Solved thermodynamics exercise

Compression under constant external pressure

Exercise 8 · Lesson 4 — The First Law of Thermodynamics

Statement

A gas is compressed from a volume $V_A$ to a volume $V_B<V_A$ under a constant external pressure $P_0$ .

Calculate the work received by the gas.
The gas returns from $V_B$ to $V_A$ against the same external pressure $P_0$ . Calculate the work received and compare it with the compression work.
Repeat the expansion from $V_B$ to $V_A$ when the gas expands into a vacuum. Interpret the three signs obtained.

Hint

Use

W=-\int P_{\mathrm{ext}}\,\mathrm{d}V

. The external pressure must be integrated, including when the transformation is not quasi-static.

Solution

Question 1. The external pressure is constant, so

W_{A\to B}=-P_0(V_B-V_A)=P_0(V_A-V_B)>0.

The gas receives work during compression.

Question 2. For the return path,

W_{B\to A}=-P_0(V_A-V_B)<0.

The two works are opposites because the transformations are carried out against the same constant external pressure.

Question 3. In a vacuum, $P_{\mathrm{ext}}=0$ , so

W_{B\to A}^{\mathrm{vide}}=0.

Thus, expansion does not automatically imply negative work: an external force must actually be displaced.