How Does Heat Capacity Change with Volume and Pressure?

[McCoy13]

Homework Statement

Let $C_{X}=T(\partial S/\partial T)_{X}$ with $X=p~or~V$. Show that $C_{p}-C_{V}=T(\partial S/\partial V)_{T}(\partial V/\partial T)_{p}$.

Homework Equations

$C_{p}-C_{V}=T[(\partial S/\partial T)_{p}-(\partial S/\partial T)_{V}]$

The Attempt at a Solution

I honestly don't know where to start. I looked at some of the Maxwell relations, but no Maxwell relation features conjugate variables in the numerator and denominator of a partial derivative. I'm not sure how what other way I might use to manipulate the thermodynamic variables.

 

[mark726]

First, let's rewrite the given equation using the definition of C_X:

C_p - C_V = T(\partial S/\partial T)_p - T(\partial S/\partial T)_V

Next, we can use the chain rule to rewrite the partial derivatives of S in terms of partial derivatives of T and V:

(\partial S/\partial T)_p = (\partial S/\partial V)_T(\partial V/\partial T)_p + (\partial S/\partial T)_V

Substituting this into the equation above, we get:

C_p - C_V = T[(\partial S/\partial V)_T(\partial V/\partial T)_p + (\partial S/\partial T)_V] - T(\partial S/\partial T)_V

Simplifying, we get:

C_p - C_V = T(\partial S/\partial V)_T(\partial V/\partial T)_p

which is the desired result.