Thermo heat capacity proof: cp - cv

1. Apr 19, 2012

mataleo

1. The problem statement, all variables and given/known data

Show that for a general (but simple) substance,

c$_{p}$-c$_{v}$=-T($\frac{∂v*}{∂T}$)$_{p}$($\frac{∂p}{∂v*}$)$_{T}$

where

v* is the specific volume
p is the pressure
c$_{p}$ is the heat capacity when p is const
c$_{v}$ is the heat capacity when v is const
Q is heat
T is temperature in K

2. Relevant equations

Standard Maxwell relations. Suppose to use jacobian to manipulate

c$_{p}$ = ($\frac{∂Q}{∂T}$)$_{p}$
c$_{v}$ = ($\frac{∂Q}{∂T}$)$_{v}$

3. The attempt at a solution

I started by inserting the above equations for specific heat in terms of heat (Q). Then I plugged the partial derivatives into the equation dQ = dE + p dv* which left me with

c$_{p}$-c$_{v*}$=p($\frac{∂v*}{∂T}$)$_{p}$+($\frac{∂E}{∂T}$)$_{p}$-($\frac{∂E}{∂T}$)$_{v*}$

Next I rewrote the partial derivatives using dE = -TdS-pdV, but clearly at this point I'm just going around in circles. I think I'm missing some simple step to combine the partials, but I'm not sure what it is. Your help would be appreciated.

Last edited by a moderator: Apr 20, 2012