Question

In a real gas, the internal energy depends on temperature and also on volume. The energy increases when the gas expands isothermally. Examining the derivation of C_p − C_v = R, find whether C_p − C_v will be more than R, less than R or equal to R for a real gas.

Answer 1

In a real gas, as the internal energy depends on temperature and volume, the derived equation for an ideal gas $(\mathrm{d}Q{)}_{\mathrm{p}}=(\mathrm{d}Q{)}_{\mathrm{v}}+nR\mathrm{d}T$ will change to ​$(\mathrm{d}Q{)}_{\mathrm{p}}=(\mathrm{d}Q{)}_{\mathrm{v}}+nR\mathrm{d}T$ + k, where k is the change in internal energy (positive) due to change in volume when pressure is kept constant. So, in the case of a real gas, for n=1 mole (say),
$$\begin{gathered} C_{\mathrm{p}}-C_{\mathrm{v}}=R+\frac{k}{\mathrm{d}T} \\ \Rightarrow C_{\mathrm{p}}-C_{\mathrm{v}}>R, \end{gathered}$$
where C_p and C_v are the specific heat capacities at constant pressure and volume, respectively.