New integration of the Gibbs-Duhem equation and thermodynamics of Pr-Zn alloys
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[1]
w h e r e X is the mole f r a c t i o n of component 1 and Yi (i = 1, 2) is any p a r t i a l m o l a r , r e l a t i v e p a r t i a l m o l a r , or e x c e s s p a r t i a l m o l a r quantity. If the c o m p o s i t i o n d e p e n d e n c e at c o n s t a n t t e m p e r a t u r e of Y~ is m e a s u r e d or known, the c o r r e s p o n d i n g quantity Yz m a y be o b t a i n e d by i n t e g r a t i o n of Eq. [1]. The i n t e g r a t i o n is u s u a l l y e x p r e s s e d as x
Y2=- f
X=O
1 -X x d y ~
[2]
G r a p h i c a l i n t e g r a t i o n is t r o u b l e s o m e b e c a u s e the o r d i n a t e , X / ( 1 - X) b e c o m e s infinite at X = 1. The r e l a t i v e p a r t i a l m o l a r free e n e r g y , A G i = R T In X i Y i , a l s o b e c o m e s i n f i n i t e a s X i a p p r o a c h e s z e r o . To c i r c u m vent s o m e of these difficulties the p a r t i a l m o l a r free e n e r g y is b r o k e n up into two p a r t s AGi = A ~ s + AGid w h e r e ACi d = R T In X i r e p r e s e n t s the p a r t i a l m o l a r f r e e e n e r g y of an ideal solution or a solution which obeys R a o u l t ' s law, and A ~ s = R T lnYi r e p r e s e n t s the e x c e s s r e l a t i v e p a r t i a l m o l a r free e n e r g y of the s o l u t i o n . H e r e y is the a c t i v i t y coefficient. All other e x c e s s p a r t i a l m o l a r q u a n t i t i e s such a s the enthalpy, AHi, the e x c e s s e n t r o p y , A ~ S , the e x c e s s v o l u m e , A V i , and so forth, follow f r o m the above definition. The a c t i v i t y of a p a r t i c u l a r component, ai,= X i Y i , and c o n s e q u e n t l y Ti a r e m e a n i n g l e s s q u a n t i t i e s u n l e s s the r e f e r e n c e s t a t e s for the c o m p o n e n t s of the solution a r e specified. T h e r e f o r e in the above definitions and in the d i s c u s s i o n to follow the r e f e r e n c e s t a t e s a r e taken to be the p u r e c o m p o n e n t s in the s a m e state of a g g r e g a t i o n of the solution in question. I n t e g r a t i o n of Eq. [2] for an ideal solution can be c a r r i e d out a n a l y t i cally and p r e s e n t s no p r o b l e m . T h e r e f o r e , the following d i s c u s s i o n will be l i m i t e d to the e x c e s s q u a n t i t i e s . P. CHIOTTI is Senior Metallurgist,Ames Laboratory of the U. S. Atomic Energy Commission,and Professor of Metallurgy,Iowa State University, Ames, Iowa 50010. Manuscript submitted May 12, 1972. METALLURGICALTRANSACTIONS
In o r d e r to f a c i l i t a t e the i n t e g r a t i o n of Eq. [1] D a r k e n and G u r r y 1 d e v i s e d the a function In yi//(1 - Xi) 2 which is a s s u m e d to be finite at all v a l u e s of X. The q u a n tity I"i//(1 - X) 2 w h e r e Y1 = A G x s = R T l n y l obviously m u s t a l s o r e m a i n finite at X = 0 and X = 1. The value of I12 is obtained by t h e i r r e l a t i o n II2 = -X(1 - X)Y1//(1 - X) 2 + f [Y,//(1 - X)Zldx The u s u a l p r o c e d u r e in e v a l u a t i n g the i n t e g r a l i s
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