How to redefine internal pressure in terms of κ and α? The formula for internal pressure of a system is:
$$\pi_T=T\left(\frac{\partial p}{\partial T}\right)_{\!V}-p$$

My professor said we can redefine this equation in terms of either the


thermal compressibility, $\kappa$
expansion coefficient, $\alpha$


How do we do this?

I was able to derive for a van der Waals gas:


$\Large\kappa = \frac{V-nb}{pV-\frac{an^2}{v}+\frac{2abn^3}{v^2}}$
$\Large \alpha= \frac{nR}{PV-\frac{an^2}{v}+\frac{2abn^3}{v^2}}$
$$dV=\left(\frac{\partial V}{\partial T}\right)_pdT+\left(\frac{\partial V}{\partial p}\right)_Tdp$$So at constant volume dV = 0 and $$\left(\frac{\partial p}{\partial T}\right)_V=-\frac{\left(\frac{\partial V}{\partial T}\right)_p}{\left(\frac{\partial V}{\partial p}\right)_T}$$

The rest is straightforward.