ChemEng
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- Help with the application of derivatives to the fundamental property relation in classical thermodynamics.
I am having some trouble following the derivation of the partial derivative of internal energy with respect to volume at constant temperature.
The fundamental property relation is given by:
But if I apply the chain rule to the -PdV term, it seems like I would also end up with a term for the partial derivative of pressure we respect to volume at constant temperature:
At a more basic level, I am confused by how the original expression for dU is itself inserted into the partial differential expression.
I know that algebraically, it looks like we just divide both sides by dV, but my understanding is that that is not quite proper use of differentials.
The fundamental property relation is given by:
- dU = TdS - PdV
- dU/dV_T = T*dS/dV_T - P
But if I apply the chain rule to the -PdV term, it seems like I would also end up with a term for the partial derivative of pressure we respect to volume at constant temperature:
- -(dP/dV_T)*dV
At a more basic level, I am confused by how the original expression for dU is itself inserted into the partial differential expression.
I know that algebraically, it looks like we just divide both sides by dV, but my understanding is that that is not quite proper use of differentials.
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