Work done by a gas and the p-V diagram

Work by a pushing gas

A gas pushes on a piston of area $A$ with force $pA$. If the piston moves out a small distance $dx$, the gas does work $pA\,dx = p\,dV$. Over a whole expansion:

which is the area under the path on a graph of $p$ against $V$. Expanding (moving right) gives positive work; squeezing gives negative work.

Two simple paths

• Steady pressure (isobaric): a rectangle, $W = p\,\Delta V$. For an ideal gas this is also $nR\,\Delta T$.
• Steady temperature (isothermal): $p = \dfrac{nRT}{V}$, a curve, and $W = nRT\ln\dfrac{V_2}{V_1}$.

You should find that the steady-pressure path gives more work, because its pressure does not fall as the gas expands.

Paths and cycles

Between the same two states, different paths enclose different areas, so the work depends on the path. Go round a closed loop, and the net work is the area inside the loop: positive if the loop runs clockwise, negative if anticlockwise.

Watch out. The area must be in $\text{Pa} \times \text{m}^3 = \text{J}$. Litres must become cubic metres.

A worked example

A gas at $3.0 \times 10^{5}\ \text{Pa}$ and $2.0\ \text{L}$ expands to $6.0\ \text{L}$. Find the work at a steady pressure and at a steady temperature.

1. Steady pressure. $3.0 \times 10^{5}\ \text{Pa} \times 4.0 \times 10^{-3}\ \text{m}^3 = 1200\ \text{J}$.
2. Steady temperature. $pV = 600\ \text{J}$, so $W = 600\ \text{J}\ln 3 = 659\ \text{J}$.
3. So the same start and end volumes give quite different work.

Practise with new numbers

Try it in Python

The program works out both paths of the example.

Remember these

• $W = \int p\,dV$: the area under the path.
• Steady $p$: $p\,\Delta V$; steady $T$: $nRT\ln\dfrac{V_2}{V_1}$.
• Work depends on the path; a cycle's net work is the area enclosed.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A gas at a steady $2.0 \times 10^{5}\ \text{Pa}$ shrinks from $5.0\ \text{L}$ to $3.0\ \text{L}$. What work does the gas do?

2. A gas is heated in a sealed rigid tank. How much work does it do?

3. A loop on a p-V graph runs anticlockwise. Is the net work done by the gas positive or negative?

4. Why does the work of an isothermal expansion depend on $\ln\dfrac{V_2}{V_1}$, not on $V_2 - V_1$?