Speed Distribution Calculator

Model thermal speed spread for gases using reliable statistical physics tools today. Get PDF, CSV, and key statistics instantly for experiments and analysis quickly.

Calculator

Temperature T

K

Use absolute temperature in kelvin.

Molar mass M

Examples: N₂ ≈ 28.0134 g/mol, Ar ≈ 39.948 g/mol.

Number of bins

Higher bins give finer table resolution.

Speed minimum v_min

Choose 0 for full distribution start.

Speed maximum v_max

The PDF and CDF are computed in SI units.

Probability range a to b

Uses the same speed unit as v_max.

Reset

Example data table

Example for nitrogen (28.0134 g/mol) at 300 K. Values are illustrative and may differ slightly by range and binning.

v (m/s)	f(v)	F(v)
0	0.000000e+0	0.000000
300	1.630467e-3	0.201354
600	1.431902e-3	0.743151
900	2.574334e-4	0.971971
1200	1.330880e-5	0.998955

Formula used

This tool uses the Maxwell–Boltzmann speed distribution for an ideal gas in three dimensions:

f(v) = 4π · ( m / (2πk_BT) )^3/2 · v² · exp( − m v² / (2k_BT) )

v is speed in m/s, T is temperature in K.
m is particle mass in kg, from m = M / N_A.
k_B is Boltzmann’s constant; N_A is Avogadro’s number.
The cumulative distribution is F(v) = ∫₀ᵛ f(u) du.

Most probable speed

v_mp = √(2k_BT / m)

Mean speed

v̄ = √(8k_BT / (πm))

RMS speed

v_rms = √(3k_BT / m)

How to use this calculator

Enter the temperature T in kelvin.
Enter the gas molar mass M and choose its unit.
Select a speed range and a practical number of bins.
Optionally enter speeds a and b for a probability query.
Press Calculate to view statistics and the distribution table.
Use Download CSV or Download PDF for reporting.

Notes and assumptions

Applies to classical ideal gases in thermal equilibrium.
At very low temperatures, quantum effects may matter.
Numerical integration uses trapezoids for stable CDF values.

Use it to compare gases and temperatures with confidence.

Accurate physics, clear outputs, and export-ready results anytime online.

Related Calculators

Boltzmann factor calculator Partition function calculator Canonical ensemble calculator Grand canonical calculator Microcanonical entropy calculator Chemical potential calculator Internal energy calculator Thermal wavelength calculator Maxwell Boltzmann speed Mean speed calculator

Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.