RT. Hence if there were no sensible energy of vibration as well, we should have the whole energy E = %R T and K v = ffl, K v = \U, and y = \ or 1-4. Now in most diatomic gases, such as oxygen (O 2 ), nitrogen (N 2 ), air, hydrogen (H 2 ), nitric oxide (NO), or carbonic oxide (CO), it is in fact found that y is equal, very nearly, to 1-4 at ordinary tem- peratures, and the inference is that the structure of their molecules is such as to give five effective degrees of freedom, namely the five that have just been described, and that their molecules do not, at ordinary temperatures, hold any considerable amount of communi- cable energy in any other form than as energy of translation and energy of rotation. But when such gases are strongly heated we know that the specific heat increases and y is reduced. This means that energy of vibration is then developed, which at high tempera- tures becomes an important part of the whole energy. In triatomic gases it may be conjectured that the three atoms of any molecule group themselves not in one straight line which would be an unstable arrangement but so that the massive centres lie at the corners of a triangle. Similarly when there are more than three atoms in the molecule, they will place themselves with their massive centres at the corners of a polyhedron. In any such triangular or polyhedral structure, considered as a rigid system, there are six effective degrees of freedom, namely three of rotation as well as three of trans- lation, for there is a finite moment of inertia about any axis, and the structure is such that the molecule can be set spinning about any axis by encounters with other molecules. As an ultimate result of many such encounters, it may be shown that each of the three degrees of freedom of rotation takes up a share of the kinetic energy equal to that of each of the three degrees of freedom of translation, namely, \RT, and consequently that the six degrees together account for a total of SRT. That is the energy which the molecules possess in virtue of their movements as rigid 360 THERMODYNAMICS [APP. structures. If there were no other way in which they could take up energy when the gas is heated, w r e should consequently find, in a triatomic or polyatomic gas, K v = 3R, K v = 472, and y = f or 1-333. The actual value of y, as experimentally measured, in the tri- atomic gases CO 2 and H 2 O, is rather less than this, and in gases of more complex constitution it is generally a good deal less. It is also found that the specific heats are greater than 3R and 4/2. The inference is that in such gases the molecule generally takes up a considerable amount of energy of vibration in addition to its energy of translation and rotation. It appears that a complex molecule can absorb energy not only by moving as a rigid body but by internal vibratory movements which arise through quasi-elastic deformation of its own structure. (Compare Art. 174.) The main part of this energy of vibration probably consists of to and fro movements on the part of the massive centres of the linked atoms. It is obvious that such a motion might occur in any molecule that is made up of more than one atom. The effect in a complex molecule is such as would occur if the lines joining the massive centres of the constituent atoms behaved like stiff springs. Thus in a diatomic molecule we might think of the "dumb-bell" as having an elastic shank which allowed the distance between the two masses to vary. The fact that in a diatomic gas at ordinary temperatures the observed specific heats are approxi- mately %R and ^R, and y is approximately 1-4, shows, however, that the diatomic molecule then behaves like a dumb-bell with a nearly inextensible shank. But when the temperature is high, the vibratory motion becomes relatively more important, and it accounts for an appreciable part of the whole energy, even in a diatomic molecule, and still more in a triatomic or polyatomic molecule. To this we must ascribe the progressive increase in specific heat, and the fall in y, which are observed when any gas is heated that has two or more atoms in the molecule. In a monatomic gas there is no possibility of this kind of vibra- tory motion, and there is no experimental evidence of any change of specific heat with temperature. The energy depends only on motion of translation, and when the gas is heated its energy increases in simple proportion to the temperature. But when diatomic, triatomic, or polyatomic gases are strongly heated, the energy increases in a more rapid ratio than the temperature. This n] MOLECULAR THEORY OF GASES 361 means that the ratio of the total energy E to the energy of trans- lation E' is not constant. In any gas that satisfies the equation PV = RT, R ^>r/ If the total energy E preserved a constant ratio to E', the specific heat would be constant, and in that case we should have y constant and equal to 1 + %E/E r , since E, reckoned from the absolute zero, is K V T, and E' is \RT. The fact, however, that y falls with rising temperature shows that the total energy does not preserve a con- stant ratio to the energy of translation, and hence that there is not equipartition of the energy among the possible modes of motion. In any gas we may write E =-- E' + E" + E'". The energy of translation E' varies as T, being equal to f RT. The energy of rotation E" bears, in any given type of molecule, a constant ratio to E' t and therefore also varies as T. If the energy of vibration E'" also bore a constant ratio to E' 9 the whole energy would vary as T, which is inconsistent with the experimental results stated in Chap. VI. 224. Energy of Vibration. The term E'" includes not only energy due to vibrations of the constituent atoms relatively to one another within the molecule (E m '") but energy due to vibrations (movements of electrons) within the constituent atoms themselves (E a " r ). It is known that E a '" is a very small part of the whole energy, even at temperatures as high as 2000 C. The vibrations that make up E a "' have much higher frequencies than those that make up E m '". It is to vibrations within the constituent atoms that one attributes the bright lines which make up the visible spectrum of an incandescent gas, and the corresponding dark lines due to absorption in the visible spectrum of light transmitted through a cold gas. The longer-period vibrations that make up E m '" emit or absorb ray's which lie in the infra-red region, beyond the range of the visible spectrum. It is these longer-period vibrations that constitute the main part of the vibratory energy when a gas is strongly heated, as in a flame or an explosion, and give rise to most of its radiant energy. From the theory that has been outlined above, of the consti- 362 THERMODYNAMICS [APP. tution of a diatomic molecule, we should expect it to have one well-marked period of free vibration, and therefore to show a strong emission band when heated, or when excited by electric discharge in a vacuum tube, and also a corresponding strong absorption band when cold. A good example is furnished by carbonic oxide (CO), whose infra-red spectrum is found to consist almost entirely of one characteristic band, the wave-length of which is about 4-7/z when the gas is emitting radiant energy, and 4-6//, when the gas is absorbing it*. The fact that these wave-lengths are so nearly the same is evidence that what may be called the stiffness of the quasi-elastic link between the atoms, due to chemical affinity, suffers little change when the gas passes from the cold to the radiant state. Again, in the infra-red spectrum of the triatomic gas CO 2 we should expect to find three prominent bands corresponding to the three modes of vibration that can be set up within a CO 2 mole- cule by relative movements of the carbon and oxygen atoms j*. This is in agreement with what is observed. There are, both in absorption and emission, three distinct infra-red bands, namely a strong band whose wave-length is about 4-4/x, a weak band with a wave-length of 2-7/z, and another with a much longer wave- length, between 14ju, and 15/z (Art. 173). This long-period vibra- tion accounts for the fact that even at ordinary temperatures the specific heat of CO 2 exceeds the value it would have if there were no vibratory energy, making y distinctly less than 1-333. For the principle holds that vibrations of long period require no more than a comparatively low temperature to excite them into taking up some considerable share of the energy, so that they then contribute substantially to the specific heat, whereas those of short period do not begin to take up an appreciable share until the gas is strongly heated. 225. Planck's Formula. This principle finds expression in a formula devised by Max Planck to connect the energy of any particular frequency of vibration with the frequency and with the temperature, when a state of equilibrium has been reached. * See W. W. Coblentz, Investigations of Infra-red Spectra, Publications of the Carnegie Institute, Washington, No. 35, 1905, No. 65, 1906, No. 97, 1908. t N. Bjerrum, Vorkandlungen der deutschen Phys. Gesellschaft, 1914, p. 737, dis- cusses the hypothetical configuration of a C02 molecule which would vibrate with periods corresponding to the three observed wave-lengths, which he takes as 2-7/i, 4-3/z, and 14-7/x. n] MOLECULAR THEORY OF GASES 363 According to Planck's theory the vibratory energy, per mol, corresponding to any particular frequency v is* Here N^ is the number of molecules per mol, namely 6-16 x 10 23 , and h is a constant, known as Planck's constant, which is the same for all gases and is approximately equal to 6-55 x 10~ 27 eg?T 'J?,"as usual, is the gas-constant, whose value per mol is 1-985 thermal units or 83-1 x 10 6 ergs, and e is the base of the Napierian logarithms, 2-71828. The frequency v is equal to c/A, where c is the velocity of light, or 3 x 10 10 cms. per second, and A is the wave-length in cms. In a gas whose molecules are capable of more than one mode of vibration the whole vibrational energy E'" would be the sum of as many terms, in the above form, as are required to express the various modes. Thus in carbonic acid, for example, there would be three terms for frequencies of vibration corresponding to the three observed wave-lengths. At any one frequency v, let the quantity N-Jiv/RT be represented bv x. Then ^ T>>T> T>>T<\ n n A n .~ n \ = ..... . = - r _ = o9/ A, x N-Jiv N-Jic and Planck's formula becomes where - is a factor the value of which depends on both v and T : for any given v it tends to an upper limit of 1 when T is indefinitely increased and to a lower limit of zero when T is indefinitely reduced. Hence, if we accept the formula as valid, it follows that when the molecules of a gas are free to vibrate in any one mode, the gas will take up, in respect of that freedom, a quantity of energy which approaches the limit RT when the gas is strongly heated. This will be true also of any other mode of free vibration which the molecules possess. When the gas is heated to any given temperature the fraction of RT which is taken up will in general be different for different modes of vibration, for it depends on the frequency, being smaller when the frequency is high. This, according to the theory, is why the high-frequency modes of vibration which are revealed by the visible spectrum do not contribute substantially to the * For a discussion of the theoretical basis of Planck's formula, see Jeans' Dynamical Theory of Oases, Chap. XVIII. 364 THERMODYNAMICS [APP. whole energy of a gas, even at temperatures such as are reached in an ordinary flame or in a gas-engine explosion, and why, in the reckoning of energy and of specific heat it is only vibrations of infra-red frequency that need be taken into account. For the same reason a gas whose molecules have one or more long-period types of vibration may, at ordinary temperatures, hold a considerable quantity of energy in the vibratory form, and have a specific heat markedly greater than the ideal (vibrationless) value. The amount by which any one mode of vibration will augment the specific heat is found by differentiating (with respect to T) the expression for the extra internal energy that is due to that mode. We may write it ^771 \x~& CK- \ _ a ^ v _ x T> \AWf ~j7ff ~~ T ^ ::TS ** dT - I) vo 0-8 0-6 0-4 02 Values of - 02 0"4 0-6 08 TO Fig. 99 V2 V4 V6 V8 2'0 Here is a factor, depending on the wave-length and the temperature, which ranges from zero to unity as the quantity I/a? is increased from zero to infinity. Fig 99 exhibits the manner in which this factor increases relatively to I/a;. It shows that there is a very rapid rise in the factor, and therefore in the specific heat, after I/a? has reached a value of about 0-1, but up to that point the effect of vibration on the specific heat is quite insignificant. At C. the value of A which corresponds to I/a; = 0-1 is 0-00052; hence it is only those modes of vibration whose wave-lengths are greater than say S/z that sensibly affect the specific heat of a gas at normal temperature. As an example, take the diatomic gas CO with its characteristic n] MOLECULAR THEORY OF GASES 365 vibration for which A is about 4-7/z or 0-00047 cm. For that wave- length the value of I/a?, at C, is 0-09; and at 2000 C. it is 0-74. The factor e x x*l(e x - I) 2 is therefore insignificantly small at C., but becomes about 0-86 at 2000 C. Hence the calculated specific heat K v , which is f 72 at C., rises, as a consequence of this vibrational energy, to (f + 0-86) R at 2000 C. ; and the correspond- ing value of y falls from 1-4 to barely 1-3. Again, take the triatomic gas CO 2 , one of whose characteristic vibrations has a wave-length of nearly 15/>t. So slow a vibration contributes substantially to the specific heat even when the gas is cold. At C. a wave-length of 15/x makes \\x = 0-24 and X X 2 /( X -1) 2 = 0-28. Hence a single mode of vibration with that frequency should bring the specific heat of the cold gas up to about (3 + 0-28) R, and reduce y from 1-333 to 1-305. When the gas is strongly heated, account has to be taken of three modes of vibration whose wave- lengths are long enough to be important. In respect of the three together, K v obviously tends, at very high temperatures, to in- crease towards a limit of 6R, and y to fall to ^, apart from any- thing that other vibrations may contribute, and apart from effects of dissociation. Though the ideas underlying Planck's theory are open to dispute, there can be little doubt that a curve more or less like that of fig. 99 does represent the way in which molecular vibration of a given type contributes to the specific heat. At first, when the gas is being heated from a cold state, the contri- bution is practically nil; then there is a sharp rise, and finally an asymptotic approach towards a limit. The temperature at which the sharp rise begins depends upon the frequency of free vibration, being higher when the frequency is high. The fact that in polyatomic gases generally the specific heats, at normal temperature, are greater than the ideal (vibrationless) values, and y is notably less than 1-333, is to be ascribed to their possessing long-period modes of vibration which are responsive to low- temperature encounters. A complex polyatomic molecule may have many such modes, each producing a substantial augmenta- tion of the specific heat. Similarly the characteristic mode of vibration in a diatomic gas may be so slow as to affect the specific heat at normal or com- paratively low temperature, making K v greater than ^R, and K p 366 THERMODYNAMICS [APP. n greater than ^R, and y less than 1-4. This is notably the case with the vapours of the halogen elements C1 2 , Br 2 , I 2 . These elements have high atomic weight, and it would seem that in each of them the pair of heavy atoms in the molecule, perhaps rather loosely held together, have a slow type of vibration, which explains the observed high specific heats and low value of the ratio y. When a hydrogen atom is substituted for one of the pair, this character- istic disappears, for the gases HC1, HBr and HI, when cold, are found to have specific heats that approximate to the normal values, with the ratio 1*4. 226. Effect of Extreme Cold on the Diatomic Molecules of Hydrogen. It has been found that when hydrogen is cooled to about 200 C. its specific heat falls progressively to a value not much greater than that for a monatomic gas, and y rises to a value not much short of that for a monatomic gas (1-667). This remarkable result, first observed in measurements of K v , has been confirmed by independent measurements of K v and of y*. It appears therefore that under extreme cold the hydrogen molecule tends to assume a different structure, becoming in effect quasi- monatomic, presumably by the coalescence of the two atoms which, at ordinary temperatures, are held apart. The pair of atoms apparently behave as if the forces which usually hold them apart what we called their fenders in Art. 223 cease to be effective in preventing the massive nuclei from coming together, to form what is virtually a single-atom molecule of double mass. It may be conjectured that this happens when the rotational speed of the diatomic molecule falls below a certain limit, and that the molecule then retains the coalesced state until its constituent atoms are forced apart by a sufficiently violent encounter. While it remains in the quasi -monatomic state it takes up energy of translation only, and when a large proportion of the molecules are in that state the gas behaves approximately as a monatomic gas in respect of its specific heats. So far as is known this action is peculiar to hydrogen ; it does not occur in oxygen, nitrogen, or carbonic oxide. * Eucken, Sitzungsberichte d. k. Preuss. Alcad., Berlin, Feb. 1912; Scheel and Heuse, do., Jan. 1913, also Ann. d. Physik, Vol. 40, p. 473, 1913; M. C. Shields, Phys. Review, Nov. 1917. APPENDIX III TABLES OF THE PROPERTIES OF STEAM Table A. Properties of Saturated Steam, in relation to the Temperature. A*. Properties of Water at Saturation Pressure. B. Properties of Saturated Steam, in relation to the Pressure. C. Volume of Steam in any Dry State. D. Total Heat of Steam in any Dry State. E. Entropy of Steam in any Dry State. F. Specific Heat, at constant pressure, of Steam in any Dry State. These Tables are based on Callendar's formulas, and will serve to illustrate his methods. The figures are, for the most part, taken from The Callendar Steam Tables published by Edward Arnold, 1915, which will be found to give much more complete particulars. 368 THERMODYNAMICS [APP. TABLE A. Properties of Saturated Steam. Temp. Cent. t Pressure, pounds per sq. inch Ps Volume, cub. ft. per Ib. V s Total Heat, Ib. -calo- ries per Ib. ^s Entropy, per Ib. 08 Latent Heat, Ib. -calo- ries per Ib. L Internal energy, Ib. -calo- ries per Ib. 0-0892 3275-9 594-27 2-17602 594-27 564-21 10 0-1788 1693-8 599-01 2-11649 589-03 567-85 20 0-3399 922-19 603-72 ! 2-06221 i 583-78 571-48 30 0-6162 525-81 608-40 2-01247 578-49 575-07 40 1-0703 312-45 613-04 1-96688 573-15 578-64 50 1-7888 192-72 617-63 1-92490 567-75 582-17 60 2-8873 122-91 622-16 1-88621 562-29 585-66 70 4-5156 80-804 626-60 1-85039 556-72 589-07 80 6-8627 54-596 630-95 1-81712 551-05 592-41 90 10-161 37-815 635-19 1-78619 545-25 595-67 100 14-689 26-789 639-30 1-75732 539-30 598-83 110 20-777 19-370 643-26 1-73027 53317 601-86 120 28-808 14-271 647-07 1-70485 526-85 604-78 130 39-213 10-696 650-72 ] -68092 520-32 607-58 140 52-482 8-1431 654-19 1-65831 513-57 610-23 150 69-150 6-2895 657-47 1-63689 506-56 612-73 160 89-800 4-9232 660-55 1-61657 499-29 615-08 170 115-06 3-9015 663-44 1-59724 491-75 617-27 180 145-59 3-1275 666-14 1-57884 483-93 619-30 190 182-08 2-5339 668-65 1-56128 475-82 621-19 200 225-24 2-0738 670-96 1-54453 467-41 622-91 210 275-78 1-7134 673-09 1-52851 458-69 624-48 220 334-38 1-4285 675-06 1-51326 449-69 625-93 230 401-89 1-2007 676-87 1 49868 440-38 627-23 240 478-74 1-0178 678-55 1-48480 430-81 628-43 250 565-63 0-8695 680-12 1-47161 420-96 629-53 Ill] TABLES OF THE PROPERTIES OF STEAM 369 TABLE A*. Properties of Water at Saturation Pressure. Temp. Cent. t Pressure, pounds per sq. inch Pw=Ps Volume, cub. ft. per Ib. v w Total Heat, Ib. -calories per Ib. *w Entropy, per Ib. fe Function G, Ib. -calories per Ib. G w = 8 0-0892 0-01602 10 0-1788 0-01603 9-98 0-03585 0-181 20 0-3399 0-01605 19-94 0-07046 0-714 30 0-6162 0-01609 29-91 0-10393 1-58 40 1-0703 0-01614 39-89 0-13631 2-78 50 1-7888 0-01621 49-88 0-16770 4-30 60 2-8873 0-01629 59-87 0-19815 6-13 70 4-5156 0-01638 69-88 0-22774 8-26 80 6-8627 0-01648 79-90 0-25652 10-68 90 10-161 0-01659 89-94 0-28454 13-38 100 14-689 0-01671 100-00 0-31186 16-36 110 20-777 0-01684 110-09 0-33853 19-60 120 28-808 0-01698 120-22 0-36460 23-10 130 39-213 0-01713 130-40 0-39011 26-86 140 52-482 0-01729 140-62 0-41511 30-86 150 69-150 0-01746 150-91 0-43963 35-10 160 89-800 0-01765 161-26 0-46373 39-58 170 115-06 0-01785 171-69 0-48743 44-29 180 145-59 0-01807 182-21 0-51078 49-22 190 182-08 0-01831 192-83 0-53381 54-38 200 225-24 0-01856 203-55 0-55654 59-75 210 275-78 0-01885 214-40 0-57904 65-33 220 334-38 0-01914 225-37 0-60128 71-12 230 401-89 0-01946 236-49 0-62332 77-11 240 478-74 0-01980 247-74 0-64517 83-30 250 565-63 0-02016 259-16 0-66687 89-68 24 870 THERMODYNAMICS [APP. TABLE B. Properties of Saturated Steam. Pressure, pounds per sq. inch P Temp. Cent. t Volume, cub. ft. per Ib. V 8 Total Heat, [b.- calories per Ib. Entropy, per Ib. 08 Latent Heat, Ib. -calories per Ib. L Function G, Ib. -calo- ries per Ib. G S = G W 0-1 1-59 2940 595-03 2-1662 593-44 0-005 0-2 11-69 1524 599-81 2-1068 588-15 0-246 0-3 17-99 1038 602-77 2-0727 584-83 0-58 0-4 22-66 790-7 604-97 2-0482 582-38 0-91 0-5 26-41 650-5 606-73 2-0299 580-40 1-23 1 38-74 333-1 612-46 9724 573-83 2-61 2 52-27 173-5 618-67 9159 566-52 4-69 3 60-83 118-6 622-53 8833 561-83 6-30 4 6723 90-54 625-38 8600 558-28 7-64 5 72-38 73-44 627-64 8422 555-38 8-81 6 76-72 61-91 62952 8277 552-92 9-86 7 80-49 53-59 631-15 8156 550-76 10-81 8 83-84 47-30 632-57 1-8049 548-82 11-69 9 86-84 42-36 633-85 1-7956 547-09 12-50 10 89-58 38-39 635-01 1-7874 545-50 13-26 12 94-44 32-37 637-02 1-7731 542-61 14-67 14 98-66 28-02 638-77 1-7611 540-12 15-94 16 102-41 24-73 640-26 1-7506 537-83 17-12 18 105-79 22-16 641-60 1-7414 535-75 18-20 20 108-87 20-08 642-82 1-7333 533-87 19-22 22 111-71 18-37 643-92 1-7258 532-09 20-18 24 114-34 16-93 644-93 1-7189 530-44 21-09 26 116-80 15-71 645-85 1-7126 528-88 21-95 28 119-11 14-66 646-74 1-7069 527-42 22-78 30 121-28 13-74 647-54 1-7016 526-02 23-56 32 123-35 12-94 648-30 1-6966 524-67 24-33 34 125-31 12-22 649-02 1-6919 523-40 25-07 36 127-17 11-59 649-69 1-6874 522-17 25-77 38 128-96 11-02 650-34 1-6831 521-00 26-45 40 130-67 10-50 650-95 6792 519-87 27-12 42 132-31 10-03 651-53 6754 518-77 27-76 44 133-89 9-603 652-08 6719 517-71 28-40 46 135-41 9-212 652-61 6685 516-68 29-00 48 136-88 8-853 653-12 6651 515-69 29-59 Ill] TABLES OF THE PROPERTIES OF STEAM 371 TABLE B (continued). Properties of Saturated Steam. Pressure, pounds per sq. inch P Temp. Cent. t Volume, cub. ft. per Ib. V 8 Total Heat, Ib. -calo- ries per Ib. Entropy, per Ib. 08 Latent Heat, Ib. -calories perlb. L Function G, Ib.- calories per Ib. 50 138-30 8-520 653-60 1-6620 514-71 30-16 60 144-79 7-184 655-77 1-6479 510-22 32-85 70 150-46 6-218 657-61 1-6359 506-23 35-30 80 155-52 5-487 659-20 1-6256 502-59 37-54 90 160-09 4-913 660-59 1-6165 499-24 39-62 100 164-28 4-451 661-82 1-6082 496-11 41-58 110 168-15 4-070 662-93 1-6007 493-18 43-40 120 171-75 3-751 663-92 1-5938 490-40 45-13 130 175-13 3-479 664-83 1-5875 487-76 46-78 140 178-31 3-245 665-69 1-5818 485-27 48-37 150 181-31 3-041 666-49 1-5765 482-90 49-89 160 184-16 2-862 667-22 1-5715 480-61 51-34 170 186-88 2-703 667-90 1-5666 478-40 52-75 180 189-48 2-562 668-53 1-5620 476-26 54-10 190 191-97 2-435 669-13 1-5577 474-20 55-42 200 194-35 2-320 669-69 1-5538 472-21 56-69 210 196-66 2-216 670-20 1-5502 470-26 57-94 220 198-87 2-120 670-70 1-5465 468-38 59-13 230 201-02 2-034 671-19 1-5429 466-55 60-31 240 203-09 1-954 671-64 1-5395 464-76 61-45 250 205-10 1-880 672-07 1-5362 463-00 62-58 260 207-04 1-811 672-48 1-5332 461-30 63-66 270 208-93 1-748 672-88 1-5303 459-65 64-72 280 210-77 1-689 673-25 1-5274 458-02 65-77 290 212-57 1-634 673-61 1-5246 456-41 66-79 300 214-32 1-583 673-96 1-5219 454-84 67-80 350 222-45 1-368 675-52 1-5096 447-44 72-57 400 229-75 1-206 676-84 1-4991 440-63 76-96 450 236-42 1-079 677-97 1-4897 434-28 81-06 500 242-57 0-977 678-97 1-4814 428-31 84-92 242 372 THERMODYNAMICS [APP, TABLE C. Volume, in cubic feet per lb., Temp. Cent. Pressure in pounds per sq. inch 20 40 60 80 100 120 140 400 35-988 17-973 11-967 8-9648 7-1632 6-0009 5-1043 350 33-295 16-617 11-058 8-2785 6-6107 5-4989 4-7048 300 30-594 15-254 10-141 7-5848 6-0509 5-0284 4-2980 290 30-052 14-981 9-9569 7-4449 5-9378 4-9330 4-2153 280 29-510 14-706 9-7718 7-3045 5-8241 4-8372 4-1323 270 28-967 14-431 9-5863 7-1636 '5-7100 4-7409 4-0487 260 28-425 14-156 9-4002 7-0221 5-5953 4-6441 3-9646 250 27-881 13-880 9-2134 6-8799 5-4798 4-5465 3-8798 240 27-337 13-603 9-0260 6-7370 5-3637 4-4481 3-7942 230 26-791 13-326 8-8376 6-5933 5-2467 4-3490 3-7078 220 26-246 13-048 8-6483 6-4486 5-1289 4-2490 3-6205 210 25-699 12-768 8-4582 6-3031 5-0101 4-1481 3-5324 200 25-150 12-488 8-2668 6-1564 4-8901 4-0459 3-4430 190 24-601 12-206 8-0743 6-0085 4-7690 3-9427 3-3524 180 24-050 11-923 7-8805 5-8592 4-6465 3-8380 3-2606 170 23-497 11-638 7-6850 5-7083 4-5224 | 3-7317 3-1670 160 22-944 11-352 7-4878 5-5558 4-3966 3-6238 3-0718 150 22-388 11-063 7-2886 | 5-4012 4-2687 3-5138 2-9745 140 21-829 10-773 7-0871 5-2443 4-1386 3-4016 2-8751 130 21-268 10-479 6-8832 5-0850 4-0061 3-2869 2-7731 120 20-705 10-183 6-6762 4-9227 3-8706 3-1791 2-6681 110 20-138 9-8840 6-4661 4-7572 3-7318 3-0482 2-5599 100 19-567 9-5809 6-2522 4-5878 3-5892 2-9235 2-4480 in] TABLES OF THE PROPERTIES OF STEAM 373 of Steam in any Dry State. Temp. Cent. Pressure in pounds per sq. inch 160 180 200 250 300 350 400 400 4-4609 3-9605 3-5601 2-8395 2-3591 2-0159 1-7586 350 4-1091 3-6459 3-2753 2-6081 2-1635 1-8458 1-6075 300 3-7501 3-3240 2-9831 2-3696 1-9605 6683 1-4492 290 3-6771 3-2585 2-9235 2-3206 1-9187 6317 1-4164 280 3-6036 3-1924 2-8634 2-2712 1-8765 5945 1-3830 270 3-5295 3-1258 2-8027 2-2212 1-8337 5568 1-3491 260 3-4550 3-0587 2-7416 2-1709 1-7894 5186 1-3148 250 3-3797 2-9908 2-6797 2-1196 1-7463 4796 1-2796 240 3-3037 2-9222 2-6171 2-0677 1-7015 4399 1-2437 230 3-2269 2-8529 2-5536 2-0149 6559 3994 1-2071 220 3-1492 2-7826 2-4893 1-9614 6094 1-3580 1-1695 210 3-0706 2-7114 2-4241 1-9068 5620 1-3158 1-1310 200 2-9908 2-6390 2-3576 | 1-8511 5135 1-2723 1-0914 190 2-9097 2-5655 | 2-2900 1-7941 4637 1-2276 1-0505 180 2-8274 2-4906 2-2210 1-7360 4126 1-1816 1-0083 170 2-7434 2-4140 2-1504 1-6760 3598 1-1339 0-9645 160 2-6578 2-3358 2-0782 1-6145 1-3054 1-0846 0-9190 150 2-5701 2-2555 2-0039 1-5508 1-2489 1-0332 0-8714 140 2-4802 2-1731 1-9273 1-4851 1-1902 0-9796 0-8217 130 2-3878 2-0881 1-8483 1-4167 1-1290 0-9235 0-7694 120 2-2924 2-0001 1-7663 1-3454 1-0649 0-8645 0-7141 110 2-1937 1-9089 1-6810 1-2708 0-9975 0-8022 0-6557 100 2-0913 1-8140 1-5920 1-1926 0-9263 0-7461 0-5934 374 THERMODYNAMICS [APP, TABLE D. Total Heat /, in lb. -calories per lb., Temp. Cent. Pressure in pounds per sq. inch 20 40 60 80 100 120 140 400 784-70 784-20 783-71 783-22 782-73 782-23 781-74 350 760-68 760-04 759-39 758-74 758-10 757-45 756-81 300 736-61 735-74 734-88 734-01 733-15 732-28 731-42 290 731-78 730-86 729-94 729-02 728-10 727-18 726-26 280 726-95 725-97 724-99 724-02 723-04 722-06 721-08 270 722-11 721-07 720-03 718-99 717-95 716-91 715-87 260 717-27 716-16 715-06 713-95 712-84 711-73 710-62 250 712-43 711-24 710-06 708-87 707-69 706-51 705-32 240 707-57 706-31 705-04 703-78 702-51 70125 699-99 230 702-71 701-36 700-01 698-66 697-30 695-95 694-60 220 697-85 696-40 694-95 693-50 692-05 690-60 689-16 210 692-97 691-42 689-86 688-31 686-76 685-20 683-65 200 688-08 68642 684-75 683-08 681-41 679-74 678-08 190 683-19 681-39 679-60 677-81 676-01 674-22 672-43 180 678-28 676-35 674-41 672-48 670-55 668-62 666-69 170 673-35 671-27 669-19 667-10 665-02 662-94 660-85 160 668-41 666-16 663-91 661-66 659-41 657-16 654-91 150 140 663-46 658-48 661-02 655-84 658-59 656-15 650-56 653-71 647-92 651-28 645-28 648-84 642-64 653-20 130 653-48 650-61 647-75 644-88 642-01 639-14 636-27 120 648-46 645-34 642-21 639-09 635-97 632-85 629-73 110 643-40 640-00 636-59 633-19 629-79 626-38 622-98 100 638-31 634-59 630-87 627-15 623-43 619-71 615-99 Ill TABLES OF THE PROPERTIES OF STEAM 375 of Steam in any Dry State Temp. Cent. Pressure in pounds per sq. inch 160 180 200 250 300 350 400 400 781-25 780-76 780-26 779-03 777-80 776-57 775-33 350 756-16 755-51 754-87 753-25 751-63 750-02 748-40 300 730-55 729-69 728-82 726-66 724-50 722-34 720-17 290 725-34 724-43 723-51 721-21 718-91 716-61 714-31 280 720-11 719-13 718-15 715-71 713-26 710-82 708-37 270 714-83 71379 712-75 710-14 707-54 704-94 702-34 260 709-51 708-40 707-29 704-52 701-75 698-98 696-21 250 704-14 702-95 701-77 698-81 695-85 692-89 689-93 240 698-72 697-46 696-19 693-03 689-87 686-70 683-54 230 693-24 691-89 690-54 687-16 683-77 680-39 677-01 220 687-71 686-26 684-81 681-19 677-56 | 673-94 670-32 210 682-10 680-54 678-99 675-10 671-22 667-34 663-45 200 676-41 674-74 673-07 668-90 664-73 660-56 656-39 190 670-63 668-84 | 667-04 662-56 658-08 653-59 649-11 180 664-76 662-83 660-90 656-07 651-24 646-41 641-58 170 658-77 656-69 654-60 649-40 644-19 638-98 633-77 160 652-66 650-41 648-16 642-53 636-91 631-28 625-66 150 646-41 643-97 641-53 635-44 629-35 623-26 617-17 140 640-00 637-36 634-72 628-12 : 621-52 614-92 608-32 130 633-41 630-54 627-67 620-50 613-33 606-16 598-99 120 626-61 623-48 620-36 612-56 604-75 596-95 589-15 110 619-57 616-17 612-77 604-26 595-75 587-24 578-73 100 612-27 608-55 604-83 595-52 586-22 576-92 1 567-62 376 THERMODYNAMICS [APP. TABLE E. Entropy of Steam Temp. Pressure in pounds per sq. inch Cent. 20 40 60 80 100 120 140 400 2-0100 .9331 1-8878 1-8556 1-8304 1-8097 1-7921 350 1-9729 8958 1-8503 1-8178 1-7924 1-7714 1-7536 300 1-9326 8551 1-8092 7764 1-7506 1-7293 1-7111 290 1-9242 8465 1-8006 7676 1-7417 1-7203 1-7021 280 1-9155 8378 1-7917 7586 1-7327 1-7112 1-6928 270 1-9067 8288 1-7826 7494 7234 1-7018 1-6833 260 1-8977 1-8197 1-7734 7401 7139 1-6921 1-6735 250 1-8885 1-8104 1-7639 7305 7041 1-6822 1-6635 240 1-8791 1-8009 1-7543 7206 6941 1-6721 1-6532 230 1-8696 1-7911 1-7443 7105 6839 1-6617 1-6426 220 1-8598 1-7812 1-7342 7002 1-6733 1-6509 1-6316 210 1-8498 1-7709 1-7238 1-6896 1-6625 1-6398 1-6204 200 1-8396 1-7605 1-7131 1-6786 1-6513 1-6284 1-6087 190 1-8291 1-7497 1-7021 1-6673 1-6397 1-6166 i 1-5966 180 1-8184 7387 1-6907 1-6557 1-6278 1-6044 i 1-5841 170 1-8074 7274 1-6791 1-6437 1-6155 1-5917 1-5711 160 1-7961 7157 1-6670 1-6313 1-6027 1-5785 1-5575 150 140 1-7845 1-7726 7037 6913 1-6546 1-6184 1-6050 1-5894 1-5755 1-5648 1-5504 1-5433 1-5285 6417 130 1-7604 1-6785 -6283 1-5911 1-5610 1-5354 1-5129 120 1-7478 1-6652 6144 1-5766 1-5458 1-5196 1-4964 110 1-7347 1-6515 5999 1-5613 1-5299 1-5029 1-4790 100 1-7213 1-6372 5848 1-5454 1-5131 1-4852 1-4605 Ill] TABLES OF THE PROPERTIES OF STEAM 377 in any Dry State, Temp. Cent. Pressure in pounds per sq. inch 160 180 200 250 300 350 400 400 1-7768 1-7633 7511 1-7250 1-7034 1-6849 6687 350 1-7381 1-7243 7118 1-6852 1-6630 1-6439 6271 300 6952 1-6810 6682 1-6406 1-6175 1-5976 5798 290 6861 1-6718 6589 1-6311 1-6077 1-5873 5696 280 6767 1-6623 6493 1-6212 1-5976 1-5771 5589 270 6670 1-6525 6394 1-6111 1-5872 1-5664 5479 260 6572 1-6425 6293 1-6006 1-5764 1-5553 5365 250 1-6470 1-6322 6188 1-5898 1-5652 1-5438 5246 240 1-6365 1-6216 6081 1-5786 1-5537 1-5319 5123 230 1-6257 1-6106 5969 1-5671 1-5417 1-5194 4995 220 1-6146 1-5993 5854 1-5551 1-5292 1-5065 1-4860 210 1-6031 1-5876 1-5735 1-5426 1-5162 1-4929 1-4719 200 5912 1-5755 1-5611 1-5296 1-5026 1-4788 1-4571 190 5789 1-5629J 1-5482 5161 1-4884 1-4639 1-4416 180 5661 ] -5497 1-5348 5019 1-4735 1-4482 1-4251 170 5527 1-5360 1-5208 4870 ] -4577 1-4316 1-4077 160 5387 1-5217 1-5060 4713 1-4411 1-4140 1-3891 150 5241 1-5067 1-4906 4548 1-4235 1-3953 1-3693 140 5088 1-4908 1-4743 4373 1-4047 1-3753 1-3482 130 4926 1-4741 1-4570 4186 1-3847 1-3539 1-3253 120 1-4755 1-4564 1-4386 3986 1-3631 1-3307 1-3006 110 1-4574 1-4375 1-4190 3772 1-3399 1-3056 1-2737 100 1-4381 1-4174 1-3980 3541 1-3147 1-2783 1-2443 378 THERMODYNAMICS [APP, in I I I I e I CJ 1 00 CO *O O CO CO F-H C^ G^l CO "^ IO CO OO ^^ C*Q IO IO IO IO 1O IO >O IO CO CO O XO O io l> CO >o o IOOO>-HOO5 -^Ol> O5O5OOO i i O IO GO QO O^ O^ O^ l> IO IO 00 IO IO IO IO GOGOGOGOO5 O5 r-H COCOOiCOt- COr-l O5 C5 O Q i I -H (N O O i I i i G<1 (NCO^COI> O5G<|-^OiT^ GOOOGOGOOO QOOOOOOOOO OOO5O5O5O ooooo oooo o OO^^ OOOOOO o O OO CO OO 33 ooooo 00 CO tj< W O TH TH T-( TH -H INDEX Absolute zero, 12, 35 Absorption bands in spectrum, 361 machinesforrefrigeration, 160 of radiation by gases, 263, 361 Adiabatic elasticity, 286 expansion, 21 effect on dryness, 81 of a fluid, 81 of a perfect gas, 22 of steam, 328 Adiathermal process, 129 After- burning, 244, 264 Aggregation, states of, 59 Air, isothermals of, 302 separation of the constituents of, 180 standard, 229 Air-engine, regenerative, 53 Joule's, 55 Air machines for refrigeration, 158 Aitken, 85, 344 Amagat, E. H., 299, 302, 310, 312 Amagat's isothermals, 299 according to Cal- lendar's equation of state, 323 Ammonia absorption machine, 161 data for, 145 use of in refrigeration, 137 Andrews, 310 Atkinson, 256 Atmosphere, pressure of standard, 8 Atomic structure, 358 Avogadro's law, 235, 355 Baly's curves, 185 Beau de Rochas, 226 Bell-Coleman refrigerating machine, 158 Bjerrum, N., 362 Bohr, N., 358 Boiling, 349 Boundary curves, 92 Boyle's law, 13, 355 Brink worth, 329 British Association Committee on gaseous explosions, 242, 244, 245, 251, 257, 265 British Thermal Unit, 9 Bubbles, equilibrium of, 350 Buckingham, E., 277 Callendar, H. L., 12, 70, 72, 73, 76, 83, 102, 108, 127, 201, 257, 298, 315, 329, 335, 338, 350, 367 Callendar and Nicholson, 207 Callendar's characteristic equation, 318 steam tables, 63, 340, 367 Calory, mean, 331 Carbonic acid, critical temperature of, 80 data for, 145 isothermal for, according to Clausius, 317 isothermals of, 300 - molecular vibration in, 362 specific heat of, 245, 364 use of, in refrigeration, 138 Carnot's cycle, conservation of entropy in, 45 of operations, 27 reversed in refrigerating machine, 138 - with a perfect gas for working substance, 30 with steam for working substance, 88 Cascade method of liquefying gases, 169 Characteristic equation, 290 Callendar's, 318 Clausius', 315 Dieterici's, 317 of a perfect gas, 290 Van der Waals', 306 Charles' law, 13 Chart of entropy and temperature, 118 total heat, 121 Charts of properties of fluids, slopes of lines in, 280 Chemical contraction, 236 Clapeyron's equation, 115, 283 application of, to changes of state, 116 Claude's apparatus for complete rectifi- cation, 188 liquefaction of air, 178 Clausius, 50, 315 Clausius' characteristic equation, 315 Clement and Desormes, 294 Clerk, Sir Dugald, 226, 229, 244, 251,259 Clerk's gas-engine, 226 Co -aggregation volume, 319 in steam, 329 Coblentz, W. W., 362 Coefficient of performance, 2, 134, 136 Coffey still, 181 Collected thermodynamic relations, 287 Combustion of gases, 235 Compound turbine, 216, 220 380 INDEX Compressibility' of a fluid, 286 Compression, advantage of, in gas- engines, 231 Condensation in expanding steam, 207 of water-vapour in air, 67 Conservation of entropy in Carnot's cycle, 45 in refriger- ating process, 135 Continuity of state, 304 Convergent-divergent nozzle, 194 Cooling effect of throttling, 127, 276, 296, 314, 318 Corresponding states, 311, 317 Co-volume, 307, 319 in steam, 328 Critical point, 80, 304, 309 in steam, 320 on the 70 chart, 123 T chart, 145 Olszewski, 277 Organs of a heat-engine, 96 Otto, 226 Oxygen, separation of, 181 Parsons' turbine, 216, 221 Partial pressures, 65 Peake, A. H., 329 Perfect and imperfect gases, 356 differential, 270 engine, criterion of, 29 engine using regenerator, 50 gas, 14, 238, 325 characteristic equation of, 290 expansion of, 22, 24 steam-engine, 88 efficiency of, 90 entropy-tempera- ture diagram of, 91 Petrol-engine, 225 Phase, change of, 283 Phases of a substance, 103 Pier, 245 Planck's constant, 363 formula for energy of vibration in molecules, 362 Polyatomic gases, specific heats, 364 molecules, 359 Porter, 318 Pound-calory, 9 Pressure, of a gas, explanation of, on the molecular theory, 352 unit of, 8 -volume diagram, 6 Preston, 295 Radiation in explosions, 260 from flames, 262 Rankine, 63, 70 cycle, 98 efficiency of, 102 for steam in any state, 104 reversibility of, 109 Rateau steam turbine, 220 Rayleigh, Lord, 294 Re-action turbine, 221 Rectification, 181 Rectification, complete, 188 Reducing- valve, 74, 126 Refrigerating machine as a means of warming, 168 coefficient of per- formance of, 2 definition of, 1 Refrigeration process, 133 Regenerative air-engine, 53 method of producing ex- treme cold, 171 Regenerator, Stirling's, 52 Regnault, 14, 62, 241, 310 Re-heat factor, 218 Report of Refrigeration Research Com- mittee, 145 Reversibility, 20 conditions of, 37 the criterion of perfection, 29 Reversible engine, efficiency of, 34 receiving heat at various temperatures, 42 heat-engine, 27 refrigerating machine, 134 Reynolds, Osborne, 198 Rontgen, 295 Rutherford, Sir E., 358 Saturated steam, 61, 62, 64, 368, 370 relation of pressure to temperature in, 336 vapour, 59 Saturation due to curvature, 345 of air with water- vapour, 66 Scale of temperature, thermodynamic, 12,39 Scheel, K., 366 Seay process, 163 Second law of thermodynamics, 26 Shields, M. C., 366 Simple turbine, 215, 220 Specification of state of any fluid, 77, 267 Specific heat of water, 67 variation of, with temp- erature, 243 heats constantinaperfectgas, 19 expressions for, 272 in Callendar's equation of state, 324 measurement of values of, 241 of a gas, 17, 239, 357 of a gas, influence of molecular vibration, 363 of gases on the molecular theory, 357 of hydrogen at very low temperatures, 365 ratio of, 23, 275, 293, 357 Stage efficiency in turbines, 218 State, specification of, 77 States of aggregation, 59 INDEX 383 Steam, Calendar's formulas for, 327, 336, 338 collected formulas for, 338 critical pressure of, 320 critical temperature of, 80 entropy of, 376 formation of, under constant pressure, 60, 67 jets, supersaturation in, 203 properties of, 62, 367 saturated, 61, 62, 64 properties of, 368 specific heat of, 378 superheated, 61 tables, 340, 367 total heat of, 374 turbine, performance of, 222 turbines, compound, 216 simple, 215 types of, 220 volume of, 372 Steam-engine working without com- pression, 94 Stirling, R., 52 Stodola, 203 Suction temperature in a gas-engine, 257 Sudden expansion, effect of, 84 Sulphurous acid, 138 Supercooling, 85, 206 Superheated vapour, 59 adiabatic expan- sion of, 83 total heat of, 73 water, 349 Supersaturation, 84 of steam discharged from a nozzle, 201 Surface tension, 342 Swann, 242 Tables of properties of steam, 367 Temperature, scales of, 10 of inversion of cooling effect, 277 thermodynamic scale of, 12,39 Temperatures in a gas-engine cylinder, 255 Tension of liquid film, 343 Thermal unit used in Callendar's tables, 331 units, 9 Thermodynamical correction of the gas thermometer, 298 Thermodynamic potentials, 103 relations, 266 collected,287 scale of temperature, 12, 39 surface, 268 Thermodynamics, science of, 1 Thomson, James, 117, 304, 316 Thomson, James, his ideal isothermal, 305, 317 Thomson, Sir J. J., 345 Thomson, W., Lord Kelvin, 16, 39, 50, 117, 345 Throttling calorimeter, 128 cooling effect of, 127, 276, 296 process, 74 Total heat, constancy of. in a throttling process, 74 of a fluid, 70, 72 - of water, 334 Triatomic gas, 261 molecules, 359 Triple point, 118 Turbines, types of, 220 Turbulence, effect of, 259 Two-stroke cycle, 226 Unit of force, 8 heat, 9 pressure, 8 work, 8 Unresisted expansion, 279 Van der Waals' characteristic equation, 306 theory of corresponding states, 311 Vapour- compression refrigerating mach- ine, 138 Vapour-pressure over a curved surface, 345 Velocity of mean square, 353, 356 Vibration of atoms in molecules, 360 Volumetric specific heats, 239 Water at saturation pressure, properties of, 369 specific heat of, 67, 335 superheating of, 349 total heat and entropy of, 334 Water- vapour refrigerating machine, 155 Watt, James, 98 Watt's indicator, 7 Weight, variation of, with latitude, 8 Wet steam, 76 Wilson, C. T. R., 85, 200, 207, 344, 349 Wimperis, H. E., 250 Witkowski, 303 Work done by change of volume, 6 - done in adiabatic expansion, 86 unit of, 8 Working substance, 2 cycle of operations 'of, 4 in refrigerating pro- cess, 137 Young, Sydney, 313 Zeuner, 63, 82 Zollv steam turbine, 220 CAMBRIDGE : PRINTED BY J. B. PEACE, M.A., AT THE UNIVERSITY PRESS UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. This book is DUE on the last date stamped below. 2lJan'49[R 3Jan'50HJG REC'D LD AN5 '65-9P ) 21-100m-9,'48(B399sl6)476 VC 12758 J347G3 THE UNIVERSITY OF CALIFORNIA LIBRARY