The thermodynamics of solution of Ti, V, Cr, Fe, and Co in liquid tin
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Element
DARBYand Jugle
r e c e n t l y r e p o r t e d the s o l u b i l i t i e s of T i , V, C r , Re, and Co in liquid tin o v e r the t e m p e r a t u r e r_ange 827 ~ to 1211~ 1 The p a r t i a l m o l a r e n t h a l p i e s , LxH, and e n t r o p i e s , /x~, of solution of T i , V, and Cr were e s t i m a t e d f r o m the s o l u b i l i t y data by l e a s t - s q u a r e s a n a l y s e s of the e q u a t i o n 2.303R logloX = & S - A H / T
[1]
w h e r e R i s the gas c o n s t a n t and X is the mole f r a c t i o n of solute in a s a t u r a t e d solution at absolute t e m p e r a t u r e T. T h e i r r e s u l t s a r e s u m m a r i z e d u n d e r the c o l u m n " E m p i r i c a l " in T a b l e I. T h i s method r e p r e s e n t s one i n t e r p r e t a t i o n of this data; in this c o m m u n i c a t i o n two o t h e r a p p r o a c h e s will be d i s c u s s e d . T h e s e a p p r o a c h e s will a s s u m e that s e v e r a l conditions apply to these s y s t e m s . F i r s t , s u p e r c o o l e d liquid solute at the t e m p e r a t u r e of i n t e r e s t is the r e f e r e n c e state. It is also a s s u m e d that the heat c a p a c i t i e s of such solute and solid solute at the g i v e n t e m p e r a t u r e a r e a p p r o x i m a t e l y e q u a l , and the a c t i v i t y c o e f f i c i e n t s a r e c o n s t a n t within the l i m i t s of c o m p o s i t i o n and t e m p e r a t u r e to be c o n s i d e r e d . The a s s u m p t i o n is also made that the dilute s o l u t i o n s are analogous to s i m p l e e u c t e c t i c - t y p e b i n a r y s y s t e m s with e s s e n t i a l l y no i n t e r a t o m i c bonding and with the solute r a n d o m l y d i s t r i b u t e d . If the e n t r o p y and e n t h a l p y of solution a r e independent of t e m p e r a t u r e in the t e m p e r a t u r e r a n g e of i n t e r e s t , Eq. [1] m a y be d i f f e r e n t i a t e d in r e s p e c t to T so that 2.303 R d[logl0X] _ - A H d [ 1 / T ] dT
dT
or dT
The p a r t i a l e n t h a l p i e s w e r e c a l c u l a t e d for all five e l e m e n t s f r o m the data by this equation a s s u m i n g that the a c t i v i t y c o e f f i c i e n t s w e r e c o n s t a n t o v e r the c o n c e n t r a tion r a n g e s of i n t e r e s t . V a r i a t i o n s in c a l c u l a t e d e n t h a l p i e s w e r e high; t h e r e f o r e , no c o r r e c t i o n f a c t o r s w e r e applied. The e n t r o p i e s were c a l c u l a t e d u n d e r the l i m i t condition by the equation zXS = • H D / T
+ 2.303R logloX
[3]
w h e r e AH D i s the c a l c u l a t e d e n t h a l p y . E n t h a l p i e s and e n t r o p i e s which were c a l c u l a t e d by this technique are given in T a b l e I in the c o l u m n m a r k e d " D i f f e r e n t i a l " . The s t a n d a r d d e v i a t i o n s a r e l a r g e , v a r y i n g f r o m 2 to 11 kcal per m o l e . T h e s e were not t e m p e r a t u r e dependent; v a r i a t i o n s o c c u r r e d throughout the
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