The Kinetic Theory of Gases relates the temperature of a gas and the average mechanical energies of its individual molecules. This law is usually stated as:

 

average translational KE_molecule = (3/2)k_BT

½v_rms² = (3/2)k_BT

 

The law allows us to use the absolute temperature of a gas as an indicator of the random, average translational kinetic energy of its molecules (atoms).

 

A derivation of this theory is given below the following example.

 

Derivation

 

To derive this equation, we will be assuming that our container holds a statistically large sample of molecules of an ideal gas [remember N_A= 6.02 x 10²³] that are moving simultaneously in three dimensions.

 

 

We can also assume that the majority of their velocities will have non-zero x-, y-, and z-components.

 

Let's begin by examining the behavior of one particle as it stikes the right (grey) wall. Since the collision is elastic, the particle will experience the following change in the x-component of its momentum.

 

 

This change in the particle's x-momentum can also be expressed as the impulse delivered by the wall on the particle during the time of the collision. By Newton's Third Law, the impulse delivered to the wall by the particle has the same magnitude but acts in the opposite direction (+). This impulse provide us with an expression for the average force exerted on the wall during the collision of one particle.

 

 

In the diagram below, the number of particles colliding with the wall during the time required for one collision are shown in the shaded box.

 

The total force exerted on the right wall can now be expressed as 

 

 

Our job now is to defined an expression for the number of colliding particles. At any given time in the container, statistically half of the particles will have a positive x-component, traveling to the right, and half of the particles will have a negative x-component, traveling to the left.

 

 

Substituting this expressions for the number of collisions, N_collisions, we can rewrite our expression for the total force as:

 

 

Simplifying and dividing through by the area gives us a new expression for the pressure on the right wall in terms of the particles' x-velocity.

 

 

But remember, that a particle's average velocity is the vector sum of its three components and that no velocity component is statistically more likely, or more prevalent, than another. Note that the phrase "average" velocity has also been replaced with the expression "root mean square" or rms velocity, v_rms.

 

 

Substituting this value for v_x² will give us an expression for the pressure exerted by a confined gas on its container entirely based on the volume of the container, the number of gas molecules in the container, and their average translational kinetic energy.

 

 

 

One final step now remains to show that the temperature of the gas can also be expressed in terms of the average translational kinetic energy of its molecules. To do this, we will "blend" our current formula for pressure with the ideal gas law, PV = nRT,

 

 

where, once again, represents Boltzmann's constant [R/N_A = 1.38 x 10^-23 J/K].