analytical expressions for the activity of the solute in binary dilute solutions in terms of associated solution model p
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I.
INTRODUCTION
A S S O C I A T E D Solution Model (ASM) has been widely used to describe the thermodynamic properties of the liquid phase in a number of metal-sulfur, II-VI, IV-VI, Ill-V, and other binary systems. 1-17Schmid and Chang TMhave recently discussed different aspects of ASM. From a practical point of view, one of the limitations of ASM is that the thermodynamic properties of the solution cannot be expressed in simple analytic forms. These have to be arrived at by solving a set of non-linear equations by iterative methods. In the age of computers, this is not a severe limitation; however, it is always desirable to have simple analytic expressions for easy usage. Quite often, people are interested only in the thermodynamic properties of the dilute solutions, which are normally expressed in terms of the activity coefficient at infinite dilution and the first and second order Wagner's self-interaction parameters. 19'2~ In the present communication, analytical expressions have been derived for the activity coefficient at infinite dilution and the first and second order self-interaction parameters in terms of the ASM parameters in the dilute solution approximation.
Kelvin. The mole fractions, y,'s, are related to actual mole fractions, xa and xB, of A and B, respectively, in the binary solution A - B by the following mass balance equations: TM Yl = XA[1 "~- (P Jr q -- 1)y3] -- PY3
[3]
Y2 = xB[1 + (p + q - 1)y3]
[41
and -
qY3
-
The excess Gibbs energy of mixing, A G xs, of the pseudoternary solution may, in general, be expressed as: 21 1 3
acx
-- Rr-
3
E E /=1 J=l
9 [W,j + (w,j - %,)yj - 4v,jy, yj]y, yj
[5]
where w,j and v,j are the solution parameters; and w, = v, = 0 and v,j = vj,. The parameters w,j and v,j are, in general, expressed as a function of temperature as: w,, = A , J T + B,j
[6]
v,j = C o / T + D U
[7]
and II.
A S S O C I A T E D S O L U T I O N M O D E L (ASM)
In the Associated Solution Model (ASM), a binary liquid solution of components A and B is modeled as a pseudoternary solution of species 'A', 'B', and 'Ap Bq', constrained by the internal equilibrium reaction. ~8 p'A'
+ q ' B ' = 'ApBq'
[ 1]
with A,j, B U, C U, and D,j being constants. The activity coefficients, f ' s , of species in the pseudoternary solution are, then, given as: 2~ 3
ln f = ~[(w,, + w , ) / 2 + ( w , , -
3
With the equilibrium constant, K, given by: K -
f3Y3 (fl yl)P(f2 y2) q
9 (y,/2 - y,) - 8v,,y, yj]yj - Z
9 [%p/2 + (%p - wp,)yp - 6vjpy/yp]y, yv
[2]
[2a]
where A and B are constants and T is temperature in degrees ROMESH C. SHARMA is Professor, Department of Metallurgical Engineering, Indian Institute of Technology, Kanpur, U.P. 208016, India. Manuscript submitted October 23, 1986.
METALLURGICAL TRANSACTIONS A
3
E
J=l p=l
wherefl,f2, and f3 are the activity coefficients and y~, y2, and Y3 are the mole fractions, respectively, of the species 'A', 'B', and 'ApBq' in the pseudoternary liquid solution. The equilibrium constant, K, may be expressed as a function of temperature as: In K
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