Treatise on Thermodynamics (1903) — part 8 of 14

since 61 and pi are absolute constants ; or

6li2g^(^" - rj.') = [hi - (e,f^^ - 2H)^v]m,,. (140)

Now, by the elimination of 8M12 and SM21, it follows, from (129), that

'^12 — '^^21 W12 — W2I

or, by (135) and (134),

Substituting this expression in (140), and replacing ,7t^

and -j7p^ by their values (136), we obtain au.2\

• M.((.).-«<|)^.(f-))].

This quantity is essentially positive, since M12, Mgi, as well

as Co, are always positive, and ^~ always negative for states

of equilibrium. There is a limiting case, when §^12 = 0, i.e. for a variation along the line of contact of the surfaces ^" and ^', as is obvious. It follows that the plane area 0" rises everywhere above the surface <^', and that ^" — 0' is never negative. This proves that the third solution within

DIFFERENT STATES OF AGGREGATION. 171

its region of validity (the fundamental triangle of the sub- stance) represents stable equilibrium.

§ 196. We are now in a position to answer generally the question proposed in § 165 regarding the stability of the equilibrium.

The total mass M, the volume V, and the energy U of

a system being given, its corresponding state of stabla

equilibrium is determined by the position of the point

V U

■y = jfj, w = srv, m the plane of Fig. 4.

If this point lie within one of the regions (1), (2), or (3), the system behaves as a homogeneous gas, liquid, or solid. If it lie within (12), (23), or (31), the system splits into two different states of aggregation, indicated by the numbers used in the notation of the region. In this case, the common temperature and the ratio of the two heterogeneous por- tions are completely determined. According to the equation (123), the point {v, u) lies on the straight line joining two corresponding points of the limiting curve. If a straight line be drawn through the given point {y, u), cutting the two branches of that curve in corresponding points, these points give the properties of the two states of aggregation into which the system splits. They have, of course, the same temperature and pressure. The proportion of the two masses, according to the equation (123), is given by the ratio in which the point {v, u) divides the line joining the corresponding points.

If, finally, the point (v, u) lie within the region of the fundamental triangle (123), stable equilibrium is charac- terized by a division of the system into a solid, a liquid, and a gaseous portion at the fundamental temperature and pressure. The masses of these three portions may then be determined by the equations (121a). It will be seen that their ratio is that of the three triangles, which the point (■y, u) makes with the three sides of the fundamental triangle.

The conditions of stable equilibrium of any substance

172 THERMODYNAMICS.

can thus be found, provided its fundamental triangle, its vaporization, fusion, and sublimation curves have been drawn once for all. To obtain a better view of the different relations, isothermal and isopiestic curves may be added to the figure. These curves coincide in the regions (12), (23), (31), and form the straight lines joining corresponding points on the limiting curves. On the other hand, the area (123) represents one singular isothermal and isopiestic (the triple point). In this way we may find that ice cannot exist in stable equilibrium at a higher temperature than the fundamental temperature (0-0074° C), no matter how the pressure may be reduced. Liquid water, on the other hand, may, under suitable pressure, be brought to any temperature without freezing or evaporating.

A question which may also be answered directly is the following. Through what stages will a body pass if subjected to a series of definite external changes ? For instance, the behaviour of a l)ody of mass M, when cooled or heated at constant volume V, may be known by observing the line

V

*' ~ M' P^^^ll^l *o ^^ ^^is of ordinates. The regions which

this line traverses show the states through which the body passes, e.g. whether the substance melts during the process, or whether it sublimes, etc.

CHAPTER III.

SYSTEM OF ANY NUMBER OF INDEPENDENT CONSTITUENTS.

§ 197. We proceed to investigate quite generally the equili- brium of a system made up of distinct portions in contact with one another. The system, contrary to that treated of in the preceding chapter, may consist of any number of inde- pendent constituents. Following Gibbs, we shall call each one of these portions, inasmuch as it may be considered physically homogeneous (§ 67), a phase. Thus, a quantity of water partly gaseous, partly liquid, and partly solid, forms a system of three phases. The number of phases as well as the states of aggregation is quite arbitrary, although we at once recognize the fact that a system in equilibrium may consist of any number of solid and liquid phases, but only one single gaseous phase, for two different gases in contact are never in equilibrium with one another.

§ 198. A system is characterized by the number of its independent constituents* in addition to the number of its phases. The main properties of the state of equilibrium depend upon these. We define the number of independent constituents as follows. First find the number of elements contained in the system, and from these discard, as depen- dent constituents, all those whose quantity is determined in each phase by the remaining ones. The number of the remaining elements will be the number of independent constituents of the system. It is immaterial which of the

• Frequently termed components.

174 THERMODYNAMICS.

constituents we regard as independent and which as depen- dent, since we are here concerned with the number, and not with the kind, of the independent constituents. The ques- tion as to the number of the independent constituents has nothing at all to do with the chemical constitution of the substances in the different phases, in particular, with the number of diiFerent kinds of molecules.

Thus, a quantity of water in any number of states forms but one independent constituent, however many associations and dissociations of H2O molecules may occur (it may be a mixture of hydrogen and oxygen or ions), for the mass of the oxygen in each phase is completely determined by that of the hydrogen, and vice versa. Should, however, an excess of oxygen or hydrogen be present in the vapour, we have then two independent constituents.

An aqueous solution of sulj)huric acid forms a system of three chemical elements, S, H, and 0, but contains only two independent constituents, for, in each phase {e.g. liquid, vapour, solid) the mass of depends on that of S and H, while the masses of S and H are not in each phase inter- dependent. Whether the molecule H2S0i dissociates in any way, or whether hydrates are formed or not, does not change the number of independent constituents of the system.

§ 199. We denote the number of independent constituents of a system by a. By our definition of this number we see, at once, that each phase of a given system in equilibrium is determined by the masses of each one of its a constituents, the temperature 6, and the pressure p. For the sake of uniformity, we assume that each of the a independent con- stituents actually occurs in each phase of the system in a certain quantity, which, in special cases, may become infinitely small. The selection of the temperature and the pressure as independent variables, produces a change in the form of the equations of the last chapter, where the tem- perature and the specific volume were considered as the independent variables. The substitution of the pressure

ANY NUMBER OF INDEPENDENT CONSTITUENTS. 175

for the volume is more convenient here, because the pressure is the same for all phases in free contact, and it can in most cases be more readily measured.

§ 200. We shall now consider the thermodynamical equilibrium of a system, in which the total masses of the a independent constituents Mi, M2, . . . M„ are given. Of the different forms of the condition of equilibrium it is best to use that expressed by equation (79)

S^ = (141)

which holds, if and f remain constant, for any change compatible with the given conditions. The function 4^ is given in terms of the entropy <l>, the energy U, and the volume V, by the equation

4^ =0 ^ —

§ 201. Now, let /3 be the number of phases in the system, then <I>, U, and V, and therefore also ^, are sums of /3 terms, each of which refers to a single phase, i.e. to a physically homogeneous body :

4^ = vp' + ^" + . . . + 4^^ . . . (142)

where the different phases are distinguished from one another by dashes. For the first phase,

^' = <I>'- ^'^^^' . . . . (143)

<!»', U', V and ^' are completely determined by 0, 'p, and the masses Mi, M2, • • • M^ of the independent constituents in the phases. As to how they depend on the masses, all we can at present say is, that, if all the masses were increased in the same proportion (say doubled), each of these functions would be increased in the same proportion. Since the nature of the phase remains unchanged, the entropy, the energy, and the volume change in the same proportion as

1 76 THERMOD YNA MICS.

the mass ; hence, also, the function ^'. In other words, ^' is a homogeneous function of the masses Mi', M2', . . . MJ of the first degree, but not necessarily linear.

To express this analytically, let us increase all the masses in the same ratio 1 + £ : 1, where t is very small. All changes are then small ; and for the corresponding change of ^' we obtain

But, by supposition, A^' = t^',

and, therefore,

*' = sl''^ + al'M^' + • • • + ah''-'- ("-'>

Various forms may be given to this Eiilerian equation

by further differentiation. The differential coefficients ^^rr „

d^' . . "^^^

^^r|-7 . . . evidently depend on the constitution of the phase,

and not on its total mass, since a change of mass changes both numerator and denominator in the same proportion.

§ 202. By (142), the condition of equilibrium becomes g^' + g^" + . . . 8^^ = . . . (145) or, since the temperature and pressure remain constant,

^,SM,' + ^,m,' +...+ ^,m;

4-

• m^^^^' + l£«^i^ + • • • + 5sVM.^=o ("«)

ANY NUMBER OF INDEPENDENT CONSTITUENTS. 177

If the variation of the masses were quite arbitrary, then the equation could only be satisfied, if all the coefficients of the variations were equal to 0. According to § 200, however, the following conditions exist between them.

Ml = M/ + Ml" + M2 = Ma' + M2" +

• Mi^
• M/

(147)

M„ = m; + M„" + . . . + M/ I

and, therefore, for any possible change of the system,

• 8M/
• SM/

= UU + SMi" +

= UU + SM2" +

= 8M„' + m:' + . . . + 8M/

(148)

For the expression (146) to vanish, the necessary and suffi- cient condition is

(149)

5M7

~ 5Mi" ~ ■

d^'

d^"

a>p^

5M2'

~ M2

• ~ m^ \

5^' 5M'

~ dM '

There are for each independent constituent (/3 — 1) equa- tions, which must be satisfied, and therefore for all the o independent constituents a(j3 — 1) conditions. Each of these equations refers to the transition from one phase into another, and asserts that this particular transition does not take place in nature. This condition depends, as it must, on the internal constitution of the phase, and not on its total mass. Since the equations in a single row with regard to a particular constituent may be arranged in any order, it follows that, if a phase be in equilibrium as regards a

N

178 THERMODYNAMICS.

given constituent with two others, these two other phases are in equilibrium with one another with regard to that constituent (they coexist). This shows that, since any system in equilibrium can have only one gaseous phase, two coexisting phases must emit the same vapour. For, since each phase is in equilibrium with the other, and also with its own vapour with respect to all constituents, it must also coexist with the vapour of the second phase. The coexist- ence of solid and liquid phases may, therefore, be settled by comparing their vapours.

§ 203. It is now easy to see how the state of equi- librium of the system is determined, in general, by the given external conditions (147), and the conditions of equilibrium (149). There are a of the former and a(/3 — 1) of the latter, a total of a/3 equations. On the other hand, the state of the /3 phases depends on («/3 + 2) variables, viz. on the a/3 masses, M/, . . . 31/, the temperature 0, and the pressure |>. After all conditions have been satisfied, two variables still remain undetermined. In general, the temperature and the pressure may be arbitrarily chosen, but in special cases, as will be shown presently, these are no longer arbitrary, and in such cases two other variables, as the total energy and the total volume of the system, are undetermined. By disposing of the values of the arbitrary variables we com- pletely determine the state of the equilibrium.

§ 204. The a/3 + 2 variables, which control the state of the system, may be separated into those which merely govern the composition of the phases {internal variables), and those which determine only the total masses of the phases (external variables). The number of the former is (a — l)/3 + 2, for in each of the /3 phases there are a — 1 ratios between its a independent constituents, to which must be added temperature and pressure. The number of the external variables is /3, viz. the total masses of all the phases.

We found that the a(/3 - 1) equations (149) contain

ANY NUMBER OF INDEPENDENT CONSTITUENTS. 179

only internal variables, and, therefore, after these have been satisfied, there remain

[(a - l)^ + 2] - [a(/3 - 1)] = „ - /-3 + 2

of the internal variables, undetermined. This number cannot be negative, for otherwise the number of the internal variables of the system would not be sufficient for the solution of the equations (149). It, therefore, follows that

/3<a + 2

The number of the phases, therefore, cannot exceed the number of the independent constituents by more than two ; or, a system of a independent constituents will contain at most (a + 2) phases. In the limiting case, where j3 = a + 2, the number of the internal variables are just sufficient to satisfy the internal conditions of equilibrium (149). Their values in the state of equilibrium are completely deter- mined quite independently of the given external conditions. Decreasing the number of phases by one increases the number of the indeterminate internal variables by one.

This proposition, first propounded by Gibbs and univer- sally known as the i^hase rule, has been amply verified, especially by the experiments of Bakhuis Roozeboom.

§ 205. We shall consider, first, the limiting case :

i3 = a + 2.

(Non-variant systems.) Since all the internal variables are completely determined, they form an [n -f- 2)-i)le foint. Change of the external conditions, as heating, compression, further additions of the substances, alter the total masses of the phases, but hot their internal nature, including tempera- ture and pressure. This holds until the mass of some one phase becomes zero, and therewith completely vanishes from the system.

If a = 1, then j3 = 3. A single constituent may split

1 8o T HER MOD YNA MICS.

into three phases at most, forming a triple point. An example of this is a substance existing in the three states of aggregation, all in contact with one another. For water it was shown in § 187, that at the triple point the tempera- ture is 0-0074° C, and the pressure 4-62 mm. of mercury. The three phases need not, however, be different states of aggregation. Sulphur, for instance, forms several modifica- tions in the solid state. Each modification constitutes a separate phase, and the proposition holds that two modifica- tions of a substance can coexist with a third phase of the same substance, for example, its vapour, only at a definite temperature and pressure.

A quadruple point is obtained when a = 2. Thus, the two independent constituents, SO2 (sulphur dioxide) and H2O, form the four coexisting phases : S02,7H20 (solid), SO2 dissolved in H2O (liquid), SO2 (liquid), SO2 (gaseous), at a temperature of 12"1° C. and a pressure of 1770 mm. of mer- cury. The question as to the formation of hydrates by SO2 in aqueous solution does not influence the application of the phase rule (see § 198).

Three independent constituents (a = 3) lead to a quin- tiple point. Thus Na2S04, MgS04, and H2O give the double salt Na2Mg(S04)24H20 (astrakanite), the crystals of the two simple salts, aqueous solution, and water vapour, at a temperature of 22° C. and a pressure of 19*6 mm. of mercury.

§ 206. We shall now take the case

/3 = a + 1,

that is, a independent constituents form a + 1 phases (Univariant systems). The composition of all the phases is then completely determined by a single variable, e.g. the temperature or the pressure. This case is generally called 'perfect heterogeneous equilibrium.

If a = 1, then /3 = 2 : one independent constituent in two phases, e.g. a liquid and its vapour. The pressure and the density of the liquid and the vapour depend on the temperature alone, as was pointed out in the last chapter.

ANY NUMBER OF INDEPENDENT COMSTlTUENTS. i8i

Evaporation involving chemical decomposition also belongs to this class, since the system contains only one independent constituent. The evaporation of solid NH4CI. is a case in point. Unless there be present an excess of hydrochloric acid or ammonia gas, there will be for each temperature a quite definite dissociation pressure.

If a = 2, then /3 = 3, for instance when the solution of a salt is in contact with its vapour and with the solid salt, or when two liquids that cannot be mixed in all proportions (ether and water) are in contact with their common vapour. Vapour pressure, density and concentration in each phase, are here functions of the temperature alone.

§ 207. We often take the pressure instead of the tem- perature as the variable which controls the phases in perfect heterogeneous equilibrium ; namely, in systems which do not possess a gaseous phase, so-called condensed systems. Upon these the influence of the pressure is so slight that, under ordinary circumstances, it may be con- sidered as given, and equal to that of the atmosphere. The phase rule, therefore, gives rise to the following proposition : A condensed system of a independent constituents forms a + 1 phases at most, and is then completely determined, temperature included. The melting point of a substance, and the point of transition from one allotropic modification to another, are examples of a = 1, /3 = 2, The point at which the cryohy- drate (ice and solid salt) separates out from the solution of a salt, and also the point at which two liquid layers in contact begin to precipitate a solid {e.g. As Bra, ^^^ H2O) are examples of a = 2, /3 = 3. We have an example of a = 3, /3 = 4 when two salts, capable of forming a double salt, are in contact with the solid simple salts, and also with the double salt.

§ 208. If

^ = a,

then a independent constituents form a phases (Divariant systems). The internal nature of all the phases depends

1 82 THERM OD YNA MICS.

on two variables, e.g. on temperature and pressure. Any homogeneous substance furnishes an example of a = 1. A liquid solution of a salt in contact with its vapour is an example of a = 2. The temperature and the pressure deter- mine the concentration in the vapour as well as in the liquid. The concentration of the liquid and either the temperature or the pressure are frequently chosen as the independent variables. In the first case, we say that a solution of given concentration and given temperature emits a vapour of definite composition and definite pressure ; and in the second case, that a solution of given concentration and given pressure has a definite boiling point, and at this temperature a vapour of definite composition may be distilled off.

Corresponding regularities hold when the second phase is solid or liquid, as in the case of two liquids which do not mix in all jiroportions. The internal nature of the two phases, in our example the concentrations in the two layers of the liquids, depends on two variables — pressure and temperature. If, under special circumstances, the con- centrations become equal, a phenomenon is obtained which is quite analogous to that of the critical point of a homo- geneous substance (critical solution temperature of two liquids).

§ 209. Let us now consider briefly the case /3 = « - 1,

where the number of phases is one less than the number of the independent constituents, and the internal nature of all phases dejiends on a third arbitrary variable, besides temperature and pressure. Thus, a = 3, /3 = 2 for an aqueous solution of two isomorphous substances (potassium chlorate and thallium chlorate) in contact with a mixed crystal. The concentration of the solution under atmospheric pressure and at a given temperature will vary according to the com- position of the mixed crystal. We cannot, therefore, speak of a saturated solution of the two substances of definite composition. However, should a second solid phase — for

ANY NUMBER OF INDEPENDENT CONSTITUENTS. 183

instance, a mixed crystal of diiFerent composition — separate out, the internal nature of the system will be determined by temperature and pressure alone. The experimental investigation of the equilibrium of such systems may enable us to decide whether a precipitate from a solution of two salts forms one phase — for example, a mixed crystal of changing concentration — or whether the two substances are to be considered as two distinct phases in contact. If, at a given temperature and pressure, the concentration of the liquid in contact were quite definite, it would represent the former case, and, if not, the latter.

§ 210. If the expressions for the functions ^F', ^", . . . for each phase were known, the equations (149) would give every detail regarding the state of the equilibrium. This, however, is by no means the case, for, regarding the relations between these functions and the masses of the constituents in the individual phases, all we can, in general, assert is that they are homogeneous functions of the first degree (§ 201). We can, however, tell how they depend upon temperature and pressure, since their differential coefficients with respect to ^ and j; can be given. This point leads to far-reaching conclusions concerning the variation of the equilibrium with temperature and pressure.

Since, for the first phase, according to (143),

4/' = 4)' _

U' + ijV

61

we have, for an infinitely small change,

u u

Under the assumption that the change is produced only by variations of 2^ and 6, and not by that of the masses Mi', M2', . . . M„', equation (60) gives

, , cW + pdY' cH) = ^ »

1 84 THERMODYNAMICS.

and, therefore,

whence

by' _ U' + yS ' ^ ^' _ _ y bQ ~ (T' ' dp ~ 6'

and for the system, as the sum of all the phases,

5^=— g/-, and^ = -^. . (150)

§ 211. These relations may be used to determine how the equilibrium depends on the temperature and pressure. For this purpose we shall distinguish between two different kinds of infinitely small changes. The notation S will refer, as hitherto, to a change of the masses Mi', Mg', . . . Mf > consistent with the given external conditions, and, therefore, consistent with the equations (148), temperature and pressure being kept constant, i.e. cO = and ^p = 0. The state, to which this variation leads, need not be one of equilibrium, and the equations (149) need not, therefore, apply to it. The notation d, on the other hand, will refer to a change from one state of equilibrium to another, only slightly different from it. All external conditions, including temperature and pressure, may be changed in any arbitrary manner.

The problem is now to find the conditions of equilibrium of this second state, and to compare them with those of the original state. Since the condition of equilibrium of the first state is

the condition for the second state is

8(^ + fZ^) = 0, hence 8<Z^ = (151)

ANY NUMBER OF INDEPENDENT CONSriTUENTS. 185 But

''* = > + 3i/i' + Xau/^^i^ + Ir/M. + • . .

where ^ denotes the summation over all the /3 phases of

the system, while the summation over the a constituents of a single phase is written out at length. This becomes, by (150),

The condition of equilibrium (151) therefore becomes

^^ dS - -^dp 4- 2^Mi 8^^

• fZMa'g— + . . . = (152)

All variations of dQ, dp, tZM/, dM.^, . . . disappear because 80 = and Sjj = 0, and because in the sum

• m^''^' + &''^ + • • •

each vertical column vanishes. Taking the first column for example, we have, by (149),

1 86 THERMO D YNA MICS.

and also, by (148),

gcZATi' + gc?31i" + . . . + U^U^

= d{llsU + g^Ii" + . . . + SMiP) = 0.

Furthermore, since, by the first law, 8U + jp^V represents Q, the heat absorbed by the system during the virtual change, the equation (152) may also be written

fdS - ^dp + S^ZMi'^J^, + d:Sl,^^, + . . . = 0. (153)

This equation shows how the equilibrium depends on the temperature, and the pressure, and on the masses of the independent constituents of the system. It shows, in tlie first place, that the influence of the temperature depends essentially on the heat effect which accompanies a virtual change of state. If this be zero, the first term vanishes, and a change of temperature does not disturb the equilibrium. If Q change sign, the influence of the temperature is also reversed. It is quite similar with regard to the influence of the pressure, which, in its turn, depends essentially on the change of volume, 8V, produced by a virtual isothermal and isopiestic change of state.

§ 212. We shall now apply the equation (153) to several special cases ; first, to those of perfect heterogeneous equili- brium, which are characterized (§ 206) by the relation

/3 = « + 1.

The internal nature of all the phases, including the pressure, is determined by the temperature alone. An isothermal, infinitely slow compression, therefore, changes only the total masses of the phases, but does not change either the com- position or the pressure. We shall choose a change of this kind as the virtual change of state. In this special case it leads to a new state of equilibrium. The internal nature of all the phases, as well as the temperature and pressure.

ANY NUMBER OF INDEPENDENT CONSTITUENTS. 187 remain constant, and therefore the variations of the functions ^IvT" Mf~" ■ ' ■ ^"^^ ^ equal to zero, since these quantities

depend only on the nature of the phases. The equation (153) therefore becomes

#_ Q_ n54^

m ~ 08V ^^^

This means that the heat eifect in a variation that leaves the composition of all phases unchanged, divided by the change of volume of the system and by the absolute tempe- rature, gives the rate of change of the equilibrium pressure with the temperature. Where application of heat increases the volume, as in the case of evaporation, the equilibrium pressure increases with temperature ; in the opposite case, as in the melting of ice, it decreases with increase of temperature.

§ 213. In the case of one independent constituent (a = 1, and /3 = 2), equation (154) leads immediately to the laws discussed at length in the preceding chapter ; namely, those concerning the heat of vaporization, of fusion, and of subli- mation. If, for instance, the liquid form the first phase, the vapour the second phase, and L denote the heat of vaporiza- tion per unit mass, we have

Q = LgM"

8V = (v" - v)m!'

where v and v" are the specific volumes of liquid and vapour, and 8M" the mass of vapour formed during the isothermal and isopiestic change of state. Hence, by (154),

L = e|K - .■).

which is identical with the equation (HI).

This, of course, applies to chemical changes as well,

l88 THERMODYNAMICS.

whenever the system under consideration contains one con- stituent in two distinct phases ; for example, to the vapori- zation of ammonium chloride (first investigated with regard to this law by Horstmann), which decomposes into hydro- chloric acid and ammonia ; or to the vaporization of ammonium carbamate, which decomposes into ammonia and carbon dioxide. Here L of our last equation denotes the heat of dissociation, and ~p the dissociation pressure, which depends only on the temperature.

§ 214. We shall also consider the perfect heterogeneous equilibrium of two independent constituents (a = 2, /3 = 3) ; for example, water (suffix 1) and a salt (suffix 2) in three phases ; the first, an aqueous solution (M/ the mass of the water, 31 2' that of the salt) ; the second, water vapour (mass 3Ii") ; the third, solid salt (mass M2'"). For a virtual change, therefore,

8M1' + SMi" = 0, and ?M2' + SM2'" = 0.

According to the phase rule, the concentration of the

solution ( jTi^, = c V as well as the vapour pressure ( j:>), is a

function of the temperature alone. By (154), the heat absorbed (0, ^, c remaining constant) is

Q = «-^.cV (155)

Let the virtual change consist in the evaporation of a small quantity of water,

SMi" = - gMi'.

Then, since the concentration also remains constant, the quantity of salt

8M2'" = - SM2' = - c8Mi' = cSMi"

is precipitated from the solution. All variations of mass have here been expressed in terms of SMi".

ANY NUMBER OF INDEPENDENT CONSTITUENTS. 189

The total volume of the system

V = ^'(Mi' + Ma') + v"Mi" + v'''^.^",

where v', v", and v"' are the specific volumes of the phases, is increased by

gv = v\uu + 8M2') + v"uu' + v"'m^"

gV = [{v" + cv'") - (1 + cy]8Mi". . . (156)

If L be the quantity of heat that must be applied to evaporate unit mass of water from the solution, and to pre- cipitate the corresponding quantity of salt, under constant pressure, temperature, and concentration, then the equation (155), since

Q = LSMi", becomes L = ^jkv" + c/v" — (1 + c)v').

A useful approximation is obtained by neglecting v' and v"', the specific volumes of the liquid and solid, in com- parison with v", that of the vapour, and considering the latter as a perfect gas. By (14),

v" = -.^-

(R = gas constant, m = the molecular weight of the vapour) and we obtain

L = 5r^ ^^ (157)

§ 215. Conversely, L is at the same time the quantity of heat given out when unit mass of water vapour combines, at constant temperature and pressure, with the quantity of salt necessary to form a saturated solution. This process may be accomplished directly, or in two steps, viz. by condensing unit mass of water vapour into pure water, and then dis- solving the salt in the water. According to the first law of thermodynamics, since the initial and final states are the

igo THERMOD YNA MICS.

same in both cases, the sum of the heat given out and the work done is the same.

In the first case the heat given out is L, the work done,

— i^wjiFT/ ; 8,nd the sum of these, by the approximation used

above, is

y-'^-v^" (158)

To calculate the same sum for the second case, we must in the first place note that the vapour pressure of a solution is different from the vapour pressure of pure water at the same temperature. It will, in fact, in no case be greater, but smaller, otherwise the vapour would be supersaturated. Denoting the vapour pressure of pure water at the tempera- ture Q by j}^, then ^j < jJq.

We shall now bring, by isothermic compression, unit mass of water vapour from pressure f and specific volume v" to pressure j^o and specific volume W'j «.e- to a state of satu- ration. Work is thereby done on the substance, and heat is given out. The sum of both, which gives the decrease of the energy of the vapour, is zero, if we again assume that the vapour behaves as a perfect gas, i.e. that its energy remains constant at constant temperature. If we then condense the water vapour of volume v^, at constant tempe- rature 61 and constant pressure jJo> into pure water, the sum of the heat given out and work spent at this step is, by equation (112),

?fl..ii^£»_y„V. . . . (159)

No appreciable external effects accompany the further change of the liquid water from pressure j;o to pressure jj.

If, finally, we dissolve salt sufficient for saturation in the newly formed unit of water, at constant temperature and constant pressure j:), the sum of the heat and work is simply the heat of solution

(160)

ANY NUMBER OF INDEPENDENT CONSTITUENTS. 191

By the first law, the sum of (159) and (160) must be equal

to (158),

Rno . A log Vq „ E„„ d log «

w dO -^ ' m do ^

or, since, by Boyle's law 2W0" = pv",

-r. d loo: —

x = Rfl^. _!£».• • • ■ (161)

m du

This formula, first established by Kirchhoif, gives the heat evolved when salt sufficient for saturation is dissolved in 1 gr. of pure water.

To express X in calories, R must be divided by the

r»

mechanical equivalent of heat, J. By (34), ^ = 1*97, and since w = 18, we have

d log ~ X = 0-1102 . _Po cal. dO

It is further worthy of notice that p, the vapour pressure of a saturated solution, is a function of the temperature alone, since c, the concentration of a saturated solution, changes in a definite manner with the temperature.

The quantities neglected in this approximation may, if necessary, be put in without any difficulty.

§216. We proceed now to the important case of two independent constituents in two phases (a = 2, /3 = 2). We assume, for the present, that both constituents are con- tained in both phases in appreciable quantity, having the masses M/, M2' in the first ; Mi", M"2, in the second phase. The internal variables are the temperature, the pressure,

192

THERMOD YNAMICS.

and the concentrations of the second constituent in both phases ;

, M2' , „ M2" c = ;^, and c =

31/

Ml"* • •

(162)

According to the phase rule, two of the variables, 0, p, c', c", are arbitrary.

Equation (153) leads to the following law regarding the shift of the equilibrium corresponding to any change of the external conditions :

Q

gv

,>54''

,^^^'

„.^"

^-m - ^dj) + rDii'g^^ + ^Ma'g^^ + ^^ir^'^^j ,f

• dM^"B~, = 0. (163)

Here, for the first phase.

^aMs^ - ^Mi'aMa'^^^' "^ 5M2^^^^2

(164)

Certain simple relations hold between the derived functions of ^' with respect to M/ and M.2'. For, since, by (144),

,d^'

,d^''

partial differentiation with respect to M/ and M2' gives

If we put, for shortness.

(165)

ANV NUMBER OF INDEPENDENT CONSTITUENTS. 193

a quantity depending only on the nature of the first phase, on B,p, and c', and not on the masses MV and Mg' individually,* we have

"M2'

(166)

Analogous equations hold for the second phase if we put

<p" = Mx"

5Mi"5M2"

§ 217. With respect to the quantities f' and f" all we can immediately settle is their sign. According to § 147, ^ is a maximum in stable equilibrium if onl y processes at constant temperature and constant pressure be considered. Hence

But

whence

S2^<0 (167)

and

(9V

d^'

d^^'

5Mi'5M;'^^^^'^^^^' "^ 5M2'2

8M,'2

aixlr'f fflyu" d^ylr"

» The general integral of ^F' = Mi' ^, + ^2& ^^ *'= ^f2'/(^)— Tk-

194 THERMODYNAMICS.

If we introdace the quantities f' and <p", then

This relation shows that the inequality (167) is satisfied, and only then, if both ^' and (p" are positive.

[ § 218. There are on the whole two kinds of changes possible, according as the first or the second constituent passes from the first to the second phase. We have, for the first,

SMi' = - gMi" ; SMa' = SMa" = ; . . (168) and for the second,

8Mi' = gMi",= 0; gMa' = - gMa".

We shall distinguish Q, the heat absorbed, and SV, the change of Tolume, in these two cases by the sufiixes 1 and 2. In the first case, the law for the displacement of the equilibrium, by (163), (164), (168), (166), and (162), reduces to

^m - ^dp - gMi"(f W - ^"dc") = and, introducing for shortness the finite quantities

i.e. the ratios of the heat absorbed and of the change of volume to the mass of the first constituent, which passes from the first to the second phase, we have

^dO - |V^J - f'dc' + fdc" = 0. . (170)

ANY NUMBER OF INDEPENDENT CONSTITUENTS. 195

Similarly, for the second constituent passing into the second phase, we get

^rf# _ iv^ + ,.|' _ y"*:' = 0. . (171)

These are the two relations connecting the four differen- tials ^0, d'p, do', do' in any displacement of the equilibrium.

§ 219. To show the application of these laws, let us consider a mixture of two liquids (water and alcohol) in two phases, the first a liquid, the second a vapour. The phase rule leaves two of the variables B, p, c', c" arbitrary. The pressure p, and the concentration e" of the vapour, for instance, are determined by the temperature and the con- centration c' of the liquid mixture. Accordingly, for any changes dd and dc' we have, from (170) and (171),

(^^, - i)e'^'dc' + (Li + c"U)de

^^ = "^ (Si + c)(i '

(1 + ly,, _ fh _ h) . ^J

„ Vsi cs^r \si §2/ B^

dc" =

ti + T^y

Of the many conclusions to be drawn from these equations, we mention only the following :

Along an isotherm {dB = 0) the equations become

(^' - l)B(p'dG'

do" = ^?i :^ . ^-r,do' . . . (173)

i+Ji ^

Si C"S2

The vapour pressure p may rise or fall with increasing

1 96 THERMO D YNA MICS.

concentration. When i) shows a maximum or minimum value, as it does according to Konowalow for a 77 : 23 mixture

of propyl alcohol and water, then ^, vanishes, and, from

equation (172), c = c", i.e. the percentage composition of the liquid and the vapour is the same, or the liquid boils at constant concentration. But if, along an isotherm, j; varies with c', the concentration of the vapour will differ from that of the liquid ; in fact, the concentration of the second constituent in the vapour will be more or less than in the liquid (c"> or <c'), according as the vapour pressure 2) rises or falls with increasing concentration. This is an immediate deduction from equation (172) if we bear in mind that p', Si, §2, and c" are always positive.

The equation (173) shows that along an isotherm the concentration of both phases always changes in the same sense.

§ 220, In the following applications we shall restrict ourselves to the case in which the second constituent occurs only in the first phase,

c" = 0,

and .:dc" = (174)

The first constituent which occurs along with the second in the first phase, and pure in the second, will be called the solvent ; the second, the dissolved suhstance. By (174), the equation (171) is identically satisfied, and from (170) there remains

j^/LQ — -^dj) — fdc = 0, , , . (175)

if we omit suffixes and dashes for simplicity.

We shall take, first, a solution of a nonvolatile salt in contact with the vapour of the solvent, and investigate the equation (175) in three directions by keeping in turn tlie concentration c, the temperature B, and the pressure p constant.

ANV NUMBER OF INDEPENDENT CONS T ITU EN TS. 1 97

§ 221. Concentration Constant: de = 0. — The relation between the vapour pressure and the temperature is, by (175),

(!)=«-. ("«)