An understanding the properties of matter depends on the knowledge about its energy and its coordinates in space and time, i.e., its conformation and dynamic properties. There are several important things we need to know:

- The physical and chemical properties can be described in detail by Schrodinger's wave function (Y).
- The distribution of a particle in space is given by the square of its wave
function |Y|
^{2}. This leads an understanding of orbitals. - Energy states are
**quantized**; thus any system has certain characteristic energy values or levels.

The energy levels represent the **characteristic states **of the molecule.
The properties of the characteristic states are related to the identity of the
molecules, to the molecular structure, and to the energetics of any chemical
processes that the molecules may undergo.

For convenience, the **ground state** is defined as the state of the lowest
energy. The ground state is the state that becomes progressively more occupied
by the molecule as matter is cooled toward absolute zero. States of higher energy
are called **excited states**.

If two or more states of the molecule have the same numerical value of energy,
then they are said to be **degenerate**. This degeneracy may be removed by
the effect of some external influence, e.g. electric or magnetic fields.

For most purposes, it is convenient to treat a molecula as if it possesses several distinct reservoirs of energy. The total energy is then given by

E_{total}=E_{translation}+E_{rotation}+E_{vibration}+E_{electronic}+E_{electron spin orientation}+E_{nuclear spin orientation}

Each * E* in the equation represents the appropriate energy as indicated
by its subscript. In soultioon, a molecule can obvious translate, rotate, and
vibrate. The energies associated with each of these are quantized. Probably
more familiar is the idea that the electrons have certain allowed energies,
which are defined by orbitals.

Although every type of energy is quantized, the separation
between neighboring translational energy is so small that for pratical purpose
we can disgard the quantization of the translational energy. The separation
between energy levels associated with the orientations that a nuclear or
electron spin can take in a magnetic field are also very small (10^{-3}
J*mol^{-1}), but they are very precise because their life-times
are long. Three or four orders of magnitude greater are the separations
between the energy levels associated with the different rates of rotation
of a moloecule (10 J*mol^{-1}). The separation between electronic
energy levels is even greater (10^{5}-10^{6} kJ*mol^{-1}). |
- | |

An illustration of the relative separation of the energy
of atoms and molecules for the different types of energy. The translational
energy levels are essential continuous. |

At any finite temperature, the molecules will be distributed among the energy
levels availabe to them because of thermal agitation. The exact distribution
will depend on the temperature (thermal energy) and on the separation between
energy levels (DE) in the energy ladder. At a given
temperature, the number of molecules in an upper state (**n**_{upper})
relative to the lower state (**n**_{lower})
is given by Boltzmann distribution law

n_{upper }/n_{lower}= exp[-DE /kT]

where *k* is the** Boltzmann constant** (*k* = 1.38*10^{-23}
J*K^{-1}). When DE applies to 1 mol, the
term on the right becomes exp[-DE / *R*T], where
R is the gas constant (*R* = 8.31 J*mol^{-1}).

The dependence of the population of energy levels
on temperature. |

It is clear that when DE << *k*T, exp[-DE
/ *k*T] approaches to1. The number of molecules in the upper and lower
levels is then equal. Conversely, when DE >>
*k*T, **n**_{upper }is neglible with
respect to **n**_{lower}.

**Q:**
Calculate the ratio of molecules in the upper to those in the lower energy level
when the separation between these is as follows. Assume the temperature is 300
K. Three type of energy involved are shown.

- 11.9 J*mol
^{-1}(rotational) - 11.9 kJ*mol
^{-1}(vibrational) - 119 kJ*mol
^{-1}(electronic)

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