# Vibrational Degrees of Freedom Formula & Definition in Chemistry

in order to consider the distribution of energies for different modes of motion, one should have an idea of the vibrational degrees of freedom.

degree of freedom is the number of independent co-ordinates required to locate all the atoms in a molecule.

the location of an atom can be specified by three coordinates.

the total degrees of freedom of a molecule = 3N

where N is the number of atoms in that molecule. translational degrees of freedom are always three. rotational degrees of freedom depends upon the fact that whether the molecule is linear or non-linear.

for a linear molecule, rotational degrees of freedom =2

for a non-linear molecule, rotational degrees of freedom = 3

after the completion of rotational and translational degrees of freedom, we should calculate the vibrational degrees of freedom.

## Degrees of freedom of a Monoatomic Molecule:

total degrees of freedom = 3×1 = 3

Translational degrees of freedom = 3

Rotational degrees of freedom = 0

Vibrational degrees of freedom = 0

so, the atoms of He, Ne, Ar, Kr, Hg, and Na vapors spend their energies only on translational motions along the x, y, and z-axis, but not on rotational and vibrational motions.

### Degrees of freedom of a diatomic molecule:

total degrees of freedom = 3×2= 6

Translational degrees of freedom = 3

rotational degrees of freedom= 3 ( diatomic molecules are always linear)

vibrational degrees of freedom= 6-3-32= 1

Energy expended for one translational degree of freedom = 1/2RT

Energy expended for one rotational degree of freedom = 1/2RT

Energy expended for one vibrational degree of freedom = 1/2RT + 1/2RT

(1/2RT is kinetic energy and 1/2RT is potential energy)

now, let us do the calculations for the total energy being expended by the diatomic linear molecule.

it means that a diatomic molecule like H_{2}, F_{2}, Cl_{2}, O_{2}, N_{2}, HCl, etc. needs the energy of 3.5 RT to maintain all their motions for six degrees of freedom. this energy is more than double those of monoatomic molecules.

#### Degrees of freedom of triatomic molecules:

total degrees of freedom = 33= 9

translational degrees of freedom = 3

rotational degrees of freedom(linear) = 2

vibrational degrees of freedom ( for linear) = 9-3-2= 4

A triatomic linear molecule like Co_{2}, CS_{2}, COS, etc. has four vibrational degrees of freedom. the total energy can be calculated as follows.

#### Tri-atomic Non-linear Molecules:

for triatomic non-linear molecules like H_{2}O, H_{2}S, H_{2}Se, SO_{2, }etc. then translational degrees of freedom are three, rotational are three, and vibrational are also three. so the total energy of such a molecule is,

it means that the total energy being expended by water is a little bit less than that ofCO2.

#### Degrees of freedom of a Tetra-atomic Molecule:

Total degree of freedom = 3 x 4 = 12

Translational degrees of freedom = 2

Rotational degrees of freedom = 3

Vibrational degrees of freedom = 12-3-3= 6

it means that energy possession goes increases for molecules having higher atomicities. the whole process is called the derivation of the law of equipartition of energies theorem.