Three moles of an ideal gas being initially at a temperature 273 K was isothermally expanded 5.0 tim — Thermodynamics and Thermochemistry Chemistry Question
Question
Three moles of an ideal gas being initially at a temperature 273 K was isothermally expanded 5.0 times its initial volume and then isochorically heated so that the pressure in the final state became equal to that in the initial state. The total amount of heat transferred to the gas during the process equals 80 kJ. The value of $C_{v,m}$ for this gas (in cal/K-mol) is
💡 Solution & Explanation
Step 1 (Isothermal): $q_1 = nRT_1 \ln(V_2/V_1) = 3 \times 8.314 \times 273 \ln(5) \approx 10955\text{ J} = 10.95\text{ kJ}$. Step 2 (Isochoric): $V$ is now $5V_1$, so to restore initial pressure $P_1$, $T_3$ must be $5T_1 = 1365\text{ K}$. $q_2 = nC_V(T_3 - T_1) = 3 C_V (1365 - 273) = 3276 C_V$. Total $q = q_1 + q_2 \implies 80000 = 10955 + 3276 C_V \implies C_V \approx 21.07\text{ J/K-mol}$. Converting to cal/K-mol: $21.07 / 4.184 \approx 5.03\text{ cal/K-mol}$. Nearest integer is 5.