312 



homogeneous functions of the first degree with regard ton, and n^, 

 we have v =: 7i^v^ -\- n,Vf. And further according to (a): 



w = n, n. 



+ ". :r^-,r +".b^-r + 



or also 



t(;=njW, — f ƒ> H i rij Aü, + p H hi, Av„ (1 ^) 



which expression will at once appear to be useful. 



Here is n^ — v^'^^Av^ and v^ — y/= Av, and evidently we have 

 Av = v—Vo ^= ("i^i + n,Vj) — {n^v^" -\- n^v,") = n^Av^ -f 7i,Ai\. 



For the differential heats ot mixing lu, = — - andtü=— - we now 



o/ii oil J 



have from (1«)'): 



J., -M V-^/ V "i^ 



or as 



d /n,\ u — njVj n,v. 



dn, V V / w' u' 



^«, = < , + ƒ> H ^^1 i 



Likewise \ . . . (2) 



_ , (V, 1/^1—^1 Va^y 



a. 





1) In these difTerentiations many parts have not been taken into account. For 

 in general v^ and Vg ^^^ still functions of n^ and w,. But as the neglected parts 



in 2<;i and mi can always be represented by z^ = ^ — and z^ ~ 3 — , in which s, 



just as w, will always be a homogeneous function of the first degree with respect 

 to rii and «3, necessarily WjSi + w-j^i will have to be = 0, n^Wy -\- n^w^ already 

 being —w according to (2). Now also n-^Zi -\- ïic^Zc^ = z^ hence z is identically = 0, 

 hence also Sj and Sg. 



It would indeed not be difficult to show directly the disappearance of the parts 

 2i and S.2, which have been left out of account. As to z^, we get the result: 



1 /^ «1 , «A / ^«1 , övA 



in which the last factor will disappear in conseqence of ^ — = 3— , as Vj is a 



homogeneous function of the Ot'i degree with respect to the molecular numbers 

 rii and «3. 



