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 xaxis is, from the impulse equation:
F 
= 
2mv_{x}
 
Dt


Our single molecule collides with this wall once every:
Dt 
= 
2L
 
v_{x}


This gives an average force of:
F 
= 
mv_{x}^{2}
 
L


This is the force from a single molecule. The total force on the wall is the sum over all the molecules:
F 
= 
S 
mv_{x}^{2}
 
L


= 
mN
 
L


S 
v_{x}^{2}
 
N


The sum represents the average value of v_{x}^{2}. The square root of this average is known as the rootmeansquare (rms) average of v_{x}, so:
F 
= 
mN
 
L


v_{x}^{2}_{rms} 
By symmetry, all directions in the box are equivalent, so:
v^{2} = v_{x}^{2} + v_{y}^{2} + v_{z}^{2} = 3v_{x}^{2}
This gives:
F 
= 
mN
 
3L


v_{rms}^{2} 
Dividing by the area, L^{2}, of the wall gives the pressure:
P 
= 
mN
 
3L^{3}


v_{rms}^{2} 
which is:
PV 
= 
N
 
3


mv_{rms}^{2} 
= 
2N
 
3


(½mv_{rms}^{2}) 
The term in brackets is K_{av}, the average translational kinetic energy of the molecules in the box.
Therefore PV 
= 
2N
 
3


K_{av} 