LEWIS. FKEE ENERGY AND EQUILIBRIUM. 15 



dv v'^ dp 



We may now write for equation (24), 



/i? „, A ^ RTdU 



(2G) 



v'^ dp 



Now, for reasons that will be obvious immediately, we will write with 



P 7? 



perfect generality in place of -^('Oj _ ^ . » where F(v) 



and f(v) are different functions of v. Then, 



ET ETdm 



^ V — fiv) v dip 



Now since the last term is independent of the temperature, — ; — R T 



dm. . 

 must be independent of the temperature. But smce — — is constant 



dm, 

 at constant temperature, R T —— is also independent of the volume. 



This quantity, therefore, is constant under all conditions to which the 



system is subjected. Let us write, — R T —^ — a, and we obtain, 



P = 77-^ - -2 ' (28) 



^ V —f{v) v^ 



This equation, which is identical with the familiar formula that van 

 der Waals has developed from purely kinetic considerations, is here 

 shown to be directly deducible from a general thermodynamic equation, 

 with the aid of two simple empirical observations, namely, the constancy 

 of the specific heat of gases at constant temperature, and the proportion- 

 ality between the cooling effect and the fall in pressure in the free 

 expansion of gases. 



Equation (28) does not define the nature of /(v), and in this respect 

 is less explicit than the corresponding term in the equation of van der 

 Waals, which is a constant, h. Nevertheless it must be borne in mind 

 that b, the volume correction in the formula of van der Waals, may only 

 be regarded as constant when the volume is larce, and that it also is in 

 reality an undetermined function of the volume. 



