Peoples Physics Book Version-3

19.2. Molecular Kinetic Theory of a Monatomic Ideal Gas http://www.ck12.org

By the Pythagorean theorem, any three-dimensional velocity vector has the following property:

v^2 =vx^2 +vy^2 +vz^2

Averaging this for thenparticles in the box, we get

(v^2 )avg= (vx^2 )avg+(vy^2 )avg+(vz^2 )avg

Since the motions of the particles are completely random (as stated in our assumptions), it follows that the averages
of the squares of the velocity components should be equal: there is no reason the gas particles would prefer to travel
in thexdirection over any other. In other words,

(vx^2 )avg= (vy^2 )avg= (vz^2 )avg

Plugging this into the average equation above, we find:

(v^2 )avg= 3 ×(vx^2 )avg= 3 ×(vy^2 )avg= 3 ×(vz^2 )avg

and

(v^2 )avg/ 3 = (vx^2 )avg

Plugging this into equation [1], we get:

PnetV=

nm(v^2 )avg

3

[2]

The left side of the equation should look familiar; this quantity is proportional to the averagekinetic energyof the
molecules in the gas, since

KEavg=

1

2

m(v^2 )avg

Therefore, we have:

PnetV=

2

3

n(KE)avg [3]

This is a very important result in kinetic theory, since it expresses the product of twostate variables, or system
parameters, pressure and volume, in terms of an average over the microscopic constituents of the system. Recall the
empirical ideal gas law from last chapter:

PV=nkT