The molar heat capacity ($C$) of an ideal gas undergoing a polytropic process defined by $PV^n = \text{constant}$ is given by the relation derived from the First Law of Thermodynamics ($q = \Delta U - w$): $C = C_V + \frac{R}{1-n}$.

1. Identify the gas type: The gas is monatomic. From the provided data, the molar heat capacity at constant volume for monatomic gases (like Helium, Argon) is approximately $12.5 \text{ J mol}^{-1} \text{ K}^{-1}$, which corresponds to $C_V = \frac{3}{2}R$ .
2. Identify the process: The given equation is $PV^3 = \text{constant}$, so the polytropic index $n = 3$.
3. Calculate C: $C = \frac{3}{2}R + \frac{R}{1-3}$ $C = \frac{3}{2}R + \frac{R}{-2}$ $C = \frac{3}{2}R - \frac{1}{2}R = \frac{2}{2}R = R$.