Molecular Kinetic Theory Model

The kinetic theory of gases is one of the most elegant parts of A-Level Physics. It derives the macroscopic gas laws from first principles by considering the microscopic behaviour of individual molecules. The central result is the equation pV = ⅓Nmc̄², which connects pressure and volume to the motion of molecules. AQA requires you to understand and be able to reproduce the key steps of this derivation.

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The Model and Its Assumptions

The kinetic theory considers a large number of identical gas molecules moving randomly inside a container. The assumptions (identical to those for an ideal gas) are:

1. The gas contains a very large number N of identical molecules, each of mass m.
2. The molecules are in constant random motion — there is no preferred direction.
3. The volume of the molecules is negligible compared to the volume of the container.
4. There are no intermolecular forces between the molecules (except during collisions).
5. All collisions (between molecules and with the container walls) are perfectly elastic.
6. The duration of a collision is negligible compared to the time between collisions.
7. The molecules obey Newton's laws of motion.

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The Derivation of pV = ⅓Nmc̄²

This derivation is one of the most important in A-Level Physics. It proceeds in several clear steps.

Step 1: Consider a Single Molecule in a Cubic Box

Consider a cubic container of side length L. A single molecule of mass m moves with velocity c. The molecule has velocity components cₓ, c_y, and c_z in the three perpendicular directions.

The speed is related to the components by:

c² = cₓ² + c_y² + c_z²

Step 2: Calculate the Momentum Change at One Wall

Consider the molecule hitting the wall perpendicular to the x-direction. The molecule approaches with x-component of velocity +cₓ and bounces back with −cₓ (elastic collision, wall is much more massive).

Change in momentum per collision:

Δp = mcₓ − (−mcₓ) = 2mcₓ

Step 3: Calculate the Time Between Successive Collisions with the Same Wall

After bouncing off the wall, the molecule must travel across the box and back — a total distance of 2L in the x-direction.

Time between collisions with the same wall:

Δt = 2L/cₓ

Step 4: Calculate the Force from One Molecule on One Wall

The rate of change of momentum gives the force (Newton's second law):

F = Δp/Δt = 2mcₓ / (2L/cₓ) = mcₓ²/L

Step 5: Sum Over All N Molecules

The total force on the wall from all N molecules is:

F_total = (m/L)(c₁ₓ² + c₂ₓ² + c₃ₓ² + ... + cₙₓ²)

F_total = (m/L) × N⟨cₓ²⟩

where ⟨cₓ²⟩ is the mean of the squared x-components of velocity for all molecules.

Step 6: Use the Isotropy of Random Motion

Because the motion is random, there is no preferred direction. Therefore the average behaviour is the same in all three directions:

⟨cₓ²⟩ = ⟨c_y²⟩ = ⟨c_z²⟩

Since c² = cₓ² + c_y² + c_z², taking the mean:

⟨c²⟩ = ⟨cₓ²⟩ + ⟨c_y²⟩ + ⟨c_z²⟩ = 3⟨cₓ²⟩

Therefore:

⟨cₓ²⟩ = ⟨c²⟩/3

Step 7: Calculate the Pressure

The force on one wall is:

F = Nm⟨cₓ²⟩/L = Nm⟨c²⟩/(3L)

Pressure = Force / Area. The area of one wall of the cube is L².

p = F/L² = Nm⟨c²⟩/(3L³)

Since L³ = V (the volume of the cube):

pV = ⅓Nm⟨c²⟩

This is the fundamental result of kinetic theory.

Important Notes on the Derivation

• The result applies to any shape of container, not just a cube. The cube is used for simplicity.
• ⟨c²⟩ is the mean square speed — the average of the squared speeds of all the molecules. It is NOT the square of the mean speed.
• The notation c̄² is sometimes used interchangeably with ⟨c²⟩.