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]

I think PChem is evil, and we should eat it.

We should eat it for dinner.