32 Mr. S. A. Shorter : Application of the Theory 



The Action of Gravity on a Solution, 



In his well-known work " On the Equilibrium o£ Hetero- 

 geneous Substances/' Gibbs* deals with the problem o£ the 

 effect of gravity on a system containing any number of 

 components, and establishes the following general theorem: — 



" When a fluid mass is in equilibrium under the influence of 

 gravity, and has the same independently variable components 

 throughout, the intrinsic potentials f for the several components 

 are constant in any given level, and diminish uniformly as the 

 height increases, the difference of the values of the intrinsic 

 potentials for any component at two different levels being equal 

 to the work done by the force of gravity ivhen a unit of matter 

 falls from the higher to the lower level" 



We will first deduce by means of this theorem an ex- 

 pression for the concentration gradient in a binary fluid 

 mixture. Afterwards we will give an elementary proof of 

 the theorem in the case of a system containing two com- 

 ponents. Suppose that at a certain point in a solution the 

 gravitational potential is G, and the concentration, pressure, 

 and temperature are s, p, and 6 respectively. Suppose that 

 at another point where the gravitational potential is G + dG, 

 the concentration and pressure are s + ds and p-\-dp respec- 

 tively. Then, according to the above theorem, we have 



/ (* + ds, p + dp, 6) —f (s, p, 0) =dG, . . (1) 

 and 



f 1 (s + ds,p + dp,6)-f 1 (s, 1 J,0) = dG. . . (2) 



We will call these equations, which express Gibbs's theorem 

 mathematically, the " Gibbs equations."" They may be 

 written in the form 



S (5, p, 6)ds-\-¥ (s, p, 0)dp = dG, . . . (3) 

 SiO, p, 6)ds + T 1 (s,p, 6)dp = dG. ... (4) 

 Now Sq + sS^O, 



Po + sPx =(1 + *)*(*,#*), 



so that we have 



v(s, p, 6)dp = dG, (5) 



which is the ordinary equation of hydrostatic equilibrium. 



* Collected AVorks, vol. i. p. 144. 

 t I. e. chemical potentials. 



