A Margules' Power Series Formula: Deriving Coefficients

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Margules proposed a power series formula to describe the activity composition variation in binary systems, represented by lnγ1 and lnγ2 equations. The derivation involves applying the Gibbs-Duhem equation while neglecting higher-order coefficients, leading to specific relationships among the coefficients, such as α1=β1=0 and β2=α2+α3. The discussion seeks clarification on the derivation of these relationships and the implications of ignoring coefficients beyond i=4. Participants also request references for Margules' original work and explanations of the terms used in the formulas. Understanding these concepts is crucial for accurately applying the Margules power series in thermodynamic calculations.
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Margules suggested a power series formula for expressing the activity composition variation of a binary system.
lnγ1=α1x2+(1/2)α2x2^2+(1/3)α3x2^3+...
lnγ2=β1x1+(1/2)β2x1^2+(1/3)β3x1^3+...
Applying the Gibbs-Duhem equation with ignoring coefficients αi's and βi's higher than i=3, we can obtain α1=β1=0, β2=α2+α3, β3=-α3

I don't know how that relationship between coefficients is derived.
Also, what would be the relationship when higher than i=4 is ignored?
Please help!
 
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Could you elaborate a bit? E.g.

a) "Margules suggested ..." where? Reference?
b) Here is explained how you can type formulas on PF: https://www.physicsforums.com/help/latexhelp/
c) "the activity composition variation of a binary system" means what? Forces? Number system?
 
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