Where gas pressure really comes from: a crowd of tiny balls hitting the walls

How to use this simulation

1. Start with the defaults: N₂ gas at 300 K with 120 molecules. Molecules are coloured by speed: blue is slow and red is fast. A wall glows orange for a moment when a molecule bounces off it.
2. Look at the pressure panel. The top bar is the pressure found by adding up the impulses of the wall hits. The bottom bar is the formula NkT/V. Wait a few seconds: the first bar creeps up to the second and then jitters around it by a few percent. That jitter is real, and it shrinks as you add molecules.
3. Raise the temperature. The molecules get faster, hit harder and more often, and both pressure bars rise. In the histogram the whole hump slides right and flattens, and the green dashed line (the rms speed) moves with it.
4. Switch the gas from N₂ to He. Same temperature, but the molecules are far faster. Helium is 7 times lighter than nitrogen, so its speeds are about 2.65 times larger. Notice the readout of mean kinetic energy does not change: that is set by temperature alone.
5. Drag the volume slider to half. The same molecules now have less room, so they reach a wall twice as often, and the pressure doubles. That is Boyle’s law, and you did not have to take it on trust.
6. Reduce N to 20 and the histogram turns ragged and the pressure bar twitches. Push N to 200 and both settle down. A gas law is a statement about crowds, not individuals.

A gas as a swarm of tiny balls

Kinetic theory starts from a picture so simple it is almost cheeky: a gas is a huge number of tiny particles flying around in straight lines, bouncing off the walls of their container. Everything else, from the pressure of a car tyre to the warmth of a room, comes from that picture.

The point of this chapter is to turn the picture into numbers. We want a formula for pressure, a meaning for temperature, and a speed for the molecules. By the end you will have all three, and each one has a reading in the simulation.

What we assume about an ideal gas

The theory works because we make a few simplifying assumptions. Learn these, because they are a standard exam question.

• A gas contains a very large number of identical molecules, in constant random motion.
• The molecules are tiny compared with the distances between them, so their own volume can be ignored.
• There are no forces between molecules except during collisions, which are perfectly elastic. No energy is lost.
• Collisions with the walls are also elastic, and the time a collision takes is negligible.
• Molecules move in straight lines between collisions. Gravity is ignored.
• All directions are equally likely, so the gas has no overall flow.

How hits on the wall become pressure

When a molecule of mass m hits a wall head-on with speed vₓ and bounces straight back, its momentum changes from +mvₓ to −mvₓ. The wall gets an impulse of 2mvₓ. One hit is nothing. But with 120 molecules, each hitting the wall again and again, the walls feel a steady push, and pressure is just that push per unit area.

Do the sum for a cubic box of side L. A molecule crosses the box and returns every 2L/vₓ seconds, so it hits that wall vₓ/2L times per second. The force from one molecule is 2mvₓ × vₓ/2L = mvₓ²/L. Add up N molecules, divide by the wall area L², and use the fact that the three directions are equal, and you get the result below.

PV = NkT: from the theory to the gas law

Experiments gave us PV = NkT (or PV = nRT with moles) long before anyone knew molecules existed. Kinetic theory explains it. Combine the pressure result P = (1/3)(N/V) m v²(rms) with the experimental law and the product PV drops out as N times a quantity that depends only on temperature.

With the constants: k = 1.381 × 10⁻²³ J/K is Boltzmann’s constant, the gas constant is R = 8.314 J/(mol·K), and N is the number of molecules. In the simulation the box is 10 nm on each side, so its volume is 1 × 10⁻²⁴ m³.

Take the default gas: N = 120 molecules at T = 300 K. Then kT = 4.14 × 10⁻²¹ J, and P = NkT/V = 120 × 4.14 × 10⁻²¹ ÷ 1 × 10⁻²⁴ ≈ 497 kPa, which is about 4.9 atmospheres. That is the value of the lower pressure bar when the volume slider is at 1.

Boyle’s and Charles’s laws, straight from the sliders

Halve the volume at the same temperature: 120 molecules at 300 K now give P = NkT/(V/2) = 994 kPa. The pressure doubled, so pressure is inversely proportional to volume. This is Boyle’s law.

Raise the temperature from 300 K to 600 K at the same volume: P = 994 kPa. The pressure is proportional to the absolute temperature, which is Gay-Lussac’s law, and at constant pressure the same reasoning gives volume proportional to T, Charles’s law.

Notice what the slider changes in each case. Smaller volume means more wall hits per second. Higher temperature means harder wall hits, and more of them, because the molecules cross the box faster.

Temperature is the average kinetic energy

Put the two expressions for pressure side by side: P = (1/3)(N/V) m v²(rms) and PV = NkT. The N and V cancel and we are left with a neat statement about a single molecule.

Mean translational kinetic energy per molecule is (3/2)kT. At 300 K that is 6.21 × 10⁻²¹ J, or 6.21 in units of 10⁻²¹ J, as the readout shows. The mass of the molecule does not appear. A helium atom and an oxygen molecule at the same temperature carry the same average kinetic energy. The heavier one is just slower.

This is also what temperature means at the microscopic level. It is not heat and it is not the total energy. It is a measure of the average kinetic energy of the molecules. At absolute zero this motion would stop (in the classical picture).

The root-mean-square speed: √(3RT/M)

Turn (1/2)m v² = (3/2)kT around and the speed pops out. Using molar mass M in kilograms per mole and the gas constant R, the root-mean-square speed is the formula below. Make sure M is in kg/mol: 28 g/mol is 0.028 kg/mol.

The name tells you the order of operations. You square each speed, take the mean, then take the square root. It is larger than the plain average speed because squaring favours the fast molecules.

Not all molecules are equally fast: Maxwell–Boltzmann

The rms speed is a typical speed, not everyone’s speed. Molecules in a gas bump into each other and the walls and swap energy, so at any moment some are slow, most are middling and a few are very fast. The histogram on the right shows the speeds of the molecules in the box. The red curve is the Maxwell–Boltzmann distribution that theory predicts.

The curve starts at zero (almost no molecules are stationary), rises to a peak, and then has a long tail to the right. It is lopsided, so the three famous speeds differ. For nitrogen at 300 K the most probable speed is 422 m/s, the mean speed is 476 m/s and the rms speed is 517 m/s.

Heat the gas and the peak moves right and sinks, because the area under the curve is always the same (every molecule is somewhere). A lighter gas at the same temperature gives the same shape stretched to higher speeds. The bars are ragged for small N and settle onto the curve as N grows.

• Most probable speed v(p) = √(2RT/M) is the peak of the curve.
• Mean speed = √(8RT/πM) is the ordinary average.
• rms speed = √(3RT/M) is the largest of the three. Their ratio is about 1 : 1.13 : 1.22.
• The tail is why a few molecules can escape or react even in a cool gas.

What the animation simplifies

Real molecules are far too small and fast to watch. A nitrogen molecule at room temperature crosses a 10 nm box in about 20 picoseconds. So the animation runs in extreme slow motion, around a hundred billion times slower than real life. The speed scale is correct relative to temperature and gas, but not in seconds on your clock.

The box is a 10 nm cube, so 120 molecules in it is a dense gas, around 4.9 atmospheres. A real room has more than 10²⁵ molecules in it, which is why real gas pressure never visibly twitches.

Common mistakes

These cost marks every year.

• Putting M in grams per mole in √(3RT/M). The formula needs kilograms per mole.
• Using Celsius for T. Gas laws need kelvin.
• Saying the molecules of a heavier gas have less kinetic energy. At the same temperature, the average kinetic energy is the same. The heavier molecule is slower.
• Treating rms speed as the average speed. They are close but not equal.
• Saying temperature is the total energy of the gas. It is related to the average energy per molecule, not the sum.
• Forgetting that pressure is caused by collisions with the walls. Without wall hits, a gas has no pressure.

Exam corner

Typical short questions ask for the assumptions of kinetic theory, the formula P = (1/3)ρv²(rms), or why heating a gas in a fixed container raises its pressure. Longer questions derive the pressure formula for a cubic box or ask you to compare rms speeds of two gases.

For a comparison question, use the ratio instead of working both out. At the same temperature, v₁/v₂ = √(M₂/M₁), as in the helium and nitrogen example.

Revision list

Say these aloud before the exam.

• Gas pressure comes from molecules hitting the walls. Each hit gives an impulse of 2mvₓ.
• P = (1/3)(N/V) m v²(rms), which together with PV = NkT gives mean kinetic energy (3/2)kT.
• Temperature measures the average kinetic energy per molecule, the same for every gas.
• v(rms) = √(3RT/M), with M in kg/mol and T in kelvin.
• At constant temperature, speed goes as 1/√M. At constant gas, speed goes as √T.
• Speeds follow the Maxwell–Boltzmann curve: v(p) < mean < v(rms).
• Boyle: halve V, double P. Gay-Lussac: double T at constant V, double P.

Frequently asked questions