Work on a Gas

Definition of Work on a Gas

Work is done on or by a gas when it changes volume against a pressure. This happens in two main ways:

• Expansion: When a gas expands, it pushes against its container or surroundings, doing work on the surroundings.
• Compression: When a gas is compressed, work is done on the gas by an external force pushing it into a smaller volume.

In both cases, energy is transferred by work. The gas either gains energy (when compressed) or loses energy (when expanding), but the focus here is on the physical work done related to volume change and pressure.

Calculating Work Done on a Gas

The work done on or by a gas during a volume change can be calculated using the formula:

Work done, $W = P \times \Delta V$

• $P$ is the pressure of the gas (in pascals, Pa)
• $\Delta V$ is the change in volume of the gas (in cubic metres, m³)
• Work done, $W$, is measured in joules (J)

Note that:

• If the gas expands, $\Delta V$ is positive, so the work done by the gas is positive (energy transferred from the gas).
• If the gas is compressed, $\Delta V$ is negative, so the work done on the gas is positive (energy transferred to the gas).

To clarify the sign convention: the formula $W = P \times \Delta V$ gives a positive value when the gas expands (work done by the gas) and a negative value when the gas is compressed (work done on the gas). The sign indicates the direction of energy transfer.

For example, if a gas at a constant pressure of 100,000 Pa expands from 0.002 m³ to 0.005 m³, the work done by the gas is:

$$W = 100,000 \times (0.005 - 0.002) = 100,000 \times 0.003 = 300 \text{ J}$$

Pressure-Volume (PV) Diagrams

A PV diagram shows how the pressure of a gas changes as its volume changes. The graph plots pressure (y-axis) against volume (x-axis).

Key points about PV diagrams:

• The area under the curve on a PV diagram represents the work done during the volume change.
• For a gas expanding, the curve moves to the right (volume increases), and the work done is by the gas on the surroundings.
• For a gas being compressed, the curve moves to the left (volume decreases), and the work done is on the gas by the surroundings.

If the pressure is constant during the process, the area under the curve is a rectangle, so work done is simply pressure × change in volume. For variable pressure, the area under the curve can be found by calculating or estimating the area under the graph.

For example, if a gas is compressed from 0.004 m³ to 0.001 m³ at a constant pressure of 200,000 Pa, the work done on the gas is:

$$W = 200,000 \times (0.001 - 0.004) = 200,000 \times (-0.003) = -600 \text{ J}$$

The negative sign shows work is done on the gas (energy transferred to the gas).

Applications and Examples

Work done on gases is important in many real-life situations, especially in engines and pistons:

• Engines: In car engines, fuel combustion causes gases to expand, pushing pistons and doing work to move the vehicle.
• Pistons: Pistons compress gases in engines or pumps, doing work on the gas to increase pressure and temperature.
• Relating to temperature: When a gas is compressed, work done on it can increase its temperature (see Energy Changes in a System topic for more details on temperature changes).

Understanding work on gases helps explain how machines convert energy and how gases behave under different conditions.

For instance, if a piston compresses 0.003 m³ of gas at 150,000 Pa to 0.001 m³, the work done on the gas is:

$$W = 150,000 \times (0.001 - 0.003) = 150,000 \times (-0.002) = -300 \text{ J}$$

This means 300 J of work is done on the gas, transferring energy to it.