Entropy of mixing formula

Question

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 $$x_i= \frac{N_i}{N} = \frac{V_i}{V} \ \ ?$$

Answer 1

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.