When a gas expands or is compressed, the surroundings can transfer energy mechanically by pushing on a moving boundary. Understanding this pressure-volume interaction is essential for describing thermodynamic work.

Pressure-volume work

Pressure-volume work occurs when the surroundings exert pressure on a system and the system’s volume changes. The usual AP Physics 2 picture is a gas in a cylinder with a movable piston, but the same idea applies to any system with a boundary that can move.

If the boundary does not move, no pressure-volume work is done. A gas may still have pressure, but pressure alone is not enough. Work requires both a force and a displacement, so the system’s volume must increase or decrease.

External pressure and the boundary

The important pressure for this subtopic is the pressure applied by the surroundings to the system boundary.

External pressure can come from atmospheric air, a piston, added weights, or any other outside agent pushing on the system. In many problems, it is constant. In others, it may change during the process, in which case an average external pressure may be used.

A helpful physical picture is a piston of area $A$ moving a distance $x$.

A weighted piston–cylinder diagram showing a gas under a movable piston, with an external load providing the surrounding pressure. It makes the “moving boundary” idea concrete: the surroundings apply $P_{ext}$ to the piston face, and piston motion produces a change in volume. This is the standard physical model behind pressure–volume work in introductory thermodynamics. Source

The surroundings exert a force with magnitude $F=P_{ext}A$. The volume change has magnitude $Ax$, so the magnitude of the work is $Fx=P_{ext}Ax=P_{ext}|\Delta V|$. This is why work by pressure is directly connected to volume change.

Equation for constant external pressure

For a constant external pressure, the work done on the system is given by

This form uses the AP Physics convention that $W$ means work done on the system. Because of that convention, the minus sign is essential.

The units are consistent with energy. Since $1 \ Pa = 1 \ N/m^2$, multiplying pressure by volume gives $Pa\cdot m^3 = N\cdot m = J$.

Expansion and compression

The sign of $\Delta V$ tells you whether the system expands or compresses, and that determines the sign of the work done on the system.

• Expansion: $V_f>V_i$, so $\Delta V$ is positive. Then $W=-P_{ext}\Delta V$ is negative.

• Compression: $V_f<V_i$, so $\Delta V$ is negative. Then $W=-P_{ext}\Delta V$ is positive.

• No volume change: $\Delta V=0$, so $W=0$.

These signs match the physical situation.

• During compression, the surroundings push inward and transfer mechanical energy to the system.

• During expansion, the system pushes outward against the surroundings, so the work done on the system is negative.

It is often useful to identify the agent doing the pushing. If the surroundings are squeezing the system smaller, the work done on the system should be positive. If the system is pushing the surroundings outward, the work done on the system should be negative.

Constant versus average external pressure

Some processes occur under a truly constant external pressure. For example, a piston may move while the outside force on it stays the same. In that case, the equation above applies directly.

Other processes do not have one unchanging value of external pressure. If the problem gives an average external pressure, you use the same structure with that average value and the total volume change. This gives the total work done on the system over the process.

Using average external pressure is an approximation or a summary description of a changing process. It is appropriate only when the problem explicitly gives that average or clearly intends you to use it.

What the equation emphasizes

This relation shows that pressure-volume work depends on:

• how strongly the surroundings push, represented by $P_{ext}$

• how much the boundary moves overall, represented by $\Delta V$

A larger external pressure means a larger force on the system boundary. A larger magnitude of volume change means the boundary has moved more. Either one increases the magnitude of the work.

The equation does not say that any increase in pressure automatically means work. A rigid container can hold a very high-pressure gas, but if its volume stays fixed, the pressure-volume work is zero. Likewise, a very small external pressure acting during a large expansion can still produce a relatively small amount of work.

Common reasoning checks

When solving problems about work done by external pressure, check these points carefully:

• Identify the system before deciding whether work is done on it or by it.

• Use the external pressure, not just any pressure mentioned in the situation.

• Decide whether the process is an expansion or a compression.

• Find $\Delta V$ from $V_f-V_i$ so the sign is correct.

• If the pressure is not constant, use an average external pressure only when the problem supports that choice.

A final conceptual check is useful: if the surroundings squeeze the system, the answer for work done on the system should come out positive. If the system expands against the surroundings, the answer should come out negative.