Kinetic Theory
Consider a cubical box, L on each side. The box contains N molecules of ideal gas, each of mass m.
All collisions are elastic. The force exerted by one molecule when it collides with a wall of the box that is perpendicular to the x-axis is, from the impulse equation:
| F |
= |
| 2mvx
|  |
| Dt
|
|
Our single molecule collides with this wall once every:
| Dt |
= |
| 2L
| |
| vx
|
|
This gives an average force of:
| F |
= |
| mvx2
| |
| L
|
|
This is the force from a single molecule. The total force on the wall is the sum over all the molecules:
| F |
= |
S |
| mvx2
| |
| L
|
|
= |
| mN
| |
| L
|
|
S |
| vx2
| |
| N
|
|
The sum represents the average value of vx2. The square root of this average is known as the root-mean-square (rms) average of vx, so:
| F |
= |
| mN
| |
| L
|
|
vx2rms |
By symmetry, all directions in the box are equivalent, so:
v2 = vx2 + vy2 + vz2 = 3vx2
This gives:
| F |
= |
| mN
| |
| 3L
|
|
vrms2 |
Dividing by the area, L2, of the wall gives the pressure:
| P |
= |
| mN
| |
| 3L3
|
|
vrms2 |
which is:
| PV |
= |
| N
| |
| 3
|
|
mvrms2 |
= |
| 2N
| |
| 3
|
|
(½mvrms2) |
The term in brackets is Kav, the average translational kinetic energy of the molecules in the box.
| Therefore PV |
= |
| 2N
| |
| 3
|
|
Kav |