Work Done by a Gas
On a p–V graph, work is an area.
Lesson at a glanceShow the key points
- ΔW = pΔV; W = ∫p dV (26.2): the area under the p–V curve; valid for any shape and for solids, liquids and phase changes.
- Isothermal ideal gas: W = nRT ln(V₂/V₁) (26.3).
- Isobaric: W = p(V₂ − V₁). Isochoric: W = 0.
- Compression gives negative work; round a cycle the net work is the enclosed area.
A gas pushing a piston does work. On a pressure–volume graph that work has a simple picture: an area.
Work by an expanding gas
Gas at pressure p pushes on a piston of area A with force pA. If the piston moves out a small distance Δx, the work done by the gas is (pA)Δx = pΔV. For a larger change the pressure may vary, so add up small steps:
Expanding (V₂ > V₁): W positive, the gas does work. Compressed: W negative, work is done on the gas.
Three standard processes
Isothermal (constant T), ideal gas
With p = nRT/V:
Isobaric (constant p)
Isochoric (constant V)
The volume doesn’t change, so W = 0.
2.0 mol of an ideal gas at 300 K expands isothermally to twice its volume. How much work does it do?
A gas goes round a cycle: at 200 kPa from 1.0 L to 4.0 L, then at constant volume down to 50 kPa, then back to 1.0 L at 50 kPa, then up at constant volume. Find the work in each part and the net work.
- 1W = ∫p dV = area under the p–V curve; positive for expansion.
- 2Isothermal ideal gas: W = nRT ln(V₂/V₁). Isobaric: W = p(V₂ − V₁). Isochoric: W = 0.
- 3Round a cycle, the net work is the enclosed area; clockwise means work done by the gas.
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