Model gas molecules and explain pressure
State the kinetic-theory assumptions and connect molecular momentum changes at container walls to gas pressure.
Section 15 · Ideal gases
A Level; Cambridge International Physics 9702, 2025–2027 syllabus, Topic 15. These original explanations and calculations are study aids; full timed assessment and human subject-teacher review remain outstanding.
Before you begin
- Use pressure, volume, temperature in kelvins, energy and scientific notation.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What physical process produces gas pressure on a container wall?
What you will learn
- State basic assumptions of the kinetic theory of gases
- Explain gas pressure using molecular collisions with the walls
- Derive and use pV = ⅓Nm⟨c²⟩
A simple kinetic model idealises molecular motion
Model the gas as a very large number of identical molecules moving randomly. Their own volumes are negligible compared with the container; intermolecular forces are negligible except during collisions; collisions between molecules and with the walls are perfectly elastic.
The molecules obey Newton's laws between collisions. The random directions mean that, on average, motion is equally distributed among the three perpendicular axes. These assumptions form a model for an ideal gas, not a claim that real molecules have no size or interaction.
Wall collisions transfer momentum and create pressure
A molecule that rebounds from a wall changes momentum. Many collisions per second transfer momentum to the wall, creating a force; pressure is force per unit area. Faster molecular motion or more molecules in the same volume produces greater pressure in this model.
For motion normal to one wall, the collision model gives pV = Nm⟨cₓ²⟩. Random isotropic motion gives ⟨cₓ²⟩ = ⟨c²⟩/3, so pV = ⅓Nm⟨c²⟩. Here m is the mass of one molecule and ⟨c²⟩ is the mean-square speed.
Pause and explain
For isotropic random motion, how is ⟨cₓ²⟩ related to ⟨c²⟩?
Worked example
A gas has N = 3.0 × 10²³ molecules, molecular mass m = 4.65 × 10⁻²⁶ kg and mean-square speed ⟨c²⟩ = 2.5 × 10⁵ m² s⁻². Find pV.
Show the worked solution
- Use pV = ⅓Nm⟨c²⟩.
- Substitute the three quantities.
- Check that the result is in joules, equivalent to Pa m³.
Answer pV = ⅓ × 3.0 × 10²³ × 4.65 × 10⁻²⁶ × 2.5 × 10⁵ ≈ 1.16 × 10³ J.
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Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Explain why collisions with the wall exert a force on it, using momentum.
Give me a hint
Use the momentum change of the molecule and Newton's third law.
Compare my reasoning
- A molecule's velocity component normal to the wall reverses in an elastic collision.
- Its momentum therefore changes; the wall receives an equal and opposite momentum change.
- Repeated momentum transfer per unit time is a force on the wall.
Wall collisions transfer momentum; the rate of transfer is a force, and force per area is pressure.
Look for these in your work
- Velocity reversal identified
- Momentum change related to wall
- Force and pressure connected
A gas has N = 2.0 × 10²³ molecules, m = 4.65 × 10⁻²⁶ kg and ⟨c²⟩ = 2.5 × 10⁵ m² s⁻². Calculate pV.
Give me a hint
Use one third of N times molecular mass times mean-square speed.
Compare my reasoning
- pV = ⅓Nm⟨c²⟩.
- Nm = 2.0 × 10²³ × 4.65 × 10⁻²⁶ = 9.30 × 10⁻³ kg.
- pV = ⅓ × 9.30 × 10⁻³ × 2.5 × 10⁵.
pV ≈ 775 J.
Look for these in your work
- Correct kinetic relation used
- N m evaluated
- Energy-equivalent unit stated
At fixed volume and particle count, the mean-square molecular speed doubles. Predict the pressure change using the kinetic-theory equation.
Give me a hint
In pV = ⅓Nm⟨c²⟩, hold V, N and m fixed.
Compare my reasoning
- p is directly proportional to ⟨c²⟩ for fixed V, N and m.
- The mean-square speed doubles.
- The pressure therefore doubles.
Pressure doubles when mean-square speed doubles under these fixed conditions.
Look for these in your work
- Fixed conditions named
- Proportionality used
- Pressure factor stated
Common mistakes
- Using speed instead of mean-square speed
- Forgetting the factor of one third for three-dimensional random motion
- Saying pressure is caused by molecules pushing continuously without collisions
What can you explain now?
What physical process produces gas pressure on a container wall?
Compare with the explanation
Molecules transfer momentum in collisions with the wall
Their momentum changes create a force on the wall.
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Make it stick
How does a molecular momentum change at a wall become macroscopic gas pressure?
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