Solution for Question 9
Initial State Analysis:
The piston is initially in equilibrium. This implies the pressure and temperature are equal on both sides ($P_A = P_B = P_0$ and $T_A = T_B = T_0$). The volumes $V_A$ and $V_B$ are not necessarily equal.
Using the Ideal Gas Law ($PV = nRT$), the ratio of moles is determined by the ratio of volumes:
$$\frac{n_A}{n_B} = \frac{P_0 V_A / RT_0}{P_0 V_B / RT_0} = \frac{V_A}{V_B}$$
This ratio $\frac{V_A}{V_B}$ is a fixed constant for the system because $n_A$ and $n_B$ are constant.
The Process:
- Heat is supplied to part A. The piston is released. It moves due to pressure difference.
- Oscillation & Damping: The piston oscillates. Although the piston material is insulating, the moving piston transfers momentum during molecular collisions. Molecules from the hotter side strike the moving piston and transfer kinetic energy to the piston, which then transfers it to molecules on the cooler side.
- Final Equilibrium: This “momentum transfer” acts as a mechanism for energy exchange. Eventually, the system reaches a steady state where pressures are equal ($P_f$) and temperatures are equal ($T_f$).
Final Position:
In the final state, we again have $P_A = P_B = P_f$ and $T_A = T_B = T_f$. Applying the Ideal Gas Law again:
$$\frac{V’_{A}}{V’_{B}} = \frac{n_A R T_f / P_f}{n_B R T_f / P_f} = \frac{n_A}{n_B}$$
The final volume ratio must be identical to the initial volume ratio ($\frac{V_A}{V_B}$). Since the total volume $V_{total} = V_A + V_B$ is fixed, the individual volumes must return to their initial values ($V’_{A} = V_A$ and $V’_{B} = V_B$).
Thus, the piston stops exactly at its initial position.