Maxwell-Boltzmann Speed Calculator

Calculate most probable, average, and RMS speeds of gas molecules from the Maxwell-Boltzmann distribution. Find fraction of molecules above any speed at a given temperature.

💨 Gas Theory📐 v_rms = √(3RT/M)🧪 Chemistry
Molar mass M (g/mol)
Temperature T (K)
Please enter valid values.

Formula & Reference

VariableSymbolFormulaUnits
Maxwell-Boltzmann Speed Calculator—v_rms = √(3RT/M)m/s

Step-by-Step Examples

Example 1
N2 at 25 C

N2 (M=28.02 g/mol) at 298 K.

  • v_rms = sqrt(3 x 8.314 x 298 / 0.02802) = sqrt(265,300) = 515 m/s
  • v_avg = sqrt(8RT/piM) = 0.9213 x v_rms = 474 m/s
  • v_mp = sqrt(2RT/M) = 0.8165 x v_rms = 420 m/s
✓ v_rms=515, v_avg=474, v_mp=420 m/s for N2
Example 2
H2 at 25 C

H2 (M=2.016 g/mol) at 298 K.

  • v_rms = sqrt(3 x 8.314 x 298/0.002016) = sqrt(3,686,000) = 1920 m/s
  • H2 much faster than N2: speed inversely proportional to sqrt(M)
✓ v_rms = 1920 m/s for H2
Example 3
Effect of Temperature

N2 at 600 K (doubled from 300 K).

  • v_rms proportional to sqrt(T)
  • Doubling T: v_rms increases by sqrt(2) = 1.414
  • v_rms(600K) = 515 x sqrt(2) = 728 m/s
✓ Speed increases by sqrt(2) when T doubles

Real-World Applications

💨
Gas Effusion
Lighter gases effuse faster (Graham law). v inversely proportional to sqrt(M).
🌡️
Thermodynamics
Average kinetic energy = (3/2)RT per mole. RMS speed quantifies this.
🧪
Atmospheric Science
Different gases have different speed distributions. Lighter H2 and He escape Earth atmosphere more easily.
⚡
Plasma Physics
Maxwell-Boltzmann applies to plasma electrons. v_rms determines thermal energy and reaction rates.

Common Mistakes to Avoid

⚠️
Molar mass must be in kg/mol for SI

M must be in kg/mol in the formula: v = sqrt(3RT/M) with R=8.314 J/molK. Divide g/mol by 1000 to get kg/mol.

⚠️
Temperature in Kelvin

Must use Kelvin. 25 C = 298 K. Never use Celsius in this formula.

⚠️
v_mp < v_avg < v_rms always

The three speeds are ordered: most probable < mean < rms. Ratios: 1 : 1.128 : 1.225 for any ideal gas.

Frequently Asked Questions

What is the Maxwell-Boltzmann distribution? ▾
The probability distribution of molecular speeds in an ideal gas at thermal equilibrium. Bell-shaped but skewed: long tail to high speeds.
What is the difference between v_mp, v_avg, v_rms? ▾
v_mp: most probable speed (peak of distribution). v_avg: arithmetic mean speed. v_rms: sqrt of mean square speed. v_rms used in kinetic energy: KE = (1/2)mv_rms^2.
How does molar mass affect speed? ▾
v inversely proportional to sqrt(M). Doubling molar mass decreases speed by sqrt(2). H2 (2 g/mol) is sqrt(28/2) = 3.74x faster than N2 (28 g/mol).
Why are lighter gases less abundant in atmosphere? ▾
H2 and He have v_rms near escape velocity (11.2 km/s). Enough molecules in the tail exceed escape velocity to gradually leave. N2 and O2 are much slower.
What fraction of molecules move above v_rms? ▾
About 32% of molecules have speed above v_rms (analogous to fraction beyond one standard deviation in a normal distribution, but distribution is skewed).

Related Chemistry Calculators

💨 Graham Law⚗️ Ideal Gas Law💨 Mean Free Path💨 Dalton Law💨 van der Waals

Formula Explorer connections

Interpretation: This relationship connects pressure, volume, temperature, amount or phase composition for gases and volatile mixtures. Assumption: Use absolute temperature and compatible pressure-volume units. Ideal behavior weakens at high pressure, low temperature, strong intermolecular attraction or near phase change.

Mean Free Path Calculator →Gas Mixture Partial Pressure Calculator →Raoult's Law Calculator →Chemistry Formula Explorer →