Picture the air in this room. Every molecule is zipping around, bumping into its neighbours billions of times each second. They do not all move at the same speed — a few crawl, most jog along in the middle, and a small number sprint. If you tagged every molecule, measured its speed, and made a bar chart of "how many molecules move this fast," you would get a lopsided hump: few slow ones, a big pile near the middle, and a long thin tail of very fast ones. That hump is the Maxwell-Boltzmann distribution. Heating the gas does not make every molecule faster — it shifts and spreads the whole hump toward higher speeds.
Two things set the shape: the temperature $T$ (how much energy the crowd shares) and the molar mass $M$ (how heavy each molecule is). Hotter means faster; heavier means slower. The handiest single speed is the most-probable speed $v_{mp}$ — the location of the peak:
For nitrogen ($M = 0.028$ kg/mol) at room temperature ($T = 300$ K): $v_{mp} = \sqrt{2(8.314)(300)/0.028} \approx 420$ m/s — faster than a cruising jet. Lighter hydrogen at the same temperature peaks near $1580$ m/s. That mass dependence is exactly why the lightest gases leak out of a planet's atmosphere first.
The full distribution gives the fraction of molecules whose speed lies between $v$ and $v+dv$:
The $v^2$ factor comes from counting directions in three-dimensional velocity space — a thin spherical shell of radius $v$ has area $\propto v^2$ — while $e^{-Mv^2/2RT}$ is the Boltzmann penalty against high energy. Because the curve is skewed, its three "averages" differ, $v_{mp} < \langle v\rangle < v_{rms}$, yet all scale the same way, as $\sqrt{T/M}$. The Temperature and Molar Mass sliders move $T$ and $M$ directly, so you can watch the peak and the width respond in real time.
| Symbol | Meaning | Unit |
|---|---|---|
| $f(v)$ | Speed distribution function (probability density) | s/m (so that $\int_0^\infty f(v)\,dv = 1$) |
| $M$ | Molar mass | kg/mol |
| $R$ | Gas constant | 8.314 J·mol⁻¹·K⁻¹ |
| $T$ | Temperature | K |
| $v_{mp}$ | Most probable speed (peak of f(v)) | m/s |
| $\langle v \rangle$ | Mean speed | m/s |
| $v_{rms}$ | Root-mean-square speed | m/s |
| Slider | Symbol | Effect on Distribution |
|---|---|---|
| Temperature T | $T$ | Higher T → flatter, broader curve shifted to higher v |
| Molar Mass M | $M$ | Higher M → sharper, narrower curve at lower v |
| Escape Velocity | $v_{esc}$ | Fraction of molecules faster than $v_{esc}$: planetary gas retention |