Treatise on Thermodynamics (1903) — part 3 of 14

The numbers of the last column are derived from those of the preceding one by multiplying by 98,100, to reduce grams to dynes, and meters to centimeters. Joule's results have been substantially confirmed by recent careful measure- ments by Rowland and others.

§ 62. The determination of the mechanical equivalent of heat enables us to express quantities of heat in ergs directly, instead of calories. The advantage of this is, that a quantity of heat is not only proportional to, but directly equal to its mechanical equivalent, whereby the mathe- matical expression for the energy is greatly simplified. This unit of heat will be used in all subsequent equations. The return to calories is, at any time, readily accomplished by dividing by 419 x 10^

§ 63. Some further propositions immediately follow from the above exposition of the principle of energy. The energy, as stated, depends on the momentary condition of the system. To find the change of energy, Ui — Ua, accompanying the transition of the system from a state 1

GENERAL EXPOSITION. 43

to a state 2, we should, according to the definition of the energy in § 58, have to measure Ui as well as U2 by the mechanical equivalent of the external effects produced in passing from the given states to the normal state. But, supposing we so arrange matters that the system passes from state 1, through state 2, into the normal state, it is evident then that Ui — TJg is simply the mechanical equiva- lent of the external effects produced in passing from 1 to 2. The decrease of the energy of a system subjected to any change is, then, the mechanical equivalent of the external effects resulting from that change ; or, in other words, the increase of the energy of a system which undergoes any change, is equal to the mechanical equivalent of the heat absorbed and the work expended in producing the change :

U2-Ui = Q + W (17)

Q is the mechanical equivalent of the heat absorbed by the system, e.g. by conduction, and W is the amount of work expended on the system. W is positive if the change takes place in the direction of the external forces. The sum Q + W represents the mechanical equivalent of all the tliermal and mechanical operations of the surrounding bodies on the system. We shall use Q and W always in this sense.

The value of Q + W is independent of the manner of the transition from 1 to 2, and evidently also of the selec- tion of the normal state. When differences of energy of one and the same system are considered, it is, therefore, not even necessary to fix upon a normal state. In the expres- sion for the energy of the system there remains then an arbitrary additive constant undetermined.

§ 64. The difference U2 - Ui may also be regarded as the energy of the system in state 2, referred to state 1 as the normal state. For, if the latter be thus selected, then Ui = 0, since it takes no energy to change the system from 1 to the normal state, and Ug - Ui = U2. The normal

44 THERMODYNAMICS.

state is, therefore, sometimes called the state of zero energy.

§ 65. States 1 and 2 may I e identical, in which case the system changing from 1 to 2 passes through a so-called cycle of ojjcrations. In this case,

U2 = UiandQ + W = . . . (18)

The mechanical equivalent of the external effects is zero, or the external heat effect is equal in magnitude and opposite in sign to the external work. This proposition shows the impracticability of perpetual motion, which necessarily presupposes engines working in complete cycles.

§ 66. If no external effects (Q = 0, W = 0) be produced by a change of state of the system, its energy remains constant (conservation of the energy). The quantities, on which the state of the system depends, may undergo con- siderable changes in this case, but they must obey the condition U = const.

A system which changes without being acted on by external agents is called a perfect system. Strictly speak- ing, no perfect system can be found in nature, since there is constant interaction between all material bodies of the universe. It is, however, of importance to observe that by an adequate choice of the system which is to undergo the contemplated change, we have it in our power to make the external effect as small as we please, in comparison with the changes of energy of portions of the system itself. Any particular external effect may be eliminated by making the body which produces this effect, as well as the recipient, a part of the system under consideration. In the case of a gas which is being compressed by a weight sinking to a lower level, if the gas by itself be the system considered, the external effect on it is equal to the work done by the weight. The energy of the system accordingly increases. If, however, the weight and the earth be considered parts of the system, all external effects are eliminated, and the

GENERAL EXPOSITION. 45

energy of this system remains constant. The expression for the energy now contains a new term representing the potential energy of the weight. The loss of the potential energy of the weight is exactly compensated by the gain of the internal energy of the gas. All other cases admit of similar treatment.

CHAPTER II.

APPLICATIONS TO HOMOGENEOUS SYSTEMS.

§ 67. We shall now apply the first law of thermodynamics as expressed in equation (17),

U2- [Ji = Q + W,

to a homogeneous substance, whose state is determined, besides by its chemical nature and mass M, by two vari- ables, the temperature and the volume V, for instance. The term liomogeneous is used here in the sense of physicaUy liomogeneous, and is applied to any system which appears of completely uniform structure throughout. The sub- stance may be chemically homogeneous, i.e. it may consist entirely of the same kind of molecules, or chemical trans- formations may take place at some stage of the process, as, for example, in the case of a vapour, which partially dissociates on being heated. The homogeneous state must, however, be a single valued function of the temperature and the volume. As long as the system is at rest, the total energy consists of the so-called internal energy U, which depends only on the internal state of the substance as determined by its density and temperature, and on its mass, to which it is evidently proportional. In other cases the total energy contains, besides the internal energy U, another term, namely, the kinetic energy, which is known from the principles of mechanics.

In order to determine the functional relation between U, d, and V, the state of the system must be changed, and the external effects of this change calculated. Equation (17) then gives the corresponding change of energy.

APPLICATIONS TO HOMOGENEOUS SYSTEMS. 47

§ 68. If a gas, initially at rest and at uniform temperature, be allowed to suddenly expand by the opening of a stopcock, which makes communication with a previously exhausted vessel, a number of intricate mechanical and thermal changes will at first take place. The portion of the gas flowing into the vacuum is thrown into violent motion, then heated by impact against the sides of the vessel and by compression of the particles crowding behind, while the portion remaining in the first vessel is cooled down by expansion, etc. Assuming the walls of the vessels to be absolutely rigid and non-conducting, and denoting by 2 any particular state after communication between the vessels has been established, then, according to equation (17), the total energy of the gas in state 2 is precisely equal to that in state 1, for neither thermal nor mechanical forces have acted on the gas from without. The reaction of the walls does not perform any work. The energy in state 2 is, in general, composed of many parts, viz. the kinetic and internal energies of the gas particles, each one of which, if taken sufficiently small, may be considered as homogeneous and uniform in temperature and density. If we wait until complete rest and thermal equilibrium have been re-estab- lished, and denote this state by 2, then in 2, as in 1, the tutal energy consists only of the internal energy U, and we have U2 = Ui. But the variables B and V, on which U depends, liave passed from B, Yi to 02> V2, where V2 > Vi. By measuring the temperatures and the volumes, the relation between the temperature and the volume in processes where the internal energy remains constant may be established.

§ 69. Joule performed such an experiment as described, and found that for perfect gases Bi = Bi. He put the two communicating vessels, one filled with air at high pressure, the other exhausted, into a common water-bath at the same temperature, and found that, after the air had ex- panded and equilibrium had been established, the change of temperature of the water-bath was inappreciable. It immediately follows that, if the walls of the vessels were

48 THERMOD YNA MICS.

non-conductiug, tlie final temperature of the total mass of the gas would be equal to the initial temperature ; for other- wise the change in temperature would have communicated itself to the water-bath in the above experiment.

Hence, if the internal energy of a nearly perfect gas remains unchanged after a considerable change of volume, then its temperature also remains almost constant. In other words, the internal energy of a ])erfect (/as depends only on the temperature, and not on the volume.

§ 70. For a conclusive proof of this important deduc- tion, much more accurate measurements are required. In Joule's experiment described above, the heat capacity of the gas is so small compared with that of the vessel and the water-bath, that a considerable change of temperature in the gas would have been necessary to produce an appreciable change of temperature in the water. More reliable results are obtained by a modification of the above method devised by Sir William Thomson (Lord Kelvin), and used by him, along with Joule, for accurate measurements. Here the outflow of the gas is artificially retarded, so that the gas passes immediately into its second state of equilibrium. The temperature Bo. is then directly measured in the stream of outflowing gas. No limited quantity of gas rushes tumultuously into a vacuum, but a gas is slowly transferred in a steady flow from a place of high pressure, pi, to one of low pressure, p^ (the atmosphere), by forcing it through a boxwood tube stopped at one part of its length by a porous plug of cotton wool or filaments of silk. The results of the experiment show that when the flow has become steady there is, for air, a very small change of temperature, and, for hydro- gen, a still smaller, hardly appreciable change. Hence the conclusion appears justified, that, for a perfect gas, the change of temperature vanishes entirely.

This leads to an inference with regard to the internal energy of a perfect gas. When, after the steady state of the process has been established, a certain mass of the gas has been completely pushed through the plug, it has been

APPLICATIONS TO HOMOGENEOUS SYSTEMS. 49

operated upon by external agents during its change from the volume, Vi, at high pressure, to the larger volume, V2, at atmospheric pressure. The mechanical equivalent of these operations, Q + W, is to be calculated from the external changes. The state of the porous plug remains the same throughout ; hence the processes that take place in it may be neglected. No change of temperature occurs outside the tube, as the material of which it is made is practically non-conducting ; hence Q = 0. The mechanical work done by a piston in pressing the gas through the plug at the constant pressure i^x is evidently ^^iVi, and this for a perfect gas at constant temperature is, according to Boyle's law, equal to the work ^2V2, which is gained by the escaping gas pushing a second piston at pressure f-i through a volume V2. Hence the sum of the external work W is also zero, and therefore, according to equation (17), 1/2= Vx. As the experimental results showed the temperature to be practically unchanged while the volume increased very con- siderably, the internal energy of a perfect gas can depend only on the temperature and not on the volume, i.e..

For nearly perfect gases, as hydrogen, air, etc., the actual small change of temperature observed shows how far the internal energy depends on the volume. It must, however, be borne in mind that for such gases the external work,

w=i)iVi-i.2y2,

does not vanish ; hence the internal energy does not remain constant. For further discussion, see § 158.

§ 71. Special theoretical importance must be attached to those thermodynamical processes which progress infinitely slowly, and which, therefore, consist of a succession of states of equilibrium. Strictly speaking, this expression is vague, since a process presupposes changes, and, therefore, disturbances of equilibrium. But where the time taken is

E

50 THERMOD YNA MICS.

immaterial, and the result of the process aloue of con- sequence, these disturbances may be made as small as we please, certainly very small in comparison with the other quantities which characterize the state of the system under observation. Thus, a gas may be compressed very slowly to any fraction of its original volume, by making the external pressure, at each moment, just a trifle greater than the internal pressure of the gas. Wherever external pressure enters — as, for instance, in the calculation of the work of compression — a very small error will then be committed, if the pressure of the gas be substituted for the external pressure. On passing to the limit, even that error vanishes. In other words, the result obtained becomes rigorously exact for infinitely slow compression.

This holds for compression at constant as well as at variable pressure. The latter may be given the required value at each moment by the addition or removal of small weights. This may be done either by hand (by pushing weights to one side), or by means of some automatic device which acts merely as a release, and therefore does not con- tribute towards the work done.

§ 72, The conduction of heat to and from the system may be treated in the same way. When it is not a question of time, but only of the amount of heat received or given out by the system, it is sufficient, according as heat is to be added to or taken from the system, to connect it with a heat- reservoir of slightly higher or lower temperature than that of the system. This small difference serves, merely, to determine the direction of the flow of the heat, while its magnitude is negligible compared with the changes of the system, which result from the process. We, therefore, speak of the conduction of heat between bodies of equal tempera- ture, just as we speak of the compression of a gas by an external pressure equal to that of the gas. This is merely anticipating the result of passing to the limit from a small finite difference to an infinitesimal difference of temperature between the two bodies.

APPLICATIONS TO HOMOGENEOUS SYSTEMS. 51

This applies not only to strictly isothermal processes, but also to those of varying temperature. One heat-reser- voir of constant temperature will not suffice for carrying out the latter processes. These will require either an auxiliary body, the temperature of which may be arbitrarily changed, e.g. a gas that can be heated or cooled at pleasure by compression or expansion ; or a set of constant-tempera- ture reservoirs, each of different temperature. In the latter case, at each stage of the process we apply that particular heat-reservoir whose temperature lies nearest to that of the system at that moment.

§ 73. The value of this method of viewing the process lies in the fact that we may imagine each infinitely slow process to be carried out also in the opposite direction. If a process consist of a succession of states of equilibrium with the exception of very small changes, then evidently a suitable change, quite as small, is sufficient to reverse the process. This small change will vanish when we pass over to the limiting case of the infinitely slow process, for a definite result always contains a quite definite error, and if this error be smaller than any quantity, however small, it must be zero.

§ 74. We pass now to the application of the first law to a process of the kind indicated, and, therefore, reversible in its various parts. Taking the volume V (abscissa) and the pressure p (ordinate) as the independent variables, we may graphically illustrate our process by plotting its suc- cessive states of equilibrium in the form of a curve in the plane of the co-ordinates. Each point in this plane corre- sponds to a certain state of our system, the chemical nature and mass of which are supposed to be given, and each curve corresponds to a series of continuous changes of state. Let the curve a from 1 to 2 represent a reversible process which takes the substance from a state 1 to a state 2 (Fig. 2). Along a, according to equation (17), the increase of the energy is

U2 - Ui = W + Q,

52

THERM OD YNA MICS.

where W is the mechanical work expended on the substance, and Q the total heat absorbed by it.

§ 75. The value of W can be readily determined. W is made up of the elementary quantities of work done on the system during the infinitesimal changes corresponding to the elements of arc of the curve a. The external pressure is at any moment equal to that of the substance, since the process is supposed to be reversible. Consequently, by the

.V

Fig. 2.

laws of hydrodynamics, the work done by the external forces in the infinitely small change is equal to the product of the pressure ]), and the decrease of the volume, — <ZV, no matter what the geometrical form of the surface of the body may be. Hence the external work done during the whole process is

W

= - />^v,

(20)

in which the integration extends from 1 to 2 along the curve a. If 'p be positive, as in the case of gases, and 'V2 >Vi as in Fig. 2, W is negative.

APPLICATIONS TO HOMOGENEOUS SYSTEMS. 53

In order to perform the integration, the curve a, i.e. the relation between 'p and V, must be known. As long as only the points 1 and 2 are given, the integral has no definite value. In fact, it assumes an entirely different value along a diiferent curve, /3, joining 1 and 2. Therefore i^dN is not a perfect differential. Mathematically this depends on the fact that p is in general not only a function of Y, but also of another variable, the temperature 0, which also changes along the path of integration. As long as a is not given, no statement can be made with regard to the relation be- tween % and V, and the integration cannot be performed.

The external work, W, is evidently represented by the area (taken negative) of the plane figure bounded by the curve a, the ordinates at 1 and 2, and the axis of abscissae. This, too, shows that W depends on the path of the curve a. Only for infinitesimal changes, i.e. when 1 and 2 are infinitely near one another and a shrinks to a curve element, is W determined by the initial and final points of the curve alone.

§ 76. The second measurable quantity is Q, the heat absorbed. It may be determined by calorimetric methods in calories, and then expressed in mechanical units by mul- tiplying by the mechanical equivalent of heat. We shall now consider the theoretical determination of Q. It is, like W, the algebraical sum of the infinitely small quantities of heat added to the body during the elementary processes corresponding to the elements of the curve a. Such an increment of heat cannot, however, be immediately calculated, from the position of the curve element in the co-ordinate plane, in a manner similar to that of the increment of work. To establish an analogy between the two, one might, in imitation of the expression — _pfZV, put the increment of heat = CfZ0, where dQ is the increment of temperature, and C the heat capacity, which is usually a finite quantity. But C has not, in general, a definite value. It does not depend, as the factor j:> in the expression for the increment of work, alone on the momentary state of the substance, i.e.

5 + THERMOD YNA MICS.

on the position of the point of the curve considered, but also on the direction of the curve element. In isothermal changes C is evidently = ± oo, because dB = 0, and the heat added or withdrawn is a finite quantity. In adiabatic changes C = 0, for here the temperature may change in any way, while no heat is added or withdrawn. For a given point, C may, therefore, in contradistinction to p, assume all values between + oo and — cc. (Cf. § 47.) Hence the analogy is incomplete in one essential, and does not, in the general case, simplify the problem in hand. We shall also find that the breaking up of the heat absorbed into the two factors B and d<P (§ 120), is permissible only in some very special cases.

§ 77. Although the value of Q cannot, in general, be directly determined, equation (17) enables us to draw some important inferences regarding it. Substituting the value of W from equation (20) in equation (17), we obtain

Q = U2-Ux+ fpdY,. . . . (21)

which shows that the value of Q depends not only on the position of the points 1 and 2, but also on the connecting path (a or /3). Garnet's theory of heat cannot be reconciled with this proposition, as we have shown at length in §§ 51 and 52.

§ 78. The complete evaluation of Q is possible in the case where the substance returns to its initial state, having gone through a cycle of operations. This might be done by first bringing the system from 1 to 2 along a, then back from 2 to 1 along f3. Then, as in all cycles (§ 65),

Q= -W.

The external work is

W = - r^dY,

the integral to be taken along the closed curve Ia2j31. W evidently represents the area bounded by the curve,

APPLICATIONS TO HOMOGENEOUS SYSTEMS. 55

and is positive if the process follows the direction of the arrow in Fig. 2.

§ 79. We shall now consider the special case where the curve a, which characterizes the change of state, shrinks into an element, so that the points 1 and 2 lie infinitely near one another. W here becomes the increment of work, — pdY, and the change of the internal energy is dXJ. Hence, according to (21), the heat absorbed assumes the value : *

Per unit mass, this equation becomes

q = clii + pdv, (22)

where the small letters denote the corresponding capitals divided by M. In subsequent calculations it will often be advisable to use as an independent variable, either in conjunction with p, or v. We shall, in each case, select as independent variables those which are most conducive to a simplification of the problem in hand. The meaning of the difi*erentiation will be indicated whenever a misunderstand- ing is possible.

We shall now apply our last equation (22) to the most important reversible processes.

§ 80. It has been repeatedly mentioned that the specific heat of a substance may be defined in very different ways according to the manner in which the heating is carried out. But, according to § 46 and equation (22), we have, for any heating process,

_ ^ _du . dv

"^h'M+Pdi (^^)

• It is Usual to follow the example of ClausiuB, and denote thia quantity by dQ to indicate that it is infinitely small. This notation, l^owever, has frequently given rise to misunderstanding, for dQ has been repeatedly regarded as the differential of a known finite quantity Q. We therefore adhere to the notation given above. Other authors use d'Q, in order to obviate the aforesaid misunderstanding.

56 THERMODYNAMICS.

In order to give a definite meaning to tlie differential coefficients, some arbitrary condition is required, which will prescribe the direction of the change. A single condition is sufficient, since the state of the substance depends on two variables only.

§ 81. Heating at Constant Volume. — Here dv = 0, c = c^, the specific heat at constant volume. Hence, accord- ing to equation (23),

^.=aa (^*)

§ 82. Heating under Constant Pressure.— Here dp = 0, c = Cp, the specific heat at constant pressure. According to equation (23),

By the substitution of

in (26), Cp may be written in the form

"^ = ('-«)„+ [(I;),, +^']d9);

or, by (24),

§ 83. By comparing (25) and (27) and eliminating u, we are led to a direct experimental test of the theory.

«^(^^)' (g).=<a-

and by (27), ^='^-n

APPLICATIONS TO HOMOGENEOUS SYSTEMS. 57

whence, differentiating the former equation with respect to V, keeping p constant, and the latter with respect to p, keeping v constant, and equating, we have

dB\ d/ dd

r _ ^_^ ^^^ ^1 _^^ ?? _ 1 /9o\ ^' ^^p ^^hpdv '^ dp' dv dv' dp~ ^^^

This equation contains only quantities which may be experimentally determined, and therefore furnishes a means for" testing the first law of thermodynamics by observations on any homogeneous substance.

§ 84. Perfect Gases. — The above equations undergo considerable simplifications for perfect gases. We have, from (14),

p = , (30)

where R = 826 x 10^ and m is the (real or apparent) mole- cular weight. Hence

e = ^pv,

and equation (29) becomes

dc„ dc„ R

P ^ ^ dp dv m

Assuming that only the laws of Boyle, Gay-Lussac, and Avogadro hold, no further conclusions can be drawn from the first law of thermodynamics with regard to perfect gases.

§ 85. We shall now make use of the additional property of perfect gases, established by Thomson and Joule (§ 70), ^ that the internal energy of a perfect gas depends only on the temperature, and not on the volume, and that hence per unit mass, according to (19),

58 THERMODYNAMICS.

The general equation,

then becomes, for perfect gases,

and, according to (24),

du = c„.de (32)

It follows from (28) that

^ = ^ + 2(%);

or, considering the relation (30),

i.e. there is a constant difference between the specific heat at constant pressure and the specific heat at constant volume. Referring the heat capacity to the molecular weight m, instead, of to unit mass, we have

mcp — mc„ = R (33)

The difference is, therefore, indei^endent even of the nature of the gas.

§ 86. Only the specific heat at constant pressure, Cj,, is capable of direct experimental determination, because a quantity of gas enclosed in a vessel of constant volume has far too small a heat capacity to produce suflScient thermal effects on the surrounding bodies. Since Cp, according to (24), like u, depends on the temperature only, and not on the volume, the same follows for Cj), according to (33), This conclusion was first confirmed by Eegnault's experi- ments. He found Cp constant within a considerable range of temperature. By (33), c^. is constant within the same range.

APPLICATIONS TO HOMOGENEOUS SYSTEMS. 59

If the molecular heats be expressed in calories, R must be divided by Joule's equivalent J. The difference be- tween the molecular heats at constant pressure and at constant volume is then

R 826-105

'^^p-^^«=J=419W

1-971

(34)

§ 87. The following table contains the specific heats

and molecular heats of several gases at constant pressure,

measured by direct experiment ; also the molecular heats

at constant volume found by subtracting 1*97, and also

c the ratio — = 7 : —

llydrogen Oxygen . Nitrogen Air . .

Specific heat at const, pressure.

3410 0-2175 0-2438 0-2375

Molecular

Molecular Molecular

weieht ' heat at const, heat at const. ° ■ pressure. volume.

6-82 ! 485 1-41

6-94 ' 4-97 1-40

6-83 I 4-86 I 1-41

6-84 I 4-87 i 1-41

The specific heat generally increases slowly on con- siderable increase of temperature. Within the range of temperature in which the specific heat is constant, equation (32) can be integrated, giving

w = c„0 -f const (35)

The constant of integration depends on the selection of the zero point of energy. For perfect gases, we consider Cp and Co as constants throughout, hence the last equation holds good in general.

§ 88, Adiabatic Process. — The characteristic feature of the adiabatic process is that g' = 0, and, according to equation (22),

= (Zw + ^dv.

6o THERMODYNAMICS.

Assuming, again, a perfect gas, and substituting the values of du from (32) and of jj from (30), we Lave

O = c/Z0 + -.-cZi;, .... (36) or, on integrating,

^v l^g ^ ^ — ^^8 ^' — const.

Replacing — according to (33) by Cp — Cy, and dividing by Cy, we get

log + (7 — 1) log 17 = const. . . (37)

{i.e. during adiabatic expansion the temperature decreases) Remembering that according to the characteristic equation (30)

log ^ + log V — log = const,

we have, on eliminating v,

— 7 log + (7 - 1) log I) = const.

{i.e. during adiabatic compression the temperature rises) ; or, on eliminating 0,

log j>' + 7 log V = const.

The values of the constants of integration are given by the initial state of the process.

If we compare our last equation in the form

pyy = const (38)

with Boyle's law 2^v = const., it is seen that during adiabatic compression the volume decreases more slowly for an in- crease of pressure than during isothermal compression, because during adiabatic compression the temperature rises. The adiabatic curves in the ^jv — plane (§ 22) are, therefore, steeper than the hyi^erbolic isotherms.

§ 89. Adiabatic processes may be used in various ways

APPLICATIONS TO HOMOGENEOUS SYSTEMS. 6i

for the determination of 7, tlie ratio of the specific heats. The agreement of the results with the value calculated from the mechanical equivalent of heat forms an important confirmation of the theory.

Thus, the measurement of the velocity of sound in a gas may be used for determining the value of 7. It is proved in hydrodynamics that the velocity of sound in a fluid is

i , where p = -, the density of the fluid. Since gases

are bad conductors of heat, the compressions and expansions which accompany sound-vibrations must be considered as adiabatic, and not isothermal, processes. The relation between the pressure and the density is, therefore, in the case of perfect gases, not that expressed by Boyle's

law ^=pv = const., but that given by equation (38), viz. —

V:

^ = const.

Hence, by differentiation

dp yp

or, according to (30),

dp Efl

dp ' m

m dp

'^"'We' dp

In air at 0°, the velocity of sound is / ~- = 33280 — : •^ ^ dp sec. '

hence, according to our last equation, taking the values of

m from § 41, and of E from § 84, and = 273,

28 8 332802 , ,,

-rr-T^ = 141.

' ~ 826 10^ 273

This agrees with the value calculated in § 87.

Conversely, the value of y, calculated from the velocity

62 THERMOD YNA MICS.

of sound, may be used in the calculation of c^ in calories, for the determination of the mechanical equivalent of heat from (33). This method of evaluating the mechanical equivalent of heat was first proposed by Robert Meyer in 1842. It is true that the assumption expressed in equation (31), that the internal energy of air depends only on the temperature, is essential to this method. In other words, this means that the difference of the specific heats at con- stant pressure and constant volume depends only on the external work. The direct proof of this fact, however, must be considered as first given by the experiments of Thomson and Joule, described in § 70.

§ 90. We shall now consider a more complex process, a reversible cycle of a special kind, which has played an important part in the development of thermodynamics, known as Carnot's cycle, and shall apply the first law to it in detail.

Let a substance of unit mass, starting from an initial state characterized by the values 0i, V, first be compressed adiabatically until its temperature rises to ^2(^2 > ^1) and its volume reduced to V2{v2<'Vi) (Fig. 3). Second, suppose it be now allowed to expand isothermally to volume v^{v2 > v-2), in constant connection with a heat-reservoir of constant temperature, B2, which gives out the heat of expansion Q2. Third, let it be further expanded adiahatically until its temperature falls to di, and the volume thereby increased to Vi. Fourth, let it be compressed isothermally to the original volume Vi, while a heat- reservoir maintains the temperature at Qi, by absorbing the heat of compression. All these operations are to be carried out in the reversible manner described in § 71. The sum of the heat absorbed by the system, and the work done on the system during this cycle is, by the first law,

Q -f W = (39)

The heat Q, that has been absorbed by the substance, is Q = Qi + Q2 (40)

APPLICATIONS TO HOMOGENEOUS SYSTEMS. 63

(Qi is here negative). The external work W may be calcu- lated from the adiabatic and the isothermal compressibility of the substance. According to (20),

r«i, H

r^z'' »2

Bl'.Ol

r^i.ei

pdv — / pdv — I 'pdv — j pdv.

These integrals are to be taken along the curves 1, 2, 3, 4 respectively ; 1 and 3 being adiabatic, 2 and 4 isothermal.

Fig. 3.

Assuming the substance to be a perfect gas, the above integrals can readily be found. If we bear in mind the relations (30) and (36), we have

W =

m

Ol

02

'dv +

Cvdd

711

«2

"'ft

^'dv (41)

Vl'

The work of the adiabatic compression in the first part of the process is equal in value and opposite in sign to that of the adiabatic expansion in the third part of the process.

64 THERMODYNAMICS.

There remains, therefore, the sum of the work in the isothermal portions —

Now, the state {v.^,^^ was developed from (i;i,0i) by an adiabatic process ; therefore, by (37),

log 02 + (7 - 1) log ^'a = log 01 4- (7 - 1) log ^'1-

Similarly, for the adiabatic process, which leads from {vi, 62) to (vi', Oi),

log 02 + (7 - 1) log Vo' = log 01 + (7 - 1) log Vi'.

From these equations, it follows that

W _ V

V2 Vi

and .-. W = - -(02 - 0i) log ^'

Vi ' "Wo'

Since, in the case considered, 02 > 0i, and — = — > 1,

the total external work W is negative, i.e. mechanical work has been gained by the process. But, from (39) and (40),

Q = Qi + Q2= - W; ... (42)

therefore Q is positive, i.e. the heat-reservoir at temperature 02 has lost more heat than the heat-reservoir at tempera- ture 01 has gained.

The value of W, substituted in the last equation, gives

Q = Q, + Q, = ^(0,-0,)log|'.. . (43)

The correctness of this equation is evident from the direct calculation of the values of Qi and Q2. The gas expands isothermally while the heat-reservoir at temperature 02 is in action. The internal energy of the gas therefore remains constant, and the heat absorbed is equal in magnitude and

APPLICATIONS TO HOMOGENEOUS SYSTEMS. 65

opposite in sign to the external work. Hence, by equating Q2 to the second integral in (41),

Q, = \log^' = \log^',

and, similarly, by equating Qi to the fourth integral in (41),

Q, = ?0,log^=-\log^,

which agrees with equation (43).

There exists, then, between the quantities Qi, Qg, W, besides the relation given in (42), this new relation —

Qi : Q2 : W = (- 0i) : 6., : (Oi - B^) . . (44)

§ 91. In order, now, to survey all the effects of the above Carnot cycle, we shall compare the initial and final states of all the bodies concerned. The gas operated upon has not been changed in any way by the process, and may be left out of account. It has done service only as a trans- mitting agent, in order to bring about changes in the surroundings. The two reservoirs, however, have undergone a change, and, besides, a positive amount of external work, W = — W, has been gained; i.e. at the close of the process certain weights, which were in action during the compression and the expansion, are found to be at a higher level than at the beginning, or a spring, serving similar purposes, is at a greater tension, etc. On the other hand, the heat-reservoir at 02 has given out heat to the amount Q2, and the cooler reservoir at 61 has received the smaller amount Qi' = — Qi. The heat that has vanished is equivalent to the work gained. This result may be briefly expressed as follows : The quantity of heat Q2, at temperature 62, has passed in part (Qi') to a lower temperature (di), and has in part (Q2 — Qi' = Qi + Q2) been transformed into mechanical work. Carnot's cycle, performed with a perfect gas, thus affords a means of drawing heat from a body and of gaining work in its stead, without introducing any changes in nature except the

F

65 THERMODYNAMICS.

transference of a certain quantity of heat from a body of higher temperature to one of lower temperature.

But, since the process described is reversible in all its parts, it may be put into effect in such a way that all the quantities, Qi, Q2, W, change sign, Qi and W becoming positive, Q2 = — Q2' negative. In this case the hotter reservoir at ^2 receives heat to the amount Q2', partly from the colder reservoir (at 0i), and partly from the mechanical work expended (W). By reversing Carnot's cycle, we have, then, a means of transferring heat from a colder to a hotter body without introducing any other changes in nature than the transformation of a certain amount of mechanical work into heat. We shall see, later, that, for the success of Carnot's reversible cycle; the nature of the transmitting agent or working substance is immaterial, and that perfect gases are, in this respect, neither superior nor inferior to other substances (c/. § 137).

CHAPTER III.

APPLICATIONS TO NON-HOMOGENEOUS SYSTEMS.

§ 92. The propositions discussed in the preceding chapter are, in a large part, also applicable to substances which are not perfectly homogeneous in structure. We shall, there- fore, in this chapter consider mainly such phenomena as characterize the inhomogeneity of a system.

Let us consider a system composed of a number of homogeneous bodies in juxtaposition, separated by given bounding surfaces. Such a system may, or may not, be chemically homogeneous. A liquid in contact with its vapour is an example of the first case, if the molecules of the latter be identical with those of the former. The beginning of a chemical reaction, inasmuch as a substance is in contact with another of different chemical constitution, is an example of the second. Whether a system is physi- cally homogeneous or not, can, in most cases, be ascertained beyond doubt, by finding surfaces of contact within the system, or, by other means — in the case of emulsions, for example, by determining the vapour pressure or the freezing point. The question as to the chemical homogeneity, i.e. the presence of one kind of molecule only, is much more difficult, and has hitherto been answered only in special cases. For this reason we classify substances according to their physical and not according to their chemical homogeneity.

§ 93. One characteristic of processes in non-homogeneous systems consists in their being generally accompanied by considerable changes of temperature, e.g. in evaporation or

68 THERMODYNAMICS.

in oxidation. To maintain the initial temperature and pressure consequently requires considerable exchange of heat with the surroundings and corresponding external work. The latter, however, is generally small compared with the external heat, and may be neglected in most chemical processes. In thermochemistry, therefore, the external effects,

Q4-W = U2-Ui, . . . . (45)