| Gas | v_rms (m/s) | v_avg (m/s) | v_mp (m/s) |
|---|
Bars are relative to the fastest gas (H₂). v_rms scales as 1/√M, so lighter gases are faster at the same temperature.
The three Maxwell-Boltzmann speeds
R = 8.314462618 J/(mol·K) (gas constant), T = absolute temperature in kelvin, M = molar mass in kg/mol. The ratios are exact constants: v_rms : v_avg : v_mp = 1 : 0.9213 : 0.8165, because v_avg/v_rms = √(8/3π) and v_mp/v_rms = √(2/3). v_rms is special: mean kinetic energy per molecule is exactly ½m·v_rms² = &frac32;kT.
Worked examples
1. Nitrogen at 25°C. M = 0.0280134 kg/mol, T = 298.15 K: v_rms = √(3 × 8.314462618 × 298.15 / 0.0280134) = 515.24 m/s = 1,854.88 km/h. Trio: 515.24 / 474.70 / 420.69 m/s.
2. Hydrogen at 25°C (fastest common gas). v_rms = 1,920.71 m/s.
3. Helium at 500 K. v_rms = 1,765.19 m/s = 6,354.68 km/h.
4. Inverse: what temperature gives nitrogen v_rms = 515 m/s? T = M·v²/(3R) = 297.87 K = 24.72°C.
Not the same as electrical RMS
"RMS" also appears in electronics (RMS voltage of an AC waveform, RMS power). That is a different concept with different formulas. This tool covers molecular RMS speed from kinetic theory only. For electrical RMS, see our boundary tools listed below.
Related tools
FAQ
What is the RMS speed formula?
vrms = √(3RT/M), where R = 8.314462618 J/(mol·K) is the gas constant, T is the absolute temperature in kelvin, and M is the molar mass in kg/mol. Example: nitrogen (N₂, M = 0.0280134 kg/mol) at 25°C (298.15 K) gives vrms = √(3 × 8.314462618 × 298.15 / 0.0280134) = 515.24 m/s.
What is the difference between v_rms, v_avg, and v_mp?
All three speeds come from the Maxwell-Boltzmann distribution: vrms = √(3RT/M) is the root-mean-square speed, vavg = √(8RT/πM) is the mean (average) speed, and vmp = √(2RT/M) is the most probable speed (the peak of the distribution curve). Their ratios are exact constants: vrms : vavg : vmp = 1 : 0.9213 : 0.8165. For nitrogen at 25°C: 515.24 / 474.70 / 420.69 m/s.
Why use RMS speed instead of average speed?
Because average kinetic energy depends on the mean of v², not the mean of v. Equipartition of energy gives exactly ½m·vrms² = &frac32;kT, so vrms is the speed that carries the mean kinetic energy of the gas. The plain average speed is about 7.87% lower than vrms.
What is the RMS speed of nitrogen at room temperature?
At 25°C (298.15 K), nitrogen (N₂) has vrms = 515.24 m/s, which equals 1,854.88 km/h or 1,152.57 mph. The full trio is vrms = 515.24, vavg = 474.70, vmp = 420.69 m/s.
Which gas has the highest RMS speed?
The lightest gas, because vrms is proportional to 1/√M. At 25°C: hydrogen H₂ = 1,920.71 m/s, helium He = 1,363.09 m/s, dry air = 506.71 m/s, argon Ar = 431.47 m/s, carbon dioxide CO₂ = 411.07 m/s.
Does pressure affect RMS speed?
No. For an ideal gas, vrms depends only on temperature and molar mass: vrms = √(3RT/M). Pressure changes how many molecules are in a given volume, not the shape of the speed distribution. (Real gases at extreme pressure deviate slightly from this ideal behavior.)
Can I plug °C or °F directly into the RMS speed formula?
No, you must convert to kelvin first: K = °C + 273.15, and K = (°F − 32) × 5/9 + 273.15. The temperature in the formula is absolute. This calculator converts your input automatically and always uses kelvin internally.
How do I find the temperature from a target RMS speed?
Rearrange the formula: T = M·vrms²/(3R). Example: for nitrogen to have vrms = 515 m/s, T = 0.0280134 × 515² / (3 × 8.314462618) = 297.87 K = 24.72°C. Use the Inverse tab above; it also shows a round-trip check back through vrms = √(3RT/M).
What does the Maxwell-Boltzmann distribution curve show?
It shows the probability distribution of molecular speeds in a gas at one temperature: f(v) on the vertical axis versus speed on the horizontal axis. The curve peaks exactly at vmp (420.69 m/s for nitrogen at 25°C), and the total area under the curve equals 1. Lighter gases or higher temperatures shift and broaden the curve to the right.
Is RMS speed the same as RMS voltage?
No. They share the word “RMS” but describe different physics: RMS speed is a molecular speed from the kinetic theory of gases, while RMS voltage is a property of an alternating-current waveform. For electrical RMS, use our peak-to-peak voltage calculator or the RMS-to-watts power tool instead.
Methodology & sources
Formulas from the kinetic theory of gases (Maxwell 1860, Boltzmann 1868). Gas constant R = 8.314462618 J/(mol·K) (CODATA). Molar masses from IUPAC atomic weights: H₂ 2.01588, He 4.002602, N₂ 28.0134, O₂ 31.9988, dry air 28.9647, CO₂ 44.0098, Ar 39.948 g/mol. Temperature conversions use the exact 273.15 K offset. Assumes ideal gas; real-gas deviations at extreme pressure are not modeled. This page runs 100% in your browser; no data leaves your device.