Kinetic theory of gases

Assumptions of the model

• A gas has a very large number of molecules moving randomly.
• The volume of the molecules is negligible compared with the volume of the gas.
• Collisions (with each other and the walls) are perfectly elastic and take negligible time.
• There are no forces between molecules except during collisions.

Pressure from molecules

A molecule of mass m hitting a wall at right angles with speed c changes momentum by 2mc. Adding up the effect of all N molecules moving in three dimensions gives

pV = (1/3)Nm<c²>

where <c²> is the mean square speed. The r.m.s. speed is sqrt(<c²>). You do not need to recall this equation, but it is the step used to derive the result below, which you must be able to derive and use.

Temperature and molecular kinetic energy

Comparing pV = (1/3)Nm<c²> with pV = NkT gives (1/3)m<c²> = kT, so

(1/2)m<c²> = (3/2)kT

The mean kinetic energy of a molecule depends only on T, not on the type of gas. At the same temperature, lighter molecules move faster.

Worked example

Find the mean kinetic energy and the r.m.s. speed of a helium atom (m = 6.64 x 10^-27 kg) at 300 K.

1. Mean E_k = (3/2)kT = 1.5 x 1.38 x 10^-23 x 300 = 6.21 x 10^-21 J
2. <c²> = 2E_k / m = 2 x 6.21 x 10^-21 / 6.64 x 10^-27 = 1.87 x 10⁶ m²/s²
3. c_rms = sqrt(1.87 x 10⁶)

Answer: 6.2 x 10^-21 J; c_rms = 1.37 x 10³ m/s

Check yourself

1. Find the mean kinetic energy of a gas molecule at 300 K, in units of 10^-21 J, to 3 s.f.
   Show answer

   6.21 x 10^-21 J. E_k = (3/2)kT = 1.5 x 1.38 x 10^-23 x 300 = 6.21 x 10^-21 J.

2. A nitrogen molecule has mass 4.65 x 10^-26 kg. Find its r.m.s. speed at 300 K, in m/s, to 3 s.f.
   Show answer

   517 m/s. <c²> = 3kT / m = 3 x 1.38 x 10^-23 x 300 / 4.65 x 10^-26 = 2.67 x 10⁵ m²/s², so c_rms = 517 m/s.

3. At what temperature is the mean kinetic energy of a gas molecule 1.0 x 10^-20 J? Answer in K, to 3 s.f.
   Show answer

   483 K. T = E_k / (1.5k) = 1.0 x 10^-20 / 2.07 x 10^-23 = 483 K.

4. Four molecules have speeds of 300, 400, 500 and 600 m/s. Find their r.m.s. speed in m/s, to 3 s.f.
   Show answer

   464 m/s. Mean of squares = (90000 + 160000 + 250000 + 360000) / 4 = 215000; r.m.s. = sqrt(215000) = 464 m/s.

5. A gas is heated from 300 K to 1200 K. By what factor does the r.m.s. speed of its molecules increase?
   Show answer

   2. (1/2)m<c²> = (3/2)kT, so <c²> is proportional to T and c_rms to sqrt(T): sqrt(1200 / 300) = sqrt(4) = 2.

6. Which is an assumption of the kinetic theory of an ideal gas?
   • Molecules attract each other strongly
   • Molecules all move at the same speed
   • Collisions between molecules are perfectly elastic
   • The molecules take up most of the volume
   Show answer

   Collisions between molecules are perfectly elastic. Elastic collisions mean no kinetic energy is lost.

7. The absolute temperature of a gas is doubled. What happens to the r.m.s. speed of its molecules?
   • It increases by a factor of sqrt(2)
   • It doubles
   • It increases by a factor of 4
   • It stays the same
   Show answer

   It increases by a factor of sqrt(2). <c²> is proportional to T, so c_rms is proportional to sqrt(T).

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Sources

• Pearson Edexcel International Advanced Subsidiary/Advanced Level in Physics specification (XPH11/YPH11), Issue 3
• Pearson qualifications: International Advanced Levels Physics (2018)
• Pearson Edexcel IAL Physics Unit 5 (WPH15) question paper, June 2023: list of data, formulae and relationships

Last updated 7 November 2026. Spotted a mistake? Tell us.