Chemical Potential to the Theory of Solutions. 935 



The Theory of Chemical Potential in a Binary System. 



Before considering the applications of the Theory of 

 Chemical Potential, we will give a brief account of Duhem's * 

 development of the theory in a biliary system. 



If the external generalized forces acting on a system are 

 constant, the condition for equilibrium may be expressed in 

 the following manner : — 



Let U denote the internal energy of the system, 6 the 

 absolute temperature, (f) the entropy, and -yfr the potential of 

 the external generalized forces (i. e. the sum of the products 

 of the forces and the corresponding generalized coordinates), 

 and let 



The necessary and sufficient condition for equilibrium is that 

 in any virtual isothermal modification of the system 



S<£>0. 



The function <X> is called by Duhem the Total Ther mo- 

 dynamical Potential of the system. 



Consider a homogeneous system containing two components, 

 S , which will be called the solvent, and S 1? which will be 

 called the solute. Let the masses of these components be 

 M and M t respectively. Let the volume of the system be 

 W, and suppose it to be subjected to a uniform pressure p. 

 Then we have 



<£ = U-i9(/HyvV. 



If we consider $ as a function of M , M 1? p, 6 it will be 

 homogeneous of the first degree in M and M l5 and can there- 

 fore be written in the form 



^M + |^M, 



Now ^p and ^-^ will be homogeneous functions of M 



oM 0-Sl\ 



and Mj of zero degree, and can therefore be written in 

 the forms / (s, p, 6) and f^s, p, 0) respectively, where 



M 



s —tuT' Since evidently 



it can easily be proved that 



~ds ~ds ' 



* La Micanique Chimique. vol. iii. pp. 1-10. 



3 Q 2 



