Solute interactions in multicomponent solutions
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I.
INTRODUCTION
A
recent paper by Sukiennik and Olesinski t questions Wagner's limiting statement of the equality of the interaction coefficients el J~ = eJ'~ for dilute solutions and suggests that use of this well-established relationship "may lead to serious errors in the thermodynamic description of the solution." However, the mathematical procedures used in arriving at this conclusion do not appear to be consistent with the GibbsDuhem equation and its consequences in dilute solutions. The problem of defining and relating interaction coefficients is in principle a matter of solving the Gibbs-Duhem relation for ternary and multicomponent systems; that is, predicting the activity or activity coefficient of one solute from known or measured activities or activity coefficients of the other. Accordingly, much of the analysis in the next two sections is based on a 1955 paper on this subject 2 which should be consulted for details. For simplicity of presentation, only a ternary solution will be considered, but the methods used are applicable to a greater number of components .3
~.~n3/.,,.2=qOn3c?n---~2,'.l \0~20n3/~,
\On--~/.v. 3 [4]
The second derivatives in Eqs. [2], [3], and [4] are not intensive thermodynamic properties, but are easily changed into a useful form based on 1 mole of solvent. The excess free energy per mole of solvent is given by
GE'
G~ + (n-~l)Gf + (n3)G~
J'lI
\hi~
= G~ + y e G ~ +y3G3e
[5]
in which solute concentrations Ya and Y3 are expressed as moles solute per mole solvent. Differentiating and invoking the Gibbs-Duhem equation, we get the perfect differential equation
d( Ge'] = G~ dyz + Gf dy3
[6]
knl/
and see by inspection that
Let G E' be the excess Gibbs free energy of a solution of nt moles solvent with n: and n3 moles of solutes 2 and 3, respectively. Then the partial molal excess free energies of the solutes are, by definition:
\ On2/,~,~3
\
On3 /~t,,:
\ On~ /.,..3
On3/~j.n2
\
OnZ3].,,.2
,63 E =
Oy3
t
/Y2
[71
The second derivatives are
OY2/Y3 = \ "
~
"JY3
[8]
/
[2] [3]
Oy3/y2 --
0~30Y2 -- \ Oy2 /y3
R SCHUHMANN, Jr. ts Ross Professor of Engineering, Emeritus, School of Materials Engineering, Purdue University, West Lafayette, IN 47907. Manuscript submitted April 24, 1985.
[10]
Equation [10] is, of course, the Maxwell relation for Eq. [6]. For the purposes of the subsequent analysis, the three partial derivatives in Eqs. [8], [9], and [10] will be related to specific interaction coefficients, o-~2~, o@, o,~2~, and o'~3~, designated as follows:
a In r,-
METALLURGICALTRANSACTIONSB
\
[1]
If the quantity of solvent (nO is held constant, the second derivatives of G E are:
Onz,.~..3
.,)
G2e =
II. ALTERNATIVE MEASURES OF SOLUTE INTERACTIONS
03' 3
1 (0Cf
R T \ 0Y3/y2
[12]
VOLUME 16B, DECEMBER 1985--807
0.(23) = o.~2) = O In 3'2 _ O in 3/3 Oy3 ay2
I (SOLVENT)
[13]
A
The specific interaction coefficients are not constants but instead measure the curvatures of the excess free energy surface in the vicinity of the composition Y2,Y3. Moreover,
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