How to use it

1. Pick a gas from the menu and set the temperature with the slider.
2. Set a speed window with the lower and upper limit sliders. Molecules whose speed is inside the window turn orange in the box, and those molecules have orange bars in the speed distribution histogram. Molecules outside the speed window are blue in the box and have blue bars in the histogram.
3. Press “Restart from one speed.” All molecules start with exactly the same speed, v_rms, and the motion is held for about a second and a half. During this pause the histogram is a single bar and every molecule in the box is the same color (orange if v_rms is inside your speed window, blue if it is outside). Then the collisions begin: watch the bars spread into the blue curve as the number of collisions per molecule grows.
4. Press “Jump to equilibrium” to start from the equilibrium distribution and “Pause” to freeze the motion. Use the animation speed and number of molecules sliders to slow the collisions or to reduce the noise in the histogram.
5. Compare the fraction of molecules in the window from the analytic curve with the fraction measured in the simulation. Move the cursor over the figure to read f(v) and the simulated value at any speed.

Watch the approach

After a restart and the brief pause, the histogram is a single spike at the rms speed. Within a few collisions per molecule it spreads into the curve. The kinetic temperature never moves: collisions redistribute energy, they do not add any.

Change the gas or T

Higher T or a lighter gas stretches the curve to higher speeds and lowers its peak, because the area under it stays 1. Compare H₂ with Xe at the same T: same mean kinetic energy, very different speeds.

Read the window

The shaded area under the curve between v₁ and v₂ is the fraction of molecules in that range. Narrow the window and it approaches f(v)·Δv.

Model notes

Molecules are hard spheres with 3-D velocities in a 2-D slice; collisions conserve momentum and energy, with the out-of-plane direction chosen at random. Speeds plotted are full 3-D speeds. Animation time is scaled so every gas moves at a watchable pace. With few molecules the histogram is noisy; the noise value in parentheses after “Distance from equilibrium” is the distance expected for a single snapshot of a gas at equilibrium.

Maxwell–Boltzmann speed distribution

The probability density for the speed \(v\) of a molecule in an ideal gas at equilibrium. The product \(f(v)\,dv\) is the fraction of molecules with speeds between \(v\) and \(v+dv\), and the density is normalized, \(\int_0^\infty f\,dv=1\). The \(v^2\) factor counts the velocity directions available at speed \(v\); the exponential is the Boltzmann factor for the kinetic energy \(\tfrac12 m v^2\).

Mass of one molecule

The molar mass \(M\) (in kg/mol) of the chosen gas divided by Avogadro’s number gives the mass of one molecule.

Characteristic speeds

The most probable speed \(v_{\mathrm{mp}}\) is the peak of \(f(v)\), \(\langle v\rangle\) is the mean speed, and \(v_{\mathrm{rms}}\) is the root-mean-square speed. They always appear in the order \(v_{\mathrm{mp}}<\langle v\rangle<v_{\mathrm{rms}}\) because the curve has a long tail toward high speeds. After a restart every molecule has the speed \(v_{\mathrm{rms}}\), so the initial kinetic energy already equals the equilibrium value.

Mean kinetic energy

All ideal gases at the same temperature have the same mean kinetic energy per molecule, so lighter molecules must move faster.

Fraction of molecules in the speed window

\(F(v)\) is the fraction of molecules with speed below \(v\). The analytic value shown on the page is \(P_{\mathrm{window}}\). The simulated value is the number of molecules with \(v_1\le v<v_2\) divided by \(N\), smoothed over time to reduce the noise from the finite number of molecules.

Kinetic temperature of the simulation

Collisions conserve energy, so \(T_{\mathrm{sim}}\) stays equal to the temperature you chose even while the shape of the distribution changes.

Distance from equilibrium

The histogram has bins \(i\), each with a simulated fraction \(p_i^{\mathrm{sim}}\) and an analytic fraction \(p_i^{\mathrm{MB}}\). One extra bin collects speeds beyond the plotted range. \(D=0\) means the histogram matches the curve exactly and \(D=1\) means no overlap. Because \(N\) is finite, even a gas at equilibrium has a small distance \(D_{\mathrm{noise}}\), which is the “noise” value shown in parentheses next to the distance.

Collisions per molecule

Each collision involves two molecules, so the average molecule has taken part in \(c\) collisions. A few collisions per molecule are enough for the distribution to relax.

Symbols