Why is heat capacity of gas less than liquid at vapor coexistence?

[subtletuna]

Homework Statement

Consider the liquid-gas coexistence curve. Show that C$$_{(g)}$$ − C$$_{(l)}$$ = T($$\stackrel{d}{dT}$$ $$\Delta$$$$_{(vap)}$$S) for T, P along the coexistence curve. Now use the fact that
$$\Delta$$(vap)S = $$\Delta$$(vap)H/T along the coexistence curve, and the fact that $$\Delta$$(vap)H is positive and decreases with increasing T, to show that C(g) < C(l) . Note that this inequality applies only at Tvap.

Homework Equations

The Attempt at a Solution

 

[Gokul43201]

Please show us what you have tried. We can not help you unless you first put in some effort.