From Eqs. 6.2-18 and 6.2-20, we have the following as definitions of the heat capacities at constant volume and constant pressure:

More generally, we can define a heat capacity subject to some other constraint X by 

One such heat capacity of special interest is C_LV, the heat capacity along the vapor-liquid equilibrium line. 

a. Show that

where αⁱ is the coefficient of thermal expansion for phase i (liquid or vapor), Cⁱ_P is its constantpressure heat capacity, Vⁱ is its molar volume, and Δ_vapH and Δ_vapV are the molar enthalpy and volume changes on vaporization.

b. Show that

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and (), Cv = T Cp = T as HT P