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 xaxis is:
F 
= 
2mv_{x}
 
Δt


The time between collisions with the right wall
is:
Δt  = 
2L
 
v_{x}


This gives the average force due to 1 molecule:
F 
= 
mv_{x}^{2}
 
L


The total force on the wall is: F 
= 
Σ 
mv_{x}^{2}
 
L


= 
mN
 
L


Σ 
v_{x}^{2}
 
N


≡ 
mN
 
L 

⟨v_{x}^{2}⟩ 
Since all directions are equivalent:
⟨v^{2}⟩=⟨v_{x}^{2} +
v_{y}^{2} + v_{z}^{2}⟩=
3⟨v_{x}^{2}⟩
we obtain: F 
= 
mN
 
3L


⟨v^{2}⟩ 
Dividing by the wall area, L^{2},
gives the pressure: P 
= 
mN
 
3L^{3}


⟨v^{2}⟩ 
Equivalently: PV 
= 
N
 
3

 m⟨v^{2}⟩ 
= 
2N
 
3


(½m⟨v^{2}⟩) 
= 
2N
 
3


ε 
&epsilon = energy per molecule