Question #42192

A thermally insulated cylinder with a movable piston contains a solid semiconductor sample of constant heat capacity $C_s = 4R$ and a gas mixture in thermal equilibrium at an initial temperature $T_0$. The gas mixture consists of a total of $4$ moles of Helium (He) and Oxygen ($\text{O}_2$). According to Dalton's law of partial pressures, the initial partial pressure of He is equal to the partial pressure of $\text{O}_2$. The system undergoes a reversible adiabatic expansion from an initial volume $V_0$ to a final volume $8V_0$. The intrinsic carrier concentration of the semiconductor is given by $n_i(T) = A T^{3/2} e^{-E_g / 2k_B T}$, where the band gap energy is $E_g = 4 k_B T_0$. Assuming the volume of the semiconductor is negligible, what is the ratio of the final intrinsic carrier concentration to the initial intrinsic carrier concentration, $\frac{n_i(T_f)}{n_i(T_0)}$?