The relationship between Gibbs free energy and the intersection of the liquidi in phase diagrams of reciprocal systems
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W E shall c o n s i d e r a r e c i p r o c a l s y s t e m A a C c - A a D d B b C c - B b D d with n e g I i g i b l e s o l u b i I i t i e s in the solid state but c o m p l e t e s o l u b i l i t i e s in the liquid state. The phase e q u i l i b r i a between the solid phases a r e c o m p l e t e l y d e t e r m i n e d by the sign of a quantity A~ s which m a y be defined by
a~
=
~
-
~
d
- ~
c.
[q
If A~ is p o s i t i v e , the B b D d + A a C c p a i r will be the stable one and t h e r e a r e two t h r e e - p h a s e t r i a n g l e s B b D d + A a C c + A a D d and B b D d + A a C c + B b C c. It is then s e l f - e v i d e n t that the liquidi of B b D d and A a C c will i n t e r s e c t and t h e r e will be no contact between the liquidi of A a D d and B b C c, Fig. 1 shows a typical case. It is quite n a t u r a l that the length of the i n t e r s e c t i o n b e tween the liquidi for B b D d and A a C c should in s o m e way r e f l e c t the s i z e of A~ s and e i t h e r one of them could be e s t i m a t e d f r o m i n f o r m a t i o n on the other one. This r e l a t i o n s h i p will now be examined.
DERIVATION OF RELATIONSHIP The i n t e r s e c t i o n will be a p p r o x i m a t e l y p a r a l l e l to the A a D d - B b C c diagonal and it is thus advantageous to e x a m i n e the f r e e e n e r g y d i a g r a m in a d i r e c t i o n p e r pendicular to the A a D d - B b C c diagonal. The two solid t h r e e - p h a s e e q u i l i b r i a will each be r e p r e s e n t e d by a t r i a n g u l a r plane and two f o u r - p h a s e e q u i l i b r i a will be found w h e r e the c u r v e d f r e e e n e r g y s u r f a c e of the liquid phase touches the planes. Let us f i r s t a s s u m e that the two f o u r - p h a s e e q u i l i b r i a lie at the s a m e t e m p e r a t u r e . Fig. 2 i I l u s t r a t e s such a c a s e and the f r e e e n e r g y s u r f a c e of the liquid phase has b e e n r e p r e s e n t e d by a contour line going through the two points of t a n gency. A s i m i l a r p i c t u r e can be obtained in the gene r a l c a s r w h e r e the two f o u r - p h a s e e q u i l i b r i a Iie at different t e m p e r a t u r e s . One should then s i m p l y d i s place one of the planes until it touches the f r e e e n e r g y c u r v e for the t e m p e r a t u r e of the other f o u r - p h a s e e q u i l i b r i u m ; s e e Fig. 3. The point of tangency is a p p r o x i m a t e l y unaffected by the d i s p l a c e m e n t if the t e m p e r a t u r e d i f f e r e n c e is not too l a r g e . The s a m e c o n s t r u c t i o n
can be m a d e even if one of the f o u r - p h a s e e q u i l i b r i a is of the p e r i t e c t i c type in which c a s e the point of t a n gency will l i e on the e x t e n s i o n of the t r i a n g u l a r f r e e e n e r g y plane of the solid p h a s e s in e q u i l i b r i u m with the liquid. Let us now e x a m i n e the slope of the f r e e e n e r g y c u r v e at the points of tangency in the d i r
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