THERMODYNAMICS OP KEVERSIBLE CYCLES IN GASES AND SATTJKATED VAPORS, FULL 87NOP8I8 OF A TEN WEEKS 1 UNDERGRADUATE COURSE OF LECTURES DELIVERED BY M. I. PUPIN, PH.D. /K , AND ED ?rp9^^BEiia, Student in Electrical Engineering, Columbia College YORK: EY & SONS, 53 lUsT TENTH STREET. 1894. 5> p, 1894, BY MAX OSTERBERG. PRINTER, NFW YORK. PREFACE. MY DEAR MR. OSTERBERG: It was quite a pleasure to examine your carefully worked out notes on my lectures. I agree with you in the opinion that the publication of these notes will be of much service to your classmates and probably to others who may be interested in the elementary features of thermodynamics. Very sincerely yours, M. I. PUPIN . COLUMBIA COLLEGE, NEW YORK, December 20, 1893. Ill SD8 ^^j V1 T I! i X II INTRODUCTION. IK this course on Theoretical Thermodynamics we shall limit our discussion to those features of the science which have a direct bearing upon the science of Caloric En- gineering. The course forms, therefore, a theoretical in- troduction to the practical course on Heat Engines. It seems desirable, however, to mould our discussion in such a way that it will serve at the same time the very important purpose of forming an introduction to the study of some of the best and most complete works on the subject. The work of R. Clausitis (Die meclianisclie Wdrmetheorie) is and very probably will always remain the classical treatise on this very important branch of exact sciences. We shall, therefore, adopt the mathematical notation and f ollow as closely as prac- ticable the method of discussion which is given in this great work of Clausius, who, as you will presently see, is one of the principal founders of the beautiful science of thermody- namics. THERMODYNAMICS OF REVERSIBLE CYCLES IN GASES AND SATURATED VAPORS. NATURE OF HEAT AND THE MATHEMATICAL STATE- MENT OF THE FIRST LAW OF THERMODYNAMICS. The Science of Thermodynamics investigates the formal or quantitative laws which underlie all physical processes by which the caloric state of material bodies is changed. By physical process we simply mean any one of the almost infinite variety of ways by which nature transforms one form of energy into another, or one form of matter into another. Heat is developed in almost every physical process, owing principally to the presence of passive or frictional resistances, unavoidable in material systems. The first question to be considered is : What is heat f It certainly must be one of the two fundamental physical quantities, that is, it must be either a form of matter or a 2 THERMODYNAMICS OF form of energy. In the first case it will be subject to the principle of conservation of matter, and in the second case it will be subject to the principle of conservation of energy. According to the old hypothesis heat is a substance ; but since the weight of a body is independent of its temperature, the heat substance, the so-called Caloric, had to be supposed to be imponderable. The experiments of Sir Humphry Davy (1799*) and of Rumford (1798 f) showed, however, that, 1st, the latent heat contained in a given body could be increased, and, 2d, the temperature of a body could be changed without a change in the quantity of heat in the surrounding bodies. The first was shown by Davy's experiment of melting two pieces of ice by rubbing them one against the other ; the sec- ond was shown by Rumford's experiment, in which heat was developed by friction, which accompanies the boring of metals. These experiments gave the death-blow to the old or substance theory of heat. They showed not only that heat cannot be a substance, but that on the contrary it has all the physical characteristics of the other fundamental physical quantity, which we call energy. As such it must be subject to the principle of conservation of energy. Joule was the first to shoiv that heat and mechanical work are not only convertible into each other, but that the * Essay on Heat, Light, and the Combinations of Light, with a new Theory of Respiration. Experiments II. and III. f Count Rumford *s essay on "An Inquiry concerning the Source of Heat which is excited by Friction," read before the Royal Society, January 25th, 1798. See Memoir of Sir Benjamin Thompson, Count Rumford, with notices of his Daughter, by George Edward Ellis. REVERSIBLE CYCLES IN GASES AND VAPORS. ratio of conversion is constant (1839-1844). Later on he ex- tended this relation to other forms of energy, like chemical energy, etc. By these experiments Joule not only proved that heat is a form of energy convertible into mechanical or any other form of energy at a constant ratio, but he also fur- nished the first exact experimental basis for the general prin- ciple of conservation of energy, first enunciated by Julius Robert Mayer about the same time (1842). It was, however, clearly stated in its most general form and mathematically elucidated by Helmholtz in 1847. The principle of Conservation of Energy can be stated as follows : Energy cannot be created nor can it be annihi- lated by any physical processes which our mind can con- ceive. The energy contained in an isolated material system must, therefore, remain constant, although all sorts of changes may be taking place within the system. These changes, however, will consist in a transformation of one form of energy into another or one form of matter into another without loss. When the energy of a body or system of bodies is changed in amount, then the change is due to the transference of energy from external bodies to the system considered, or vice versa. In the first case, work is done upon the system; in the second, the system does work upon external bodies. If we heat a body, one part of the heat energy put into it will appear as heat, another part as work done against ex- ternal forces, like surface pressure, and another part as potential energy overcoming the internal forces, like forces of cohesion, chemical affinity, etc. The common unit of meas- ure of all these forms of energy is the unit of mechanical 4 THERMODYNAMICS OF work, that is, the kilogramme-meter, The unit of heat is one Tdgr. calorie. DEF. One Tdgr. calorie is the heat necessary to raise the temperature of a Iclgr. of a standard body one degree. The standard body being- pure distilled water at 4 C., and at nor- mal pressure (760 mm.). The mechanical equivalent of heat is the number of Tdgr. meters which must be spent to generate that much heat. It was found by experiment that one klg. calorie = 424 klg. m. (very nearly). The most distinguished experimental- ists in this line being Kobert Mayer, Joule, Him,* Kowlaud. To express, say, 890 klgr. meters in calories, divide 890 by one J (" Joule ") or 424 (/ being the symbol of the mechani- cal equivalent of heat). STATEMENT OF THE FIRST LAW OF THERMODYNAMICS FOR A PARTICULAR CLASS OF PHYSICAL PROCESSES. "We will now limit ourselves to that class of bodies and processes in which the external work done is mechanical work, and in that class of processes we shall select those processes where the mechanical work is done in overcoming a certain amount of pressure on the surface of a body to which we apply the heat, and consider that the pressure is uniform and normal at all points of the surface. *Hirn, Recherches sur 1'equivalent mecanique de la chaleur, p. 20; Also, Theorie mecanique de la chaleur, 2d edition, part 1st, p. 35. REVERSIBLE CYCLES IN GASES AND VAPORS. 5 Expression for External Work due to Infinitesimal Expansion. Consider an infinitesimal area of the surface du square meters ; if the pressure per sq. m. is p kg., the total pressure is pdu. Now suppose the body to expand, overcoming the surface pressure : the work done by the act of displacing du is equal to the normal displacement dn meters times the pressure, that is pdudn kg. meters. To find the total work done over the whole surface, we must integrate (remembering that p is uniform all over the surface). / pdudn pi dudn pdv. That is to say, for infinitely small increments of volume, dur- ing which the surface pressure may be considered constant, the external work is equal to the surface pressure multiplied by the increment in volume. Intrinsic Energy and Co-ordinates of a Body. Let us now communicate an infinitely small quantity of heat dQ to a body. Experience tells us that the following changes in the body may be looked for: change in tempera- ture, internal aggregation, chemical constitution, electrical and magnetic state, volume, etc., etc. If we now call the energy which a body contains in consequence of its tempera- ture, internal aggregation, chemical constitution, electrical and magnetic state, etc., etc., the intrinsic energy of the body, and denote it symbolically by the letter U, and if we denote by W 6 THERMODYNAMICS OF the external work which a body does against the surface pressure in expanding from a given volume to some other volume, then it is evident that the above changes in tempera- ture, aggregation, chemical constitution, electrical and mag- netic state, volume, etc., etc., imply a certain small change dU in the intrinsic energy and a certain small external work dW. According to the Principle of Conservation of Energy the mechanical equivalent of dQ is equal to the mechanical equivalent of dU plus the mechanical equivalent of dW. Hence This is the most general form of the first law of thermody- namics. It is a formal statement of the Principle of Conser- vation of Energy for processes which involve a transformation of a certain amount of heat. Experimental Physics and Experimental Chemistry teach us how to measure the physical quantities by means of which we describe the various states of physical bodies, these quan- tities being temperature, forces of cohesion, chemical affinity, quantities of various chemical elements contained in a body, magnetic and electrical intensity in the various parts of the body, volume, pressure, etc., etc. The energy contained in a body depends on its state and not on the processes by means of which the body has been brought into that state. Hence the physical quantities just mentioned which describe the state of the body will also determine completely the value of the intrinsic energy U which the body has in that state. Tit is is what we mean when we say that U is an analytical function of the above physical quantities. REVERSIBLE CYCLES IN OASES AND VAPORS. 7 The external work which a body does against surface press- ure while expanding from a given volume to some other volume may or may not be independent of the process by means of which this expansion is effected. To be general, we shall suppose that it is not independent, that is to say, we shall suppose that W is such a function of the physical quantities which define the state of the body, that in passing from one set of values of these quantities to another set the change in the value of JFwill depend on the physical processes by means of which we passed from the first to the second set. This will be illustrated further on by actual examples. The physical quantities, like temperature, magnetic and electrical intensity, pressure, volume, chemical affinity, quantity of mat- ter, etc., which define the state of a body are called the co- ordinates of a body. We say then that 7 and JFare functions of the co-ordinates of the body. It is evident that the more complex the state of a body the more co-ordinates shall we need to describe its state. For the present we shall limit ourselves to those physical bodies and physical processes in which temperature, volume, and pressure suffice to describe completely the slate of the body, and the processes by means of which that state is varied. U will therefore be a function of the pressure p, the volume v, and the temperature t ; the same is true of W. We shall measure p in kilogrammes per square meter, v in cubic meters, and t in degrees centigrade. It must be observed, however, that these three co-ordi- nates of the body are not independent of each other. Thus the surface pressure and the temperature of a body being given, its volume is also determined. Varying thr nTBgiiirp ^ - -' -: :~ '2 8 THERMODYNAMICS OF without varying the temperature, we can pass from any volume to any other volume (within certain definite limits). We can therefore express U in terms of two of these. In what follows U will be considered a function of t and v unless the contrary is stated. We shall therefore have in the case of these bodies and these physical processes Since the pressure is normal and uniform and constant for infinitely small variations of volume and temperature, dW=pdv; Hence the first law of thermodynamics in the case of bodies and processes, which we are considering, says : If we com- municate an infinitely small amount of heat to a body under pressure, a part of the heat is used up in increasing the internal energy of the body, part of which is potential and part heat energy, and the rest appears as mechanical work done against the surface pressure. If a finite quantity of heat Q is communicated to a body, causing the same to pass from a given state, in which its tem- perature is t l , surface pressure p l , and total volume #, , to an- other state, in which these quantities are 2 , p a , # 2 , then Q=U,-U t REVERSIBLE CYCLES IN GASES AtfD VAPORS. 9 That is, the total quantity of heat is equal to the total in- crement of internal energy plus the total external work. The increment in internal energy is due to two changes: 1st, change of temperature; 2d, change in the arrangement of matter in the body. The first change produces a Tariation in the thermometric heat; the second, in the latent heat of the body. If pressure is constant, then external work =p(v 9 v t ). But since this is not always the case, we must, in order to find the numerical value of the integral fpdv, consider first the law of variation of p with v. For infinitely small variations of volume the pressure can be considered constant. II. APPLICATION OF THE FIRST LAW OF THERMO- DYNAMICS TO PERFECT GASES. We pass now to the application of the First Law of Ther- modynamics to the study of the simplest thermodynamic processes that can be performed upon the simplest class of physical bodies, that is, we pass to the application of the First Law of Thermodynamics to the study of the process of ex- pansion and compression of perfect gases. These bodies may justly be considered the simplest bodies, because experimental research tells us that their co-ordinates describe the state of these bodies by the simplest formal rela- tions that we know of, namely, Relation (1) contains what is known as Boyle's law, viz. : That is to say, if a given quantity of a perfect gas, say a kilo- 10 THERMODYNAMICS OF gramme of dry air at a given temperature, occupies a volume v t when it is under a pressure p t , then the same quantity of the gas at the same temperature when it is under pressure p^ will occupy the volume v a as given in above equation. This relation is true lor a large number of gases within a large interval of pressure and temperature. Relation (2) contains what is known as the Mariotte-Gay- Lussac law, viz. : pv=pj>.(l + ot) (2) Here p is pressure in kg. per square meter and v is volume in cubic meters of a unit weight (that is a kilogramme) of a perfect gas when it is under the pressure p and when its temperature is t deg. centigrade above the standard freezing- point. p Q and v denote the corresponding quantities of the same weight of the gas at the freezing-point, a is called the temperature coefficient of expansion. It is found to be very nearly constant for all temperatures within a large interval, and to be = -^^ (very nearly) for all perfect gases. The Mariotte-Gay-Lussac law can also be written w _V 973 i A 2^0 T RT P 273 l ' 273 Here T is called the absolute temperature of the gas and R is a constant which depends on the nature of the gas. It can .be calculated for every gas from experimental data, as will be shown presently. Choosing Tand v as the co-ordinates of the gas, we have REVERSIBLE CYCLES IN GASES AND VAPORS. 11 ON THE RATES OF VARIATION OF THE INTRINSIC ENERGY OF A PERFECT GAS. When a small quantity of heat dQ is communicated to a body, then a part of it will appear as an increment in the in- trinsic energy of the body, and the other part will appear as external work. The first part consists of two terms, namely, -~ dT and dv. Now ^jPis the increment of intrinsic energy due to the increase of the body's temperature by dT. The factor 7^ is the rate at which internal energy varies with the teln- et perature, the volume being constant. If that rate were con- stant, then the factor = would mean the increment of O-L internal energy due to a rise of temperature of 1. This part -^TpdT appears as sensible heat of the body, i.e., heat which can be measured by a thermpmeter. The other part ^ TJ -^dv is the increment of intrinsic energy due to the change of the body's volume, the temperature remaining constant. The factor -- is the rate at which this increment takes dv place; if this rate remained constant for finite variations' of volume, then would mean the increment of intrinsic 'dv energy for a unit increment in volume, the temperature re- constant^ 12 THERMODYNAMICS OF We have every evidence in favor of the hypothesis that heat is a mode of motion of the molecules of a body. Its temperature is due to the kinetic energy of this motion; hence ^p^dT denotes the increment of the kinetic part of the O-i intrinsic energy. The other part of the increment of the internal o 77" energy, namely, dv, will be work done against the forces between these molecules. These forces will depend on the distances of the molecules from each other, and therefore on the volume of the body. The work done against these forces, (when the temperature, and consequently the kinetic energy of the body, remains constant), will therefore appear as potential energy. Whenever there are no internal forces between the molecules, the term -dv will be zero. That is, the intrin- dv sic energy of the body will be independent of the volume. If such a body expands without doing work against external pressure, its temperature will remain constant, if no heat is communicated from without. For, from the equation we see that when dQ = 0, there being no heat communicated from without, and pdv = 0, there being no external work done, then -dT== 0. That is, dT = 0, since - cannot = 0. REVERSIBLE CYCLES IN OASES AND VAPORS. 13 Vice versa, if when the body expands without doing external work, and without receiving heat from without, its tempera- ture remains constant, then This result, compared with the preceding one, gives an identity. It tells us that the entire amount of heat put into the gas was utilized to do external ivork ; for since the temper- ature remains constant, none of the heat supplied goes to in- crease the intrinsic energy of the gas. Isothermal and Adiabatic Curves. If we make the supposition that during the expansion of the gas the temperature remains constant, we have from Ma- riotte-Gay-Lussac's law pv = O. Eepresent this result graphically, taking p for ordinates and v for abscissae. The resulting curve is an equilateral hyper- bola referred to its asymptotes. Such a curve is called an iso- thermal curve. (Fig. 1.) There is evidently an isothermal curve for every tempera- ture; hence an infinite number of them, but everyone of them is an equilateral hyperbola in the case of a perfect gas. 22 THERMODYNAMICS OF The isothermal curves of a perfect gas form a system of curves which never intersect. For suppose there is a point where two curves do intersect, this would have to be a point where at different temperatures the pressure is the same for the same volume, which is absurd. FIG. 1. FIG. 2. v Let us suppose again that we diminish the pressure without introducing additional heat. The piston will go up, hence the temperature will diminish; and as soon as a point is reached where p^v^ = RT^ the expansion will cease. If we plot a curve expressing a relation between p and v at any moment during this expansion, we get what is called AN ADIABATIC Or ISENTEOPIC CURVE. (Fig. 2.) The equation of this curve will be deduced presently. Equation of the Adiabatic Curve of a Perfect Gas. Since no heat is communicated during an adiabatic expan- sion, the following relation must exist at any moment between the physical constants of the gas: - 0,,) log - = HE VERBID IE CYCLES IN GASES AND VAPORS. 23 or Let us illustrate this by an example. Suppose the initial temperature is 0. above the freezing- point, and a certain initial . volume of a kg. of air is com- pressed to \ the volume; calculate the rise in temperature. T, = 273, ^- = 2. .-. JL = (2). = 1.329, or 7; = 363 C . Hence t z = T 2 273 =90 C. above the freezing-point. Compressing the initial volume to 5, we find t = 209 C. Compressing the initial volume to T V, we find t = 429 C. We shall now calculate the external work done, when the gas expands adiabatically. From pv = RTwe get But we found = p /V \ k n v fc Hence ~ = \~7l ' OY ' more generally, p = - L ^-. 24 THERMODYNAMICS OF The last equation is the equation of an adiahntic of a perfect gas. It is evidently of the form const. To calculate the external work done during an adiabatic expansion from volume v 1 to v^ , we have w I 7 k I * c ^ v Pi v i k j 1 1 ) ^4 family of adiabatic curves never intersect eacli other, but every adiabatic curve intersects every isothermal curve. The axes of p and v are asymptotes to the adiabatics of a perfect gas, but the adiabatic passing through a given point is more steeply inclined to the p axis than the isothermal passing through the same point. Summary. It has been shown so far that heat is a form of energy, and that therefore it obeys the Principle of Conservation of Energy. The constant ratio at which it is convertible into mechanical energy is 424; that is to say, if the quantity of heat which must be supplied to the unit mass of the standard substance (one cubic decimeter of pure distilled water at 4 C. and 760 mm. barometric pressure) in order to raise its temperature l c C. be called a unit of heat, then this unit of heat is equivalent to 424 mechanical units of work, the weight of the unit mass at REVERSIBLE CYCLES IN OASES AND VAPORS. 25 a definite place, that is, the kilogramme weight, being taken as the unit of force and the meter as the unit of length. Briefly stated, the mechanical equivalent of a kilogramme calorie is 424 kilogramme-meters. After that it was shown that if we add a quantity of heat to a body that heat will appear partly as an increment of the intrinsic energy of the body and partly as external work. For very small quantities of heat this can be stated as follows : which is the most general form of the First Law of Thermo- dynamics. In this equation dU l is the increment of the in- trinsic energy due to increase of the sensible heat of the body, and d 7 2 is the increment of the intrinsic energy due to the increment of internal potential energy of the body. The ex- ternal work dW was shown then to be pdv, as long as we limit ourselves to a particular kind of external work, that is, work done by the body in expanding against a uniform, normal surface pressure. Limiting ourselves to such physical processes in which the state of the body, which is the seat of these processes, can be described completely at any moment by temperature, vol- ume, and pressure, it was shown that the first law of thermo- dynamics can be stated as follows : atr fdi #+i The application of the first law of thermodynamics to the study of such physical processes in various classes of physical 26 THERMODYNAMICS OF bodies was then in order. AVe commenced with the simplest class of physical bodies, that is, with perfect gases. Starting with the Boyle and the Mariotte-Gay-Lussac Laws, and re- membering that the intrinsic energy of a perfect gas is inde- pendent of its volume, we showed that the First Law of Ther- modynamics for perfect gases can be stated as follows : when C v is the specific heat of the gas at constant volume, measured in mechanical units. By the relations C p = C v -{- R and pv = RT we then gave the various forms of statement of the above equation, these various statements differing from each other in the selection of the independent variables p, v, and T, and the physical constants C p , C v , JR, and k. An important observation was then made, and that was, that ac- cording to Regnault's experiments, C p and therefore C v , R, and Tc, are the same for all temperatures, volumes, and press- ures at which the gases retain their characteristic properties of a perfect gas. The application of these various forms of statement of the First Law of Thermodynamics for perfect gases was then made, and the isothermal and adiabatic changes of such gases dis- cussed. We ended with the discussion of isothermal and adia- batic or isentropic curves of a perfect gas. We could, following the same method of discussion, apply now the first law of Thermodynamics to the study of physical processes of above description in the case of any other class of physical bodies. It is evident, however, that since the fun- damental relation between pressure, volume, and temperature REVERSIBLE CYGLES IN GASES AND VAPORS. 27 in the case of physical bodies in general is far from being as simple as in the case of perfect gases, it is evident, we say, that our discussion would lead to very complicated equations, involving physical constants which, in very many cases, have not as yet been determined experimentally. We shall there- fore abstain from a general discussion, but pass on to the application of the first law of thermodynamics, to the study of reversible processes in vapors. This class of bodies is ex- tremely important, particularly from an engineering stand- point, because it is the action of heat upon these bodies that is generally employed as a means of transforming heat energy into mechanical energy, and vice versa. Before making this next step, it is advisable to deduce another general law, which, like the first law of thermody- namics, underlies all reversible heat processes in nature. It is called the Second Law of Thermodynamics. We close now this part of our course with a discussion of the various methods of expressing the specific heats of gases, and the relations which exist between them. ON THE VAKIOUS WAYS OF EXPRESSING THE SPECIFIC HEATS OF PERFECT GASES. r> Consider the relation c p = c v -j j . J This relation says : Given the spec, heat of a gas at con- stant pressure, the spec, heat at constant volume can be calculated, if the constant R of the gas, and the mechanical equivalent of heat, are known. We mentioned previously that we could detfl0jSnJb4i& ^ 2 THERMODYNAMICS OF ratio of the spec, heats by means of the velocity of sound. This gives another method of calculating c v when c p is known. But it is very difficult in some gases to determine (k) by velocity measurements. In such cases we must take our recourse to the above formula. R can be calculated from the formula R - ~ 273' It is possible, nowever, that the gas cannot oe well observed at the freezing temperature; hence we must look for another method of calculating R. We can write in every case At the same temperature and pressure we shall have for the same weight of a standard gas, say well-dried air, R'=. Hence R=R'-. v The fraction is the reciprocal value of the spec, weight of the gas, that is of its density, the density of air being taken is equal to unity. Calling this spec, weight, that is, the density, d } we have KEVEliSIBLE CYCLES IN GASES AND VAPORS. 29 By substituting this value of R in the first equation, we get R' 1 _ r R' in our case is 29.27. Example. Calculate c v for air. on cm d-l f hence c v = .2375 - -~- = -1684. That is to say, to raise the temperature one degree, a kilog. of air requires only about \ of the heat that a kilog. of stand- ard water does. The spec, heats denoted by c p and c v refer to a unit weight of the gas, and their unit is that quantity of heat which is necessary to raise the temperature of a unit mass of standard water under standard conditions one degree; that is, in other words, the gas is compared calorically by weiglit, to water. While, as a matter of fact, it is sometimes more convenient to compare in this respect gases with air, by considering equal volumes, i.e., to determine the spec, heat in such a manner, that the amount of heat necessary to raise the unit volume of a gas one degree, is compared with the quantity of heat necessary to raise a unit volume of air through one degree (at the same temperature and pressure). We can employ that -method of comparison for both specific 30 THERMODYNAMICS OF heats, considering in one case that the gas and the air are heated at constant pressure, and in the other that the gas and air are heated at constant volume. Let the volume of a unit weight of gas he v. The amount of heat which a unit volume must get to raise its temperature one degree at constant pressure must there- fore be Let the volume of a unit weight of air v 1 , then the amount of heat necessary to raise a unit volume through one degree under the same conditions will be The ratio of these quantities evidently gives us y p ; that is, the quantity of heat necessary to raise the temperature of the unit volume of a gas at constant pressure measured in terms of the heat necessary to raise the temperature of the unit volume of air under the same conditions. Hence - Y_ - _fk -. C P Similarly, c v , y v =--,*. REVERSIBLE CYCLES IN OASES AND VAPORS. 31 TABLE OF SPECIFIC HEATS OF GASES, -A ^ Of i Density. Specific Heat at Const. Pressure. Specific Heat at Const. Volume. C P IP c v Vu Air 2 ~ Ni H 2 C1 2 CO HC1 CO, H 2 O 1. 1.1056 0.9713 0.0692 2.4502 0.9673 1.2596 1.5201 0.6219 0.5894 1.5890 2.5573 0.2375 0.21751 0.24380 3.40900 0.12099 0.2450 0.1852 0.2169 0.4805 0.5084 0.4534 4797 1. 1.013 0.997 0.993 1.248 0.998 0.982 1.39 1.26 1.26 3.03 5.16 0.1684 0.1551 0.1727 2.411 0.0928 0.1736 0.1304 0.172 370 0.391 0.410 0.453 1. 1.018 0.996 0.990 1.350 0.997 0.975 1 55 1.36 1.37 3.87 6.87 Oxvfifcn Nitrogen Hydrogen Chlorine Curb, monoxide. . Hydroch. ucid gas. Carbonic acid Steam .... Ammonia NH 3 C 2 H 6 C 4 H 10 Alcohol Ether III. SECOND LAW OF THERMODYNAMICS. CARNOT'S REVERSIBLE ENGINE AND CARNOT'S CYCLE. Let ns start with a process of expansion and compression of a perfect gas enclosed in a cylinder which is impermeable to heat. The piston which shuts the gas in, can glide up and down without friction. The piston being under, a given pres- sure, the gas will at a given temperature occupy a definite volume. Heat can be communicated to the gas through plug p. Let the length oe on the abscissa denote the original volume v 1 of a gas, and the ordinate ea the initial pressure jo,. Evidently these two conditions determine the tempera- ture of the gas at that moment. Let the gas expand isother- 32 THERMODYNAMICS OF mally. To accomplish this imagine it connected to a large reservoir A (Fig. 4) of temperature T l , the same as that of the gas. By gradually diminishing the pressure and expanding / / FIG. 3. FIG. 4. very slowly, the temperature will remain constant and equal to T l . Plotting the curve expressing the relation between p and V) we get a part of an isothermal curve. Point b indicates the volume and pressure at the end of the isothermal expan- sion. Now cut off the heat reservoir A, and let the gas ex- pand slowly still further (by diminishing the pressure), but of course adiabatically, that is to say, the gas expands and decreases in temperature on account of loss of heat due lo external work done. The curve of expansion, be, will be a part of an adiabadic. Call the temperature, when volume v z has been reached, T 3 . From now on compress the gas, but having first connected the cylinder to a reservoir B of tem- perature T^ and of so large a capacity that it will take up all the heat from the compressed" gas without changing its temperature perceptibly. The heat generated in the gas by compression will be given off to B, hence the compression will be isothermal. The curve of compression will again be REVERSIBLE CYCLES jtf GA8ES AND VAPORS. 33 a part of an isothermal. At this point we impose a particular condition, namely, that the gas be compressed until a point is reached where the isothermal cuts that adidbatic curve ivhich goes through the starting point a. Arrived at that particular point, we disconnect the reservoir B and continue the compression. From there on the gas is compressed adiabatically, the temperature rises, and when the original temperature T l has been reached the gas will evi- dently have the same volume and will be under the same pressure as at the beginning. These four operations con- stitute what is called a Carnot Cycle. Now we shall consider the work done by the gas and upon the gas during the various operations in the cycle. The work done during the first expansion can be represented by the area abfe, that during the second by I cgf. During the compressions work has been spent upon the gas equal to the areas cdhg and hdae respectively. The difference between the work done and the work expended is the work gained. This gain must evidently be at the expense of heat given up to the gas by the reservoir A. Heat was taken up from A by the gas during the first operation; let us call that Q lS and during the isothermal compression heat was given out from the gas to the reservoir j5; let us call that 2 . Since on the whole a certain quantity Q was transformed into work represented by the area abed, it is evident that Let us illustrate that method by an example, 34 THERMODYNAMICS OF Say that we have .5 kilogr. of air, and an initial pressure of 10 atmospheres, that is equal to 7600 mm. barometric pressure or 103,330 kilogr. per square meter. Let the initial volume be cubic m. v l = .1 cm. jo, = 103,330 klg. per sq. m. Weight = .5 kg. 103330 X .1 X 2 29.27 _ " * Suppose -now that we work between the reservoirs A and B at 706 and 293 respectively. Hence T v = 708, ?; = 293. We will proceed by allowing the gas to expand isothermally, until the volume is equal to v a , and suppose for convenience of calculation ^ = e = 2.72. Work done by the gas during this expansion = / pdv W /** ion = / pdv *./ Vi REVERSIBLE CYCLES IN GASES AND VAPORS. 35 But in isothermal expansion pv RT l constant. * 1 RT, ^- = RT l log V -= 29.27 X 706 kg. meters. ty v l v .1 29.27 X 706 . Heat taken up from A will be = - -- kg. calories. Passing to the second operation we expand the gas from now on adiabatically, until its temperature is 293. The final volume will be v 3 . Let the work done during this operation be denoted by PF a , then PT 3 = / pdv. \s v 2 K But during this operation p = ~-?. = 29,480 kg. meters. Compress now isothermally until the point is reached where the isothermal cuts the adiabatic which passes through the starting point p l v l and denote this work of compression by W\. Then if v t be the final volume, 36 THERMODYNAMICS OF (v\ k ~ l T fv\ k ~ l T . . But since ( j = ~, also ( M = ~, it follows that Wj/ J. a Wj/ -/ 2 Similarly the work done during the last adiabatic compression will be The total external work represented by the area abed is W= W W W W w Efficiency = == T 1 * * This relation expresses the second law of thermodynamics, but only in a very limited form ; that is to say, this relation being simply a mathematical expression for the efficiency of a particular machine, Carnot's engine, operating in a perfectly definite manner, that is, by simple reversible cycles, upon a particular class of bodies, namely, perfect gases, this relation should not be considered to be anything more than it really is, REVERSIBLE CYCLES IN OASES AND VAPORS. 37 namely, a law which holds true for a particular class of physi- cal processes taking place in a particular class of physical bodies. It was in this form that the great French engineer, Sadi Carnot, first discovered the law, seventy years ago, in 1824. But he did more than that. He also pointed out that it is only a particular form of a more general law, that is to say, a law which holds true for a class of physical processes much larger than the class we have just described. It was by a train of reasoning first suggested by Carnot's great essay, "Beflexions sur la puissance motrice du feu et snr les machines propres a la developper," that subsequent inves- tigators, and foremost among them Clausius and Eankine, were able to extend the above relation into a general law, called the second law of thermodynamics. Before proceeding any further in our discussion towards the generalization of the above efficiency relation into the second law of thermodynamics let us first examine carefully the foundation on which this relation rests. . Consider each one of the four operations which taken to- gether constitute Garnet's cycle. Any one of them when reversed will produce just the opposite effect. Take for in- stance the second operation, that is, the first adiabatic expan- sion. If at the completion of this operation we reverse the operation, we can bring the gas back into the state which it had at the beginning of this operation. The reversed opera- tion will cost us mechanical work of compression equal to the area ~bcgh. That work will appear as intrinsic energy in the gas, that is, as heat, in consequence of which the temperature of the gas is increased from T % to T^ If the first operation be 38 THERMODYNAMICS OF called the direct and the second the reversed, then we can say that the direct and the reversed operation just neutralize each other. The same is true of every other operation of the cycle. It follows, therefore, that the whole cycle can be reversed and the reversed cycle will produce just the opposite effect of the direct cycle, that is to say, it will deprive the reservoir B of Q^ units and add Q 1 units of heat to reservoir A, and it will also transform Q l Q^ = Q units of /work into heajj In the first cycle heat passes from a hot bo^jToTaTcold body, but some of this heat in its journey to the cold body is transformed into mechanical work and therefore never reaches the cold body. In the reversed cycle mechanical work is expended, in order to transfer 2 units of heat from the cold body to the hot, and in addition generate Q units of heat which are also given up to the hot body. Such a cycle is called a reversible cycle. The relation Q _ r. - T, Qf After performing cycle I in the direct order, by which a quantity of heat Q is transformed into mechanical work and $ 2 transferred to the colder reservoir, employ cylinder B to perform cycle II in the reverse order, so that the mechanical work obtained by cycle I is retrans- formed into heat, and a quantity of heat equal to QJ is trans- REVERSIBLE CYCLES IN GASES AND VAPORS. 41 f erred from the cold to the hot reservoir. At the end of the two cycles the only change that still remains is the transfer of a quantity of heat equal to Q 9 ' Q^ from the cold to the hot reservoir, that is, a transference of heat from a body of lower to a body of higher temperature without any compensa- tion. This is contrary to the axiom of Clansins, hence the supposition that Q 9 ' > 2 must be dismissed. Next suppose that Q y ' < Q 9 . We can prove that this sup- position also violates our axiom by going through cycle II in the direct and cycle I in the reverse order. Hence Olausius' Axiom leads to the conclusion that QJ = Q 9 . That is to say, ivhencver mechanical work is obtained at /he expense of heat ~by a reversible cyclic operation, heat is transferred from a hot to a cold body and the ratio of the work obtained to the quantity of heat transferred from the hot to fhe cold body is independent of the substance which is operated upon. That ratio will therefore be the same for a perfect gas as for any other substance. We proceed to calculate it for a perfect gas. Let the temperatures of the two reservoirs be 7\ and 7\ and suppose that 7\ > 7* a . We found previously that in the case of a perfect gas or Q " Q Q " T, - T,> 42 THERMODYNAMICS OF We see therefore that this ratio in the case of a perfect ga and therefore also in the case of any other substance depend on the two temperatures between which the cycle is performed A slight modification of this relation will give us anothe form of. stating the second law of thermodynamics applicabl to simple reversible cycles. From this relation we obtain Q T u T l T^ We obtained previously Q hence Q l _ T ~ Combining this with the above we obtain =., or f?-a = ft or |' + ^ = 0. *1 ^3 -*! ^3 -^1 -*3 Q^ is the heat given off to the cold reservoir ~by the sul stance operated upon, hence ( QJ or + Q* can be said t be the heat received by the substance from the cold reservoir With that mental reservation we can write down T, REVERSIBLE CYCLES IN OASES AND VAPORS. 43 This is the more compact and at the same time more flexible form of stating the second law of thermodynamics applicable to simple reversible cycles. Tho English translation of this mathematical statement is : If in a simple, reversible, cycle the heats received by the substance operated upon be divided by the temperatures at which these heats were received and the quo- tients thus obtained be added together, then will the sum be equal to zero. This statement of the second law of a simple reversible cycle is certainly not nearly as clear nor physically as in- telligible as the other statement, which involved the efficiency of an ideal reversible engine, but, as will be presently seen, it enables us to extend in a very simple manner the applicability of this law to complex reversible cycles and thus give the second law of thermodynamics a much more general form. GENERAL FORM OF THE SECOND LAW OF THERMO- DYNAMICS. Complex Cycles. So far we have considered cycles between two temperatures only. We now proceed to extend this to a so-called complex cycle, that is, a cycle of operations between several reservoirs of different temperatures. We commence with three reser- voirs of temperature T l9 T^, and T 3 . Let ab (Fig. 5) denote an isothermal expansion at the tem- perature T l , be an adiabatic expansion down to the tempera- ture T^ , cd an isothermal expansion at the temperature T^ , de an adiabatic expansion down to the temperature T 3 , e/an isothermal compression at the temperature T a , the isothermal 44 THERMODYNAMICS 0V compression to continue until the isothermal curve ef cuts the adiabatic through a, and lastly fa an adiabatic compres- sion to the starting point. During the expansions ab and cd the substance operated upon takes up the quantities of heat Q t , Q 2 from the reser- voirs A and B at temperatures T t and T t , and during the FIG. 5. isothermal compression ef it takes up the negative quantity of heat Q 3 from the reservoir C at temperature T 3 . The external work done during the cycle is represented by the area abcdefa. We proceed to show now that a similar relation holds true in this case as in the simple cycle. To do that, let us produce the adiabatic be until it cuts the isothermal fe at g ; then will the whole complex cycle be divided into two simple cycles dbg fa and cdegc. The negative quantity Q a consists of q 9 and q/ , given off to the coldest reservoir during the isothermal compressions gf and eg respectively. We can now write down the symbolical REVERSIBLE CYCLES IN GASES AXL VAPORS. 45 statement of the second law for each one of these two simple cycles, viz. : + -TF = for cycle dbcgfa ; 7FT + % = " " C%C. Adding these two we obtain or, since g, + #/ = rp -* i In the same manner we can prove this relation for a com- plex cycle divisible into any number of simple cycles and obtain If the cycle be of infinite complexity, that is to say, if it be representable by an infinite number of infinitely small elements of aduibatic and isothermal curves, it is evident that the relation ^~, =-. will still hold true. Of course each one of 46 THERMODYNAMICS OF the Q's will be infinitely small and the sum will be an in- finite sum. We can therefore express this relation in the notation of the infinitesimal calculus, thus : A reversible cycle of this complexity will evidently in tne limit be representable by a continuous closed curve. The area bounded by this curve will represent the external work done during the cycle if the cycle be performed in the direct sense, otherwise it will represent the external work spent in order to transfer a certain amount of heat to the hot reservoir, a part of which comes from the cold reservoir. The statement of the second law of thermodynamics (appli- cable to reversible cycles) as given by the last equation, though very general, gives us but little information about the various elements of which the cycle is composed. It gives us no in- formation about the progress of the various physical processes which taken together constitute the cycle. It has the defect which is common to all integral laws. For such laws give us the total effect of a series of physical processes without giving us the separate effect of each process of the series. Take for instance the integral laws of Keppler which describe the motion of the planets around the sun. They tell us all about the areas traced out by radii vectores of the planets during a finite time, about the periodic times, and about the orbits of the planets, but they do not tell us anything about the action which is going on at any and every moment between the sun REVERSIBLE CYCLES IN GASES AND VAPORS. 47 and the planets to produce these integral effects. It took no less a genius than that of a Newton to infer these differential effects from the integral effects discovered by Keppler. The inference was the law of gravitation. To draw that inference Newton had to invent the infinitesimal calculus. We may well assert that he had to invent it ; for a simple consideration will convince you that the process of passing from an integral physical law to the differential law that is, the law which de- scribes a physical process as it progresses from point to point through infinitely small intervals of space and time is exactly the same as the mathematical process of passing from an in- tegral to the differential of that integral. We are now ready todiscuss the differential of the integral T Consider any one of the infinitely small terms -^ in the -* 10 FIG. 6. infinite series j ~. The subscripts 12 (Fig. 6) denote that dQ lz is the heat taken up during the infinitely small element 48 THERMODYNAMICS OF 12 of the cycle and that T lt was the mean temperature of the substance operated upon at that particular moment. It is evi- dent that clQ iy depends on the instantaneous state of the sub- stance operated upon at that moment, and also upon the nature of the operation represented by the element 12. In other words, dQ^ depends on the volume, pressure, and tem- perature, and the method of their variation during the opera- tion represented by the element 12. Now all these things are completely defined by the co-ordinates of the extremities of the element, hence dQ^ is a function of these co-ordinates. The same is true of the infinitely small quantity -~ 2 . ' 12 Consider now a quantity 0, and suppose it to be a finite, continuous, and singly-valued function of the pressure, volume, and temperature of the substance operated upon.

= B - <{> A and / d$ = B J(AB) l J(AB)^ A> Denote now the integral from B to A along the second part of the cycle by subscript (BA); then evidently (BA) (AB) 9 REVEHS1BLE CYCLES IN GASES AND VAPORS. 51 It follows therefore that /W-f /d0= defy- I d=((f> B - A )-( B - A )=0, J (AB)j. J (BA) c/UB)i c/(AB)o as it should be, since -j- / dcf) integral all around the cycle. / dcf) i J(BA) We can therefore say that the integral all around the cyclic diagram vanishes because the value of the integral between any two points of the diagram depends on the position of the two points and not on the shape of the diagram between these two points. The only other way in which that integral could vanish would be that dcp is zero for every element of every cycle, in which case would be a constant. This is contrary to our supposition, for we supposed that it is a variable function of volume, pressure, and temperature. A careful inspection of the series which in the limit gave the integral J d$ will convince you that this integral be- tween any two points of any cyclic diagram depends on the position of these two points and on nothing else, because the quantity under the integral sign is a perfect differen- tial of a function which is finite, continuous, and singly valued at every point of any cyclic diagram. We conclude, therefore, since the integral i -^ taken around a diagram 52 THERMODYNAMICS OF representing any reversible cyclic process vanishes, that -~ is the complete differential of a function S which is a finite, continuous, and singly-valued function of the volume, press- ure, and temperature of the substance upon which we oper- ate. We express this by writing This is the most general form of the second laiv of thermo- dynamics. This statement of the second law bears about the same relation to the other statement, as the Newtonian law of universal gravitation bears to the three laws of Keppler which describe the motion of planets around the sun. It tells us all about the progress of the infinite number of infinitely small processes which, taken together, constitute the whole reversible cyclic process of which the cyclic diagram is a geometrical representation. We shall illustrate this statement by reference to a par- ticular physical process when we come to the discussion of the application of the two laws of thermodynamics to the study of saturated vapors, which we are now ready to take up. This statement of the second Liw of thermodynamics of reversible processes was first given by Clausius. The func- REVERSIBLE CYCLES IN GASES AND VAPORS. 53 tion S was named by him the Entropy of the system. It is impossible to give a general and yet complete definition of the physical meaning of the entropy function, because this meaning will vary considerably with the nature of the process that we may be considering. Hence it can be dis- cussed only in connection with the discussion of each par- ticular process under consideration. It is desirable, however, to illustrate its meaning by considering a few simple pro- cesses. 1st. Suppose a body expands reversibly at constant tem- perature from volume v l and pressure p l to volume v^ and pressure p z . Let the heat that must be supplied to the body be Q, then Q=T(S,-$ l ); i that is, the heat is equal to the absolute temperature multi- plied by the increase of the entropy. In the case of a perfect gas that heat is also equal to the external work done by the gas during the expansion, and this work we v found to be equal to RT log . Hence for isothermal expansions of perfect gases we have 2d. If the gas expands adiabatically, then, since no heat is supplied during such expansion, we shall have = TdS, hence dS = 0, 54 THERMODYNAMICS OF That is to say, the entropy remains constant. It is on this account that adiabatic expansion is called sometimes an isentropic expansion, the word isentropic meaning that the entropy remains constant. IV. APPLICATION OF THE TWO LAWS OF THERMODY- NAMICS TO SATURATED VAPORS. We are now ready to take up the discussion of the rever- sible processes in the next simplest class of physical bodies, that is, saturated vapors. Our discussion will be framed after the model of our discourse on the reversible cyclic pro- cesses in the case of perfect gases. We shall therefore com- mence with an appeal to experience for information about the physical properties of sat united vapors. We did the same thing in the case of perfect gases by taking account of the laws of Boyle and Mariotte-Gay-Lussac, and of the experimental researches of Regnault on the specific heats of perfect gases. We shall then put the two laws of thermo- dynamics into a suitable form by taking proper account of this information about the physical properties of saturated vapors. This will give us two mathematical statements of these laws involving certain physical constants of saturated vapors. Some of these constants are known from experi- mental researches just as in the case of perfect gases; others we shall have to calculate with the assistance of the two laws. Hence after modelling properly our two fundamental laws, REVERSIBLE CYCLES IN GASES AND VAPORS. 55 we shall proceed, just as we did in the case of perfect gases, to calculate and tabulate the various physical constants of saturated vapors. Having done that, we shall then be ready to pass on to the discussion of isothermal and adiabatic expansions of saturated vapors, calculating in each case the external work done and the heat supplied to the expanding vapor. Since water vapor is the substance most generally employed in our vapor-engines, it is advisable to concentrate our attention as much as possible upon that vapor. The discussion will nevertheless lose very little of its generality, since in dealing with saturated water-vapor we have to consider the same questions that would have to be con- sidered in any other vapor. PHYSICAL PROPERTIES OF SATURATED VAPORS. Consider a cylinder A (Fig. 8) in which a piston B can s^de np and down without friction. Let the volume enclosed by Pro. 8. the piston be denoted by v, and let a part a of this volume contain a volatile liquid. The other part b will be vapor of 56 THERMODYNAMICS OF that liquid. Let the temperature be uniform throughout and equal to T. Experience tells us that this vapor, as long as it is in contact ivitli its own liquid, lias a definite tension which depends on the nature of the liquid and on its tempera- ture and on nothing else. Let the pressure on the piston be just such as to be in equilibrium with the tension of the vapor at the temperature T. Suppose now that the piston is pushed down through a very small distance ; the volume of the vapor is diminished ; experience tells us that the vapor will condense. Vice versd, if the piston is slightly raised, keeping the temperature constant, then some more liquid will evaporate so as to keep the density in the vapor volume the same as before. We infer, therefore, that vapor when in contact with its own liquid has the maximum density for that temperature ; in other words, it is saturated. Both the tension and the density of the saturated vapor increase with temperature. I Saturated vapors do not obey the Mariotte-Gay-Lussac law, 'not even approximately, except for low temperatures. So, for instance, this law cannot be applied to saturated water-va- pors at any temperature which is more than only a very few degrees centigrade above the freezing-point. When a liquid is brought into a space which does not con- tain the vapor of this liquid, then the liquid will evaporate until its vapor has the maximum density in every part of the surrounding space. The process of evaporation will be slow if the space contains a gas at considerable pressure, but rapid if the space is a vacuum. This is due to the slowness with which the vapor diffuses through a gas of considerable tension. Experience tells us that evupqration is accompanied REVERSIBLE CYCLES Itf OASES AND VAPORS. 57 by a cooling off of the liquid, and the cooling is the more rapid the quicker the process of evaporation. You are acquainted with the well-known experiment of freezing mercury by bringing it in contact with a volatile liquid which is made to evaporate very rapidly by the action of a pump. If we wish to maintain the temperature of an evaporating liquid constant, then heat must be supplied to it. Conversely, if the saturated vapor be compressed ffireontaet-w4th^-Us~owft-4i^i^ ; heat will be developed, and if we wish to keep the temperature constant, then this heat must be taken away. The quantity of heat which is developed by compressing a kilogramme of saturated vapor to liquid at the same temperature and pressure is called the latent heat of saturated vapor at that temperature. This is a very important physical constant and has been deter- mined experimentally for a large number of vapors. When the temperature of a liquid has reached that point at which the tension of its vapor is the same as the pressure of the surrounding space, then the vaporization will be rapid, because it will take place not only on the surface of the liquid, but also in the interior parts, especially if the liquid, as is usually the case, contains small bubbles of a gas, like air, dis- solved in it. When this takes place we say that the liquid is boiling. The boiling point of pure distilled water at 760 mm. barometric pressure is 100 C., and it rises with the rise of barometric pressure. The boiling point of water is raised by the solution of foreign substances in it. Sea-water, for in- stance, has a considerably higher boiling point than pure dis- tilled water at the same barometric pressure. As long as the boiling process continues, the temperature of the liquid re- 58 THERMODYNAMICS Off maiws constant; the heat supplied to the liquid is all utilized to transform the liquid into steam. If, however, the liquid does not contain any dissolved gases, then the process of boiling is considerably retarded. There is superheating and the boiling is accompanied by the well- known " bumping " or concussive boiling. When the contact of a saturated vapor with its own liquid is cut off and the vapor allowed to expand slowly, it will cool off, of course, and its density will diminish. The ratio of the diminution of the density to the diminution of temperature is different for different vapors. In some vapors the tem- perature diminishes more rapidly than the density. Such vapors will condense during adiabatic expansion. If, how- ever, the temperature diminishes more slowly than the dens- ity, then the vapor will be superheated, that is to say, it will not remain saturated during adiabatic expansion. In the first case we have superheating and in the second we have condensation during adiabatic compression. OK THE PHYSICAL CONSTANTS OF SATURATED VAPORS. Let the total weight of the liquid and its saturated vapor ill the above-mentioned cylinder be M. Denote by T the absolute temperature of the whole mass, and by #the volume. Let m = weight of the vapor in kilogrammes; p = tension of the saturated vapor in kgr. per sq. m. ; a = specific volume of the liquid; s = " >< vapor. REVERSIBLE CYCLES IN OASES AND VAPORS. 59 We shall have then ms + (M m) J =c > T = h > hence we obtain from (I), (II), and (III) by these substitu tions __ + c _; i = ___, .... (I a > Time three equations are our fundamental equations in the thermodynamics of reversible processes in saturated vapors. We now proceed to the next part of our programme, and REVERSIBLE CYCLES IN OASES AND VAPORS. 71 that is the numerical calculation of the physical constants of saturated vapors. NUMERAL CALCULATION OF THE PHYSICAL CONSTANTS OF SATURATFED VAPORS. Our three fundamental equations (I a ), (II a ), and (IIP) repre- sent two independent relations [since (IIP) has been deduced from (I a ), that is the first law, and (II a ), that is the second law of thermodynamics] between five unknown quantities, viz., r, h, u, c, and ^-. Hence they enable us to calculate any two of these quantities when the remaining three are known. That is to say, three of these physical constants of saturated vapors must be determined experimentally and the other two can then be calculated by means of the fundamental laws of thermodynamics. What is meant by experimental determination is simply this: It must be found by experi- mental investigation in what way these physical constants depend on the variable co-ordinates of the body. In our dis- cussion of the general physical properties of saturated vapors we saw that all the physical constants of such a vapor depend on the temperature only, hence the object of an experimental investigation will be to express these physical constants as functions of the temperature T. Three of the above con- stants having been experimentally determined, the remaining two can then be also expressed as functions of the temperature by simple calculation, starting with our fundamental equa- tions (I a ), (II a ), and (IIP). In most cases r, c, and p have been determined experiment- 72 THERMODYNAMICS OF ally, so that it is generally // and u, that is, the specific heat and the specific volume of saturated vapors, that we have to calculate. We now proceed to carry out these calculations for a par- ticular vapor, and select water vapor for reasons given above. Regnault in his classical researches (Relation des experiences, Mem. de 1'Acad. t. xxi, 1847, etc.) made the quantities c, r, and JP, that is, the specific heat of water, the heat of evapora- tion, and the tension of saturated vapor of water, the subject of very careful experimental investigations. His results have been verified by all investigators in that field of physical re- search. We shall, therefore, use the values obtained from the results of his investigations. The specific heat of water is expressed, according to Regnault's experimental data, by the following formula : c = 1 + .00004^ + .0000009*", - where t is the temperature in degrees centigrade above the freezing point. For the heat of evaporation experimental data furnish the following relation : r = 606.5 + .305^ - C cdt. t/O Substituting the value of c and carrying out the integration, we obtain r = 606.5 - .695^ - .00002? - .0000003?. REVERSIBLE CYCLES IN OASES AND VAPORS. 73 For this long formula, Clansius (Mechanische Warmetheorie, vol. i. p. 137) substitutes the following: r= 607 -.708*. To show the agreement between the values of r obtained from these two formulae the following table is given for com- parison : t 50 100 150 200 T according to Regnault's form .... 606 5 571 6 586 5 500 7 464 3 T " " Clciusius' form 607 571 6 536 2 500 8 465 4 The agreement is excellent. We shall, therefore, employ Clausius* simpler expression. Taking the experimental data of Regnault's researches, Clausius* constructed the following table for the tension^? of saturated water- vapor at various temperatures: t in C. by Air Therm. p in mm. of Hg. t in C. by Air Therm. p in mm. ofHg. -20 .91 110 1073.7 -10 2.08 120 1489 4.60 130 2029 10 9.16 140 2713 20 17.89 150 3572 30 31.55 160 4647 40 54.91 170 5960 50 91.98 180 7545 60 148.79 190 9428 70 283.09 200 11660 80 854.64 210 14308 90 52545 220 17390 100 760 230 20915 * Mechanische Warmetheorie, vol. i. p. 149. 74 THERMODYNAMICS OF OBSERVATION 1. To transform p from pressure in mm. Hg to pressure in kg. per sq. m. we simply remember that 760 mm. barometric pressure is equivalent to 10,333 kg. per square meter. If we now wish to know how many kg. per sq. m. correspond to, say, 11,660 mm. barometric pressure, we simply put down the proportion p: 10,333 :: 11,660:760, where p is the pressure in kg. per sq. m., corresponding to 11,660 mm. barometric pressure. 10333 X 1166 p = - - Io8,530 kg. per sq. m. (about 229 pounds per square inch). This, then, is the pressure of saturated water-vapor at 200 C. OBSERVATION 2. If we refer the tension p and the tem- perature t of the above table to a set of rectangular axes, measuring off t as the abscissae and the corresponding p's as the ordinates, we obtain the curve as in diagram on opposite page. The value of the differential coefficient (which is evi- dt \ dently the same as ~j at any temperature represents the tangent of the angle which the tangent line to the curve at the point corresponding to that temperature makes with the axis of t. An inspection of the curve shows that this angle remains practically constant for considerable intervals of temperature, certainly for intervals of 10, IRBVERSTELB CYCLES IN GASES AND VAPORS. 75 I dp d ' a 15 30 45 60 75 9 10 ' Consider now any interval of 10 and let -JT = OL for that interval. at . = ~ for tlie sam ^ interval 76 THERMODYNAMICS OF This relation enables us to calculate in a very simple manner the quantity - ~ for any temperature. This is a quantity which we shall need presently. Suppose we wish to calculate ( ~] , that is, the value of - -j- for the temperature of \p dt /25 p dt t = 25. We select, therefore, that 10 interval the mean temperature of which is 25, that is, the interval between 20 and 30. We obtain log />.. -log A. = / d l. P The mean value of p within the limits of integration is very nearly the value of p at 25, that is, p w i <* ' log A. - log p,, = ~- I dt = ---. But =m ; 1 fdp\ logffso-logAo = Log ^ 30 - Log ao 10 J/10 ' where Log stands for Briggs' logarithm, and Mis the modulus of that system. We are ready now to take up the calculation of the re- maining physical constants, that is, to find the values of the REVERSIBLE CYCLES IN GASES AND VAPORS. 77 specific heat and the specific volume of saturated water-vapor for various temperatures within those limits within which the other physical constants were studied experimentally. 1. Calculation of h. From the second law ^ r i TJ - r we obtain dr . r Kegnault's experiments gave us r = 606.5 + .305^- / cdt .-. ~ = . at Therefore 78 THERMODYNAMICS OF In the last term of the expression for li we shall substitute for r the simple expression given by Clausius and obtain Ji = .305 - 607 -.708; 273 T~* The following table contains the calculated values of li for several temperatures: t 50 100 150 200 h - 1.916 - 1.465 - 1.133 -.879 -.676 The physical meaning of this negative specific heat will be considered further on in connection with adiabatic expansion of saturated water-vapor. We pass on to the calculation of the specific volume of saturated water-vapor. 2. Calculation of the Specific Volume of Saturated Water- vapor. The first question to consider is, whether in its saturated state water- vapor follows the law of Mariotte-Gay-Lussac, and if it does not, to what extent and in what way deviations occur. Making use of the fundamental relation (III a ), we find, after replacing (s a) for u, _ T(s a) dp T~ ~dt REVERSIBLE CYCLES IN GASES AND VAPORS. 79 or 273r Mp - ^v =pd(mu) = d(mup) mu-^d'j O-L From our fundamental equation (III) we have . mp .:pdv=d(mup)---dT. 90 THERMODYNAMICS OF Since v changes adiabatically, we can make use of the re- lation deduced in the last paragraph when we considered the process of condensation during adiabatic expansion. The re- lation is or ^-dT - d(mp) = MCdT. .\pdv = d(mup) d(mp) MCdT. .V W = / d(mup) - / d(mp) - MC dT J Vl J Vl JT, = mup - m 1 u 1 p 1 - [mp - w^pj -f MC(T l - T). Since the quantities p and C are given, by the experimental formulae discussed above, in kg.-calories, it is preferable to ex- press W thus: W= mup - vn l u l p l J{(mr m^) Mc(T^ T)\. This formula enables us to calculate the external work W in kg.-meters when a certain initial quantity m l of saturated water-vapor at temperature T 1 , being in contact with a quan- tity M m^ of its own liquid at the same temperature, ex- pands adiabatically until its temperature has gone down to a temperature T. Thus: REVERSIBLE CYCLES IN OASES AND VAPORS. 91 Since T l and T are given, p l and p can be found by a refer- ence to Kegnault's table of vapor tensions. Since m 1 and M are also given, JMc(T l T) and Jmj\ can be easily calcu- lated. There still remain mr and mu to be calculated. The quantity mr we have already calculated in the preceding paragraph and found mr = The quantity mu can be easily found thus : From the fundamental relation (IIP) we have T dp Substituting this value of mr in the last equation, we ob- tain J (m l r l '. W The quantity -, is the tangent of the angle which the tangent line to the curve of vapor tension, given above, at the point corresponding to temperature T = 273 -f- t, makes with the axis of t, and can be easily found either graphically or by calculation, thus: Suppose we wish to calculate - , for. the temperature 92 THERMODYNAMICS OF t = 110. On account of the gradual ascent of the curve of vapor tension, we have .~ 10 Referring now to Regnault's table we find p^ in mm. = 1489 and p ll9 in mm. = 1073.7. Hence, according to the ex- planation given before, in order to transform these pressures into kg. per square m. we have to divide by 760 and mul- tiply by 10,333. We obtain (J) = (1489 ~ yWH* = 564 . 64 . It is seen, therefore, that the two laws of thermodynamics in connection with the experimental data on saturated water- vapor enable us to calculate all the quantities which enter into the expression for the external work done by the adia- batic expansion of such a vapor. It should be noticed that this expression is very much different from the corresponding expression obtained for the adiabatic expansion of a perfect gas. This difference would not exist if saturated vapors obeyed the law of Mario tte-Gay- Lussac. The following table is taken from Clausius*; its verifica- tion is recommended as a very useful exercise. It contains the external work per kg. of saturated vapor when the initial temperature is t = 150 C. and the initial quantity of the liquid part is zero, hence M = ?H V * Mech. Warmetheorie, vol. i. p. 167. REVERSIBLE CYCLES IN OASES AND VAPORS. 93 t 150 125 100 75 50 25 F TTZi 11,300 23,200 35,900 49,300 63,700 Thus the external work done by a kg. of saturated water- vapor in expanding adiabatically from initial temperature of t = 150 C. until its temperature has sunk down to 100 C. is 23200 kg. meters. CBS. 1. In evaporating a kg. of water at 150 to a kg. of steam at 150 work must be done against the vapor pressure corresponding to that temperature. This work can be easily found to be 18,700 kg. meters. OBS. 2. In the above calculation Clausius assumed J^ 423.55. (y) External Work done during Isothermal Expansion. During isothermal expansion pressure remains constant, since temperature remains constant. The heat supplied is utilized to compensate two distinct processes which, taken together, constitute the operation of isothermal expansion. The two processes are: separation of the vapor from the liquid mass, and overcoming of the external pressure. Ac- cording to our definition of p, that is, the latent heat of evaporation, the total heat necessary to evaporate m kilo- grammes of vapor is mp. That part of this total heat which does the external work is given by W = fpdv. 94 THERMODYNAMICS OF But v = mu -j- M3