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# Concept of Partial Derivative and Thermodynamic Derivations

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Question is attached. It is about illustrating a mathematical relation that has both partial derivatives with p and T constant in a graph. And also about the derivation for the joule-Thomson effect.

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https://brainmass.com/chemistry/physical-chemistry/concept-partial-derivative-thermodynamic-derivations-555432

#### Solution Preview

Q1. Interpretation of (MB2.3a)

A1. Here you have mistaken between 'variables' and 'axes'. Remember x, y and z are variables in (MB2.3a), not axes. It can be a, b and c instead of x, y and z.

Q2. In (2.46) why do they have ?
A2. Since U is independent of V for a perfect gas, the partial derivative (∂U/∂T)v for Cv becomes an ordinary derivative, dU/dT. [Physical Chemistry by Levine, page 56]
As Cv of perfect gas only depends on T, a constant P condition can then be stipulated.

But I like more, the way the derivation is shown in Levine's book. Please consult pages 52 and 56 of that book (Google book has these pages available).
In page 52, they are deriving: Cp - Cv = [(∂U/∂V)T + P] (∂V/∂T)P
And then they are coming to the perfect gas in page 56, and deriving Cp - Cv = nR in a better ...

#### Solution Summary

This solution offers clear and rational discussion along with pertinent references and corrects the common misconceptions that a student may have. This solution will help one to gain better understanding about the thermodynamic derivations related to partial derivatives.

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