660 



Equilibria of n components in n phases, between which a phase- 

 reaction may occur, at variable T and P. The turning-line Er. 



For the equilibrium E= F, -j- E"^ . . . -\- Fu the equations (2) [XVII] 

 and (3) [XVII] are true. When the equilibrium E passes into an 

 equilibrium Er, then r, v, • • • «^'^ //, • • ■ etc. have also to satisfy (13) 

 [XVII]. From those latter equations which we once more mention 

 here sub (2). 



K-\-K^ + >-« = j 



^1 .Vi + ^, .V, + • • • + ^'" .V" = I 



follows one single relation between the variables j.\y^ . . .x^j/, . .; 

 we are able to find this relation by eliminating from (2) ?.^ . . . A„. 

 We shall call this relation, resulting from (2), which we may write 

 also in the form of a determinant, equation (2). 



Now we have n' -}- 1 equations and n'* -\- 2 variables; conse- 

 quently the equilibrium Er is monovariant; it is represented, 

 therefore, in the P, T-diagram by a curve, e.g. curve e f in fig. 2 

 (XVI) and fig. 4 (XVI). 



From (2) [XVII] now follow the n equations (7) [XVII], they 

 are of the form : 



_ F.LP 4- H,LT+z', [d{x\ -f . . .] + y, [cl{y), -f . . .] -f. . . 

 . . . -\- i d'Z, + i d'Z, -\- . . . = — AK 



From (3) [XVII] follow the n {n—1) equations (8) [XVII]; the\ 

 are of the form : 



d{x), + i d'-(x\ + . . . = d(x), + i d\x\ -\- ... = ... = LK^ 

 d{y\ + i d'iu\ + • • • = d{y\ + i d'{y\ -\- ... = ... = LKy 

 etc. When we differentiate the equation resulting from (2), then we 

 obtain another relation between Lx, Ly^ . . . Lx^ Ly^ . . . For our 

 purpose we may find this in the following way. It follows viz. from (2) 



Lx, -f A/, -f . . . + a;,. = j 



X, LX, -f X, a;, + . . . + Xn A;.„ + ;., Lx, + . . . + An A^„ = . (5) 



y, AA, -}- y, A^, + . . . + .V» ^'^-n + K ^.Vi + • • • + ^" ^y- = 0' 



etc. As relation (2) exists between .r, ,«/,..., vve may eliminate 



AX^ . . . AX„ from (5). For this we add the ?i equations (5) after 



having multiplied the 1^' by (i,, the 2"^ by (u,, etc. Now we may 



put : 



/^i -h f^, -2^1 -h M, 2/1 + • • • = j 



fi, 4- fi, '^i + f^, .V» + ■ •• = ^ i 



(3) 



(4) 



(16) 



etc. Then we have n relations between the n — 1 ratio's Mi • ■ • M" 



