Maxwell-Boltzmann Speed Distributions

The horizontal axis is particle speed and the vertical axis is the relative number, or fraction, of particles at that speed. The curve rises from zero, passes through a peak at the most probable speed, and trails off in a long tail to the right because a few particles are always moving very fast.

Speed is a nonnegative scalar, so this is not a plot of positive and negative velocity components. A curve's height is a density of particles per speed interval, not the number of particles at one exact speed. To find the fraction between two speeds, use the area over that interval; the area to the right of a threshold is the fraction exceeding it. The position of the peak identifies the most probable speed, not the total particle count.

Heating the sample gives every particle more energy on average. The peak moves right to a higher most probable speed and drops in height, so the curve becomes broader and flatter. The total area stays the same because you still have the same number of particles.

For a normalized probability-density axis, the total area is one for every curve. If the vertical axis instead represents actual particle counts per speed interval, the area represents sample size and is unchanged only when the same number of particles is compared. A lower peak therefore does not by itself mean fewer particles. Heating broadens the distribution and increases the high-speed fraction, while particles still occupy a range of speeds with overlapping cold and hot distributions. It does not make every heated particle faster than every particle in the original sample.

At a fixed temperature, a heavier gas has the same average kinetic energy but a smaller average speed. Its curve peaks further left and is taller and narrower. Compare helium with xenon at 300 K and the helium curve is a wide, low sprawl while xenon is a tall, tight spike.

At the same T, all gases have the same mean translational kinetic energy, (3/2)k_B T per particle. Their speeds differ because kinetic energy contains mass: E_k = (1/2)mv². A larger mass corresponds to a lower characteristic speed. Peak height is comparable only if the curves use the same normalization and axis scales; otherwise a visual comparison can be misleading.

For an ideal gas at temperature T and molar mass M, use M in kg/mol with R = 8.314 J mol⁻¹ K⁻¹:

For one gas, v_mp < v_mean < v_rms. These formulas are quantitative enrichment for interpreting the distribution; the central graph-reading skill is to distinguish peak, area and spread. A graph of speed cannot be read as an energy graph by keeping the same horizontal scale.

For the same gas, v₂/v₁ = √(T₂/T₁) for any of these characteristic speeds. Raising T from 300 K to 600 K therefore raises the characteristic speed by √2 ≈ 1.41, not by two. To double it requires four times the absolute temperature.

For different gases at one temperature, v_A/v_B = √(M_B/M_A). H₂ compared with O₂ gives √(32/2) = 4. Use the mass of the other gas in the numerator when solving the ratio this way, and check that the lighter gas comes out faster. For an absolute speed calculation, 32.00 g/mol must become 0.03200 kg/mol. This conversion makes RT/M have units m²/s², so its square root is a speed.