A monoatomic ideal gas undergoes a process in which the ratio of P to V at any instant is constant and equal to unity. The molar heat capacity of the gas is

Let P₁V₁ be the pressure and volume of monoatomic gas at temperature T.
$\begin{array}{l}{\mathrm{P}}_1{\mathrm{V}}_1=\mathrm{RT}\\ {\mathrm{P}}_2\left({\mathrm{V}}_1+\mathrm{dV}\right)=\mathrm{R}(\mathrm{T}+1)\\ \therefore {\mathrm{P}}_2^2=\mathrm{RT}+\mathrm{R}\left(\because \frac{{\mathrm{P}}_2}{{\mathrm{V}}_1+\mathrm{dV}}=1\right)\end{array}$
or $2{\left(\frac{\mathrm{\partial }\mathrm{P}}{\mathrm{\partial }\mathrm{T}}\right)}_{\mathrm{v}}=\mathrm{R}$
$\therefore {\left(\frac{\mathrm{\partial }\mathrm{P}}{\mathrm{\partial }\mathrm{T}}\right)}_{\mathrm{v}}=\frac{\mathrm{R}}{2}$
$\because \mathrm{C}={\mathrm{C}}_{\mathrm{v}}+{\left(\frac{\mathrm{\partial }\mathrm{P}}{\mathrm{\partial }\mathrm{T}}\right)}_{\mathrm{v}}$for a process
$=\frac{3}{2}\mathrm{R}+\frac{\mathrm{R}}{2}=\frac{4\mathrm{R}}{2}$