The molar specific heat of the process $\mathrm{P}=\frac{\mathrm{a}}{\mathrm{T}}$ for a monoatomic gas, a being constant , is $\frac{b}{2}R$. The value of b is

We know that 

$\begin{array}{l}\mathrm{dQ}=\mathrm{dU}+\mathrm{dW}\\ \mathrm{specific} \mathrm{heat} \mathrm{C}=\frac{\mathrm{dQ}}{\mathrm{dT}}=\frac{\mathrm{dU}}{\mathrm{dT}}+\frac{\mathrm{dW}}{\mathrm{dT}} .......\left(\mathrm{i}\right)\\ \mathrm{Since}, \mathrm{dU}={\mathrm{C}}_{\mathrm{V}}\mathrm{dT} ..........\left(\mathrm{ii}\right)\\ \mathrm{C}={\mathrm{C}}_{\mathrm{V}}+\frac{\mathrm{dW}}{\mathrm{dT}}\\ ={\mathrm{C}}_{\mathrm{V}}+\frac{\mathrm{pdV}}{\mathrm{dT}}\\ \mathrm{pV}=\mathrm{RT}\\ \therefore \mathrm{For} \mathrm{the} \mathrm{given} \mathrm{process}, \mathrm{V}=\frac{\mathrm{RT}}{\mathrm{p}}=\frac{{\mathrm{RT}}^2}{\mathrm{a}}\\ \frac{\mathrm{dV}}{\mathrm{dT}}=\frac{2\mathrm{RT}}{\mathrm{a}}\end{array}$

$\begin{array}{l}\therefore \mathrm{C}={\mathrm{C}}_{\mathrm{V}}+\mathrm{p}\left(\frac{2\mathrm{RT}}{\mathrm{a}}\right)\\ ={\mathrm{C}}_{\mathrm{V}}+2\mathrm{R}\\ =\frac{3}{2}\mathrm{R}+2\mathrm{R}=\frac{7}{2}\mathrm{R}\end{array}$