Maxwell-Boltzmann Speed Distribution Calculator

What Is the Maxwell-Boltzmann Distribution? The Maxwell-Boltzmann distribution describes how the speeds of gas molecules are spread out at a given temperature. Not all molecules move at the same speed, because they collide constantly and exchange energy, producing a range of speeds. James Clerk Maxwell derived the distribution in 1860 in Scotland. Ludwig Boltzmann extended the statistical mechanics framework in 1872 in Austria. This distribution is foundational to kinetic theory and statistical mechanics.

The Three Characteristic Speeds Most probable speed (v_p): The peak of the distribution, where the most molecules are found. v_p = √(2kT/m) = √(2RT/M) Mean speed (v_avg): The arithmetic average speed of all molecules. v_avg = √(8kT/πm) = √(8RT/πM) RMS speed (v_rms): Square root of mean squared speed, used in kinetic energy calculations. v_rms = √(3kT/m) = √(3RT/M) Where: k = Boltzmann constant (1.381×10⁻²³ J/K), T = temperature (Kelvin), m = molecular mass (kg), R = 8.314 J/(mol·K), M = molar mass (kg/mol).

Relationship Between the Three Speeds v_p : v_avg : v_rms = 1 : 1.128 : 1.225 v_p < v_avg < v_rms, always in this order. The distribution is asymmetric (skewed right), because there is no upper limit to speed, but the probability decays exponentially at high speeds.

The Distribution Function f(v) = 4π(m/2πkT)^(3/2) × v² × exp(−mv²/2kT) This gives the probability density of finding a molecule with speed v. The area under the entire curve is always 1 (all molecules have some speed). As temperature increases, the peak shifts right (faster molecules) and the curve flattens (wider distribution).

Temperature Dependence All speeds scale as √T, so doubling absolute temperature increases speeds by √2 ≈ 41%. At room temperature (293 K), nitrogen (N₂) has v_rms ≈ 511 m/s, which is about Mach 1.46 against the 349 m/s speed of sound in nitrogen itself. At the surface of the sun (~5800 K), atomic hydrogen has v_rms ≈ 12,000 m/s. Pick H₂ in the dropdown at that temperature and you get 8,470 m/s instead, because a hydrogen molecule is twice the mass of an atom, and at 5800 K the sun’s hydrogen is not in molecules anyway. Hydrogen at the top of Earth’s atmosphere can exceed escape velocity (11,200 m/s), explaining why Earth loses hydrogen to space. Note that this happens long before the average molecule is that fast: it is the fast tail of the distribution that leaks, and the tail is refilled by collisions, so the loss keeps going.

Real-World Applications Effusion and diffusion: lighter gases effuse faster, at a rate proportional to 1/√M. That is Graham’s law, and it is why a helium balloon goes soft in a day while an air-filled one holds for a week. Atmospheric retention: planets with low gravity or high temperatures lose light gases. Reaction rates: only molecules above activation energy threshold can react (Arrhenius equation). Laser cooling: lasers slow atomic speeds to microkelvin temperatures.

Common Gas Molar Masses H₂ (hydrogen): 2.016 g/mol. He (helium): 4.003 g/mol. N₂ (nitrogen): 28.014 g/mol. O₂ (oxygen): 31.998 g/mol. CO₂ (carbon dioxide): 44.01 g/mol. Ar (argon): 39.948 g/mol.