Work done from a $P$-$V$ diagram

The work done by a gas is $W = \int P\,dV$ which is equal to the area under the curve in the $P$-$V$ diagram.

Path $A \to B \to C \to D$

From the given diagram, the process consists of horizontal and vertical segments.

• Along a horizontal segment, pressure is constant, so $W = P\Delta V$
• Along a vertical segment, volume is constant, so $W=0$

Evaluate each segment

From the diagram:

• $A \to B$ is an expansion at pressure $P_0$ from volume $V_0$ to $3V_0$. So, $W_{AB} = P_0(3V_0 - V_0) = 2P_0V_0$

• $B \to C$ is vertical, hence $W_{BC}=0$

• $C \to D$ is a compression at pressure $2P_0$ from volume $3V_0$ to $V_0$. So, $W_{CD} = 2P_0(V_0 - 3V_0) = -4P_0V_0$

Total work done

$W = W_{AB}+W_{BC}+W_{CD}$ $W = 2P_0V_0 + 0 - 4P_0V_0$ $W = -2P_0V_0$

Compare with options

The calculated result is $\boxed{-2P_0V_0}$

This does not match any of the given options.

Comparison with stored answer

Stored correct answer is B: $-3P_0V_0$.

My derived answer is $-2P_0V_0$, so I do not agree with the stored answer. This suggests either the diagram values were read differently or there may be an issue in the stored answer / reproduced figure.