618 On the Osmotic Theory of Solutions. 



comparison with the osmotic pressure, an integration from 

 A to B gives, 



osmotic pressure at B = pressure of solution at B 



-Jyfr-nO.Wj/fci.dfc • (32) 



ill which the upper limit of the integral is the given concen- 

 tration at B, and the lower limit is the unknown to be found 

 from the equation. 



The values of tjPi/Ucs (subject as below to a small correction) 

 are obtained on differentiating the curve which expresses the 

 relation between concentration (in grams per gram of 

 solution) and the corresponding equilibrium osmotic pres- 

 sures ; for the equilibrium pressures are all measured with 

 the same pressure (one atmosphere) on the solvent, and 

 accordingly they fulfil the condition which we imply by the 

 notation tfPj/tog. But each fcPi/tt^ thus derived from the 

 curve corresponds to a definite hydrostatic pressure on a 

 solution of given concentration, and in the column the 

 pressure associated with that concentration will be different, 

 except just at the level in contact with B. 



To apply the correction we may first find p approximately 

 in terms of c 2 by integrating (31) without taking into account 

 the effect of pressure on the factors s 1 /(s 1 — w) and &Pi/ttc 2 *. 

 The values of p thus obtained are amply accurate for com- 

 puting the corrections required, since the whole pressure- 

 correction for the strongest solution here considered only 

 amounts to 0*8 per cent. 



For the more dilute solutions a simple approximate formula 

 suffices : namely 



osmotic pressure at B — pressure of solution at B 



= Oi]/([>i]-M)x[PiA- 2 ]xA<- 2 



= h]/([s.]-H)xAP„ • • (33) 



where the brackets [] indicate mean values and /\c 2 and 

 ^Px are the total changes in value in c 1 and P 1? respectively, 

 as we pass from the top to the bottom of the column. 



* The correction to the factor JjP L /tfc 2 is given by 



dCWVttc,,) ftp = - i/ Sl x (WyircJ x d vd/> - l/*i x d^/d^, 



an exact relation, which, is not difficult to verify. The correction to 

 sj(s r —w) involves extrapolation from Tait's results for the compres- 

 sibility of sugar solutions (see Landolt and Bernstein's Tables, 

 p. 62, 3rd Ed.). 



