Kinetic Theory

The assumptions for an ideal gas for kinetic theory are:

• the gas contains a large number of particles
• the particles move rapidly and randomly
• the motion of the particles follows Newton’s laws
• collisions between particles themselves or at the walls of a container are perfectly elastic
• there are no attractive forces between particles
• any forces that act during collisions are instantaneous
• particles have a negligible volume compared with the volume of the container

A particle moves with speed, c, in a square box of length L

The components of c in three dimensions are u, v and w.

You can show that c² = u² + v² + w² - An extension of pythagoras in 3 dimensions.

Particle of mass, m, and velocity, u₁.

Momentum = m u₁

Change in momentum = -2m u₁

F = change in momentum / time = -2m u₁ / t

u₁ = 2L / t

t = 2L / u₁

F = -m u₁² / L

P = F / A = (m u₁² / L) / A = m u₁² / V

Total Pressure = (m / V) (u₁² + …...+ u_N²) = m N u² / V

u² is only in one direction so c is in 3 direction which gives u² + v² + w²

c_rms² = u_rms^{2 +} v_rms^{2 +} w_rms²

u_rms² = ⅓ c_rms²

P = ⅓m N c­_rms² / V

P V = ⅓m N c­_rms²

©2011 Grant Dwyer