phase, a i ~ dF,! is \v!int iiiaj- he called ihe chemical potential of the $11 compoiient i i i the 2-tli pliase. 8T121and 8H2, are the changes in tlie \,olunie aiid the eiitrop- of tlie sj-stem due to a virtnal cliaiige at coiistaiit teiiiperatiire a i d under constant pressnre, such that the iiiass of the first phase is increased by a small mass 6 X r n.hicli has been taken from the second pliase without changing the concentrations of that phase. Similarly, 6VIZand AHI2denote the changes i n tlie voliiine and the entropy of the .jystein due to a virtual cliaiige at coiistaiit teiiiperature and iiiider constant pressure, sucli that the iiiass of the second phase is iiicrea.sed by a s i n a ~ mass l 6112, x-liich has been taken from the first phase without changing the concentrations of that phase. If one of the phases of the bi\.ariaiit tem, say the second, 1ia1.e a fixed composition, then it can he slionn that equations I and II are to lie replaced 11~.a single eqiiatioii of the form I. Consider now a t tlie teniperatnre '1' and under the pressure I1 a uiii~-ariaiithiliar!. s!.steiii consisting of the phases I , 2 , 3. Consider also the bivai-inut s!.steiii foi-tned the phases I , 2 aiid the liivnriaiit system foriiietl 1 ) ~ -the phases, I. 3. Under tlie coiiataiit pi-essiii-e I1 the slope of tlie conceiitration ciir've of tlie phase I in the system of phases I , 2 is given 1)y the equation : ~
From equations I and 2 we find at once for the ratio of the slopes of the two concentration curves at their point of intersection :
T h i s equation may be put into a more suggestile form. For this purpose consider a reversible change, zit tlie temperature T and under the pressure II,which coniists in incren+ ing the mass of the phase I at the expense of the phases 2 and 3 If the masses of the phases be increased by 8111) 8 M 2 , ;>I7, \Y? have the following relations :
811,t 81127-8M3= 0 , 7/2118x1- ??2-18111,1-7?/,18>1,== 0 , 7?~,,8>1, - i / ~ ~ , 8 >TI ~?/~,,8llI~ == 0 . From the last two equations x e get
1
831, ?jL-,,
77222
l?lll,
711 32
I 1 -_ __
811,~. 741,
71/32
77Zlll
?12,,
1 =p;I 811,
1111.
f n-
If then we take &I2 and 6313 equal to the determinants ~ l i i c l i appear beneath them in these fractions, equation 3 takes tlie simple form
If we denote "y Q,,and Q3' the quantities o heat absorbed during the virtual changes to which 8H2: and 6H31 refer, \ r e have Q,, = T8H2,, Q,, = T8H,,
anti equation 3 takes tlie foriii
lye thiis olitaiii the folloxiiig theorem, which is dne to Le Chatelier : Coiisider a t tlie temperature 'r and under the 1)ressnre 11, a iuii\.arinnt binary system coiisistiiig of the p1i:uer I , 2, 3. Coiisider also, iiiicler the presstire IT3 tlie two li\-ariaiit hiiiai-J- s!xteins formed hy tlie pliases I , 2 aiitl I . 3. If the phase I i.; a phase of variable concentratioii, to each of these .\-stenis there will correspond a curve xi\.ing tlie relation betn-een tlie teuiperatiire and tlie coiiceiitratioii of the phase I . These tu-o c1iri.v'; intersect at a point ivliicli corresponds to t h e unii-ariant s\,stviii. '!?he ratio of tlie slopes of the coiiceiitratioii cnrves at that point is equal to the ne,q:ati\-e of tlie ratio of t h e qiiaiitities of heat :ilisorl~etldiiriiig- tiyo T.irtual cliaiiges nliicli ,+eparatel!. 1ear.e tlie coiiceiitratii)ns of the phases 2 and 3 1111chaiigetl, and Jyliicli taken tog-etlier coiistitute a i-eI.ersilile c1inng.e ior tlie 1111i T-ariaiit .s;!,iteiii. It n-ill lit. oliser\-ecl that tlie tleinotistratioii ashlimes tliat the phase I is a phase of \-arinlilc conceiitr:itiiin, but that no assumption is made concerning tlie phase.; 2 anti 3. In conclusion, it ilia!. lie observed tliat a demonstration eiitirely analogous to that u.
