# van der Waals and the difference Cp - Cv

I would appreciate assistance with the following question please.

Show that for a gas obeying van der Waals' EOS, $$\left( p +\frac{a n^2}{V^2}\right)(V-nb) = nRT,$$

and whose internal energy is given by $$U = cT - \frac{a n^2}{V} + d,$$

that $$C_p-C_v = nR(1 + (2n^2 a)/pV^2))$$ where $$a$$, $$b$$, $$c$$, and $$d$$ are constants.

My thoughts:

Expanding the vdW EOS gives $$nRT = pv - nbp + an^2/V - abn^3/V^2$$ and this final term can be assumed to be negligible for the purposes of this question.

From the expression for $$U$$,
\begin{align} C_v &= (\partial U/\partial T)_v = c\\ H &= U + pV\\ C_p &= (\partial H/\partial T)_p = (\partial U/\partial T)_p + (\partial (pv)/\partial T)_p \end{align} However, how are these results combined to give $$C_p-C_v$$ in the required form?

• @Poutnik- yes I have put my apostrophe of possession in the wrong place in the OP, it should have been van de Waals' so thanks for pointing that out Commented Feb 7 at 14:45

Start with $$dU=C_vdT-\left[P-T\left(\frac{\partial P}{\partial T}\right)_V\right]dV$$ and $$dH=dU+PdV+VdP$$and show that $$C_p-C_v=T\left(\frac{\partial P}{\partial T}\right)_V\left(\frac{\partial V}{\partial T}\right)_P$$