25

Physics Factsheet January 2002

Number 25

Molecular Kinetic Theory This Factsheet will explain: • the theory behind the kinetic model for gases; • how the theory agrees with experimentally determined results; • how to use the theory to do simple calculations.

To simplify the ideas consider a molecule of mass m travelling parallel to the x axis with speed vx in a cubic box of side l. • The molecule’s momentum is mvx in the positive x direction. • After an elastic collision with the wall, its momentum will be mvx in the negative x direction, so the change of momentum is 2mvx. • It will travel across the box and back, a distance of 2l, at speed vx , so it will take time 2l seconds.

Before studying this Factsheet, you should be familiar with the simple qualitative ideas of how particles are arranged in solids, liquids, and gases.

vx

•

You should know that: • • •

Thus the molecule will hit the face

vx 2l

times per second. vx 2l

Change of momentum per s = 2mvx ×

pressure = force area momentum = mass × velocity kinetic energy = ½ mv2 .

= mvx

2

l

To generalise from this one molecule to average values for all the molecules, consider a molecule with speed v in a random direction.

The theory makes several assumptions to simplify the model: • There are many molecules in the gas, so that average figures are representative of the whole. • The molecules are moving rapidly and randomly. • The behaviour of the molecules can be described by Newtonian mechanics. • The molecules are far apart(compared until their diameters)and have no pull on each other. • Any forces acting, do so only during collisions and then only for a very short time compared with the time between collisions. • The collisions are elastic (kinetic energy is conserved). • The total volume of the molecules is small compared to the volume of the gas as a whole.

y v x z It will have components vx, vy, vz. By Pythagoras: v2 = vx2 + vy2 + vz2 Since the directions are random, there is no preference for any particular direction, so we can assume that on average for all the molecules: = = so = 13

Exam Hint: You will need to know these assumptions of the model.

To understand the basis of the theory Consider a rectangular box of gas:

So the average change in momentum per s for n molecules on this one face, 2 becomes: 13 nm l

pressure = force area , so the pressure due to all the molecules 2 > P = 13 nm