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Does $$ \left.\frac{\partial {\cal E}}{\partial\rho}\right|_T=?\;\; {\rm or} \;\; \left.\frac{\partial E}{\partial V}\right|_T=? $$ where $E$ is energy, ${\cal E}$ is energy density, $V$ is volume, $\rho$ is (mass) density have a name? Is there a simple relation ship to "standard" quantities, like the speed of sound, volume expansion coefficients, or specific heats?

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$$dE = T\,dS-P\,dV$$ $$\begin{align*}\left(\frac{\partial E}{\partial V}\right)_T&=T\left(\frac{\partial S}{\partial V}\right)_T-P \\ &=T\left(\frac{\partial P}{\partial T}\right)_V-P \\ &=-T\left(\frac{\partial P}{\partial V}\right)_T\left(\frac{\partial V}{\partial T}\right)_P-P \\ &=T\alpha_V\kappa-P \end{align*}$$ where $\alpha_V$ is the volumetric coefficient of linear expansion and $\kappa$ is the bulk modulus. To convert the partial derivatives, I used a Maxwell relation and then the triple product rule.

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  • $\begingroup$ Thanks. One small thing: In the third equality $(\partial V)/(\partial P)$ is presumable upside down. $\endgroup$
    – Thomas
    May 30 '17 at 18:44

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