UC Posted November 4, 2009 Share Posted November 4, 2009 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] Link to comment Share on other sites More sharing options...

Genecks Posted November 7, 2009 Share Posted November 7, 2009 I think PChem is evil, and we should eat it. We should eat it for dinner. Link to comment Share on other sites More sharing options...

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