Root Mean Square Velocity Calculator — Ideal Gas RMS Speed
Calculate the root-mean-square speed of gas molecules from temperature and molar mass — plus mean speed, most probable speed and molar kinetic energy.
°C
Gas
√(3RT/M) — the root-mean-square speed of gas molecules
- 1
Temperature in Kelvin
25 + 273.15 = 298.15 - 2
Molar mass in kg/mol
28.97 ÷ 1000 = 0.02897 - 3
3 × R × T
3 × 8.3145 × 298.15 = 7,436.87 - 4
Divide by M (kg/mol)
7,436.87 ÷ 0.02897 = 256,709.39 - 5
v_rms = √(3RT/M)
√(256,709.39) = 506.7
How does this calculator work?
v_rms = √(3RT/M) where R = 8.314 J/(mol·K), T is temperature in Kelvin (°C + 273.15), and M is molar mass in kg/mol. At 25 °C, air molecules (28.97 g/mol) have v_rms ≈ 509 m/s; hydrogen (2.016 g/mol) reaches ≈ 1926 m/s — lighter and hotter means faster.
Formula
How this is calculated
Kinetic-molecular theory models an ideal gas as a large number of identical, randomly-moving particles. The speeds of the individual molecules follow the Maxwell–Boltzmann distribution. Three useful averages emerge from this distribution:
The root-mean-square (RMS) speed v_rms = √(3RT/M) comes from setting the mean kinetic energy (½mv²) equal to (3/2)kT per molecule; for one mole, R (the universal gas constant, 8.314 J/(mol·K)) replaces Boltzmann's constant k and M (kg/mol) replaces the mass of a single molecule. The result is in m/s when R is in J/(mol·K) and M in kg/mol.
The mean speed v̄ = √(8RT/πM) is the simple average of all molecular speeds. It is slightly lower than v_rms because squaring before averaging gives extra weight to the faster molecules. The most probable speed v_p = √(2RT/M) is the peak of the Maxwell–Boltzmann curve — the speed most molecules are nearest to.
All three expressions assume an ideal gas: no intermolecular attractions, elastic collisions, and negligible molecular volume. Real gases deviate at high pressures and low temperatures. The calculator converts °C to Kelvin (T = T_°C + 273.15) and g/mol to kg/mol internally.
Frequently asked questions
v_p (most probable) < v_mean (average) < v_rms (root mean square). They are all averages of the Maxwell–Boltzmann distribution, but weighted differently: v_p is the distribution peak, v_mean is the arithmetic mean, and v_rms is the square root of the mean of squared speeds (emphasising the faster tail).
At the same temperature every gas has the same average kinetic energy (3/2 RT per mole). Since KE = ½Mv², a lighter gas (smaller M) must have a higher speed to carry the same energy. H₂ (2 g/mol) is 14× lighter than N₂ (28 g/mol), so its v_rms is √14 ≈ 3.7× higher.
The formula is exact only for ideal gases. Air behaves near-ideally at normal conditions (20–25 °C, 1 atm), so the result is a good approximation. At very high pressures or very low temperatures, molecular interactions become significant and the ideal-gas assumption breaks down.
Also known as
TG we-Calculate Editorial Team. (2026). Root Mean Square Velocity Calculator — Ideal Gas RMS Speed [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/root-mean-square-velocity-calculator
TG we-Calculate Editorial Team. "Root Mean Square Velocity Calculator — Ideal Gas RMS Speed." TG we-Calculate. 2026. https://we-calculate.com/calculator/root-mean-square-velocity-calculator.
TG we-Calculate Editorial Team, "Root Mean Square Velocity Calculator — Ideal Gas RMS Speed," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/root-mean-square-velocity-calculator
@misc{wecalculate_root_mean_square_velocity_calculator, title = {Root Mean Square Velocity Calculator — Ideal Gas RMS Speed}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/root-mean-square-velocity-calculator}}, year = {2026}, note = {TG we-Calculate} }
Did this calculator help you?