﻿624 Mr. Bernard Cavanagh on 



where 



and 



^ = (6l + ^l), ^'=[^2 + ^1+^3 + ^3], ") /g 4 x 



V = (*rW + \e 2 + 1 2 ), t; = [e Y r 3 + ^(^ + W) + ^1 + i^ + l J ' 

 The Gibbs fundamental relation can readily be applied at 

 this point as a check on the detail. 



A Special Procedure, sometimes necessary. 



A difficulty arises, however, in the practical application of 

 this result, as it stands, at high concentrations. 



We have seen that the series obtained for a s may be 

 expected, up to sufficiently high concentrations, to " converge 

 practically " in the few terms given, and the same clearly 

 applies to the ^-series of (54), (55), and (56). 



In passing to " experimental ; ' concentrations, however, we 



M ' M / 



have introduced the r- and Z-series for -=^- and log^p- 



respectively, which, when a and perhaps also ft assume 

 large values, will frequently fail at high concentrations to 

 " converge practically " in so few terms. A glance at the 

 values of the r- and /-coefficients will show this, and also that 

 the difficulty would vanish if « and ft were small, since 

 7 and 8 occur by themselves only in the first power. 



Now since (62) can be used at lower concentrations (with 

 the t and t' series cut down to two terms, see below) to 

 determine a and ft experimentally, we are at liberty to assume 

 on turning our attention to the higher concentrations that a 

 close estimate of a and at least a rough estimate of ft can 

 be made, though neither, of course, will be accurate enough 

 for these higher concentrations. Our difficulty can then be 

 overcome by the following procedure. 



Let u ', ft', y', 8' be estimated values of ™ , ft, 7, 8. (The 

 procedure is stated in a general form — if 7 and 8 are not 

 estimated 7' and 8' will be zero.) Then there will be an 

 estimated value (M ") of M , and "estimated" concentrations 

 (c s ", C") referred to it, such that 



Cj T _ (T _ W 

 cj ~ C " Mo" 



= [l + *o'C" + ftV" 2 + v'C' / *+'8 , C"*'}. . . (65) 



