8 PROCEEDINGS OF THE AMERICAN ACADEMY. 



dWi dc 



Therefore — — = ; moreover, -~ = 0. This is experimentally 



proved in the case of gases, and may be shown in the case of solutions 



as follows. 



,dWi dm 



dc^ ^ ^,/r ^ d'-m ^ " dv 

 dv do dvdT~ dT 



(9) 



now, ^'^^ 



dm ^ ~d^ _ , c?c, - 



— — = ; — ™- = ; and -^ =: 0. 

 dv dT dv 



Equation (8) becomes 



V dv 



R T 



But since we know that p = is the characteristic equation of 



V 



d'^ 



the perfect gas and the dilute solution, -^ = 0. 



In these two simple cases, therefore, we find that m, c^, P^, are all 

 independent of the volume. If then, equation (6) is applied to a reaction 

 between gases and dilute solutions, the quantities 5Ei, c,.j, P^i ; Wi,2i c^^, 

 ^-2, etc. will not at any given temperature change with changing volume, 

 and the quantities U, C^^ — O^^, H, will be constant, however the ini- 

 tial and final volume conditions of the system are changed. For any one 

 temperature equation (6) may be written, 



n J n\ 



A = RT\xi V^^,>"' ' ' + G (a constant). (10) 



Vi 'v J 'i . . . 



This equation, moreover, obviously applies to any system which con- 

 tains, besides gases and dilute solutions, any constituents participating in 

 the reaction, whose molecular volume is not changed appreciably by a 

 change . in the conditions of volume or pressure in the system. Thus 

 any solid phase of definite constitution in a heterogeneous system may be 

 considered constant in its molecular volume, as well as in its functions 

 5E, C„, f^, when the pressure of the system is varied through limits not 

 too wide. 



Since equation (6) gives an expression for the change of free energy 

 in any isothermal process, we may derive immediately a general equation 

 for equilibrium in any system. Let us consider a system such as is ordi- 

 narily studied, upon which the only external force is a uniform pressure, 



