Root Mean Square Velocity Calculator
Enter a temperature and select a gas to instantly find the root mean square velocity (v_rms) of its molecules, along with the mean speed, most probable speed, and average kinetic energy per molecule. Switch between Kelvin, Celsius, and Fahrenheit and between m/s, km/h, mph, and ft/s. The steps panel shows every calculation, and the chart traces how RMS speed changes across a temperature range.
What is root mean square velocity?
Root mean square (RMS) velocity is a statistical measure of the speed of molecules in a gas. In a sample of gas, molecules move in all directions at a wide range of speeds described by the Maxwell-Boltzmann distribution. Rather than tracking every molecule, kinetic theory summarises the distribution with three characteristic speeds. The most probable speed (v_mp) is the speed at the peak of the distribution. The mean speed (v_avg) is the ordinary arithmetic average. The RMS speed (v_rms) is the square root of the average of all squared speeds. Because kinetic energy scales with v squared, v_rms is directly connected to thermal energy and is used most often in thermodynamic calculations. At any given temperature, the three speeds follow the ratio v_mp : v_avg : v_rms = 1 : 1.128 : 1.225, so v_rms is always the largest of the three.
The RMS speed formula and how to use it
The formula for RMS speed is v_rms = sqrt(3RT / M), where R = 8.314 J mol-1 K-1 is the universal gas constant, T is the absolute temperature in Kelvin, and M is the molar mass of the gas in kg/mol. For mean speed the numerator coefficient changes to 8/pi, giving v_avg = sqrt(8RT / (pi x M)). For most probable speed the coefficient is 2, giving v_mp = sqrt(2RT / M). To use the calculator: select a gas from the preset list (or enter a custom molar mass), choose your temperature unit, and enter the temperature. For reverse problems, switch to one of the reverse modes to find the temperature or molar mass that corresponds to a known RMS speed. Temperature must always be in Kelvin for the formula, but this calculator handles the conversion automatically.
Relationship between RMS speed and temperature
Because T appears under a square root, doubling the temperature does not double v_rms. It increases it by a factor of sqrt(2) (about 1.41). To double v_rms you must quadruple the absolute temperature. This square-root relationship also means that light gases move much faster than heavy ones at the same temperature. Hydrogen (2.016 g/mol) has an RMS speed about 3.7 times that of oxygen (32 g/mol) at 25 C, because sqrt(32/2.016) approximately equals 3.98. This principle explains why hydrogen and helium can escape a planet's atmosphere over geological time: a small fraction of their molecules exceed escape velocity, and those molecules are lost. The chart below shows how the three characteristic speeds grow with temperature for the selected gas.
Average kinetic energy and its independence from gas identity
The average translational kinetic energy per mole is E_k = (3/2)RT, and per molecule it is E_k = (3/2)k_B x T, where k_B = 1.3806 x 10-23 J/K is the Boltzmann constant. Crucially, this energy depends only on temperature, not on the identity of the gas or its molar mass. At 25 C (298.15 K), every ideal gas has exactly the same average kinetic energy per molecule: approximately 6.17 zJ (6.17 x 10-21 J). Heavier molecules achieve that same energy by moving more slowly, while lighter molecules move faster. This equipartition of energy is one of the cornerstones of statistical mechanics.
RMS speeds of common gases at 25 C (298.15 K)
| Gas | Molar mass (g/mol) | v_rms (m/s) | v_avg (m/s) | v_mp (m/s) |
|---|---|---|---|---|
| Hydrogen (H2) | 2.016 | 1934 | 1782 | 1580 |
| Helium (He) | 4.003 | 1373 | 1265 | 1121 |
| Water vapour (H2O) | 18.015 | 647 | 596 | 529 |
| Neon (Ne) | 20.18 | 611 | 563 | 499 |
| Nitrogen (N2) | 28.014 | 518 | 477 | 423 |
| Air | 28.97 | 509 | 469 | 416 |
| Carbon monoxide (CO) | 28.01 | 518 | 477 | 423 |
| Oxygen (O2) | 31.998 | 484 | 446 | 396 |
| Argon (Ar) | 39.948 | 433 | 399 | 354 |
| Carbon dioxide (CO2) | 44.01 | 413 | 380 | 337 |
| Propane (C3H8) | 44.097 | 412 | 380 | 337 |
| Chlorine (Cl2) | 70.906 | 325 | 299 | 265 |
| Krypton (Kr) | 83.798 | 299 | 275 | 244 |
| Xenon (Xe) | 131.293 | 239 | 220 | 195 |
Calculated from v_rms = sqrt(3RT/M) with R = 8.314 J mol-1 K-1. Lighter gases move faster.
Frequently asked questions
What is the difference between RMS speed, mean speed, and most probable speed?
All three describe the Maxwell-Boltzmann distribution but from different angles. The most probable speed (v_mp) is the speed at the peak of the distribution curve. The mean speed (v_avg) is the straightforward average of all molecular speeds. The RMS speed (v_rms) is the square root of the average of squared speeds, which weights faster molecules more heavily and is the relevant measure for kinetic energy. The fixed ratio is v_mp : v_avg : v_rms = sqrt(2) : sqrt(8/pi) : sqrt(3), which gives roughly 1 : 1.128 : 1.225.
Does RMS speed depend on pressure or volume?
No. For an ideal gas, v_rms depends only on the temperature and the molar mass of the gas: v_rms = sqrt(3RT/M). Pressure and volume do not appear in the formula. Two gas samples at the same temperature have the same RMS speed regardless of how much pressure they are under.
Why do we use Kelvin and not Celsius or Fahrenheit?
The formula v_rms = sqrt(3RT/M) requires absolute temperature in Kelvin. At 0 K there is no thermal energy and molecules would have zero speed, so the formula would give zero. At 0 degrees Celsius (273.15 K), molecules still have substantial kinetic energy. Using Celsius or Fahrenheit would give a physically incorrect result. This calculator converts your input to Kelvin automatically.
Which gas has the highest RMS speed at room temperature?
Hydrogen (H2, molar mass 2.016 g/mol) has the highest RMS speed of any diatomic molecule at roughly 1934 m/s at 25 C. Among noble gases, helium (4.003 g/mol) is fastest at about 1373 m/s. The lightest possible gas overall would be atomic hydrogen (1.008 g/mol), but it is not stable under normal conditions.
How do I calculate RMS speed in mph or km/h?
Use this calculator's speed unit selector to switch the output to km/h, mph, or ft/s. To convert manually: 1 m/s = 3.6 km/h = 2.237 mph = 3.281 ft/s. For example, the RMS speed of nitrogen at 25 C (approximately 515 m/s) is about 1854 km/h or 1152 mph.
What is the RMS speed of air at room temperature?
Air has a molar mass of about 28.97 g/mol. At 20 C (293.15 K), v_rms = sqrt(3 x 8.314 x 293.15 / 0.02897) approximately equals 502 m/s (1807 km/h or 1123 mph). At 25 C (298.15 K) the result is approximately 509 m/s.
How is RMS speed related to kinetic energy?
The connection is direct: kinetic energy of one mole of gas is E_k = (1/2)Mv_rms^2 = (3/2)RT. Because E_k equals (3/2)RT, the average kinetic energy is the same for all ideal gases at the same temperature, regardless of their molar mass. The RMS speed squared is proportional to energy, which is why v_rms rather than v_avg appears in the kinetic energy formula.