Gas Molecule SimulatorBeta

About Gas Molecule Simulator

Every dot here is a hard disk of the same mass. Disks bounce perfectly elastically off each other and off the walls, and nothing else acts on them. On restart every molecule gets the same speed, √(2kT), in a random direction, so the first histogram is a single spike. Collisions then smear that spike into the Maxwell–Boltzmann curve drawn on the right. Energy is conserved the whole time: nothing is added, the same energy is just reshuffled among the molecules.

The simulation is two-dimensional, which changes the numerical factors you may remember from a textbook. All numbers below are for 2D. Temperature is shown in kelvin for convenience, but internally 300 K is mapped to kT = 10,000 px²/s², so the units are arbitrary screen units.

The 2D Maxwell–Boltzmann distribution and its three speeds

If the x and y velocity components are each normally distributed with variance kT/m, the speed follows f(v) = (mv/kT)·exp(−mv²/2kT). In 3D the prefactor is v²; in 2D it is just v. Three characteristic speeds come out of this curve: the most probable speed √(kT/m), the mean speed √(πkT/2m) and the root-mean-square speed √(2kT/m). The three dashed lines on the histogram are these, left to right.

At 300 K (kT = 10,000) they are 100, 125.3 and 141.4 px/s. The ratio 1 : 1.253 : 1.414 does not depend on temperature, because heating does not change the shape of the curve, only stretches the speed axis by √T. Note that the starting speed √(2kT) equals the RMS speed exactly. The initial spike and the final curve carry the same average kinetic energy; only the spread differs.

Equipartition: what the thermometer actually reads

Classical statistical mechanics gives each quadratic degree of freedom an average energy of ½kT. A disk moving in x and y has two, so its mean kinetic energy is kT (a monatomic gas in 3D would have 3/2 kT). The readout labelled measured T runs this backwards: it averages ½v² over all molecules and calls the result kT.

The disks never spin. Real diatomic molecules such as N₂ and O₂ have two rotational degrees of freedom as well and store more energy at the same temperature, which is why their heat capacity is higher than a monatomic gas. This model leaves rotation out entirely.

Experiment 1: from a spike to a curve

With the defaults (2,500 molecules, radius 2.5 px), press restart and watch the histogram. The spike near 141 px/s collapses within seconds, spreading both left and right, because in any collision one partner usually speeds up while the other slows down. Drop the count to 500 and relaxation is visibly slower, since collisions are rarer, and the histogram is noisier. Statistical scatter in a bin shrinks only like 1/√N.

Slow the playback and press on the canvas to follow one molecule. Its path is a chain of short straight segments with random kinks, the same structure as the Brownian motion of a dust grain in air.

Experiment 2: why PA/NkT drifts above 1

For an ideal gas, pressure times area equals NkT. The tool measures pressure from the momentum delivered to the walls and shows the ratio. Real disks have size, though: no disk can sit where another already is. The free area is smaller than the box, so molecules hit the walls more often and the ratio rises above 1.

A convenient measure is the packing fraction η = Nπr²/A. On an 800×580 px canvas with default settings, η ≈ 2,500 × 19.6 / 464,000 ≈ 0.11. The Henderson approximation for hard disks, Z = (1 + η²/8)/(1 − η)², then predicts about 1.25. Cut the radius to 1.5 px and η ≈ 0.04, Z ≈ 1.08, close to ideal. Your canvas size depends on your screen, so plug in your own width and height and see how close the readout comes. If a combination would push η above 0.7 (say radius 8 px × 8,000 molecules), the tool places only as many as fit and shows a note: any denser and the disks overlap, so position corrections rather than collisions generate the pressure and PA/NkT stops meaning anything. Near 0.7 the Henderson formula overshoots (about 17), while the simulation reads around 9.

Scaling up to real air

In 3D the three speeds are √(2kT/m), √(8kT/πm) and √(3kT/m). For nitrogen at 300 K (one molecule is 4.65×10⁻²⁶ kg, kT = 4.14×10⁻²¹ J) they come to roughly 422, 476 and 517 m/s, faster than the speed of sound in air. A smell still takes a while to cross a room because at atmospheric pressure a molecule collides every few tens of nanometres and loses its direction. The kinked path in tracking mode is that process in miniature.

Worked examples

How to use it

1. Press restart so every molecule starts at the same speed.
2. Watch the histogram bars settle onto the theory curve.
3. Change temperature, count and radius, and compare the dashed speed markers and the PA/NkT readout.
4. Press on the canvas to trace a single molecule.

Frequently asked questions

Notes

Z = (1 + η²/8)/(1 − η)² is Henderson's empirical equation of state for hard disks; it is most accurate at low packing fractions.

There is no attraction between molecules and no rotation, so condensation and van der Waals effects do not appear.