A gas is undergoing an adiabatic process. At a certain stage A, the values of volume and temperature $({\mathrm{V}}_0, {\mathrm{T}}_0)$. From the details given in the graph, find the value of adiabatic constant $\mathrm{\gamma }$

We have, for an adiabatic process, in relation with temperature and volume
${\mathrm{TV}}^{\mathrm{\gamma }-1}$ = constant
Differentiating with respect to T.
${\mathrm{V}}^{\mathrm{\gamma }-1}.1 + \mathrm{T}.(\mathrm{\gamma }-1){\mathrm{V}}^{\mathrm{\gamma }-2}\frac{\mathrm{dV}}{\mathrm{dT}} = 0$

$\frac{\mathrm{dV}}{\mathrm{dT}} = \frac{\mathrm{V}}{\mathrm{T}(\mathrm{\gamma }-1)}$

But from the graph ${\left|\frac{\mathrm{dV}}{\mathrm{dT}}\right|}_{(T_0,V_0)}= \tan (\mathrm{\pi }-\mathrm{\theta }) = -\tan \mathrm{\theta }$

$\frac{{\mathrm{V}}_0}{{\mathrm{T}}_0(\mathrm{\gamma }-1)} = \tan \mathrm{\theta } \Rightarrow \mathrm{\gamma }-1 = \frac{{\mathrm{V}}_0}{{\mathrm{T}}_0\tan \mathrm{\theta }}$

$\mathrm{\gamma } = \frac{{\mathrm{V}}_0}{{\mathrm{T}}_0\tan \mathrm{\theta }}+1$