Question #42536

A rigid container holds an ideal diatomic gas. The container loses heat to the surrounding environment (at a constant temperature $T_{\text{env}} = 300\text{ K}$) at a rate $\frac{dQ_{\text{loss}}}{dt} = H(T - T_{\text{env}})$, where $H = 1\text{ W/K}$ and $T$ is the instantaneous gas temperature.

A semiconductor heater inside the container is powered by a constant voltage source $V$. The resistance of the heater depends on the gas temperature as $R(T) = R_0 e^{\beta/T}$, where $\beta = 600 \ln 2\text{ K}$ and $\frac{V^2}{R_0} = 600\text{ W}$. Assuming the heater is always in thermal equilibrium with the gas, and the gas is initially at $300\text{ K}$, what is the final steady-state temperature of the gas?