Energy is a very important concept in science as the behaviour of atoms and molecules are very much dependent on the amount of energy they possess. We have seen how the energy determines the physical state of the molecules.

Although the energy possessed by the compound (or element) is dependent on the temperature of their surrounding, not all molecules in the sample (or system) will have the same amount of energy. Have you ever wonder how a beaker of water left in the room will eventual evaporate completely? This should not happen as water should only vapourise at 100�C at atmospheric pressure. Well actually a very small fraction (in the region of less than 1 percent) of water molecules in the beaker has extremely high energy.

There is a distribution of molecules with different amount of energies in the sample, even though the total amount of energy will remain constant. So there is a dynamic equilibrium between the molecules to ensure that the sum total energy for the sample is maintained.

L. Boltzmann

The fraction of molecules with energy state ε(i), p_i,, is given by the Boltzmann's Distribution function (after Ludwig Boltzmann).

p_i   =   e^-ε(i)/kT / q

where q is the total amount of energy for the sample given by

q   =   ∑ n_ie^-ε(i)/kTq

The summation is from zero to ∞. k is the Boltzmann's constant = 1.38 x 10‾�� JK‾�, and T is the temperature in Kelvin.

So molecules, in a liquid state, with sufficient energy can escape from the attraction of other neighbours and leave to be in a gaseous state. The ratio of those left will readjust itself to the new reality. The process continues until all the water is vapourised.

Tutorial 1

• If there are 11 energy states in a sample - 0, 1kT, 2kT, 3kT .... and 10kT - plot the Boltzmann's distribution for the various energy states (p_i vs ε(i)).
• Plot the distribution function for molecules with the following speed;
  s�/(2kT/m) = 0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 2.0, 3.0, 4.0, 5.0, 6.0, and 7.0.
• If the mean speed of the molecule, ĉ, is given by ĉ=∫ sf(s)ds   from 0 to ∞, show that ĉ = (8RT / πM)^�.      Answer

Please do not focus too much on the mathematics. The important lesson here is that the energies for the molecules in a sample is not uniform. It is a dynamic process given a constant sum total energy value for the particular situation. Once we come to systems where we need to apply the distribution functions the functions will be better understood.

Some features of Boltzmann's Distribution function worth knowing.

• At T=0 (absolute zero temperature), p_i = 0.
  That is, there is only one energy state = zero energy.

• At T = ∞ ;   p_i = 1. That is all energy states are equally populated.

• The energy states considered in the function is the total energy of the sample - the vibrational, rotational, electronic, translational, etc.