I Can internal energy be calculated from equation of state?

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1. Oct 15, 2016

arpon

We know,
$$dU=TdS-PdV$$
$\int PdV$ can be calculated if the equation of state is given.
I tried to express $S$ as a function of $P ,V$ or $T$ (any two of those).
$$dS=\left(\frac{\partial S}{\partial V}\right)_T dV+\left(\frac{\partial S}{\partial T}\right)_V dT$$
$$=\left(\frac{\partial P}{\partial T}\right)_V dV+\left(\frac{\partial S}{\partial P}\right)_V \left(\frac{\partial P}{\partial T}\right)_V dT~~~ [Using ~~Maxwell's~~ relation]$$
$$=\left(\frac{\partial P}{\partial T}\right)_V dV-\left(\frac{\partial V}{\partial T}\right)_S \left(\frac{\partial P}{\partial T}\right)_V dT~~~[Using ~~Maxwell's~ ~relation]$$
All the terms except $\left(\frac{\partial V}{\partial T}\right)_S$ can be calculated using the equation of state.
Any suggestion will be appreciated.

2. Oct 15, 2016

Staff: Mentor

You can't do it solely in terms of the equation of state. You need to use the heat capacity as well.

3. Oct 15, 2016

arpon

Thanks. That's exactly what I wanted to know.

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