Calculate RMS, mean, and most probable molecular velocities from the Maxwell-Boltzmann distribution. Includes kinetic energy, mean free path, and gas speed reference.
In any gas at thermal equilibrium, molecules move at a range of speeds described by the Maxwell-Boltzmann distribution. Three characteristic velocities summarize this distribution: the most probable speed (peak of the curve), the mean speed (arithmetic average), and the RMS speed (root-mean-square, relevant to kinetic energy). These velocities depend on temperature and molecular mass, and they explain phenomena from atmospheric escape to diffusion rates.
This Particle Velocity Calculator computes all three characteristic speeds, average kinetic energy per molecule and per mole, mean free path, and an estimated speed of sound. Presets cover common gases at 300 K, and a text-based Maxwell-Boltzmann distribution chart visualizes the speed distribution. A gas reference table compares molecular speeds across species from hydrogen to xenon.
Students studying thermodynamics, kinetic theory, and statistical mechanics use this calculator to explore how temperature and molecular mass determine gas behavior at the microscopic level. Check the example with realistic values before reporting.
Kinetic theory connects microscopic molecular motion to macroscopic properties like pressure, temperature, and diffusion. This calculator makes the Maxwell-Boltzmann distribution concrete — revealing why lighter gases move faster, why hot gases exert more pressure, and why Earth retains nitrogen but not hydrogen. Keep these notes focused on your operational context.
Most Probable Speed: vₚ = √(2kT/m) Mean Speed: v̄ = √(8kT/πm) RMS Speed: v_rms = √(3kT/m) Average KE per molecule: ⟨KE⟩ = ³⁄₂ kT Mean Free Path: λ = kT / (√2 π d² P) Where: k = 1.381 × 10⁻²³ J/K (Boltzmann constant) T = temperature (K) m = molecular mass (kg) d ≈ 3.7 × 10⁻¹⁰ m (effective diameter) P = pressure (Pa)
Result: v_rms ≈ 517 m/s, v̄ ≈ 476 m/s, vₚ ≈ 422 m/s
Nitrogen (N₂, M = 28 g/mol) at 300 K has an RMS speed of 517 m/s. The mean speed is 476 m/s, and the most probable speed (distribution peak) is 422 m/s. Average kinetic energy per molecule is 6.21 × 10⁻²¹ J, independent of molecular mass — only temperature matters for average KE.
James Clerk Maxwell and Ludwig Boltzmann independently derived the speed distribution of molecules in an ideal gas at thermal equilibrium. The distribution arises from two competing factors: the Boltzmann factor exp(−E/kT), which favors low energies, and the density of states (proportional to v²), which favors higher speeds. The product creates the characteristic asymmetric curve that peaks at vₚ and has a long tail toward high speeds.
Molecular speed distributions govern reaction rates (only molecules with sufficient kinetic energy can react — Arrhenius equation), effusion rates (Graham's law), thermal conductivity, viscosity of gases, and atmospheric escape of planetary atmospheres. In vacuum technology, mean free path determines whether gas flow is viscous or molecular.
The Maxwell-Boltzmann distribution applies to classical ideal gases. At very low temperatures or very high densities, quantum effects become important: fermions follow the Fermi-Dirac distribution (electrons in metals), and bosons follow the Bose-Einstein distribution (photons, superfluid helium). The Maxwell-Boltzmann distribution is the high-temperature, low-density limit of both.
The most probable speed is where the distribution peaks. The mean speed is the simple arithmetic average. The RMS speed is the square root of the mean of v², which is relevant because kinetic energy depends on v². The ordering is always vₚ < v̄ < v_rms.
No. Average kinetic energy per molecule is ³⁄₂kT — it depends only on temperature. At the same temperature, all gas molecules have the same average KE, but lighter molecules move faster to achieve it.
A gas molecule can escape Earth if its speed exceeds escape velocity (~11.2 km/s). N₂ at ~500 m/s is well below this, but H₂ at ~1,900 m/s has a significant tail of the distribution above escape velocity, so hydrogen gradually escapes over geological time.
Mean free path is the average distance a molecule travels between collisions. At atmospheric pressure, it is about 68 nm for air. It increases in vacuum (lower pressure, fewer collision partners).
Higher temperature broadens the distribution and shifts the peak to higher speeds. The distribution becomes flatter and wider, meaning more molecules have higher speeds.
It is the probability distribution of molecular speeds in a gas at thermal equilibrium. It arises from random collisions that distribute energy among molecules according to statistical mechanics. The distribution function is f(v) = 4π(m/2πkT)^(3/2) v² exp(-mv²/2kT).