./ W. Gibbs — Equilibrmm of Heterogeneous Substances. 221 



masses by a diaphragm which is permeable to the liquid. It will 

 tlien be easy to maintain the liquid at any constant pressure which is 

 not greater than that in the gas. The potential in the liquid for the 

 substance which it yields as gas will then remain constant, and there- 

 fore the potential for the same substance in the gas and the density 

 of this substance in the gas and the part of the gaseous pressure 

 due to it will not be affected by the other components of the gas. 



But when the gas and liquid meet under ordinary circumstances, 

 i. e., in a free plane s\irfi\ce, the pressure in both is necessarily the 

 same, as also the value of the potential for any common component 

 aSj. Let us suppose the density of an insoluble component of the gas 

 to vary, while the composition of the liquid and the temperature 

 remain unchanged. If we denote the increments of pressure and of 

 the potential for S^ by djj and c?/<j, we shall have by (272) 



\dp lt,m \dmjt,p,m 



the index (l) denoting that the expressions to which it is affixed refer 

 to the liquid. (Expressions without such an index will refer to the 

 gas alone or to the gas and liquid in common.) Again, since the gas 

 is an ideal gas-mixture, the relation between p^ and /u^ is the same 

 as if the component aS'j existed by itself at the same temperature, 

 and therefore by (268) 



(/// J = a J t d log p^. 

 Therefore 



(dv \^^^ 

 - — I dp. (285) 



dmjt,p,m ^ 



This may be integrated at once if we regard the differential coeffi- 

 cient in the second member as constant, which will be a very close 

 approximation. We may obtain a result more simple, but not quite 

 so accurate, if we write the equation in the form 



-^ dp, (286) 



dm ^ ft, 2), m 



where ;/j denotes the density of the component /S^ in the gas, and 

 integrate regarding this quantity also as constant. This wdll give 



where jt?/ and p/ denote the values of ^j and p -when the insoluble 

 component of the gas is entirely wanting. It will be observed that 

 p—p' is nearly equal to the pressure of the insoluble component, in 

 the phase of the gas-mixture to which pi relates. /S', is not neces- 



