The number of degrees of freedom of molecules in a gas whose molar heat capacity is 29 J/mol-K in th — Thermodynamics and Thermochemistry Chemistry Question

💡 Solution & Explanation

$PT = \text{const} \implies P(PV/R) = \text{const} \implies P^2V = \text{const} \implies PV^{1/2} = \text{const}$. This is a polytropic process with $n=1/2$. Molar heat capacity $C = C_V + \frac{R}{1-n} = C_V + \frac{R}{1 - 0.5} = C_V + 2R$. Given $C = 29\text{ J/K-mol}$ and $R = 8.314\text{ J/K-mol}$: $C_V + 2(8.314) = 29 \implies C_V = 29 - 16.628 = 12.372\text{ J/K-mol}$. Equating $C_V = \frac{f}{2}R \implies \frac{f}{2}(8.314) = 12.372 \implies f \approx 3$.