(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that for a general (but simple) substance,

c[itex]_{p}[/itex]-c[itex]_{v}[/itex]=-T([itex]\frac{∂v*}{∂T}[/itex])[itex]_{p}[/itex]([itex]\frac{∂p}{∂v*}[/itex])[itex]_{T}[/itex]

where

v* is the specific volume

p is the pressure

c[itex]_{p}[/itex] is the heat capacity when p is const

c[itex]_{v}[/itex] 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[itex]_{p}[/itex] = ([itex]\frac{∂Q}{∂T}[/itex])[itex]_{p}[/itex]

c[itex]_{v}[/itex] = ([itex]\frac{∂Q}{∂T}[/itex])[itex]_{v}[/itex]

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[itex]_{p}[/itex]-c[itex]_{v*}[/itex]=p([itex]\frac{∂v*}{∂T}[/itex])[itex]_{p}[/itex]+([itex]\frac{∂E}{∂T}[/itex])[itex]_{p}[/itex]-([itex]\frac{∂E}{∂T}[/itex])[itex]_{v*}[/itex]

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.

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# Homework Help: Thermo heat capacity proof: cp - cv

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