Thermodynamics of binary and ternary solutions containing one interstitial solute
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[1]
H e r e y~ and Ye a r e the a t o m f r a c t i o n s in the b a s e 1-2 b i n a r y and Y3 the r a t i o of i n t e r s t i t i a l to l a t t i c e a t o m s . The v a r i a b l e Y3 is u s e f u l in m o d e r a t e l y dilute s o l u t i o n s , for example w h e r e l e s s than half of the i n t e r s t i t i a l s i t e s a r e occupied. The G i b b s - D u h e m equation may be w r i t t e n : yldGl + y2dG2 + y3dG3 = 0
[2]
The m o l a r v a l u e of G m u s t be taken p e r m o l e of l a t t i c e a t o m s , G L r a t h e r t h a n the m o r e f a m i l i a r G M p e r / JOHN CHIPMAN, Fellow of the MetallurgicalSociety and Fellow of the American Society for Metals, is Professor Emeritus, Department of Metallurgyand Materials Science, Massachusetts Institute of Technology, Cambridge, Mass. Manuscript submitted April 23, 1971. METALLURGICALTRANSACTIONS
m o l e of solution. It may be noted that, where both t e r m s r e f e r to the t e r n a r y solution: G L = G M ( 1 + Y3)
[3]
and, in t e r m s of the r e l a t i v e p a r t i a l m o l a r p r o p e r t i e s , C L -- ylG1 + y2G2 + ysG3
[4]
S i m i l a r equations may be w r i t t e n for the enthalpy, exc e s s f r e e e n e r g y , and e n t r o p y . C e r t a i n f e a t u r e s of the u s e of the atom r a t i o , r a t h e r than the a t o m f r a c t i o n of the i n t e r s t i t i a l component, lead to d i f f e r e n c e s in the e q u a t i o n s r e l a t i n g the p a r t i a l m o l a r p r o p e r t i e s . Simple r e l a t i o n s b e t w e e n the two v a r i a b l e s x3 and ya for the i n t e r s t i t i a l solute include: Y3 = x3/(1 - x3); x3 = 23/(1 + Y3); (1 - x3)(1 + 23) = 1; d.v3 = dy3/(1 + Y3)2 ; dr3 = dxa/(1 - x3)2. The g e n e r a l r e lation which is the b a s i s for the " t a n g e n t i n t e r c e p t " method f o r evaluating p a r t i a l p r o p e r t i e s in a b i n a r y solution m a y be applied to a q n a s i b i n a r y of c o n s t a n t x2/xl t h u s : Gs = C M + (1 - x 3 ) \ a x 3
/x~/xl
[5]
By m e a n s of Eq. [3] and the s i m p l e r e l a t i o n s m e n tioned above, t h i s b e c o m e s
[6] E q u a t i o n s for the p a r t i a l m o l a r p r o p e r t i e s of the s u b s t i t u t i o n a l c o m p o n e n t s a r e m o r e complex. THE IDEAL DILUTE INTERSTITIAL SOLUTION An ideal b i n a r y solution m a y be defined a s one having the following p r o p e r t i e s . The l a t t i c e a t o m s a r e a r r a n g e d on a p e r f e c t l a t t i c e . The i n t e r s t i t i a l component is r a n d o m l y d i s t r i b u t e d on the a v a i l a b l e i n t e r s t i t i a l s i t e s . The p a r t i a l m o l a r e n e r g y and v o l u m e of the components are independent of concentration. It has been shown that in such a solution the activity of the interstitial component is proportional to the ratio of filled to unfilled interstitial sites. Thus, for an ideal solution of hydrogen in a "good absorber" M at a fixed temperature, Fowler and Guggenheim I derive VOLUME 3, APRIL 1972
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