Root Mean Square Speed

v_{rms} = \sqrt{\frac{3k_B T}{m}} = \sqrt{\frac{3RT}{M}}

RMS speed of gas molecules.

Variables

v_rms

RMS speed

Root mean square velocity

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

From ½m⟨v²⟩ = (3/2)kT: v_rms = √(3kT/m).

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_rms = 4.0778579 m/s

Other thermodynamics formulas with the same quantities

Most Probable Speed (Kinetic Theory) shares Boltzmann constant, Temperature, Molecular mass, Gas constant, Molar mass
Mean Speed (Kinetic Theory) shares Boltzmann constant, Temperature, Molecular mass, Gas constant, Molar mass
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

Where this formula is listed

Thermodynamics formulas, under Kinetic Theory, alongside Pressure (Kinetic Theory).
In the thermodynamics list it comes after Average Kinetic Energy (Gas) and before Mean Free Path.
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 Root Mean Square Speed formula?

Root Mean Square Speed is written v_rms = √((3k_B T)/m) = √(3RT/M). RMS speed of gas molecules. It belongs to Thermodynamics, under Kinetic Theory.

What are the units in v_rms = √((3k_B T)/m) = √(3RT/M)?

In SI units, v_r ms is rms 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 Root Mean Square Speed formula derived?

From ½m⟨v²⟩ = (3/2)kT: v_rms = √(3kT/m). The result is v_rms = √((3k_B T)/m) = √(3RT/M).