12 PROCEEDINGS OF THE AMERICAN ACADEMY. 



If we consider the special case of a system whose volume is unchanged 

 by the reaction to which the equilibrium equation refers, then V = 0, 

 and 



dlnK f 1 8F \ _ U 



/>T O (J 

 "' „ — - d T — H, was shown to be con- 



stant under the conditions which led to equations (10) and (12), when 



the conditions of equilibrium changed at constant temperature. In such 



?>F 

 cases, therefore, —-j — p = 0, and the equation becomes 

 o In A 



d\nK _ U 

 dT ~R^'' ^'^^ 



This equation applies to both homogeneous and heterogeneous equilib- 

 rium. When applied to the former it is identical with the well known 

 equation of van't HofF, sometimes called the equation "isochore." This 

 equation, however, has been used to express the change of equilibrium 

 with the temperature, not merely in those systems in which the reaction 

 causes no change in volume, but in general. That this use is justifi- 

 able in the cases for which equation (20) was developed may be readily 

 shown. For, in systems subject to moderate pressure, the only consider- 

 able isothermal changes in volume are those of the gaseous phase. The 

 volume of the gases is, at a given pressure and temperature, proportional 

 to the total number of gram-molecules of the various gases present. If, 

 duriug the reaction to which the equation refers, there is a change of n 

 gram-molecules in the gaseous phase, then the total change in volume is, 



,, nRT PV 



V= p , or —^ = nR. 



IF 

 In the case under discussion, where ^ — v,=: 0, equation (18) may be 



written, 



d\x\K _ 

 dT 



P V 



Since — =- =. n R, a constant, 



dh^K _ U 



~d~T~ ~ Wr-' 



