Analysis
The frequency of collisions of gas atoms per unit area of the wall ($f$) is proportional to the number density ($n/V$) and the average speed ($v_{av}$).
$$ f \propto \frac{N}{V} v_{av} \propto \frac{1}{V} \sqrt{T} $$The problem states that this frequency remains constant.
$$ \frac{\sqrt{T}}{V} = k \implies T \propto V^2 $$Process Identification
For a polytropic process $TV^{x-1} = \text{constant}$. Here we have $TV^{-2} = \text{constant}$.
$$ x – 1 = -2 \implies x = -1 $$So the process is defined by $PV^{-1} = \text{constant}$.
Molar Heat Capacity
Using the formula $C = C_V + \frac{R}{1-x}$ for a mono-atomic gas ($C_V = \frac{3R}{2}$):
$$ C = \frac{3R}{2} + \frac{R}{1 – (-1)} = \frac{3R}{2} + \frac{R}{2} = \frac{4R}{2} = 2R $$Correct Option: (b) 2R