436 J. W. Gibbs — Equilibrium of Heterogeneous Substances. 



ively. Hence, since the snrface receives the increment of entropy //g, 

 the total quantity of entropy will be increased by 



r r 



which by equation (580) is equal to 



\~di)p' 



Therefore, for the quantity of heat Q imparted to the surface, we 



shall have 



Q=^t(^\=-(-p-\. (593) 



^ \dtJ2) \d\ogt/p ^ ^ 



We must notice the diiFerence between this formula and (58V). In 

 (593) the quantity of heat Q is determined by the condition that the 

 temperature and pressures shall remain constant. In (587) these 

 conditions are equivalent and insufficient to determine the quantity 

 of heat. The additional condition by which Q is determined may be 

 most simply expressed by saying that the total volume must I'emain 

 constant. Again, the differential coefficient in (593) is defined by 

 considering p as constant ; in the differential coefficient in (587) p 

 cannot be considered as constant, and no condition is necessary to 

 give the expression a definite value. Yet, notwithstanding the differ- 

 ence of the two cases, it is qviite possible to give a single demonstra- 

 tion which shall be applicable to both. This may be done by con- 

 sidering a cycle of operations after the method employed by Sir 

 William Thomson, who first pointed out these relations.* 



The diminution of volume (per unit of surface formed) will be 



and the work done (per unit of surface formed) by the external 

 bodies which maintain the pressure constant will be 



-=-(|) = -(^|.),- <-> 



Compare equation (592). 



The values of Q and W may also be expressed in terms of quanti- 

 ties relating to the ordinary components. By substitution in (593) 

 and (595) of the values of the differential coefficients which are given 

 by (581), we obtain 



* See Proc. Roy. Soc, vol. ix, p. 255, (June, 1858) ; or Phil. Mag., Ser. 4, vol. xvii, 

 p. 61. 



