# Using Maxwell Relations

## The Prompt

Compare the following two expressions. What do you find? $$dU=\left(\frac{\partial U}{\partial S}\right)_{V,N}dS+\left(\frac{\partial U}{\partial V}\right)_{S,N}dV+\left(\frac{\partial U}{\partial N}\right)_{S,V}dN$$ $$dU=TdS-pdV+\mu dN$$ Write down the equivalent two statements for the Helmholtz free energy equation. Compare the statements and find corresponding relationships.

This SWBQ

## Wrap Up

When comparing the experimentally-shown physics form of the Helmholtz free energy to the mathematical form, valuable relations between the variables in the physics form and the partial differentials in the mathematical form can be found. These expressions can eventually be used by students to find one of the Maxwell relations. The same technique can be applied to the expressions for the internal energy, enthalpy, and Gibbs free energy to find the remaining Maxwell relations.

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