Kinetic Theory

Jim Daly
January 26, 2018

 Figure 1   A cube of side L is shown. An atom bounces off one side of the cube.
Using Newton's second law, F = ma, and the ideal gas law we can show that the average kinetic energy of a monatomic gas atom is,

 (1)
where k is Boltzmann's constant and T is the absolute temperature.

Consider the force exerted on the atom in the cube in Figure 1 when it bounces off the wall. From Newton's second law the average force exerted on the atom by the wall is,

 (2)
Only the x component of the velocity changes.
 (3)
This assumes an elastic collision where no energy is lost. The magnitude of the velocity is unchanged by the collision. The x component of the velocity reverses direction.

The time over which this force is averaged is the round trip transit time. It is the time it takes the atom to bounce off the wall then bounce around the cube and return to the wall. Only the x component of the velocity determines the time it takes the atom to return to the wall.

 (4)
Pluggiing Equations 3 and 4 into Equation 2 yields an expression for the average force exerted by the wall on the atom.
 (5)
On average the x, y, and z components of the velocity are equal. From the Pythagorean Theorem, the square of the total velocity is three times the square of the x component.
 (6)
Using Equation 6 to eliminate vx from Equation 5 results in the average force exerted by the wall on a atom.
 (7)
Since there are N atoms in the cube, the total force exerted on the gas by the wall of the cube is,
 (8)
The pressure is the Force divided by the area of the wall, L2 .
 (9)
Since L3 is the volume of the cube.
 (10)
The product of the pressure and the volume is,
 (11)
Recall the ideal gas law,
 (12)
where k is Boltzmann's constant and T is the absolute temperature.

Equating Equations 11 and 12,

 (13)
The expression for the kinetic energy of an individual atom has been used.
 (14)
From Equation 13 it follows that
 (1)
The average kinetic energy of a monotonic atom is proportional to the absolute temperature. Since the atom is monatomic it will not have any rotational kinetic energy.