On the relationship between interaction coefficients
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I.
INTRODUCTION
THERMODYNAMIC analysis of liquid dilute solutions, particularly the iron-based alloys, frequently utilizes the formalism introduced by Wagner and Chipman. ~,2,3The activity coefficient, Y. of the solute (its mole fraction x~ ~ 0, i -- 2 . . . . . s) is presented in this formalism as a Taylor series expansion about the point (xl -- 1, x~ = 0, i = 2, . . . . s). Assuming Henrian ideal solution as the reference state, the expansion takes the following form: In% =
~ (O- ln %) -
j=2\
C3Xj / x k ~ j
~,=l
X)
II. DIFFERENTIATION OF THERMODYNAMIC FUNCTIONS W I T H R E S P E C T TO MOLE FRACTIONS
The first terms of this series are called the first order interaction coefficients and are denoted by elJ~: e~j) =
(a In yi]
1
[2]
The second derivatives of series (1) are known as the interaction coefficients of the second order, el j,~>. The relationships among the interaction coefficients have been investigated quite extensively, a,5 A major controversy still seems to surround the popular relationship involving the first order coefficients, namely: E (J) i = ~.)i)
[31
Probably the most elegant demonstration of relation [3] is based on the following thermodynamic identity:
Zi = Z' + OZ____~'_~ XF(OZ'I OXi
r=2
[4]
\OXr/xk~x r
where Z' denotes a molar property of the phase and Z~ is the partial molar property of the component i:
njCni
Z: property of the whole phase n~: number of moles of component i Applying Eq. [4] to the partial excess Gibbs free energy of the components i and j, then differentiating with respect M.Z. SUKIENNIK is Associate Professor, Academy of Mining and Metallurgy, Cracow, Poland. R.W. OLEStNSKI is Research Associate, Department of Materials Science and Engineering, University of Florida, Gainesville, FL 32611. Manuscript submitted December 12, 1983. METALLURGICALTRANSACTIONS B
to xj and xe, respectively, and comparing (OZi/Oxl)x~,,
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