The Clausius-Clapeyron equation gives us an expression for dP/dT. Now we will derive an analog to the Clausius-Clapeyron equation by obtaining an expression for dT/dµ when two phases are at equilibrium. For simplicity, let’s derive this for vapor-liquid (V-L) equilibrium.
a. Write three conditions for V-L equilibrium.
b. Obtain an expression for dP in terms of dT and dµ.
c. Combine (some) of your results from part (a) and your result from part (b) to find an expression for dT/dµ in terms of T, ρL, ρV (ρ = N/V), SL/VL and SV/VV.