State the equation relating the average kinetic energy of a molecule to the absolute temperature, and define every symbol.

½m⟨c²⟩ = (3/2)kT, where m is the mass of one molecule in kg, ⟨c²⟩ the mean square speed in m² s⁻², k the Boltzmann constant of 1.38 × 10⁻²³ J K⁻¹ and T the absolute temperature in K.

What does the equation ½m⟨c²⟩ = (3/2)kT tell you about the meaning of temperature?

It shows that the absolute temperature of a gas is a direct measure of the mean random kinetic energy of one of its molecules, whatever gas it is.

Calculate the root mean square speed of a nitrogen molecule of mass 4.65 × 10⁻²⁶ kg at 300 K.

⟨c²⟩ = 3kT/m = (3 × 1.38 × 10⁻²³ × 300)/(4.65 × 10⁻²⁶) = 1.242 × 10⁻²⁰/(4.65 × 10⁻²⁶) = 2.67 × 10⁵ m² s⁻², so the root mean square speed is 5.2 × 10² m s⁻¹.

By what factor must the absolute temperature of a gas be increased to double the root mean square speed of its molecules?

The temperature must be increased four times, because ⟨c²⟩ is proportional to T and the root mean square speed is proportional to √T.

CCEA A-Level Physics 1210

Physical Quantities, Scalars and Vectors

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Thermal Physics

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Thermal Physics

Average molecular kinetic energy and temperature

3 things to know
Unit A2 1

Overview

Recall questions on ½m⟨c²⟩ = (3/2)kT and what it says about the meaning of temperature, its derivation from pV = ⅓Nm⟨c²⟩ and pV = NkT, worked calculations of mean kinetic energy and root mean square speed, comparing helium with oxygen at one temperature, and why light gases escape a planet's atmosphere.

What you need to know

Answer each one before you reveal it.

Your recallNot started

Numbers

Calculate the mean kinetic energy of a gas molecule at 300 K.

Key term

How is the total kinetic energy of one mole of an ideal gas obtained from the average kinetic energy of a molecule?

Numbers

Calculate the total internal energy of 1.0 mol of an ideal gas at 300 K.

Questions from this subtopic

Each one with the model answer underneath.

1
State the equation relating the average kinetic energy of a molecule to the absolute temperature, and define every symbol.
Model answer
½m⟨c²⟩ = (3/2)kT, where m is the mass of one molecule in kg, ⟨c²⟩ the mean square speed in m² s⁻², k the Boltzmann constant of 1.38 × 10⁻²³ J K⁻¹ and T the absolute temperature in K.
2
What does the equation ½m⟨c²⟩ = (3/2)kT tell you about the meaning of temperature?
Model answer
It shows that the absolute temperature of a gas is a direct measure of the mean random kinetic energy of one of its molecules, whatever gas it is.
3
Calculate the mean kinetic energy of a gas molecule at 300 K.
Model answer
½m⟨c²⟩ = (3/2)kT = 1.5 × 1.38 × 10⁻²³ × 300 = 6.2 × 10⁻²¹ J.
4
Calculate the root mean square speed of a nitrogen molecule of mass 4.65 × 10⁻²⁶ kg at 300 K.
Model answer
⟨c²⟩ = 3kT/m = (3 × 1.38 × 10⁻²³ × 300)/(4.65 × 10⁻²⁶) = 1.242 × 10⁻²⁰/(4.65 × 10⁻²⁶) = 2.67 × 10⁵ m² s⁻², so the root mean square speed is 5.2 × 10² m s⁻¹.
5
By what factor must the absolute temperature of a gas be increased to double the root mean square speed of its molecules?
Model answer
The temperature must be increased four times, because ⟨c²⟩ is proportional to T and the root mean square speed is proportional to √T.
6
How is the total kinetic energy of one mole of an ideal gas obtained from the average kinetic energy of a molecule?
Model answer
Multiply the average molecular kinetic energy by the Avogadro constant, giving a total of (3/2)Nₐ kT, which is the same as (3/2)RT.

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