# Temperature and Average Kinetic Energy

We will now study what the velocity of the gas molecules is as a function of the temperature. First we introduce the average squared velocity <v2>. This is obtained by taking the square of the velocities of each molecule in the gas and averaging (indicated by the angular brackets). One can show that in this case <v2> = <v>2.

The relationship between pressure, volume, and average velocity square is:

pV = 1/3 $\cdot$ N m <v>2

Here N is the number of molecules in the volume V, and m is the mass of one of them. Combining this result with the ideal gas law gives:

(3/2) kT = (1/2) m <v2>

Thus we find that temperature is a measure of the average kinetic energy of the molecules of the gas (or wall). Boltzmann's constant is just a unit conversion from K to Joules. If you mix a gas, eventually all the molecules have the same kinetic energy on the average, independent of their mass or the temperature at which they were introduced. This is called equilibrium. This means that a light molecule like hydrogen must move much faster than a heavy one like oxygen.

In the simulation, a hot gas (represented by red balls) is mixed with a cold gas (represented by the blue balls). As time passes, the gases interact with each other and reach equilibrium.