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Could someone give me a proof for the entropy of mixing formula, $$\Delta S_{{mix}}=-R(x_{1}\ln x_{1}+x_{2}\ln x_{2}),$$ with $$x_i= \frac{N_i}{N} = \frac{V_i}{V} \ \ ?$$

Níckolas Alves
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    For ideal gases the mixing entropy was found by Gibbs, "On equilibrium of heterogeneous substances": he used Dalton's law. On Wikipedia you can find also the modern "statistical mechanics" derivation https://en.wikipedia.org/wiki/Entropy_of_mixing . See also: https://physics.stackexchange.com/q/78860/226902 https://physics.stackexchange.com/q/705599/226902 https://physics.stackexchange.com/q/261522/226902 https://physics.stackexchange.com/q/12627/226902 – Quillo Jun 11 '22 at 09:30
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    Start with Vant Hoff Equation( ∆G = -RT*lnK) Use the Equilibrium constant with appropriate variables, remember to use dH = TdS and also take the difference of free energies before and after mixing. –  Jun 11 '22 at 10:11

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Herbert B. Callen - Thermodynamics And An Introduction To Thermostatistics-Wiley (1985) has two proofs, one more theoretical and the other one is a simple thought experiment. The first start at page 66 in the paragraph 3-4 THE SIMPLE IDEAL GAS AND MULTICOMPONENT SIMPLE IDEAL GASES, the second one right after at page 69.

Mattia
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