Ideal gases: (c) pressure in terms of this model

The idea

Pressure in the kinetic theory model is the force per unit area exerted by molecular collisions with the container walls. Each collision changes a molecule's momentum; the rate of change of momentum gives the force. Summing over many molecules and dividing by the wall area gives pressure. For a gas in a box, the pressure depends on the number of molecules, their mass, the mean square speed and the volume: p = (1/3)ρ⟨c²⟩, where ρ is density and ⟨c²⟩ is mean square speed. Since ⟨c²⟩ is proportional to absolute temperature, this links microscopic motion to measurable pressure.

What earns the marks

• Pressure is force per unit area exerted by molecules colliding with the walls.
• Each collision involves a change in momentum; the rate of change of momentum gives the force on the wall.
• The total force is the sum of forces from many collisions per unit time.
• Pressure depends on molecular mass, number density and mean square speed.
• The equation p = (1/3)ρ⟨c²⟩ relates pressure to density and mean square speed.

Common mistakes and how to put them right

• Thinking pressure is due to static molecules pushing.

  Put it right: It arises from dynamic collisions.

• Forgetting that momentum change is 2mv for a head-on elastic collision with a wall.

• Using average speed instead of mean square speed in the pressure equation.

Exam tips

• Derive pressure from momentum change per collision and collision rate.
• Use the correct expression p = (1/3)ρ⟨c²⟩ and define each symbol.
• Relate mean square speed to temperature using kinetic energy.