42 On the Theory of Chemical Potential. 



be negative. In general this choice of an imaginary com- 

 pound as one of the components of the system does not result 

 in any simplification. In the case of the three two-phase 

 systems we have been considering, however, Ave can make a 

 great simplification in this way. This is due to the fact that 

 in these systems, the only possible virtual modification is the 

 passage of a certain mass of the solvent from one part of 

 the system to the other. Now this modification does not 

 change the relative proportions of the various solutes, so that 

 if we choose for the compound S a mixture of S 1? S 2 , ... S» 

 in the proportions in which they exist in one particular state 

 of the system, so as to make the concentrations of the re- 

 maining (n — 1) solutes zero in that state, these concentrations 

 will remain zero throughout all modifications of the system. 

 Hence the formulae we have obtained for a binary system 

 will apply to a solution containing any number of involatile 

 solutes. 



Many of these formulae involve differentiation with respect 

 to the concentration. We must, therefore, find out what is 

 meant by differentiation with respect to the concentration of 

 the imaginary compound. The concentration s of the com- 

 pound is related to the concentrations of the separate solutes 

 by the equation 



s = s 1 + s 2 -r ... +5 H . 



In order to obtain a solution of concentration s + Ss it is 

 necessary to increase the concentrations of s 1? s 2 , ... , s n by 

 the respective amounts 



8si = st — (i=l, 2, ... ,7l). 



s 



Hence if V be any function of the composition of the solution 

 we have 



so that 



e=l OSi S i=1 OSi 



TsY i=n ?*V 



OS j = i QSi 



If we compare the equations giving the value of P , X, and 

 /(/ for a binary solution with the corresponding equations in 

 the present section, we see that this simple method of 

 generalizing the equations leads to results in agreement with 

 those obtained previously. 



The University, Leeds, 

 Sept. 3, 1912. 



