# Maxwell–Boltzmann distribution The **Maxwell–Boltzmann distribution** is the probability distribution of molecular speeds in a classical gas at [[Thermodynamic_equilibrium|thermal equilibrium]]. For a [[Molecule|molecule]] of mass m at [[Temperature|temperature]] T it reads `f(v) = 4·pi·(m/(2·pi·k_B·T))^1.5·v^2·exp(−m·v^2/(2·k_B·T))`, with `dN = N·f(v)·dv` molecules in the speed band from v to v + dv.[^os-mb] Two factors fight over the shape: a `v^2` from the number of ways to point a velocity in three dimensions, which pushes the curve up from zero, and a falling exponential in the [[Kinetic_energy|kinetic energy]], which pulls it back down. Nothing about the gas enters but m and T. In the microsim below the reader slides T on a logarithmic axis from 20 to 20,000 K and picks the gas — [[Hydrogen|H₂]], [[Helium|He]], [[Nitrogen|N₂]], [[Oxygen|O₂]] or [[Xenon|Xe]]. Three vertical lines move together as the curve slides and flattens, because the most probable, mean and root-mean-square speeds keep the fixed ratio √2 : √(8/π) : √3 for every gas and every temperature.[^os-mb] An Earth/Moon preset then shades everything beyond the [[Escape_velocity|escape speed]] — 11.1 km/s for [[Atmosphere_of_Earth|Earth]], 2.38 km/s for the [[Moon|Moon]] — and reads the logarithm of the shaded fraction from a baked lookup table.[^os-escape] At 250 K the escaping fraction for helium is 10⁻⁵⁰·⁵ and for nitrogen 10⁻³⁵⁹ (derived): three hundred orders of magnitude between two gases in the same air, which is why Earth kept its N₂ and lost its H₂ and He. A readout carries the equipartition results `<KE> = (3/2)·k_B·T` and `C_V/R = d/2` alongside.[^os-kinetic][^os-cv] On the [[Physics]] flagship this page serves Part II — Core theories, in the section *The Maxwell–Boltzmann distribution* (row P25), which Physics owns and shares with the chemistry spine. Upstream is [[Kinetic_theory_of_gases|kinetic theory]], whose own live microsim tallies wall impacts to recover the pressure this distribution assumes; downstream is the [[Boltzmann_distribution|Boltzmann distribution]] over energy levels, of which this is the continuous, three-dimensional special case. ## Distribution function Written in terms of the most probable speed `v_p = sqrt(2·k_B·T/m)` the law loses its constants and becomes a pure shape: `f = (4/sqrt(pi))·(v^2/v_p^3)·exp(−v^2/v_p^2)`.[^os-mb] Every Maxwell–Boltzmann curve is therefore the *same* curve, stretched horizontally by `v_p` and squashed vertically to keep unit area. That is the single most useful fact about it, and the microsim is built on it: the drawn polyline never changes, only the axis under it. The scaling explains the two controls at once. Because `v_p` goes as `sqrt(T/m)`, raising T by a factor of four doubles every characteristic speed and halves the peak height, while switching from He (4 g/mol) to Xe (131 g/mol) at fixed T divides every speed by `sqrt(131/4)` = 5.7 (derived). Real numbers make the scale concrete: helium at 250 K has `v_p` = 1,019 m/s, and argon at 273 K has `v_rms` = 413 m/s (derived).[^os-escape] Nitrogen's room-temperature `v_rms` is reflected in the [[Sound|speed of sound]] in air, about 340 m/s — sound cannot travel much faster than the molecules that carry it.[^os-escape] The distribution is normalized, so `integral(f(v)·dv)` from 0 to ∞ is 1, and the area under any band is the fraction of molecules in it. That is the only reading the microsim ever makes: every number it reports — a fraction above the escape speed, a fraction within 10 % of `v_p` — is an area. One caution belongs with the formula. The [[Probability_distribution|distribution]] of a single velocity *component* is a Gaussian centred on zero with standard deviation `sqrt(k_B·T/m)`, and it has a different normalization from the three-dimensional speed law above; mixing the two is the commonest error in the subject.[^raven-doppler] ## Relaxation to the 2D Maxwell–Boltzmann distribution The distribution is not an assumption about how a gas starts but a description of where it ends up. Set a two-dimensional box of disks moving at identical speeds in random directions and let them collide elastically; within a few collisions per particle the speed histogram converges on the two-dimensional form `f(v) = (m/(k_B·T))·v·exp(−m·v^2/(2·k_B·T))`, whose `v^1` prefactor replaces the three-dimensional `v^2` because a two-dimensional velocity space has circumference rather than surface area (derived). The kinetic-theory derivation of pressure is untouched by those collisions: molecular collisions do not spoil it, because they redistribute momentum among molecules without changing the total delivered to the walls.[^os-mfp] There is a practical trap here that the sibling [[Kinetic_theory_of_gases|kinetic theory]] microsim documents. A simulation in which molecules bounce only off walls never thermalizes at all — elastic wall reflections preserve every speed — so such a model must either be started from a Maxwell–Boltzmann draw or given molecule–molecule collisions before its histogram means anything.[^os-mfp] Relaxation needs a mechanism, and the [[Mean_free_path|mean free path]] `lambda = k_B·T/(sqrt(2)·4·pi·r^2·p)` sets how long it takes.[^os-mfp] ## Typical speeds Three speeds are quoted, and the choice between them matters more than it looks. The most probable speed `v_p = sqrt(2·k_B·T/m)` is the peak of the curve. The mean speed `v_avg = sqrt(8·k_B·T/(pi·m))` governs collision rates and [[Effusion|effusion]]. The root-mean-square speed `v_rms = sqrt(3·k_B·T/m) = sqrt(3·R·T/M)` is the one tied to energy, since `(1/2)·m·<v^2> = (3/2)·k_B·T` exactly.[^os-mb][^os-kinetic][^raven-kt] Their ratio is fixed: √2 : √(8/π) : √3, or 1 : 1.128 : 1.225 (derived). No gas and no temperature changes it, which is why the microsim can draw all three as rigidly linked markers. The ordering `v_p < v_avg < v_rms` follows from the tail: the distribution is skewed to the right, so averaging pulls above the peak, and averaging the square pulls further still. The energy statement behind `v_rms` is [[Equipartition_theorem|equipartition]], which assigns `(1/2)·k_B·T` to every quadratic degree of freedom, giving `<KE> = (3/2)·k_B·T` for translation and a molar heat capacity `C_V = (d/2)·R`.[^os-kinetic][^os-cv] The measured values confirm it where the classical picture holds: helium, neon and [[Argon|argon]] all give `C_V/R` = 1.50, and carbon monoxide 2.50, with two rotational [[Degrees_of_freedom_(physics_and_chemistry)|degrees of freedom]] added.[^os-cv] [[Diatomic_molecule|Diatomic]] hydrogen is the famous exception: d = 3 below about 60 K, d = 5 from just under 300 K to about 600 K, and d = 7 above about 3,000 K, a staircase that no classical argument can produce and that the [[Boltzmann_distribution|Boltzmann distribution]] over quantized levels explains.[^os-cv] [[Noble_gas|Monatomic]] gases have no rotational contribution at all, because their [[Moment_of_inertia|moment of inertia]] is negligible.[^os-kinetic] ## Limitations The law is classical, non-relativistic, and about equilibrium, and it fails on each count in turn. It ignores [[Quantum_mechanics|quantum]] statistics, so it breaks down when the thermal wavelength of a particle approaches the spacing between particles — cold, dense systems need [[Bose–Einstein_statistics|Bose–Einstein]] or Fermi–Dirac statistics instead, and a [[Fermi_gas|Fermi gas]] of electrons in a metal is nowhere near Maxwellian at room temperature. It assumes speeds far below the [[Speed_of_light|speed of light]], which fails in a hot [[Plasma_(physics)|plasma]] or a fusion device. And it assumes equilibrium, which a gas being stirred, shocked or illuminated does not have. Two limitations bear directly on the microsim's escape preset. The tail it shades is a tail of an *equilibrium* distribution, so the model implicitly assumes that molecules lost from the tail are replaced by collisions as fast as they leave; a real [[Atmospheric_escape|escaping atmosphere]] depletes its own tail, and the escape flux is set by conditions at the altitude where collisions stop, not at the ground. The second is numerical. The escaping fraction must be stored as a logarithm, because 10⁻³⁵⁹ underflows even a double-precision float, and the lookup table therefore holds `log10(F)` over the reduced variable `x = v_esc/v_p` rather than F itself (derived). For nitrogen at 250 K that x is 28.8, far outside any range a naive integration would survive. ## Derivation and related distributions The speed law is the last step of a chain, not the first. The chain begins with a statement about energy — that a state of energy E is occupied in proportion to `exp(−E/(k_B·T))` — and ends, four changes of variable later, with `f(v)`. ### Maxwell–Boltzmann statistics [[Maxwell–Boltzmann_statistics|Maxwell–Boltzmann statistics]] describes distinguishable classical particles with no restriction on how many may share a state: the average occupation of a state of energy E is proportional to the [[Boltzmann_distribution|Boltzmann factor]] `exp(−E/(k_B·T))`, normalized by the [[Partition_function_(statistical_mechanics)|partition function]]. James Clerk Maxwell obtained the velocity law in 1860 from a symmetry argument — that the three components must be independent and the distribution isotropic, which forces a Gaussian — and Ludwig Boltzmann later derived the same result from collision dynamics and generalized the exponential factor to any energy.[^maxwell1860][^boltzmann1877] The statistics are the classical limit of both quantum families, valid whenever occupation numbers are small. ### Distribution for the momentum vector In momentum variables the kinetic energy is `p^2/(2·m)`, so the Boltzmann factor becomes `exp(−p^2/(2·m·k_B·T))`, a product of three independent Gaussians in `p_x`, `p_y` and `p_z` each of standard deviation `sqrt(m·k_B·T)` (derived). This is the cleanest form of the law, because the three components genuinely are independent: knowing that a molecule is moving fast along x says nothing about its motion along y. Every later form is this one rewritten. ### Distribution for the energy Changing variable from momentum to kinetic energy, with `E = p^2/(2·m)`, gives `f(E) = 2·sqrt(E/pi)·(1/(k_B·T))^1.5·exp(−E/(k_B·T))` (derived). The `sqrt(E)` is the density of states in three dimensions, and the mean of this distribution is `(3/2)·k_B·T`, recovering equipartition.[^os-kinetic] The energy form is the one used in [[Chemical_kinetics|reaction-rate]] theory, where the fraction of molecules above an [[Activation_energy|activation energy]] carries the exponential that the [[Arrhenius_equation|Arrhenius equation]] fits. ### Distribution for the velocity vector Dividing momentum by mass turns the three Gaussians into velocity components of standard deviation `sqrt(k_B·T/m)` — the form used to draw a [[Doppler_broadening|Doppler]] profile, where only the component along the line of sight matters and a Monte-Carlo draw of `v_par` from that Gaussian reproduces the observed line shape.[^raven-doppler] The bookkeeping is worth watching: at 400 K a molecule of mass 2.33×10⁻²⁶ kg has a component standard deviation of 486.7 m/s, giving a velocity full width at half maximum of 1,146 m/s and, at 940 nm, a Doppler width of 1.22 GHz (derived).[^raven-doppler] ### Distribution for the speed The last step throws away direction. Integrating the velocity-vector distribution over the sphere of radius v multiplies it by the surface area `4·pi·v^2`, which is where the `v^2` prefactor in the headline formula comes from.[^os-mb] That prefactor is the whole reason the curve starts at zero: there is exactly one way for a molecule to be at rest and an ever-growing number of ways to be moving at speed v, until the exponential overwhelms the count. ## In <i>n</i>-dimensional space The construction generalizes immediately. In n dimensions the sphere of radius v has area proportional to `v^(n−1)`, so the speed distribution is proportional to `v^(n−1)·exp(−m·v^2/(2·k_B·T))`, and the most probable speed becomes `v_p = sqrt((n−1)·k_B·T/m)` (derived). In one dimension the prefactor vanishes and the distribution peaks at zero — a half-Gaussian in |v| — which is the Doppler case above. In two dimensions it is linear in v, the case the relaxation section describes. Equipartition scales with it: the mean kinetic energy is `(n/2)·k_B·T`, so the same argument that gives `(3/2)·k_B·T` in a gas gives `(1/2)·k_B·T` per direction in a two-dimensional film or a surface-adsorbed layer.[^os-kinetic] ## Extension to real gases Two idealizations are separable, and only one of them fails first. The Maxwell–Boltzmann *speed* distribution survives interactions remarkably well, because the kinetic energy of a classical system is a separate, quadratic term in the Hamiltonian whatever the potential energy does; the equation of state fails long before the speed distribution does. So a dense gas obeying the [[Van_der_Waals_equation|van der Waals equation]] `P + a·(N/V)^2 = N·T/(V − N·b)`, with its excluded volume and its attractive term, still has Maxwellian speeds.[^likharev-vdw] Where the distribution genuinely stops applying is at the [[Critical_point_(thermodynamics)|critical point]] and below, where the gas is no longer a gas: water condenses at 647.4 K and 219.0 atm, nitrogen at 126.2 K and 33.6 atm, helium at 5.3 K and 2.27 atm.[^os-ideal] Below those temperatures a liquid phase appears, [[Intermolecular_force|intermolecular forces]] dominate the energy budget, and the ideal relations that make the microsim's readouts exact — `p·V = N·k_B·T`, 22.4 L for a mole at standard conditions — cease to hold.[^os-ideal] The speed distribution is the last of the ideal-gas results to fail and the first to be recovered, which is why it remains the working description of the atmosphere, of a [[Molecular_dynamics|molecular-dynamics]] thermostat, and of the feedstock in [[Gaseous_diffusion|gaseous diffusion]] and [[Isotope_separation|isotope separation]], where the mass dependence of `v_avg` is the whole separation mechanism. ## See also - [[Equipartition_theorem]] - [[Atmospheric_escape]] - [[Ideal_gas_law]] - [[Kinetic_theory_of_gases]] - [[Boltzmann_distribution]] - [[Maxwell–Boltzmann_statistics]] - [[Hydrogen]] - [[Helium]] ## Notes Explanatory material is carried in the body rather than in separate notes; every footnote definition on this page, bibliographic and explanatory alike, is collected under References below. Values marked "(derived)" were computed for this article from the inputs its sources print, not read off a source's own printed answer. The sub-manual records that nearly every displayed equation in Portal Book 078's kinetic-theory chapter was lost in text extraction and was restored from standard forms, so the equations quoted here should be checked against the printed pages before they are relied on for anything beyond the microsim. ## References [^os-mb]: Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 2*. OpenStax. Chapter 2 "The Kinetic Theory of Gases", p. 101 (`f(v) = 4·pi·(m/(2·pi·k_B·T))^1.5·v^2·exp(−m·v^2/(2·k_B·T))`, with `dN = N·f·dv`), p. 102 (`v_avg = sqrt(8·k_B·T/(pi·m))`) and p. 103 (`v_p = sqrt(2·k_B·T/m)` and the reduced form `f = (4/sqrt(pi))·(v^2/v_p^3)·exp(−v^2/v_p^2)`). The displayed equations of this chapter were lost in the sub-manual's text extraction and supplied in standard form; the ratio √2 : √(8/π) : √3 = 1 : 1.128 : 1.225 is derived. Portal Book 078, https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-2 [^os-kinetic]: Sanny and Ling (2016), *University Physics Volume 2*, Chapter 2, p. 88 (`KE_avg = (3/2)·k_B·T`) and p. 89 (`v_rms = sqrt(3·k_B·T/m) = sqrt(3·R·T/M)`; monatomic gases have no rotational contribution because their moment of inertia is negligible). Portal Book 078. [^os-escape]: Sanny and Ling (2016), *University Physics Volume 2*, Chapter 2, pp. 90–91 (escape speeds of 11.1 km/s for Earth and 2.38 km/s for the Moon; a high-altitude temperature of about 250 K; nitrogen's root-mean-square speed reflected in a sound speed of about 340 m/s) and p. 95 (the argon inputs of Example 2.7, 0.0399 kg/mol at 273 K, whose printed answer was lost). Derived from those inputs: helium's `v_p` = 1,019 m/s at 250 K, argon's `v_rms` = 413 m/s at 273 K, escaping fractions of 10⁻⁵⁰·⁵ for He and 10⁻³⁵⁹ for N₂ at 250 K, the reduced variable `x = v_esc/v_p` = 28.8 for N₂, and the temperature 1.98×10⁴ K at which helium's `v_rms` reaches 11.1 km/s. Portal Book 078. [^os-cv]: Sanny and Ling (2016), *University Physics Volume 2*, Chapter 2, p. 97 (`C_V = (d/2)·R`; hydrogen's d = 3 below about 60 K, d = 5 from just under 300 K to about 600 K, and d = 7 above about 3,000 K) and p. 98, Table 2.3 (measured `C_V/R` of 1.50 for He, Ne and Ar and 2.50 for CO; the gas names of seven further rows were lost in extraction). The book gives the plateaus but not the curve between them. Portal Book 078. [^os-mfp]: Sanny and Ling (2016), *University Physics Volume 2*, Chapter 2, pp. 85–88 (`p·V = (1/3)·N·m·<v^2>` from elastic wall impulses and isotropy, under the assumptions of a rigid box, point molecules, elastic collisions and no gravity; collisions between molecules do not spoil the derivation) and pp. 94–95 (mean free path `lambda = 1/(sqrt(2)·4·pi·r^2·(N/V)) = k_B·T/(sqrt(2)·4·pi·r^2·p)` and mean free time `tau = lambda/v_rms`). That a walls-only simulation never thermalizes, and must therefore be seeded from a Maxwell–Boltzmann draw or given molecule–molecule collisions, is the sibling kinetic-theory microsim's design note. Portal Book 078. [^os-ideal]: Sanny and Ling (2016), *University Physics Volume 2*, Chapter 2, pp. 76 and 81 (`p·V = N·k_B·T = n·R·T`, restored in standard form after extraction loss), p. 82 (a molar volume of 22.4 L at standard temperature and pressure) and p. 85, Table 2.1 (critical points: water 647.4 K / 219.0 atm, N₂ 126.2 K / 33.6 atm, He 5.3 K / 2.27 atm, among eight substances). Portal Book 078. [^raven-kt]: Raven, Will (2025). *Atomic Physics for Everyone*. Chapter 4 "Atoms in Motion", p. 83 (`k_B` = 1.38×10⁻²³ J/K; the worked conversion of Maxwell–Boltzmann fractions 0.726, 0.061 and 0.347 into counts of 3,630, 305 and 1,735 out of 5,000 atoms) and p. 87 (`(1/2)·m·<v^2> = (3/2)·k_B·T`). Portal Book 046, https://open.umn.edu/opentextbooks/textbooks/atomic-physics-for-everyone-an-introduction-to-atomic-physics-quantum-mechanics-and-precision-spectroscopy-with-no-college-level-prerequisites [^raven-doppler]: Raven (2025), *Atomic Physics for Everyone*, Chapter 4, pp. 79–86 (Doppler broadening built from N atoms, with the line-of-sight component `v_par` drawn from a Gaussian of standard deviation `sqrt(k_B·T/m)`; the one-dimensional velocity law has a different normalization from the three-dimensional speed law). Derived for the book's Figure 4.7 parameters — m = 2.33×10⁻²⁶ kg, 400 K, 940 nm — are σ = 486.7 m/s, a velocity FWHM of 1,146 m/s and a frequency FWHM of 1.22 GHz. Portal Book 046. [^likharev-vdw]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part SM: Statistical Mechanics*. Chapter 4, pp. 108–113 (the van der Waals equation `P + a·(N/V)^2 = N·T/(V − N·b)`; `P_c = a/(27·b^2)`, `V_c = 3·N·b`, `T_c = 8·a/(27·b)`; the model holds only for `N·b` much less than V and cannot say whether the condensed phase is liquid or solid; the Maxwell equal-area construction). Portal Book 075, https://open.umn.edu/opentextbooks/textbooks/part-sm-statistical-mechanics [^maxwell1860]: Maxwell, James Clerk (1860). "Illustrations of the Dynamical Theory of Gases." *Philosophical Magazine*, 4th series, volumes 19 and 20. [^boltzmann1877]: Boltzmann, Ludwig (1877). "Über die Beziehung zwischen dem zweiten Hauptsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung respektive den Sätzen über das Wärmegleichgewicht." *Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften in Wien*, volume 76. ## Further reading - Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 2*. OpenStax. Portal Book 078 — Chapter 2 is the source of every kinetic-theory number on this page, including the escape speeds and the heat-capacity table. - Raven, Will (2025). *Atomic Physics for Everyone*. Portal Book 046 — Chapter 4 builds a Doppler profile from a Maxwell–Boltzmann velocity draw, atom by atom. - Likharev, Konstantin (2013). *Essential Graduate Physics, Part SM: Statistical Mechanics*. Portal Book 075 — the statistical-mechanical setting, from the Gibbs distribution to the van der Waals gas and its critical point. ## External links - [University Physics Volume 2](https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-2), OpenStax — Portal Book 078 - [Atomic Physics for Everyone](https://open.umn.edu/opentextbooks/textbooks/atomic-physics-for-everyone-an-introduction-to-atomic-physics-quantum-mechanics-and-precision-spectroscopy-with-no-college-level-prerequisites), Will Raven — Portal Book 046 - [Part SM: Statistical Mechanics](https://open.umn.edu/opentextbooks/textbooks/part-sm-statistical-mechanics), Konstantin Likharev — Portal Book 075 - The Wikipedia pair's external links list further resources, including interactive plots of the distribution <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Maxwell–Boltzmann_distribution.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Maxwell–Boltzmann distribution* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Maxwell–Boltzmann_distribution.html" data-title="Maxwell–Boltzmann distribution"></div> *Built from `MICROSIM_GUIDE/specs/sims/Maxwell–Boltzmann_distribution.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Maxwell–Boltzmann_distribution) : [Wikitube](https://en.wikitube.io/wiki/Maxwell–Boltzmann_distribution) · pinned revision [1362212677](https://en.wikipedia.org/w/index.php?oldid=1362212677) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Physics]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Physics row P25 · sim pending (matter/Maxwell–Boltzmann_distribution).*