Mean Speed

v_{mean} = \sqrt{\frac{8k_B T}{\pi m}} = \sqrt{\frac{8RT}{\pi M}}

Average speed of gas molecules from Maxwell-Boltzmann distribution.

Variables

v_mean

Mean speed

Average molecular speed

Allowed values: greater than 0

m/s

k_B

Boltzmann constant

1.38×10⁻²³

In SI units: kg·m²/(s²·K)

Allowed values: greater than 0

J/K

Temperature

Absolute temperature

Allowed values: greater than 0

Molecular mass

Mass per molecule

Allowed values: greater than 0

Gas constant

8.314

In SI units: kg·m²/(s²·mol·K)

Allowed values: greater than 0

J/(mol·K)

Molar mass

Mass per mole

Allowed values: greater than 0

kg/mol

Derivation

v_mean = ∫v f(v)dv = √(8kT/πm). Order: v_p < v_mean < v_rms.

Your notes

0/5000

Constants in this formula

k_B

Boltzmann Constant

1.380 649 × 10⁻²³. Exact by definition.

J/K

Molar Gas Constant

8.314 462 618…. Exact by definition.

J/(mol·K)

Check value

Round inputs run through this page's calculator, with each constant at its tabulated value. Use it to test your own working or a rearranged form. It is a test case, not a real-world example.

R = 8.314 462 618… J/(mol·K) (Gas constant)
T = 2 K (Temperature)
M = 3 kg/mol (Molar mass)

Result: v_mean = 3.7570028 m/s

Other thermodynamics formulas with the same quantities

Ideal Gas Law (Kinetic Theory) shares Gas constant, Temperature
Ideal Gas Law (Molecular) (Kinetic Theory) shares Boltzmann constant, Temperature
Mean Free Path (Kinetic Theory) shares Boltzmann constant, Temperature
Work (Isothermal Process) (First Law) shares Gas constant, Temperature
Internal Energy (Ideal Gas) (First Law) shares Gas constant, Temperature
Stefan-Boltzmann Law (Heat & Temperature) shares Temperature

Where this formula is listed

Thermodynamics formulas, under Kinetic Theory, alongside Pressure (Kinetic Theory).
In the thermodynamics list it comes after Most Probable Speed and before Entropy Change (Ideal Gas).
Thermodynamics formula sheet, one of 34 formulas printed with symbol keys and SI units.
The full formula index, grouped by category and topic.

Common questions

What is the Mean Speed formula?

Mean Speed is written v_mean = √((8k_B T)/(πm)) = √(8RT/(πM)). Average speed of gas molecules from Maxwell-Boltzmann distribution. It belongs to Thermodynamics, under Kinetic Theory.

What are the units in v_mean = √((8k_B T)/(πm)) = √(8RT/(πM))?

In SI units, v_m ean is mean speed, measured in m/s; k_B is boltzmann constant, measured in kg·m²/(s²·K); T is temperature, measured in K; m is molecular mass, measured in kg; R is gas constant, measured in kg·m²/(s²·mol·K); M is molar mass, measured in kg/mol.

How is the Mean Speed formula derived?

v_mean = ∫v f(v)dv = √(8kT/πm). Order: v_p < v_mean < v_rms. The result is v_mean = √((8k_B T)/(πm)) = √(8RT/(πM)).