Temperature & Molecular Speed

Temperature measures the average kinetic energy of the molecules.

11 minUpdated 22 October 2026kinetic interpretation of temperatureBoltzmann constantrms speed

Lesson at a glanceShow the key points

For a given gas T = (273.16 K/v²_tr) v²_rms (24.5): T ∝ total translational KE.
In a mixture, ½m₁v₁² = ½m₂v₂² (24.6): heavier molecules are slower.
pV = NkT, k = 1.38 × 10⁻²³ J/K (24.7); pV = nRT, R = N_A k (24.8).
v_rms = √(3kT/m) = √(3RT/M₀) (24.9–24.10); average KE per molecule (3/2)kT; U = (3/2)nRT (24.11–24.12).
Mean speed v̄ = √(8RT/πM₀), slightly below v_rms.
Bulk motion of the gas doesn't affect its temperature.

What a thermometer really measures, according to kinetic theory, is how fast the molecules are flying about.

Temperature is kinetic energy

For an ideal gas the internal energy is just the translational kinetic energy of its molecules, ½Mv²_rms. At fixed volume, pV = ⅓Mv² gives p/p_tr = v²/v²_tr; and on the absolute scale p/p_tr = T/273.16 K. So

(24.5)

T = \left(\frac{273.16\ \text{K}}{v_{tr}^2}\right)v_{rms}^2

for a given gas, T ∝ v²_rms, so T ∝ the total translational kinetic energy

Mixtures

In a mixture, collisions pass energy from faster to slower molecules until, in equilibrium, the average kinetic energy is the same for every kind:

(24.6)

\tfrac12m_1v_1^2 = \tfrac12m_2v_2^2

at the same temperature heavier molecules move more slowly

Worked example

Helium versus argon

At some temperature helium atoms (4 g/mol) have rms speed 1350 m/s. What is the rms speed of argon atoms (40 g/mol) at the same temperature?

Boltzmann’s constant and the gas equation

Putting v² = (v²_tr/273.16 K)T into pV = ⅓Nmv² gives pV = N(⅓mv²_tr/273.16 K)T. The bracket contains ½mv²_tr, the same for every gas at 273.16 K, so it is a universal constant, k:

Boltzmann form(24.7)

pV = NkT, \qquad k = 1.38\times10^{-23}\ \text{J K}^{-1}

With N = nN_A (N_A = 6.02 × 10²³ mol⁻¹):

Equation of state of an ideal gas(24.8)

pV = nRT, \qquad R = N_Ak = 8.314\ \text{J K}^{-1}\text{mol}^{-1}

Worked example

How many molecules in a cubic metre?

How many molecules are in 1 m³ of gas at 1.0 × 10⁵ Pa and 0°C?

Speeds and energies in terms of T

Comparing pV = ⅓Nmv²_rms with pV = NkT:

rms speed(24.9–24.10)

v_{rms} = \sqrt{\frac{3kT}{m}} = \sqrt{\frac{3RT}{M_0}}

M₀ = molar mass

Kinetic energy(24.11–24.12)

\tfrac12mv_{rms}^2 = \tfrac32kT, \qquad U = \tfrac32nRT

average per molecule, and total for n moles

The average speed is a little smaller than the rms speed:

\bar v = \sqrt{\frac{8kT}{\pi m}} = \sqrt{\frac{8RT}{\pi M_0}}

Worked example

Nitrogen in a warm room

Find the rms speed of nitrogen molecules (28 g/mol) at 300 K.

What to remember

1For a given gas, T ∝ v²_rms; in a mixture all molecules share the same average kinetic energy.
2pV = NkT (k = 1.38 × 10⁻²³ J/K); pV = nRT with R = N_A k.
3v_rms = √(3kT/m) = √(3RT/M₀); v̄ = √(8RT/πM₀).
4Average KE per molecule (3/2)kT; total U = (3/2)nRT.
5Systematic (bulk) motion doesn't raise temperature.

Check yourself

To double the rms speed of a gas's molecules, its absolute temperature must be…

Hydrogen and oxygen are at the same temperature. Their molecules have the same…

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

ReadPressure, temperature and rms speedOpenStax University Physics Vol. 2 §2.2 ReadKinetic temperatureHyperPhysics ReadIdeal gas lawHyperPhysics ReadBoltzmann constantWikipedia ReadRoot-mean-square speedWikipedia

PreviousPressure from Molecules Next The Gas Laws from Kinetic Theory

Temperature & Molecular Speed

Temperature measures the average kinetic energy of the molecules.

11 minUpdated 22 October 2026kinetic interpretation of temperatureBoltzmann constantrms speed

Lesson at a glanceShow the key points

For a given gas T = (273.16 K/v²_tr) v²_rms (24.5): T ∝ total translational KE.
In a mixture, ½m₁v₁² = ½m₂v₂² (24.6): heavier molecules are slower.
pV = NkT, k = 1.38 × 10⁻²³ J/K (24.7); pV = nRT, R = N_A k (24.8).
v_rms = √(3kT/m) = √(3RT/M₀) (24.9–24.10); average KE per molecule (3/2)kT; U = (3/2)nRT (24.11–24.12).
Mean speed v̄ = √(8RT/πM₀), slightly below v_rms.
Bulk motion of the gas doesn't affect its temperature.

What a thermometer really measures, according to kinetic theory, is how fast the molecules are flying about.

Temperature is kinetic energy

(24.5)

T = \left(\frac{273.16\ \text{K}}{v_{tr}^2}\right)v_{rms}^2

for a given gas, T ∝ v²_rms, so T ∝ the total translational kinetic energy

Mixtures

In a mixture, collisions pass energy from faster to slower molecules until, in equilibrium, the average kinetic energy is the same for every kind:

(24.6)

\tfrac12m_1v_1^2 = \tfrac12m_2v_2^2

at the same temperature heavier molecules move more slowly

Worked example

Helium versus argon

At some temperature helium atoms (4 g/mol) have rms speed 1350 m/s. What is the rms speed of argon atoms (40 g/mol) at the same temperature?

Boltzmann’s constant and the gas equation

Putting v² = (v²_tr/273.16 K)T into pV = ⅓Nmv² gives pV = N(⅓mv²_tr/273.16 K)T. The bracket contains ½mv²_tr, the same for every gas at 273.16 K, so it is a universal constant, k:

Boltzmann form(24.7)

pV = NkT, \qquad k = 1.38\times10^{-23}\ \text{J K}^{-1}

With N = nN_A (N_A = 6.02 × 10²³ mol⁻¹):

Equation of state of an ideal gas(24.8)

pV = nRT, \qquad R = N_Ak = 8.314\ \text{J K}^{-1}\text{mol}^{-1}

Worked example

How many molecules in a cubic metre?

How many molecules are in 1 m³ of gas at 1.0 × 10⁵ Pa and 0°C?

Speeds and energies in terms of T

Comparing pV = ⅓Nmv²_rms with pV = NkT:

rms speed(24.9–24.10)

v_{rms} = \sqrt{\frac{3kT}{m}} = \sqrt{\frac{3RT}{M_0}}

M₀ = molar mass

Kinetic energy(24.11–24.12)

\tfrac12mv_{rms}^2 = \tfrac32kT, \qquad U = \tfrac32nRT

average per molecule, and total for n moles

The average speed is a little smaller than the rms speed:

\bar v = \sqrt{\frac{8kT}{\pi m}} = \sqrt{\frac{8RT}{\pi M_0}}

Worked example

Nitrogen in a warm room

Find the rms speed of nitrogen molecules (28 g/mol) at 300 K.

What to remember

1For a given gas, T ∝ v²_rms; in a mixture all molecules share the same average kinetic energy.
2pV = NkT (k = 1.38 × 10⁻²³ J/K); pV = nRT with R = N_A k.
3v_rms = √(3kT/m) = √(3RT/M₀); v̄ = √(8RT/πM₀).
4Average KE per molecule (3/2)kT; total U = (3/2)nRT.
5Systematic (bulk) motion doesn't raise temperature.

Check yourself

To double the rms speed of a gas's molecules, its absolute temperature must be…

Hydrogen and oxygen are at the same temperature. Their molecules have the same…

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

PreviousPressure from Molecules Next The Gas Laws from Kinetic Theory

Temperature is kinetic energy#

Mixtures

Boltzmann’s constant and the gas equation#

Speeds and energies in terms of T#

Check yourself

Keep exploring

Temperature is kinetic energy#

Mixtures

Boltzmann’s constant and the gas equation#

Speeds and energies in terms of T#

Check yourself

Keep exploring

Temperature is kinetic energy

Boltzmann’s constant and the gas equation

Speeds and energies in terms of T

Temperature is kinetic energy

Boltzmann’s constant and the gas equation

Speeds and energies in terms of T