LIBRARY 
 
 OF THE 
 
 UNIVERSITY OF CALIFORNIA. 
 Cla&s 
 

 
ENTKOPY; 
 
 OK, 
 
 rHERMODYNAMICS FROM AN ENGINEER'S 
 STANDPOINT. 
 
ENTRO PY; 
 
 OR. 
 
 THERMODYNAMICS FROM AN ENGINEER'S 
 STANDPOINT, 
 
 AND THE 
 
 REVERSIBILITY OF THERMODYNAMICS, 
 
 BY 
 
 JAMES SWINBUKNE. 
 
 or THJ* 
 UNIVERSITY 
 
 NEW YORK: 
 E. P. DUTTON & CO. 
 
PEEFACE. 
 
 MY reason for adding another little book on 
 thermodynamics to those in existence is that it is 
 wanted. As far as I am aware there is not any 
 work on the steam- or gas-engine in this country 
 that gives a correct definition of entropy. I do 
 not know about foreign text-books on this subject. 
 This is rather a serious indictment; and when it 
 is added that most treatises on Physics, English 
 and foreign, contain incorrect definitions of entropy, 
 and that even some of those specially on thermo- 
 dynamics are at fault, especially in classing entropy 
 as a factor of heat, there is no need for any apology, 
 at least, for the appearance of a book like this. 
 
 Instead of treating the subject in the orthodox 
 way, however, I have ventured on a new method 
 of explaining it. The reasons for adopting this 
 method are given in the paper, " The Rever- 
 sibility of Thermodynamics," at the end of the 
 book. This paper was written to be criticised by 
 
 ) O f\ f* ,- 
 
 Ioub77 
 
vi PREFACE. 
 
 those already well up in the subject, and was read 
 before Section A. of the British Association this 
 year, 1903. Though the paper was circulated in 
 proof there was no relevant discussion. It gives 
 a sort of syllabus of the method of exposition 
 adopted in the body of the book, and the reasons 
 for its adoption. 
 
 The definition of heat, which makes what is 
 usually denoted by U the heat of the body, is an 
 entirely new departure, as far as I know, though 
 it is quite consistent with the treatment of chemical 
 energy in thermodynamics. If it were not it would 
 be wrong ; but its being consistent with correct 
 usage does not mean that it is not new. Another 
 new development is in connection with the entropy 
 temperature diagram. It is assumed in treatises 
 on this subject that the entropy temperature dia- 
 gram is of the same area as the indicator diagram. 
 The essence of the treatment here followed is that 
 it is not, and that the difference is an indication 
 of the badness of the engine. I am not the first, 
 however, to call attention to the inaccuracies 
 current about the entropy temperature diagram. 
 Marchis (" Comptes Eendus," 132, p. 671) has 
 criticised the ordinary error, and the late Bryan 
 Donkin, in the preface to his translation of 
 
PKEFACE. vii 
 
 Boulvin's exhaustive book, " The Entropy Dia- 
 gram," says it " ought only to be applied to 
 reversible cycles"; but he is not at all clear. 
 
 The main portion of this book appeared as a 
 series of articles in Engineering, August 28th , 
 1903, and the following issues. I have to thank 
 the proprietors of Engineering for permission to 
 republish the articles. 
 
 JAMES SWINBUKNE. 
 
 82, VICTORIA STREET, 
 
 WESTMINSTER, 
 October, 1903. 
 
CONTENTS. 
 
 CHAPTER I. 
 
 INTRODUCTION. 
 
 PAGE 
 
 Origin of Common Error as to Entropy Personal 
 
 Irreversible Changes not Considered Properly . 1 
 
 CHAPTER II. 
 
 ENTROPY. 
 
 Work, Heat, and Waste "What Entropy is Not, and 
 What it Is Three Kinds of Perpetual Motion 
 Three Laws of Thermodynamics Reversible 
 Cycle Entropy as a Factor of Incurred Waste 
 Reversibility Localisation of Entropy More 
 Convenient Definition of the Entropy of a Body . 6 
 
 CHAPTER III. 
 
 THE TEMPERATURE ENTROPY DIAGRAM. 
 
 Irreversible or Free Expansion Difference of Areas 
 of 6 <f> and p v Diagrams Ungained Work Dia- 
 gram for Steam Real Factors of Heat . . .73 
 
x CONTENTS. 
 
 CHAPTER IV. 
 
 CONDUCTION. 
 
 PAGE 
 
 Movement of Entropy Conduction of Heat Volume 
 Growth of Entropy in Conduction Unit of En- 
 tropy Physical Meaning of Entropy Conclusion 94 
 
 APPENDIX. 
 
 THE REVERSIBILITY OF THERMODYNAMICS. 
 
 Present or Orthodox Treatment Skeleton of Proposed 
 Treatment of Thermodynamics Waste Reversi- 
 bility Equilibrium Stabilities Conduction 
 Units Conclusion. 113 
 
ENTROPY; 
 
 OR, 
 
 THERMODYNAMICS FROM AN ENGINEER'S 
 STANDPOINT, 
 
 CHAPTER I. 
 
 INTRODUCTION. 
 
 IT may be as well to begin by stating that I have 
 no peculiar theories as regards entropy, and that I 
 am merely trying to explain, in as lucid a way as I 
 can, the underlying principle, using the term as it 
 is employed in the thermodynamics of reversible 
 and irreversible changes. The discussion of entropy 
 involves writing a sort of introduction to thermo- 
 dynamics, so it may be worth while to do so from 
 an engineer's point of view. The treatment of the 
 subject may thus be novel, but novelty is not 
 important ; the real aim is to make the theory 
 clear. 
 
 Books on thermodynamics do not usually give 
 any idea of any sort of physical meaning of entropy, 
 and only define it so that its magnitude can be 
 
 E. B 
 
2 INTRODUCTION. 
 
 calculated. Many otherwise scientific men, includ- 
 ing some engineers, have, unfortunately, got hold 
 of an inaccurate definition of entropy, derived from 
 books which treat exclusively, or almost exclusively, 
 of reversible processes ; and as no properly consti- 
 tuted engineer can use mathematical symbols with- 
 out having an underlying physical idea, a physical 
 interpretation has been forced on to the inaccurate 
 definition, and entropy has been defined as " heat- 
 weight," or as the quantity factor of heat-energy, 
 as if temperature corresponded with head or differ- 
 ence of level, and entropy with weight of water in 
 hydraulics, so that the product of the two, in the 
 case of constant level or temperature, gives energy 
 or heat. Similarly, the entropy temperature is 
 nearl} 7 always called a e< heat " diagram, whereas its 
 area is not proportional to any particular quantity 
 of existing heat. 
 
 In spite of the inaccuracies of thought and state- 
 ment that are, unfortunately, very common, the 
 < diagram is of the utmost value to steam and 
 gas engineers, and my object is to try to get clearer 
 ideas as to the real meaning of the diagram. 
 Neither is the <#> diagram as commonly used 
 seriously wrong. Most writers, after defining 
 entropy incorrectly, unconsciously depart from their 
 definitions in dealing with the <j> diagram, so that 
 the two errors approximately cancel out. There is, 
 however, still an error in the ordinary treatment of 
 
PERSONAL. 3 
 
 this diagram. But perhaps the chief evil is the 
 confusion of mind into which readers are led by 
 inaccuracy and inconsistency on the part of writers 
 on the steam-engine. 
 
 Personal. As a young man I tried to read ther- 
 modynamics, hut I always came up against entropy 
 as a brick wall that stopped my further progress. 
 I found the ordinary mathematical explanation, of 
 course, but no sort of physical idea underlying it. 
 No author seemed even to try to give any physical 
 idea. Having in those days great respect for text- 
 books, I concluded that the physical meaning must 
 be so obvious that it needs no explanation, and that 
 I was especially stupid on that particular subject. 
 (Everyone who studies by himself knows he is par- 
 ticularly stupid in certain directions, and is con- 
 stantly realising new limitations.) After a few 
 3^ears I would tackle the subject again, and always 
 I was brought up dead by the idea of entropy. I 
 asked people, but I never met any one who could 
 tell me, and I met one, an engineer, who admitted 
 he did not know. Not only could I get no physical 
 idea of entropy, but the definition of entropy, and 
 the statements about it, did not make sense as 
 soon as one tried to understand irreversible changes. 
 Later on, instead of making the common mistake 
 that elementary books are easy to understand, I 
 got into the stud} 7 of irreversible thermodynamics 
 by the road of physical chemistry, and found that 
 
 B2 
 
4 INTRODUCTION. 
 
 my previous troubles were due to inaccurate defini- 
 tions and faulty analogies on the part of writers 
 who had an incomplete grasp of thermodynamics. 
 Having once got accurate and consistent definitions, 
 it is not so difficult to get some sort of physical 
 idea of entropy. I hope I may be pardoned these 
 personal reminiscences ; they are to show that when 
 I write to correct errors in scientific text-books and 
 papers on engineering, written by able men, I do 
 not do so in any spirit of superiority. 
 
 Irreversible Changes not Considered. The source 
 of the whole trouble is the obscurity of the mathe- 
 matical writer, especially when he is not of the first 
 rank. The early books on thermodynamics, and most 
 of the treatises specially on thermodynamics of the 
 present day, are so anxious to prove the second law 
 with the help of the ideal reversible cycle, and to 
 explain Kelvin's absolute thermometer scale, that 
 they discuss reversible changes almost exclusively, 
 and either neglect irreversible changes, or treat only 
 one case that of conduction of heat, and that in a 
 meagre way so that the reader gets quite a wrong 
 notion of the subject. Worse than this, the writer 
 on thermodynamics makes statements which are 
 true of reversible but untrue of irreversible changes, 
 as if they were quite general. The reader who is 
 already acquainted with his subject can mentally 
 insert " in the ideal case of reversibility only," 
 every here and there, when he studies thernio- 
 
IRREVERSIBLE CHANGES. 5 
 
 dynamics ; but that there is great confusion on the 
 subject, especially of entropy, is almost entirely the 
 fault of the mathematical writer on thermodynamics. 
 He makes general statements which would only be 
 true in an ideal and impossible case, and makes no 
 attempt to give any physical ideas of the quantities 
 his symbols represent. To an address to the 
 Institution of Electrical Engineers I added a foot- 
 note calling attention to the fact that most engineer- 
 ing and physical, as opposed to thermodynamical, 
 writers use a definition of entropy that is numerically 
 accurate in the hypothetical case of reversibility 
 only, and is, in fact, always wrong. I thought 
 merely calling attention to the error and its source 
 would correct it at once, but instead of that it led 
 to an extraordinary correspondence, which aston- 
 ished me beyond measure by showing that many 
 able men have the most confused notions on the 
 subject. I wrote an article on Entropy in the 
 Electrical Review of January 9th, 1903, attempting 
 to give a general explanation so as to cover chemistry 
 as well as physics and engineering ; but the article 
 was written under pressure, and was therefore short, 
 so perhaps it may do no harm to write more fully, 
 dealing with the subject from an engineer's point 
 of view, and discussing the meaning of the </> 
 diagram. In this way but little of the same ground 
 is covered, and it is always well to approach a 
 subject from two different points of view. 
 
CHAPTER II. 
 
 ENTROPY. 
 
 Work, Heat, and Waste. Energy is indestruc- 
 tible ; but it exists in many forms, such as potential, 
 kinetic, electric, magnetic energy on one hand, and 
 heat on the other. All of these, except heat, are 
 interchangeable, with a slight change into heat, 
 which can be diminished, in fact, till very small, 
 and in idea to nothing. All can be changed into 
 heat ; but heat can only be partly changed into the 
 other forms. For want of a better word, and for 
 simplicity, as " energy " includes heat, we m:y call 
 the high-grade energy, such as potential, kinetic, 
 electric, magnetic, &c., " work," and low-grade 
 "heat." We need not concern ourselves with such 
 things as energy of radiation or chemical energy 
 here. It is the aim of the engineer, when dealing 
 with steam, gas, oil, or compressing engines, to 
 avoid converting or degrading energy into heat. 
 The term " dissipation of energy " is generally used 
 to denote degradation of work into heat. As Ber- 
 trand points out, the energy is not dissipated, as 
 it is not annihilated it is degraded, and we may 
 therefore adopt his word, though dissipated does 
 
WOKK, HEAT, AND WASTE. 7 
 
 \ 
 
 not really mean annihilated. When work is con- 
 verted into heat, it may be called "degraded"; 
 but generally some of the heat can be elevated back 
 into work if the rest is given out at a lower tempera- 
 ture. The portion of the heat that must at least 
 be finally given out as heat at the lowest available 
 temperature is of no use : it is waste. I will there- 
 fore call the part of the heat that is eventually and 
 unavoidably produced or left at the lowest available 
 temperature the "waste." 
 
 The idea of degradation of energy- that is, con- 
 version of work into heat is quite familiar and 
 clear. Thus all movement against friction means 
 degradation. But there may be a change during 
 which no work is actually converted into heat, but 
 which cannot be unmade, so as to bring the " work- 
 ing substance " back to its original state without 
 involving degradation. Such a change thus " lets 
 us in " for some degradation at some future date, 
 so that the degradation is, as it were, a liability 
 incurred, which must be eventually liquidated. 
 
 Wliat Entropy is Not, and What it Is. In order 
 to get a clear idea as to what entropy is, it is best 
 to begin by stating clearly what it is not. It is not 
 any form of energy, nor a quantity of the dimen- 
 sions of energy.* It is not heat- weight. It is not 
 
 * The misapplication of the term " entropy " by Maxwell 
 and others, with no confusion of idea or inaccuracy of 
 thought, has been corrected long ago, and may be dismissed. 
 
8 ENTROPY, 
 
 equal to fd H/0, as in fact it is essentially greater. 
 It is not heat taken in by the substance divided by 
 the absolute temperature ; it is always greater. 
 It is not a factor of heat. It is not, in fact, a 
 function of the heat taken in by the substance. 
 The temperature-entrop} r diagram is not a heat 
 diagram at all. Its area does not represent the 
 heat of the substance, nor the heat taken in by the 
 substance it is necessarily greater nor the energy 
 of the substance, nor the energy taken in by the 
 substance. The statements here contradicted, with 
 others equally misleading, are continually made, 
 not in advanced treatises on thermodynamics, but 
 in books on mathematical physics, and treatises on 
 the steam-engine. If anyone who has not a clear 
 idea of entropy will first banish all preconceived 
 notions as to heat-weight, heat diagrams, or heat 
 divided by temperature, and will think of entropy 
 as the measure of waste actually effected, or inevit- 
 ably to be effected subsequently, he will have a clear 
 idea. Entropy may be defined thus : Increase of 
 entropy is a quantity which, when multiplied by the 
 lowest available temperature, gives the incurred 
 waste. In other words, the increase of entropy 
 multiplied by the lowest temperature available gives 
 the energy that either has been alread} T irrevocably 
 degraded into heat during the change in question, 
 or must, at least, be degraded into heat in bringing 
 the working substance back to the standard state. 
 
WHAT IT IS, ANDT'WT. 9 
 
 The last part of this definition must not be omitted. 
 The entropy of a body may increase while the 
 entropy of another body that gives it heat decreases 
 equally. In this case there is no increase of entropy 
 of the whole system and no new incurred waste. 
 By increase of entropy here is meant the increase 
 of entropy as a whole in any isolated system, or 
 region, which has no exchange of energy with any- 
 thing outside it during the increase of entropy. 
 When the increase of entropy of one body is exactly 
 balanced by the corresponding decrease in another, 
 the increase of entropy of the body is called " com- 
 pensated," and there is no real increase of entropy 
 and no further incurred waste beyond that due to 
 the original production of the entropy. At first 
 sight anyone may say, " if the object is to measure 
 the work irrevocably wasted as heat, why should 
 we not have a quantity of the same dimensions as 
 energy, which would give us the work or heat right 
 off, instead of a quantity like entropy, which is 
 not energy, and has to be multiplied by a tempera- 
 ture to give the energy wasted?" The answer is, 
 that the practical value of heat depends on the 
 temperature, so that unless the lowest available 
 temperature is known, the energy that at least must 
 be run to waste as heat is not known. Suppose by 
 some process, say, 10,000 units of heat were pro- 
 duced at, say, 1,000 absolute temperature ; it does 
 not in the least follow that 10,000 units of work are 
 
10 ENTROPY. 
 
 wasted as heat, for 1,000 may not be the lowest 
 available temperature. Suppose 500 is, then, as will 
 be shown directly, half this heat can theoretically 
 be converted back again into work, and 10 X 500 
 = 5,000 units of work are really irrevocably degraded 
 into heat or wasted. Again, if 250 is the lowest 
 available temperature, three-quarters of the 10,000 
 units can, theoretically, be got back into work, 
 10 X 250 being wasted. It is clear, therefore, that 
 the total quantity of work wasted cannot be deter- 
 mined unless we know the lowest temperature avail- 
 able. What we can know, however, is the increase 
 of entropy, a quantity which, when multiplied by the 
 lowest available temperature, gives the work wasted. 
 In this case we might call the entropy 10. 
 
 Suppose by some other process a change took 
 place without the actual degradation of work into 
 heat, but of such a nature that to get the substance, 
 for instance, some wire-drawn steam, back into its 
 original state, 10,000 British thermal units would 
 eventually have to be converted into heat ; if this 
 heat is given out into a body at a temperature of 
 1,000, and 500 is available as a lowest tempera- 
 ture, half can again be theoretically saved, and 
 so on. In this case, however, the increase of 
 entropy takes place when the steam is wire-drawn ; 
 its growth is not delayed until the heat is actually 
 produced. 
 
 As will be more fully explained directly, we can 
 
PERPETUAL MOTION. 11 
 
 only discuss increase of entropy, or decrease of 
 entropy of a body. We cannot evaluate the 
 whole entropy of a bod}'. The entropy of a 
 body is therefore measured by comparison with a 
 standard state. For instance, if water at 32 Fahr. 
 is taken as a standard or zero for water and 
 steam, the entropy of steam is really the difference 
 between its entropy and that of the same weight of 
 water at 32 Fahr. 
 
 Three Kinds of Perpetual Motion. We can 
 certainly take it as a fundamental principle that 
 perpetual motion is impossible. The original idea 
 of perpetual motion was a mechanism that never 
 stopped. It was not necessarily a mechanism that 
 created energy, or gave out energy. We may also 
 take it that energy is conserved, or invariable 
 in quantity. The idea of perpetual motion that 
 involves creation of energy thus contradicts the 
 law of conservation of energy; but a frictionless 
 mechanism, which would run for ever, is merely 
 unrealisable. It is not an absurdit} r : it is a theo- 
 retical abstraction. Every mechanism has friction. 
 But imagine a mechanism in a case through which 
 no energy passes, so that the energy inside the case 
 is constant. Once started, the friction would con- 
 vert work into heat ; but if the mechanism could 
 convert the heat so produced completely into work 
 again, there would be no contradiction of the law 
 of conservation of energy, as the energy is constant. 
 
12 ENTROPY. 
 
 There would be no stoppage by friction, as all tbe 
 work converted into heat would be converted into 
 work again, as kinetic energy of motion. We 
 would thus have a form of perpetual motion, not 
 involving frictionless mechanism, and not involving 
 creation of energy. Does this form of perpetual 
 motion rank with the creation of energy form, in 
 which the mechanism gives out work and creates 
 energy; or is it a theoretical abstraction, which 
 cannot be realised only because of the imperfection 
 of our workmanship, like the frictionless mechanism ? 
 The answer is that it is not a mere theoretical 
 abstraction, but an absurdity of the same order 
 as the energy-creating form of perpetual motion. 
 It has been called " perpetual motion of the second 
 class." It is perpetual motion which does not 
 involve creation of energy, but involves the opposite 
 or complete undoing of the degradation of energy. 
 The friction degrades work into heat. Some of the 
 heat might be converted back into work if the rest 
 is given out at a still lower temperature ; but to 
 make this mechanism work continuously in spite of 
 its friction, all the heat would have to be converted 
 back into work. 
 
 The waste may be defined as the residual heat, 
 of which none can be elevated back into work ; that 
 is to say, it is the heat that must still remain at the 
 lowest available temperature. Perpetual motion of 
 a mechanism with friction would involve the whole 
 
PERPETUAL MOTION. 13 
 
 of the heat produced by friction being elevated back 
 into work. It would thus involve the reduction or 
 diminution of waste once incurred. This abundant 
 experience shows it to be an absurdity of the same 
 order as the creation of energy. We may then 
 say: 
 
 Energy is Conservative or constant in quantity in, 
 the universe, or in any part of it which neither 
 takes in or gives out energy. This is the first law 
 of thermodynamics, as it involves the equivalence 
 of heat and work. 
 
 Waste, once incurred, cannot be diminished in the 
 universe, or in any part of it, which neither takes 
 in nor gives out energy. This is the second law of 
 thermodynamics. 
 
 The relations of perpetual motion to thermo- 
 dynamics may be summed up shortly. 
 
 1. Perpetual motion where an otherwise isolated 
 system gives out energy continually is impossible, 
 and not even approximately realisable. This, with 
 the proviso that an otherwise isolated system cannot 
 absorb energy without increasing its internal energy, 
 is the principle of the conservation of energy, or 
 first law of thermodynamics. 
 
 2. Perpetual motion of an isolated system, such 
 as a mechanism with friction, is impossible, and not 
 approximately realisable. This is the second law 
 of thermodynamics. 
 
 3. Perpetual motion of a frictionless mechanism 
 
14 ENTROPY. 
 
 is unrealisable. This may be called the third law 
 of thermodynamics. 
 
 As every change in existence either involves 
 some friction or some analogous cause of waste, 
 every change or process causes some increase of 
 waste. A change or process which has no friction, 
 and nothing analogous to friction, is an ideal 
 abstraction only, not an absurdity, like the creation 
 of energy, but a useful hypothetical case which can 
 be approximately realised. A frictionless mechanism 
 is the simplest example. 
 
 The notion of degradation and of waste is here 
 introduced, not as being convenient for practical 
 use, but as a good way of getting at the significance 
 of entrop3 T . The difficulty with waste is that you 
 cannot give it a value unless you know the lowest 
 available temperature. 
 
 Reversible Cycle. It is shown in text-books that 
 if an imaginary thermodynamic engine, with a per- 
 fect gas as working substance, working a Carnot 
 cycle without friction or other cause of waste, takes 
 in heat HI from a reservoir or body so large that it 
 can give up this heat without fall of temperature, at 
 temperature O l measured with a perfect gas thermo- 
 meter, and gives out heat H 2 to another reservoir 
 at 2 , the engine returning to its original condition, 
 so that it has performed a cycle, a certain proportion 
 of the heat HI will be elevated into work, and the 
 rest must be rejected as H 2 at the lower temperature 
 
REVERSIBLE CYCLE. 15 
 
 the engine has, by hypothesis, returned to its 
 
 i TT f\ 
 
 \ state. It is also shown that - = - 1 in 
 
 particular case. 
 
 It must be noticed that H here is not the in- 
 1 crease of heat of the gas; it is the decrease of 
 heat of the reservoir ; or, to put it more generally, 
 it is the heat that comes into the gas from the 
 outside. This heat may be converted into work 
 at once, so that the heat of the gas is not in- 
 creased. Again the heat of the gas may be varied 
 without any heat passing in or out through its case 
 or envelope ; for instance, by adiabatic compression 
 or expansion. 
 
 is here the temperature of the envelope the 
 heat H passes through. In reversible changes like 
 this the reservoir and gas are at the same tempera- 
 ture, and so is the envelope, so is the temperature 
 of all three. But if the temperature of the working 
 substance and the body that supplies heat H are 
 not uniform, denotes the temperature of the sur- 
 face of the working substance through which the 
 heat passes. This is important, and laxity of state- 
 ment in books on thermodynamics has given rise to 
 much confusion in this connection. When is 
 mentioned hereafter as the temperature of the 
 body, the body and envelope are at the same 
 temperature. 
 
 The machine in the ideal cycle with perfect gas, 
 
16 
 
 ENTROPY. 
 
 starting from A, in the p v diagram, Fig. 1, where 
 the gas is at temperature 0. and pressure and 
 volume pi Vi takes in heat HI, while it slowly 
 expands still at temperature 6\. When it has 
 expanded to B, it has thus taken in heat HI, and 
 converted it all into external work. The gas has 
 expanded at constant temperature, and Joule's 
 experiment showed that a real gas needs practically 
 no heat to expand it ; a perfect gas is supposed to 
 absorb no heat in expansion. From B to C the 
 
 o "" 
 
 f G H 
 
 .gas expands without taking in any heat, and as it 
 does external work its temperature falls to 2 . From 
 C to D it is compressed, giving out heat H 2 at 
 temperature 2 , and D is chosen so that from it the 
 gas is compressed without giving out any heat until 
 it arrives at A. As the gas has the same internal 
 energy, Ui, at A and B, and the same, U 2 , at C and 
 D, as it merely changes volume, but not tempera- 
 ture, from A to B, and C to D, the difference 
 between the internal energy of A B and C D is the 
 same, whether the change is by the path B C or 
 
REVERSIBLE CYCLE. 
 
 17 
 
 AD. The work done by the gas from B to C is 
 therefore equal to that from A to D. Suppose, 
 therefore, that when the gas has reached B at tem- 
 perature 0i, instead of cooling it by letting it give 
 out its energy as work, we cool it by abstracting 
 heat, without expanding it ; it will then fall to E 
 (Fig. 2), as the internal heat energy of a perfect gas 
 varies directly as the temperature. (To make this 
 
 Jf 
 
 quite accurate I have assumed a perfect gas thermo- 
 meter, whose readings differ very slightly from a 
 mercury instrument.) The height of E will be to 
 that of B as 0% is to 0i. Now compress the gas to 
 F, giving out heat H 2 ; then give back exactly the 
 heat that was abstracted from B to E, and the 
 temperature rises to 6\ at A again. It is clear that 
 the height of F is to that of A also as 2 is to 0i, 
 and that is true of any pressure lines we may draw. 
 Any vertical slice I J K L of the area A B G H is 
 E. c 
 
18 ENTROPY. 
 
 therefore cut by the curve F E, so that the relation 
 of the whole to the lower part is #i/0 2 , and the ratio 
 of the area A B G H or HI to F E G H, or H 2 is 
 as 01/02, and that of A B E F the balance of external 
 work Wz, to A B G H or HI is (^ - 2 )/0i, Garnet's 
 ratio. As A B G H is equal to the work done on 
 expansion, and therefore to HI, and E F H G to 
 the work put in on compression, given out as H 2 , 
 A B E F is equal to the balance of external work 
 done, or to the heat, HI H 2 converted into work. 
 The output of the engine cycle is thus, where W b is 
 the balance of work done : 
 
 = H! - H a = 
 
 and the efficiency 
 
 W* _ O l - 02 
 Hi" #i 
 
 and 
 
 Hi_0i 
 H 2 ~~ 2 ' 
 
 The proof is not conclusive as it stands, because 
 though the heat given out by the gas from B to E 
 is the same as that taken in from F to A, it might 
 be said that the temperatures vary, so that the heat 
 might not be taken in and given out at the same 
 temperature. The lines E B, F A can be divided 
 into a number of equal corresponding parts, each 
 part of E B having a corresponding part of F A. 
 Each pair of parts then represents a corresponding 
 step in temperature due to the same increase or 
 
REVERSIBLE CYCLE. 
 
 19 
 
 decrease of heat. A 
 perfect gas thermometer 
 may be made which reads 
 in pressure of gas at 
 constant volume. If the 
 gas in the cylinder is so 
 used, the lines F A and 
 G B would be divided 
 up into equal parts, 
 and F A would have 
 0i 0% equal parts, and 
 so would E B. The 
 heat capacity of a perfect 
 gas being constant, each 
 of these degrees means 
 the same heat taken in 
 at the same temperature. 
 The proposition can be 
 proved otherwise. Fig. 
 3 shows a double ther- 
 modynamic engine, with 
 two cylinders which may 
 be arranged back to 
 back. The working sub- 
 stance is in the cylinder 
 to the right. First the 
 gas G expands, without 
 
 using the left-hand engine, at constant temperature, 
 the piston moving from the dotted position A, 
 
 c2 
 
20 ENTROPY. 
 
 to B. Then the heat reservoir E is removed from 
 the cover of the cylinder, and the second cylinder S 
 applied. The gas in S is then slowly expanded, 
 taking heat from the working substance, and also 
 supplying heat of its own. All this heat is 
 converted into work. When the substance is thus 
 cooled to 0%, the second cylinder is removed, and a 
 cold reservoir 'applied. The piston is pushed in 
 from B to A, giving out heat H 2 . The other 
 cylinder is then applied, and the gas G heated to 
 its original state by compressing the gas in S to its 
 original state. The double engine has then taken 
 in heat HI, and given out H 2 and work Wz>, and there 
 is no question about the two intermediate steps. 
 The artist has designed these engines without 
 guides or brasses, because hypothesis prevents there 
 being any friction anyhow. 
 
 We have thus one specimen ideal machine which 
 will convert heat from a body at #1 into work, 
 
 rejecting at a lower temperature 2 with an efficiency 
 
 f\ c\ 
 of ] .. - : it is now to be shown that this one 
 
 *i 
 specimen will give us information as to the 
 
 possibilities of all such machines. The machine 
 will obviously work equally well in either direction. 
 If it is reversed, it takes in heat H 2 at the lower 
 temperature, converting it all into work, and takes 
 in more work at 0\ giving out HI. Suppose there 
 were any other possible kind of thermodynamic 
 
REVERSIBLE CYCLE. 21 
 
 engine which could work with a higher efficiency 
 
 - f) 
 than l - 3 , and suppose it took in HI and gave 
 
 out more balance of work than Wj, say W, and 
 therefore rejected less heat than H 2 , say H 3 ; so 
 that HI H 3 = W. Imagine these machines coupled 
 together. The new machine would drive the old 
 one round backwards, as W > W&, and at every turn 
 the hotter reservoir would give as much heat to the 
 new engine as the old engine gave it. But the 
 cooler reservoir would only get H 3 from the new 
 engine every turn, while the old engine would take 
 away H 2 . The difference between W and Wj might 
 then be devoted to overcoming friction, being con- 
 verted into heat, and given to the lower reservoir. 
 We would then have a case of second-class perpetual 
 motion namely, a mechanism, with friction, which 
 could be shut up in a case, and go on for ever. 
 The second law of thermodynamics, which, in the 
 form I have put it, is a denial of the possibility of 
 perpetual motion of a machine which has any 
 friction, thus shows that the maximum ideal or 
 
 r\ _ s\ 
 
 theoretical efficiency of a heat engine is -^~ -. 
 
 v i 
 In the example given of a thermodynamic machine 
 
 with a higher efficiency working the machine of 
 
 f\ A 
 
 efficiency backwards, we supposed the 
 *i 
 balance of work per cycle to go into friction. 
 
 Suppose, however, the friction were reduced a little ; 
 
22 ENTROPY, 
 
 the pair of engines could take all the heat from the 
 cooler and put it into the hotter reservoir. 
 
 Entropy as a Factor of Incurred Waste. Suppose 
 that we had produced the heat Ui* by degrading 
 work into heat, say by friction, the heat being 
 produced in a body at temperature B lt and we want 
 to find the waste incurred. If 0% is the lowest 
 available temperature, and as the thermodynamic 
 engine is frictionless in its widest sense, so that 
 there is no increase of waste incurred by its working, 
 the incurred waste is, when the heat Ui is received 
 by the engine as HI. 
 
 So that altering the lowest available temperature 
 alters the final waste in proportion. So the incurred 
 waste corresponding to the wasteful conversion of a 
 given quantity of work into an equivalent quantity 
 of heat at any given temperature varies inversely as 
 the temperature at which the heat is formed, and 
 directly as the lowest -available temperature. But 
 
 * The symbol U is generally used to denote more than 
 mere sensible heat, but it includes sensible hi-iit, and it is 
 used to denote the increment of heat of the body, while II 
 denotes the heat that passes through the envelope. On 
 page 122 it is urged that U should be regarded as the heat of 
 the substance. This involves a new definition of heat, which, 
 though consistent with orthodox' thermodynamics, is itself 
 unorthodox. It is, therefore, not introduced here. 
 
A FACTOR OF INCURRED WASTE. 2B 
 
 the entropy has been defined as a quantity whose 
 increase, when multiplied by the lowest available 
 temperature, gives the inevitable incurred waste. 
 Therefore, still assuming the frictionless thermo- 
 dynamic engine to be used, the entropy is inde- 
 pendent of the lowest available temperature 
 
 as 2 <I> = H 2 
 
 by definition, 
 
 and TT TT TJ 
 
 H 2 _ HI HI 
 
 ~Z~~ 'a ' IP* 
 
 t/2 "l "1 
 
 where H is the heat taken in, and the temperature 
 of the envelope through which it passes in this 
 case the same as that of the gas and reservoir. If, 
 therefore, we know the amount of heat Ui or HI 
 produced, and the temperature 0\, we can get 3>, 
 and we need not trouble about the ultimate actual 
 waste. Similarly, if a change has taken place which 
 has, in fact, produced no heat, but is of such a 
 nature that to get the working substance, whatever 
 it may be, back to its original state, work must be 
 degraded into heat, we have no information as to 
 the waste so caused. But if we know the highest 
 temperature Q\ at which the inevitable heat can be 
 given out, we get Hi/0i = $, the entropy ; and that 
 gives us the incurred waste corresponding to any 
 lower temperature that may eventually be available. 
 The incurred waste thus varies directly as the 
 lowest available temperature, and directly as the 
 
24 ENTROPY, 
 
 entropy ; and if the lowest temperature is fixed, the 
 waste incurred depends simply on the entropy. If, 
 therefore, a change of any sort involved no increase 
 of entropy, it would involve no increase of waste. 
 Waste can be diminished by obtaining a lower 
 available temperature, but not by any other means. 
 If it were possible, we could then have perpetual 
 motion of the second class, the degradation of work 
 by friction being wholly undone by the mechanism 
 itself. This leads to a most important law ; as in 
 all real changes there is either friction, or some 
 analogous source of waste, every change whatever 
 involves waste, small though it may be ; and as the 
 change does not affect the lowest available tempera- 
 ture, there is always an increase of entropy. The 
 entropy of the working substance may be increased 
 or diminished or remain constant, but the entropy 
 of the substance and its externals increases. The 
 entropy of an isolated system that is to say, a 
 system which as a whole neither takes in nor gives 
 out any energy increases with every change. 
 Entropy once created is therefore permanent for all 
 time. The entropy of the universe tends towards a 
 maximum, and is always growing towards it. 
 Changes take place in Nature in which heat is 
 produced, as by friction. Changes take place in 
 which heat disappears, or is elevated into work, as 
 when a gas expands adiabatically doing external 
 work, or when a photographer dissolves what he calls 
 
A FACTOR OF INCURRED WASTE. 25 
 
 hypo in water and gets a mild freezing mixture. In 
 this case the heat is not converted into external 
 work, but into another form of heat. But in Nature 
 no change takes place which does not either produce 
 heat, and thus give rise to waste, or incur waste, 
 without actually producing the heat there and then. 
 Clausius's general statement that the energy of the 
 universe remains constant, and the entropy strives 
 towards a maximum, may therefore be paraphrased 
 thus: "The energy of the universe is constant, 
 but no change takes place without incurring waste 
 of energy." 
 
 It is the business of the engineer to design his 
 machinery to utilise energy to the utmost ; and to 
 get as little as possible into the waste form of heat 
 at low temperatures. If he takes care of his entropy, 
 the waste will take care of itself. He should design 
 his steam- or gas-engine so that the originarentropy 
 of the fuel and oxygen is increased as little as 
 possible. He is not responsible for the original 
 entropy, but he is for all increase of it ; and the 
 more he keeps down the increase of entropy, and 
 the corresponding incurred waste, the more near 
 perfection is his engine. 
 
 The theory of thermodynamics is therefore vitally 
 important, and the use of the 3> diagram should be 
 to the engineer what the balance-sheet is to the 
 financier. The difference is that the financier can 
 make either profit or loss, but the engineer can 
 
26 ENTROPY. 
 
 only, and must, make loss : it is his business to 
 keep it as small as possible. 
 
 Reversibility. We may now consider the question 
 of reversible and irreversible changes and reversible 
 and irreversible cycles. 
 
 If the working substance were to change from the 
 state A to the state B in such a way that the change 
 could be reversed, so that the substance went from 
 B to A in such a way that every part of the change 
 from B to A were exactly the same as the corre- 
 sponding change from A to B, but in the opposite 
 direction, the change or process would be reversible. 
 There are no reversible changes in Nature ; a rever- 
 sible change is an ideal, or an abstraction, which 
 can only be approximately reached. There is no 
 such thing as a circle or a square or a straight line. 
 They are mathematical abstractions which can be 
 approached, and are exceedingly convenient for 
 reasoning purposes. In a reversible change, not 
 only must the substance retrace exactly the same 
 path, but everything that is involved in the change 
 must retrace the same path. The perfect gas in 
 the thermodynamic engine is the simplest case 
 of reversibilit}' ; each of the changes A, B ; B, C ; 
 C, D ; and D, A being reversible. Heat was taken 
 in and work given out from A to B. From B to 
 A, therefore, heat is given out and work taken in, 
 in exactly the reverse way. 
 
 When the working substance starts from any 
 
REVERSIBILITY. 27 
 
 state, and, after change, returns to the original 
 state, it is said to have performed a cycle. If 
 every part of the cycle is made up of changes 
 of which each is reversible, the whole cycle is 
 reversible. 
 
 A reversible cycle can cause no incurred waste ; 
 for if it did, it would reduce incurred waste if 
 worked the other way, and that would give us 
 second-class perpetual motion. Thus a thermo- 
 dynamic engine working a reversible cycle might 
 convert a certain quantity of heat HI, taken from a 
 reservoir at Oi into work W, rejecting H 2 at a lower 
 temperature. If it were coupled to the wasteful 
 reversible engine of a size to give out HI at Oi, each 
 revolution, that engine, when working backwards, 
 and reducing incurred waste, must take in more 
 than H 2 at the lower temperature to give out HI at 
 the highest ; so it needs less work each cycle to 
 drive it as H 2 + W = HI. The balance of energy 
 per cycle might then be devoted to overcoming the 
 friction, unavoidable in a real engine, and any 
 balance might do other work, and we would have 
 second-class perpetual motion. 
 
 A reversible change is, therefore, a change in 
 which there is no waste incurred, and in a reversible 
 cycle there is no waste incurred. 
 
 But we have denned entropy as a quantity whose 
 increase, when multiplied by the lowest available 
 temperature, gives the incurred waste. There is 
 
28 ENTROPY. 
 
 thus, then, no increase of entropy during a rever- 
 sible change or a reversible cycle. 
 
 The entropy of a working substance may, of 
 course, increase or diminish during a reversible 
 change, but the entropy of an isolated system 
 that is to say, the working substance and whatever 
 is affected by the change, remains constant. 
 
 Localisation of Entropy. There has already been 
 incurred an enormous amount of waste in the 
 universe. From the beginning of things there has 
 been degradation of work into heat which can be 
 elevated only partially into work by rejecting some 
 of it at a lower temperature. There is thus an 
 enormous total entropy in the universe, and it is 
 always increasing. More than that, as work gets 
 degraded into heat the lowest available temperature 
 is raised, so that the incurred waste is increasing, 
 not only in proportion to the growth of entropy, 
 but also in proportion to the lowest available 
 temperature. But how should entropy be localised ? 
 When heat is produced, say, by friction, it means 
 that something gets hotter, or else the heat is 
 absorbed and rendered latent. To get the hotter 
 substance back to its original state it must either 
 give out the heat as heat, or it may expand, doing 
 external work, and cooling to its old temperature, 
 being then compressed to its original volume and 
 rejecting heat, which is degraded energy. The 
 increased waste is clearly connected with the hot 
 
LOCALISATION OF ENTROPY. 29 
 
 body. The hot body may be moved about, or taken 
 into different surroundings ; but to get it back to 
 its original state involves its giving out waste heat. 
 We therefore talk of the entropy of the substance. 
 If the substance is homogeneous in every way, it 
 may be divided up into parts, say seven, and each 
 part will have to give out one-seventh of the waste 
 heat that the whole has to give out in returning to 
 its original state. Entropy is thus additive, like 
 weight, or mass, or volume not qualitative, like 
 temperature, hardness, or pressure ; two tons of 
 water at a given temperature has twice the entropy 
 of one ton, and so on. We thus talk of the entropy 
 of a substance for instance, the entropy of 17 lb. 
 of steam under given pressure, and volume, and 
 temperature. We can also talk of the entropy per 
 unit mass, or specific entropy of a body. For 
 instance, in working <j> diagrams, the diagram is 
 generally drawn for a pound of steam, and then 
 shows the specific entropy. The term " specific 
 entropy " is not often used, but it is as well to adopt 
 it, to distinguish between the entropy of a mass of 
 steam and its entropy per pound. The example of 
 Professor Planck may well be followed. He uses 
 capitals for volume, entropy, and so on, and small 
 letters for specific volume, and entropy. 
 
 It has been shown that during any reversible 
 change there is no incurred \vaste, and therefore no 
 increase of entropy in the system concerned. Thus, 
 
30 ENTROPY, 
 
 if an isolated system consists of some heat reser- 
 voirs, a perfect gas-engine, and a flywheel ; if there 
 is a reversible change, there is no increase of 
 entropy of the system. The perfect gas may have 
 its entropy increased, however, but in that case the 
 entropy of some of the reservoirs is decreased by 
 exactly the same amount. 
 
 More Convenient Definition of the Entropy of a 
 Body. So far we have nearly always discussed the 
 entropy, or the increase of entropy, of a whole 
 isolated system. In thermodynamics this course is 
 inconvenient, and it is much more useful to discuss 
 the entropy of the working substance alone. When 
 the entropy of the working substance is increased 
 by exactly the amount the entropy of the rest 
 of the isolated system is diminished, the increase 
 is called "compensated." If the working substance 
 increases its entropy by any reversible change, then 
 its increase of entropy is compensated by exactly 
 equal decrease of entropy in other things which are 
 involved in the change. Thus if there were two 
 bodies, say vessels of partly melted lead, A and B, 
 and A were able, through an infinitesimal difference 
 of temperature, to communicate one thermal unit 
 to B, in an infinite time, the change would be 
 sensibly reversible, so the entropy of A and B 
 together would be unchanged. 
 
 But if we now proceed to reduce B to the tempera- 
 ture of ice, so as to compare it with the standard 
 
OF A BODY. 31 
 
 state, the first thing it does is to allow the extra 
 lead thawed by the unit of heat to solidify, giving 
 out a thermal unit at the temperature of melting 
 lead, and the waste corresponding to this unit of 
 heat, divided by the lowest available temperature, 
 was the increase of entropy of B. It has already 
 been explained that this is equal to the heat taken 
 in at the melting-point of lead, divided at the 
 absolute temperature at which it is taken in 
 namely, 1/0 of a unit of entropy, where is the 
 melting-point of lead. The entropy is thus localised. 
 We may now give another definition of entropy, 
 which isjnpre convenient than that in terms of the 
 incurred waste, because it refers to the entropy of 
 the working substance, without taking the whole 
 isolated system into account, and because it is not 
 given in terms of the lowest available temperature 
 or the incurred waste. Thejiew__definitioiLjnay be 
 arrived at by considering reversible changes. If two 
 bodies A and B are in contact, and are at the same 
 temperature, and there is a reversible change 
 so that the sum of the entropies of A and B 
 remains constant, B can only increase its entropy 
 by a corresponding decrease of the entropy of A. 
 But the only way A can have its entropy reduced is 
 by losing heat. For of the heat in A a part is avail- 
 able, and the rest, depending on the lowest available 
 temperature, is waste. This waste can only be 
 reduced by abstracting some of the heat of A. A 
 
32 ENTROPY, 
 
 might lose work, but that would not reduce its 
 entropy. We thus find that the entropy of a body 
 can only be reduced by its losing heat. But it does 
 not follow that if a body's heat is decreased it loses 
 entropy. For if a gas is expanded in a cylinder so 
 as to do external work without being supplied with 
 heat from the outside, its heat is decreased. Its 
 entropy is not, however, reduced, for, as already 
 explained, the waste, once incurred, cannot be 
 reduced ; so that the entropy of a body cannot be 
 reduced without at least an equivalent increase ol 
 the entropy of something else. The entropy of A 
 can thus only be reduced by its losing heat to some 
 other body. Merely diminishing its heat by con- 
 verting it into work, or giving out the work, does 
 not decrease its entropy. The heat must be given 
 out so as to cause at least an equal simultaneous 
 increase of entropy elsewhere. This practically 
 means that for the entropy of A to decrease, heat 
 must pass out across the boundary or envelope 
 of A.* 
 
 * The only exception that comes to mind is the case where 
 a thermo-circuit is arranged with a junction inside A. li 
 the other junction is outside A, say in B, and B is colder, 
 heat disappears from A and the entropy of A is reduced 
 without heat passing across the boundary of A. But the 
 other statement is still true : the heat is given out so as tc 
 cause at least an equal simultaneous increase of entropy else- 
 where. There cannot be a single thermo- junction elevating 
 heat into work with no corresponding degradation elsewhere 
 in the circuit. 
 
OF A BODY. 33 
 
 It may be similarly shown that the entropy of a 
 body can only be increased reversibly by its taking 
 in heat through its envelope. Consider an isolated 
 system undergoing a reversible change, so that its 
 total entropy remains constant while the entropy of 
 one part A is reduced, and that of B increased by 
 the same amount. A must give out heat, and if B 
 increases its entropy without taking in that heat, 
 that heat must go into some third body C, and 
 increase its entropy to at least the extent that the 
 entropy of A fell. There would then be an increase 
 of the total entropy of the system, so that the 
 change is not reversible. To make the change 
 reversible B must increase in entropy only in pro- 
 portion to the heat it takes through its envelope.* 
 It must not be for a moment supposed that because 
 the entropy of a body can only be increased 
 reversibly by heat coming from outside through the 
 envelope, it cannot be increased irreversibly other- 
 wise. The entropy of a body cannot be decreased 
 without giving out heat, either reversibly or irre- 
 versibly, and it cannot be increased reversibly 
 
 * The thermo- junction case is omitted from consideration 
 here. B could in theory increase its entropy reversibly by 
 means of a thermo -contact inside it ; but even then, though 
 its heat is not taken in as heat through its envelope, its 
 increase of entropy, in a reversible change, is exactly 
 balanced by the decrease of entropy of another body which 
 does not lose the corresponding heat through its envelope as 
 heat. 
 
 E. D 
 
34 ENTROPY. 
 
 without taking in heat; but it can be increased 
 irreversibly without taking in the corresponding 
 heat through its envelope. 
 
 As the entropy of a body is decreased by giving 
 out heat only, its entropy, compared with that of 
 the same body in the standard state can be found 
 by bringing it to the standard state by a reversible 
 change or a series of reversible changes, and finding 
 the heat given out and the temperature at which it 
 is given out, and the corresponding entropy. If the 
 heat H can all be given out at one temperature 0, 
 then H/0 is the entropy, taking that of the standard 
 state as zero. If the heat is given out at various 
 temperatures, the entropy isfdR/0. If the entropy 
 of the body is less than in the standard state, as in 
 the case of a block of ice at its melting-point, the 
 entropy can be measured by finding the heat taken 
 in during a reversible change to the standard state, 
 and giving it the negative sign. 
 
 Definition. We may therefore define the entropy 
 of a body in state B, compared with its standard 
 state A, as being numerically equal to the heat that 
 would have to be taken in to get it from A to B by 
 reversible changes, divided by the absolute tempera- 
 ture ; or H/0, if the heat is taken in (or given out 
 if the entropy is less than in state A) at constant 
 temperature 0, or fdH/0 if the temperature varies. 
 Thus the entropy of the body in state B is not a 
 function of the heat actually taken in during its 
 
DEFINITION. 35 
 
 change from A to B, as the change must have been 
 partially, and may have been wholly, irreversible ; 
 but it can be measured as a function of the heat 
 which would have to be taken in to change from A 
 to B reversibly, or which would have to be given 
 out if the substance were changed from B to A 
 reversibly, which amounts to the same thing. 
 
 It is, perhaps, not unnecessary to point out 
 again that though the entropy is measured in 
 terms of heat that would have to be taken in, and 
 the temperature at which it would come in in 
 changing from A to B reversibly, the entropy is, 
 in fact, always greater than H/0 or J dH/0, as a 
 reversible change never takes place. In a real 
 change 3> > J dYLjO ; and while 4> is positive, 
 fdH/0 may even be negative that is to say, the 
 body may change from state A to state B, increasing 
 its entropy all the time, and giving out heat all the 
 time too. As every text-book on the steam- and 
 gas-engine (except Kankine's*) defines entropy as 
 fdH/O instead of as being measured by fdH/0, 
 
 * Rankine is not clear about his ' ' thermodynamic func- 
 tion." His first definition makes it the same as entropy, but 
 lie also makes it equal tofdIL/0 without limiting the change 
 to reversible change. He certainly did not develope the idea 
 of entropy and its relation to waste, which forms the basis of 
 this book. No doubt a man of his ability, if he had written 
 on steam-engines somewhat later, would have been not only 
 perfectly correct, but also clear and unambiguous in his 
 statements and definitions. 
 
 D2 
 
36 ENTKOPY. 
 
 if the change is imagined by a reversible path from A 
 to B, this point cannot be insisted on too strongly. 
 The incorrect definition makes the conception of 
 entropy impossible to the reader ; and if the writer 
 did not unconsciously depart from his own definition, 
 it would make nonsense of thermodynamics. 
 
 As the entropy of a bod} 7 can be measured by 
 bringing it back to the standard state by rever- 
 sible changes that is to say, changes which 
 involve no waste, and therefore no increase of 
 entropy in the body and its surroundings the 
 entropy in state B, compared with state A, is 
 measured by seeing how much entropy comes out 
 on changing it from B to A without creating any 
 entropy during the change. 
 
 The entropy of a body in a given state, therefore, 
 depends only on the state, and not on its past 
 histor} r ; thus a quantity of steam in a given state as 
 to pressure, temperature, and volume must give out 
 the same entropy on being brought by reversible 
 change to the state of ice-cold water, in whatever 
 wa} r it was made into steam. 
 
CHAPTER III. 
 
 THE 6<f> DIAGKAM. 
 
 WE may now deal with the < diagram, taking a 
 simple case to start with, and discussing it from 
 the point of view put forward in this article. 
 
 We may begin with the case of a perfect gas. 
 
 Suppose we start with a quantity, say, a pound, 
 so that we deal with the specific volume and entropy 
 
 of a perfect gas in a cylinder with heat-tight sides, 
 and let the area of the piston be unit}', so that its 
 pressure and volume are given by the point A in 
 Fig. 4, where the height is the pressure on the 
 piston, and the horizontal distance the volume of 
 the gas. We may take the state A as standard for 
 simplicity that is, we take its entropy as zero. 
 In the Ocf> diagram, to correspond with Fig. 4, we 
 
38 
 
 THE 6$ DIAGRAM. 
 
 therefore start with A to correspond with A in 
 Fig. 4, marking off the height O A to correspond 
 with the temperature 6 1 of the gas to start with, and 
 on the line of zero entropy. Let the gas be now 
 allowed to expand, doing external work MI and taking 
 in heat h lt at temperature O l the expansion being 
 carried on reversibly to the volume and pressure 
 represented by B, Fig. 4. 
 
 Consider the position at B. The gas has taken 
 in heat 7^, but it is no hotter ; that is to say, some- 
 
 thing external, say a heat reservoir, has given out 
 heat h l9 which has crossed the boundary or envelope 
 of the gas, and has then ceased to exist as heat, 
 being converted into work ?r, which has gone outside, 
 say, into a flywheel. But as the change is rever- 
 sible, so that the entropy of the gas and the reser- 
 voir is constant, and as the reservoir has lost heat 
 7t x at temperature P and the gas has taken in heat 7^ 
 at temperature 6 l9 and has given out no heat else- 
 where, its entropy must have increased by Ii\j6\ 
 In Fig. 5, from A draw A B, so that A B 
 
DIAGRAM FOB GAS. 
 
 39 
 
 represents MV l9 or <. The height representing the 
 temperature' is constant, so A B is a horizontal 
 straight line. 
 
 That the entropy of something outside, which 
 supplied the heat h l reversibty, diminished, may 
 seem an insufficient reason for saying the entropy 
 of the gas is increased. But the gas cannot be 
 compressed to its old state without giving out heat 
 at some temperature, of which part must be waste. 
 
 The next step is to expand the gas from B to C 
 reversibly, without taking any heat, but doing 
 external work, so that the heat of the gas is drawn 
 upon to do the external work. The entropy remains 
 constant during the change, as the process is rever- 
 sible. It could not reduce the entropy during this 
 change, as no heat comes out across the envelope, 
 and that is the only way the entropy could be 
 reduced. As the change from B to C is reversible 
 by hypothesis, the entropy of the gas cannot have 
 
40 
 
 THE 0<l> DIAGRAM. 
 
 increased, else it would have to give out heat on 
 reversal to get back to the state of B. From B to 
 C then the temperature falls to, say, 2 , while the 
 entropy remains constant. From B on Fig. 5 draw 
 B C, so that the height of C represents 2 , the 
 entropy D C being equal to A B. Let the gas be 
 compressed isothermally and reversibly from C to 
 D, giving out heat /i 2 at temperature 0% until the 
 
 G 
 
 c 
 
 entropy is reduced to zero that is to say, its 
 original value. 
 
 On Fig. 5, the state point therefore moves along 
 C D to D, where the temperature is 2 and the 
 entropy zero, as at A. The gas is then compressed 
 reversibly without any passage of heat across the 
 boundary surface, until the gas reaches its old 
 temperature O lf If the temperature is 6 lt the new 
 state point must be on the curve A B, and as the 
 entropy is zero, the new point must be A, which is 
 the original starting point. We have thus gone 
 
6+ DIAGRAM FOR GAS. 
 
 41 
 
 through the ordinary Carnot cycle, and Fig. 4 is 
 the p v diagram, and Fig. 5 the 6 <f> diagram of it. 
 As the cycle described is reversible, the curve A B, 
 Fig. 4, is for a perfect gas, p v = R 0, where, as 
 the expansion is isothermal, is constant, and 
 R is constant anyhow, so the curve is part of 
 rectangular hyperbola, and is easily described with 
 the help of a slide-rule. For instance, if R is 
 found on one of the scales, and the slide turned 
 round, and the 1 of the same scale put opposite 
 
 0,; V 6' H J 
 
 R 6, the corresponding values of p and v are read 
 off without moving the slide. As the height is equal 
 to the pressure on the piston, and the length to the 
 volume, the area A B H F is equal to w\, the external 
 work done during expansion ; and the area is also 
 proportional to h lt the heat taken in at tempera- 
 ture BI. In the B < diagram, as the change from A 
 to B is reversible, and at constant temperature 19 
 the entropy, &i/#i, multiplied by 19 is obviously 
 equal to the heat h l9 so the area A B H (Fig. 5) 
 is also, in this case, equal to the heat taken in 
 
42 THE 00 DIAGRAM. 
 
 and to the work done. It is probably this that 
 has misled people into supposing the 6<j> is a 
 heat diagram, or that its^^rea is proportional to 
 some existing heat or to some work. Suppose the 
 expansion from A to B had been carried out irrever- 
 sibly without any heat be4ng taken in by the gas, 
 and without doing any external work., For instance, 
 a partition might be put across the C3 T linder at A, to 
 hold the gas, while the piston is moved to B. If a 
 hole is now made in the partition, the gas will 
 blow through into the vacuum, and eventually the 
 gas will be, according to Joule's experiment, at the 
 same temperature as before, and therefore at the 
 same pressure, as the volume is the same. The 
 gas has therefore got to state B without giving out 
 any work, and without taking in any heat. It 
 must be remembered that F A and H B represent 
 the pressures on the piston, so that in the last case 
 A B H F was equal to the external work 7^. In 
 this case there was no pressure on the piston 
 during its movement. The passage from A to B 
 was really not by the curve that joins them, but 
 from A down to F, then to H, and up to B. There 
 is apt to be some confusion as to what is meant by 
 the pressure of the gas during such an irreversible 
 change ; but the only pressure that has to do with 
 external work, which is the matter in question, 
 on an indicator diagram, such as Fig. 4, is the 
 pressure on the piston. We might have had an 
 
0<t> DIAGRAM FOR GAS. 
 
 43 
 
 indicator on the cylinder which would, if put on the 
 gas side of the partition we inserted, have read 
 pressure as F A throughout the movement of the 
 piston. In a steam- or gas-engine the indicator is 
 merely to tell the piston pressure. An indicator 
 put on this cylinder, unless put on the piston side 
 of the division, would only give incorrect infor- 
 mation. In a p v diagram we are only concerned 
 with the real piston pressure, not with any state of 
 the gas. 
 
 If instead of using a partition, and opening a 
 hole to let the gas through after the movement of 
 the piston, we had used a very light piston and 
 moved it suddenly so as to allow the gas to expand 
 freely from A to B, immediately after the move- 
 ment of the piston, we can imagine the piston 
 moved so suddenly and quickly that during the 
 movement there is no pressure on it, and therefore 
 no external work is done. The gas then rushes 
 into the vacant space, and the work of expansion 
 of each of the little parts of the gas is spent on 
 
44 THE 0$ DIAGRAM. 
 
 imparting kinetic energy to other parts of the 
 gas. This is degraded again into heat, and the 
 final result is that when the gas settles down to a 
 uniform temperature and loses its kinetic energy, 
 the pressure on the piston is H B, and the tempera- 
 ture O l9 and the entropy A B. It has been urged 
 by men of science too, that in this case the increase 
 of entropy is due to the production of this heat by 
 friction in wiping out the kinetic energy ; and in 
 support of this contention it must be noted that if 
 the gas puts all its work of expansion into kinetic 
 energy, and that is then degraded into heat, the 
 heat so produced will be equal to the heat /i x that 
 would have been taken in during the expansion if 
 it had been reversible and isothermal. But the 
 whole of the heat that is produced by friction in 
 the reduction of kinetic energy in the irreversible 
 expansion has previously disappeared from the gas 
 in producing the kinetic energy. If the heat pro- 
 duced inside the gas by friction is to be counted as 
 generated, the equal heat lost in producing the 
 kinetic energy must also be counted, and the total 
 internal generation of heat is then zero. 
 
 It has been urged that the increase of entropy in 
 irreversible expansion is still due to the gas receiv- 
 ing heat, and the increase of entropy is still fdh/0, 
 even in an irreversible change. This is due to a 
 misunderstanding of the meaning of the symbols. 
 h is not the heat of the substance, nor the heat 
 
0<f> DIAGRAM FOR GAS. 45 
 
 gained by the substance. It is the heat taken in by 
 the substance from the outside. It is the loss of 
 heat suffered by some external reservoir or body. 
 If you give h a different meaning, so as to include 
 heat generated by any means inside the body with- 
 out coming through the envelope or case as heat, h 
 increases when a gas is compressed adiabatically 
 that is to say, without any heat passing through the 
 envelope and decreases during adiabatic expansion. 
 And if the equation f d]i\Q = $ were still true when 
 dh meant the increase of heat of the body, the 
 reversible adiabatic compression of a gas would not 
 be isentropic. On the other hand, the isothermal 
 reversible expansion of a perfect gas would be 
 isentropic, because the heat taken in would be 
 exactly balanced by the internal disappearance of 
 heat which is converted into external work. It is 
 important to remember that in these equations h, or 
 H stands for the heat passed into the working 
 substance, from outside through the envelope, and 
 not for the heat of the substance, or for heat 
 generated inside by friction, by internal combustion, 
 by an electric resistance worked from outside, by 
 compression or otherwise. 
 
 It may be asked what meaning the curve A B, 
 Fig. 4, has when the expansion is quite irreversible 
 that is to say, all the increase of entropy of the 
 substance is uncompensated by reduction of entropy 
 elsewhere, and there is no external work. If the 
 
46 
 
 THE *+ DIAGRAM. 
 
 piston is moved suddenly to B, after a little, the 
 temperature, which varied throughout the volume of 
 
 the gas during the rush, comes to $ again, and the 
 pressure to H B. 
 
 If the piston were first moved suddenly to a point, 
 saj, half way along, and stopped, the pressure would 
 correspond to the vertical height, say K E, Fig. 6. 
 
 The piston might divide its whole stroke into a 
 series of little quick steps, each being so quick that 
 the gas exerts no pressure on the piston during its 
 motion. The diagram win then be a sort of comb 
 of vertical lines, the tops being in the curve A B, 
 
*4> DIAGRAM FOR GAS. 
 
 and there will be no area and no outside work. If 
 these steps are made more numerous and smaller, 
 there will still be no area, enclosed nd no external 
 work. The malW each step is made, the "1W 
 will be the variation of temperature throughout the 
 during the little rashes of giy, and 
 
 tion of pressure i 
 of the cylinder. 
 If the jumps are 
 
 my, 
 
 the 
 
 indefinitely <anall 1 the 
 
 of the 
 
 the pressure on the 
 remains constantly zero, so 
 done, and there is no area 
 of the gas on the walk of the cylinder follows the 
 OTTO A B (Fig. 4) as if the expansion we 
 and iftfttJifi ""*! 
 
 Turning to the 0+ diagram (Fig. 5), 
 lessoning apply ? At B the temperature is 
 the entropy A B; bat the 
 
48 
 
 THE 6$ DIAGRAM. 
 
 state point come along the straight line A B, or 
 does it come round by O and H, so that 110 area is 
 enclosed ? or does it do something between the two ? 
 To come round by O and II would mean that the 
 gas had no temperature, or was at absolute zero 
 during the change, which is absurd. If the whole 
 energy of expansion were first devoted to producing 
 kinetic energy of the gas, equal to the external work 
 
 that might have been done by isothermal expansion, 
 the temperature would fall to # 2 ; and as the kinetic 
 energy became degraded into heat, there would be 
 an increase of temperature and entropy, so that the 
 state point would move by a sloping curve from D 
 to B. If the piston only makes small jumps, the 
 state point will move much as in Fig. 7. The 
 temperature will fall a little. It will be different 
 throughout the mass of gas, but in no part will it 
 
6^ DIAGRAM FOR GAS. 49 
 
 fall much. If the expansion takes place in a succes- 
 sion of jumps, the state point will make a succession 
 of saw teeth ; and if the jumps are indefinitely small, 
 the state point moves along the line A B, tracing 
 one side of the area A B H O. 
 
 We are justified, therefore, in drawing A B 
 straight as representing the movement of the state 
 point in the $<}> diagram ; but this long explanation 
 has been entered into rather to clear up difficulties 
 than anything else. The area of the 0$ diagram is 
 not energy, but the area is important, for the excess 
 of the area of the <f> diagram over the area of the 
 p v shows the badness of the engine from a thermo- 
 dynamical point of view. 
 
 There is considerable confusion often as to the 
 meaning of a p v diagram ; that is to say, as tojyhat 
 p means in an irreversible change. As a p v or 
 Watt diagram is to tell the work given out by the 
 expansion, p is the pressure on the piston ; so that 
 the external work is w = fp dv. In any reversible 
 expansion the curve is also a diagram of the state of 
 the gas ; and people have been misled into supposing 
 that the p v diagram refers to the state of the gas in 
 an irreversible change. This is the usual confusion 
 about reversible and irreversible changes. 
 
 In Fig. 6 the dotted curve A E B is the diagram 
 of the equivalent state of the gas, and if p is taken 
 as the height of this curve at any volumey^p dv>w, 
 and the area does not represent external work. 
 
 E. B 
 
50 
 
 THE 6$ DIAGRAM. 
 
 Also dh<^p dv. There are, in fact, two p v dia- 
 grams one in which p is the pressure on the 
 piston, which is not a diagram of the state of the 
 gas at all, and another which is unconsciously used 
 instead of it, which gives the equivalent state of the 
 gas, but not the external work. In reversible 
 changes the diagrams coincide ; in irreversible they 
 do not, and we have two sets of pressures. Writers, 
 for instance, often state that dh/0 d<j>, and that 
 
 dh/0 and d<f> are both complete differentials ; dh/0 is 
 not a complete differential in terms of the p and v 
 of the diagram of the state of the gas in which 
 pv = RO; rf<, on the other hand, is a complete 
 differential in terms of the ordinates of the state 
 diagram in which p v = R 0, but it is not a complete 
 differential with reference to the external work or 
 piston co-ordinates of the Watt diagram. dhlO = d<{> 
 would be true in the ideal case of reversibility when 
 the two diagrams coincide. 
 
 As the change of entropy follows the state of the 
 
DIAGEA 
 
 51 
 
 gas, and not the external work, the </> diagram is 
 the same as before. At B the entropy is A B 
 (Fig. 5), for at B the gas is in a given state, as to 
 pressure, temperature, and volume, and, as already 
 fully explained, the entropy depends on the state of 
 the gas, not on how it got into that state. 
 
 It is therefore incorrect to say that the area 
 A B H O, Fig. 5, represents heat, for no heat was 
 taken in. It is equally incorrect to say it represents 
 
 Ilg.6. 
 
 external work, for no external work was done. 
 Thus entropy is not a factor of heat ; it is not heat- 
 weight, and it is not a factor of energy. It is not 
 hi/tii, for no heat has been taken in, and the 
 temperature has been constant. Neither is it equal 
 tofdhi/9. It is, to repeat, a quantity which, when 
 multiplied by the lowest temperature available, gives 
 the incurred waste. Thus, if, for example, 6% is 
 taken as the lowest available temperature, the gas 
 may be expanded reversibly without change of 
 
 E2 
 
52 
 
 THE 6$ DIAGRAM. 
 
 entropy from B to C, and may then be compressed 
 isothermally along C D, Figs. 4 and 5, giving out 
 the heat /* 2 at temperature 2 . This is represented 
 by the area D C J G, Fig. 4, and D C H 0, Fig. 5. 
 The area D C H is thus the waste that must at 
 least result from bringing the substance from a state 
 where its entropy is A B or C D back to the 
 standard or zero state. 
 
 The waste has not yet taken place when the gas 
 has got to B, even irreversibly ; but to get it to A 
 
 Fly. 4. 
 
 again, work must be degraded into heat, of which 
 O D, the lowest available temperature, multiplied 
 by the entropy A B, gives the waste D C H O. That 
 is why the expression " incurred waste " has been 
 used. The waste does not take place necessarily 
 when the entropy increases ; but as it must take 
 place eventually, it is called " incurred." There is 
 still another reason for the expression. In reality 
 reversibility is unattainable, so in bringing the gas 
 back from B to A a little more waste must result. 
 But this waste is not " incurred " when the gas is 
 at B ; it is a subsequent product, due to the 
 
DIAGRAM FOR GAS. 
 
 53 
 
 difficulty of approaching reversibility in the changes 
 necessary to bring the gas back to A. 
 
 The other definitions given also fit the case. 
 
 The entropy of the gas at B is numerically equal 
 to the heat that must be given out on returning to 
 A by reversible changes, divided by the tempera- 
 ture of the surface at which the heat is given out.* 
 Thus, if the gas is brought back isothermally along 
 B A, Fig. 4, the heat given out at temperature #1 is 
 
 proportional to the area A B H 0, Fig. 5, and that 
 divided by O A is A B. The same reasoning applies 
 to the path B C D A, Fig. 4, where the heat H 2 or 
 D C H 0, Fig. 5, is given out. 
 
 * In this sense entropy might be called a factor of heat, but 
 that is not at all the sense in which it is so termed. If a 
 body has entropy * to get it the standard state where *=0, 
 heat H, so that H/0= * or/r!H/0=* must at least be given 
 out, but that heat may have to come from external work, 
 and the entropy of the body is not a factor of some heat of 
 unknown amount that is going to be .produced at a future 
 date from some external energy or other. 
 
54 
 
 THE 6$ DIAGRAM. 
 
 The curve B C, Fig. 4, may.be easily drawn with 
 the help of a slide-rule and proportional compasses. 
 Its equation is p v? = constant. This is shown in 
 the text-hooks, so it need not be gone over again, as 
 
 the object of this book is rather to be supplementary 
 and explanator}', and not a complete self-contained 
 treatise on elementary thermodynamics. 
 
 As we know the value of p v at B, we can find the 
 
 JKf.fi. 
 
 Q "- Jf LK 
 
 value of p itf there, if we know y. For a perfect 
 gas, 7 is 1.6 ; we therefore find p vY. Starting from 
 B, Fig. 2, for instance, we multiply the corre- 
 sponding p by the corresponding vY, and note the 
 
6$ DIAGRAM FOR GAS. 55 
 
 number on the slide-rule scale. The proportional 
 compass is set at 1.6. For every pressure the 
 remainder of the scale up to pv7 is spanned 
 by the long legs of the compass ; the short legs then 
 give the corresponding v on the same scale. 
 
 The word " adiabatic " means without passing 
 through, denoting that during an adiabatic change 
 there is no crossing of heat into or out of the gas. 
 The heat of the gas may vary, of course, but by 
 changing to or from work. If it were reversible, 
 an adiabatic would be an isentropic change. But 
 the entropy can increase without any heat coming 
 in, and, in fact, always does ; so an adiabatic change 
 never is, in fact, isentropic. 
 
 Text-books frequently treat isentropic and adia- 
 batic as synonymous terms a very widespread 
 error, arising, like the others, from ignoring real as 
 opposed to hypothetical thermodynamics. 
 
 It will be noticed, on comparing the 6<f> and the 
 p v diagram for the Carnot cycle of Figs. 4 and 5, 
 that if the diagrams are drawn to suitable scale, 
 while the work point in the p v diagram traces the 
 first step from A to B, the state point on the 6 </> 
 traces A B. The area A B H F on Fig. 4 is then 
 equal to the heat hi taken in at 0, and to the external 
 work done from A to B. The heat of the perfect 
 gas remains constant. The corresponding area, 
 A B H O,.Fig. 5 is also equal to the heat hi taken 
 in at temperature fy. But the area A B H O does 
 
56 
 
 THE 6$ DIAGEAM. 
 
 not represent the increase of heat of the gas. The 
 diagram is not a heat diagram, in fact. From B to C 
 the area B C J H represents the work done l>y the 
 gas, but there is no heat taken in or given out ; but 
 Fig. s. 
 
 the gas loses heat, which goes out as work. The 
 state point in the 0<j> diagram merely descends. 
 The other two steps are similar, but in the opposite 
 direction. The result is that the area A B C D is 
 
 equal on these diagrams. The efficiency in Fig. 4 
 is the ratio of the area A B C D to the area A B H F. 
 The efficiency in Fig. 5 is the ratio of the area 
 ABCD to ABFO, and is much easier to see, 
 
6$ DIAGRAM FOR STEAM. 57 
 
 and is obviously proportional to (#1 - O^/Oi. But 
 that is only true of the reversible cycle ; if the cycle 
 is irreversible, for instance, on account of the 
 expansion from A to B being free, Fig. 5 does 
 not give the efficiency. It is worthy of remark 
 that in the reversible cycle the entropy alters 
 during the changes in which the heat of the gas 
 remains constant, while the entropy remains con- 
 stant during steps two and five, in which the heat 
 varies. 
 
 0(f> Diagram for Steam. Instead of taking a 
 perfect gas, steam may be taken. Water at the 
 temperature of ice is generally taken as standard of 
 comparison. The water-steam diagram may there- 
 fore be started from 32 Fahr., or 493 F.A., or 
 Fahrenheit absolute. The unit of heat used by 
 English engineers is the heat necessary to raise 
 1 Ib. of water 1 Fahr., namely, from 59 to 60. 
 It is a pity there is a special unit of heat. It is a 
 survival of the times when people did not fully 
 realise that heat is energy ; but there is no reason 
 why we should still have a separate unit of heat. 
 But as we have the foot-pound and, I think, the 
 poundal,* as units of energy, perhaps a special 
 unit of heat saves confusion. The heat necessary 
 to raise 1 Ib. of water 1 Fahr. is called the British 
 
 * I think poundals are energy, but they may be force, or 
 a kind of yellow cakes. I do not remember, and am not 
 anxious to know. 
 
58 THE 00 DIAGRAM. 
 
 thermal unit ; and a quantity of heat is measured 
 in British thermal units.* 
 
 The Institution of Civil Engineers has recently 
 recommended officially that " British thermal unit" 
 be shortened into B.Th.U. as opposed to B.T.U., 
 which stands for "Board of Trade Unit," an irregular 
 and barbarous name of 3,600 joules. No doubt 
 British thermodynamicians are getting so efficient 
 in manipulating their units that inaccuracies due 
 to the interchangeable use of the British thermal 
 and the Board of Trade unit are becoming per- 
 ceptible. The letters B.Th.U. will, therefore, be 
 used to denote these things. 
 
 To raise water 1 at a given temperature, or 
 sensibly at that temperature, needs 1 B.Th.U. 
 
 per lb., so the increase of entropy per degree is - or 
 
 ~ = ; provided the specific thermal capacity, 
 d9 
 
 inaccurately called the " specific heat," of the water 
 remains constant. The 0$ curve for water is 
 
 * Lord Kelvin and others have pointed out that this use of 
 the word "units" is incorrect. The guinea, pennyweight, 
 perch, quartern, bushel, acre, kilderkin, hand, baker's dozen, 
 ream, em, month of Sundays, and blue moon are British 
 units, so that a statement as to, say, 31,416 British units 
 would refer to a sort of museum that could easily be collected. 
 But it is no use being too pedantic over such things. What 
 is wanted is a name for the British thermal unit, such as the 
 therm, then we could say that the water took in so many 
 therms. It would be simpler still to say so many foot- 
 pounds. 
 
DIAGRAM FOE STEAM. 
 
 59 
 
 thus sucli that the rate of increase of the 
 horizontal line is inversely as the height. As 
 
 ^ = I/O, = f l dO = loge + const. The 
 6<f> for a substance whose thermal capacity remains 
 
 constant is thus a logarithmic curve, and the con- 
 stant is chosen so that the curve cuts the ordinate 
 O P, Fig. 8, at the height corresponding to 493 
 Fahr. This curve can be easily plotted by calcula- 
 tion, and more easily copied from a steam table or 
 Sankey's valuable steam chart. The steam chart, 
 for which engineers are deeply indebted to Captain 
 
60 
 
 THE 
 
 DIAGRAM. 
 
 Sankey, shows properties of steam graphically, being 
 a 6<f> diagram with curves giving every possible 
 information carefully plotted on it. The statement 
 that a square of a given size is a British thermal 
 unit need not be regarded.* 
 
 It need hardly be mentioned that it is much easier 
 to work all thermodynamical calculations on the 
 metric and especially the C. G. S. system ; but 
 there seem to be no convenient tables available. 
 
 * For accurate work a correct steam table or chart is 
 necessary. Probably the National Physical Laboratory will 
 soon check Kegnault's experimental figures. The present 
 steam tables are in urgent need of careful revision. 
 
e$ DIAGRAM FOR STEAM. 61 
 
 It is one of the strongest arguments against the 
 adoption of the metric system in this country that 
 Continental people, even scientific people, who could 
 use it without any difficulty, generally try to avoid 
 its use. A steam table, with the temperatures in 
 degrees from freezing-point, the pressures in milli- 
 metres of mercury, atmospheres, or kilogrammes, 
 instead of megadynes per square centimetre, the 
 heats in calories instead of joules, and the entropies 
 not given at all, is not convenient; and it is easier, 
 even if it is desired to work on the metric system, to 
 use the English or rather American steam tables, 
 such as Peabody's or Reeve's. Peabody's and 
 Reeve's can both be got separately,* and Reeve's 
 are especially complete and convenient. The 
 expression " Entropy of the Heat-Energy " at 
 the head of the table does not make the data 
 inaccurate. 
 
 Starting from 492.8 F.A., Fig. 8, the curve 
 during the heating of the water is practically 
 logarithmic. It would be accurately logarithmic if 
 the specific thermal capacity, or specific "heat" of 
 the water were constant. The curve is easily got 
 from the temperature T, and entropy N, columns,! 
 
 * Messrs. Wiley & Co. and Messrs. Macmillan & Co. 
 
 t It seeins a pity to introduce a new letter, N, for entropy, 
 Eankine used <p for the thermodynamic function, and Clausius 
 used S for entropy. Maxwell used and $ for temperature 
 and entropy. Gibbs used i\ for entropy. T and t are con- 
 
62 
 
 THE 6$ DIAGRAM. 
 
 and as the curvature is very small, a few points are 
 enough. Suppose the water is heated up to 850 
 F.A., omitting decimals, before steam begins to 
 form. The table gives 220 Ib. per square inch as 
 the pressure, and the volume is inappreciable. The 
 indicator or p v diagram may be drawn at the same 
 time (Fig. 9) with the lettering to correspond. The 
 0< diagram, so far, is a piece of a logarithmic 
 curve, and at 850 F.A. the entropy is 0.552. This 
 
 means that, to get the water back to the standard 
 temperature of 493 F. A., 493 X .552 = 272 British 
 
 venient for absolute and thermometer temperatures, but T 
 is already in use for kinetic energy, and this gives rise to 
 confusion in dealing with thermodynamics and the kinetic 
 theory of gases together. The Germans often use $ for 
 temperature. Quantity of heat taken in by a body is 
 generally written Q, but as Q, in other branches of physics 
 denotes the quantity factor of energy, and not energy itself, 
 H is more accurate. 0, <j>, and H are therefore used here, 
 following Maxwell and others, though T, S, and Q are more 
 general, especially in scientific books and papers. 
 
6$ DIAGRAM FOR STEAM. 63 
 
 thermal units must be wasted, if 493 F.A. is the 
 lowest available temperature. It does not follow 
 that this waste has been caused by warming this 
 water. If the 0.552 was compensated entropy 
 that is to say, if the increase of entropy of the water 
 was exactly balanced by an equal decrease of 
 entropy of whatever supplied the heat there was 
 no increase of total entropy, and therefore no new 
 waste incurred. If, on the other hand, the water 
 was sensibly at the same temperature throughout 
 its volume at each instant, but was heated by gases 
 at a much higher temperature, there was a growth 
 of uncompensated entropy outside the water during 
 the heating. The growth of entropy due to heat 
 conduction will be discussed presently. In the case 
 of heating water, practically all the energy taken 
 in, whether as heat, or as work which is degraded 
 into heat in the body of the liquid, is in the form 
 of sensible heat, and it might appear that the 
 entropy was a factor of the heat ; for though^ dh/0 
 < </>, where h is the heat taken in as heat, if the 
 process is not reversible, fdu\B = is very nearly 
 true where duis the differential of the sensible heat 
 of the body. It is not quite true, however. The 
 water can do external work during the expansion, 
 and then even jdujO < <j>. It is quite clear that the 
 entropy of the water at B that is to say, at 850 
 F.A. is 0.522, however the heating has been carried 
 out. Text-books on the steam-engine define the 
 
64 THE e$ DIAGRAM. 
 
 entropy &sfdh/0, but if the water is heated in any 
 other way than by taking in heat from the outside 
 for instance, by heating by Foucault currents pro- 
 duced in the water by a rapidly changing electrical 
 field fdh\B might be zero if the water took in no 
 heat, and negative if it gave out some as it heated. 
 It cannot be too often urged that the entropy of a 
 body depends on its condition and not on its history. 
 Unless all the changes are reversible, fdh\ '6 depends 
 on the history, not on the condition of the sub- 
 stance. If the writers of the text-books stuck to their 
 definition of entropy when working with irreversible 
 processes, they would come to grief at once ; but 
 they unconsciously avoid the abyss that ought to 
 yawn for them by taking their data from steam 
 tables or Sarikey's chart, which is correct, because 
 the entropy then depends on the state of the 
 substance and not onfdh/0. 
 
 Continuing the diagrams, suppose all the water 
 vaporised at pressure 220 Ib. and temperature 850 
 F.A. The p v diagram is traced by the constant 
 pressure line B C to 2.1 cubic feet. The state 
 point on the 6$ diagram goes to < = 1.537. At 
 the point C the p v diagram shows the external 
 work done by the steam in expanding. It does not 
 matter whether the steam is in the boiler or the 
 cylinder for the moment ; the work is done outside 
 the steam during expansion. We may assume the 
 steam to be in a hypothetical cylinder. Part of the 
 
6<t> DIAGRAM FOR STEAM. 
 
 65 
 
 heat taken in is converted into work namely, 84.8 
 British thermal units, and this is shown on the p v 
 diagram. 753 British thermal units have been 
 devoted to vaporising the water. This energy is 
 still in the steam. The allocation of the energy 
 
 taken in by the water in evaporating, which we will 
 assume was taken in as heat, might be shown on 
 <f> curve. The area A B B 1 then represents the 
 sensible heat which raised the temperature of the 
 water, or it so nearly represents the area that it may 
 be taken to represent it. The variations of specific 
 thermal capacity of water are small. The vertical 
 
66 THE 00 DIAGRAM. 
 
 ordinate is temperature, the horizontal entropy. 
 Take now the area B C C l B l ; it does not represent 
 the increase of heat of the water from B to C, so it 
 is not a heat diagram ; neither does it represent the 
 increase of energy of the water from B to C, so it is 
 not an energy diagram of the steam. It is a 6 <f> 
 diagram, and the length B C, multiplied by the 
 lowest available temperature, gives the waste. Also 
 the area B C C 1 B 1 gives the heat that must at 
 least be given out at temperature BI B, or 850 
 F.A., to get the steam back into water at the same 
 temperature ; but this heat would not all be pro- 
 vided by the steam; some of it must come from 
 outside, generally as mechanical work. As frequently 
 explained, entropy is not a factor of heat. Suppose 
 we adopt a suitable factor x> so that x is * ne 
 increase of energy in the form of heat in the body 
 itself if the temperature has been constant, and 
 y#dxis the increase of heat from the standard state 
 when the temperature varies. It must be pointed 
 out that splitting heat into factors 6 and x> so that 
 f6 dx is energy, is unorthodox. There has for a 
 long time been a sort of hazy idea in thermo- 
 dynamics that heat should be split up into a tension 
 and a quantity factor, like other forms of energy. 
 Taking temperature as one factor, capacity for heat 
 has been proposed as the other ; but that is obviously 
 absurd, as, if C is the capacity, the energy would 
 bey C dO, not J dC, Besides C is capacity for 
 
6$ DIAGRAM FOR STEAM. 67 
 
 energy not for a quantity factor, so that capacity 
 for heat is by no means the analogue of capacity 
 in, say, hydraulics or electricity. As has been 
 frequently mentioned already, the error that entropy 
 is a factor of energy, is very widespread, not only 
 among physicists and engineers, but even among 
 many who write specially on thermodynamics. I 
 brought a paper before the Physical Society about 
 two years ago, in which it was pointed out that, 
 although in a reversible change <j> is numerically 
 equal to the factor of energy lost as heat by the 
 external reservoir, not gained as energy by the 
 working substance corresponding with tempera- 
 ture, fQ d<f> is not energy in a real change. The 
 paper discussed the question of splitting heat into 
 factors of energy, and compared the advantages and 
 drawbacks. The paper was rejected, however, and 
 until scientific men generally attend to real partly 
 irreversible processes, the subject cannot come 
 forward for discussion. It is fair to point out as 
 far as possible where my treatment is not orthodox, 
 so that the reader does not willingly absorb without 
 criticism, as established thermodynamics, what are 
 really the heterodox ideas of the writer. 
 
 As we have to deal with sensible heat, the heat 
 that makes things hot, and latent heat, which 
 changes their physical state, for instance, the dif- 
 ference of internal energy between steam and water 
 at the same temperature, we may use suffixes, so 
 
 F2 
 
68 THE 6$ DIAGRAM. 
 
 that x* is the quantity factor of sensible heat, and 
 Xp that of latent heat of physical change.* W e 
 may now plot the heat energy of the pound of water 
 from A to C. From A to B we get the figure 
 A B B 1 O, Fig. 10, in which the height is and the 
 breadth x-?> so that the area isfOdxs, giving the 
 sensible heat of the water at 850 F.A., neglecting 
 any question of variation of specific capacity. From 
 water to steam at B 850 the water takes in 
 837.7 British thermal units of energy, of which 
 753 remain as latent heat, and 84.7 go out as work. 
 
 * The expression "latent heat " is used absurdly in thermo- 
 dynamics. Thus, if ice is melted, heat is taken in and remains 
 in the water. The alteration of volume is minute. If, on 
 the other hand, water is evaporated into steam doing external 
 work, the external work is included in the so-called " latent 
 heat." There is the same confusion as regards the mis- 
 called " specific heat." The heat absorbed per unit mass at 
 constant volume may be measured by a specific thermal 
 capacity at constant volume; but to measure the heat 
 absorbed at constant pressure or temperature by a ' ' specific 
 heat at constant pressure" or a "specific heat at constant 
 temperature " is absurd. In the case of a perfect gas, for 
 instance, the difference between K p and K v is entirely 
 external work, and not heat at all ; and the " specific heat at 
 constant temperature " is entirely external work. The whole 
 nomenclature of thermodynamics demands re-modelling. 
 
 By latent heat in connection with XP i s meant that 
 fedxp, or generally e xp> is the heat taken in and kept by the 
 substance itself during a physical change such as fusion or 
 evaporation; and it does not include any external work 
 Similarly fedxc is the heat of chemical change. We are not 
 concerned with that at present. 
 
DIAGRAM FOR STEAM. 
 
 69 
 
 As is here constant, \p = 0.9, and we may add 
 a rectangle B C C 1 B 1 , to represent \p> to Fig. 10. 
 As this rectangle is the same height as the larger 
 rectangle in the $ diagram, we might cut an equal 
 area off that area B P P 1 B 1 . This area would 
 
 B' CJ)' 
 
 represent the increase of latent heat of the water 
 from B to C, and the remaining area P C C 1 P 1 
 would represent the external work. This division 
 of the < area into two areas representing xp 
 and w is not really legitimate. The quantities have 
 no real place on a </> diagram ; and it is only 
 because the temperature is constant that they can 
 be superposed that way. The diagram would look 
 
70 THE 6$ DIAGRAM. 
 
 as if the latent heat had been completed first, and 
 as if there were a state P in which all the water 
 was evaporated, and no external work yet done. 
 More than this, there might have heen no work 
 corresponding to the small area P C C 1 P 1 . For 
 instance, the water might have been evaporated at 
 a pressure of 220 Ib. and blown through a small 
 hole into a space big enough to hold all the steam 
 at the same pressure and temperature. This pro- 
 cess would be irreversible, and the area P C C 1 P 1 
 would represent external work that might have 
 been done, but was not. The entropy would be the 
 same in both cases, so would \p* The work that 
 might have been done, but was not, has been called 
 " uncompensated work," I think, first by Duhem, 
 to correspond with the idea of uncompensated 
 entropy. The term is not happy, because the work 
 does not exist. Uncompensated entropy is a quan- 
 tity that is produced, without the redeeming com- 
 pensation of reduction of entropy elsewhere. Uncom- 
 pensated work is not produced at all. Entropy is a 
 kind of drawback, so uncompensated entropy is an 
 unmitigated drawback. But work is the other way. If 
 a British workman unit is paid for work he does 
 not do, it would not express the employer's feelings 
 to say the workman did uncompensated work. It is 
 the wage that is uncompensated. It is on such 
 grounds that I venture to differ from so eminent an 
 authority as Duhem. The uncompensated work is 
 
6$ DIAGRAM FOR STEAM. 71 
 
 thus the difference betweeny# d<j> and h. " Uncom- 
 pensated " work is thus work that might have been 
 gained, but was not. " Ungained work" might be 
 a better term for it. It is the product of the 
 uncompensated entropy and the temperature if that 
 is constant, ovfOd<j> u when it is not, <% being the 
 uncompensated entropy. The ungained work at 
 any temperature 6\ is to the corresponding incurred 
 waste as 0i/#2> where 2 is the lowest available 
 temperature. Where a reversible change takes 
 place there is no uncompensated entropy and no 
 ungained work. The change is then said to be 
 made at constant " thermodynamic potential " 
 another rather unfortunate term. The reversibility 
 of a change can be expressed wholly in terms of 
 the co-ordinates of the substance, such as the 
 pressure, temperature, volume, entropy, latent heat, 
 chemical energy, &c., without reference to the out- 
 side conditions. These expressions were worked 
 out by Helmholtz, Massieu, and Gibbs, and are 
 now generally known as Gibbs's functions or 
 thermodynamic potentials. If these functions were 
 expressed clearly in words, instead of being treated 
 as obscure results of pages of differential equations, 
 they would be quite useful to engineers. If the 
 common statement that the area of the $ is the 
 same as, or proportional to that of the p v diagram 
 were correct, there would be no such thing as 
 ungained work, and there would be no such thing 
 
72 
 
 THE 6$ DIAGRAM. 
 
 as waste, and all steam and gas-engines would have 
 an efficiency of (Q\ O^jO^ and there would be no 
 need to study their thermodynamics, as the} r would 
 be incapable of improvement. 
 
 Let the steam now be superheated to, say, 
 
 1,000 F.A. at the same pressure, and reversibly. 
 The p v diagram shows a corresponding increase of 
 volume, and the state point rises to D. A new area 
 under C D is thus added to the (j> diagram. The 
 added heat is thus partly converted into external 
 work, and partly converted into sensible heat, 
 making the steam hotter. We may add a curve 
 
0<i> DIAGRAM FOB STEAM. 
 
 73 
 
 G D to the 6 x diagram, Fig. 10, so that the height 
 is 6 and the breadth x* and the area /^ d\s is the 
 increase of sensible heat during the superheating. 
 The area under C D, Fig. 10, is less than that 
 under C D in the < diagram, Fig. 8, because the 
 breadth of the heat diagram is x, the factor of heat, 
 
 and of the <f>, < the entropy, which is always, in 
 fact, greater. The difference in area between the 
 heat diagram and the <j> diagram is thus the 
 external work if the change has been reversible, and 
 the external and " uncompensated " or " ungained " 
 work if it has been wholly or partly irreversible. 
 What was said about its not being legitimate to 
 
74 THE 6$ DIAGRAM. 
 
 divide the area B C C 1 B 1 of the <}> diagram into 
 two areas B P P 1 B 1 and P C C 1 P 1 , one represent- 
 ing heat and the other work, is now clearer. The 
 6 $ area under C D, Fig. 8, with its curved top, 
 cannot be divided into a heat area C D D 1 C 1 of the 
 heat diagram, and an external work area by any 
 vertical line. In fact, the heat diagram, Fig. 10, is 
 itself unlawful, for it has two separate kinds of 
 heat on the same area. Thus, if the steam did not 
 behave as a perfect gas during heating, but stored 
 up some latent heat, the area C D D 1 C 1 on the 
 heat diagram would really represent^ d\s -\-J d Xp> 
 and the area could not be divided out into latent 
 and sensible heat. The area of the heat diagram 
 (Fig. 10) above the lowest available temperature, 
 M S, is the available energy of the steam in relation 
 to the state point M. That energy can be got as 
 work by going round from D to M by S reversibly. 
 From D suppose the steam to be expanded 
 isothermally, but partly reversibly and partly irre- 
 versibly. If the expansion were wholly irreversible, 
 no heat would be taken in and no work done, and 
 the trace of the p v diagram would be D L K E ; 
 whereas if the change were reversible, the trace 
 would be by the bit of hyperbola D E shown dotted. 
 If it is partly reversible, the pressure will be less 
 than for the hyperbola, and it may be as in the full 
 curve D E. The area on the 6 <f> diagram under 
 D E is the same whether the path in the p v is by 
 
0$ DIAGRAM FOR STEAM. 75 
 
 the dotted or full line, and is equal to the area 
 D E K L of the p v diagram, taking the dotted line, 
 and the difference is the "ungained" work. No 
 part of the area under D E of the < diagram is 
 thus heat, assuming the superheated steam to be a 
 perfect gas. In reality, allowing for steam not 
 being a perfect gas, the increase of entropy D E 
 should be a little more, and the small difference 
 should cover an area 6 xp which should be added to 
 the heat diagram. That may be neglected. 
 
 The superheated steam may now be expanded in 
 a heat-tight non-conducting cylinder, in a cylinder 
 which gives up heat to the steam as it expands, or 
 in a cylinder that abstracts heat. In order to 
 make the discussion as complete as possible we 
 may take some of each. Then the steam may 
 expand reversibly doing the full complement of 
 external work, or wholly irreversibly doing no ex- 
 ternal work, or partly irreversibly doing some 
 external work. The reversible and irreversible 
 expansion of a perfect gas have already been 
 discussed. 
 
 If the steam now expanded without doing any 
 external work, its state point would move to the 
 right of E on the <j> diagram, and the p v diagram 
 would go down E K, and then the volume would 
 increase, at zero pressure. If the cylinder gave up 
 enough heat to keep the steam at the constant 
 temperature of 1,000 F.A., and the expansion was 
 
76 
 
 THE 6<l> DIAGRAM. 
 
 reversible, the p v diagram would be a continuation 
 of the dotted isothermal D E. Let the expansion 
 be adiabatic and reversible, the cylinder giving up 
 no heat to the steam, the curve from E is p v? = 
 K E X (M K)7. During this change the state 
 
 point of the < diagram goes down vertically, the 
 entropy of the steam remaining constant. If the 
 expansion were reversible and the cylinder supplied 
 some heat, but not enough to keep the steam at 
 1,000 F.A., the p v line would go between the 
 isothermal and adiabatic, but for simplicity this 
 part will be taken as both reversible and adiabatic, 
 
00 DIAGRAM FOR STEAM. 77 
 
 and therefore isen tropic. From F the steam will 
 be supposed to be expanded in another cylinder, 
 which abstracts heat during the expansion ; so that 
 the p v curve F H bends down below the reversible 
 adiabatic line ; and the state point on the 6 </> dia- 
 gram moves from F to H, showing fall of tempera- 
 ture and decrease of entropy going on together. 
 This curve F H will cut the saturation line at G, 
 so that at H about 0.2 of the steam has been con- 
 densed. From H to I let the expansion be neither 
 adiabatic nor isothermal, so that the cylinder walls 
 give up heat to the steam, and the entropy thus 
 increases, and the temperature falls. At I let the 
 exhaust open to a condenser at lower temperature 
 than the steam, say, as represented by the height 
 of the line M J. When the steam has all fallen to 
 this temperature, the entropy will be reduced, and 
 the state point will be at the point L. How it gets 
 from I to L will be discussed presently. At L we 
 have nearly a pound of water and a little steam 
 occupying a large volume. As the piston comes 
 back the steam is all condensed to water, and we get, 
 not to A, the freezing-point, but to M, which is 
 not at zero entropy. 
 
 It need hardly be said that the p v and 6 <f> dia- 
 grams here discussed are not practical at all; they 
 are merely put together on paper to illustrate the 
 theory of entropy. In practice the indicator diagram 
 and the steam per half stroke is the starting point, 
 
78 
 
 THE 00 DIAGEAM. 
 
 and there is only a very rough approximation to 
 knowledge of what happens . In the present examples 
 the <f> diagram is made up from steam data and an 
 imaginary p v diagram, as if of a single cylinder 
 with no clearance or cushioning, and many pecu- 
 
 liarities not met with in practice. A diagram 
 with straight lines and corners, and a right-hand 
 side like Fig. 8, is not found in steam work. 
 
 The 6 % diagrams need not be discussed further, 
 as it is beyond the scope of this book to discuss 
 factors of heat ; they were introduced here to show 
 the difference between the 6 <f> and real heat diagrams. 
 
6$ DIAGRAM FOE STEAM. 79 
 
 In order to keep the diagram clear, vertical lines 
 are drawn dotted from the various state points to 
 the base, and the corresponding letters with accents 
 are put on the base line for reference. As it is 
 tedious to follow a diagram with polygonal areas 
 denoted by numerous corner letters, areas are 
 marked with small letters, each letter denoting the 
 area enclosed by hard lines. Thus a + b is the area 
 MBCDEFHIJLM. 
 
 At E the entropy is at the maximum value 
 touched during the cycle. To find the incurred 
 waste at E the increase of entropy is multiplied by 
 the lowest available temperature. For simplicity 
 the same pound of water may be supposed to be 
 used over and over again. So the waste is not the 
 distance P E multiplied by the lowest available 
 temperature, as the entropy never falls below E M, 
 corresponding to water at the temperature of the 
 condenser. 
 
 Making P Q = R M, Q E is the maximum increase 
 of entropy during the cycle. Choosing 32 Fahr., 
 or 493 F. A. as zero of entrop} T , is purely arbitrary ; 
 in this case water at temperature O E might have 
 been taken as the standard that is, as having zero 
 entropy. Taking Q E as the increase of entropy at 
 E, the next question is, What is the lowest avail- 
 able temperature ? The lowest temperature reached 
 is at L M, so the waste incurred up to the point E is 
 the area e + d ; but the waste is really considerably 
 
80 
 
 THE 00 DIAGKAM. 
 
 greater owing to the subsequent treatment, as 
 the temperature of the condenser is not really 
 available, except for the small outgiving of waste 
 heat between L and M. At E, however, the waste 
 so far incurred is the area d + e. A perfect engine 
 
 working from E by reversible changes to M, without 
 taking in any more heat, would move the state 
 point vertically to S, and then across to M. 
 
 The path from E to F calls for no remark, as it 
 is assumed to be reversible. It may be as well to 
 point out, however, that a vertical line on the <f> 
 diagram does not necessarily mean reversible isen- 
 
6$ DIAGRAM FOE STEAM. 81 
 
 tropic adiabatic change. If the change is reversible 
 and isentropic, it is also adiabatic, and the path is 
 vertical; but suppose the steam were wire-drawn 
 between E and F, its entropy from that cause 
 would increase ; but if, at the same time, it lost 
 heat to the valve or other metal-work so that its 
 entropy was reduced by loss of heat exactly enough 
 to counterbalance the increase by irreversible expan- 
 sion, the trace would be vertical. These doubly 
 irreversible disturbing causes would have additive 
 effects in reducing the area of the p v diagram 
 relatively to the <f>. 
 
 Matters are complicated in Fig. 8 by the left- 
 ward slope F H, and the next slope H I. It need 
 hardly be repeated that such a corner as that at H 
 does not occur; it is shown simply as an example. 
 At E the incurred waste was d + e. From F to H 
 the entropy is reduced, and at H the incurred 
 waste is M T H 1 M 1 . It might therefore be sup- 
 posed that from F to H the actual waste is the 
 difference in entropy of F and H multiplied by 
 M 1 M. But M 1 M is not the lowest available tem- 
 perature here at all. The entropy from F to H 
 has been reduced by giving up heat at a varying 
 higher temperature, not to a condenser, or to 
 anything where it can be utilised, but to cylinder 
 walls, where it may be conducted away and wasted 
 absolutely. The waste corresponding to the change 
 F H is thus the area H F E 1 H 1 . If the heat given 
 
 . G 
 
82 
 
 THE 6$ DIAGRAM. 
 
 to the cylinder walls were used in a little engine, 
 some of this would be recovered, and the waste 
 would be less. The waste may also be reduced by 
 some of the heat being given back to the steam 
 later on. 
 
 PTTC' 
 
 From H to I the steam is receiving heat from 
 the cylinder walls. If this heat is supplied from 
 the furnace, for instance, by a jacket, the incurred 
 waste at I exceeds that at H by the area T J I 1 H 1 ; 
 we thus have a vertical strip of waste area 
 T J I 1 H 1 which has to be reckoned twice. If the 
 heat supplied between H and I was some of that 
 
6$ DIAGRAM FOR STEAM. 83 
 
 parted with between F and H to the cylinder walls, 
 the waste due to F H, which was taken as H F E 1 H 1 , 
 has been over-estimated, and instead of the narrow 
 strip T J I 1 H 1 being reckoned twice, it must be 
 reckoned once only. The result is that the waste 
 from F to H that is, the heat now useless is the 
 area c + d, so that the strip under H I is only 
 counted once, and the little area H I J T would be 
 saved, or utilised if the lower line M J of the closed 
 cycle were the lowest available temperature. If the 
 curve I K L is the lowest available temperature, the 
 area HUT is reduced to a little triangular piece 
 at the top. 
 
 The next point is the passage from I to L. The 
 difficulty is that while the steam is blowing through 
 the exhaust, the part that is condensed is at con- 
 denser temperature, while the part that is in the 
 cylinder begins by being at a higher temperature, 
 and the temperature gradually falls. There is no 
 one value for the temperature during the change. 
 The best thing to do, therefore, is to make some 
 assumption which will cause no error in deriving 
 results from the 0<f> and p v diagrams, and will at 
 the same time give a meaning to during this 
 change. One method is to assume that the steam, 
 instead of rushing off to a condenser, is cooled to 
 condenser temperature at constant volume. This 
 gives the curve I K L (Fig. 8). Another assump- 
 tion might be that the steam expands adiabatically 
 
84 
 
 THE e<l> DIAGRAM. 
 
 and reversibly to the condenser temperature, and is 
 then compressed isothermally to L. This involves 
 external work, and is therefore incorrect. The 
 condensation at constant volume is correct, as it 
 gives the right result. By opening the exhaust at 
 
 I the lowest available temperature is raised to I, 
 and falls along the path I K L. The area I K L J 
 is thus added to the actual waste. 
 
 The meaning of this slice off the < diagram is 
 not difficult to catch. As has been explained 
 already, the statement generally made that the area 
 of the < cycle is the same as that of the indicator 
 
6$ DIAGRAM FOR STEAM. 85 
 
 diagram is wrong. It is always greater, and the 
 difference is the ungained work and the "lost 
 work," which has not yet been discussed. Cutting 
 off a corner like I K L, therefore, looks like re- 
 ducing the area of the 6 < cycle, so as to lessen the 
 difference of area, and therefore lessen the ungained 
 work. It might therefore appear that the 0<f> 
 diagram showed that exhausting above condenser 
 temperature is rather a good thing than otherwise, 
 and therefore the 6<i> diagram is misleading. This 
 would be, however, an incorrect view. The ungained 
 work is not lessened by raising the lower side of the 
 0< cycle. Thus, if there was some ungained work 
 during the increase of entropy from B to C, it 
 might be regarded as represented by an area of 
 height B B or C C, and width equal to the uncom- 
 pensated entropy. It stretches right down to the 
 bottom of the diagram. This is an area by which 
 the 6 (f> cycle exceeds the p v cycle, therefore, what- 
 ever the height of the lower line of the 0$ cycle. 
 If the lower side of the #< cycle were raised until 
 it coincided with the upper, and the ungained work 
 were as before, the area of the p v cycle would be 
 negative, and numerically equal to the ungained 
 work ; that is to say, the steam would expand doing 
 no external work, and the engine would do work 
 equal to the ungained work in compressing it into 
 water again. 
 
 The theory of ungained work may be shown 
 
THE 04, DIAGKAM. 
 
 better by a simple diagram. Suppose water at A 
 in the 0<f> and p v diagrams, Fig. 11, were heated 
 reversibly to B, then expanded to C reversibly and 
 isothermally, and then expanded to D reversibly 
 and adiabatically, and finally compressed to A 
 again, also reversibly. Assume the p r diagram, 
 Fig. 11, to be drawn to correspond in scale with 
 the <, and let the areas be cut up by the vertical 
 and horizontal lines into small areas which do not 
 overlap, each with its letter of reference. Taking 
 
 the four steps, and putting down the intake of heat 
 and output of work, we have : 
 
 Step. 
 A to B 
 B C 
 C D 
 D A 
 Balance 
 
 Heat Taken in. 
 a -f d 
 
 1) + e + c +f 
 
 -(/+<? -M) 
 a -f- b -f c a 
 
 Work Done. 
 
 
 c+f=fj+j 
 a 4- ft = k + i 
 /= (i - 
 
Heat Taken in. 
 
 Work Done. 
 
 
 a + d 
 
 
 
 
 b+ e 
 
 
 
 
 
 
 a 4. b = h + i 
 
 ( 
 
 f+e+d) 
 
 -/ = - (i+; 
 
 a +&-/ a+ i-f=h-j 
 
 6<j> DIAGRAM FOR STEAM. 87 
 
 If, however, the expansion were wholly irre- 
 versible, to take an extreme case for simplicity, the 
 statement would be : 
 
 Step. 
 A to B 
 B C 
 C D 
 D A - 
 
 The area of the < cycle would be, as before, 
 a + b + c, but thepv cycle would be smaller by the strip 
 c -f- /. If the lowest available temperature is raised, 
 it does not reduce this difference, as it is on both sides 
 of the line of the lowest available temperature.* 
 
 Turning back to Fig. 8, cutting off the area by 
 the curve I K L does not make the efficiency higher, 
 according to the 6$ diagram, though the waste is 
 the difference of areas of the p v and 6 < cycles, and 
 though cutting off this corner lessens the 6 < cycle. 
 If the path had been by I J L, the steam would have 
 been expanded further, and more work would have 
 been obtained. If the expansion from I to J and the 
 compression from J to L were reversible, both p v 
 and 6<j> cycles would be equally enlarged by the 
 
 * For simplicity it is here assumed that at the lower tem- 
 perature U = d -f- e ; it is in fact a little less. This does not 
 reduce the area c -f /, so the proof is not affected by the 
 slight inaccuracy of the assumption. 
 
88 
 
 THE 
 
 DIAGRAM. 
 
 course taken. In cutting off the corner what is 
 really done is to increase the waste \>y raising the 
 lowest available temperature. This is another form 
 of the same thing as occurred from G to H. Thus 
 at I the waste so far incurred is e, part of the area 
 
 N I J M being still to be had as work ; but if the 
 exigencies of the case call for a raising of the lowest 
 available temperature, there is a reduction of the 
 efficiency of the engine, and an increase of the waste, 
 but not an increase of difference of cycle areas. The 
 corner area I J L K is not ungained work in the 
 
00 DIAGKAM FOR STEAM. 89 
 
 sense the term has been employed (fo d(f> h), 
 due to irreversibility ; it is increased waste due to 
 raising the lower temperature limit. 
 
 The irregular area b + c + d + e is the total 
 reversible waste, assuming that the entropy from H 
 to I was increased by some of the heat given out 
 between F and H. It must not be supposed, how- 
 ever, that the waste is necessarily no greater than 
 this. 
 
 So far irreversible compression has not been dis- 
 cussed. If a cylinder of gas or steam is compressed 
 by using up external work, it is difficult to think of 
 any form of irreversible compression which would 
 use up more work and give out more heat than 
 would account for the decrease of the steam's 
 entropy. In a turbine acting as a pump, however, 
 the waste would be considerably greater than the 
 area under the 6<f> cycle. This is pointed out to 
 prevent any such mistake as supposing the area 
 under the 0<t> cycle to be the only waste. The 
 waste must be at least that area, and it may be 
 more. This increased waste appears, however, as 
 a reduction of the p v cycle area, as the increased 
 waste due to irreversible compression increases the 
 work put into the working substance by raising the 
 lower side of the p v diagram's cycle. It thus shows 
 the difference of area of the 6 < and the p v cycles. 
 The excess of area of the 6 < over the p v cycle is 
 thus made up of two parts : the ungained work, due 
 
90 THE 6$ DIAGEAM. 
 
 to irreversible expansion, and lost work, due to 
 irreversible compression. 
 
 In practical engineering, in the case of a recipro- 
 cating engine, we have some approach to the p v 
 diagram given by the indicator ; but there is un- 
 certainty as to the value of v at the beginning of 
 the stroke that is, until the admission is finished. 
 Again, there is uncertainty if the exhaust opens 
 gradually. Moreover, the steam may not be at the 
 same temperature throughout its volume at any 
 given time. An approximately true p r diagram is 
 made, and the 0$ inferred from it by means of 
 tables or Sankey's chart. The difference of areas 
 of the cycles should then be measured, to give the 
 ungained work, and the wasted work, and the lower 
 boundary of the cycle should be examined to see 
 whether the waste can be reduced anywhere by 
 keeping down the lower temperature limit. 
 
 Ungained work due to irreversible expansion in 
 the cylinder is too minute to matter in a recipro- 
 cating engine, but in the turbine it may be a very 
 serious matter. The steam turbine may in the 
 future be very much more carefully studied thermo- 
 dynamically than the reciprocating engine. But 
 there is a good deal of ungained work in an ordinary 
 engine in wire-drawing through valves and ports. 
 
 The 0<f> cannot well be compared with the 
 indicator diagram in the case of a turbine. It is in 
 fact somewhat difficult to find out what is going on 
 
6$ DIAGRAM FOR STEAM. 91 
 
 inside the case. If it is driving a dynamo the output 
 and the steam supply are known and the pressure, 
 temperature, and dryness, of the steam as it enters. 
 The temperature can be taken by inserting small 
 thermometers or thermo-couples at various points. 
 The velocity may be not great enough to prevent 
 the pressures being taken from point to point ; and 
 perhaps some conclusions as to the dryness at 
 various vanes can be obtained. 
 
 If the condition of the steam entering is known, 
 and the temperature of the condenser, the maximum 
 or ideal efficiency is known, and an ideal 6<j> diagram 
 can be plotted. If the turbine is jacketed the jacket 
 steam must be included. From the pressure tem- 
 perature and wetness of the exhaust steam, its 
 specific entropy can be found and marked on the 
 0<J> curve, and an equivalent exhaust curve drawn, 
 as in Fig. 8. If the turbine, whether jacketed or 
 not, does not lose heat perceptibty by radiation and 
 convection, the state point for the exhaust will be 
 to the right of that for admission, and lower in 
 temperature of course. From the diagram so 
 obtained the " ungained work " can be found. 
 The inefficiency will be the result of irreversible 
 expansion, and conduction of heat along the metal- 
 work of the turbine. This tells the result of the 
 whole turbine, and when several turbines are used 
 in series that is specially valuable, corresponding 
 roughly to the information as to each of the 
 
92 THE 00 DIAGRAM. 
 
 cylinders in a reciprocating engine. If the necessary 
 information can be obtained from point to point 
 throughout the turbine, the localisation of blame is 
 possible, and the designer can find out where the 
 steam is rushing through too easily without doing 
 its proper work. 
 
 Into the details of the application of the 0<f> 
 diagram to actual practice it is not my object to 
 go. That may be left for a future opportunity, or 
 the work may be done better by others. But the 
 first thing is to get the theory of the 9 < diagram 
 cleared of the common errors about it, so that its 
 real meanings can be brought out. 
 
 The present position of the <j> diagram in engi- 
 neering is that it is discussed in college text-books 
 and in treatises for engineers ; and it is said to be 
 very instructive, and to give great insight into the 
 economy of the steam-engine. But it has made 
 little or no progress in real engineering. One 
 engineer, of high rank, has been said to have used 
 it, and got benefit from it. But so long as it is 
 supposed that entropy isj dh/0, or heat-weight, or 
 a factor of heat, or that the 6$ is a heat diagram, 
 or that its area is in British thermal units, or that 
 the < diagram is of the same area as the p v cycle, 
 it seems likely to remain practically useless. It 
 will be treated with great respect, because people 
 always respect what they do not understand, espe- 
 cially if it looks at all mathematical. Engineers 
 
w^-v 
 OF THE 
 
 UNIVERSITY 
 
 OF 
 
 DIAGRAM FOR STKfflrT^ 93 
 
 will not like to admit that they do not understand 
 it; or, at most, those who have neglected mathe- 
 matics will regret that the calculus was not taught 
 apprentices in their young days, and recommend 
 young men to take the <f> diagram to their bosoms 
 and thank their stars for it. They may even tackle 
 one of the numerous and excellent books of potted 
 mathematics with which the benighted engineer is 
 now befriended. If they can digest the potted 
 mathematics with any sort of comfort, they are 
 certain to be saddened internally by the <{> diagram 
 as at present constituted. But they feel it is a 
 wonderful thing, which ought to be swallowed. 
 
CHAPTER IV. 
 
 CONDUCTION. 
 
 Movement of Entropy. When one body is in 
 contact with another and loses heat to it, it is usual 
 to say that the heat moves from the hotter to the 
 colder body, just as if the heat were caloric, that 
 moved from one place to another. But what really 
 happens is that the heat of one body increases as 
 the other diminishes. Similarly, energy is said to 
 move. Light, electricity, sound and waves on water 
 are all regarded as moving. When a body is in 
 contact with another, and decreases its entropy at 
 the same rate as the other increases, it is most con- 
 venient to think of entropy as moving from one to 
 the other, just as we think of heat moving. As the 
 entropy of a body cannot decrease without heat 
 coming out, while it can be increased without heat 
 coming in, it is easy to think of entropy as moving 
 from one body to another, and as increasing or 
 growing in any body, but never as disappearing. 
 Thus entropy may move, and may grow, but never 
 disappears. There are sources, but no sinks. I 
 do not know if entropy is considered as moving 
 in any treatises on thermodynamics, but think 
 
CONDUCTION OF HEAT. 95 
 
 not ; but that does not make the idea any less 
 useful or valuable. 
 
 Conduction of Heat. Conduction of heat is a 
 very important irreversible process. Suppose a hot 
 body, say a vessel containing 10 Ibs. of water at 
 650 F.A., its entropy being 2.79, is put in contact 
 with another of the same size, but at a temperature 
 of 550 F.A., its entropy being 1.09, the two will 
 come to the mean temperature of 600 F.A., and 
 their entropies will be 1.97 each, or 3.94 together. 
 
 The total entropy has thus gone up from 3.88 to 
 3.94, showing incurred waste of 0,06 multiplied by 
 the lowest available temperature. Suppose the 
 lowest available temperature to be 550, the waste 
 is thus 33 British thermal units. 
 
 If the 10 Ib. of water at 650 were put in 
 connection with a thermodynamic engine which 
 could take heat at a gradually falling temperature 
 and return the waste heat at 550, a certain output 
 in work would be obtained. Of course the water 
 cannot give out all its heat at 650; the mutivity 
 (#i ~~ 0a)'0i> or changeability of the heat into work, falls 
 with the temperature. The water loses as much heat 
 between 650 and 600 as between 600 and 550, 
 but more of the heat given out between 650 and 
 600 is convertible into work. If one vessel gives 
 heat to a thermodynamic engine from 650 down to 
 550, there will thus be more work obtainable than 
 if two gave out heat from 600 to 550. 
 
96 
 
 CONDUCTION. 
 
 This is shown clearly by the < diagram. Fig. 12 
 is the 0$ diagram for 10 Ib. of water from 550 to 
 650 F. A. The area A C F I represents Ui - U 2 , or 
 U at 650 F.A., taking 550 as zero. It is obviously 
 equal to 10 X 100, or 1,000 British thermal units. 
 The available part of the energy is 
 
 W= 
 
 The first term is the whole of the heat needed 
 to raise the temperature from 550 to 
 650, or 1,000 British thermal units, 
 and is shown by the area A C F I. 
 The second term is the waste A D F I, 
 for A D = $ = 10 log. OjOto and 3 is 
 I A. So the available part is the area 
 A C D. If a vertical line B G is 
 drawn so that G B = 600, A E is 
 the entropy of 10 Ib. of water at that 
 temperature, taking 550 as zero ; and 
 A B G I is 500 British thermal units. 
 One vessel of water thus has the heat 
 
 I Q 
 
 A C D available or convertible into 
 work to start with, while the other has none. On 
 equalising temperatures there are two vessels, each 
 with only ABE available ; and A C I) is obviously 
 greater than twice ABE. As the area A B G I is 
 equal to B C F G, A E is greater than E D, so the 
 entropy A D of the vessel at 650 is less than 
 that of the two vessels at 600, or twice A E, 
 
CONDUCTION OF HEAT. 97 
 
 and the waste, twice A E G I, is greater than 
 AD FI. 
 
 When a hot body communicates heat to one of a 
 lower temperature, there is thus incurred waste or 
 increase of entropy of the two. Earlier in this book 
 reversible changes have been discussed in which 
 heat went from a reservoir to a perfect gas and 
 expanded it, and so on. The provision was made, 
 however, that the difference of temperature was 
 sensibly nothing, so as to give no perceptible down 
 grade of temperature, and no increase of entropy 
 due to irreversible conduction. 
 
 It has been shown that if a hot and cold body 
 are put in contact with each other and allowed to 
 equalise their temperature, there is a growth of en- 
 tropy ; but only the final result has been considered. 
 It may be as well to discuss what happens during 
 the change. Suppose a reservoir of heat that is 
 to say, an ideal bod} 7 so large that it can give out 
 heat without falling in temperature is put in con- 
 tact with, say, a mass of iron at a lower temperature. 
 Suppose for the moment that the conductivity of 
 the ideal body is so enormous that the part of it 
 against the iron remains at the same temperature 
 as the rest of the reservoir, say O lt The face of the 
 part of the iron block touching the reservoir is then 
 at temperature O lm If the block, to begin with, is 
 all at uniform temperature, as soon as contact takes 
 place the block is at #1 on one face, and at lower 
 
 E. H 
 
98 CONDUCTION. 
 
 temperatures elsewhere, and the temperature of the 
 block is no longer uniform. In a given time the 
 reservoir has lost heat, say HI, at constant tempera- 
 ture QI, so its entropy has been reduced by HI/#I. 
 The block has received heat HI at 6\, as #1 is the 
 temperature of the face or part of the envelope 
 through which the heat HI passes, so its entropy 
 increase, as far as heat from outside is concerned, 
 is HI/#I. But at the end of the process, when the 
 block has come to temperature O lt and taken in HI, 
 its increase of entropy is greater than HI/#I. There 
 has been a growth of entropy in the volume of the 
 block. This volume growth is evidently the un- 
 compensated entropy due to the irreversible nature 
 of conduction. If the block takes in heat HI at 
 temperature 0i, its increase of entropy is thus greater 
 than HI/#I ; and if the temperature of the face varies 
 during the change, the entropy taken in isy d H/0 
 the compensated entropy ; but the entropy of the 
 block is greater ihaufd H/0 by the uncompensated 
 entropy, or $ >J d H/0, as in all irreversible or 
 real changes. It must be borne in mind that is 
 the temperature of the face of the block in contact 
 with the reservoir, not the temperature of the block. 
 The temperature of the block varies from part to 
 part. To find how the entropy grows during con- 
 duction, imagine a bar of, say, copper of uniform 
 section, say, a square inch. Let it be kept hot at 
 one end and cold at the other, and be wrapped in 
 
CONDUCTION OF HEAT. 99 
 
 heat-insulating lagging along its length. Imagine 
 a surface cutting this bar across so that all this 
 surface is at one temperature 0i. The copper on 
 each side of this surface is either hotter or colder, 
 but the surface is at 1} and remains at i9 because 
 the copper is in a steady state, with heat flowing 
 along steadily and continually at uniform rate. An 
 inch along toward the colder end imagine a second 
 surface to cut the bar at right angles, and let its 
 temperature be 0* Let O l be 132 Fahr., and 2 be 
 32 Fahr. Then the flow of energy, calculated 
 *rom the thermal conductivity of copper, is about 
 0.5 British thermal unit per second. Through the 
 hotter face we thus have each second H = 0.5 and 
 #1 = 593 F.A., so there is an intake of entropy of 
 0.00083 per second. At the cold face there is an 
 outflow of H = 0.5 at 2 = 493, and the outflow of 
 entropy per second 0.00103. There is thus a 
 volume growth of entropy of 0.0092 per second in 
 the cubic inch of copper. If freezing-point is the 
 lowest temperature, the waste due to the irreversi- 
 bility comes out at 0.14 British horse-power. 
 
 Whenever there is conduction of heat there is 
 thus growth of entropy throughout the volume of 
 the conducting body. Thus the fall of temperature 
 between the immediate surface of the hot coal and 
 the water is all by continuous temperature gradients 
 that is to say, by conduction. 
 
 There is great growth of entropy in the furnace 
 
 ii 2 
 
100 CONDUCTION. 
 
 gas between the spot where the conduction takes 
 place, and the boiler flues or tubes. There is 
 growth of entropy in the external scale or dirt ; a 
 little growth in the metal separating the gases from 
 the water ; considerable growth in the scale. Boiler 
 scale is a sort of entropy hot-bed. There is more 
 growth of entropy in conduction in the boiler ; but 
 this is not serious. Then there is growth in the 
 metal of the valve that separates steam at different 
 temperatures. The growth of incurred waste in a 
 large slide valve, such as used to be employed in 
 marine engines, may amount to about 10 British 
 horse-power, though there is no loss of heat. Then 
 there is waste in conduction from the jacket to the 
 cylinder, from the steam to the metal walls and 
 back again, and so on. All this growth of entropy 
 means waste of energy that it is the engineer's 
 object to keep down. All this entropy has nothing 
 to do with the waste of heat by radiation and con- 
 duction. When heat is radiated from a pipe, for 
 example, at the rate of, say, enough British thermal 
 units per second to amount to 10 British horse- 
 power, whatever that number may be, it does not 
 deduct 10 British horse-power from the engine 
 output, because the engine would only have used 
 some of the heat anyhow. 
 
 The growth of entropy in conduction of heat is 
 not generally treated in this way ; in fact, the method 
 may be original. Writers on thermodynamics 
 
CONDUCTION OF HEAT. 101 
 
 even specialists do not usually give any indication 
 of what they mean by or p in irreversible changes. 
 In discussing conduction in reversible changes, is 
 the temperature of the reservoir or of the working 
 substance, or both, as it is the same, and the entropy 
 equation isJdH/0 = 3>. Then in the case of 
 irreversible change we are frequently told that 
 J (H{/0 = ^ is still true, which is nonsense anyhow, 
 or that fd H/0 < 3>, which is vague if we are not 
 told what 6 is. There may be no reservoir, as the 
 heat H may come from a body whose temperature 
 is not uniform. If the working substance is also of 
 varying temperature, cannot mean the temperature 
 of the working substance. Personally, I have 
 always read 6 to be the temperature of the bounding 
 surface through which the heat H passes, as that 
 is the only reading which conveys any physical 
 meaning to me ; but I have since found I am 
 rather peculiar in this. What idea is conveyed by 
 J d H/0 < $ to those who do not think of as the 
 temperature of the bounding surface I cannot con- 
 ceive. There seems to be a large class of people 
 who are mathematically trained only to the point 
 of having facility in the blind manipulation of 
 mathematical symbols, and they have the extra- 
 ordinary faculty of being able to read, and even 
 write, mathematical symbols, and to get correct 
 results without having any clear idea of what they 
 are doing. Of course, this involves a very degraded 
 
102 CONDUCTION. 
 
 type of mathematics. Sometimes it is held that 
 still means the temperature of the conducting body, 
 and the body must be split up into elements each 
 so small that its temperature is sensibly constant. 
 Thus, if there are two bodies at different tempera- 
 tures, but each at uniform temperature through- 
 out, the hot one may give heat slowly to the cold 
 one. When they are equalised, the total entropy has 
 increased. Of each it is true that fd H/# = <I>, 
 where is the temperature of the body. It is, 
 therefore, it is said, true of each element of a body, 
 if the body is not at uniform temperature ; but the 
 elements are so small that each is at a uniform 
 temperature, so that for the whole body fd H/0 
 = <. Of course, this is dreadful confusion ; H and 
 6 have different meanings in the two cases. The 
 underlying idea is also confused. If a hot and a 
 cold bod} 7 are each at a uniform temperature, and 
 the hot gives up heat to the cold, there must be 
 something between them in which the conduction 
 of heat and temperature gradient takes place. That 
 something, in a given time, takes in H at O l and 
 gives out H at 0. 2 , if not itself absorbing or giving 
 out heat. The entropy growth is therefore going 
 on in it. The growth goes on when there is con- 
 duction of heat and a down grade of temperature ; 
 and to assume that a conducting body can be divided 
 up into elements, each uniform in temperature, 
 involves the body being made up of something like 
 
THE UNIT OF ENTROPY. 103 
 
 the bricks of a house with hadly conducting mortar 
 between them. The mortar may be thinned with- 
 out limit, but if the down gradient of temperature 
 is there and a flow of heat across, the uncompen- 
 sated entropy grows entirely in the mortar. To 
 neglect the mortar is to make out that a body con- 
 ducting heat is made up of elements, each of which 
 is undergoing a reversible change, while the whole 
 undergoes an irreversible change. The argument 
 is like saying that a path down a hill can be divided 
 up into elements so small that the height of each is 
 uniform. Water running over each, therefore, 
 does no work, so water running down the hill does 
 no work. This point is emphasised here because 
 it is not generally realised that there is a volume 
 growth of uncompen sated entropy wherever there 
 is a temperature gradient. This is a very convenient 
 idea, as it shows that all conduction or radiation of 
 heat involves waste, the power wasted depending 
 solely on the temperature and its rate of decrease, 
 and the conductivity of the material. 
 
 The Unit of Entropy. Nomenclature is one of 
 the weak points of thermodynamics. There is no 
 name for the unit of temperature, and there are two 
 thermometer scales used in England and two on the 
 Continent. None reads exactly in accordance with 
 the Kelvin or perfect gas scale. The unit of heat, 
 which is quite an unnecessary nuisance, has no name, 
 for British thermal unit is not a name ; it is an 
 
104 CONDUCTION. 
 
 opprobrious epithet. The unit of entropy has not 
 even that. It might be called the British entropic 
 unit. I have elsewhere used the term " entrop " to 
 denote the unit, also the term " claus," short for 
 Clausius, for the practical unit of entropy on the 
 C.G.S. system. It is usual to derive the practical 
 unit on the C.G.S. system from a man's name, as 
 the Ohm, Volt, Joule, &c. The claus is the entropy 
 which, when multiplied by the lowest available 
 temperature in absolute Centigrade degrees, gives 
 the incurred waste in joules, not in calories, either 
 of the big kind or of the little kind, or of the middle- 
 sized, if there is one yet. The British entropic 
 unit, as used in this book, is the entropy which, 
 when multiplied by the lowest available absolute 
 temperature, gives the incurred waste in British 
 thermal units. " Entrop " is shorter and more con- 
 venient than " British entropic unit." It does not 
 matter which thermometer scale is used in the 
 case of the entrop, for if a body say 10 Ib. of 
 water at the ' temperature of boiling water has 
 3.14 entrops the waste, if all the 180 British 
 thermal units are given out at freezing temperature, 
 will be 180 by 3.14 British thermal units. But if 
 degrees Centigrade are used, there will only be 
 100 thermal units, whose nationality it may be 
 difficult to get an}' civilised country to acknowledge, 
 so the waste will be 100 X 3.14, which is the same 
 heat as before. The claus refers to absolute Centi- 
 
PHYSICAL MEANING OF' ENTEOPY. 105 
 
 grade degrees only, as the alteration of the thermo- 
 meter scale, though it would change the magnitude 
 of each of the various calories, would not affect 
 the joule. 
 
 It may seem strange that a standard state has to 
 be taken for zero entropy, instead of simply finding 
 the total entropy of a body and stating it. The 
 usual reason given is that the total entropy of a body 
 is unknown. The real reason is that the entropy of 
 a body is necessarily infinite. 
 
 If a body of unit mass is imagined at zero tem- 
 perature, and that is taken as zero entropy, because 
 as no heat can be given out no further reduction of 
 entropy is possible ; to heat it up to any finite 
 temperature 0, the increase of entropy necessary is 
 y^ d0/0 = <. So < = log. $i - log. ; but log. or 
 log. is negative infinity, so the entropy needed 
 to raise the bod} 7 to any finite temperature is 
 infinite. 
 
 Physical Meaning of Entropy. So far entropy has 
 been treated as a function of the co-ordinates, or of 
 the incurred waste ; but the question arises as to 
 whether it has any further physical meaning. For 
 instance, if a body is heated, we picture the change 
 of temperature as being accompanied by change of 
 molecular motion. The difficulty with entropy is 
 that it is a quantity whose increase denotes incurred 
 waste, and it is thus very general, so that entropy is 
 increased in different ways in different cases, and 
 
106 CONDUCTION. 
 
 there is no one molecular change that corresponds 
 with increase of entropy. 
 
 Take first the case of a perfect gas. The tem- 
 perature there corresponds with the kinetic energy 
 of the molecules. Suppose, for simplicity, the 
 molecules or atoms are all moving at the same 
 speed, and their total kinetic energy is the same 
 as in the perfect gas, with its molecules at various 
 speeds, the equivalent uniform speed may he called 
 the " equivalent speed." It is not the mean speed, 
 because kinetic energy varies as the square of the 
 speed. Then if the gas is heated at constant 
 volume, the entropy varies as the logarithm of the 
 equivalent kinetic energy, or of the square of the 
 equivalent speed. In this case the entropy increases 
 as a function of the heat of the gas. Suppose now 
 the gas is expanded isothermally, its internal energy 
 and equivalent speed remain the same as before, 
 but the entropy increases as the logarithm of the 
 volume. This has nothing to do with the kinetic 
 energy of the molecules or equivalent speed, as they 
 are not altered. What is altered is the infrequency 
 of collision and the mean path, both in the same 
 sense and to the same degree. The entropy is thus 
 proportional to the logarithm of the length of the 
 equivalent free path of the molecules. Thus when 
 a perfect gas is compressed adiabatically, its entropy 
 is not increased, though it gets hot, as the increase 
 of the logarithm of the square of the equivalent 
 
PHYSICAL MEANING OF ENTROPY. 107 
 
 speed is exactly balanced by the decrease of the 
 logarithm of the equivalent free path. 
 
 In heating a substance which undergoes no 
 physical change, such as a solid, the entropy- is 
 again proportional to the logarithm of the square 
 of some sort of equivalent molecular speed. When 
 the solid melts, or a liquid vaporises, there appears 
 to be some change of kinetic energy by regrouping 
 the parts of the moving systems so that they still 
 communicate the same energy to neighbouring 
 bodies by battering them, but have some addi- 
 tional energy, such as of rotation, which is not com- 
 municated by battering. The entropy increase of 
 physical change is thus still proportional to the 
 logarithm of the square of an equivalent speed. 
 
 One reason why there is obscurity as to the 
 physical meaning of entropy is that it is treated in 
 orthodox thermodynamics as one quantity. Some 
 heterodox observations here may do no harm. 
 Entropy is really the sum of several quantities. 
 In the case of a pound of a perfect gas increasing 
 its entropy by being heated at constant volume, the 
 increase of entropy is proportional to the logarithm 
 of the equivalent momentum of the particles. That 
 is one distinct quantity, which I have called ^, or 
 the quantity factor of sensible heat. If a pound of 
 ice is melted, the increase of entropy is again the 
 logarithm of some equivalent momentum. This 
 quantity I have called xp, or the quantity factor of 
 
108 CONDUCTION. 
 
 latent physical heat. What is often called the dis- 
 gregation energy of evaporating water, that is to 
 sa} r , the increase of internal energy not including 
 any external work done, is thus OXP* If chemical 
 work is done by taking in heat, its quantity factor is 
 yc, where \^ i g > again, the logarithm of some equi- 
 valent momentum of the particles forming the new 
 combination. Return now to the perfect gas, and 
 let it expand isothermally. It increases in volume, 
 and the increase of entropy is proportional to the 
 logarithm of the volume. We thus have X A >, X]>> X r > 
 and a function of the expansion which may be called 
 e, which is proportional to the logarithm of the 
 volume. The increase of entropy is the sum of 
 these four, or 
 
 <j> = x- s> + X/> + X* + - 
 
 We need not consider chemical changes here, so 
 we have 
 
 fe d 4 =fe d x* +fo d X p +fo dc. 
 
 Uncompensated entropy thus occurs in engineer- 
 ing in two ways. If heat is conducted from a hot 
 to a cold body the bodies average their energy up. 
 This means increasing the equivalent momentum of 
 their particles. Thus, if a shot going at 2,000 feet 
 per second and another at 1,000 were to collide and 
 glance off at equal velocities, so that the total 
 energy remains constant, as the initial energies 
 are as 4 to 1, the final energies are 2J times that 
 
PHYSICAL MEANING OF ENTROPY. 109 
 
 due to 1,000 feet per second, or the speed is as the 
 root of 2J, or nearly 1,600 feet per second, instead of 
 1,500, which would be the average speed. The mo- 
 menta before collision were as 2 and 1, the sum being 3, 
 whereas after collision the sum is in proportion of 3'2. 
 In averaging up their energy by conduction of heat 
 there is thus an increase of equivalent momentum, 
 and the logarithm of this is the increase of x* or of 
 <, as far as conduction goes. This increase is 
 obviously uncompensated, corresponding to an 
 irreversible change. It is clearly impossible to 
 reduce this equivalent momentum by making one 
 part hot and the other cold again without taking in 
 heat at a. high temperature and giving it out at a 
 lower, that is to say, the change incurs waste. 
 
 When the perfect gas expands isothermally, whether 
 it does external work or not, it is only changed as 
 regards the free path of its particles, but to bring it 
 back to its original state needs compression by 
 external work, and if a particle rebounds from a 
 wall that is approaching it, it increases its velocity, 
 so that if the compression is to bring the gas back 
 to its original state, heat must be given out and 
 work taken in, that is to sa} r , there must be degra- 
 dation of work into heat, of which a portion must 
 be waste. 
 
 Thus x and e have this in common, that their 
 increase means that to get the substance back to its 
 standard state, even reversibly, involves giving out 
 
110 CONDUCTION. 
 
 heat. When a gas expands isentropically its increase 
 of c is exactly balanced by its decrease of x> as it 
 cools, and the entropy remains constant. Splitting 
 entropy up into the sum of a number of quantities 
 which have nothing in common except their relation 
 to waste is new and unorthodox ; but if the reader 
 wants to have a physical idea of entropy I believe it 
 is quite necessary. In thermodynamics the entropy 
 is a sum of quantities having only one property 
 in common, and has therefore no single physical 
 meaning except as regards the results of its 
 increase. 
 
 It may be urged that it is highly artificial 
 for x and c and their sum < to vary as the 
 logarithm of the equivalent momentum, the 
 logarithm of the volume or free path, and so on ; 
 whereas volume and momentum are factors of 
 energy. The anstfer is that we have by a pure 
 accident taken temperature so that it varies as the 
 square of the equivalent velocity of the particles. 
 If we take as the tension factor T, so that T = Jt 9 
 and TT as the corresponding factor of heat, we 
 get KB, irj) t and ire, varying, not with the logarithms 
 of momentum, but with the equivalent momentum ; 
 and i], corresponding with for the expansion func- 
 tion, which varies directly as the volume, and not as 
 its logarithm. Splitting up heat into factors, and 
 abolishing the idea of entropy except as the sum of 
 the quantity factors and the expansion function 
 
FACTORS OF HEAT. Ill 
 
 will, I feel convinced, make the study of thermo- 
 dynamics more simple. 
 
 Conclusion. Though everything in this book is, 
 I believe, consistent with modern thermodynamics, 
 the whole method of treatment is practically 
 new. It has, as far as possible, been pointed out 
 when anything may be unorthodox, so that the 
 reader who is new to the subject may be on his 
 guard. 
 
 As far as possible, I have tried to put a difficult 
 matter in plain English, and not in mathematical 
 symbols. This is a very thankless attempt, for if 
 one were to succeed in making a difficult subject 
 appear clear and simple, the reader would think the 
 subject itself easy, or that it has only been treated 
 in a very elementary way, and very often that the 
 writer's knowledge is also equally elementary, 
 perhaps untrustworthy. The sa"me subject treated 
 in an involved way, with a quantity of obscure 
 mathematics a first-rate man's mathematics are 
 always clear, however, and his writing as simple as 
 the case allows is supposed to be more advanced 
 and deeper, and to have much more in it. 
 
 Few realise that ease of reading varies inversely 
 about as the square of easy writing, and if one 
 wants kudos, the right thing is to make the subject 
 as complicated and difficult as possible, and espe- 
 cially to use as many mathematical symbols as can be 
 crowded in. It is quite the correct thing in science, 
 
112 CONDUCTION. 
 
 after making calculations, to " remove the scaffold- 
 ing," that is to say, to publish without giving the 
 intermediate steps, so that others should think the 
 feat greater. 
 
 In dealing even with an old subject in a novel 
 way, which involves criticism of existing methods, 
 it is necessary for a writer to rely on his own 
 reasoning from step to step, as he cannot rest on 
 others whose way is different, and, in his opinion, 
 inaccurate. It is therefore most likely there are 
 many slips and false steps in this essay. I am 
 not young enough to be at all infallible ; one would 
 have to be ver} r young indeed to be infallible in 
 thermodynamics, which is perhaps the most slippery 
 branch of science there is. It is hoped anyone who 
 notices any slips will write to me, and advantage 
 will be gratefully taken of any corrections or sug- 
 gestions, if a second edition should be demanded. 
 
APPENDIX. 
 
 THE REVERSIBILITY OF THERMO- 
 DYNAMICS. 
 
 A NEW way of looking at an old subject can do no 
 harm, and may do, and generally does, a great deal 
 of good. In this appendix it is submitted that the 
 ordinary discussion of the science of thermody- 
 namics makes the subject unnecessarily obscure, and 
 leads to all sorts of errors ; and my object is to urge 
 a different treatment which, it is hoped, will make 
 the science more easily understood. 
 
 There is no branch of physical science so com- 
 monly misunderstood, not only by students, but 
 also by scientific men of considerable standing, as 
 thermodynamics. Some time ago I asked an 
 eminent authority on the Continent to join in a dis- 
 cussion on entropy which had arisen out of a note 
 to the presidential address I had the honour of 
 delivering before the Institution of Electrical En- 
 gineers. He replied that though I was quite right 
 in my statement that most writers on physics had 
 got hold of a wrong idea of entropy, the matter 
 was merely pedagogic, and therefore he would not 
 
 E. I 
 
114 APPENDIX. 
 
 discuss it in print. A question of pedagogy would 
 be outside the province of an engineer ; but as it is 
 not really a question of method of teaching young 
 men at colleges, but of giving clear ideas to scientific 
 men who are not specialists in thermodynamics, the 
 subject is of real scientific importance ; and I will- 
 ingly acceded to the request of the Committee of the 
 Mathematical and Physical Section of the British 
 Association to open a discussion on thermodynamics, 
 and this appendix consists essentially of the paper 
 written for that purpose. 
 
 Present or Orthodox Treatment. The study of 
 thermodynamics suffers from the historical develop- 
 ment of the science. During the first half of last 
 century the principle of the conservation of energy 
 was on its trial, and, though he soon discovered his 
 mistake, Carnot had supposed that in his cycle as 
 much heat came out of the working substance as 
 went in. I pretend to no knowledge of the history 
 of thermodynamics, but it looks as if Clausius, 
 realising that the heat is not conservative in a Carnot 
 cycle, found a function S, which is. Then came the 
 question of the second law arid the best way of 
 formulating it, and the whole discussion centred 
 round the Carnot cycle with dQ/T a complete dif- 
 ferential. Instead of S, Q and T, I have used 3>, H 
 and 6 in this book, as Q is generally used in other 
 branches of physics for the quantity factor of energy, 
 not for energy, and T for kinetic energy ; and if $ is 
 
ORTHODOX TREATMENT. 115 
 
 used for entropy, may well be used for tempera- 
 ture, as by Rankine, Maxwell and others. 
 
 The first matter generally is to state the law of 
 the conservation of energy and to define " heat." 
 Here there is a complete failure. The definitions 
 and explanations really come down to saying in a 
 more or less roundabout and complicated way that 
 heat is what makes things hot. In its way this 
 would be quite a nice and all-sufficing definition, 
 but it is not adhered to at all. The domination of 
 a name comes in. The increase of internal energy 
 necessary to produce a change of (physical) state, 
 such as melting ice, was called heat before thermo- 
 dynamics began. Objectors to the caloric theory, I 
 think, refused to call this energy heat, in order to 
 annoy their opponents ; but when the caloric advo- 
 cates disappeared by dying out, or perhaps even by 
 being occasionally convinced, increase of internal 
 energy became latent heat again ; and then the 
 increase of internal energy and external work 
 together became "latent heat," having a symbol 
 Cp or Kp to itself, so that increase of U where there 
 is no chemical change is heat, and the external 
 work done per degree rise of temperature is part of 
 the latent heat of the body, so the definition goes by 
 the board, and heat that makes things hot is called 
 " sensible," perhaps as a stigma on the other kinds 
 of heat. The definition of heat is thus wanting. I 
 will quote from some correspondence I had with one 
 
 i2 
 
116 APPENDIX. 
 
 of our leading mathematical authorities on thermo- 
 dynamics. " What is wanted is a definition of heat. 
 Can you define heat ? If you can give a definition 
 which will make it perfectly clear what is heat and 
 what is not heat, you will be doing good work. But 
 you must cover all possible cases." The italics are 
 his. Now what does my correspondent want ? 
 There are two classes of definition in science. One 
 is a mere verbal distinction which fits pre-existing 
 ideas and conveys no information. For instance, 
 everyone of my readers has, no doubt, a perfectly 
 clear idea of what energy is, and also knows what is 
 energy and what is not, in any given case ; but to define 
 energy so as to convey any information to anybody 
 who has no pre-existing notion of it is no easy task. 
 But perhaps what he really meant was something like 
 this. " Is latent heat, heat ? Surely external work 
 is not heat. Is chemical energy heat ? How far is 
 radiation, thermal or luminous, heat ? Are electron 
 flights ever heat ? If so, when are they kinetic energy, 
 and when heat ? When are electro-magnetic waves 
 heat ? Is the energy of pedesis heat ? If so, and 
 the little particles are made to jolt bigger particles, 
 and so on till they finally shake lumps, is that heat 
 or directed kinetic energy, and where is the line of 
 demarcation ? At what stages in the electrostatic 
 or electro-magnetic hysteresis cycles does the 
 entropy increase ? " I do not know whether a 
 definition which would answer these questions would 
 
ORTHODOX TREATMENT. 117 
 
 satisfy my friend, but some such definition is 
 wanted. Many of these questions are, no doubt, 
 answered by inference in papers on those particular 
 subjects written from a thermod} 7 namical standpoint; 
 but an ordinary treatise on thermodynamics does 
 not touch them, or answer my friend's question in 
 the least. 
 
 Then there is a difficulty about thermometry. 
 The Fahrenheit scale takes the expansion of mercury 
 one-hundredth per cent, as a " degree," the Centi- 
 grade divides the difference of two arbitrary tempera- 
 tures into 100 parts, and is therefore regarded as 
 more scientific, and the Reaumur exists in real life 
 but not in science. Much trouble is therefore devoted 
 to elucidating Kelvin's scale, and it is then gene- 
 rally assumed that the Fahrenheit and Centigrade 
 thermometers read in the Kelvin scale, only with 
 different zeros and different units of temperature. 
 The main point to which attention is especially 
 directed, however, is, that writers are so eager 
 about showing that production of work means dis- 
 appearance of heat, and so anxious to explain 
 Kelvin's scale and to discuss the second law and 
 Carnot's cycle, that the} r discuss the whole subject 
 only in connection with ideal reversible changes. 
 Then the fact of dH/0 being a complete differential 
 in reversible changes is very attractive to many 
 people, because they can then get a lot of un- 
 deserved happiness in stringing together partial 
 
118 APPENDIX. 
 
 differential equations tracing the relations of every 
 conceivable thermoclynamic quantity with every other, 
 down to the most remote cousinships. Thermo- 
 dynamics is thus apt to degenerate into nothing 
 better than an exercise in differential equations. 
 
 In discussing the principle of entropy, the entropy 
 of the working substance is treated almost exclu- 
 sively, and that in connection with reversible changes 
 and cycles alone. Irreversible changes are dis- 
 cussed as a sort of curious exception, but only very 
 slightly. In many cases two bodies are considered 
 before and after equalising their temperatures, and 
 their increase of total entropy is discussed, but 
 there is no explanation of how the entropy increases; 
 only the initial and final stages are considered. 
 
 But the chief ground of the accusation I have the 
 honour to bring is that anxiety to discuss the Carnot 
 cycle and its bearing on the second law, and to work 
 with a quantity 4> which could be assumed conserva- 
 tive, has led to a very elaborate mathematical 
 development of the theory of reversible changes, 
 and has relegated real thermodynamics to an 
 undeserved background. 
 
 My historical notions may be wrong, but it looks 
 as if Clausius, when Carnot's reasoning had been 
 vitiated by his original mistake as to conservation of 
 heat, looked for something that was conservative in 
 the Carnot cycle and found entropy. He got the 
 " tropie " from the Greek, and put on the "en" 
 
ORTHODOX TREATMENT. 119 
 
 to make the word sound something like its conserva- 
 tive analogue " energie." Then in discussing real 
 changes he found that entropy is not conservative, 
 and founded the principle of modern thermo- 
 dynamics, that the entropy of the universe strives to 
 increase ; a principle which is the very backbone of 
 the science. In spite of this, the ordinary treatise, 
 though it mentions Clausius's principle, and perhaps 
 touches on Kelvin's work of the same period, says 
 very little about it. By "ordinary treatise," I 
 need hardly say, I do not mean to include such 
 works as those of Planck or Duhem, or those written 
 by the abler modern physical chemists. 
 
 Entropy is denned with reference to reversible 
 changes only by the equations 
 
 fdR/e = <*> and (/)dR/0 = 0, 
 
 which give no sort of notion what entropy is, and. 
 which are only numerically correct in the case of rever- 
 sible changes and cycles respectively, and, in fact, 
 are never even numerically true. The foundation 
 of thermodynamics is that neither of these equations 
 is true. I know of no writer who has tried to give 
 any sort of explanation of what is meant by entropy, 
 except that it is the quantity factor of heat, which is 
 obviously nonsense. It is not meant that specialists 
 on thermodynamics do not understand their own 
 subject or write inaccurately ; my complaint is that 
 the treatment of thermodynamics is obscure and 
 
120 APPENDIX. 
 
 misleading, and has lead to great confusion and 
 much error. 
 
 Nearly all writers on physics, for instance, define 
 entropy by the equation dH/6 = d, or its equivalent. 
 That is to say, they are led to suppose that what 
 would be true if there were any reversible changes 
 is true in fact. dH/0, or d, is treated as a typical 
 complete differential. An adiabatic change is sup- 
 posed to be isentropic. (y*)cZH/0 = is supposed 
 to be the second law of thermodynamics. Entropy 
 is said to be a factor of heat, and so on, a whole 
 tissue of errors and misconceptions thus arising 
 from the impression given by looseness of diction. 
 When any man of ordinary intelligence and the 
 necessary training cannot understand what has been 
 written, it may be taken for granted it is the fault 
 of the writer. Generally, people are of an opposite 
 opinion, perhaps thoughtlessly, and if the treatment 
 of a subject proves to be puzzling, the author is 
 rather to be admired for being master of such a 
 difficult subject, and not blamed for not making it 
 simple ; whereas, if he put the matter simply and 
 clearly, he would get but little credit for his work. 
 This is especially true of subjects that can be 
 treated mathematically ; the scientific world has an 
 immense reverence for anything put in mathematical 
 language, though it needs much more ability and a 
 clearer head to put it in words. Was it not Maxwell 
 who said that mathematics is a shorthand, and that 
 
ORTHODOX TEEATMENT. 121 
 
 anything that can be put in symbols can be put in 
 words, if the writer really understands it ? Nobody 
 really understands unless he can express himself in 
 words. Thermodynamics seems to be peculiarly 
 unfortunate in being a vehicle for blind mathematics. 
 It is not for a moment hinted that science cannot 
 be clear if it is mathematical ; quite the reverse. The 
 great writers on physics use mathematics in such a 
 way that they are merely shorthand methods of 
 explaining physical ideas. Any reader grasps the 
 physical ideas at once, unless his^mathematical equip- 
 ment is too limited ; but even then he feels the 
 physical ideas are there, and very often finds the study 
 a good way of learning mathematics. On the other 
 hand, another writer will make the simplest matter 
 unintelligible by mathematical treatment. Thus 
 such an expresion 'dsfdH/0 = $ is a perfectly clear 
 statement of fiction. But when we come to fact we 
 are told that^dH/0<$. Now what does that mean ? 
 Being an engineer, in coming across such an 
 expression I sought an interpretation that gave a 
 physical meaning. could not be the temperature 
 of the reservoir, because there may be no reservoir ; 
 for instance, the body ceding heat may not be at 
 uniform temperature, for example, in considering 
 part of a conductor of heat where there is a tempera- 
 ture gradient. Then cannot generally be the 
 temperature of the body because that is not usually 
 uniform. To make sense, is the temperature of 
 
122 APPENDIX. 
 
 the bounding surface, and H is a flux through the 
 surface. But I find I appear to be rather peculiar 
 in this interpretation. What then does the orthodox 
 writer mean by H and in irreversible changes, and 
 what idea is really in the mind of the reader as he 
 glides over such expressions? That there is a 
 tendency towards blind mathematics and fogginess 
 of idea in thermodynamics is, I submit, proved by 
 the inaccuracy of the terms used, the absence of 
 named units, and the absence of physical ideas. 
 How many chemists have any clear-cut idea of 
 what the " Thermodynamic Potentials " really 
 are? 
 
 The entropy principle was formulated half a 
 century ago. Before and since then the question 
 of what determines the direction of chemical action 
 was eagerly discussed. The entropy principle was 
 supposed to be familiar to every scientific man, but 
 it was not assimilated in the least. After about 
 twenty years the chemical puzzle was solved by Horst- 
 mann, who pointed out that the increase of entropy 
 is the criterion. Bayleigh, doubtless independently, 
 said the same in 1875 ; then followed Massieu, 
 Gibbs, and Helniholtz. Rayleigh said it in plain 
 English ; the others, especially Massieu, said it in 
 mathematics. But even that has produced but 
 little effect, and now, after more than half a century, 
 the principle of increase of entropy is by no means 
 generally understood by chemists, nor by engineers, 
 
OKTHODOX TREATMENT. 123 
 
 though they are both exceedingly anxious to under- 
 stand and use thermodynamics. This might be 
 the fault of chemists, engineers, and other scientific 
 men, but it seems much more likely to be due to 
 obscurities in thermodynamics. 
 
 It is easy to get the reputation of a reformer by 
 finding fault with existing conditions. It is more 
 to the purpose to show what one thinks is an 
 improvement by a specimen. I have, therefore, 
 attempted a short elementary exposition, especially 
 as far as it concerns mechanical, apart from chemical, 
 engineers. This forms the body of this book. In 
 this is exemplified a treatment of elementary thermo- 
 dynamics, which depends largely on the reversal of 
 the ordinary method. For instance, dissipation and 
 waste are explained before reversibility and the 
 Carnot cycle ; entropy is defined without any refer- 
 ence at first to heat passed into a body ; the increase 
 of entropy is treated as normal, and reversible 
 changes as a purely ideal exception ; the second law 
 of thermodynamics is merged in the impossibility of 
 perpetual motion ; the entropy of the universe and 
 of the isolated system is treated before that of the 
 working substance ; and the growth of entropy 
 during conduction is treated later. For that reason 
 this essay is headed as it is. To put the matter 
 succinctly, for easy grasp, a sort of sketch will now 
 be set out, which also summarises the body of the 
 previous chapters. 
 
124 APPENDIX. 
 
 SKELETON OF THERMODYNAMICS. 
 
 Energy can be divided broadly and clearly into 
 two kinds, which may be called work and heat. 
 Different kinds of work can be converted one into 
 another wholty in idea, and nearly wholly in fact ; 
 different kinds of heat can be converted wholly into 
 one another. Work can be wholly converted into 
 heat, but heat can only be partially converted into 
 work if the converting substance returns to its 
 original state, the rest of the heat being degraded 
 into a less available form. This definition of heat 
 includes the heat that makes things hot, and loco- 
 motive heat in general, and it also includes "latent 
 heat " at constant volume, but only part of any mis- 
 named "latent heat" that includes any form of 
 external work. It includes latent heat of fusion, of 
 vaporisation apart from external work, and of allo- 
 tropic modification. AVliat is most heterodox is 
 that it includes chemical energy. 
 
 There are three classes of perpetual motion. 
 The} r are all impossible, and two of them are not 
 approximately attainable, but third-class perpetual 
 motion is approximately possible. / 
 
 First-class perpetual motion is when an otherwise 
 isolated system can give out energy continually, 
 or without decreasing its own. The impossibility 
 of first-class perpetual motion, coupled with the 
 impossibility of an energy sink is the principle 
 
DEFINITION OF ENTKOPY. 125 
 
 of conservation of energy, and gives the first law of 
 thermodynamics . 
 
 Second-class perpetual motion is when an isolated 
 system, in spite of friction or its equivalent, goes on 
 continually. Second-class perpetual motion would 
 not contradict the principle of the conservation of 
 energy. Its impossibility is the second law of 
 thermodynamics. 
 
 Third-class perpetual motion is when an isolated 
 system a mechanism, for instance has no friction, 
 or no equivalent of it, and therefore goes on or 
 changes continually. The impossibility of third- 
 class perpetual motion is the third law of thermo- 
 dynamics. It is not called the third law, but it 
 deserves that rank. 
 
 Waste. Work is valuable to man, and the portion 
 of any heat that can be converted into work, leaving 
 the converting substance in its original state, is 
 valuable ; but the rest of the heat is waste. When 
 work is degraded into heat, a portion of it is avail- 
 able as work, the rest is waste. The third law of 
 thermodynamics states, in other words, that no 
 change in nature takes place without incurring waste. 
 
 Definition of Entropy. The increase of entropy 
 of an isolated system, multiplied by the lowest 
 available temperature, is the incurred waste. The 
 waste, therefore, depends on the lowest available 
 temperature and the entropy. As any given change 
 can only increase the lowest available temperature 
 
126 APPENDIX. 
 
 infinitesiinally, the second and third laws may be 
 stated in the form, " The entropy of the universe 
 strives to increase." So does the waste. Its maxi- 
 mum will be reached when all heat is unavailable and 
 there is no work left. The incurred waste increases 
 faster than the entropy of the universe, because 
 the lowest available temperature is always rising. 
 The limiting expression " incurred " is used before 
 waste, advisedly. Suppose, in an isolated system, 
 some gas expands doing some work external to the 
 gas. The heat of the isolated system is diminished, 
 the work increased, and the total energy unchanged. 
 The heat being diminished without any correspond- 
 ing rejection at lower temperature, the waste portion 
 is decreased. There is thus actual decrease of 
 waste. But, to get the gas back to its original 
 state, work must be degraded into heat again, and, 
 according to the third law, there is then increase of 
 waste. Had the process been reversible, there 
 would have been no increase of incurred waste in 
 the system, but, as far as the gas alone is concerned, 
 its own increase of entropy multiplied by the lowest 
 available temperature would be the waste involved 
 in bringing it back to its original state. 
 
 When during a change the entropy of a body 
 forming part of the system diminishes, the entropy 
 of another part must increase to the same extent 
 in an ideal change, and to a greater extent in a 
 real change. In the ideal change the increase of 
 
DEFINITION OF ENTROPY. 127 
 
 entropy of the body is called compensated, in a real 
 change the part of the increase equal to the 
 corresponding decrease elsewhere is compensated 
 entropy, <&, and the balance is uncompensated 
 entropy, u , where &u = Q fdH/Q. 
 
 When the heat of one body diminishes as that of 
 another in contact with it increases, it is usual to 
 say that the heat moves from one to the other. 
 Similarly, when the entropy of one body diminishes 
 while that of another in contact increases to the 
 same or a greater extent, it is convenient to say 
 the entropy moves and grows also. From the idea 
 of the entropy of an isolated system and its increase 
 in every change, it is easy to pass to the idea of the 
 increase or decrease of the entropy of a body. In 
 the case of reversibility the entropy of the isolated 
 system is conservative, so that the discussion of 
 the behaviour of any body or bodies may be treated 
 as in dynamics, by examining the changes and inter- 
 relations of the external co-ordinates ; and the 
 conservation of entropy, in the ideal case of reversi- 
 bility, is a very convenient assumption, especially 
 as it specially helps the treatment as in dynamics, 
 the conservation of entropy corresponding with the 
 conservation of energy. The exaggeration of the 
 importance of this treatment has done much to 
 foster incomplete understanding of thermodynamics. 
 
 By treating the increase of entropy of an isolated 
 svstern as the fundamental idea and variations of 
 
128 APPENDIX. 
 
 the entropies of the parts as derived ideas it is sub- 
 mitted that a much clearer notion is obtained. It 
 is more convenient in practice, however, to consider 
 the entropy of some substances or volumes which 
 form parts of isolated systems. In this connection 
 we have the following propositions : 
 
 Reversibility. A . reversible change is an ideal 
 change which could take place, with everything 
 involved, in the other direction. 
 
 When a reversible change takes place in an 
 isolated system there is no increase of incurred 
 waste, for if there were, on reversing there would 
 be decrease of incurred waste, and second-class 
 perpetual motion would be possible. 
 
 When a reversible change takes place, therefore, 
 the entropy of the isolated system remains constant. 
 \A reversible cycle is a series of reversible changes 
 which brings the working substance back to its 
 original condition. The entropy of an isolated 
 system is not altered by a reversible cycle being 
 performed inside it. 
 
 In any reversible change, the total entropy of the 
 system being constant, the increase of entropy of 
 any part is compensated by an equal decrease of 
 another part. The decrease of entropy of any body 
 can only take place by a process which involves at 
 least equal increase of entropy of the rest of an 
 isolated system. Thus the entropy of a body may 
 be reduced without its giving out heat b} r a therm o- 
 
REVERSIBILITY. 129 
 
 electric circuit, but that involves at least equal 
 increase of entropy elsewhere in the electric circuit. 
 Except in the case of a thermo-electric circuit, the 
 entropy of a body can only be reduced by its giving 
 out heat, for if it could be reduced by giving out 
 work, there would be no corresponding increase 
 outside, and second-class perpetual motion would 
 be possible. 
 
 The reversible increase of entropy of a body can 
 only take place by its taking in heat from outside 
 (except by a thermo circuit), otherwise the entropy 
 of an isolated system would be increased, and the 
 change would not be reversible. 
 
 The entropy of a body, compared with its entropy 
 in a standard state, is the entropy that must at least 
 come out of it in bringing it back to its standard 
 state ; or it is a quantity which, when multiplied by 
 the lowest available temperature, gives the waste 
 portion of the heat given out in bringing the body 
 to its standard state by reversible changes. 
 
 As the waste portion, or loss L, of heat given out 
 through the envelope, H, is H0 2 /#i where 6\ is the 
 temperature of the envelope, and 6% the lowest 
 available temperature; and as by definition 02$ = L, 
 < = H/0i, or if the temperature of the envelope varies 
 during the output of the heat H, &=fdHI0. Thus 
 the entropy of a body in state B compared with 
 state A is v A=Jdfi/0, where H is the heat that 
 would be given out or taken in if the change were 
 
 E. K 
 
130 APPENDIX. 
 
 reversible. The third law shows that in fact 
 <l> 3>A>ySH/#i, the excess being uncompensated 
 entropy & u . 
 
 The entropy of a body thus depends on its state, 
 and not on its past history. 
 
 Equilibrium. If any small change imagined in 
 an isolated system involves decrease of entropy, 
 that change is impossible, in accordance with the 
 second law. If it causes no change of entropy, there 
 is equilibrium as far as that change is concerned, and 
 no change will take place in either direction. If the 
 change would increase the entropy, the change is pos- 
 sible and may take place, but only in that direction. 
 
 A possible change thus involves increase of 
 entropy, or uncompensated entropy. 
 
 The uncompensated entropy of the working sub- 
 stances can be expressed in terms of the co-ordinates 
 of each working substance alone, without other 
 reference to the rest of the isolated system. Thus 
 in the case of a working substance, where <?,, is the 
 uncompensated entropy, H the heat taken in, 6 the 
 temperature of the envelope, U the heat of the body, 
 according to the definition given above, and W the 
 external work done. 
 
 d*u = d* - dH/0 and dR = dU + dW, 
 
 therefore d*u = d* - dU/6 - rZW/0, 
 and if there is equilibrium, 
 
EQUILIBRIUM. 131 
 
 If there is uncompensated entropy on making the 
 chane 
 
 The integral of this quantity is therefore written 
 with the sign reversed and disguised under the 
 name of a " Thermodynamic Potential," so that 
 increase of uncompensated entropy decreases instead 
 of increasing the quantity. Uncompensated entropy 
 is a drawback, and thermodynamic potential is an 
 advantage. It is with great respect to Professor 
 Planck or Professor van Laar * that it is submitted 
 that thermodynamic potential is a bad name for the 
 function 
 
 The third law shows that heat can never be 
 changed into work to the extent it would be by a 
 reversible cycle. The work that would be gained 
 by a reversible cycle, and is not gained in a real 
 change, may be called the " ungained work." 
 Professor Duhem t calls it the "uncompensated 
 work." The ungained work, W=y&W>, where 
 6 is, if the working substance is of uniform 
 temperature, the temperature of the envelope or 
 body. If the temperature is not uniform, 6 is the 
 temperature of each volume element in which the 
 uncompensated entropy grows, and integration is 
 
 * Or whoever is responsible for the expression. 
 t I think the term is originally due to Professor Duhem , 
 but my memory may be at fault. See p. 70. 
 
132 APPENDIX. 
 
 carried out throughout the volume ; but as the 
 expression is important only when W u = 0, the 
 meaning of 6 otherwise does not matter for the 
 moment. 
 
 If a small change involves decrease of ungained 
 work it is impossible ; if the ungained work is zero, 
 the substance and its externals are in equilibrium 
 as far as that change goes ; if the ungained work is 
 positive the change is possible, and may take place 
 in that direction only. As the ungained work does 
 not, unfortunate!}', exist outside the substance, nor 
 anywhere else as work, it is best not to call it W w , 
 but A or D. Then, if the ungained work is zero 
 during a proposed change, the " thermodynamic 
 potential " is said to remain constant. If the un- 
 gained work increases, the thermodynamic potential 
 is said to decrease. The ungained work due to 
 a small change may be represented in terms of 
 the co-ordinates of the working substance. 
 
 The external work, W, is general. It may be 
 done electrically, or by expansion. If by expansion 
 
 <ZW =pdv = dpi: vdp. 
 Then d A = 6d$> - dU + dW = d6* - &dO - dU + dW 
 
 (1) 
 
 (2) dt* =-</U-<Z\V 
 
EQUILIBRIUM. 183 
 
 (3) 
 
 (4) dp* =0d&-dU-dp 
 
 (5) <fc,.A=,/0$-dU 
 
 (6) 
 (7) 
 (8) 
 
 If the various partial differentials are taken as 
 zero, the condition of equilibrium, and the right- 
 hand terms, integrated without adding any constants, 
 and reversed in sign so that their decrease corre- 
 sponds with increased or ungained work, eight more 
 thermodynamic potentials are involved. Part of 
 No. 1 gives what is called "free energy." It is not 
 energy, and is in no way remarkable for freedom of any 
 kind. The results of three of these are the Gibbs's 
 functions, which have been called, first by Professor 
 Duhem, I believe, thermodynamic potentials, no 
 doubt because they are constant for equilibrium, 
 like potential in mechanics. Again, it is submitted 
 with much deference, that the name is misleading. 
 If the signs were not reversed, stabilities, or stabilit} r 
 functions, might be a better name. In non-thermal 
 reversible mechanics, equilibrium or maximum 
 stability is reached when the potential is a minimum. 
 Maximum stability can then be called minimum 
 potential ; but, surety, it does not follow from 
 this that maximum stability is always synonymous 
 with minimum potential, and that thermodynamic 
 
134 APPENDIX. 
 
 potential is a synonym for the negative of stability, 
 especially when applied to quantities that are not 
 energy, though of the same dimensions as energy. 
 Besides, the term potential is getting worn out. 
 Counting the eight potentials just given as one 
 specimen, I can think of nine kinds of potential used 
 in physics, and a collector could doubtless find many 
 more. They have various meanings and dimensions. 
 Four or five of them belong to thermodynamics. If 
 we have very many more potentials, the term may 
 become a little difficult to define accurately and 
 succinct!} 7 , though it is always impressive. 
 
 By defining entropy first in terms of the waste, 
 and dealing with the increase of entropy of -an 
 isolated system first, and coming to the relation of 
 increase of entropy to heat taken in reversibly later, 
 the chance of confusing entropy withy^H/0 generally, 
 and supposing entropy is a factor of heat is avoided. 
 This prepares the way for discussing the real factors 
 of heat, such that ^/&ZX = U, where # is the tension 
 and X the quantity factor. In other branches 
 of physics, not only fadb = W, where a and b are 
 tension and quantity factors, but where c is capacity 
 for the quantity, so that db/da = c, facda W, or 
 when c is constant, W = Ja 9 c. In that case the 
 tension factor is \/0 = T, an d the corresponding 
 quantity factor H, so thatyrdII = U. In a change 
 where there is no external work done and no 
 ungained work X is numerically equal to 3>, and is 
 
EQUILIBRIUM. 185 
 
 the logarithm of the square of some sort of equiva- 
 lent velocity of the particles. Thus in the case of 
 a perfect gas the entropy depends on the logarithm 
 of the square of the equivalent speed, and the loga- 
 rithm of the equivalent free path of the particles of 
 the gas if there is one, or of each if there are more. 
 
 The frequency of collision has to do with external 
 work, and X depends only on the speeds. The 
 physical meaning of r and n is much simpler; r 
 varies as the equivalent speed, and n as the equiva- 
 lent momentum. 
 
 Take the case of chemical energy as one of the 
 forms of heat. U can be divided up into Us, Up, U c 
 for sensible, latent physical and chemical heat. 
 f 6dTls = Us ; frdHc = Uc and so on. At present 
 heats * of combustion or combination are given in 
 the lump. In the future we may have tables giving 
 the 6 and x or the r and TT of each free element, and 
 their change on reaction ; or even their and </>. 
 Then we would know beforehand in which direction 
 each reaction would go. The question of the 
 factors of heat need not be discussed here ; it is a 
 large subject by itself. 
 
 * This is another example of the extraordinary looseness 
 of terminology that runs through thermodynamics. They 
 ought to be colds of combustion or combination, for the heat 
 of fusion or vaporisation is the heat taken in during the 
 change. The so-called heat of chemical change is given out 
 during the change. 
 
136 APPENDIX. 
 
 Heat Conduction. If a body is conducting heat 
 steadily, for instance, a uniform bar with one end 
 kept hot and the other cold and no side leakage, the 
 bar can be cut up into lengths by cross cuts imagined 
 so that each cut is at uniform temperature. Let 
 one be at #1 and the next at 2 > and let heat H pass 
 per second. Every second H/0j. comes in and H/0 2 
 comes out, so that there is a volume growth of 
 entropy depending on the conductivity and the 
 temperature gradient. Any body whatever which is 
 not at uniform temperature can be divided up into 
 elements bounded by two sides each with no tempera- 
 ture gradient, and a bounding surface parallel to 
 the flow of heat, so that there is no passage of heat 
 across. If the flux of heat is H joules per second, 
 
 and + d6 and & the temperatures of the two 
 faces, the increase of uncompensated entropy per 
 
 TT 7/j 
 
 second, is y- per unit volume, and the un- 
 6 as 
 
 gained work is Hdti/ds per second, or the ungained 
 power is - ~H.d6/ds watts ; dBjds being negative. 
 Integration throughout the volume gives the total 
 ungained power. 
 
 Units. The fact that the units in thermodynamics 
 have no names goes to show that the science is not 
 fully developed. Measurement is an essential of 
 science. 
 
 There is no name for the unit of difference of 
 
UNITS. 137 
 
 temperature. Trigonometry and heat alone use 
 " degrees." Marks or notches would not be less 
 barbarous. Then there is no name for differences 
 of temperature according to the absolute Kelvin 
 scale. 
 
 The practical unit of heat, on the C.G.S. system, 
 is the joule, but this is hardly ever used in thermo- 
 dynamics. The calorie is a survival of the days 
 when it was not fully realised that heat is energy. 
 It involves an unnecessary and troublesome co-effi- 
 cient, and people are putting big calories and little 
 calories, and, perhaps, intermediate calories into 
 circulation. 
 
 There is no unit of entropy. I would suggest the 
 claus ; a claus being the entropy which incurs a 
 waste of 1 joule at a lowest available temperature of 
 unity ; e.g., if the lowest available temperature is 
 200 absolute, and the entropy 10, the incurred 
 waste is 2,000 joules. 
 
 Conclusion. Enough has now been said to show 
 the importance of making an inherently difficult 
 subject as easy to understand as possible ; and it is 
 hoped that the somewhat novel way of arranging 
 and treating the subject-matter of the groundwork 
 of thermodynamics may meet with the approval of 
 those who specially deal with that science. 
 
INDEX. 
 
 ADIABATIC Curve 
 Board of Trade Unit 
 British Thermal Unit 
 Carnot 
 
 Cycle 
 
 PAGE 
 ... 54 
 ... 58 
 ... 58 
 ... 114 
 ... 14 
 104, 137 
 
 Clausing 25,61,114 
 
 Compensated Entropy ... 9 
 Degradation of Energy ... 6 
 Dissipation of Energy ... 6 
 Duhem ... 70, 131, 133 
 
 Energy, Free 133 
 
 Entrop 104 
 
 Entropy, Definition 8, 34, 125 
 Factor of In- 
 curred Waste... 22 
 Localisation of... 28 
 Movement of ... 94 
 not a Factor of 
 
 Heat 7 
 
 Specific ... ... 29 
 
 Equilibrium ... ... 130 
 
 Factors of Heat ... 67,134 
 First Law of Thermo- 
 dynamics ... ... 13 
 
 Free Energy 133 
 
 Gibbs ... 61,71,122,133 
 
 Heat, Definition 124 
 
 Diagram ... 8, 73 
 
 Factors of ... 67,134 
 
 Heat- Weight 2 
 
 Helmholtz 71, 122 
 
 Horstmann ... 122 
 
 Incurred Waste ... 22, 52 
 Irreversible Expansion ... 42 
 Isentropic 55 
 
 PAGE 
 
 Kelvin 53 
 
 van Laar ... ... 131 
 
 Latent Heat ... 67, 115 
 Localisation of Entropy ... 28 
 
 Lost Work 85 
 
 Massieu 7^ 122 
 
 Maxwell ... ... ...7 61 
 
 Peabody 61 
 
 Perpetual Motion... 11. 124 
 Planck ... ... 131 
 
 Rankine 35, 61 
 
 Ilayleigh 122 
 
 Reeve 6i 
 
 Reversibility 26 
 
 Sankey 59 
 
 Second-class Perpetual Mo- 
 tion 12 
 
 Second Law of Thermo- 
 dynamics ... 13 
 
 Sensible Heat H5 
 
 Specific Entropy 29 
 
 Heat ... 58, 67 
 
 Stabilities '133 
 
 Thermodynamic Function 35 
 Potential... 71, 131 
 
 Thermodynamics, Laws of 13 
 Third Law of Thermo- 
 dynamics ... ... 13 
 
 Turbine 91 
 
 Uncompensaled Work 70, 131 
 Ungained Work ... 70,85 
 
 Waste 7,52,125 
 
 Watt and State Diagrams 49 
 
 Work 6 
 
 Lost 85 
 
 ,, Ungained ... ... 70 
 
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