Kinetic Theory: Connection between Energy and Pressure
Goal: Relate energy and pressure in an
ideal gas, and then derive E=3NkT/2.
Consider a cube of linear size L with N ideal gas molecules, each of mass
m.
The force exerted by one molecule when it collides with a wall of the box
that is perpendicular to the x-axis is:
| F |
= |
| 2mvx
|  |
| Δt
|
|
| The time between collisions with the right wall
is:
Δt | = |
| 2L
| |
| vx
|
|
| This gives the average force due to 1 molecule:
F |
= |
| mvx2
| |
| L
|
|
| The total force on the wall is: F |
= |
Σ |
| mvx2
| |
| L
|
|
= |
| mN
| |
| L
|
|
Σ |
| vx2
| |
| N
|
|
≡ |
| mN
|  |
| L |
|
⟨vx2⟩ |
Since all directions are equivalent:
⟨v2⟩=⟨vx2 +
vy2 + vz2⟩=
3⟨vx2⟩
| we obtain: F |
= |
| mN
| |
| 3L
|
|
⟨v2⟩ |
| Dividing by the wall area, L2,
gives the pressure: P |
= |
| mN
| |
| 3L3
|
|
⟨v2⟩ |
| Equivalently: PV |
= |
| N
| |
| 3
|
| m⟨v2⟩ |
= |
| 2N
| |
| 3
|
|
(½m⟨v2⟩) |
= |
| 2N
| |
| 3
|
|
ε |
&epsilon = energy per molecule