Explain the concept of positive and negative work with varying pressure. Draw corresponding p-V diagrams.

Answer:

When a gas in a cylinder fitted with a movable piston changes its volume, work is done by or on the gas. If the pressure changes as the volume changes from \(V_i\) to \(V_f\), the total work done by the gas is

which is equal to the area under the p-V curve between \(V_i\) and \(V_f\).

Positive work (expansion): When the gas expands (\(V_f \gt V_i\)), \(dV\) is positive, so \(W\) is positive: work is done by the gas on the surroundings.

[Figure (a): p-V diagram with p on the y-axis and V on the x-axis. A curve goes from point i (high p, small \(V_i\)) down to point f (lower p, larger \(V_f\)) with the arrow pointing towards increasing volume. The area under the curve, between \(V_i\) and \(V_f\) and the V-axis, is shaded and marked W > 0.]

Negative work (compression): When the gas is compressed (\(V_f \lt V_i\)), \(dV\) is negative, so \(W\) is negative: work is done on the gas by the surroundings.

[Figure (b): p-V diagram in which the curve goes from point i (low p, large \(V_i\)) up to point f (higher p, smaller \(V_f\)) with the arrow pointing towards decreasing volume. The area under the curve is shaded and marked W < 0.]

The magnitude of the work in each case is the area under the curve; its sign depends on the direction of the process. Since the area depends on the path between the same end states, work done depends on the path followed, not only on the initial and final states.