Mathematical representation of thermodynamic properties in binary systems and solution of Gibbs-Duhem Equation
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A. D. PELTON
The s t o r a g e , r e t r i e v a l , and manipulation of thermodynamic data with the aid of a computer r e q u i r e s a c c u r a t e a n a l y t i c a l r e p r e s e n t a t i o n of t h e r m o d y n a m i c p r o p e r t i e s of solutions. In the p r e s e n t p a p e r , a c r i t i c a l a s s e s s m e n t is made of s i m p l e power s e r i e s expansions and their l i m i t a t i o n s in r e p r e s e n t i n g t h e r m o d y n a m i c p r o p e r t i e s over the e n t i r e c o m p o s i tion range of a b i n a r y s y s t e m . The advantages of c e r t a i n orthogonal s e r i e s as an a l t e r n a tive method of r e p r e s e n t a t i o n is also d i s c u s s e d . P a r t i c u l a r e m p h a s i s is p l a c e d upon s e r i e s r e p r e s e n t a t i o n s which use Legendre polynomials due to t h e i r s i m p l i c i t y and the fact that t h e i r functional form is consistent with e m p i r i c a l o b s e r v a t i o n s of solution b e havior. Since the coefficients of orthogonal s e r i e s a r e independent of each o t h e r when the e n t i r e composition range of a b i n a r y s y s t e m is r e p r e s e n t e d , any t h e r m o d y n a m i c p r o p e r t y can be fitted to any d e s i r e d d e g r e e of a c c u r a c y by a finite n u m b e r of t e r m s without the n e c e s s i t y of s t o r i n g a l a r g e number of significant digits. Also, b e c a u s e the coefficients a r e u n e o r r e l a t e d , they a r e amenable to m a t h e m a t i c a l interpolation and extrapolation as well as to p h y s i c a l i n t e r p r e t a t i o n . The r e l a t i o n s h i p s between coefficients of s e r i e s e x pansions of all p a r t i a l and i n t e g r a l p r o p e r t i e s for the g e n e r a l case have been d e r i v e d using the Gibbs-Duhem equation.
MODERN
advances in computer technology have c r e ated an i n c r e a s i n g n e c e s s i t y to have an a n a l y t i c a l r e p r e s e n t a t i o n of the t h e r m o d y n a m i c p r o p e r t i e s of b i n a r y s y s t e m s for p u r p o s e s of: i) data s t o r a g e and r e t r i e v a l , and ii) data manipulation, as for example in phase d i a g r a m , Gibbs-Duhem, and c h e m i c a l equilibrium c a l c u lations. Until now, s i m p l e power s e r i e s expansions in t e r m s of mole f r a c t i o n s have been used a l m o s t e x c l u s i v e l y for these p u r p o s e s . The fact that the coefficients of a power s e r i e s expansion a r e highly interdependent means that a l a r g e number of significant digits must be r e t a i n e d in s t o r a g e and in the calculations for all but the s i m p l e s t s y s t e m s . F u r t h e r m o r e , for many s y s t e m s in which the t h e r m o d y n a m i c p r o p e r t i e s v a r y in a m o r e c o m p l i c a t e d way with composition it is i m p o s s i ble to adequately r e p r e s e n t the p r o p e r t i e s by a s i m p l e power s e r i e s at all. This p r o b l e m may be p a r t i a l l y r e c
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