Treatise on Thermodynamics (1903) — part 6 of 14

As a further example, we shall take a mixture of hydrogen and oxygen which has been exploded by means of an electric spark. The spark acts only the secondary part of a release, its energy being negligible in comparison with the energies obtained by the reaction. The work of the chemical affinities in this process is equal to the mechanical work that might be gained by chemically com- bining the oxygen and hydrogen in some reversible and

GENERAL DEDUCTIONS. 113

isothermal way. Dividing this quantity by the number of oxidized molecules of hydrogen, we obtain a measure of the force with which a molecule of hydrogen tends to become oxidized. This definition of chemical force, however, has only a meaning in so far as it is connected with that work.

§ 144. In chemical processes the changes of the first term, U, of the expression for the free energy (71), fre- quently far surpass those of the second, 0<I>. Under such circumstances, instead of the decrease of F, that of U, i.e. the heat effect, may be considered as a measure of the chemical work. This leads to the proposition that chemical reactions, in which there is no external work, take place in such a manner as to give the greatest heat etfects (Berthelot's principle). For high temperatures, where 0, and for gases ^ and dilute solutions, where $ is large, the term 04> can no longer be neglected without considerable error. In these cases, therefore, chemical changes often do take place in such a way as to increase the total energy, i.e. with the absorption of heat.

§ 145. It should be borne in mind that all these pro- positions refer only to isothermal processes. To answer the question as to how the free energy acts in other processes, it is only necessary to form the differential of (71) viz. :

and to substitute in the general relation (70). We have then

dF<W - ^de

for any physical or chemical process. This shows that, with change of temperature, the relation between the external work and the free energy is far more complicated. ^ This relation cannot, in general, be used with advantage.

§ 146. We shall now compute the value of the free energy of a perfect gas. Here, according to (35),

U = M« = M(c„0 + const.),

1 1 4 THE R MOD YNA MICS.

and, by (52),

(p = M(^ = M(c^ log + " log i; + const.).

Substituting in (71), we obtain

F = M{^^0(const - log 0) log v + const} (74)

which contains an arbitrary linear function of 0.

For isothermal changes of the gas, we have, by § 142,

cZF < W, or, by (74), since is const.,

m V ^ —

If the change be reversible, the external work on the gas is AV = — ^jrfV, but if it be irreversible, then the sign of inequality shows that the work of compression is greater, or that of expansion smaller, than in a reversible process.

§ 147. Case III. Isothermal -isopiestic Process.— If, besides the temperature 0, the external pressure p be also kept constant, then the external work is given by the formula,

W = -i^dN,

and the left-hand side of (69) becomes a complete differ- ential :

<*-E±i^)>o

In this case, it may be stated that for finite changes the function,

<,_£+i^ = ^, .... (75)

GENERAL DEDUCTIONS. 115

must increase, and will remain constant only in tlie limit when the change is reversible.*

§ 148. Conditions of Equilibrium.— The most general condition of equilibrium for any system of bodies is derived from the proposition that no change can take place in the system if it be impossible to satisfy the condition necessary for a change.

Now, by (69), for any actual change of the system,

aO g > 0.

The sign of equality is omitted, because it refers to ideal changes which do not actually occur in nature. Equilibrium is, therefore, maintained if the fixed conditions imposed on the system be such that they will permit only changes in which

Here S is used to signify a virtual infinitely small change, 4 in contrast to d, which corresponds to an actual change.

§ 149. In most of the cases subsequently discussed, if any given virtual change be compatible with the fixed con- ditions of the system, its exact opposite is also, and is represented by changing the sign of all variations involved. This is true if the fixed conditions be expressed by equa- tions, not by inequalities. Assuming this to be the case, if we should have, for any particular virtual change,

• Multiplying (75) by — 6, we get P. Duliem's thermodynamic potential at constant pressure,

U + pV - 0*,

for which, so long as remains constant, the same propositions hold as for the function *. However, the equation (153) in § 211, which is important for the dependence of the equilibrium on temperature and pressure, can be more conveniently deduced from the function * than from the thermo- dynamic potential.

1 16 THERMOD YNAMICS.

whicli, by (69), would make its occurrence in nature im- possible, its opposite would conform to the condition for actual changes (69), and could therefore take place in nature. To ensure equilibrium in such cases, it is necessary, therefore, that, for any virtual change compatible with the fixed conditions,

m _ W

S$ - ^ = . . . . (76)

This equation contains a condition always sufficient, but, as we have seen, not always necessary to its full extent, for the maintenance of equilibrium. As a matter of experience, equilibrium will occasionally subsist when equation (76) is not fulfilled, even though the fixed conditions permit of a change of sign of all variations. This is to say, that occasionally a certain change will not take place in nature, though it satisfy the fixed conditions as well as the demands of the second law. Such cases lead to the conclusion that in some way the setting in of a change meets with a certain resistance, which, on account of the direction in which it acts, has been termed inertia resistance, or passive resist- ance. States of equilibrium of this description are always unstable. Often a very small disturbance, not comparable in size with the quantities within the system, suffices to produce the change, which under these conditions often occurs with great violence. We have examples of this in overcooled liquids, supersaturated vapour, supersaturated solutions, explosive substances, etc. We shall henceforth discuss mainly the conditions of stable equilibrium de- ducible from (76).

This equation may, under certain circumstances, be expressed in the form of a condition for a maximum or minimum. This can be done when, and only when, the conditions imposed upon the system are such that the left- hand side of (76) represents the variation of some one function. The most important of these cases are dealt with separately in the following paragraphs. They correspond exactly to the propositions which we have already deduced

genMral deductions. Hi

for special cases. From these propositions it may at once be seen whether it is a case of a maximum or a minimum.

§ 150. First Case (§ 141). — If no exchange of heat take place with the surrounding medium, the first law gives

gU = W, hence, by (76), g<I) = (77)

Among all the states of the system which can proceed from one another by adiabatic processes, the state of equilibrium is distinguished by a maximum of the entropy. Should there be several states in which the entropy has a maximum value, each one of them is a state of equilibrium; but if the entropy be greater in one than in all the others, then that state represents absolutely stable equilibrium, for it could no longer be the starting-point of any change what- soever.

§ 151. Second Case (§ 142). — If the temperature be kept constant, equation (76) passes into

<--?)

• ^=0,

and, by (71), - 8F = -W.

Among all the states which the system may assume at a given temperature, a state of equilibrium is characterized by the fact that the free energy of the system cannot^ decrease without performing an equivalent amount of work. If the external work be a negligible quantity, as it is when the volume is kept constant or in numerous chemical processes, then W = 0, and the condition of equilibrium becomes

SF = 0,

i.e. among the states which can proceed from one another by isothermal processes, without the performance of external work, the state of most stable equilibrium is distinguished by an absolute minimum of the free energy.

ii8 ^ THERMODYNAMICS.

§ 152. Third Case (§ 147). — Keeping the temperature and the pressure ]) constant and uniform, we have

W = -jjgV, (78)

and the condition of equilibrium (76) becomes

or, by (75), g4^ = (79)

i.e. at constant temperature and constant pressure, the state of most stable equilibrium is characterized by an absolute maximum of the function "9^.

AYe shall now proceed to consider, in succession, states of equilibrium of various systems by means of the theorems we have just deduced, going from simpler to more compli- cated cases.

PART IV.

Applications to Special States of Equilibrium.

CHAPTER I.

HOMOGENEOUS SYSTEMS.

§ 153. Let the state of a homogeneous system be deter- mined, as hitherto, by its mass, M ; its temperature, ; and

V

either its pressure, ^, or its specific volume, v = ^. For

the present, besides M, let and v be the independent variables. Then the pressure |:>, the specific energy w = ^,

and the specific entropy ^ — ^ are functions of and v, the definition of the specific entropy (61) being

On the other hand.

Therefore, since dO and dv are independent of each other,

I20 THERMODYNAMICS.

These two equations lead to an experimental test of the second law ; for, diiferentiating the first with respect to v, the second with respect to 0, we have

dOdv ~ 6 ' dddv ~ e 02

i^)r'>(^^r" («")

By this and equation (24), the above expressions for the differential coeflScients of tp become :

§ 154. Equation (80), together with (28) of the first law, gives the relation :

'--="('!) ('a • • • («^)

which is useful either as a test of the second law or for the calculation of e^. when Cp is given. But since in many cases

igo) cannot be directly measured, it is better to introduce the relation (6), and then

As (1)^ is necessarily negative, e, is always greater than .„

except in the limiting case, when the coefficient of expansion is = 0, as in the case of water at 4° C, ; then Cp — c,, = 0.

^ an example, we may calculate the specific heat at constant volume, Cc, of mercury at 0"^ C. from the following data:

Cj, = 0-0333; = 273=; <5^A _ _ 1014000 Cdv). ~ 0-00000295 . V

HOMOGENEOUS SYSTEMS. 121

where the denominator is the coefficient of compressibility in atmospheres (§ 15) ; the numerator, the pressure of an

atmosphere in absolute units (§ 17) ; v = yi^a' *^^ volume

of 1 gr. of mercury at 0° C. ; (^^j = 0-0001812. -y, the

coefficient of thermal expansion (§ 15).

To obtain c„ in calories, it is necessary to divide by the mechanical equivalent of heat, 419 x 10^ (§ 61). Thus we obtain from (83)

273 X 1014000 X 000181 2^ ^nan-A ""P ""' ~ 0-00000295 X 13-6 x 419 x 10^ ~ ^'^^^^'

whence, from the above value for c^,

c„ = 0-0279. _

§ 155. This method of calculating the difference of the specific heats Cp — c„, applicable to any substance, discloses at the same time the order of magnitude of the different influences to which this quantity is subject. According to equation (28) of the first law, the difference of the specific heats is

"--'="= tiv), + ^'idil

The two terms of this expression, -^)^a) *^^ H ^) '

depend on the rate of change of the energy with the volume, and on the external work performed by the expansion respectively. In order to find which of these two terms has the greater influence on the quantity Cp — c^, we shall find the ratio of the first to the second : *

1 /3w\ '^

, f KdvJe

o'.by(80). |*(|-1 (84).

122 THERMODYNAMICS.

A glance at the tables of the coefficients of thermal expansion and of the compressibility of solids and liquids shows that, in general, the first term of this expression is a large number, making the second, —1, a negligible quantity. For mercury at 0°, e.g., the above data give the first term to be

^^^ ^^ 00001812 _^.. ^^^ ^ 0-00000295 = •^^^^"•

Water at 4° C. is an exception.

It follows that, for solids and liquids, the difference Cp — Cr, depends rather on the relation between the energy and the volume than on the external work of expansion. For perfect gases the reverse is the case, since the internal energy is independent of the volume, i.e. —

©. = "•

During expansion, therefore, the influence of the internal energy vanishes in comparison with that of the external work ; in fact, the expression (84) vanishes for the charac- teristic equation of a perfect gas. With ordinary gases, however, both the internal energy and the external work must be considered.

§ 156. The sum of both these influences, i.e. the whole expression Cp — c„ may be said to have a small value for

most solids and liquids ; thus the ratio -? = y is but slightly

greater than unity. This means that in solids and liquids the energy depends far more on the temperature than on the volume. For gases, 7 is large ; and, in fact, the fewer the number of atoms in a molecule of the gas, the larger does it become. Hydrogen, oxygen, and most gases with diatomic molecules have 7 = 141 (§ 87). The largest value of 7 ever observed is that found by Kundt and Warburg for the monatomic vapour of mercury, viz. 1666.

§ 157. For many applications of the second law it is

HOMOGENEOUS SYSTEMS. 123

convenient to introduce p instead of v as an independent variable. We have, by (61),

,^ du--pdv /du\ . /dv]d6 , /du\ , /dv\ ]dp

^ r- = lUi + n,5e)JT + W, + Am

On the other hand,

''* = (i)/'' + (|)A

whence, (^-|)^ = _^-^ :

and r^^l - WA"^Hv,

Differentiating the first of these with respect to jj, the second with respect to 0, we get

dddp 9 6 6|2

The differential coefficients of ^ become, then, by (26),

(lel = t=''°HS),= -(55);

Finally, differentiating the first of these with respect to p, the second with respect to 0, and equating, we have

0r-C^ («^)

This equation contains only quantities that can be directly measured, and establishes a relation between the rate of change of the coeflficient of thermal expansion of the sub- stance with temperature {i.e. the deviation from Gay-Lussac's

124 THERMODYNAMICS.

law), and the rate of change of the specific heat with pressure.

§ 158. By means of the relations furnished by the second law we may also draw a further conclusion from Thomson and Joule's experiments (§ 70), in which a gas was slowly pressed through a tube plugged with cotton wool. The interpretation in § 70 was confined to their bearing on the properties of perfect gases. It has been mentioned that the characteristic feature of these experiments consists in giving to a gas — without adding or withdrawing heat * — an increase of volume, Va — Vi, or v-2 — Vi per unit mass, while the external work done per unit mass is represented by

This expression vanishes in the case of perfect gases, since then the temperature remains constant. In the case of actual gases we may put

Ih = p, Ih = i^ + ^P (A p < 0) Vi = V, V2 = V + ^v ( A v > 0)

whence W = — ^ij)v),

and by the first law, since Q = 0,

Aw = W + Q = - M^v).

For the sake of simplicity we shall assume Aj:> and Av to be small, and we may then write the above equation :

or, by (24), (82), and (80),

• Whether this condition is actually fulfilled may be ascertained by measurements in the medium surrounding the tube through which the gas flows.

HOMOGENEOUS SYSTEMS. 125

and, by (6)

^dv

A0 = J£^^£ Ap . . .• (86)

'p

By means of this simple equation, the change of tempera- ture (A0) of the gas in Thomson and Joule's experiments, for a difference of pressure AjJ, may be found from its specific heat, c,,, and its deviation from Gay-Lussac's law. If, under constant pressure, v were proportional to 6, as in Gay-Lussac's law, then, by equation (86), A0 = 0, as is really the case for perfect gases.

§ 159. Thomson and Joule embraced the results of their observations in the formula

A0 = ^Ap,

where a is a constant. If we express p in atmospheres, we have, for air,

a = 0-276 X (273)2.

No doubt the formula is only approximate. Within the region of its validity we get, from 86,

and, differentiating with respect to B,

d^v\ _ a /dep\ _ 2aCp

^ ~Q\Te)p~~W

<w\

whence, by the relation (85),

/5cp\ ,a(dcp\ 2acp^

126 THERMODYNAMICS.

The general solution of this differential equation is

where / denotes an arbitrary function of its argument,

If we now assume that, for small values of 'p, the gas, at any temperature, approaches indefinitely near the ideal state, then, when 'p = 0, Cp becomes a constant = 4"^ (for ^ir, Sp' = 0*238 calorie). Hence, generally,

or . =— iL_ (88)

This expression for e^^ will serve further to determine v in terms of and p. It follows from (87) that

de\eip= B'-i^ -3ap)^

whence

^ = T,:^(Vl-?3' +

ft). . (89)

This is the characteristic equation of the gas, and /3, the constant of integration, may be determined from its density at 0° C. and atmospheric pressure. Equations (88) and (89), like Thomson and Joule's formula, are valid only within certain limits. It is, however, of theoretical interest to see how the different relations necessarily follow from one another.

§ 160. A further, theoretically important application of the second law is the determination of the absolute tem- perature of a substance by a method independent of the

HOMOGENEOUS SYSTEMS. 127

deviations of actual gases from the ideal state. In § 4 we defined temperature by means of the gas thermometer, but had to confine that definition to the cases in which the readings of the different gas thermometers (hydrogen, air, etc.) agree as nearly as the desired accuracy of the result requires. For all other cases (including mean temperatures, when a high degree of accuracy is desired) we postponed the definition of absolute temperature. Equation (80) enables us to give an exact definition of absolute tempera- ture, entirely independent of the behaviour of special substances.

Given the temperature readings, t, of any arbitrary thermometer (mercury-thermometer, or the scale deflection of a thermo-element, or of a bolometer), our problem is to reduce the thermometer to an absolute one, or to express the absolute temperature as a function of t. We may by direct measurement find how the behaviour of some appro- priate substance, eg. 9. gas, depends on t and either v or 'p. Introducing, then, t and v as the independent variables in (80) instead of and v, we obtain

0^r<

where f^),^, and („^j represent functions of t and v,

which can be experimentally determined. The equation can then be integrated thus :

If we further stipulate that at the freezing-point of water, where ^ = ^0,^ = ^0 = 273, then.

128 THERMODYNAMICS.

This completely determines as a function of t. It is evident that the volume, v, no longer enters into the expression under the sign of integration.

§ 161. The numerator of this expression may be found directly from the characteristic equation of the substance. The denominator, however, depends on the amount of heat which the substance absorbs during isothermal reversible expansion. For, by (22) of the first law, the ratio of the heat absorbed during isothermal reversible expansion to the change of volume is

(!>), = (5-"),+^'-

§ 162. Instead of measuring the quantity of heat absorbed during isothermal expansion, it may be more convenient, for the determination of the absolute temperature, to experi- ment on the changes of temperature of a slowly escaping gas, according to the method of Thomson and Joule. If we introduce t (of § 160) instead of into equation (86), which represents the theory of those experiments on the absolute temperature scale, we have

(

(It Wp ~ \dtjp ' cW

dt , dt

_ / q\ _ f 9\ dt _ ,dt ~ \de)p ~ \dt)p ' dO ~ ^PdO

where c?^ is the specific heat at constant pressure, determined by a ^ thermometer. Consequently, by (86),

/'dv\ dt

and again, by integration,

log|=|-^^=J-- • • (90)

HOMOGENEOUS SYSTEMS. 129

The expression to be integrated again contains quantities which may be measured directly with comparative ease.

§ 163. The stipulation of § 160, that, at the freezing- point of water, 6 = Bq = 273, implies the knowledge of the coefficient of expansion, a, of perfect gases. Strictly speak- ing, however, all gases show at all temperatures deviations from the behaviour of perfect gases, and disagree with one another. To rid ourselves of any definite assumption about a, we return to our original definition of temperature, viz. that the difference between the absolute temperature of water boiling under atmospheric pressure (0i), and that of water freezing under the same pressure (Oo), shall be

01 _ 00 = 100 (91)

Now, if ti be the boiling-point of water, measured by means of a ^ thermometer, then, by (90),

and, eliminating Bo and Bi from (90), (91), and (92), we find the absolute temperature :

B = ^,^^ (93)

From this we obtain the coefficient of thermal expansion of a perfect gas, independently of any gas thermometer,

„-l = ^-l:ii (94) ■

" - 00 100 ^ ^

Since, in both J and Ji, the expression to be integrated depends necessarily on t only, it is sufficient for the calcula- tion of the value of the integral to experiment at different temperatures under some simplifying condition, as, for instance, always at the same pressure (atmospheric pressure).

§ 164. The formula may be still further simplified by

K

1 30 THERMOD YNA MICS.

using as thermometric substance in the t thermometer the same gas as that on which Thomson and Joule's experi- ments are being performed. The coefficient of expansion, o', referred to temperature t, is then a constant, and if, as is usual, we put Iq = 0, and ti = 100,

V = ^0(1 + at),

Vq being the specific volume at the melting-point of ice under atmospheric pressure. Also

U = "''

Hence, by (90),

and, by (92),

J = ■•' «''"

J} + "

J I a'dt

Ji =

Vo A})

In the case of an almost perfect gas (e.g. air), At is small,

c ' At and the term -^ • -— acts merely as a correction term, Vo A})

and, therefore, no great degree of accuracy is required in

the determination of Cp and vq. For a perfect gas we should

have A^ = 0, and, from the last two equations,

J = log (1 + a't), Ji = log (1 + 100a') ; therefore, by (93),

a

and, by (94),

1 « = TT = a,

»0

as it should be.

HOMOGENEOUS SYSTEMS. 131

As soon as accurate measurement of even a single substance has determined as a function of t, the question regarding the value of the absolute temperature may be considered as solved for all cases.

The absolute temperature may be determined not only by experiments on homogeneous substances, but also from the theory of heterogeneous substances (cf. § 177).

CHAPTEK II.

SYSTEM IN DIFFERENT STATES OF AGGREGATION.

§ 165. We shall discuss in this chapter the equilibrium of a system which may consist of solid, liquid, and gaseous portions. We assume that the state of each of these portions is fully determined by mass, temperature, and volume; or, in other words, that the system is formed of but one independent constituent (§ 198). For this it is not necessary that any portion of the system should be chemi- cally homogeneous. Indeed, the question with regard to the chemical homogeneity cannot, in general, be completely answered (§ 92). It is still very uncertain whether the molecules of liquid water are the same as those of ice. In fact, the anomalous properties of water in the neighbour- hood of its freezing-point make it probable that even in the liquid state its molecules are of different kinds. The decision of such questions has no bearing on the investiga- tions of this chapter. The system may even consist of a mixture of substances in any proportion ; that is, it may be a solution or an alloy. What we assume is only this : that the state of each of its homogeneous portions is quite definite when the temperature and the specific volume v are definitely given, and that, if the system consists of different substances, their proportion is the same in all portions of the system. We may now enunciate our problem in the following manner : —

Let us imagine a substance of given total mass, M, enclosed in a receptacle of volume, V, and the energy, U, added to it by heat-conduction. If the system be now isolated and left to itself, M, V, and U will remain con- stant, while the entropy, <P, will increase. We shall now

DIFFERENT STATES OF AGGREGATION. 133

investigate the state or states of equilibrium which the system may assume, finding at the same time the conditions of its stability or instability. This investigation may be completely carried through by means of the proposition expressed in equation (77), that of all the states that may adiabatically arise from one another, the most stable state of equilibrium is characterized by an absolute maximum of the entropy. The entropy may in general, however, as we shall see, assume several relative maxima, under the given external conditions. Each maximum, which is not the absolute one, will correspond to a more or less un- stable equilibrium. The system in a state of this kind {e.g. as supersaturated vapour) may occasionally, upon appropriate, very slight disturbances, undergo a finite change, and pass into another state of equilibrium, which necessarily corresponds to a greater value of the entropy.

§ 166. We have now to find, first of all, the states in which the entropy <!> becomes a maximum. The most general as- sumption regarding the starte of the system is that it consists of a solid, a liquid, and a gaseous portion. Denoting the masses of these portions by Mi, M2, Mg, but leaving open, for the present, the question as to which particular portion each suffix refers, we have for the entire mass of the system Ml + M2 + M3 = M. All the quantities are positive, but some may be zero. Further, since the state under discussion is to be one of equilibrium, each portion of the system, also when taken alone, must be in equilibrium, and therefore of uniform temperature and density. To each of them, there- fore, we may apply the propositions which were deduced in the preceding chapter for homogeneous substances. If Vij •^2, "ysj denote the specific volumes, the given volume of the system is

MiVi + M2V2 + M3V3 = V. Similarly, the given energy is

MiWi + M2W2 + MgWg = U,

where Wi, %, W3 denote the specific energies of the portions.

134

THERMO D YNAMICS.

These three equations represent the given external conditions.

§ 167. For the entropy of the system we have

*^i5 ^2i i>3 heing the specific entropies.

For an infinitesimal change of state this equation gives

Since, by (61), we have, in general,

^ hi + ])Sv dfp =■ ^ .

we obtain

g4> = ;gM^^ + ^M^l^^^ + S^,8M. . (95)

By

6>i

These variations are not all independent of one another. In fact, from the equations of the imposed (external) con- ditions, it follows that

^mi = ]

SMig«;i + SvigMi = . . . (96) " -2MigHi+ 2»igMi = J

With the help of these equations we must eliminate from (95) any three variations, in order that it may contain only independent variations. If we substitute in (95), for instance, the values for S3l2, 3^2, and hi^ taken from (96), the equation for S4' becomes

(^ A ^^2 - ^3 1{V^ - Vs)^^r

(97)

DIFFERENT STATES OF AGGREGATION. 135

Since the six variations occurring in this expression are now independent of one another, it is necessary that each of their six coefficients should vanish, in order that S<I> may be zero for all changes of state. Therefore

' Oi = 02 = ^aC = 0)

fl — <P2 —

<^2 - </'3 =

(98)

These six equations represent necessary properties of any state, which corresponds to a maximum value of the entropy, i.e. of any state of equilibrium. As the first four refer to equality of temperature and pressure, the main interest centres in the last two, which contain the thermodynamical theory of fusion, evaporation, and sublimation.

§ 168. These two equations may be considerably simpli- fied by substituting the value of the specific entropy ^, which, as well as u and ^j, is here considered as a function of and V. For, since (61) gives, in general,

, du -- pdv cl(p = g >

we get, by integration,

,1 du + 2>dv

^1 — 02 =

e

where the upper limit of the integral is characterized by the values Bi, Vi, the lower by O2, V2. The path of integration is arbitrary, and does not influence the value of ^i — ^2. Since, now, B^ = O2 = 6 (by 98), we may select an isothermal path of integration (B const.). This gives

Ml - W2 , 1 P 7

'^ = —IT- + tilf"-

1 36 THERM OD YNA MICS.

The integration is to be taken along an isotherm, since f is a known function of and v determined by the charac- teristic equation of the substance. Substituting the value of <pi — (f)2 in the equations (98), we have the relations:

J V2

p / ^ ) • • • • (99)

to which we add jh = jh = 2h

With the four unknowns B, Vi, v.j, i% we have four equations which the state of equilibrium must satisfy. The constants which occur in these equations dej^end obviously only on the chemical nature of the substance, and in no way on the given values of the mass, M, the volume, V, and the energy, U, of the system. The equations (99) might therefore be called the system's internal or intrinsic conditions of equili- brium, while those of § 166 represent the external conditions imposed on the system.

§ 169. Before discussing the values which the equations (.99) give to the unknowns, we shall investigate generally whether, and under what condition, they lead to a maximum value of the entropy and not to a minimum value. It is necessary, for this purpose, to find the value of S'^<I>. If this be negative lor all virtual changes, then the state considered is certainly one of maximum entropy. .\ ^ From the expression for §<!> (97) we obtain S^4», which J^ may be greatly simplified with the heljj of the equations (98). The equations of the imposed external conditions/^ and the equations (96) further simplify the result, and we obtain, finally,

This may be written

0gM> = -2Mx(g^ig0i - ^iHcih)

DIFFERENT STATES OF AGGREGATION, 137

To reduce all variations to those of the independent variables, and v, we may write, according to (81),

and 8, = (1)^.9 + (II);.

/. 0gM> = - 2Mi(^''J^'80i' - (^') Wy . (100)

Obviously, if the quantities {c„)i, (6^)2, {cv)3 be all posi- tive, and the quantities ( /-M ... all negative, 8^<t> is

negative in all cases, and $ is really a maximum, and the corresponding state is a state of equilibrium. Since c„ is the specific heat at constant volume, and therefore always positive, the condition of equilibrium depends on whether

( „- ) is negative for all three portions of the system or not.

In the latter case there is no equilibrium. Experience immediately shows, however, that in any state of equi- librium ^ is negative, since the pressure, whether positive

or negative, and the volume always change in opposite directions. A glance at the graphical representation of p, as an isothermal function of v (Fig. 1, § 26), shows that

dp . there are certain states of the system in which ^- is posi- tive. These, however, can never be states of equilibrium, and are, therefore, not accessible to direct observation. If,

on the other hand, ^^~ be negative, it is a state of equi- librium, yet it need not be stable; for another state of equilibrium may be found to exist which corresponds to a greater value of the entropy.

We shall now discuss the values of the unknowns, d, ly i'2, V3, which represent solutions of the conditions of equi- librium (98). Several such systems may be found. There- after, we shall deal (beginning at § 189) with the further

1 38 THERMOD YNA MICS.

question as to which of the different solutions in each case represents the most stable equilibrium under the given external conditions; i.e. which one leads to the larjiest value of the entropy of the system.

't>^

§ 170. First Solution. — If we put, in the first place,

Vx = V.2 = l'3( = V)

all the equations (98) are satisfied, for, since the tempera- ture is common to all three portions of the system, their states become absolutely identical. The entire system is, therefore, homogeneous. The state of the system is deter- mined by the equations of § 166, which give the imposed conditions. In this case they are

Ml + M2 + 3I3 = 31

vQ,h + M2 + 3I3) = Y «(Mi + M2 + M3) = U

.-, V = ^ and u = ^^ M M

From V and u, 6 may be found, since u was assumed to be a known function of and v.

This solution has always a definite meaning ; but, as we saw in equation (100), it represents a state of equilibrium

only when ^- is negative. If this be the case, then, the

equilibrium is stable or unstable, according as under the external conditions there exists a state of greater entropy or not. This will be discussed later.

§ 171. Second Solution. — If, in the second case, we put

'l^l < v.2, V.2 = V3,

the states 2 and 3 coincide, and the equations (98) reduce to

Pi = V-2 I

ui -u<i+ 2)1(^1 - V2) \ . (101) (l>i - <t>-2 = ^ J

DIFFERENT STATES OF AGGREGATION. 139 or, instead of the second of these equations,

I -pdv = jhivi - V2) . . . . (102)

J V2

In this case two states of the system coexist ; for instance, the vapour and the liquid. The equations (101) contain three unknowns, B, Vi, v^ ; and hence may serve to express Vi and ^2? consequently also the pressure pi = 2h> aid the specific energies Ui and U2, as definite functions of the temperature 0. The internal state of two heterogeneous portions of the same substance in contact with one another is, therefore, completely determined by the temperature. The temperature, as well as the masses of the two portions, may be found from the imposed conditions (§ 166), which are, in this case,

M, + (M2 + M3) = M )

M^v, + (M2 + Ms)v, = V . . (103)

Mi«i -f (M2 + Ms}ii.2 = U )

These equations serve for the determination of the three last unknowns, 6, Mi, and M2 + M3. This completely determines the physical state, for, in the case of the masses M2 and M3, it is obviously sufficient to know their sum. Of course, the result can only bear a physical interpretation if both Ml and M2 + M3 have positive values.

§ 172. An examination of equation (102) shows that it can be satisfied only if the pressure, }>, which is known to have the same value ( j^i = 2h) for both limits of the integral, assume between the limits values which are partly larger and partly smaller than jh. Some of these, then, must correspond to unstable states (§ 169), since in certain places

p and V increase simultaneously ( ^ > Oj. The equation

admits. of a simple geometrical interpretation with the help of the above-mentioned graphical representation of the characteristic equation by isotherms (Fig. 1, § 26). For

the integral / ^cZy is represented by the area bounded by

I40 THERMO D YNA MICS.

the isotherm, the axis of abscissae, and the ordinates at Vy and V2, while the product ih{vi — v^) is the rectangle formed by the same ordinates (^i = p^, and the length {vi — Vg). We learn, therefore, from equation (102) that in every isotherm the pressure, under which two states of aggregation of the substance may be kept in lasting contact, is repre- sented by the ordinate of the straight line parallel to the axis of abscissae, which intercepts equal areas on both sides of the isotherm. Such a line is represented by ABC in Fig. 1. We are thus enabled to deduce directly from the characteristic equation for homogeneous, stable and unstable, states the functional relation between the pressure, the density of the saturated vapour and of the liquid in contact with it, and the temperature.

Taking Clausius' equation (12) as an empirical expres- sion of the facts, we have, for the specific volume Vi of the saturated vapour, and v.2 of the liquid in contact with it, the two conditions

m _ G ^ m e

Vx — a d{vi 4- by ~ V2 — a 0(v2 + b)'^' and, from (102),

By means of these Vu ^2, and 2h = 2^ ^^J ^ expressed as functions of 0, or, still more conveniently, Vi, v^, jh, and 6 as functions of some appropriately selected independent variable. With Clausius' values of the constants for carbon dioxide (§ 25), this calculation furnishes results which show a satis- factory agreement with Andrews' observations. According to Thiesen, however, Clausius' equation is by no means the general form of the characteristic equation.

§ 173. We shall now follow the interpretation of the equation (101) in other directions. If we put, for shortness,

u-e<l>=f (104)

(free energy per unit mass, by equation (71)),

DIFFERENT STATES OF AGGREGATION. 141 the equations (101) become, simply,

jPi=^-2 (105)

f^-fi=Pi{v,-v^) (106)

The function / satisfies the following simple conditions.

By (79a), (1)^ = - ^ (107) «/

Also, by (104),

and, by (80) and (81),

dfl = -^ ('^'^^

The conditions of equilibrium for two states of aggrega- tion in mutual contact hold for the three possible com- binations of the solid and liquid, liquid and gaseous, gaseous and solid states. In order to fix our ideas, how- ever, we shall discuss that solution of those equations which corresponds to the contact of a liquid with its vapour. Denoting the vapour by the subscript 1, the liquid by 2, Vi is then the specific volume of the saturated vapour at the temperature 0; pi = jh^ its pressure ; V2 the specific volume of the liquid with which it is in contact. All these quantities, then, are functions of the temperature only, which agrees with experience.

§ 174. Further theorems may be arrived at by the differentiation of the conditions of equilibrium with respect to 0. Since all variables now depend only on 0, we shall

use -j^ to indicate this total differentiation, while partial do

differentiation with regard to 6 at constant v will be expressed, as hitherto, by ^-

1 42 THERM OD YNA MICS

Equations (105) and (106), thus differentiated, give

d^ ~ cW

But, by (107) and (108), we have

de dd~\de)^^\dvJ, do KdoK \dv}, d~e

dv.2 , dvi

whence, by substitution,

or, finally, by (101),

(wi - u,) + 2h(vx - V,) = B{v, - v,f^. (109)

Here the left-hand side of the equation, according to equation (17) of the first law, represents the heat of vaporization, L, of the liquid. It is the heat which must be added to unit mass of the liquid, in order to completely change it to vapour under the constant pressure of its saturated vapour. For the corresponding change of energy is Wi — %, and the external work performed, here negative, amounts to

W = - 2hivi - V2) :. L = «i - U2+ 2h{vi - V2), . . (110)

whence L = %i - -yg)^ (Ill)

This equation, deduced by Clapeyron from Carnot's theory, but first rigorously proved by Olausius, may be used for the determination of the heat of vaporization at any temperature, if we know the specific volumes of the saturated vapour and the liquid, as well as the relation between the pressure of the saturated vapour and the

DIFFERENT STATES OF AGGREGATION. 143

temperature. This formula has been verified by experiment in a large number of cases.