214 J. W. Gibbs — Equilibriiini of Heterogeneous Substances. 



The preceding fundamental equations all apply to gases of constant 

 composition, for which the matter is entirely determined by a single 



Now if a unit of the gas unites with the quantity /i of the first solid to form the 

 quantity \ + l of the second it will be necessary for equilibrium (see pages 121, 122) 



that 



^3+A^, =(1 +A)//.,. 



Substituting the values of /i,, //._,, //;j, given above, we obtain after arranging the 

 terms and dividing by at 



when 



loo; — = A- Bios t + D — , 



^ a ^ t I ' 



_ H+AH' - {l+l)H"-c-a-lk' + {l + 'A)k' 



„ (WA)Jc"-lk'-c~a 

 a 



E+lE'-{\+l)E" (1+A)F"-AF' 



— , L) — ■ . 



a a 



We may conclude from this that an equation of the same form may be applied to 

 an ideal gas in equilibrium with a liquid of wliich it forms an independently variable 

 component, wlien the specific heat and density of the liquid are entirely determined 

 by its composition, except that the letters A. B, C, and D must in this case be under- 

 stood to denote quantities which vary with the composition of the liquid. But to 

 consider the case more in detail, we have for the liquid by (a) 



- - —u=ki-kt loo; t-H't+ Vp + E', 

 m 



where k, H% V, E' denote quantities which depend only upon the composition of the 

 liquid. Hence, we may write 



C = bt-kt log (~m + V]) + E, 



where k, H, V, and B denote functions of m^, rwj, etc. (the quantities of the several 



components of the liquid). Hence, by (92), 



dk ^ dk , dH. dV dE 



// , = ^T—t- -—t log If — -- — 1+ -^—2}+ 1— . 

 am I dm, dm-i dm, dm, 



If the component to which this potential relates is that which also forms the gas, we 

 shall have by (269) 



•p H—c—a c + a, /"j— -^ 



log ^ = + log t+'-^-- . 



a a a at 



Eliminating /^ i , we obtain the equation 



log^=^-51ogi-y+ i>-y-, 



in which A, B, C, and D denote quantities which depend only upon the composition 



of the liquid, viz : 



\ I d-H. rfk 



A = —[ H- c-a-\-^,— 



a \ dm, dm,. 



B=L(^-c-a) 



a \dm, )' 



