of Mixed Solutions. 277 



ft)' is the thermodynamic potential per unit volume at the point 

 considered. The thermodynamic potential for the whole system 



(4) o3=\\\ (jd'dxdydz. 



ft)' being a homogeneous function of the first degree*, 



(5) ft)' =f,M,n, -^f,M,n, + . .JrMrUr 



= (/>oWo + ^1 Hi + . . . ^r^r- 



/,,yi ...y";. are the thermodynamic potentials per unit mass of 

 the various components. 



(j)fi,(j)i ... ^r are the thermodynamic potentials per gram- 

 molecule of the various components, and 



(6) ilf,/,= <^,= |^; 



(f)i being homogeneous of degree with regard to no, iii ... riy, 



s=r 



and since 



(7a) ^'^^\ = 0, 



5=0 07ls 



(75) d^^d^^ 



(7c) ¥^^.i, = 0. 



Let ft) be the thermodynamic potential for constant pressure. 

 Then 



(8a) -^ = (volume) = 1, 



or by means of (5) 



r 



(86) 2/AiWi = l, 







where fxi is the molecular volume of the component {i) under the 

 conditions present. Let hn^, Sti^ ... Sfiy be simultaneous variations 

 of 7?i, ^2 ... Ur. Then from (86) and (7a) 



r 



(9a) XfiiSrii = 0. 







The expression to the left in equation (86) is a function of 

 {xyz), and its total differential must be equal to zero, but as {xyz) 

 are independent variables, the partial differentials with regard to 



* See Duhern, Mecanique Chimique, tome iii. livre vi. 



