Show that the following equation holds, relating the Helmholtz free energy to the constant-volume heat capacity of a material.(∂^2A/∂T^2)v=-(∂S/∂T)v=-(cv/T)

Question

Show that the following equation holds, relating the Helmholtz free energy to the constant-volume heat capacity of a material.

[latex]\left\lgroup\frac{\partial^2A}{\partial T^2} \right\rgroup _v=-\left\lgroup\frac{\partial S}{\partial T} \right\rgroup_v=-\left\lgroup\frac{c_v}{T} \right\rgroup [/latex]

Accepted Answer

A = U - TS

[latex]-S=\left\lgroup\frac{\partial A}{\partial T} \right\rgroup_v=\left\lgroup\frac{\partial U}{\partial T} \right\rgroup_v-T\left\lgroup\frac{\partial S}{\partial T} \right\rgroup _v-S [/latex]

Thus, [latex]\left\lgroup\frac{\partial U}{\partial T} \right\rgroup_v=T\left\lgroup\frac{\partial S}{\partial T} \right\rgroup_v=c_v [/latex]

We saw that [latex]\left\lgroup\frac{\partial A}{\partial T} \right\rgroup _v=-S [/latex]

[latex]\left\lgroup\frac{\partial^2 A}{\partial T^2} \right\rgroup _v=-\frac{\partial S}{\partial T} =-\frac{c_v}{T} [/latex]

Question 6.10.2: Show that the following equation holds, relating the Helmhol...

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