Solution thermodynamic quantities represented by modified legendre polynomials
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P. M. NOVOTNY,
AND
In a r e c e n t p a p e r by Bale and P e l t o n , 1 it was shown that a m o d i f i e d L e g e n d r e p o l y n o m i a l s e r i e s has m a n y a d v a n t a g e s over a s i m p l e power s e r i e s i n the r e p r e s e n t a t i o n of s o l u t i o n t h e r m o d y n a m i c q u a n t i t i e s as a function of c o m p o s i t i o n . The m a i n r e a s o n for t h e s e a d v a n t a g e s is the o r t h o g u n a l i t y of the m o d i f i e d L e g e n d r e p o l y n o m i a l s over the i n t e r v a l 0 -< x - 1. Bale and P e l t o n d e m o n s t r a t e d how a l e a s t s q u a r e s r e g r e s s i o n a n a l y s i s can be u s e d to d e t e r m i n e the L e g e n d r e coefficients which will e x p r e s s the b e h a v i o r of e x c e s s t h e r m o d y n a m i c p r o p e r t i e s o v e r the e n t i r e c o m p o s i t i o n r a n g e of a b i n a r y s y s t e m . They r e c o m m e n d that t h e s e coefficients be obtained by f i r s t d r a w ing the b e s t c u r v e " b y e y e " to the a p p r o p r i a t e e x p e r i m e n t a l data points and then f i t t i n g this s m o o t h e d c u r v e to the a p p r o p r i a t e L e g e n d r e e x p a n s i o n . We have c o n f i r m e d that, for a given s e t of data, a l e a s t s q u a r e s L e g e n d r e s e r i e s r e p r e s e n t a t i o n and a l e a s t s q u a r e s power s e r i e s r e p r e s e n t a t i o n y i e l d the s a m e function, that is, both r e p r e s e n t a t i o n s compute the s a m e v a l u e at a given c o m p o s i t i o n . A c c o r d i n g to Bale and P e l t o n , this is to be expected s i n c e the L e g e n d r e s e r i e s can obviously be r e d u c e d to a s i m p l e power s e r i e s by a r e a r r a n g e m e n t of t e r m s . It should be noted, however, that d i s c r e p a n c i e s i n c o m p a r i s o n s of the two s e r i e s r e p r e s e n t a t i o n s will a p p e a r at high o r d e r ( a p p r o x i m a t e l y 9th or higher) p o w e r s e r i e s r e p r e s e n t a t i o n s b e c a u s e of roundoff e r r o r s a s s o c i a t e d with e v a l u a t i n g the power s e r i e s . T h i s r e s u l t s f r o m t a k i n g d i f f e r e n c e s b e t w e e n the higher o r d e r t e r m s which u s u a l l y have a l a r g e a b s o l u t e v a l u e . T h e r e is no roundoff e r r o r e n c o u n t e r e d in the e v a l u a t i o n of the L e g e n d r e s e r i e s b e c a u s e of the e x i s t e n c e of a r e c u r sion formula. T h i s c o m m u n i c a t i o n c o n s i d e r s m e t h o d s of c o n v e r s i o n f r o m a power s e r i e s r e p r e s e n t a t i o n to a L e g e n d r e s e r i e s r e p r e s e n t a t i o n . L e a s t s q u a r e s a n a l y s e s for s e v e r a l types of s e r i e s r e p r e s e n t a t i o n s of s o l u t i o n t h e r m o d y n a m i c q u a n t i t i e s a s a f u n c t i o n of c o m p o s i t i o n can be found in the l i t e r a t u r e . E s d a i l e 2 has shown that m a n y of
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