Kinetic Theory — Quick Summary

Mr Toogood's Physics · Thermal physics

AQA 3.6.2.2
pV = ⅓Nmc̄²
Kinetic theory eqn.
p = ⅓ρc̄²
Using density
c_rms = √c̄²
RMS speed
E = 3/2 kT
Mean KE / particle

From one particle to a gas

A single gas particle bouncing between the walls of a box

One particle bouncing in a box — the derivation's starting point.

Time between collisions with a wall: t = 2l/v. Newton's 2nd law and the momentum change (−2mv, elastic collision) combine to give the force on the wall:
F = mv²/l
Dividing by area (l²) gives pressure from one particle:
p = mv²/V

Summing over N particles and using the mean square speed, c̄²:

p = Nmc̄²/V
Many gas particles moving randomly in three dimensions

Generalising to random 3D motion.

Real particles move randomly in 3D, so c̄² = 3v̄ₓ². Substituting the factor of 3 gives the final kinetic theory equation:

pV = ⅓Nmc̄²  (or p = ⅓ρc̄²)

Root mean square speed

The rms speed is the square root of the mean of the squared speeds — not the mean speed itself:

c_rms = √[(v₁²+v₂²+⋯+vₙ²)/n]
Speed vs. velocity: the mean velocity of gas particles is zero (random directions cancel as vectors), but the mean speed is always positive.

The Boltzmann equation

Boltzmann speed distribution graph showing spread of particle speeds at different temperatures

Particles spread over a range of speeds; higher T widens and shifts the distribution.

Combining pV = nRT with the kinetic theory equation, and using Ē = ½mc̄² for mean kinetic energy per particle, gives:

E = 3RT/(2N_A)

Since the Boltzmann constant k = R/N_A, this simplifies to the key result:

E = 3/2 kT

Exam essentials

Key equations

  • pV=⅓Nmc̄²   p=⅓ρc̄²
  • E=½mc̄²=3/2 kT

Key definitions

  • N = number of particles; c̄² = mean square speed.
  • c_rms = √c̄² (rms speed).
  • k = Boltzmann constant = R/N_A.

Common slips

  • Mean square speed (c̄²) ≠ mean speed squared ((c̄)²) — don't average then square.
  • The factor of ⅓ comes directly from 3D motion (x, y, z equally likely) — don't drop it.
  • E=3/2 kT is per particle, not per mole — multiply by N for total KE.