For any equation of state,
$$
c_P-c_V
=\frac{P}{\rho T}\frac{\chi_T^2}{\chi_\rho},
\qquad
\Gamma_3-1=\frac{P\chi_T}{\rho Tc_V}.
$$
Eliminating $P/(\rho Tc_V)$ gives the requested ratio of specific heats:
$$
\boxed{
\gamma=\frac{c_P}{c_V}
=1+(\Gamma_3-1)
\frac{\left(\partial\log P/\partial\log T\right)_\rho}
{\left(\partial\log P/\partial\log\rho\right)_T}}.
$$


Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 317 / 3 / iii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-317/3/iii/solution