Chemistry

Law of Equipartition of Energy and Degrees of Freedom : Definition and Theorem

Elaborate the Law of Equipartition of Energies

in physical chemistry, for a monoatomic gas, the translational energy associated with one mole of the molecule at any temperature is 3/2RT.

According to the law of equipartition of energy, 1/2RT is expended along the x-axis, 1/2RT along the y-axis and 1/2 RT along the z-axis.

law of equipartition energy equation

we have already considered that if the component velocities are u, v and w, then the kinetic energies along x, y and z-axis are,

kinetic energies along x,y and z-axis

U , Vand  Ware the average velocities along x, y, and z-axis,

if we want to apply this law on the equipartition of energy for a single molecule, then equation (1) and (2) can be written as (3) and (4) respectively.

equipartition of energy for a single molecule

where ‘K’ is Boltzmann constant.

or,       modified form of energy of equipartition of a single molecule

The degree of Freedom and Atomicity of a gas:

in order to consider the distribution of energies for different modes of motion, one should have the idea of the degree 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 for translational motions along x, y, and z-axis, but not for rotational and vibrational motions.

equipartition of energy for a single molecule

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 molecule 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 total energy being expended by the diatomic linear molecule.

total energy being expended by the diatomic linear molecule

it means that a diatomic molecule like H2, F2, Cl2, O2, N2, HCl etc. need the energy of 3.5 RT to maintain all their motions for six degrees of freedom. this energy is more than double than 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 Co2, CS2, COS, etc. has four vibrational degrees of freedom. the total energy can be calculated as follows.

total energy of a triatomic linear molcule

Tri-atomic Non-linear Molecules:

for triatomic non-linear molecules like H2O, H2S, H2Se, SOetc. then translational degrees of freedom are three, rotational are three and vibrational are also three.  so total energy of such a molecule is,

total energy of a triatomic non linear molcule

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

total energy for a tetra-atomic molecule

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

Read Also: Heat capacities of Gases : ( at constant Volume and Temperature)

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