- #1
JorgeMC59
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Homework Statement
Using basic thermodynamic relations, show that the Joule Coefficient is given by:
Homework Equations
[tex] {\left(\partial T \over \partial V \right)_U}=-{T^2 \over C _v}{\partial \over \partial T} \left(P \over T \right)_V [/tex]
The Attempt at a Solution
I started with the cyclic thermodynamic relation:
[tex] {\left( \partial T \over \partial V \right)_U \left( \partial V \over \partial U \right)_T \left (\partial U \over \partial T \right)_V}=-1 [/tex]
Rearranging the equation:
[tex] {\left( \partial T \over \partial V \right)_U}={-\left( \partial U \over \partial V \right)_T \left( \partial T \over \partial U \right)_V} [/tex]
Knowing that: [itex] C_V= \left( \partial U \over \partial T \right)_V [/itex] → [itex] {1 \over C_V}= \left( \partial T \over \partial U \right)_V [/itex] I get:
[tex] {\left( \partial T \over \partial V \right)_U}={-{1 \over C_V} \left( \partial U \over \partial V \right)_T} [/tex]
And from there I don't know how to continue.
I hope someone here can help me, thanks in advance.