The Particle Box: Kinetic Molecular Theory

A box of atoms you can play with

Everything about a gas (its temperature, its pressure, whether it’s even a gas at all) comes from one simple picture: tiny particles in constant motion, colliding. That’s the kinetic molecular theory, and this is it, live. The box above isn’t an animation playing back a recording; it’s a real simulation of a few hundred particles bouncing. The pressure and speed distribution are measured straight from those collisions, and the temperature is held at whatever you set on the slider (like a thermostat).

• Temperature is a measure of the particles’ average kinetic energy, set by you.
• Pressure is how hard and how often they hit the walls.
• Colour shows each particle’s speed: blue is slow, yellow is fast.

What temperature really measures: kinetic energy

Kinetic energy is the energy a thing has because it is moving. A particle of mass m moving at speed v carries a kinetic energy of

KE = ½ m v²

so a particle that moves twice as fast carries four times the kinetic energy. In this box every atom has the same mass, so kinetic energy comes down to speed alone: the fast yellow atoms carry the most, the slow blue ones the least, and the colour ramp is really an energy map.

Here is the key idea the whole simulator rests on: temperature is the average kinetic energy per particle. Not the energy of any one atom (some are always fast, some slow) but the average across all of them. That’s why dragging the Temperature slider works the way it does: it gently rescales every atom’s speed until the average kinetic energy, the number in the Temperature readout, matches what you asked for. Slide it all the way down and every atom stops dead. That point is absolute zero: zero average kinetic energy, zero motion, zero pressure.

The ideal gas law, one variable at a time

Drag the sliders and the gas laws stop being formulas to memorize and become things you can watch. All of them are pieces of a single equation, the ideal gas law:

P V = n R T

Every symbol in it is something you can see or set in the box:

• P (pressure) is the Pressure readout: how hard and how often the atoms hit the walls.
• V (volume) is the Volume slider: the size of the box (its area here, since the box is 2D).
• n (amount of gas) is the Particles slider: how many atoms are in the box.
• T (temperature) is the Temperature readout: the average kinetic energy per particle.
• R (the gas constant) is a fixed number that just makes the units balance. You never touch it.

Read together, the law says P V is proportional to n T. So if you hold two of those quantities still and change a third, the fourth has no choice but to follow. That is exactly how Boyle, Gay-Lussac, and Avogadro each found their gas law: change one thing, hold the rest fixed, watch what moves. Do the same here.

These three results aren’t separate facts to memorize; they are the single equation P V = n R T taken apart one variable at a time. (The box counts individual particles while the textbook law counts moles; that only changes the constant, not the relationship, because chemists simply bundle Avogadro’s number of particles into each mole. That bundling is the whole point of the mole.)

See all of these worked out in real units (atmospheres, liters, kelvin), with a pick-a-law simulator, in the gas laws guide.

Why all speeds aren’t equal: the Maxwell–Boltzmann curve

Temperature is the average kinetic energy, but at any moment the atoms share a whole range of speeds, because every collision reshuffles energy between them. That spread is the Maxwell–Boltzmann distribution, the histogram below the box. Start the box and watch it grow from a single spike (every atom set to one speed) into the lopsided curve, with most atoms near a typical speed and a long tail of fast ones. (Because this box is 2D, the speed curve is the Rayleigh form of Maxwell–Boltzmann: it starts at zero and peaks at a nonzero speed, a little different from the familiar 3D textbook curve, but built from the same Gaussian velocity components.)

From gas to liquid to solid

Real particles tug on each other a little, the same attractions behind bond polarity and intermolecular forces. Turn up the Cohesion slider and that pull switches on. While the particles are hot they ignore it and stay a gas; cool them down and the attraction wins. These are the three states of matter, and the energy it takes to switch between them is the same contest of energy versus attraction that drives the water cycle, where heat boils water into vapour and cooling condenses it back to liquid.

An honest model

This is a 2D teaching model in relative units, but the physics is real, not faked. The atoms move by velocity-Verlet integration under a genuine pair potential: a soft repulsive core plus a short-range attractive well (Lennard-Jones in spirit). The underlying dynamics are conservative (velocity-Verlet on that potential conserves energy), and a thermostat then gently adds or removes energy each step to hold the temperature you set (that’s also how the box sheds its “latent heat” as it condenses). Everything you watch then emerges from those rules: pressure from wall impacts, the Maxwell–Boltzmann speed curve, the gas laws (P V ∝ N T), and condensation itself, because thermal motion genuinely competes with the attraction. We verify all of it in the build. The honest caveats: it’s 2D and in relative units (not a predictor of real pressures or melting points); the solid-vs-liquid label is read from how tightly the atoms pack and how cold they are; and “absolute zero” here means the classical limit where all motion stops (real matter keeps a little quantum zero-point motion). When the gas condenses, the pressure correctly drops toward zero: the clustered atoms stop hitting the walls (that’s the point, not a glitch).

It’s free to embed on your own site or LMS. Build a molecule over in the molecule builder, or see what happens when those molecules react in balancing equations.