A gas has the equation of state

\[\frac{p v_{m}}{\Re T}=1+N p+M p^{2}\]

where \(N\) and \(M\) are functions of temperature. Show that the equation of the inversion curve is

\[p=-\frac{\mathrm{d} N}{\mathrm{~d} T} / \frac{\mathrm{d} M}{\mathrm{~d} T}\]

If the inversion curve is parabolic and of the form

\[\left(T-T_{0}\right)^{2}=4 \mathrm{a}\left(p_{0}-p\right)\]

where \(T_{0}, p_{0}\) and a are constants, and if the maximum inversion temperature is five times the minimum inversion temperature, show that \(\mathrm{a}=\frac{T_{0}^{2}}{9 p_{0}}\) and give possible expressions for \(N\) and \(M\).

\(\left[M=T ; N=-p_{0} T+\frac{\left(T-T_{0}\right)^{3}}{12 a}+c\right]\)