Treatise on Thermodynamics (1903) — part 7 of 14

§ 175. As an example, we shall calculate the heat of vaporization of water at 100' C, i.e, under atmospheric pressure, from the following data : —

= 273 + 100 = 373.

Vi = 1658 (volume of 1 gr. of saturated water vapour at 100° in c.c, according^to Wiillner). V2 — 1 (volume of 1 gr. of water at 100° in c.c).

-~^ is found from the experiments of Regnault. Saturated

water vapour at 100° C. gave an increase of pressure of 27*2 mm. of mercury for a rise of 1' C. In absolute units,

by §7,

W - 760 "^ lOldboO. Thus, the required latent heat of vaporization is ^ 373 X 1657 X 27-2 x 1013650 _

^ - 76rxTi9 >rio^ ~ ^"^^ ''*^-

By direct observation Regnault found the heat of vaporiza- tion of water at 100° C. to be 536.

§ 176. As equation (110) shows, part of the heat of vaporization, L, corresponds to an increase of energy, and part to external work. To find the relation between these two it is most convenient to find the ratio of the external work to the latent heat of vaporization, viz.

jJi(^i - V2) _ Pi . L fApi

In the above case 2^ = 760 mm., = 373, ,^ = 272 mm., and therefore,

pijvi - V2) _ 76 _ ^.^„^ L " 373 X 27-2 ~ ^ '^-

144 THERMODYNAMICS.

This shows that the external work forms only a small part of the value of the latent heat of vaporization.

§ 177. Equation (Hi) also leads to a method of calculating the absolute temperature %, when the latent heat of vaporization, L, as well as the pressure and the density of the saturated vapour and the liquid, have been determined by experiment in terms of any scale of temperature t (§ 160). We have

and therefore may be determined as a function of t. It is obvious that any equation between measurable quantities, deduced from the second law, may be utilized for a deter- mination of the absolute temperature. The question as to which of those methods deserves preference is to be decided by the degree of accuracy to be obtained in the actual measurements.

§ 178. A simple approximation formula, which in many cases gives good, though in some, only fair results, may be obtained by neglecting in the equation (111) the specific volume of the liquid, v^, in comparison with that of the vapour, Vi, and assuming for the vapour the characteristic equation of a perfect gas. Then, by (14),

vi = — , m'px

where R is the absolute gas constant, and m the molecular weight of the vapour. Equation (111) then becomes

L = ?.'^'.fj .... (112) m pi do ^ ^

For water at 100° C. we have K = 1 971 ; m = H2O = 18 ;

DIFFERENT STATES OF AGGREGATION. 145

= 373 ; iH = 760 mm. ; ^^' = 272 mm. Hence the latent heat of vaporization is

^ 1-971 X 3732 X 27-2 ..^ , ^ = 18 X 760 = ^^^/'^l-

This value is somewhat large (§ 175). The cauip of this lies in the fact that the volume of saturated water vapour at 100° is in reality smaller than that calculated from the characteristic equation of a perfect gas of molecular weight 18. But, for this very reason, accurate measurement of the heat of vaporization may serve as a means of estimating from the second law the amount by which the density of a vapour deviates from the ideal value.

Another kind of approximation formula, valid within the same limits, is found by substituting in (109) the value of the specific energy Wi = c„0 + const., which, by (39), holds for perfect gases. We may put the specific energy of the liquid «2 = ^^2^ + const., if we assume its specific heat, c.2, to be constant, and neglect the external work. It then follows from (109) that

(cv - cM + const. H = - . — . /^.

If we multiply both sides by rp> tliis equation may be

integrated, term by term, and we finally obtain with the help of (33)

Pi

where a and h are positive constants ; Cp and Ca the specific heats of the vapour and the liquid, at constant pressure. This relation between the pressure of the saturated vapour and the temperature is the more approximately true, the further the temperature lies below the critical temperature of the

vapour.

L

1 16 THERMOD YNA MICS.

For mercury vapour, for example, according to a calcula- tion by H. Hertz, if jji be given in mm, of mercury,

a = 3-915 X 10^0; b = 7695; ^=^(Cj> - c^) = - 0-847.

§ 179. Equation (111) is applicable to the processes of fusion and sublimation in the same manner as to that of evaporation. In the first case L denotes the latent heat of fusion of the substance, if the subscript 1 correspond to the liquid state and 2 to the solid state, and j^i the melting pressure, i.e. the pressure under which the solid and the liquid substance may be in contact and in equilibrium. The melting pressure, therefore, just as the pressure of evapora- tion, depends on the temperature only. Conversely, a change of pressure produces a change in the melting point :

dO d(vi - Va) ... , o^

dpr L (^^"^

For ice at 0° C. and under atmospheric pressure, we have

L = 80 X -119 X 10^ (heat of fusion of 1 gr. of ice in C.G.S.

units) ; 6 = 213;

Vi = 1-0 (vol. of 1 gr. of water at 0° C. in c.c.) ; ^2 = 1 09 (vol. of 1 gr, of ice at 0° C. in c.c).

d9 To obtain -5 — in atmospheres we must multiply by 1,013,650 :

dS 273x0-09x1013650 ^^^^, ,,,,,

dj, = - 80 X 419 X 10^ = - ^'^^^^- ^11^)

On increasing the external pressure by 1 atmosphere, the melting point of ice will, therefore, be lowered by 0074' C. ; or, to lower the melting point of ice by 1° C, the pressure must be increased by about 130 atmospheres. This was first verified by the measurements of W. Thomson (Lord Kelvin). Equation (113) shows that, conversely, the melting point of substances, which expand on melting, is raised by an increase

DIFFERENT STATES OF AGGREGATION. 147

of pressure. This has been qualitatively and quantitatively verified by experiment.

§ 180. By means of the equations (101) still further important properties of substances in different states may be shown to depend on one another. From these, along with (110), we obtain

L

Differentiating this with respect to 0, we have

^'dB 0'~\dei'^\dvJe'de \de)v \dv)g' dO' or, by (81),

_ (cvh. , (dpiX dvx _ (c^ _ /5/)2\ d^

~ e ^\de)/ de e \dti)/ do'

We now introduce Cp, the specific heat at constant pressure, for Cv, that at constant volume, and obtain by (82), on multiplying by 6,

or, since for both states, according to (6),

dL L . f^idvi. ndpi. ,(dji\ dvn

-fe)^+«(li-[(MHla-t]-

Now, the expressions within the brackets are identical with

d2h _ (hh ^ _

148 THERMODYNAMICS.

This gives, finally,

This equation, which is rigorously true, again leads to a test of the second law, since all the quantities in it may be observed independently of one another.

§ 181. We shall again take as example saturated water vapour at 100° C. (atmospheric pressure), and calculate its specific heat at constant pressure. We have the following data:

(cp)2 = 103 (= spec, heat of liquid water at 100^ C) ; L = 536 ; = 373 ;

Tn— 0*708 (decrease of the heat of vaporization with

increase of temperature, according to Kegnault's observations).

We determine v, and (n/iM from the observations of

Him, who found the volume of 1 gr. of steam, at 100° under atmospheric pressure, to be 16504 c.c. ; and at 1185°, 1740 c.c, whence

K'-nt /"^M 1740-16504 ,,,,„

Vi = lbo04 ; -^ ) = ^^-- = 4-843.

' V^^/p 18-0

5Vo

Also .,= l-0;(,^^)^ =

These values, substituted in (115), give

(c^)i - (c>)2 = - 0-56, or, (c^)i = (cp)2 - 0-56 = 103 - 0-56 = 047.

By direct measurement, Kegnault found the mean specific heat of steam under atmospheric pressure for temperatures somewhat higher than 100° C. to be 048.

DIFFERENT STATES OF AGGREGATION. 149

§ 182. The relation (115) may be simplified, but is in- accurate, if we neglect the volume v.^ of the liquid in com- parison with V, that of the vapour, and apply to the vapour the characteristic equation of a perfect gas.

Then

Vx

RO

mpi

and equation (115) becomes, simply,

In our example,

(Op)i - (^p)2 = - 0-71 (cp)i = 1-03 - 0-71 = 0-32,

a value considerably too small.

§ 183. We shall now apply the relation (115) to the melting of ice at 0° C. and under atmospheric pressure. The subscript 1 now refers to the liquid state, and 2 to the solid state. The relation between the latent heat of fusion of ice and the temperature has probably never been measured. It may, however, be calculated from (115), which gives

dL in which

= («,)i - {e,).2 + ^ - :^l{^\ - (^OJ

(cp)i = 1 (spec, heat of water at 0° C.) ; (cp)2 = 0*505 (spec, heat of ice at 0° C.) ; L = 80; = 273; t^i = 1 ; 1^2 = I'OO ;

(^^\ = _ 0-00006 (therm. coeflf. of expansion of water at O^C); ^^A = 0-00011 (thermal coeflf. of expansion of ice at 0° C).

ISO THERMOD YNA MICS.

Hence, by the above equation,

i.e., if the melting point of ice be lowered 1° C. by an appropriate increase of the external pressure, its heat of fusion decreases by 0*64 cal.

§ 184. It has been repeatedly mentioned in the early chapters, that, besides the specific heat at constant pressure, or constant volume, any number of specific heats may be defined according to the conditions under which the heating takes place. Equation (23) of the first law holds in each case:

_ du dv

In the case of saturated vapours special interest attaches to the process of heating, which keeps them permanently in a state of saturation. Denoting by //i the specific heat of the vapour corresponding to this process (Clausius called it the specific heat of " the saturated vapour "), we have

,, = f| + 4'. .... (116)

No oif-hand statement can be made with regard to the value of //i ; even its sign must in the mean time remain uncertain. For, if during a rise of temperature of 1' the vapour is to remain just saturated, it must evidently be compressed while being heated, since the specific volume of the saturated vapour decreases as the temperature rises. This compression, however, generates heat, and the question is, whether the latter is so considerable that it must be in part withdrawn by conduction, so as not to superheat the vapour. Two cases may, therefore, arise : (1) The heat of compression may be considerable, and the withdrawal of heat is necessary to maintain saturation at the higher temperature, i.e. hi is negative. (2) The heat of compres- sion may be too slight to prevent the compressed vapour.

DIFFERENT STATES OF AGGREGATION. 151

without the addition of heat, from becoming supersaturated. Then, li\ has a positive value. Between the two there is a limiting case {hi = 0), where the heat of compression is exactly sufficient to maintain saturation. In this case the curve of the saturated vapour coincides with that of adiabatic compression. Watt assumed this to be the case for steam.

It is now easy to calculate hi from the above formulae. Calling /i2 the corresponding specific heat of the liquid, we have

^ = T0+^^zr ' • • • (^^^>

During heating, the liquid is kept constantly under the pressure of its saturated vapour. Since the external pressure, unless it amount to many atmospheres, has no appreciable influence on the state of a liquid, the value of Ag practically coincides with that of the specific heat at constant pressure,

h={c„h (118)

Subtracting (117) from (116), we get

d{ui - W2) d{vi- V,) hi-h,- ^g + lh~^- -

But (110), differentiated with respect to Q, gives

fZL _ d {ui - n.2) d {vi - v^ ) .dpi

m - — m — ^ ^''~~do^ + ^''^ "''^cW ,./,,_ A, = ^^(^,^,)^\

or, by (118) and (111),

For saturated water vapour at 100°, we have, as above, dL

de

(c,,)2 = 1-03; '^' = - 0-71; L = 536; = 373;

152 THERMODYNAMICS,

whence

hy = 1-03 - 0-71 - Ifl = - 1-12.

Water vapour at 100° C. represents the first of the cases described above, i.e. saturated water vapour at lOO'^ is superheated by adiabatic compression. Conversely, satu- rated water vapour at 100^ becomes supersaturated by adiabatic expansion. The influence of the heat of compres- sion (or expansion) is greater than the influence of the increase (or decrease) of the density. Some other vapours behave in the opposite way.

§ 185. It may happen that, for a given value of 6, the values of Vi and v.2, which are fully determined by the equation (101), become equal. Then the two states which are in contact with one another are identical. Such a value of is called a critical iemj^erature of the substance. From a purely mathematical point of view, every substance must be supposed to have a critical temperature for each of the three combinations, solid-liquid, liquid-gas, gas-solid. This critical temperature, however, will not always be real. The critical temperature 6 and the critical volume Vi = v.2, fully determine the critical state. We may calculate it from the equations (101) by finding the condition that Vi — V2 should vanish. If we first assume Vi — V2 to be very small, Taylor's theorem then gives for any volume v, lying between Vi and V2,

P = 1"- + (1)/" - «.) + i(^(- - nf . . . (119) and therefore the first equation (101) becomes

and equation (102), by the integration of (119) with respect to V, gives

M'^i - '"■^) + 'J^J'^i'^i - ^2)' + 2^(a^^) («^i - ■y2)'=i>2(vi - v.2).

DIFFERENT STATES OF AGGREGATION. 153

The last two equations give, as the conditions of the critical state,

(E = «.

These conditions agree with those found in § 30. They are there geometrically illustrated by the curve of the critical isotherm. In the critical state the compressibility is infinite ; so are also the thermal coefficient of expansion and the specific heat at constant pressure ; the heat of vaporization is zero.

For all temperatures other than the critical one, the values of t'l and v.2 are different. On one side of the critical isotherm they have real, on the other imaginary values. In this latter case our solution of the problem of equilibrium no longer admits of a physical interpretation. Several reasons may be given for assuming that not only for evaporation but also for fusion in the case of many sub- stances there exists a real critical temperature at which the solid and liquid states are identical (§ 31 and § 191).

§ 186. Third Solution. — In the third place, we shall assume that in the conditions of equilibrium (98)

Vl < «'2 < V^.

We have then, without further simplification,

h-h = g J . . (120)

h- h =

These refer to a state in which the three states of aggrega- tion are simultaneously present. There are four equations, and these assign definite values to the four unknowns 0, Vi, V2, Vs. The coexistence of the three states of aggregation in equilibrium is, therefore, possible only at a definite tempe- rature, and with definite densities; therefore, also, at a

r54 THERMODYNAMICS.

definite pressure. We shall call this temperature the fundamental temperature, and the corresponding pressure the fundamental 'pressure of the substance. According to equations (120), the fundamental temperature is character- ized by the condition that at it the pressure of the saturated vapour is equal to the pressure of fusion. It necessarily follows, by addition of the last two equations, that this pressure is also equal to the pressure of sublimation.

After the fundamental temperature and pressure liave been found, the external conditions of § 166 —

Ml + M.2 + M3 = i\r )

Miwi + M2i*2 + M3M3 = TJ )

uniquely determine the masses of the three portions of the substance. The solution, however, can be interpreted physically only if Mi, Mg, and M3 are positive.

§ 187. Let us determine, e.g., the fundamental state of water. 0° C. is not its fundamental temperature, for at 0' C. the maximum vapour pressure of water is 4-62 mm., but the melting pressure of ice is 760 mm. Now, the latter decreases with rise of temperature, while the maximum vapour pres- sure increases. A coincidence of the two is, therefore, to be expected at a temperature somewhat higher than 0" C. According to equation (114), the melting point of ice rises by 00074° C. approximately, when the pressure is lowered from 760 mm. to 462 mm. The fundamental temperature of water is, then, approximately, 0*0074° C. At this temperature the maximum vapour pressure of water nearly coincides with the melting pressure of ice, and, therefore, also with the maximum vapour pressure of ice. The specific volumes of water in the three states are, therefore,

vi = 205,000 ; v.^ = 1 ; V3 = 1*09.

For all temperatures other than the fundamental tem- perature, the pressure of vaporization, of fusion, and of sublimation differ from one another.

DIFFERENT STATES OF AGGREGATION. 155

§ 188. We return once more to the intrinsic conditions of equilibrium (101) which hold for each of the three combinations of two states of aggregation. The pressure pi, and the specific volumes of the two portions of the substance, in each case depend only on the temperature, and are determined by (101). It is necessary, however, to distinguish whether the saturated vapour is in contact with the liquid or the solid, since in these two cases the functions which express the pressure and specific volume in terms of the temperature are quite different. The state of the saturated vapour is determined only when there is given, besides the temperature, the state of aggregation with which it is in contact, whether it is in contact with the liquid or solid. The same applies to the other two states of aggregation. If we henceforth use the suffixes 1, 2, 3, in this order, to refer to the gaseous, liquid, and solid states, we shall be obliged to use two of them when we refer to a portion of the substance in a state of saturation. The first of these will refer to the state of the portion considered, the second to that of the portion with which it is in con- tact. Both the symbols Vxi and ^13 thus denote the specific volume of the saturated vapour, V12, in contact with the liquid, and V^ in contact with the solid. Similarly v^^^, and V'lx, %i and ^32, represent the specific volumes of the liquid, and of the solid in a state of saturation. Each of these six quantities is a definite function of the temperature alone. The corresponding pressures are

Of vaporization. Of fusion. Of sublimation.

V\z = Vi\ Piz = Pm P31 = Pi3

These are also functions of the temperature alone. Only at the fundamental temperature do two of these pressures become equal, and therefore equal to the third. If we represent the relation between these three pressures and the temperature by three curves, the temperatures as abscissae and the pressures as ordinates, these curves will meet in one point, the fundamental point, also called the triple point.

1 56 THERM OD YNA MICS.

The inclination of the curves to the abscissa is given by the differential coefficients

^JP12 #^ ^i^i

We have, therefore, according to equations (111), ^£12 _ L12

m ~ e{vi - V2)'

'dS ~ i){v.2 - V3)'

#31 __ L31

dd ~ e(V3 - Vi)'

where v refers to the fundamental state, and, therefore, requires only one suffix. We can thus find the direction of each curve at the fundamental point if we know the heat of vaporization, of fusion, and of sublimation.

Let us compare, for example, the curve of vaporization, 2Ji-2, of water, with its curve of sublimation, jhs, near the fundamental point, 0-0074'' C. We have, in absolute units,

L12 = 604 X 419 X 10^ (heat of vaporization of water at 0-0074° C.) ;

L13 = - L31 = (80 + 604) X 419 X 10^ (heat of sublima- tion of ice at 0-0074° C.) ; vi = 205000 ; V2 = l; vs = 1-09 (§ 187) ; 6 = 273.

Hence

dpi2 _ 604 X 419 X 10^ X 760 _ ^ ooq

de ~ 273 X 205000 x 1013650 ~ '

dp^ _ 684 X 419 X 10^ X 760 _

dB ~ 273 X 205000 x 1013650 ~ '

in millimeters of mercury. The curve of the sublimation pressure ^hzt is steeper at the fundamental point than the curve of the vaporization pressure ih%- For temperatures above the fundamental one, therefore, ih^ > 2h2 ; for those below it, P12 > 2h3- Their difference is

#13 _ #12 _ ^(i>i3 - P12) _ f..^^

d:e dti ~ de'~ ~ ^^

DIFFERENT STATES OF AGGREGATION. 157

If, therefore, the maximum vapour pressure of water be measured above the fundamental point, and of ice below it, the curve of pressure will show an abrupt bend at the funda- mental point. This change of direction is measured by the discontinuity of the differential coefficient. At — 1° C, {dQ = — 1), we have, approximately,

P13 - Pn = - 0-045 ;

i.e. at — 1° C. the maximum vapour pressure of ice is 0'045 mm. less than that of water. This has been verified by experiment. The existence of a sharp bend in the curve, however, can only be inferred from theory.

§ 189. We have hitherto extended our investigation only to the different admissible solutions of the equations which express the intrinsic conditions of equilibrium, and have deduced from them the properties of the states of equilibrium to which they lead. We shall now consider the relative merit of these solutions, i.e. which of them represents the state of greatest stability. For this purpose we resume our original statement of the problem (§ 165), which is briefly as follows :

Given the total mass M, the total volume V, and the total energy U, it is required to find the state of most stable equi- librium, i.e. the state in which the total entropy of the system is an absolute maximum. Instead of V and U, however, it is

V

often more convenient to introduce v = ^, the mean specific

volume of the system, and u = ^, the mean specific energy

of the system.

We have found that the conditions of equilibrium admit, in general, of three kinds of solution, according as the system is split into 1, 2, or 3 states of aggregation. When we come to consider which of these three solutions deserves the preference in a given case, we must remember that the second and third can be interpreted physically only if the values of the masses, as given by the equations (103) and

158

THERMOD YNA MICS.

(121), are positive. This restricts the region of validity of these two solutions. AVe shall first establish this region of validity, and then prove that within its region the third solution is always preferable to the other two, and, similarly, the second is preferable to the first.

A geometrical representation may facilitate, a general survey of the problem. We shall take the mean specific

V . U

volume, ^ = iTij and the mean specific energy, u = , ,, of the

system as the rectangular co-ordinate axes. The value of M is here immaterial. Each point of this plane will.

Critical point ^^^ ^^"-^-f^^'b^

/ ^^-^ -"'"" ^^'>>c>

II

/ --'' .^^""""'^ X '^ ^^

3»

/.--'' ^ /

Oj

V.

»

^ y^ ^'

V

r^^/rf /

•1^

to

i\ ""V y""'"'

ll

flC

^ 1 \ \ ^ y

S

.0/ \ \ y

\ / » ^^'

,^

H

^ y 1 s

"<<5

\ 'jAsAs

Critical point "^ — ^ j^

<

/"§

A F Axis of mean specific rolume : v = -j.

M

■<%

Fig. 4.

then, represent definite values of u and v. Our problem is, therefore, to find the kind of stable equilibrium which will correspond to any given point in this plane.

§ 190. Let us consider the region of validity of the third

DIFFERENT STATES OF AGGREGATION. 159

solution. The values of the masses given by the equations (121) are

il 1 11 Mi:M2:M3:M= i; ^2^3:

|MW2%!

1 1 li 11 1 11

MW3M1! k<.WiW2

111

Vl V2 V3 Wl W2 %

(121a)

where Vi, ^2, v^^ iii, u^, %, refer, as hereafter, to the special values which these quantities assume in the fundamental state.

It is obvious from this that the values of Mi, M2, M3 can be simultaneously positive only when the point (v, u) lies within the triangle formed by the points (vi, Ui) (v^, u^) and (vs, Uq). The area of this triangle then represents the region of validity of the third solution, and may be called the fundamental triangle of the substance. In Fig. 4 this triangle is represented by (123). The diagram is based on a substance for which, as for water,

-^1 > ^3 > V2 and Ui > W2 > Us.

§ 191. We shall now consider the region of validity of the second solution contained in equations (101) and (103). These equations furnish three sets of values for the three possible combinations, and no preference can be given to any one of these. If we consider first the combination of liquid and vapour, the equations referred to become, under our present notation,

(122)

^12 = 021

lh2 = Ihi

W12 - 1/21 + Pn{vi2 - ^21) ,

U - hi = 0^^ J

M12 + M21 = M )

M12V12 + M21V21 = V = Mv . (123)

M12W12 + M2itt21 = U = Mw J

In order to determine the area within which the point {v, «) must lie so that M12 and M21 may both be positive, we shall find the limits of that area, i.e. the curves represented

i6o THERMODYNAMICS.

by the conditions M12 = 0, and M21 = 0. The introduction of the latter (no liquid mass) gives M12 = 31 and

y = Via, u = W12. (124)

Since ^12 and W12 are functions of a single variable, the conditions (124) restrict the point (v, u) to a curve, one of the limits of the region of validity. The curve passes through the vertex 1 of the fundamental triangle, because, at the liindameutal temperature, I'la = ^i, and 1112 = Ui. To follow the path of the curve it is necessary to find the

differential coefficient -j^. We have

du\9 /du\ , /du\ dO

dv 12 \d V )i2 \dOJi^v 12

The partial differential coefficients here refer to the independent variables and v. It follows from (80) and (24) that

By means of this equation the path of the curve (124) may be experimentally plotted by taking 6I12, or V12, or some other appropriate quantity as independent parameter.

Similarly, the condition M12 = (no vapour) gives another boundary of the region of validity, viz. the curve,

V = t'2i, U = U.2U

which passes through the vertex 2 of the fundamental triangle, and satisfies the differential equation

dii^ a /dp\ . , . dOi2

^, = M.p;2i~^'"+^'^^"^'

since B21 = 0i2 and 2hi = lh-2-

The two limiting curves, however, are merely branches of one curve, since they pass into one another at the critical point (fia = V21) without forming an angle or cusp at that

DIFFERENT STATES OF AGGREGATION. i6i

point, as a further discussion of the values of ,— and

dvi2

du'

dv.

—will show. We may, therefore, include the two branches

21

under the name of the vaiiorization curve. Every point (^i2j W12) of one branch has a corresponding i)oint {v2, W21) on the other, since these two represent the same temperature 012 = 02ij and the same pressure pi^ = p2i- This co-ordina- tion of points on the two branches is given by the equations (122), and has been indicated on our diagram (Fig. 4) by drawing some dotted lines joining corresponding points. In this sense the vertices 1 and 2 of the fundamental triangle are corresponding points, and the critical point is self-corresponding.

This vaporization curve bounds the region of validity of that part of the second solution which refers to liquid in contact with its vapour. Equation (123) makes it obvious that the region of validity lies within the concave side of the curve. The curve has not been produced beyond the vertices 1 and 2 of the fundamental triangle, because we shall see later, that the side 12 of that triangle bounds the area within which this solution gives stable equilibrium. There may be found, quite analogous to the vaporization curve, also a fusion curve the two branches of which are represented by

V = V23, U = ^23)

and V = v^,u = W32,

and a sublimation curve represented by

V = V31, U = «3i,

and V = Vi3, u = 1*13.

The former passes through the vertices 2 and 3, the latter through 3 and 1, of the fundamental triangle. The region of validity of the three parts of the second solution have been marked (12), (23), and (31), respectively, in Fig. 4. The relations which have been specially deduced for the area (12) apply to (23) and (31) as well, only with a corresponding

M

1 62 THERMODYNAMICS.

interchange of the suflSxes, Some pairs of corresponding points have again been joined by dotted lines. On the fusion curve a critical point has been marked, on the assumption that, with falling temperature, the latent heat of ice decreases by 0-6-4 calorie per degree Centigrade (§ 183). If, as a rough approximation, we assume this same ratio to hold for much lower temperatures, the latent heat of fusion would be zero at about —120" C, and this would be the critical point of the fusion curve. The pressure here would be about 17,000 atmospheres, and water and ice would become identical. AVe might imagine this to be the result of a considerable increase in the viscosity of water and in the plasticity of ice, as they both approach this state.

§ 192. Having thus fixed the region of validity for the second solution, we find that for all points {y, u) outside this region only the first solution admits of physical interpretation. It follows that for such points the first solution re2)resents the stable equilibrium. The areas where such is the case have been marked (1), (2), and (3) in our figure, to signify the gaseous, liquid, and solid states respectively.

§ 193. We have now to consider the following question : Which of the different states of equilibrium, that may correspond to given values M, v, u (or to a given point of the figure), gives to the system the greatest value of the entropy ? Since each of the three solutions discussed leads to a definite state of the system, we have for each given system (M, r, ?<) as many values of the entropy as there are solutions applying to it. Denoting these by O, $', and 4»", we get for the first solution

$=:M0 (125)

for the second

$' = 31^' = Mi2^i2 + Mai^ai • • (126) terchange of the suffixes 1, 2, 3) ; for tl

a>" = M.^" = Mi.^1 + M2f2 + M3^3 . . (127)

(or a cyclic interchange of the suffixes 1, 2, 3) ; for the third

DIFFERENT STATES OF AGGREGATION. 163

All these quantities are fully determined by the given values of M, v, and u. Now, we can show that for any system (M, y, u) we have <I>" ><!>'> $, or «^" >((>'> (f), provided all the partial masses are positive. It is more convenient to deal with the mean specific entropies than with the en- tropies themselves, because the former, being functions of v and w alone, are quite independent of M.

As a geometrical representation, we may imagine, on the plane of our figure (Fig. 4), perpendiculars erected at each point (y, u), proportional in length to the values of (p, <p', and «/)" respectively, at that point. The upper ends of these perpendiculars will generate the three surfaces of entropy, (^, <p', and 0".

§ 194. We shall show that (/»' — </> is always positive, i.e. that the surface of entropy, </>', lies everywhere above the surface ^.

While the value of (^ may be taken directly from (61), which contains the definition of the entropy for homogeneous substances, ^' may be found from (126), (122), and (123), in terms of v and u. The surface ^' forms three sheets corre- sponding to the three combinations of two states of aggre- gation. We shall in the following refer to the combination of vapour and liquid.

With regard to the relative position of the surfaces <^ and ^', it is obvious that they have one curve in common, the projection of which is the vaporization curve. At any point on the vaporization curve we have v = Vi^, u = «i2> and for the first entropy surface, ^ = ^12 ; for the second we have, from (123),

M2i = 0,Mi2 = M (128)

and, from (126),

In fact, for all points of the vaporization curve, both solutions coincide. The curve of intersection of the surfaces «/> and <p' is represented by

V = V12, U = U12, (j) = <pl2>

1 64 THERMOD YNA MICS.

where v, u, and ^ are the three rectangular co-ordinates of a point in space, fia, U12, ^12 depend on a single variable parameter, for example, the temjDerature, 612 = B^i- This curve passes through the point (vi, Ui, <pi), which has the vertex 1 for its projection. A second branch of the same curve is given by the equations

V = V.21, U = U.21, (p = ^21,

and these branches meet in a point whose projection is the critical point. Each point of one branch has a corresponding point on the other, since both correspond to the same temperature, 612 = On, and the same pressure, 2^2 = Ihi- Thus, (t'l, «i, 0i) and {v^,, ih, ^2) are corresponding points.

It is further obvious that the surface ^' is a ruled surface and is developable. The first may be shown by considering any point with the co-ordinates

_ XVl2 + flVjl _ X^1 2 + At'Mai , _ A ^ 12 + fJl(p -2l ,

'"- X -f- /. ' " - X + fM ' '^- X + ^^ '

where X and fj. are arbitrary positive quantities. By giving A and fx all positive values, we obtaLn all points of the straight line joining the corresponding points (ri2, Ui2, ^12) and (v2i> ^*21j 02i)- But this line lies on the surface 0', since all the above values of {v, u, <p) satisfy the equations (123) and (126) if we put M12 = A and M21 = fx. The sur- face (j)', then, is formed of the lines joining the corresponding points on the curve in which the surfaces <p' and <p meet. One of these is the line joining the points (vi, ui, (pi) and {v2, u-2, i>i), the projection of which is the side 12 of the fundamental triangle. At the critical point, the line shrinks to a point, and here the surface ^' ends. The other two sheets of the surface are quite similar. One begins at the line joining {v2, U2, ^2) and (^3, u^, ^3), the other at the line joining {v^, u-s, ^3) and {vi, «i, ^1).

The developability of the surface <p' may best be inferred from the following equation of a plane : —

2h2{v - Via) -I- (u - H12) - Oi2i(p - ^12) = 0,

DIFFERENT STATES OF AGGREGATION. 165

where v, u, <^ are variable co-ordinates, while pi2, Via, W12, 012, «^i2j depend, by (122), on one parameter, e.g. On. This plane contains the point (1^12, U12, (^12), and by the equations (122) the point (^21, u^i, 021), which are cor- responding points, and hence also the line joining them. But it also, by (61), contains the neighbouring corresponding points

(vi2 + dvi2, W12 + dui2y 012 + ^012) and

(^21 + ^^21, U21 + Chl2i, 021 + d(fi2i)

hence also the line joining them. Therefore, two consecutive generating lines are coplanar, which is the condition of developability of a surface.

In order to determine the value of <p' — 0, we shall find the change which this difference undergoes on passing from a point (v, u) to a neighbouring one (v + Sv, u + ^u). During this passage we shall keep M = M12 4- M21 constant. This does not affect the generality of the result, since </> and 0' are functions of v and u only. From (126) we have

Mg0' = M128012 + M2ig02l + 012?Mi2 + 02lgM2i,

and, by (61),

. ^^ = — e —

But, by (123),

gMi2 + gMsi = ) Mi28«;i2 + M218V21 + ^i2SMi2 + V21SM21 = Mgv (129) M128W12 + M21SW21 + W128M12 -}- W21SM21 = Mg« )

Whence, by (122),

g , ^ du+p,^ (130)

"12

.„d «(■-)=(;^->+(«^«>■ ^''''>

i66 THERMODYNAMICS.

If we now examine the surfaces ^ and 0' in the neigh- bourhood of their curve of contact, it is evident from the last equation that they touch one another along the whole of this curve. For, at any point of the vaporization curve, we have v = Vi^ and u = Ui2 ; therefore also

6 = Oio, and 2J = 2h2 • • • • (132)

and hence, for the entire curve, S(^' — 0) = 0.

To find the kind of contact between the two surfaces, we form B^' — (f) from (131), and apply it to the same points of tlie curve of contact. In general.

According to (132) we have, at the points of contact of the surfaces,

02g2(<^' _ 0) = hl{^e- g0i2) + ^^'(%12 - % - P^0i2 + p^o),

or, by (Gl),

0g2(^' _ ^) = (w - Bei2)H + i^pi2 - h')^v . (133)

All these variations may bo expressed in terms of ^d and Bv, by putting

i^^'pe + f^iv (by 81),

We have now to express ^0i2 in terms of W and Sv. Equations (129), here simplified by (128), give

Swi2 — Sw _ Svia — ^v

Ul2 — W2I '^^12 — '^21

DIFFERENT STATES OF AGGREGATION. 167 In these we put

SWi2 = ^^^^12, SVl2 = ^^^^^12 • (134)

av

11*- Sfl vat; i;i2 - v^^J

and obtain cWia = , j

criti2 _ -?ti2 - M2 I «i^

^(?12 '«^12 - '^21 f^^l2

If we consider that, by (109),

= "12 7/1 ~ i^l2 .... l,A'''->;

V12 - '2^21 «C>12

that, by (80), dv = ^de^i'

and that

fZwi2 _ /3w\ /5?(\ fZvi2

V56>A2 \c

also that ^^^^ = m-^dv-M:

we obtain S0i2 =

dv ddi2

dvKdOi

Equation (133), with all variations expressed in terms of ^6 and Sv, finally becomes

dp C„ \aU12 ^

" ^dvKdeJ

This expression is essentially positive, since c„ is positive

1 68 THE R MOD YNA MICS.

on account of its physical meaning, and -^ is negative for any state of equilibrium (§ J.69). There is a limiting case,

when ^^IQ -lv = (),

for, then, ^<p' - 0) = 0.

In this case the variation (SO, Si') obviously takes place along the curve of contact (O12, t'la) of the surfaces. Here we know that (p' = <}).

It follows that the surface (p', in the vicinity of all points of contact with 0, rises above the latter throughout, or that ^' — is everywhere >0. This proves that the second solution of the conditions of equilibrium, within its region of validity, i.e. in the areas (12), (23), and (31), always re- presents the stable equilibrium.

§ 195. Similarly, it may be shown that the third solution, within its region of validity, is preferable to the second one. The quantities r and v being given, the value of the mean specific entropy, <j)", corresponding to this solution is uniquely determined by the equations (127) and (121). The quantities Vi, v.2, '^3, '?'i, ih, ^'3> aiid therefore also ^1, ^2» 03> have definite numerical values, given by equations (120).

In the first place, it is obvious that the surface 0" is the plane triangle formed by the points {vi, Wi, «/»i), (v^, 1(2, ^2), and (v^, t*3, ^3), the projection of which on the plane of the figure is the fundamental triangle, since any point with the co-ordinates

_ X^i + ju^a + Wa X + ft + 1'

X + jU + V

X^l 4- fl<l>2 + V<b3

^= X + ;« + v '

(X, /u, V may have any positive values) satisfies the equations (121) and (127). To show this, we need only put Mj = X,

u

DIFFERENT STATES OF AGGREGATION. 169

M2 = /i, M3 = V. This plane meets the three sheets of the developable surfa(3e ^' in the three lines joining the points (vi, wi, «^i), (^2, W2j ^2), (%, %», ^3). In fact, by making V = 0, i.e., by (121), M3 = 0, the third solution coincides with the second ; for, then,

Ml = M12 ; M2 = M21 ; Vi = V12 ; Ui = W12 v-2 = ^21 ; 01 = 012 ; etc.

'} (l^-^^)

If we also put fi = 0, then we have M2 = 0, Vi = v, ui = u, which means the coincidence of all three surfaces, <})", (f>', and (p.

In order to find the sign of rp" — f', we again find S(^" — (f)') in terms of 8m and Sv. Equation (127) gives

Mg(^" = .^iMi + 02SM2 + .^3gM3. . (138)

where, by (121),

nil + 8M2 + 8M3 =

W18M1 + W2gM2 + WggMa = M8w

Multiplying the last of these by v^, the second by 4^, and adding to (138), we obtain, with the help of (120),

80 =—01

This, in combination with (130), gives

if the surface 0' is represented by the sheet (12). This equation shows that the surface 0" is a tangent to the sheet (12) along the line joining (vi, Ui, ^i) and (v^, u^, 02)> for all points of this line have 0i = 0i2, pi — pvi, so that 8(0" — </)') vanishes. Thus, we find that the plane 0" is a tangent plane to the three sheets of the surface 0'. The curves of contact are the three straight lines which form

1 70 THE R MOD YNA MFCS.

the sides of the plane triangle ^". We have, from (139), for any point of contact