Consider a fully adiabatic piston-cylinder arrangement as shown in the figure. The piston is massless and cross-sectional area of the cylinder is 𝐴. The fluid inside the cylinder is air (considered as a perfect gas), with γ being the ratio of the specific heat at constant pressure to the specific heat at constant volume for air. The piston is initially located at a position 𝐿₁. The initial pressure of the air inside the cylinder is 𝑃₁ ≫ 𝑃₀, where 𝑃₀ is the atmospheric pressure. The stop S1 is instantaneously removed and the piston moves to the position 𝐿₂, where the equilibrium pressure of air inside the cylinder is 𝑃₂ ≫ 𝑃₀.

What is the work done by the piston on the atmosphere during this process?

Correct Answer:

The correct answer is $P_{0}A(L_2-L_1)$.

Analysis of the Figure:

Based on the schematic diagram of the fully adiabatic piston-cylinder system, we observe the following details:

1. The cylinder has a constant cross-sectional area, denoted as $A$.

2. The cylinder initially holds "Air, pressure $P_1$" on the left side of the piston.

3. The piston is initially held at the "Initial position of the piston" at a distance of $L_1$ from the closed end of the cylinder, secured by the stop $S_1$.

4. The stop $S_1$ is removed, allowing the piston to travel to the "Final position of the piston" at a distance of $L_2$, where it is stopped by $S_2$.

5. The right side of the piston is open to the external environment, labeled "Atmosphere, pressure $P_0$".

Step-by-Step Derivation and Logical Reasoning:

Step 1: Identify the relevant system boundary
We need to calculate the work done by the piston on the atmosphere. It is crucial to distinguish this from the work done by the expanding air inside the cylinder. The expansion of the air is an irreversible process, but the boundary of the atmosphere is pushed back at a constant pressure.

Step 2: Formulate the work done on the atmosphere
The atmosphere behaves as a constant pressure boundary with a pressure of $P_0$. The work done by any system (in this case, the piston) pushing against a constant pressure environment is given by the constant external pressure multiplied by the change in volume of that environment:

$$W=P_{\text{ext}}·\Delta V$$

Here, the external pressure is $P_{\text{ext}}=P_0$.

Step 3: Determine the volume change
The piston moves from position $L_1$ to position $L_2$ within a cylinder of cross-sectional area $A$. The volume displacement of the piston $\Delta V$ is:

$$\Delta V=A(L_2-L_1)$$

Step 4: Calculate the final work
By substituting the volume displacement $\Delta V$ into the work formula, we find:

$$W=P_{0}A(L_2-L_1)$$

This matches the correct option provided.