In a gas made of tiny hard billard balls temperature is proportional to mean kinetic energy of the particles. But certainly, nobody expects all molecules to have the same speed. The image (Maxwellian distribution) shows the distribution of molecular speeds for three different temperatures.
 

Maxwellian distribution of particle speeds
The Maxwellian distribution of molecular speeds for three different temperatures. Even at low temperature, there is a small fraction of molecules having high speed. This fraction increases with temperature, while the fracition of molecules with low speed becomes smaller but does not vanish

Obviously, the distribution of molecular speeds depends on temperature. Conversely, temperature is determined by the ratio of the number of molecules at high speed to the number of molecules at low speed. A model can help us understand this relationship.

To keep things simple, I will assume a system that can have but two levels of energy, E(hi) and E(lo): the particles either have energy or they have not.

When the system is heated, energy is transferred to it; the particles must somehow accomodate this energy. In this model, this can only happen if some change from a low to a high energy state.

What is the number of particles that will be on the i-th energy level? The answer is given by the Boltzmann distribution:

N(i) = C*exp (-E(i)/kT),

where

C

a constant (at a given temperature)

N(i)

the number of particles with energy E(i)

E(i)

is the energy portion, according to our simplifying assumption it is either nought or it is something

k

the Boltzmann constant

T

absolute temperature

For only two levels of energy the ratio of the population of these levels is:

N(hi)/N(lo) = exp (-DeltaE/kT)

where

DeltaE = E(hi)-E(lo)

- DeltaE/k
T = -------------------
ln [N(hi)/N(lo)]


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