Two moles of an ideal monoatomic gas are expanded adiabatically and completely reversibly from an in — Thermodynamics and Thermochemistry Chemistry Question
Question
Two moles of an ideal monoatomic gas are expanded adiabatically and completely reversibly from an initial temperature of $300 \text{ K}$ down to $200 \text{ K}$. Given the molar heat capacity at constant volume $C_v = 12.5 \text{ J K}^{-1} \text{ mol}^{-1}$, calculate the net magnitude of the expansion work executed by the gas in Joules.
💡 Solution & Explanation
By definition of an adiabatic process, there is zero heat transfer across the system boundaries ($q = 0$). The First Law of Thermodynamics simplifies to $\Delta U = q + w = w$. The change in internal energy for an ideal gas is $\Delta U = n C_v \Delta T = n C_v (T_2 - T_1)$. Substituting parameters: $w = 2 \text{ mol} \times 12.5 \text{ J K}^{-1}\text{mol}^{-1} \times (200 \text{ K} - 300 \text{ K}) = 25 \times (-100) = -2500 \text{ J}$. Work done *by* the gas equals the absolute magnitude, which is $2500 \text{ J}$.