8.3 Particle model and pressure — Learn

Prerequisites: Section 8.2, especially internal energy and the link between temperature and particle motion; Section 2.4, especially pressure and $p = \dfrac{F}{A}$.

By the end of this section you can: explain gas pressure using the motion of the particles; explain how temperature and volume change the pressure of a gas; calculate a pressure or a volume using $pV = \text{constant}$; and calculate the force a gas exerts on a surface. At Higher tier you can also explain how doing work on a gas raises its temperature.

1. Particles in a gas

The particles in a gas are in constant random motion. They move in straight lines until they hit something, then bounce off in a new direction.

The temperature of the gas is related to the average kinetic energy of the particles. A hotter gas has particles with a higher average kinetic energy, so they move faster.

2. What causes gas pressure

When a gas particle hits the wall of its container, it changes direction. A force is needed to change its direction, and the particle pushes back on the wall. Each collision gives the wall a tiny push.

A gas contains a huge number of particles, and they hit the walls many times every second. The total push on each unit of area is the pressure of the gas.

The pressure acts at right angles to the wall, or to any surface the gas touches. A gas in a sealed container pushes outwards on every wall at once.

Pressure is calculated from force and area.

where $p$ is the pressure in pascals, $F$ is the force in newtons and $A$ is the area in square metres.

Worked example 1. A gas pushes on a window of area $0.015\,\text{m}^2$ with a force of $3000\,\text{N}$. Calculate the pressure of the gas on the window.

3. Temperature and pressure at constant volume

Suppose a gas is sealed in a rigid container, such as a metal cylinder. The volume cannot change.

If the gas is heated, the particles gain kinetic energy and move faster. They hit the walls more often, and each collision is harder. The pressure increases.

If the gas is cooled, the particles slow down. They hit the walls less often and less hard, so the pressure decreases.

Worked example 2. A sealed metal can contains air. The can is placed in a freezer. Explain what happens to the motion of the air particles and to the pressure inside the can.

A model answer is:

• The air particles cool, so their average kinetic energy falls and they move more slowly.
• They collide with the walls of the can less often and with less force.
• The pressure inside the can therefore decreases.

4. Volume and pressure at constant temperature

A gas can be compressed or expanded by a pressure change. When you push a piston into a cylinder, the gas is squashed into a smaller volume.

For a fixed mass of gas held at a constant temperature, the pressure and the volume are linked by this relationship.

This means that the pressure multiplied by the volume does not change. For two sets of conditions on the same gas:

where $p_1$ and $V_1$ are the starting pressure and volume, and $p_2$ and $V_2$ are the new pressure and volume.

If the volume halves, the pressure doubles. If the volume doubles, the pressure halves.

Worked example 3. A fixed mass of gas at $100\,\text{kPa}$ fills a volume of $0.30\,\text{m}^3$. The gas is compressed at constant temperature until its volume is $0.10\,\text{m}^3$. Calculate the new pressure.

The pressure rises to $300\,\text{kPa}$.

Worked example 4. A fixed mass of gas at $100\,\text{kPa}$ fills $0.60\,\text{m}^3$. It expands at constant temperature to a pressure of $150\,\text{kPa}$. Calculate the new volume.

5. Doing work on a gas (Higher tier)

Work is the transfer of energy by a force. When you push a piston into a cylinder, you do work on the gas inside.

Doing work on a gas increases its internal energy. The particles gain kinetic energy, so the temperature of the gas can rise.

A bicycle pump is a good example. When you push the handle, you do work on the air trapped in the pump. The air gets warmer, and so does the pump barrel.

Worked example 5. A student pushes the handle of a bicycle pump quickly. The air inside the pump gets warm. Explain why, using energy ideas.

A model answer is:

• Pushing the handle does work on the air inside the pump, because a force moves through a distance.
• The work done is transferred to the internal energy of the air.
• The particles of air gain kinetic energy, so the temperature of the air rises.

The wording above uses "the work done is transferred to the internal energy". You can also say that the work done increases the internal energy of the gas.

6. Pressure and volume

The figure shows a fixed mass of gas at a constant temperature inside a cylinder. You can change the volume with the slider. The pressure changes to keep $pV$ constant.

Predict first. Do not move the slider yet. Write down your answers.

1. If the volume is halved, what happens to the pressure?
2. What happens to the value of $pV$ as you change the volume?

Test. Move the volume slider. Watch the pressure and the value of $pV$. Try to halve the volume and check the pressure.

Explain. A smaller volume means that each particle hits the walls more often. Explain, in your own words, why the pressure rises when the volume falls at a constant temperature. Write your answer in two or three sentences.

Watch out. $pV = \text{constant}$ applies only to a fixed mass of gas at a constant temperature. If gas leaks out, or if the temperature changes, the relationship does not hold. Also check the units of pressure and volume before you substitute: both pressures must have the same unit, and both volumes must have the same unit. The constant only has a meaning when the temperature does not change. Re-check by asking whether the amount of gas and the temperature are both fixed.

Progress check

Return to the chapter problem. The weather balloon holds $2.0\,\text{m}^3$ of gas at $100\,\text{kPa}$ when it is filled in the valley. High above, the pressure is $25\,\text{kPa}$ and the temperature of the gas is the same.

1. Calculate the volume of the gas at the higher altitude.
2. Explain, using the particle model, why the balloon expands as the pressure outside falls.

You should now be able to

• describe the random motion of the particles in a gas;
• explain gas pressure in terms of collisions with the walls;
• explain how heating a gas at constant volume changes its pressure;
• use $p = \dfrac{F}{A}$ to find a force or a pressure;
• use $pV = \text{constant}$, or $p_1V_1 = p_2V_2$, for a fixed mass of gas at constant temperature;

• explain how doing work on a gas increases its internal energy and can raise its temperature.