The particle model. A gas is made of tiny particles, very far apart compared with their size, flying about in straight lines and bouncing off each other and off the walls. The faster they move, the hotter the gas. The temperature is a measure of the average kinetic energy of the particles: for a gas in two dimensions the average kinetic energy of a particle is kT (and (3/2)kT in three dimensions), where k is the Boltzmann constant. So the typical speed goes up as the square root of the temperature:

v_typical ∝ √T

Double the speed and you need four times the temperature (in kelvin).

Diffusion. If you open a bottle of perfume in a room, the smell reaches the other side after some minutes. The particles are moving at hundreds of metres per second, so why so slow? Because each particle collides with others about ten billion times a second, and bounces in a new direction every time. It follows a zigzag path, and gets away very slowly. The net result is that the particles spread from where there are many to where there are few, until they are evenly spread. No force pushes them: it is just that there are many more ways to be spread out than to be bunched up. The spreading is faster at a higher temperature, because the particles are faster. (Try it with the hot, room and cold gas.)

The speeds. At the start, all the particles start with the same speed. After a few collisions they have a range of speeds: some are slow and some are fast. The collisions share the energy at random, and the speeds end up in a pattern that does not depend on how they started, the Maxwell-Boltzmann distribution. For particles in two dimensions the fraction with speed v is proportional to v exp(−mv²/2kT). The orange curve shows this, and the bars show the particles in the box. The speed axis of the graph is fixed, so you can compare temperatures: when the gas is colder the peak moves to lower speeds and the curve becomes narrower and taller; when it is hotter the peak moves to higher speeds and the curve spreads out. The area under the curve stays the same, because the number of particles does not change. The dashed white curve shows room temperature (300 K) for comparison.

What the model leaves out. The particles are hard discs in two dimensions, with perfectly elastic collisions and no forces between them. Real particles are much smaller, much more numerous and move in three dimensions. The units of the time and the speed are chosen so that you can see them. The “temperature” sets the average kinetic energy in the model in the same way as for a real gas.

Things to try. (1) Open the barrier, and watch how long it takes for the numbers on the two sides to become equal. (2) Do it again at a lower and a higher temperature. How does the time change? (3) Look at the speeds: how does the pattern change when you double the temperature? (4) Tick Follow one particle. Why does it take a zigzag path?