Treatise on Thermodynamics (1903) — part 10 of 14

§ 252. If the dilution is not sufficient to warrant the use of this very simple form of the function U, a more accurate relation may be obtained by expanding Taylor's series still further,

• = Wo + Wi- + . . . + Un{ - ) + 2?fi2- • — + Waal — ) + • • •

Wo «0 VWo/ Wo 72-0 VWo^

The coefficients Wu, U12, Waa, . . . refer to the influence of the interactions of the dissolved molecules with one another. This, in fact, is the only practicable way of obtaining a rational thermodynamical theory of solutions of any con- centration.

§ 253. We shall here keep to the simple form, and write

U = iiqUq + WiWi + Waita + • • • ) /.^qqx

and V = WoVo + Wi^i + w^-^a + • • • 3 * *

How far these equations correspond to the facts may be determined by the inferences to which they lead. If we dilute the solution still further by adding one molecule of the solvent in the same state of aggregation as the solution, keeping meanwhile the temperature and the pressure p constant, the corresponding change of volume and the heat effect may be calculated from the above equations. One molecule of the pure solvent, at the same temperature and pressure, has the volume Vq and the energy wq. After dilu- tion, the volume of the solution becomes

V = (Wo + 1) ^0 + «1^1 + %^'2 + . . .

and the energy

U' = (wo + 1) Wo + wiWi 4- W2W2 + . . .

The increase of volume brought about by the dilution is therefore

V - (V + vo),

i.e. the increase of volume is zero. The heat absorbed is, by the first law (47),

U'-(U + Wo)+i){V'-(V + 2;o)}.

226 THERMODYNAMICS.

This also vanishes. These inferences presuppose that the number of molecules of the dissolved substances remain unchanged, i.e. that no chemical changes {e.g. changes of the degree of dissociation) are produced by the dilution. If such were the case, the number of molecules of the dissolved substances would have values in the equations for U' and V different from those in the equations for U and V, and there- fore would not disappear on subtraction. We may therefore enunciate the following proposition : Further dilution of a dilute solution, if no chemical changes accompany the process, produces neither an appreciable change of volume nor an appreciable heat effect; or, in other words, any change of volume or any heat effect' pjroduced hy further dilution of a dilute solution is due to chemical transformations among the molecules of the dissolved substances.

§ 254. We now turn to the calculation of the entropy *P of a dilute solution. If the number of molecules no, ni, n.2, . . .he constant, we have, by (60),

j^ dV + pdV d<P = gi^,

and, by (209),

7 duo + pdvo , dui + pdvi , dv^ + pdv^ , d^ = no j^ h ui ^^-^ + ih-^—0- — + . . .

Since u and v are functions of and p only, and not of n, each of the coefficients of uq, Wi, %, . . ., must be a perfect differential, i e. there must be certain functions 0, depending only on and p, such that

, diia + pdvo '] "90 = 2

, dui + pdvi

«^i = — gr —

, du2 + pdv2

«^2 =

(210)

DILUTE SOLUTIONS, 227

We have, then,

• = Wo<^0 + %^l + W202 + . . . + C, (211)

where the integration constant C cannot depend on Q and jp, but may be a function of the number of molecules. C may be determined as a function of Wo) wi, n2, . . . for a particular temperature and pressure, and this will be the general ex- pression for C at any temperature and pressure.

We shall now determine C as a function of n taking the particular case of high temperature and small pres- sure. By increasing the temperature and diminishing the pressure, the solution, whatever may have been its original state of aggregation, will pass completely into the gaseous state. Chemical changes will certainly take place at the same time, i.e. the number of molecules n will change. But we shall assume that the process takes place in such a way as to leave the number of the different kinds of molecules unaltered, because C remains constant only in this case. Only an ideal process can accomplish this, since it passes through unstable states. There is, however, no objection to its use for our present purpose, since the above expression for 4> holds not only for stable states of equilibrium, but for all states characterized by quite arbitrary values of B, p, uq, 7ii, W2, . . . Stable equilibrium is a special case, satisfying a further condition to be established below.*

• Hr. Cantor maintains (Ann. d. Phys., 10, p. 205, 1903) that it is not per- missible to suppose that the gaseous state may be reached in this way. " It must be proved that this represents a possible state of the substance, that it may be at least a momentary state. But no theoretical proof of this has been advanced, and direct experience does not at all justify such an assump- tion." In reply, it has first of all to be pointed out that the possibility of varying the temperature and the pressure, keeping the number of molecules constant, merely depends on the fact that the number of molecules together with the temperature and the pressure form the independent variables which are necessary for the unique determination of the state of solution under consideration. The variables are not subject to any limitations, except that the number of atoms must remain on the whole unchanged. This does not concern us here, and is chemically self-evident. Therefore, from a general thermodynamical point of view, nothing stands in the way of letting the pressure diminish and the temperature rise in any way, keeping the number of molecules constant, if the formation of a new phase is prevented. When

228 THERMOD YNA MICS.

At a sufficiently high temperature, and a sufficiently low pressure, any gaseous system possesses so small a density, that it may be regarded as a mixture of perfect gases (§ 21, and § 43). We have, therefore, by (194), bear- ing in mind that here the first kind of molecule is denoted by the suffix 0,

<I> = %(ct.o log + K log -)

• %(c„, log + R log ?)+...+ C. (212)

The constant C is independent of and 'p, and has the value given in (198). On comparing this with (211), it is seen that the expression for <{>can pass from (211) into (212) by mere change of temperature and pressure, only if the constant C is the same in both expressions, i.e. if, by (198),

C = ?io (A<) - R log Co) 4- wi (ki - R log ci) + . . .

Here /.;o, A^i, h-2, . . . are constants, and the concentrations are

llQ 111

^ = — ; . — : '•> ^1 =

no + ui + th- • ' no -- Hi + 112. . .

By (211), the entropy of a dilute solution becomes

$ = 7io(<^o+^-Rlog<'o) + ni(0i + ^i-RlogCi)+ . . . (213)

If we put, for shortness, the quantities which depend only on 6 and p,

, no + 2JVo _ ^

0i + ;fci-^^^ = ^J. . . (214)

<pi + f^2 = <?

this is recognized, it requires only the hypothesis that by continuing this process the ideal gaseous state is finally reached — a supposition which scarcely anyone can object to, and which Hr. Cantor does not, at least directly, contra- dict (Planck, Ann. d. Phyg. 10, p. 436, 1903).— Tb.

DILUTE SOLUTIONS. 229

we have, finally, from (75), (213) and (209),

^ = % (f - R log Co) + nx (<pi - R log ci)

• W2(^2-RlogC2)+ ... (215)

This equation determines the thermodynamical properties of a dilute solution.

§ 255. We may now proceed to establish the conditions of equilibrium of a system consisting of several phases. As hitherto, the different kinds of molecules in the phase will be denoted by suffixes, and the different phases by dashes. For the sake of simplicity the first phase will be left with- out a dash. The entire system is then represented by

7?o ?'io» ^1 ''ii, '^2 »^2, • • • i '^0' "io'j Wi' mi', n-i nii, . . .

j Hq Mo", )h" mi", W2" WI2", . . . I . . (216)

The number of molecules are denoted by u, and the molecular weights by m, and the individual phases are separated by vertical lines. In the general formula we signify the summation over the different kinds of molecules of one and the same phase by writing the individual terms of the summation ; the summation over the different phases,

on the other hand, by the symbol S.

In order to enable us to apply the derived formulae, we shall assume that each phase is either a mixture of perfect gases or a dilute solution. The latter designation will be applied to phases containing only one kind of molecule, e.g. a chemically homogeneous solid precipitate from an aqueous solution. One kind of molecule represents the special case of a dilute solution in which the concentrations of all the dissolved substances are zero.

§ 256. Suppose now that an isothermal isopiestic change be possible, corresponding to a simultaneous variation 8%, Swi, Sri2> • . • ^f^o, ^ni, SW2', ... of the number of mole- cules %, «i, n-i, . . . no, 111, rii, . . . ; then, by (79), this change

230 THERMODYNAMICS.

will not take place, if at constant temperature and pressure

or, by (215), if

^(^0 - R log co)Srio 4- (?i - R log Ci)5?ii -f (^2 - K log c^hi-i

• . . . + ^?ioS(fo - R log Co) + n^{^x - R log ci)
• 112^^2 - R log C2) 4- . • . =

The summation ^ extends over all the phases of the

system. The second series is identically equal to zero for the same reason as was given in connection with equation (200). If we again introduce the simple integral ratio

= I'o : vi : V2 : • • • • vj : n' '. v^ '. ... (217) then the equation of equilibrium becomes

]^(^0-R log Co)vo-(^l-R log (^1)1'! + (^2 -R log 62)1^2-'- . . .

= or

^vo log Co 4- vi log Ci 4- V2 log C.2 4- . . . = jj^i'ofo 4- vi<pi4- . . .

= logK. . (218)

K like <po5 fi> p-2, is independent of the number of mole- cules n.

§ 257, The definition of K gives its functional relation to Q and 'p.

3 log K 1 ^ a^o , 5^1 a^pa , a log K 1 ^ 5^0 , a^i a^.2

DILUTE SOLUTIONS. 231

Now, by (214), we have for an infinitely small change of B and p

u

and therefore, by (210), From this it follows that

de~ e^ ' d[3~ 6'

Similarly

d(pi _ Wi 4- pvi dfi _ vi

M ~ ¥ ' d^ ~ ~ 9' Hence

d W K 1 ^. . . N / , , \

— ^ — = j^Zr(*'o^o + ''1**1 +...)+ ^^(voi^o + viVi +...),

5 log K 1 ^

—^ = - H^2»'o^o + '^i^i + • • •

Denoting by s the increase of volume of the system, and by L the heat absorbed, when the change corresponding to (217) takes place at constant temperature and pressure, then, by (209),

S = ^VqVq 4- Vi-^i + V2V2 + . . .

and, by the first law of thermodynamics,

L = ^{vqUq + viWi +...)+ iK^oVq + viVi + . . .) ;

therefore ~dt = KP ^"^^^^

232 THERMODYNAMICS.

^■"^ ^-=-K9 (220)

The influence of the temperature on K, and therewith on the condition of equilibrium towards a certain chemical reaction, is controlled by the heat effect of that reaction, and the influence of the pressure is controlled by the corre- sponding change of volume of the system. If the reaction take place without the absorption or evolution of heat, the temperature has no influence on the equilibrium. If it produce no change of volume the pressure has no influence. The former equations (205) and (206) are particular cases of (219) and (220), as may be seen by substituting for log K the special value obtained from (218) and (204) :

7 n

log K = log a - ^ + (I'l + V.2 + . . .) log -.

§ 258. By means of equation (218) a condition of equili- brium may be established for each possible change in a given system subject to chemical change. Of course, K will have a different value in each case. This corresponds to the requirements of Gibbs's phase rule, which is general in its application (§ 204). The number of the different kinds of molecules in the system must be distinguished from the number of the independent constituents (§ 198). Only the latter determines the number and nature of the phases ; while the number of the different kinds of molecules plays no part whatever in the aj^plication of the phase rule. If another kind of molecule be introduced the number of the variables increases, to be sure, but so does the number of the possible reactions, and therewith, the number of the conditions of equilibrium by the same amount, so that the number of independent variables is quite independent thereof.

§ 259. Equations (218) shows further that, generally speaking, all kinds of molecules possible in the system

DILUTE SOLUTIONS. 233

will be present in finite numbers in every phase ; for instance, molecules of H2O must occur in any precipitate from an aqueous solution. Even solid bodies in contact must partially dissolve in one another, if sufficient time be given. The quantity K, which determines the equilibrium, possesses, according to the definition (218), a definite, in general, a finite value for each possible chemical change, and none of the concentrations c can, therefore, vanish so long as the temperature and the pressure remain finite. This prin- ciple, based entirely on thermodynamical considerations, has already served to explain certain facts, e.g. the impossibility of removing the last traces of impurity from gases, liquids, and even solids. It also follows from it that absolutely semi- permeable membranes are non-existent, for the substance of any membrane would, in time, become saturated with the molecules of all the various kinds of substances in contact with one side of it, and thus give up each kind of substance to the other side.

On the other hand, this view greatly complicates the calculation of the thermodynamical properties of a solution, since, in order to make no mistake, it is necessary to assume from the start the existence in every phase of all kinds of molecules possible from the given constituents. We must not neglect any kind of molecule until we have ascertained by a particular experiment that its quantity is inappreciable. Many cases of apparent discrepancy between theory and experiment may probably be explained in this way.

We shall now discuss some of the most important particu- lar cases. They have been arranged, in the first place, accord- ing to the number of the independent constituents of the system ; in the second, according to the number of the phases.

§ 260. One Independent Constituent in One Phase. —

According to the phase rule, the nature of the phase depends on two variables, e.g. on the temperature and the pressure. The phase may contain any number of different kinds of molecules. Water, for instance, will contain simple, double,

234 THERMODYNAMICS.

and multiple HaO-molecules ; molecules of hydrogen and

oxygen, H2 and O2 ; electrically charged ions H, HO, and

O, etc., in finite quantities. The electrical charges of the ions do not play any important part in thermodynamics, so long as there is no direct conflict between the electrical and the thermodynamical forces. This happens when and only when the thermodynamical conditions of equilibrium call for such a distribution of the ions in the different phases of the system as would lead, on account of the constant charges of the ions, to free electricity in any phase. The electrical forces strongly oppose such a distribution, and the resulting deviation from the pure thermodynamical equi- librium is, however, compensated by differences of potential between the phases. A general view of these electromolecular phenomena may be got by generalizing the expressions for the entropy and the energy of the system by the addition of electrical terms. We shall restrict our discussion to states which do not involve electrical phenomena, and need not consider the charges of the ions, which we may treat like other molecules.

In the case mentioned above, then, the concentrations of all kinds of molecules are determined by and ^. The calculation of the concentrations has succeeded so far only

in the case of the H and OH ions (the number of the

O ions is negligible), in fact, among other methods, by the

measurement of the electrical conductivity of the solution,

which depends only on the ions. Kohlrausch and Heydweiller

found the degree of dissociation of water, i.e. the ratio of

the mass of water split into H and OH ions to the total

mass of water to be, at 18^ C,

14-3 X 10-1".

This number represents the ratio of the number of dis- sociated molecules to the total number of molecules. We may determine by thermodynamics the change of the dis- sociation with temperature.

DILUTE SOLUTIONS. 235

The condition of equilibrium will now be established. The system is, by (216),

iIq H2O ; ui H ; 7^2 OH. Let the total number of molecules be

n = Uq + Hi + )1.2,

the concentrations are, therefore,

Wn % ih

<'o = - ; ^'i = - ; ^^2 = — •

" w n n

The chemical reaction in question,

I'o • I'l : v-2 = Silo : Sill : S)i-2,

consists in the dissociation of one H2O molecule into H and OH.

1^0= -1;_ I'l = 1 ; 1/2 = 1;

and therefore, by (218), in the state of equilibrium

• log Co + log Ci + log C2 = K,

or, since Ci = C2, and Co = 1 nearly,

2 log Ci = log K.

This gives, by (219), the relation between the concentration and the temperature :

o^ log ci _ 1 L

de ~ R 02"

(221)

According to Arrhenius, L, the heat necessary for the disso- ciation of one molecule of H2O into H and OH, is equal to the heat of neutralization of a strong monobasic acid and base in dilute aqueous solution. J. Thomsen's experiments give for mean temperatures :

r 4045000 I

L = cal.

u

236 THERMODYNAMICS.

On reducing calories to C.G.S. units, we get

3 log ci _ 1 4045000

50 ~ 2 X 1-971 ^ 03 •

On integrating, we have

, „^ 4045000 1 513000 ,

The value of the constant C is found from the degree of dissociation at 18° C. (0 = 291) ;

ci = c2 = 14-3xl0-'» .\ C = 6-1 X 10-'

Hence the degree of dissociation for any temperature is,

_ 5i:iOuo

c'l =6-le "' X 10 -^

This agrees well with the electrical conductivity of pure water when measured at different temperatures. Only at the absolute zero of temperature does the dissociation, and with it the conductivity, vanish. On the other hand, it does not increase indefinitely with temperature, but reaches a maximum value C.

§ 261. One Independent Constituent in Two or Three Phases. — The main features of these cases have already been discussed in Chapter II., §§ 205 to 207, and §213.

§ 262. Two Independent Constituents in One Phase — (A substance dissolved in a homogeneous solvent). According to the phase rule, one other variable besides the pressure and the temperature is arbitrary, e.<j. the number of the molecules dissolved in 1 litre of the solution, a quantity which may be directly measured. The values of

DILUTE SOLUTIONS. lyj

these three variables determine the concentrations of all kinds of molecules, whether they have their origin in dis- sociation, association, formation of hydrates, or hydrolysis of the dissolved molecules. Let us consider the simple case of a binary electrolyte, e.cj. acetic acid in water. The system is represented by

% H2O, wi CH3.COOH, 7H &, % CH3.COO.

The total number of molecules,

n = n^ + Wi 4- ^^2 + *«3>

is only slightly greater than %. The concentrations are

" w n n n

The reaction to be considered is represented by

vo : VI : i'2 : V3 = Swq : Swi .: hi-2 : 8%,

and consists in the dissociation of one molecule of CH3.COOH into its two ions.

vq = ; vi = - 1 ; V2 = 1 ; V3 = 1.

Therefore, in equilibrium,

• log Ci + log C2 + log Cg = log K;

(222)

or, since

C2 =

C3,

'1 =

Cl

K

Now,

we

may

regard

the sum

Cl + C2

= c

as known, since the total number (wi + W2)of the undissociated and the dissociated molecules of the acid, and the total number of water molecules, which may be put = n, are

238 THERMODYNAMICS.

measured directly. Hence Cx and c.2 may be calculated from the last two equations.

Ci 74i

C ni + 7l2

G ~ ni + «2

With increasing dilution (decreasing c), the ratio - increases

in a definite manner approaching the value 1, i.e. complete dissociation. This also gives for the electrical conductivity of a solution of given concentration Ostwald's so-called law of dilution of binary electrolytes* which has been experi- mentally verified in numerous cases. In a manner quite similar to that of § 260, the heat effect of the dis- sociation shows how the degree of dissociation depends on the temperature. Conversely, as was first shown by Arrhenius, the heat of dissociation may be calculated from the rate of change of the dissociation with temperature.

§ 263. Usually, however, in a solution, not one, but a large number of reactions will be possible. Accordingly, the complete system contains many kinds of molecules. As another example, we shall discuss the case of an electro- lyte capable of splitting into ions in several ways, viz. an aqueous solution of sulphuric acid. The system is repre- sented by

• ---

Wo H2O, wi H2SO4, 112 H, «3 HSO4, n^ SO4.

The total number of molecules is

11 = no + ni + 112 + W3 -1- W4 (nearly equal to Wo).

• K = '

where X^ is the molecular conductivity at dilution v ; X^c the molecular con- ductivity at infinite dilution ; and /• the molecular volume of the electrolyte. — Tb.

DILUTE SOLUTIONS. 239

The concentrations are

tir. nx n^ Uq n*

" n n n n n

Here two different kinds of reactions

VQ : vi : va : Vi = driQ : Sw^ : Sn^ • ^% • ^^4 must be considered ; first, the dissociation of one molecule of H2SO4 into H and HSO4 :

vo = ; 1^1 = — 1 ; V2 = 1 ; i'3 = 1 ; V4 = ;

• • —

second, the dissociation of the ion HSO4 into H and SO4

vo = ; VI = ; i'2 = 1 ; 1/3 = — 1 ; v^ = 1.

Hence, by (218), there are two conditions of equilibrium :

• log Ci + log Ca + log C3 = log K and log C2 - log Cg + log C4 = log K' ;

or ^ = K

and ^^ = K'.

C3

This further condition must be added, viz, that the total number of SO4 radicals (#1 + n^ + n^) must be equal to half the number of H atoms (2ni + n^ + %) ; otherwise the system would contain more than two independent con- stituents. This condition is

2^4 + C3 = C2.

Finally, the quantity of sulphuric acid in the solution is supposed to be given :

Cl + C3 + ^4 = c.

HO THERMODYNAMICS.

The last four equations determine o^^ Ci, r^, e^, and hence the state of equilibrium is found.

For a more accurate determination it would be necessary to consider still other kinds of molecules. Every one of these introduces a new variable, but also a new possible reaction, and therefore a new condition of equilibrium, so that the state of equilibrium remains uniquely determined.

§ 264. Two Independent Constituents in Two Phases. — The state of equilibrium, by the phase rule, depends on two variables, e.g. temperature and pressure. The wide range of cases in point makes a subdivision desirable, according as only one phase contains both constituents in appreciable quantity, or both phases contains both constituents.

Let us first take the simpler case, where one (first) phase contains both constituents, and the other (second) phase contains only one single constituent. Strictly speaking this never occurs (by 259), but in many cases it is a suffi- cient approximation to the actual facts. The application of the general condition of equilibrium (218) to this case leads to different laws, according as the constituent in the second phase plays the part of dissolved substance or solvent (§ 249) in the first phase. We shall therefore divide this case into two further subdivisions.

§ 265. The Pure Substance in the Second Phase forms the Dissolved Body in the First. — An example of this is the absorption of a gas, e.g. carbon dioxide in a liquid of comparatively small vapour pressure. The system is represented by

n H2O, 111 CO2 I uq CO2.

The concentrations of the diiferent kinds of molecule of the system in the two phases are

_ Uq _ 111 '%'!

''~% + ni' ""' - n, + wi' ''o - ^ - ^-

. DILUTE SOLUTIONS. 241

The reaction

'o : v\ '• vo = Suq : Sni : ^hq

consists in the evaporation of one molecule of carbon dioxide from the solution, therefore,

Vo = 0, Vi = — 1, Vo' = 1.

The condition of equilibrium

vo log Co + vi log Ci + vo' log Co' = log K. is, therefore,

• log ci = log K, . . . . (223)

or, at a given temperature and pressure (for these deter- mine K), Ci the concentration of the gas in the solution is determined. The change of concentration with pressure and temperature is found by substituting (223) in (219) and (220) :

''f^' = ir • • • • (^2*)

s is the increase of volume of the system, L the heat absorbed during isothermal-isopiestic evaporation of one gram molecule of CO2. Since s represents nearly the volume of one gram molecule of carbonic dioxide gas, we may, by (16), put

m

s = — , P

and equation (224) gives

d log ci _ 1

242 THERMOD YNA MICS.

On integrating, we have

log Ci = log 2> + const. or ci = C;j (226)

i.e. the concentration of the dissolved gas is 'proportional to the jjressure of tlie free gas on the solution (Henry's law). The factor C, which is a measure of the solubility of the gas, still depends on the temperature, since (225) and (226) give

3 1ogC _ _ 1 L 56> ~ K ■ Fa*

If, therefore, heat is absorbed during the evaporation of the gas from the solution, L is positive, and the solubility decreases with increase of temperature. Conversely, from the variation of C with temperature, the heat effect pro- duced by the absorption may be calculated ;

T - _ ?^' ^^ CM'

According to the experiments of Naccari and Pagliani, the solubility of carbon dioxide in water at 20^ {0 = 293), (expressed in a unit which need not be discussed here), is 0*8928, its temperature coefficient — 0'02483 ; therefore, by (34),

T 1-971 X 2932 ^ 0-02483 _,,^ ,

^ = mm =^ '^'^^ '^^-

Thomsen found the heat effect of the absorption of one gram molecule of carbon dioxide to be 5880 cal. The error (according to Nernst) lies mainly in the determination of the coefficient of solubility. Of the heat effect, the amount

m = 1-97 X 293 = 586 cal.

corresponds, by (48), to external work.

§ 266. A further example is the saturation of a liij^uid

DILUTE SOLUTIONS. 243

with an almost insoluble salt ; e.g. succinic acid in water. The system is represented by

CH2 - COOH ^0 H2O, Wi I

CH2 - COOH,

w,

CH2 - COOH

I CH2 - COOH,

if the slight dissociation of the acid in water be neglected. The calculation of the condition of equilibrium gives, as in §223,

• log ci = log K,

ci is determined by temperature and pressure. Further, by (219),

L = -R0^^-i|^ .... (227)

Van't Hofif was the first to calculate L by means of this equation from the solubility of succinic acid at 0° C. (2*88) and at 8-5° C. (4-22)

B log ci log, 4-22 - log, 2-88

°— ! — ^^ °l — 0-04494

This gives, for = 273, L = - 1-971 X 273^ x 0-4494 = — 6600 cals. ; i.e. on the precipitation of one molecule of the solid from the solution, 6600 cals. are given out. Berthelot found the heat of solution to be 6700 cals.

If L be regarded as independent of the temperature, which is permissible in many cases as a first approximation, the equation (227) may be integrated with respect to 0, giving

log ^1 = p^ + const.

§ 267. The relation (227) becomes inapplicable if the salt in solution undergoes an appreciable chemical trans- formation, e.g. dissociation. For then, besides the ordinary

m THERMODYNAMICS.

molecules of the salt, the products of the dissociation are present in the solution; for example, in the system of water and silver acetate,

7«oH20, WiCHaCOOAg, »2Ag, M3CH3TCOO | ^^o'CHaCOOAg.

The total number of molecules in the solution :

n = ?io + »i + "2 + «3 (nearly = »o).

The concentrations of the different molecules in both phases are

" n n n n ^ n^

The reactions,

vq : vi : i'2 : vs : vq = hi^ : hii : S??2 : his : S"o',

are :

(1) The precipitation of a molecule of the salt from the solution :

Vq = 0, n = - 1, V.2 = 0, V3 = 0, Vq = 1.

(2) The dissociation of a molecule of silver acetate :

Vq = 0, Vi = - 1, V2 = 1, V3 = 1, Vq = 0.

Accordingly, the two conditions of equilibrium are :

(1) - log ci = log K

(2) - log ci + log Ca + log Cs = log K' ;

or, since C2 = C3,

= K'.

el

At given temperature and pressure, therefore, there is in the saturated solution of a salt a definite number of undis- gociated molecules ; and the concentration (C2) of the dissociated molecules may be derived from that of the

DILUTE SOLUTIONS. 245

undissociated (ci) by the law of dissociation of an electrolyte, as given in (222),

Now, since by measuring the solubility the value of ci + C2, and by measuring the electrical conductivity the value of 6-2, may be found, the quantities K and K' can be calculated for any temperature. Their dependence on tempe- rature, by (219), serves as a measure of the heat effect of the precipitation of an undissociated molecule from the solution, and of the dissociation of a dissolved molecule. Jahn has thus given a method of calculating the actual heat of solution of a salt, from measurements of the solubility of the salt and of the conductivity of saturated solutions at diiferent tempe- ratures ; i.e. the heat effect which takes place when one gram molecule of the solid salt is dissolved, and the fraction

is dissociated into its ions, as is actually the case in

C2

Ci -f C2

the process of solution.

268. The Pure Substance occurring in the Second Phase forms the Solvent in the First Phase.— This case is realized when the pure solvent in any state of aggregation is separated out from a solution of another state of aggre- gation, e.g. by freezing, evaporation, fusion, and sublimation. The type of such a system is

The question whether the solvent has the same molecular weight in both phases, or not, is left open. The total number of molecules in the solution is

n = no -I- wi -f- 11.2 + «3 + . . . (nearly = ?io).

The concentrations are

n„

A possible transformation,

VQ : vx . . . : v'o = Swo : on\ . . . • cUq^

246 THERM OD YNA MICS.

is the passage of a molecule of the solvent from the first phase to the second phase, i.e.

V(, = - 1 ; i^i = ; 1^2 = ; . . . vo' = --H . (228)

Equilibrium demands, by (218), that

• log ^0 + 5 log < = log K, '"o

and, therefore, on substituting the above values of Cq and c^, log — = log K.

n - , Wi + n^ + ns+ . . .

But - = 1 4-

and, therefore, since the fraction on the right is very small, ^^^ + ^ + ^^ + --- = logK. . . (229) By the general definition (218), we have

log K = ^(vqPq + Vxfi + 1'2?2 + . . . + vJ(Pq'),

and, therefore, on substituting the values of v from (228),

This expression shows that log K also has a small value.

Suppose for the moment that log K = 0, i.e. that the pure solvent takes the place of the solution

#1 + 7^2 + . . . = 0,

then, by (230),

DILUTE SOLUTIONS. 247

Since spg and (^^ depend only on 0, 'p and the nature of the solvent, and not on the dissolved substances, the above equation asserts a definite relation between temperature and pressure, which is, in fact, the condition which and p must fulfil, in order that the two states of aggregation of the pure solvent may exist in contact. On substituting the values of fo and spo' from (214), we return immediately to the condition of equilibrium (101) which we deduced in the second chapter. The pressure (vapour pressure) may be taken 6is depending on the temperature, or the temperature (boiling point, melt- ing point) as depending on the pressure.

Eeturning now to the general csise expressed in equa- tion (230), we find that the solution of foreign molecules, ni, «2> ^^3> • • • affects the functional relation between and jp, which holds for the pure solvent. The deviation, in fact, depends only on the total number of dissolved molecules, and not on their nature. To find its amount in measurable quantities, we may introduce either jj^, the pressure which would exist in the system at the given temperature 0, if there were no dissolved molecules (lowering of the vapour pressure), or the temperature Q^ which would exist at the given pressure jj, if there were no dissolved molecules (elevation of the boiling point, depression of the freezing point). If we take the second alternative, — 6q will be very small, and we may, therefore, put

log K = "^(0 - %),

dS or, by (219), log K = ^ • ^(0 - ^o),

and

Wi + ?l2 + «3 + . • • L /fl « \

or 61 -6lo = ^(ni + ?i2 + % + .. 0(231)

By this formula the elevation of the boiling point may be calculated directly from the number of the dissolved molecules, the temperature, and the heat of vaporization.

248 THERMODYNAMICS.

Since L refers to the evaporation of one gram molecule of the liquid, the product MqL depends only on the mass, and not on the molecular weight i^m^ of the liquid solvent. If L is to be expressed in calories, we must put R = 1*97 (by 34). For instance, for one litre of water under atmospheric pressure,

«oL = 1000 X 536 cal. (approximately), = 373,

and, therefore, the elevation of the boiling point is

. . 1-97 X 3732 ^-^" = r 000x53(/ ^'^ + "^ + ---> = 0-51(wi + W2 + . . .)° C.

§ 269. Let us now compare equation (231) with the relation (183), also referring to the elevation of the boiling point, but deduced from more general principles inde- pendent of any molecular theory. The equation is

6I_0^ = ^^_ (232)

Here c denotes the ratio of the mass IMs of the dissolved non-volatile substance to the mass Mi of the solvent. In the present notation,

^ _ n-QUx + %??i2 + . ■ . /23o>|

L, in (232), is the heat of vaporization per unit mass of the solvent ; therefore, in the present notation,

^ (234)

m,

The equation (232), therefore, becomes

• e = ('^i^»i + n^m^ +

DILUTE SOLUTIONS. 249

Comparison with (231) shows that the two theories will agree perfectly only if

R(% + ^2 + ._^)^^ . . . (235) nxmx + W2?n2 + . . .

The molecular theory here set forth specializes the previous more general theory in such a way as to assign the particular value (235) to the quantity f, formerly defined by (165).

§ 270. The quantity <^ was found to be of importance for a whole series of other properties of solutions besides the elevation of the boiling point. These relations may at once be specialized for dilute solutions by substituting the value of c<p from (233) and (235),

B(^i + % + %+ ^.J^ . . (236) and for L and s, by (234), the values

— and— (237)

In this way, for the lowering of the vapour pressure of dilute solutions, we deduce, from (180),

i'o-i^ = ^%i + ^2 + %-l-...)- . (238)

If the vapour of the solvent form a perfect gas, and the specific volume of the solution be negligible in comparison with that of the vapour, then s (the change of volume of the system produced by the evaporation of a gram molecule of the liquid) is equal to the volume of the vapour formed. By (228),

. = 11^,. ^, mo ly

250 THERMOD YNA MICS.

therefore, by (238),

_ <j>(wi + ?t2 + . . .)

or, the relative lowering of the vapour pressure,

i^o^^' = (,,,4- ^^a + %+... )^-

This relation is frequently stated thus : — The relative loivei'- ing of the vapour pressure of a solution is equal to the ratio of tlie number of the dissolved molecules (ui + »-2 + "3 4- • • .) to the number of the molecules of the solvent ()Iq), or, what is the same thing in dilute solutions, to the total number of the molecules of the solution. This proposition holds only, as is evident, if rn^ = m^, i.e. if the molecules of the solvent pos- sess the same molecular weight in the vapour as in the liquid. This, however, is not generally true, as, for example, in the case of water. It may be well therefore to emphasize this fact, that nothing concerning the molecular weight of the solvent can be inferred from the relative lowering of the vapour pressure, any more than from its boiling point, freez- ing point, or osmotic pressure. Measurements of this kind will not, under any circumstances, lead to anything but the total number {ni + n.2 + . . .) of the dissolved molecules. Thus, in the last equation the product riQinQ is immediately determined by the mass of the liquid solvent, and the mole- weight, m^, of the vapour by its density.

§ 271. For the depression of the freezing point of a dilute solution, it follows from (186), (236), and (237), that

6i;-0'= ^-f,(ni + 7^2 + «3 + ...).

L' being the heat of solidification of a gram molecule of the solvent. The product, h^L', is given by the mass of the solvent ; it is independent of its molecular weight. To express L' in calories we must put R = 1 97 (by 34).

DILUTE SOLUTIONS. 251

Take water as an example : For 1 litre of water under atmospheric pressure, n^' = 1000 x 80 cal. approximately. 6q = 273, and therefore the depression of the freezing point is

^»' -^'= IGWVW^'*^ + n, + ...) = 1-84(,H + n,+ .. .fC.

§ 272. Finally, for the osmotic pressure P we have, from

(190),

V is the specific volume of the solution, and therefore the product }y>?(,i' is approximately its whole volume V. Hence

P = ^(Wi + ^2 + M3 + . . .),

an expression identical with the characteristic equation of a mixture of perfect gases with the number of molecules,

Ml, n.2, «3, • • .

§ 273. Each of the theorems deduced in the preceding paragraphs contains a method of determining the total number of the dissolved molecules in a dilute solution. Should the number calculated from such a measurement disagree ^ith the number calculated from the percentage composition of the solution on the assumption of normal molecules, some chemical change of the dissolved molecules must have taken place by dissociation, association, hydrolysis, or the like. This inference is of great importance in the determination of the chemical nature of dilute solutions. The number and nature of the different kinds of molecules are uniquely determined by the total number of mole- cules only in quite special cases, viz. when the dissolved substance undergoes a chemical change only in one way. In this case the total mass of the dissolved substance and the total number of molecules formed by it in the

252 THERMOD YNAMICS.

solution are sufficient for the calculation of the number of all the different kinds of molecules present. This case is exceptional, however, for we have seen (§ 259) that all the molecules a substance is capable of forming necessarily occur in the solution in finite quantities. As

• — soon as two reactions {e.g. H2SO4 = 2H + SO4 and

• —

H2SO4 = H + HSO4) must be considered, the analysis of