8 PROCEEDINGS OP THE AMERICAN ACADEMY. 



d cl de 



riii-n-iore = ; moreover. - 0. This is experimentally 



</ v dv 



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

 as follows. 



de, " ,i r d»« ,/,■ 



(9) 



dv do dvd T d T ' 



now, j f L^^ 



-7— = 0; -—— = 0; and ~ = 0. 

 dv d T d v 



Equation (8 1 becomes 



/*' T m d% 



7? V 1 



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



v 



the perfect gas and the dilute solution. ; • = 0. 



d <• 



In these two simple cases, therefore, we find that ZX- e„ li), arc all 

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

 between gases and dilute solutions, the quantities til,. c M . Ii), ; ?!._../• . 

 Ii).,, etc. will not at any given temperature change with changing volume. 

 and the quantities U, C ri — (',.. If, will be constant, however the ini- 

 tial and final volume conditions of the 83 stem are changed. For any one 

 temperature equation (G) may be written, 



V "■ r' "'= 



A=RT\n V. - + tf (a constant). (10) 



Px '»i 1 • • ■ 



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 Bystem. Thus 

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

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

 55, C„ li), when the pressure of the Bystem 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 Bystem. Let U8 consider a Bystem such as is ordi- 

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



