Maxwell relations between thermodynamic quantities
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For thermoelastic materials, show that the following relations hold:
![{\displaystyle {\frac {\partial \psi }{\partial {\boldsymbol {E}}}}={\cfrac {1}{\rho _{0}}}~{\hat {\boldsymbol {S}}}({\boldsymbol {E}},T)~;~~{\frac {\partial \psi }{\partial T}}=-{\hat {\eta }}({\boldsymbol {E}},T)~;~~{\frac {\partial g}{\partial {\boldsymbol {S}}}}={\cfrac {1}{\rho _{0}}}~{\tilde {\boldsymbol {E}}}({\boldsymbol {S}},T)~;~~{\frac {\partial g}{\partial T}}={\tilde {\eta }}({\boldsymbol {S}},T)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42e5acaabefb96b0291d00a73a8207257d496793)
where is the Helmholtz free energy and is the Gibbs free energy.
Also show that
![{\displaystyle {\frac {\partial {\hat {\boldsymbol {S}}}}{\partial T}}=-\rho _{0}~{\frac {\partial {\hat {\eta }}}{\partial {\boldsymbol {E}}}}\qquad {\text{and}}\qquad {\frac {\partial {\tilde {\boldsymbol {E}}}}{\partial T}}=\rho _{0}~{\frac {\partial {\tilde {\eta }}}{\partial {\boldsymbol {S}}}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f88dcf04f04a72144232cf8a12a832c25cdaf43)
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Proof:
Recall that
![{\displaystyle \psi ({\boldsymbol {E}},T)=e-T~\eta ={\bar {e}}({\boldsymbol {E}},\eta )-T~\eta ~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39a258d4b872a307b3aec78d3db297dbae593df8)
and
![{\displaystyle g({\boldsymbol {S}},T)=-e+T~\eta +{\cfrac {1}{\rho _{0}}}~{\boldsymbol {S}}:{\boldsymbol {E}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a60118ead2458f8308d2db03e3c0552dc3c6ed67)
(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
![{\displaystyle {\frac {\partial \psi }{\partial {\boldsymbol {E}}}}={\frac {\partial {\bar {e}}}{\partial {\boldsymbol {E}}}}={\cfrac {1}{\rho _{0}}}~{\boldsymbol {S}}~;\qquad {\frac {\partial \psi }{\partial T}}=-\eta ~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c02ed01c69b295b061b53beb144d32604be0b360)
and
![{\displaystyle {\frac {\partial g}{\partial {\boldsymbol {S}}}}={\cfrac {1}{\rho _{0}}}~{\frac {\partial {\boldsymbol {S}}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {E}}={\cfrac {1}{\rho _{0}}}~{\boldsymbol {E}}~;\qquad {\frac {\partial g}{\partial T}}=\eta ~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db980f9f88df23da43124a844ab4f7a1301f3a55)
Hence,
![{\displaystyle {{\frac {\partial \psi }{\partial {\boldsymbol {E}}}}={\cfrac {1}{\rho _{0}}}~{\hat {\boldsymbol {S}}}({\boldsymbol {E}},T)~;~~{\frac {\partial \psi }{\partial T}}=-{\hat {\eta }}({\boldsymbol {E}},T)~;~~{\frac {\partial g}{\partial {\boldsymbol {S}}}}={\cfrac {1}{\rho _{0}}}~{\tilde {\boldsymbol {E}}}({\boldsymbol {S}},T)~;~~{\frac {\partial g}{\partial T}}={\tilde {\eta }}({\boldsymbol {S}},T)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b124b398ba2f90446c46456b39c0fda8e1500a7)
From the above, we have
![{\displaystyle {\frac {\partial ^{2}\psi }{\partial T\partial {\boldsymbol {E}}}}={\frac {\partial ^{2}\psi }{\partial {\boldsymbol {E}}\partial T}}\qquad \implies \qquad -{\frac {\partial {\hat {\eta }}}{\partial {\boldsymbol {E}}}}={\cfrac {1}{\rho _{0}}}{\frac {\partial {\hat {\boldsymbol {S}}}}{\partial T}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4a32a6c0390093441fb28c8ab4ce2621a523f48)
and
![{\displaystyle {\frac {\partial ^{2}g}{\partial T\partial {\boldsymbol {S}}}}={\frac {\partial ^{2}g}{\partial {\boldsymbol {S}}\partial T}}\qquad \implies \qquad {\frac {\partial {\tilde {\eta }}}{\partial {\boldsymbol {S}}}}={\cfrac {1}{\rho _{0}}}{\frac {\partial {\tilde {\boldsymbol {E}}}}{\partial T}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/180f46f409ebbfc3b12ce0dbe5476a687ad74010)
Hence,
![{\displaystyle {{\frac {\partial {\hat {\boldsymbol {S}}}}{\partial T}}=-\rho _{0}~{\frac {\partial {\hat {\eta }}}{\partial {\boldsymbol {E}}}}\qquad {\text{and}}\qquad {\frac {\partial {\tilde {\boldsymbol {E}}}}{\partial T}}=\rho _{0}~{\frac {\partial {\tilde {\eta }}}{\partial {\boldsymbol {S}}}}~.}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb65075b51eb259d05df2d92b1670899da692aa4)
More Maxwell relations
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For thermoelastic materials, show that the following relations hold:
![{\displaystyle {\frac {\partial {\hat {e}}({\boldsymbol {E}},T)}{\partial T}}=T~{\frac {\partial {\hat {\eta }}}{\partial T}}=-T~{\frac {\partial ^{2}{\hat {\psi }}}{\partial T^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e5e72b50d143c59241af30b2ad70b45342d2068)
and
![{\displaystyle {\frac {\partial {\tilde {e}}({\boldsymbol {S}},T)}{\partial T}}=T~{\frac {\partial {\tilde {\eta }}}{\partial T}}+{\cfrac {1}{\rho _{0}}}~{\boldsymbol {S}}:{\frac {\partial {\tilde {\boldsymbol {E}}}}{\partial T}}=T~{\frac {\partial ^{2}{\tilde {g}}}{\partial T^{2}}}+{\boldsymbol {S}}:{\frac {\partial ^{2}{\tilde {g}}}{\partial {\boldsymbol {S}}\partial T}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc9cb44b12a7ebd56d631f2ca8e9845a4bd3aad)
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Proof:
Recall,
![{\displaystyle {\hat {\psi }}({\boldsymbol {E}},T)=\psi =e-T~\eta ={\hat {e}}({\boldsymbol {E}},T)-T~{\hat {\eta }}({\boldsymbol {E}},T)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef3236d454593c5ee2103f5b26c373ecdf32f2ff)
and
![{\displaystyle {\tilde {g}}({\boldsymbol {S}},T)=g=-e+T~\eta +{\cfrac {1}{\rho _{0}}}~{\boldsymbol {S}}:{\boldsymbol {E}}=-{\tilde {e}}({\boldsymbol {S}},T)+T~{\tilde {\eta }}({\boldsymbol {S}},T)+{\cfrac {1}{\rho _{0}}}~{\boldsymbol {S}}:{\tilde {\boldsymbol {E}}}({\boldsymbol {S}},T)~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4296390c3bd568a4d3e60e405a82bf3f2589058)
Therefore,
![{\displaystyle {\frac {\partial {\hat {e}}({\boldsymbol {E}},T)}{\partial T}}={\frac {\partial {\hat {\psi }}}{\partial T}}+{\hat {\eta }}({\boldsymbol {E}},T)+T~{\frac {\partial {\hat {\eta }}}{\partial T}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb74c5fcd24f7d0b7488481e353c3ace4fd1e430)
and
![{\displaystyle {\frac {\partial {\tilde {e}}({\boldsymbol {S}},T)}{\partial T}}=-{\frac {\partial {\tilde {g}}}{\partial T}}+{\tilde {\eta }}({\boldsymbol {S}},T)+T~{\frac {\partial {\tilde {\eta }}}{\partial T}}+{\cfrac {1}{\rho _{0}}}~{\boldsymbol {S}}:{\frac {\partial {\tilde {\boldsymbol {E}}}}{\partial T}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/587fb4cc56334b9c9adc5059dce0c80533579260)
Also, recall that
![{\displaystyle {\hat {\eta }}({\boldsymbol {E}},T)=-{\frac {\partial {\hat {\psi }}}{\partial T}}\qquad \implies \qquad {\frac {\partial {\hat {\eta }}}{\partial T}}=-{\frac {\partial ^{2}{\hat {\psi }}}{\partial T^{2}}}~,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d8efb7a5134fe70b2362154391e7388e9d5099d)
![{\displaystyle {\tilde {\eta }}({\boldsymbol {S}},T)={\frac {\partial {\tilde {g}}}{\partial T}}\qquad \implies \qquad {\frac {\partial {\tilde {\eta }}}{\partial T}}={\frac {\partial ^{2}{\tilde {g}}}{\partial T^{2}}}~,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e05b47f01a1231b77f3f26d1ecd44c93afb69f2)
and
![{\displaystyle {\tilde {\boldsymbol {E}}}({\boldsymbol {S}},T)=\rho _{0}~{\frac {\partial {\tilde {g}}}{\partial {\boldsymbol {S}}}}\qquad \implies \qquad {\frac {\partial {\tilde {\boldsymbol {E}}}}{\partial T}}=\rho _{0}~{\frac {\partial ^{2}{\tilde {g}}}{\partial {\boldsymbol {S}}\partial T}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3abd158adfa5eb696c7cdc9af22ed899a8d48c19)
Hence,
![{\displaystyle {{\frac {\partial {\hat {e}}({\boldsymbol {E}},T)}{\partial T}}=T~{\frac {\partial {\hat {\eta }}}{\partial T}}=-T~{\frac {\partial ^{2}{\hat {\psi }}}{\partial T^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8724386db5dd2f01f6c53028683717e7ef46fee)
and
![{\displaystyle {{\frac {\partial {\tilde {e}}({\boldsymbol {S}},T)}{\partial T}}=T~{\frac {\partial {\tilde {\eta }}}{\partial T}}+{\cfrac {1}{\rho _{0}}}~{\boldsymbol {S}}:{\frac {\partial {\tilde {\boldsymbol {E}}}}{\partial T}}=T~{\frac {\partial ^{2}{\tilde {g}}}{\partial T^{2}}}+{\boldsymbol {S}}:{\frac {\partial ^{2}{\tilde {g}}}{\partial {\boldsymbol {S}}\partial T}}~.}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98f304efcdc30bb24a3cc816a5fc8f994345653d)