of Chemical Potential to the Theory of Solutions. 33 



Eliminating dp between equations (3) and (5) we obtain the 

 equation 



da~ ~" s («, p, e) *■■•-« .w 



This equation is clue to Duhem *, who deduced it from first 

 principles. It has also been established as an approximate 

 result by Vegard f . It may be regarded as the fundamental 

 equation for the effect of gravity on a binary mixture. 



This equation may be established in the following elemen- 

 tary manner in the case of a solution of an involatile solute. 

 Suppose that the solution is separated from the solvent 

 vapour by a vertical partition pierced by a number of 

 horizontal capillary channels whose walls are not wetted by 

 the liquid. Suppose that one of the channels is at a level 

 where the gravitational potential, concentration, pressure in 

 the solution, and vapour-pressure are G, s, p, and P respec- 

 tively. The equilibrium equation is 



Ms,p,6)=v (p,e). 



Suppose that there is another channel at a level where the 

 above quantities are respectively Gr + dGr, s + ds, p + dp, and 

 P + dP. The equilibrium equation is 



f (s + ds,p + dp, 0)=F o (P + rfP, 6). 



Hence we have 



So(s,p, 0)ds + Y o (s,p, 0)dp = V(P, 0)dP. 



Now the equation of hydrostatic equilibrium of the vapour is 



V(P, 0)dP = dG, 

 so that we have 



S (s, p, 0)ds + Y (s, p, 6)dp — dQ. 



This is one of the Gibbs equations. The remaining Gibbs 

 equation and equation (6) may readily be deduced from this 

 equation and the equation of hydrostatic equilibrium of the 

 solution. 



Devices similar to the above have been extensively used 

 for establishing in an elementary manner various relations. 



* Journal de Physique, 2 e serie, vii. 1888, p. 391. This paper contains, 

 in a note at the end, a proof of equation (9) of Part I. of the present 

 work (connecting the vaponr pressures of two solutions with the pressures 

 under which they coexist in osmotic equilibrium). This important 

 result seems to have been overlooked by subsequent writers. The method 

 of proof adopted in the present work is rather simpler than that adopted 

 by Duhem. 

 * t Phil. Mag. vol. xiii. p. 589, May 1907. 



Phil. Mag. S. 6. Vol. 25. No. 145. Jan. 1913. D 



