States, and the Thermodynamic State-Equation. 157 



For investigating the two-phase conditions near the 

 critical point, for which £ = 0, these functions are expanded 

 in powers of t. The first three terms of these expansions 

 are given in the Table, in small print, below the respective 

 functions. 



Finally, we reiterate, for comparison, the reduced equations 

 used : — 



6 1 



van der Waals : 7r= -t — r — -70 • 



Berthelot : it — — -w-rs . 



<p — 1 u<p~ 



CM 1 



(JJausius : it — 



r 



<£-l 0(4> + x ) 2 ' 

 To each of these equation?, written for the two values 

 and v 2 , the equation 



is added, the integration being carried out with it given as 

 function of <f> at constant temperature, by the respective 

 equations. 



The solutions, in terms of the parameter 



« = (&-*»)/(&- 1)> 



are then as follows (p. 156). 



The equations of van der Waals and Berthelot give the 

 same expressions for <£,, (f>. 2 , 7^/0, 7t$ 2 /#. Clausius' 

 equation gives for it and 6 the expressions given by Berthelot' s 

 equation, divided by (1 + X)* and (1 + X)* respectively. 

 These two equations give the same expressions for dlp/dVE, 

 M?*/RT. It is interesting to note that the relation of the 

 latter quantities as calculated from Clausius' equation does 

 not involve the specific constant \ ; this 4-eonstant equation 

 in p, v, T gives an exact correspondence of states with 

 respect to dlpjdlT and Mr/RT. 



For purposes of reference, the values of the different 

 quantities, as obtained from van der AVaals' and Berthelot's 

 equations, have been calculated for values of t ranging from 

 to 10 7 , and are given as common logarithms i n the 

 following Table * (p. 1G0). Corresponding states in the two- 

 phase system are of course defined by equal values of the 

 parameter t. 



* The calculation is made with the slide-rule, and there is a slight 

 uncertainty in the last place of decimals ; for any likely applications 

 these values will, however, be sufficientlv accurate. 



