# 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.