Mean particle speed v = √(8kT / πm) from temperature and mass.
The Average Particle Speed Calculator applies the Maxwell–Boltzmann distribution to a single particle, returning the mean thermal speed v̄ = √(8kT / πm) for any temperature and mass.
This is the microscopic counterpart of the molecular-speed calculation: rather than working with a mole of gas, it works with one particle of mass m. Useful for plasma physics, semiconductor physics, atmospheric science, and any context where individual particle kinematics matter.
Average Particle Speed is calculated from absolute temperature T (in kelvin) and particle mass m (in kg): v̄ = √(8kT / πm), with k = 1.380649 × 10⁻²³ J/K. The Average Particle Speed Calculator reports this in m/s, km/h, mph.
Atoms in air at 300 K average 400–500 m/s. Micron-sized aerosols Brownian-average a few mm/s. Electron thermal speed at 10 000 K is 6×10⁵ m/s.
Average Particle Speed is a statistical / derived quantity rather than a measured distance ÷ time. It describes the average behaviour of a population — gas molecules, orbiting bodies, fluid parcels or rotating points — at equilibrium.
The Average Particle Speed formula is v̄ = √(8kT / πm), with k = 1.380649 × 10⁻²³ J/K. Symbolically: v̄ = √(8kT / πm).
The formula has these rearrangements that solve for any unknown:
The output unit depends on the input units. SI inputs (kelvin, kg/mol or kg, m³/s, m²) produce m/s.
To calculate average particle speed:
Worked example: For an oxygen molecule (m = 5.31 × 10⁻²⁶ kg) at 300 K, v̄ = √(8 × 1.38e−23 × 300 / (π × 5.31e−26)) = 445 m/s.
Three steps:
Enter absolute temperature (K) and particle mass (kg). The calculator works from single atoms to dust grains.
Substitute the physical inputs directly into the formula. Unlike distance / time tools, Average Particle Speed doesn't need a measured trip — it derives from a state variable (temperature, mass, radius).
Worked example: For an oxygen molecule (m = 5.31 × 10⁻²⁶ kg) at 300 K, v̄ = √(8 × 1.38e−23 × 300 / (π × 5.31e−26)) = 445 m/s.
Switch input units freely; the calculator does the conversion before substituting.
Rearrange the formula to solve for any input given the output. The calculator inverts v̄ = √(8kT / πm) for you.
Worked example: Given v̄, the equipartition temperature is T = π m v̄² / (8k). For an electron (m = 9.11×10⁻³¹ kg) to average 10⁶ m/s, T = 8.6 × 10⁴ K.
This is useful for planning — e.g. finding the temperature required to reach a target average particle speed, or the orbital radius for a target m/s speed.
For a population of particles with different masses, the bulk mean speed is the mass-fraction-weighted sum of √(8kT/πmᵢ).
For example, comparing two different operating conditions side-by-side highlights the inverse-square / square-root scaling that governs particle kinetic theory.
Particle speed is an instantaneous statistical quantity — there is no elapsed time input. Add a separate per-collision time only if you need mean-free-path / collision frequency.
Where a derived time (e.g. orbital period, mean-free-path) is requested, convert units in the calculator's results panel rather than in the input form.
A particle changes direction every collision. Multiple-leg averaging therefore reduces to the same equilibrium mean speed regardless of how the path is segmented.
Segment-by-segment analysis is most useful when the input variable changes — e.g. temperature ramp, orbital perihelion → aphelion, pipe diameter step-down.
Average Particle Speed is normally reported in m/s. Common alternative units:
The calculator handles all conversions automatically.
Average Particle Speed is a scalar — magnitude only. Velocity is a vector that adds direction.
For a particle in a thermal / isotropic situation, the average velocity is zero (directions cancel out) even though the average speed is large. Average Particle Speed reflects the typical magnitude that matters for kinetic energy, mean free path or transport.
Average Particle Speed is the population / time-averaged value. Instantaneous average particle speed is the value at one moment for one particle.
For gas molecules, instantaneous speeds follow the Maxwell–Boltzmann distribution; the average is the central tendency, not the value any individual molecule has.
Constant average particle speed means the value doesn't change with time. Average Particle Speed can equal that constant if conditions are steady (constant T, constant r, constant Q), but the moment one input drifts, the average drifts with it.
For practical purposes treat Average Particle Speed as a snapshot of the system's current state — re-evaluate whenever an input changes.
Use the Maxwell–Boltzmann speed distribution: P(v) ∝ v² exp(−mv² / 2kT). The mean v̄ is the centroid.
The "graph" for a physics-style average average particle speed is usually a distribution plot rather than a v(t) trace — the average corresponds to the area-weighted centroid of the distribution.
Replace a velocity-time graph with the Gaussian velocity distribution along each axis: standard deviation = √(kT/m).
For isotropic systems, plotting one Cartesian component of velocity yields a Gaussian centred at zero whose width sets the magnitude of Average Particle Speed.
There are several common mistakes when computing average particle speed. Click each card below to expand the explanation.
Practice the following worked average particle speed problems. Click "Show Solution" to reveal the step-by-step answer.
v̄ = √(8 × 1.38e−23 × 295 / (π × 6.63e−26)) = √(1.56 × 10⁵) ≈ 395 m/s.
T = π m v̄² / (8k) = (π × 1.673e−27 × 10¹⁰) / (8 × 1.38e−23) = 477 K.
v̄ ∝ 1/√m. m_e / m_p ≈ 1 / 1836 → electrons are √1836 ≈ 42.8× faster: ~1.07 × 10⁵ m/s vs 2500 m/s.
m ≈ 10⁻¹⁵ kg makes v̄ ≈ a few mm/s. Brownian motion is visible but barely on human timescales.
For a single particle of mass m at absolute temperature T, the mean thermal speed is v̄ = √(8kT / πm), where k = 1.381 × 10⁻²³ J/K is the Boltzmann constant.
Molecular speed uses molar mass M and the gas constant R. Particle speed uses individual particle mass m and the Boltzmann constant k. Both give the same answer for any specific gas (since R = N_A × k).
Yes — enter the particle mass in atomic mass units (u) or kilograms. The calculator handles electrons (~0.000549 u), protons (~1.007 u), and any individual atom or ion.
Only on absolute temperature T and the particle's mass m. Pressure, volume and the presence of other gases do not affect the thermal speed of a single particle.
Yes. Most probable: v_p = √(2kT/m). Mean (average): v̄ = √(8kT/πm). RMS: v_rms = √(3kT/m). The ratio v_p : v̄ : v_rms is approximately 1 : 1.128 : 1.225.
At room temperature, free electrons in a conductor have a thermal speed of about 100 km/s (just from kT), and a Fermi speed of roughly 1500 km/s — far higher than any classical particle.