Figure shows the adiabatic curve for n moles of an ideal gas; the bulk's modulus for the gas corresponding to the point P will be

Bulk's modulus K = $\frac{-\mathrm{dP}}{{\displaystyle \frac{\mathrm{dV}}{\mathrm{V}}}} = \mathrm{\gamma P} [{\mathrm{PV}}^{\mathrm{\gamma }} = \mathrm{constant}]$

For the curve ${\mathrm{TV}}^{\mathrm{\gamma }-1} = \mathrm{k} \left(\mathrm{constant}\right)$

or $\mathrm{V} \mathrm{dT} + \mathrm{T}(\mathrm{\gamma }-1)\mathrm{dV} = 0$

or $\frac{\mathrm{dV}}{\mathrm{dT}} = \frac{-\mathrm{V}}{(\mathrm{\gamma }-1)\mathrm{T}} = -1 (\mathrm{slope} \mathrm{at} \mathrm{point} \mathrm{P})$

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

$\therefore \mathrm{K} = \mathrm{\gamma }\left[\frac{{\mathrm{nRT}}_0}{{\mathrm{V}}_0}\right] = \mathrm{nR}(1+\frac{{\mathrm{T}}_0}{{\mathrm{V}}_0})$