﻿Molecular Thermodynamics. 631 



Ranges of Applicability of Approximations. 



Our four-term expansion of <x s leads to (62) or (76) or 

 (80) which may be called a " fifth-approximation " and 

 from which, by curtailing the series one term at a time, 

 the fourth, third, second, and first approximations are 

 obtained. 



Owing to the diversity of possible magnitudes of the 

 quantities a So , a , /3 S , etc., precise statement of the ranges 

 of applicability of these successive approximations is not 

 possible, particularly in the case of the higher ones, but some 

 indication of probable range of the lower, for aqueous 

 solution, may be given. 



The third (as far as t 2 ) will commonly suffice as far as 

 Molar (M.) or 2M. solution (but will often go further), the 

 second (as far as t{) probably up to iM. or M. (sometimes 

 further), while the first in which the pseudo-general terms J 

 are omitted altogether cannot be assumed to be accurate to 



/ M \ 

 0*2 per cent, above hundredth-molar I -r-^r ) concentration. 



When hydrates higher than the decahydrate are not im- 

 probable, the upper limit for the first approximation must 

 be set still lower. 



Examples of Application. 



Detailed application of the results of these papers to 

 existing data must be postponed, but it may be of interest 

 to cite one or two of the simpler instances. Consider, first, a 

 perfect solution (Gr vanishing) of middle concentration, for 

 which our " second approximation" will suffice, so that 



||-=^ I + RC[l + < 1 C] (88) 



If P be the osmotic pressure, defined and measured so 

 that the pressure on the pure solvent is relatively negligible, 

 and ft the coefficient of compressibility of the pure solvent, 



then writing (with van't Hoff) V for — ~- and considering 



the smallness of t\G relative to unity , we easily obtain 



p[v -^ + |^ET\]=RT, . . . (89) 



where, if we can neglect the effect of the pressures used on 

 the solvation of the solute, the quantity in round brackets is 

 a constant at constant temperature. 



