Maxwell relations between thermodynamic quantities
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For thermoelastic materials, show that the following relations hold:
where is the Helmholtz free energy and is the Gibbs free energy.
Also show that
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Proof:
Recall that
and
(Note that we can choose any functional dependence that we like, because
the quantities , , are the actual quantities and not any
particular functional relations).
The derivatives are
and
Hence,
From the above, we have
and
Hence,
More Maxwell relations
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For thermoelastic materials, show that the following relations hold:
and
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Proof:
Recall,
and
Therefore,
and
Also, recall that
and
Hence,
and