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Start: [math] dU=TdS-PdV [/math]


Divide across by dV and hold T constant.


[math] (\frac{\partial U}{\partial V})_T=T(\frac{\partial S}{\partial V})_T-P [/math]


Use the maxwell relation [math] (\frac{\partial P}{\partial T})_V=(\frac{\partial S}{\partial V})_T [/math] to get:



[math] (\frac{\partial U}{\partial V})_T=T(\frac{\partial P}{\partial T})_V-P [/math]


Starting from [math] PV=nRT [/math], take the differential form and hold V constant:


[math] d(PV)=d(nRT) \rightarrow VdP=nRdT \rightarrow (\frac{\partial P}{\partial T})_V=\frac{nR}{V} [/math]


Substitute that in to get:


[math] (\frac{\partial U}{\partial V})_T=\frac{nRT}{V}-P [/math]


Simple rearrangement of the ideal gas law gives the value of the right side of that equation:


[math] PV=nRT \rightarrow P=\frac{nRT}{V} \rightarrow 0=\frac{nRT}{V}-P [/math]


So, overall:


[math] (\frac{\partial U}{\partial V})_T=0 [/math] [Valid only for ideal gases]

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