Maxwell-Boltzmann Distribution

10 min read Updated Mar 26, 2026

Not all molecules in a gas sample move at the same speed. Some are barely crawling, some are zipping along at extreme velocities, and most are somewhere in between. The Maxwell-Boltzmann distribution describes exactly how molecular speeds are spread out in a gas sample, and it explains why raising the temperature makes reactions faster.

The Distribution Curve

The Maxwell-Boltzmann distribution is a plot of “number of molecules” (y-axis) vs. “molecular speed” (x-axis). The curve:

Starts at zero (no molecules have zero speed)
Rises to a peak at the most probable speed
Has a long tail extending to the right (a few molecules move very fast)
Is asymmetric - the tail always extends further to the right than the left

Maxwell-Boltzmann speed distribution curves for four noble gases at room temperature: helium-4 (broadest, flattest curve at highest speeds), neon-20, argon-40, and xenon-132 (narrowest, tallest peak at lowest speeds). The x-axis shows speed in meters per second, and the y-axis shows probability density. — Maxwell-Boltzmann speed distributions for noble gases at room temperature. Lighter gases (He) have broader distributions shifted to higher speeds. Heavier gases (Xe) have narrower, taller peaks at lower speeds. All four gases have the same average kinetic energy at the same temperature. *Credit: Wikimedia Commons, public domain*

Three Measures of Speed

The distribution defines three characteristic speeds, all slightly different:

Speed	Definition	Relative Magnitude
Most probable (v(mp))	Speed at the peak of the curve	Smallest
Average (v(avg))	Arithmetic mean of all speeds	Middle
Root mean square (v(rms))	sqrt of the mean of v²	Largest

The order is always: v(mp) < v(avg) < v(rms). For MCAT purposes, you mainly need to know v(rms) = sqrt(3RT/M) and that these three speeds are close to each other.

Effect of Temperature

Increasing temperature changes the Maxwell-Boltzmann curve in three ways:

The peak shifts to the right (higher most probable speed)
The peak gets shorter (fewer molecules at the most probable speed)
The curve broadens (wider range of speeds)

The total area under the curve stays the same (same number of molecules), but the distribution flattens out and shifts toward higher speeds. This means that at higher temperatures, a larger fraction of molecules exceed any given speed threshold - including the activation energy threshold for a chemical reaction.

Effect of Molar Mass

At the same temperature, lighter gases have:

A peak shifted further to the right (higher speeds)
A broader, flatter distribution
Higher v(rms) values

Heavier gases have a narrower, taller curve centered at lower speeds. Both have the same average kinetic energy (same temperature), but the lighter gas converts that energy into more speed because KE = $\frac{1}{2}$ mv².

Temperature vs. Molar Mass Effects - Summary

Change	Effect on Speed Distribution
Increase T (same gas)	Peak shifts right, curve broadens and flattens
Decrease T (same gas)	Peak shifts left, curve narrows and gets taller
Lighter gas (same T)	Peak shifts right, curve broadens and flattens
Heavier gas (same T)	Peak shifts left, curve narrows and gets taller

Notice that increasing temperature and decreasing molar mass have the same qualitative effect on the distribution. Both shift the curve to higher speeds.

A sample of N2 gas is heated from 300 K to 600 K. What happens to the Maxwell-Boltzmann distribution curve?

Click to reveal answer

The curve shifts to the right, becomes shorter, and broadens. The peak moves to a higher speed (molecules are faster on average). The peak height decreases and the curve spreads out over a wider range of speeds. The total area under the curve stays the same (same number of molecules). More molecules now exceed any given energy threshold, which is why higher temperature increases reaction rates.

At the same temperature, which gas has a higher average molecular speed: He (M = 4) or N2 (M = 28)?

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He. v(rms) = sqrt(3RT/M). Since He has a much smaller molar mass than N2, its molecules move much faster at the same temperature. Specifically, v(He)/v(N2) = sqrt(

\frac{28}{4}

) = sqrt(7) is roughly 2.6, so He molecules move about 2.6 times faster. Both gases have the same average kinetic energy (same T), but He converts that energy into more speed because its molecules are lighter.