Kinetic Theory Gas Simulator

The gas law is not assumed here

Every gas law calculator takes PV = nRT as given and rearranges it. This page contains no gas law at all. It contains atoms with positions and velocities, walls they bounce off, and Newton’s laws. Pressure is measured by adding up the momentum the atoms deliver to the walls. Temperature is measured from their mean kinetic energy. Nothing connects the two.

Then read the top figure. PA/NkT, where k is the Boltzmann constant, sits at about 1, and it stays there when you change the temperature, the box, or the gas. That is the whole content of kinetic theory: the gas law is not a separate law of nature, it is what mechanics and statistics produce when you have enough particles to average over.

Where pressure comes from

An atom hits a wall and rebounds. Its momentum along that direction goes from +mv to −mv, so the wall receives 2mv. One impact is a kick, not a pressure. What a wall actually feels is an enormous number of these arriving in quick succession, and the average force is the total momentum delivered divided by the time it took.

That is exactly what the simulation does, which is why the pressure readout says “measuring” for a moment when you change something: it needs a window of time to average over, because most individual timesteps contain no wall impact at all.

Why this gas is flat, and what that changes

The screen is two-dimensional, so the gas is too. That is an honest simplification rather than a hidden one, and it changes several formulas that are usually quoted in their three-dimensional form.

• Equipartition. Mean kinetic energy is kT per atom here, one half kT per degree of freedom with two of them. In three dimensions it is 3kT/2.
• The speed distribution. f(v) ∝ v·e^(−mv²/2kT), with one power of v from the circumference of a ring in velocity space. The familiar v² comes from the surface area of a sphere and does not belong in a flat gas.
• Pressure. A wall in two dimensions is a line, so pressure is force per unit length, quoted here in millinewtons per metre. PA = NkT still balances dimensionally.
• Characteristic speeds. Most probable √(kT/m), mean √(πkT/2m), rms √(2kT/m). All three differ from the 3D values.

Using the three-dimensional formulas on this simulation would put the temperature out by 50 percent and the peak of the distribution out by a factor of √2, and it would look entirely plausible while doing it.

Watch a distribution assemble itself

Set the starting speeds to “All identical”. Every atom now has the same speed in a random direction, which is a state no real gas can hold, and the histogram is a single spike while the whole box is one colour.

Press play. Collisions begin redistributing energy between pairs, and within a few collisions per atom the spike has spread out and settled onto the Maxwell-Boltzmann curve. Two things are worth noticing while it happens. The temperature does not change, because elastic collisions conserve total energy, so the gas is thermalising rather than heating. And it never goes back: the distribution is where a randomly colliding population ends up, and there is nothing to push it off again.

Where the ideal gas law starts to fail

Now push the atom count up and watch PA/NkT climb. With argon, the default gas, 400 atoms in the smallest box cover about 20 percent of it and the ratio reaches about 1.6, and it is not an error.

The ideal gas law assumes atoms are points. Real ones occupy space, so when they fill an appreciable fraction of the container each atom has less room than the box area suggests, and it reaches the walls more often. For hard discs the leading correction is PA/NkT = 1 + 2φ, where φ is the fraction of the area the atoms cover, which is the “area filled” readout. At φ of 8 percent that predicts 1.16, and the simulation measures 1.21. The difference is what the formula leaves out: the next term in the series, and the thin strip along each wall that no atom’s centre can reach.

This is the same physics as the b term in the van der Waals equation. Turn the atom count back down and the ratio returns to 1, because the ideal gas law is the dilute limit and nothing more.

Mean free path, and why it decides everything else

The bar under the box is the mean free path drawn to scale: the average distance an atom covers between collisions, 1/(2√2 n d), with n atoms per unit area and d the diameter of one. The 2d is the flat version of a collision cross-section: a moving atom hits any other whose centre comes within one diameter of its path, so it sweeps a strip two diameters wide.

Compare it with the box. When it is longer than the container, atoms cross without meeting each other and only the walls matter, which is the regime a vacuum chamber operates in and why a thermos flask works. When it is a small fraction of the box, atoms are interrupted constantly, which is why a dense gas diffuses slowly and why smells take so long to cross a still room despite molecular speeds of several hundred metres per second.

Why helium is the one that escapes

Switch between the four gases at a fixed temperature and watch the speeds. Temperature fixes the mean kinetic energy, not the speed, so speed goes as one over the square root of mass. Helium at 300 K averages about 990 metres per second where xenon manages about 170. Those are this flat gas’s speeds; a real three-dimensional gas moves faster, about 1,260 and 220 metres per second, in the same ratio.

That single fact explains a balloon going soft overnight, why Earth has kept its nitrogen and lost almost all of its primordial hydrogen and helium, and why separating uranium isotopes by gaseous diffusion is possible at all despite a mass difference under one percent.

What this model leaves out

The atoms are hard discs: they collide when they touch and ignore each other completely otherwise. Real atoms repel steeply when close, much as these discs do, but they also attract weakly at a distance, which is the a term in van der Waals and the reason gases can be liquefied at all. Nothing here can condense, no matter how cold or crowded you make it.

Collisions are also perfectly elastic, and the walls are perfectly smooth and rigid: each one returns an atom at the speed it arrived with, so no heat passes between the gas and its container. There is no rotation, no vibration, and no internal structure, so this is a monatomic gas and its heat capacity would come out accordingly. And it is flat, which is the simplification with the most consequences, all of them listed above.

Common mistakes

• Using the three-dimensional distribution on a flat gas. The v² prefactor belongs to a sphere. Here it is v.
• Reading a ratio above 1 as a broken simulation. It is the excluded-volume correction, and the area-filled readout tells you how large to expect it.
• Confusing the three characteristic speeds. The peak of the histogram is the most probable speed, which is below the mean, which is below the rms. Only the rms one appears in the energy.
• Thinking thermalisation heats the gas. Starting from identical speeds, the distribution changes shape at constant temperature. Elastic collisions cannot add energy.
• Expecting a faster gas to be a hotter one at a glance. Compare helium and xenon at the same temperature: wildly different speeds, identical mean kinetic energy.
• Assuming pressure is instantaneous. It is an average over many impacts. Any single moment either has an impact or does not.