Maxwell–Boltzmann distribution
If you could freeze a gas for an instant and read off the speed of every molecule, you would not find them all alike. A few crawl, a few tear along, and most cluster around a middling speed. The Maxwell–Boltzmann distribution is the curve that tells you what fraction of molecules has each speed — the full statistical portrait behind the single rms number.
Plotted as a graph, it rises from zero, climbs to a hump at the most probable speed, then trails off in a long tail toward the very fast molecules. As you heat the gas the whole curve flattens and slides to the right: the typical speed rises and the spread widens, while the area under the curve always stays one because every molecule has some speed.
This distribution matters far beyond gas speeds. Its long high-energy tail explains why only a small, temperature-sensitive fraction of molecules ever has enough energy to react, which underlies the steep temperature dependence of reaction rates. It was one of the first triumphs of treating matter statistically rather than tracking each particle.
Raise the temperature of a gas and the distribution's hump shifts right and flattens, while the high-speed tail swells — the same shift that, in a beaker, lets a modest rise in temperature roughly double a reaction's rate.
Heating shifts the speed curve right and fattens its high-energy tail.
The curve is named for James Clerk Maxwell and Ludwig Boltzmann. It describes the distribution of speeds in a classical ideal gas at thermal equilibrium; quantum effects modify it for very cold or very dense systems.