An ideal gas, $$\bar{C}_v = \frac{5}{2}R$$, is expanded adiabatically against a constant pressure of 1 atm until it doubles in volume. If the initial temperature and pressure is $$298 \text{ K}$$ and $$5 \text{ atm}$$, respectively then the final temperature is _______ K (nearest integer). [$$\bar{C}_v$$ is the molar heat capacity at constant volume]

Adiabatic expansion against constant external pressure (irreversible).

$$q = 0$$, so $$\Delta U = w = -P_{ext}(V_2 - V_1)$$.

$$nC_v(T_2 - T_1) = -P_{ext}(V_2 - V_1)$$. Using ideal gas: $$PV = nRT$$.

$$V_1 = nRT_1/P_1 = nR(298)/5$$, $$V_2 = 2V_1 = 2nR(298)/5$$.

$$nC_v(T_2 - 298) = -1 \times (2V_1 - V_1) = -V_1 = -nR(298)/5$$.

$$\frac{5R}{2}(T_2 - 298) = -\frac{R(298)}{5}$$

$$T_2 - 298 = -\frac{2 \times 298}{25} = -23.84$$

$$T_2 = 298 - 23.84 = 274.16 \approx 274$$ K.

The answer is 274.