🫧Kinetic Theory Gas Simulator - Maxwell–Boltzmann, Pressure, Graham's Law

How to use Kinetic Theory of Gases Simulator

1. 1Pick the molecule box, Graham effusion, gas mixing or the Maxwell distribution calculator
2. 2Choose the gases and set the number of particles, temperature and particle size
3. 3Play the simulation, drag the piston, heat or cool, or remove the partition
4. 4Compare the histogram with theory and read the temperature, pressure, mean free path and rate ratios

FAQ

Why does the simulated speed distribution peak further left than the textbook Maxwell curve?

The simulation is two-dimensional, so each molecule has two translational degrees of freedom and the speed distribution is f(v) = (m/kT)·v·e^(−mv²/2kT) with v_p = √(kT/m). A 3D gas has one more degree of freedom, giving a v² factor instead of v and v_p = √(2kT/m), so the peak moves right. Both come from each velocity component being Gaussian; the tool plots the 2D curve for the simulation and the 3D curve in the calculator.

Why is the measured PA/NkT a little above 1?

The ideal gas law treats molecules as points. The simulated molecules are disks with real size: they take up part of the area and pass momentum across their diameter when they collide, so the pressure ends up slightly higher. The Henderson equation for hard disks, PA/NkT = (1 + η²/8)/(1 − η)² with η the area fraction, captures this well — bigger, denser particles deviate more. It is the same physics as the b term in the van der Waals equation.

What does Graham's law say, and why is uranium enriched as UF₆?

In effusion through a small hole, the number of molecules escaping per second is proportional to number density times mean speed, and at a given temperature the mean speed goes as 1/√M, so the rates of two gases are in the ratio √(M_B/M_A). For ²³⁵UF₆ and ²³⁸UF₆ the mass ratio is only 1.0086, giving a separation factor of about 1.0043 per stage, so gaseous diffusion plants needed over a thousand stages. UF₆ is used because it turns to gas near room temperature and fluorine has a single stable isotope, so it adds no mass spread of its own.

How fast do air molecules move, and how far do they travel between collisions?

At room temperature the RMS speed of N₂ is about 517 m/s — faster than sound — and H₂ about 1,930 m/s. Yet at atmospheric pressure an N₂ molecule travels only about 70 nm between collisions and collides some seven billion times a second, which is why smells spread by slow random diffusion. The high-speed tail also explains why Earth has lost its hydrogen and helium: a small fraction of those molecules always exceeds escape velocity.