Show that the Joule-Thomson coefficient, (mu), is given by [mu=frac{1}{c_{p}}left(Tleft(frac{partial v}{partial T} ight)_{p}-v ight)] Hence or otherwise

Show that the Joule-Thomson coefficient, \(\mu\), is given by

\[\mu=\frac{1}{c_{p}}\left(T\left(\frac{\partial v}{\partial T}\right)_{p}-v\right)\]

Hence or otherwise show that the inversion temperature \(\left(T_{\mathrm{i}}\right)\) is

\[T_{\mathrm{i}}=\left(\frac{\partial T}{\partial v}\right)_{p} v\]

The equation of state for air may be represented by

\[p=\frac{\Re T}{v_{m}-0.0367}-\frac{1.368}{v_{m}^{2}}\]

where \(p=\) pressure (bar), \(T=\) temperature \((\mathrm{K})\), and \(v_{m}=\) molar volume \(\left(\mathrm{m}^{3} / \mathrm{kmol}\right)\).

Determine the maximum and minimum inversion temperature and the maximum inversion pressure for air.

[896 K; 99.6 K; 339 bar]