Treatise on Thermodynamics (1903) — part 9 of 14

Here L may be called briefly the heat of vaporization of the solution. If, instead of regarding L as the ratio of two infinitely small quantities, we take it to be the heat of vaporization per unit mass of the solvent, then the mass of the solvent must be assumed so large that the concen- tration is not appreciably altered by the evaporation of unit mass. The quantity s may generally be put = v, the specific volume of the vapour. Assuming, further, that the laws of Boyle and Gay Lussac hold for the vapour, we get

s = V = (177)

m p ^ ^

and. by (170), I. 4>f-^0,

On the other hand, L is also the quantity of heat given out when unit mass of the vapour of the solvent combines at constant temperature and pressure with a large quantity of a solution of concentration e. This process may be per- formed directly, or unit mass of the vapour may be first condensed to the pure solvent and then the solution diluted with it.

In the first case the sum of the heat given out and the work spent is

^-^^-A-ri-^^-

In the second case, by the method used in § 215 we obtain, as the sum of the heat given out and the work spent during condensation and dilution,

R^2^ log 2h , .

m elf)

198 THERMODYNAMICS.

where 'p^ is the pressure, v^ the specific volume of the vapour of the solvent in contact with the pure liquid solvent, A the heat of dilution of the solution, i.e. the heat given out on adding unit mass of the solvent to a large quantity of the solution of concentration c. Both the above expressions being equal according to the first law, we obtain, on applying Boyle's law,

^ = M-^7' • • • • (1^^)

m \ dd /c

which is Kirchhoffs formula for the heat of dilution.

The quantities here neglected, by considering the vapour a perfect gas, and its specific volume large in comparison with that of the liquid, may readily be taken into account when necessary.

The similarity of the expressions for A, the heat of dilution, and for X, the heat of saturation (161), is only external, since in this case the solution may be of any con- centration, and therefore may be differentiated with respect to the temperature, e being kept constant, while in (161) the concentration of a saturated solution changes with temperature in a definite manner.

§ 222. Since A is small for small values of c (dilute solu- tions, § 97), then, according to (178), the ratio of the vapour pressure of a dilute solution of fixed concentration to the vapour pressure of the pure solvent is practically independent of the temperature (Babo's law).

§ 223. Temperature Constant : dd = 0. — The relation between the vapour pressure (p) and the concentration (c) of the solution is, according to (175),

(^=-T- • ■ • • <™> Neglecting the specific volume of the liquid in comparison

ANY NUMBER OF INDEPENDENT CONSTITUENTS. 199

with that of the vapour, and considering the latter a perfect gas of molecular weight m, equation (177) gives

or

/9 log: «\ m

Since ip is always positive (§ 217), the vapour pressure must decrease with increasing concentration. This proposition furnishes a means of distinguishing between a solution and an emulsion. In an emulsion the number of particles suspended in the solution has no influence on the vapour pressure.

So long as the quantity <p is undetermined, nothing further can be stated with regard to the general relation between the vapour pressure and the concentration.

§ 224. As we have jj = |)o when c = (pure solvent),^ — |>o is small for small values of e. We may, therefore, put

5p _ p — po _ p — p o dc ~ c — ~ c

Hence, by (179),

Po-P^'^ (180)

and substituting for s, as in (177), the specific volume of the vapour, considered a perfect gas, we get

Pojz± = 'J^ (181)

This means that the relative decrease of the vapour pressure is proportional to the concentration of the solution ( Wiillner's law). For further particulars, see § 270.

§ 225. Pressure Constant : dp = 0. — The relation

200 THERMODYNAMICS.

between the temperature (boiling point) and the concen- tration is, by (175),

i~\ =

(182)

Since p is positive, the boiling point rises with increasing concentration. By comparing this with the formula (179) for the decrease of the vapour pressure, we find that any solution gives

(f) : fl) = _ »^ \oc/p \ac^ L

i.e. for an infinitely small increase of the concentration the rise in the boiling point (at constant pressure) is to the decrease of the vapour pressure (at constant temperature) as the product of the absolute temperature and the specific volume of the vapour is to the heat of vaporization of the solution.

Kemembering that this relation satisfies the identity

\dcL • ydel - Wh

we come immediately to the equation (176).

§ 226. Let 00 be the boiling point of the pure solvent (c = 0), then, for some values of c, the difference between B and 0o will be small, and we may put

dO _ 0-00 ^ 0-0 dc~ c — i) ~~ c

whereby the equation becomes

0-00 = "^ (183)

This means that the elevation of the boiling point is pro- portional to the concentration of the solution. For further details, see § 269.

ANY NUMBER OF INDEPENDENT CONSTITUENTS. 201

§ 227. Let the second phase consist of the pure solvent in the solid state instead of the gaseous state, as happens in the freezing of an aqueous salt solution or in the pre- cipitation of salt from a saturated solution. In the latter case, in conformity with the stipulations of § 220, the salt will be regarded as the first constituent (the solvent), and water as the second constituent (the dissolved substance). The equation (175) is then directly applicable, and may be discussed in three different ways. We may ask how the freezing point or the saturation point of a solution of definite concentration changes with the pressure {dc = 0) ; or, how the pressure must be changed, in order that a solution of changing concentration may freeze or become saturated at constant temperature (clB = 0) ; or, finally, how the freezing point or the saturation point of a solution under given pressure changes with the concentration (dp = 0). In the last and most important case, if we denote the freezing point or the saturation point as a function of the concen- tration by 6', to distinguish it from the boiling point, equa- tion (175) gives

\dc)j,- L'

L being the heat absorbed when unit mass of the solvent separates as a solid (ice, salt) from a large quantity of the solution of concentration c. Since L is often negative, we may put L = — L' and call L' the Jieat of solidification of the solution or the Jieat of 2>^ecipitation of the salt. We have, then,

The heat of solidification (L') of a salt solution is always positive, hence the freezing point is lowered by an increase of concentration e. On the other hand, if the heat of pre- cipitation (L') of a salt from a solution be positive, the saturation point 0' is lowered by an increase of the mass of water, or rises with an increase of the concentration of

202 THERMODYNAMICS.

the salt. If L' be negative, the saturation point is lowered by an increase of the concentration of the salt. Should we prefer to designate by e, not the amount of water, but the amount of salt in a saturated solution, then, according to the definition of e in (162) and of <p in (165), we should

have ~ replacing c in (184) and c^ replacing f, and therefore

\ =^. (185)

Here c and ^ have the same meaning as in equation (184), which refers to the freezing point of a solution.

§ 228. Let 0o' be the freezing point of the pure solvent (c = 0), then, for small values of c, 6' will be nearly = 0'o, and we may put

dO' ^ e^Oo' ^ 0' - Op'

do c — ~ c Equation (184) then becomes

0o' - 0' = ^ (186)

which means that the lowering of the freezing point is proportional to the concentration. For further particulars, see § 271.

§ 229. The positive quantity, f , which occurs in all these formulae, has a definite value for a solution of given c, 6, and p, and is independent of the nature of the second phase. Our last equations, therefore, connect in a perfectly general way the laws regarding the lowering of the vapour pressure, the elevation of the boiling temperature, the depression of the freezing point, and the change of the saturation point. Only one of these phenomena need be experimentally investigated in order to calculate f, and by means of the value thus determined the others may be deduced for the same solution.

ANY NUMBER OF INDEPENDENT CONSTITUENTS. 203

We shall now consider a further case for which <p is of fundamental importance, viz. the state of equilibrium which ensues when the pure liquid solvent forms the second phase, not in contact with a solution, for no equilibrium would thus be possible, but separated from it by a membrane, permeable to the solvent only. It is true that for no solution can perfectly semipermealle membranes of this cha- racter be manufactured. In fact, the further development of this theory (§ 259) will exclude them as a matter of prin- ciple, for in every case the dissolved substance will also diffuse through the membrane, though possibly at an extremely slow rate. For the present it is suflficient that we may, without violating a law of thermodynamics, assume the velocity of diffusion of the dissolved substance as small as we please in comparison with that of the solvent. This assumption is justified by the fact that semipermeability may be very closely approximated in the case of many substances. The error committed in putting the rate of diffusion of a salt through such a membrane equal to zero, falls below all measurable limits. An exactly similar error is made in assuming that a salt does not evaporate or freeze from a solution, for, strictly speaking, this assumption is not admissible (§ 259). The condition of equilibrium of two phases separated by a semipermeable membrane is contained in the general thermodynamical condition of equilibrium (145),

W + 8^" = 0, .... (187)

which holds for virtual changes at constant temperature and pressure in each phase. The only difference between this case and free contact is, that the pressures in the two phases may be different. Pressure always means hydrostatic pressure as measured by a manometer. If, in the general equation (76), we put

W = - p'W - i)"8V",

it immediately follows that (187) is the condition of equili- brium. The further conclusions from (187) are completely

204 THERMO D YNA MICS.

analogous to those which are derived, when there is a free surface of contact. Corresponding to (163) we have for any displacement of the equilibrium

P^ - ^#' - ^#" + ^Mi'g^^ + ^M^'g^l-, + . . . = 0.

Since the constituent 2 occurs only in the first phase, we get, instead of (175),

pfZe - ^(Z^y - yif - fde = 0. . . (188)

Here, as in § 221, L is the "heat of removal" of the solvent from the solution, i.e. the heat absorbed when, at constant temperature and constant pressures p and ^/', unit mass of the solvent passes through the semipermeable membrane from a large quantity of the solution to the pure solvent. The change of volume of the solution during this process is s' (negative), that of the' pure solvent s" (positive). In the condition of equilibrium (188), three of the four variables 0, p', I)", c remain arbitrary, and the fourth is determined by their values.

Consider the pressure ^f in the pure solvent as given and constant, say one atmosphere, then dp" = 0. If, further, we put dS = and dc not equal to zero, we are then considering solutions in which the concentration varies, but the tem- perature and the pressure in the pure solvent remains the same. Then, by (188),

Since <p > 0, and s' < 0, ^' the pressure in the solution increases with the concentration.

The difference of the pressures in the two phases, 2j' — p' = P, has been called the osmotic pressure of the solution. Since jj" has been assumed constant, we may write

m,-'! (-)

A XY NUMBER OF INDEPENDENT CONSTITUENTS. 205

Thus the laws of the osmotic pressure have also been expressed in terms of ^, wliich controls those of the depression of the freezing point, the elevation of the boiling point, etc. Since <p is positive, the osmotic pressure increases with in- creasing concentration, and also, since j>' — p" vanishes when c = 0, the osmotic pressure is necessarily positive.

For small values of c,

3P ^ P - ^ P

5c ~ c — ~ c

and — s' is nearly equal to v the specific volume of the solution. It therefore follows from (189) that

P = '^. (190)

A further discussion of this question will be found in

§ 272.

§ 230. In the preceding paragraphs we have expressed the laws of equilibrium of several systems, that fulfil the conditions of § 220, in terms of a quantity <p which is cha- racteristic for the thermodynamical behaviour of a solution. Starting from the two equations (170) and (171), we find that all the relations in question depend on f' and f". A better insight into the nature of these quantities is gained by extending to the liquid state the idea of the mole- cule, hitherto applied only to gases. This step is taken in the next two chapters, and it appears that the manner in which the idea applies is uniquely determined by the propositions of thermodynamics, which have been given.

§ 231. Just as the conditions of equilibrium (170) and (171) were deduced for two independent constituents in two phases from the general relation (153), so in the same way an entirely analogous deduction may be made in the general case.

We shall conclude this chapter by giving, briefly, the

2c6 THEN MOD YNA MICS.

results for a system of a independent constituents in /3 phases.

Denoting the concentrations of the independent con- stituents, relative to one fixed constituent 1,. by

Mo'

MV

M/

Ml'

—

6-2 ;

Ml'

—

^';

Ml'

—

6'/;...

M,"

Ms"

M4"

Ml"

<h!\

Ml"

t^a";

Ml"

ci" ; . . .

the condition that, by any infinitely small change of the system : d%, dj), dc2, dc^, del, . . . de^, dc^", del', • • • 5 the equilibrium may remain stable with regard to the passage of the constituent 1 from the phase denoted by one dash to the phase denoted by two dashes is

^1^0 _ ^p + (^^'de^' _ ^^dc^') + (^3"^C3" - f^de,') + . . . = 0, where, analogous to (165),

/« ' — Af ' , ^ . - ' _ AI '

aMi'aMa" ^' ^a^ii'SMg"* ••

^2 - ^^^ 5Mi"5M2" ' ^^ - ^^ 5Mi"5M3" ' ' ' '

and Li, Si denote the heat absorbed, and the increase of volume of the system during the isothermal and isopiestic transference of unit mass of constituent 1 from a large quantity of the phase denoted by one dsish to a large quantity of the phase denoted by two dashes. The corre- sponding conditions of equilibrium for any possible passage of any constituent from any one phase to any other phase may be established in the same way.

CHAPTER IV.

GASEOUS SYSTEM.

§ 232. The relations, which have been deduced from the general condition of equilibrium (79) for the different pro- perties of thermodynamical equilibria, rest mainly on the relations between the characteristic function 4^, the tempera- ture, and the pressure as given in the equations (150). It will be impossible to completely answer all questions regard- ing equilibrium until ^ can be expressed in its functional relation to the masses of the constituents in the different phases. The introduction of the molecular weight serves this purpose. Having already defined the molecular weight of a chemically homogeneous gas as well as the number of molecules of a mixture of gases by Avogadro's law, we shall turn first to the investigation of a system consisting of one gaseous phase.

The complete solution of the problem consists in express- ing ^ in terms of 0, f, and wi, ?*2> %> • • • » the number of all the different kinds of molecules in the mixture.

Since we have, in general, by (75),

T ^ U + pV

we are required to express the entropy $, the energy U, and the volume V as functions of the above independent variables. This can be done, in general, on the assumption that the mixture obeys the laws of perfect gases. Such a restriction will not, in most cases, lead to appreciable errors. Even this assumption may be set aside by special measurement of the

2o8 THERMOD YNA MICS.

quantities <J>, U, and V, as is given later. For the present, however, perfect gases will be assumed.

§ 233. The laws of Boyle, Gay Lussac, and Dalton deter- mine the volume of the mixture, for equation (16) gives

V = |%i + ,^, + . . .) = ^^n,. . (191)

By the first law of thermodynamics, the energy U of a mixture of gases is given by the energies of its constituents, for, according to this law, the energy of the system remains unchanged, no matter what internal changes take place, provided there are no external effects. Experience shows that when diffusion takes place between a number of gases at constant temperature and pressure, neither does the volume change, nor is heat absorbed or evolved. The energy of the system, therefore, remains constant during the process. Hence, the energy of a mixture of perfect gases is the sum of the energies of the gases at the same temperature and pressure. Now the energy Ui of Ux molecules of a perfect gas depends only on the temperature ; it is, by (35),

Ui = »ii((J.,0 + /ii), (192)

where c^^ is the molecular heat of the gas at constant volume, and li\ is a constant. Hence the total energy of the mixture is

U = 5%(c.,0 + Ai) (193)

§ 234. We have now to determine the entropy ^ as a function of 0, j), and Wi, n.j, . . . the number of molecules. <I>, in so far as it depends on ^ and ^j, may be calculated from the equation (60),

a

GASEOUS SYSTEM. 209

where the differentials correspond to variations of Q and "p, but not of the number of molecules. Now, by (193),

and, by (191),

clY = R;gM(3)'

and, by integration,

iP = 5wi(^^,. log + E log 3) + C. . (194)

The constant of integration C is independent of 6 and j;, but may depend on the composition of the mixture, i.e. on the numbers ui, n^, %. . . . The investigation of this relation forms the most important part of our problem. The deter- mination of the constant is not, in this case, a matter of definition. It can only be determined by applying the second law of thermodynamics to a reversible process which brings about a change in the composition of the mixture. A reversible process produces a definite change of the entropy which may be compared with the simultaneous changes of the number of molecules, and thus the relation between the entropy and the composition determined. If we select a process devoid of external effects either in work or heat, then the entropy remains constant during the whole process. We cannot, however, use the process of diffusion, which leads to the value of U ; for diffusion, as might be expected, and as will be shown in § 238, is irreversible, and therefore leads only to the conclusion that the entropy of the system is thereby increased. There is, however, a reversible process at our disposal, which will change the composition of the

p

2 lo THERMOD YNA MICS.

mixture, viz. the separation by a semipermeable membrane, as introduced and established in § 229.

§ 235. Before we can apply a semipermeable membrane to the purpose in hand, we must acquaint ourselves with the nature of the thermodynamical equilibrium of a gas in contact with both sides of a membrane permeable to it. The membrane will act like a bounding wall to those gases to which it is impermeable, and will, therefore, not introduce any special conditions. Experience shows that a gas on both sides of a membrane permeable to it is in equilibrium when its partial pressures (§ 18) are the same on both sides, quite independent of the other gases present. This proposition is neither axiomatic nor a necessary consequence of the preceding considerations, but it commends itself by its simplicity, and has been confirmed without exception in the few cases accessible to direct experiment.

A test of this kind may be established as follows: Platinum foil at a white heat is permeable to hydrogen, but impermeable to air. If a vessel having a platinum wall be filled with pure hydrogen, and hermetically sealed, and the platinum be then heated, the hydrogen must completely diffuse out against atmospheric pressure. As the air cannot enter, the vessel must finally become completely exhausted.*

• This inference was tested by me in the Physical Institute of the University of Munich in 1883, and was confirmed witliin the limits of experimental error as far as tlie actual deviation from ideal conditions might lead one to expect. As this experiment has not been published anywhere, I shall briefly describe it here. A glass tube of about 5 mm. internal diameter, blown out to a bulb at the middle, was provided with a stop-cock at one end. To the other end a platinum tube 10 cm, long was fastened, and closed at the end. The whole tube was exhausted by the mercury pump, filled with hydrogen at ordinary atmospheric pressure, and then closed. The closed end of the platinum portion was then heated in a horizontal position by a Bunsen burner. The connection between tlie glass and platinum tubes having been made by means of sealing-wax, had to be kept cool by a continuous current of water to prevent the softtning of the wax. After four hours the tube was taken from the flame, cooled to the temperature of the room, and the stop-cock opened under mercury. The mercury rose rapidly, almost completely filling the tube, proving that the tube had been very nearly exhausted.

GASEOUS SYSTEM.

211

§ 236. We shall make use of the properties of semi- permeable membranes to separate in a reversible and simple manner the constituents of a gas mixture. Let us consider the following example : —

Let there be four pistons in a hollow cylinder, two of them, A and A', in fixed positions ; two, B and B', movable in such a way that the distance BB' remains constant, and equal to AA'. This is indicated by the brackets in Fig. 5.

Fig. 5.

Further, let A' (the bottom), and B (the cover) be im- permeable to any gas, while A is permeable only to one gas (1), and B' only to another one (2). The space above B is a vacuum.

At the beginning of the process the piston B is close to A, therefore B' close to A', and the space between them contains a mixture of the two gases (1 and 2). The con- nected pistons B and B' are now very slowly raised. The gas 1 will pass into the space opening up between A and B, and the gas 2 into that between A' and B'. Complete separation will have been eifected when B' is in contact with

2 1 2 THERMOD YNA MICS.

A. We shall now calculate the external work of this pro- cess. The pressure on the movable piston B consists only of the pressure of the gas 1, upwards, since there is a vacuum above B ; and on the other movable piston, B', there is only the partial pressure of the same gas, which acts downwards. According to the preceding paragraph both these pressures are equal, and since the paths of B and B' are also equal, the total work done on the pistons is zero. If no heat be absorbed or given out, as we shall further assume, the energy of the system remains constant. But, by (193), the energy of a mixture of gases depends, like that of pure gases, on the temperature alone, so the temperature of the system remains constant throughout.

Since this infinitely slow process is reversible, the entropy in the initial and final states is the same, if there are no external effects. Hence, the entropy of the mixture is equal to the sum of the entropies which the two gases would have, if at the same temperature each by itself occu- pied the whole volume of the mixture. This proposition may be easily extended to a mixture of any number of gases. The entropy of a mixture of gases is the sum of the entrojjies which the individual gases would have, if each at the saiw temperature occupied a volume equal to the total volume of the mixture. This proposition was first established by Gibbs.

§ 237, The entropy of a perfect gas of mass M and mole- cular weight m was found to be (52)

M/"- log + - log V + const.),

where c„ is the molecular heat at constant volume, as in (192). By the gas laws (14), the volume of unit mass is

E e

V = — • -i m p

whence the entropy is

n{e^ log + K log -^ + Jc), . . . (195)

GASEOUS System. 213

• M

where n = — , the number of molecules, and the constant Tc

in

E ^

includes the term log — . Hence, according to Gribbs's pro- position, the entropy of the mixture is

a

$ = ^wi(c,j log + R log — + ^1),

jpi being the partial pressure of the first gas in the mixture. Now, by (8), the p^sure of the mixture is the sum of the partial pressures, Spi = p, and, by § 40, the partial pressures are proportional to the number of molecules of each gas,

Ih'- Ih ''•■• = ni: n.2 : . . .

Hence m = — ; ^ p

^ Wi + Wa + . . .

_ %

or, if we introduce the concentrations of the different gases in the mixture,

^1 = - — ^-T — , ; ^2 =

wi + W2 + . . . Wi 4- W2 + • • • Ih = Oi2) ; 2h = (^2P (196)

Thus the expression for the entropy of a mixture as a function of 0, p, and n the number of molecules, finally becomes

• = 2**i(^«'i log ^ + R log -^ + h). . (197)

Comparing this expression with the value of the entropy of

214 THERMODYNAMICS.

a gas mixture given by (194), we see that the constant of integration which was left undetermined is

C = ;^«i(A;i - E log ci) . . . (198)

§ 238. Knowing the value of the entropy of a gas mixture, we may answer the question which we discussed in § 234, whether and to what extent the entropy of a system of gases is increased by diifusion. Let us take the simplest case, that of two gases, the number of molecules being % and %, diffusing into one another under common and constant pressure and temperature. Before diffusion begins, the entropy of the system is the sum of the entropies of the gases, by (195),

ni{e^, log + E log - + h) -{- W2(c,, log + E log - + ki).

After diffusion it is, by (197),

ni{cv., log + E log 1- h) + rh{cv., log + E log f- ^2)-

Therefore, the change of the entropy of the system is, by (196),

— Wi E log ci — W2 E log C2

an essentially positive quantity. This shows that diffusion is always irreversible.

It also appears that the increase of the entropy depends solely on the number of the molecules ni, n^, and not on the nature — e.g. the molecular weight, of the diffusing gases. The increase of the entropy does not depend on whether the gases are chemically alike or not. By making the two gases the same, there is evidently no increase of the entropy, since no change of state ensues. It follows that the chemi- cal difference of two gases, or, in general, of two substances, cannot be represented by a continuous variable; but that

GASEOUS SYSTEM. 215

here we can speak only of a discontinuous relation, either of equality or inequality. This fact involves a fundamental distinction between chemical and physical properties, since the latter may always be regarded as continuous.

§ 239. The values of the entropy (197), the energy (193), and the volume (191), substituted in (75), give the function ^,

^ = 2ni(c., log + K log J-^ + h - e,, - | - E) ;

or, putting the quantity, which depends on p and 9, and not on the number of molecules,

Cn log - ^' + R log - + ^1 - c., - R = ^1, (199)

§ 240. This enables us to establish the condition of equilibrium. If in a gas mixture a chemical change, which changes the number of molecules Ui n^ . . . by 8ni, hi^ . . . be possible, then such a change will not take place if the condition of equilibrium (79) be fulfilled, i.e. if, when 80 = and S^ = 0,

or 2(^1 - R log (?i)a»i + ^nxl{fx - R log Ci) = 0. (200) The quantities fi, <p2 • • • depend on B and j^ only, therefore

Further,

Wi<N , W2^

wiS log ci 4- W2S log C2 + . . . = — 8ci + — 8^2 + • . •

Cl C2

and, by (196),

= {ni + n.2+ .. .){hi + 8c2 +•••) = 0, since Ci + C2 + . . . = 1.

2i6 THERMODYNAMICS.

The condition of equilibrium, therefore, reduces to

^(fi - K log c,)hH = 0.

Since this equation does not involve the absolute values of the variations Swi, but only their ratios, we may put

hh : ^112 : . . . = VI : V2 : (201)

and take n, V2 . . . to denote the number of molecules simultaneously passing into the mass of each constituent. They are simple integers, positive or negative, according as the gas in question is forming, or is being used up in the formation of others. The condition of equilibrium now becomes

]^((pi - E log Ci)vi = 0, or v: log ., + ., log .,+ ... = "'^' + v^^ + . . . .

The right-hand side of the equation depends only on tem- perature and pressure (199). The equation gives a definite relation between the concentrations of the different kinds of molecules for given temperature and pressure.

§ 241. We shall now substitute the values of fi, f2- • • • If, for shortness, we put the constants

XM^i - \ - '^) . g = log ci

?^ = 6 (202)

XV

^^ = e (203)

It

GASEOUS SYSTEM. 217

tnen

i>iiogCi-Lv2loge.2 + ... = loga + {vi + v-2 + ...)log — ^ + clog9, or Ci-'iCa"^ . . . = al-j e eS"

§ 242. This condition may be further simplified by making use of the experimental fact (§ 50) that the atomic heat of an element remains unchanged in its combinations. By equation (203) He is the change of the sum of the molecular heats of all the molecules of the system during the reaction. The sum of the molecular heats, however, being the sum of the atomic heats, remains unchanged, hence c = 0, and the equation becomes

Jlci"! = ae cf-J

§ 243. According to this equation the influence of the pressure on the equilibrium depends entirely on the number Si'i, which gives the degree to which the total number of molecules, therefore also the volume of the mixture, is increased by the reaction considered. Where the volume remains unchanged, as, e.g., in the dissociation of hydriodic acid, considered below, the equilibrium is independent of the pressure.

The influence of the temperature depends further on the constant h, which is closely connected with the heat eifect of the reaction. For, by the first law,

Q = gU + p^Y, which, by (193) and (191), 6 and p being constant, becomes

Q = ^(c,, e + h + JXB) g/ii.

2i8 THERMODYNAMICS.

If we refer the heat effect to the finite numbers v, instead of the infinitely small numbers Sw, then the heat absorbed is :

L = 2(s 61 + /ii + m)vx,

and by (202) and (203), again putting c = 0,

L = KJ + EOSpi, or, L = 1-97 (& + eSi^i) cal.

The term containing h refers to the heat spent in the increase of the internal energy ; the term containing to that spent in external work.

§ 244. Before proceeding to numerical applications, we shall enumerate the principal equations. Suppose that in a gaseous system

wi mi ; % m2 ; % W3 ; . . .

(w the number of molecules, m the molecular weight) any chemical change be possible, in which the simultaneous changes of the number of molecules are

hi\ : ^^-2 '- ^^3

= 1^1 : V2 : V3

{v simple, positive or negative integers) then there will be equilibrium, if the concentrations

^ ~ Til + Wa + ' ^ ~ wi + M2 +

satisfy the condition

-y/Bxn + vi + n.

rici"! = ae

-l/Bx--

%)^' (204)

GASEOUS SYSTEM. 219

The heat absorbed during the change at constant temperature and pressure is

L = l-97 {6 + (vi + v2 + ...)0) = l'97(6 + 0Si;i)cal. (205)- while the change of volume is

s = K(vi + v2 + ...)^ = R^Sri. . . (206)

§ 245. Dissociation of Hydriodic Acid. — Since hydriodic acid gas splits partly into hydrogen and iodine vapour, the system is represented by three kinds of molecules :

rii HI; Wa H2; n^ T2;

The concentrations are :

ni W2 %

^1 = 7. — r^. — v~zr ; ^i = :;:; — n:, — r-zr 5 ^3 =

ni + ?i2 + % ' ^1 + W2 + % ' ni + n2 + n^

The reaction consists in the transformation of two molecules of HI into one of H2 and one of I2 :

VI

= -2; V2 = 1; V3 = 1.

By (204), therefore, in the state of equilibrium,

_b

Ci~Wc3^ = ae If

'^ = '^ = ae-l .... (207)

Since the total number of atoms of hydrogen {iix + 2/^2) and of iodine {ux + 2%) in the system are supposed to be known, equation (207) is suflScient for the determination of the three quantities, Wi, n^, and n^, at any given temperature. The pressure has in this case no influence on the equilib- rium. Any two measurements of the degree of dissociation are sufficient for the calculation of a and &. From Boden- stein's measurements we have for

= 273 + 448 = 712; ^ = 0-01984;

220 THERMOD YNA MICS.

and for = 273 + 350 = 623 ; ^ = 0-01494.

Hence, by (207),

a = 0-120 ; I = 1300.

Thus the equilibrium of any mixture of hydriodic acid, hydrogen, and iodine vapour at any temperature, even when the hydrogen and the iodine are not present in equivalent quantities, is determined by (207). Equation (205) gives the heat of dissociation of two molecules of hydriodic acid into a molecule of hydrogen and a molecule of iodine vapour :

L = 1-971 X 1300 = 25G0 cal.

§ 246. — Dissociation of Iodine Vapour. — At high tem- peratures iodine vapour appreciably decomposes, leading to a system of two kinds of molecules :

«i I2 ; n.2 1.

The concentrations are

til 7U

ni + «2 wi + W2

The reaction consists in a splitting of the molecule I.2 into two molecules I,

.*. vi = — 1 ; i'2 = 2 ;

and in equilibrium, by (204),

crW = ,\ ^ = «'e"^- - • . (208) ni{ni + 11.2) f ^ '

a and V may be calculated from data given by Fr. Meier and Crafts. AYhen p = 728 mm. of mercury,

,^ — = 0-145 when 6 = 273 + 940 = 1213,

Zni 4- U2

and = 0-662 when B = 273 + 1390 = 1663.

GASEOUS SYSTEM. 22 r

This gives, if p be measured in millimeters of mercury,

a' = 9375; V = 14690;

from which the equilibrium of dissociation may be deter- mined for any temperature and pressure.

The heat of dissociation of a molecule of iodine is, by (205),

L = 1-97 (14690 + 61) = 28900 + 1 970 cal.

It will be seen that at such temperatures the external work, on which the second term depends, has an appreciable influence. At 1500" C. {B = 1773) it amounts to 3.500 cal., making the heat of dissociation

L = 32400 cal.

§ 247. Graded Dissociation. — Since, by equation (208), the concentration c^ of the monatomic iodine molecules does not vanish even at low temperatures, the decomposition of the iodine vapour should be taken into account in deter- mining the dissociation of hydriodic acid. This will have practically no influence on the results of § 245, but never- theless we shall give the more rigorous solution on account of the theoretical interest which attaches to it.

There are now four kinds of molecules in the system :

ui HI ; 712 H2 ; % I2 ; % I. Two kinds of chemical changes are possible :

(1) vi = — 2 ; V2 = 1 ; V3 = 1 ; 1/4=0; and

(2) vi' = ; V2' = ; v^ = — 1 ; V4' = 2.

There will be equilibrium for each of these, if, by (204),

/, N C2 Cq % % _-

(1) crcpc^^^c,^^ = -y = -A_3 = ,,e ^

222 THERMODYNAMICS.

and

2 ^2 v^ Q

(2) Ci-^VVV*' = ^ — / — \ \ , — x =a!e~^ ■ -.

^ ^ Ca = %('h + n.2 + 113 + Hi) 2^

The constants a, h, a', h' have the values calculated above. The total number of hydrogen atoms (ni + 2)1.2) and of iodine atoms (tii + 2% + n^) being known, we have four equations for the complete determination of the four quantities Wi, n^, iH, W4.

§ 248. The general equation of equilibrium (204) also shows that at finite temperatures and pressures none of the concentrations, e, can ever vanish ; in other words, that the dissociation can never be complete, nor can it completely vanish. There is always present a finite, though perhaps a very small number of all possible kinds of molecules. Thus, in water vapour at any temperature at least a trace of oxygen and hydrogen must be present (see also § 259). In a great number of phenomena, however, these quantities are too small to be of any importance.

CHAPTER V.

DILUTE SOLUTIONS.

\ 249. To determine ^ as a function of the temperature 0, the pressure 'p, and the number n of the different kinds of molecules in a system of any number of constituents and any number of phases, we may use the method of the pre- ceding chapter. It is necessary first to find by suitable measurements the volume V, and the internal energy U of each single phase, and then calculate the entropy ^ from the definition (60). A simple summation extending over all the phases gives, by (75), the function ^ for the whole system. On account of incomplete experimental data, how- ever, the calculation of ^ can be performed, besides for a gaseous phase, only for a dilute solution^ i.e. for a phase in which one kind of molecule far outnumbers all the others in the phase. We shall in future call this kind of molecule the solvent, the other kinds the dissolved substances. This differs from the definition of § 220. If Wo be the number of molecules of the solvent, Wi, %, %, . . . the number of molecules of the dissolved substances, then the solution may be considered dilute if wo be large in comparison with each of the numbers Wi, n^, %. . . . The state of aggrega- tion of the substance is of no importance, it may be solid, liquid, or gaseous.

§ 250. We shall now determine by the above method the energy U and the volume V of a dilute solution. The important simplification, to which this definition of a dilute solution leads, rests on the mathematical theorem, that a finite, continuous, and differentiable function of several

224 THERMODYNAMICS.

variables, which have very small values, is necessarily a linear function of these variables. This determines U and V as functions of »0) «b ^^2> • • • Physically speaking, this means that the properties of a dilute solution, besides depending on the interactions between the molecules of the solvent, necessarily depend only on the interactions between the molecules of the solvent and the molecules of the dis- solved substances, but not on the interactions of the dissolved substances among themselves, for these are small quantities of a higher order.

§ 251. The quotient — , i.e. the internal energy divided

by the number of molecules of the solvent, remains un- changed if the numbers, %, Wi, «2 • • • be varied in the same proportion ; for, by § 201, IT is a homogeneous function of the number of molecules n^, %, n-i, . . ., oi the first degree.

• is, therefore, a function of the ratios —,—,., ., and also a

'llQ tin Wo

linear function, since these ratios are small, and the function is supposed to be diiferentiable. The function is, therefore, of the form

U lu , n.j

— = »o + Wi h Wa h . . .

Wt) Hq Hi)

where, Hq, Wi, U2 are quantities depending, not on the number of molecules, but only on the temperature B, the pressure p, and the nature of the molecules. In fact, hq depends only on the nature of the solvent, since the energy reduces to ?«o Uot when wi = = W2 = . . ., and Ui only on the nature of the first dissolved substance and the solvent, and so on. Ho, therefore, corresponds to the interactions between the molecules of the solvent, Ui to those between the solvent and the first dissolved substance, and so on. This contains a refutation of an objection, which is often raised agaiust the modern theory of dilute solutions, that it treats dilute solu- tions simply as gases, and takes no account of the influence of the solvent.

DILUTE SOLUTIONS. 225