How Can You Prove the Grunberg-Nissan Equation?

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SUMMARY

The Grunberg-Nissan equation, represented as ln(η) = Σ[x[SIZE="1"]i*ln(η[SIZE="1"]i)], is a semiempirical relationship used in thermodynamics. The discussion emphasizes the need to prove the approximation ln(η) ≈ x[SIZE="1"]1*ln(η[SIZE="1"]1) + x[SIZE="1"]2*ln(η[SIZE="1"]2) mathematically. Participants suggest using the properties of logarithms and the constraint Σ(x[SIZE="1"]i)=1 to derive the proof. The equation's applicability is primarily validated through experimental testing rather than strict mathematical proof.

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Homework Statement



Prove:

ln(η) = Σ[xi*ln(ηi)]

2. The attempt at a solution

I tried using the relationship:

Σ(xi)=1

and the rules for adding logarithms, but I seem to be totally failing at this. I'm not asking to have it worked out, but any hints on which direction I should go?
 
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From what I understand this is a semiempirical equation, so there is nothing to prove - you can at best test its applicability experimentally.
 
I was told to, specifically, prove the relationship:

ln(η) ≈ x1*ln(η1) + x2*ln(η2)

is true mathematically. I'm not really sure how to go about that.
 

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