Particle Velocity Calculator — Kinetic Theory of Gases
Find the root-mean-square speed, mean speed and most probable speed of gas molecules at any temperature, using kinetic theory. Select a gas or enter a custom molar mass — results update instantly.
°C
Gas
Root-mean-square speed — the most common measure of particle velocity in a gas
- 1
Temperature in Kelvin
25 + 273.15 = 298.15 K - 2
Molar mass in kg/mol
28.014 ÷ 1000 = 0.02801 kg/mol - 3
3 × R × T ÷ M
3 × 8.31446 × 298.15 ÷ 0.02801 = 265,469.7 m²/s²R = 8.314 J/(mol·K) is the molar gas constant. - 4
RMS speed v_rms = √(3RT/M)
√(265,469.7) = 515.2
How does this calculator work?
Gas particles' root-mean-square speed is v_rms = √(3RT/M), mean speed v̄ = √(8RT/πM), and most probable speed v_p = √(2RT/M), where R = 8.314 J/mol·K, T is temperature in Kelvin, and M is molar mass in kg/mol. Speed rises with temperature and falls with molecular weight.
Formula
How this is calculated
Kinetic theory models an ideal gas as a vast number of molecules in random motion, colliding elastically with each other and the container walls. The Maxwell–Boltzmann distribution describes the statistical spread of molecular speeds. From it, three characteristic speeds emerge: the most probable speed v_p = √(2RT/M) (the peak of the distribution), the mean speed v̄ = √(8RT/πM) (the arithmetic average), and the root-mean-square speed v_rms = √(3RT/M) (the square root of the average squared speed, directly linked to kinetic energy). All three grow as √T and shrink as √M — light molecules at high temperature move fastest.
The mean translational kinetic energy per molecule is (3/2)k_BT regardless of mass, where k_B is the Boltzmann constant. Per mole it is (3/2)RT. This is the equipartition result for three translational degrees of freedom. Note that real gases deviate from these ideal formulas at high pressure or low temperature (near condensation), and the formulas apply strictly to monatomic or diatomic gases in translational modes only — rotational and vibrational energy add further contributions to heat capacity.
Molar masses used for the preset gases are IUPAC 2021 standard atomic weights rounded to four significant figures.
Frequently asked questions
Use v_rms when relating speed to kinetic energy (KE = ½mv_rms²). Use v̄ for effusion rate calculations (Graham's law). Use v_p for the most likely single-molecule measurement. For everyday gas behaviour comparisons, v_rms is the most quoted.
All gases at the same temperature have the same average kinetic energy per molecule (3/2)k_BT. Since KE = ½mv², a lighter molecule (small m) must have a larger v to hold the same energy. Hydrogen molecules are ~14× lighter than nitrogen, so they move ~3.7× faster.
No — kinetic theory in this ideal-gas form applies only to gases where inter-molecular forces are negligible. In liquids and solids, strong intermolecular attractions dominate and particles move in constrained or vibrating patterns, not freely.
Also known as
TG we-Calculate Editorial Team. (2026). Particle Velocity Calculator — Kinetic Theory of Gases [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/particles-velocity-calculator
TG we-Calculate Editorial Team. "Particle Velocity Calculator — Kinetic Theory of Gases." TG we-Calculate. 2026. https://we-calculate.com/calculator/particles-velocity-calculator.
TG we-Calculate Editorial Team, "Particle Velocity Calculator — Kinetic Theory of Gases," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/particles-velocity-calculator
@misc{wecalculate_particles_velocity_calculator, title = {Particle Velocity Calculator — Kinetic Theory of Gases}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/particles-velocity-calculator}}, year = {2026}, note = {TG we-Calculate} }
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