The equation of state for a certain gas is

\[v_{m}=\frac{\Re T}{p}+\frac{k}{\Re T}\]

where \(k\) is a constant. Show that the variation of temperature with pressure for an isenthalpic process from 1 to 2 is given by

\[T_{1}^{2}-T_{2}^{2}=-\frac{4 k}{c_{p} \Re}\left(p_{1}-p_{2}\right)\]

If the initial and final pressures are 50 bar and 2 bar respectively and the initial temperature is \(300 \mathrm{~K}\), calculate

a. the value of the Joule-Thomson coefficient at the initial state, and

b. the final temperature of the gas, given that \[\begin{aligned}
& k=-11.0 \mathrm{~kJ} \mathrm{~m}^{3} /(\mathrm{kmol})^{2} \\
& c_{p, m}=29.0 \mathrm{~kJ} / \mathrm{kmol} \mathrm{K.}
\end{aligned}\]
\(\left[3.041 \times 10^{7} \mathrm{~m}^{3} \mathrm{~K} / \mathrm{J} ; 298.5 \mathrm{~K}\right]\)