Thermo heat capacity proof: cp - cv

[mataleo]

Homework Statement

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

Homework Equations

Standard Maxwell relations. Suppose to use jacobian to manipulate

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

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.

 

[mark726]

Hello,

Thank you for your post. Let's start by writing out the definitions of c_p and c_v in terms of Q:

c_p = (dQ/dT)_p
c_v = (dQ/dT)_v

Next, we can use the fact that dQ = dE + pdV and substitute this into the definitions of c_p and c_v:

c_p = (dE + pdV)/dT|_p
c_v = (dE + pdV)/dT|_v

Now, let's use the Maxwell relation dE = TdS - pdV to rewrite the partial derivatives in terms of entropy and volume:

c_p = T(dS/dT)_p + p(dV/dT)_p
c_v = T(dS/dT)_v + p(dV/dT)_v

Finally, we can use the chain rule to rewrite the partial derivatives in terms of specific volume (v*) and pressure (p):

(dS/dT)_p = (dS/dv*)_{p} * (dv*/dT)_p
(dS/dT)_v = (dS/dp)_{v*} * (dp/dT)_v

Substituting these into the equations for c_p and c_v, we get:

c_p = T(dS/dv*)_{p} * (dv*/dT)_p + p(dV/dT)_p
c_v = T(dS/dp)_{v*} * (dp/dT)_v + p(dV/dT)_v

Now, we can rearrange these equations to get c_p - c_v on one side:

c_p - c_v = T[(dS/dv*)_{p} * (dv*/dT)_p - (dS/dp)_{v*} * (dp/dT)_v] + p[(dV/dT)_p - (dV/dT)_v]

Finally, we can use the fact that (dV/dT)_p = (dV/dT)_v to simplify the last term, and we get the desired result:

c_p - c_v = T[(dS/dv*)_{p} * (dv*/dT)_p - (dS/dp)_{v*} * (dp/dT)_v]

I hope this helps! Let me know if you have any further questions.