Kinetic gas in a piston

Pressure is not a thing the gas "has": it is the average push of billions of molecules hitting the wall. Here each dot stands for 0.4 millimole of molecules.

Experiments

Controls

Gas

Heavier molecules move slower at the same temperature, but the pressure stays the same.

Heat exchange

Insulated: squeezing the gas heats it (the piston does work on the molecules).

Live numbers

Pressure-volume diagram

● live measured pressure · ━ isotherm P = nRT/V · ━ adiabat P V^5/3 = const

Molecular speeds

bars: simulated molecules · line: Maxwell–Boltzmann distribution at the measured temperature

Independent numerical checks

The science

Each molecule of mass m bounces elastically. A wall hit with normal speed v_y transfers momentum 2 m v_y. Averaging the hits over the wall area gives P = (N m <v²>) / (3 V). Kinetic temperature is defined by (3/2) k_B T = <½ m v²>, so P V = N k_B T = n R T.

Boyle: at fixed T, halving V doubles P (twice as many hits per second on half the area... of a shorter flight).
Gay-Lussac: at fixed V, P is proportional to T (faster molecules hit harder and more often).
Charles: V proportional to T at fixed P: read it off the diagram by following a horizontal line.
Adiabatic: the moving piston adds energy 2 m u v per hit; for a monatomic gas this gives T V^2/3 = const and P V^5/3 = const.

Simplifications (honest list): molecules are points with only 3 translational degrees of freedom (real N₂ also rotates, giving 7/5 instead of 5/3); they collide with each other by random pair scattering that conserves momentum and energy but ignores where they are (DSMC-like), which is what spreads energy between the three directions; there are no intermolecular forces; the thermostat is a smooth velocity rescaling (Berendsen); the piston moves at most 8 m/s (about 1.5 % of the thermal speed), so compression is almost, but not exactly, reversible — a faster piston would heat the gas more. Time is slowed by about 1000 times so you can see the molecules. Pressure from 100–400 simulated molecules fluctuates by a few percent; the numbers in the table average over many milliseconds of simulated time.

Not a substitute for a lab: ideal gas, no gravity, no leaks. Model constants: k_B = 1.380649e-23 J/K, N_A = 6.02214076e23 /mol.

3D rig modelled in Blender (table, heater, frame, piston, gauge) with baked textures; physics in plain JavaScript; Three.js r160.