Question #43669

A monatomic ideal gas is maintained in a container of initial volume $V_0$ at an initial absolute temperature $T_0$. The atoms of the gas have a ground state energy $E_0 = 0$ and a first excited state energy $E_1 = \Delta E$, where $\Delta E \gg k_B T_0$ ($k_B$ is the Boltzmann constant). The gas undergoes a reversible adiabatic expansion to a final volume $V_f = 8 V_0$. Assuming the total emission intensity $I$ (total power radiated by the entire gas) of the spectral line corresponding to the transition $E_1 \rightarrow E_0$ is directly proportional to the total number of atoms in the excited state, which of the following expressions represents the ratio of the final emission intensity $I_f$ to the initial emission intensity $I_0$?