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The number of independent variables or coordinates needed to describe the motion of a particle is called the degree of freedom.

- Total number of DOF = 3N (where N = atomicity)
- No. of translational DOF (f
_{t}) = 3 (for all gases) - No. of rotational DOF (f
_{r}) = 0 (for**monoatomic gas**)

No. of rotational DOF (f_{r}) = 2 (for**diatomic gas**)

No. of rotational DOF (f_{r}) = 2 (for**polyatomic linear molecules**)

No. of rotational DOF (f_{r}) = 3 (for**polyatomic nonlinear molecule**) - No. of vibrational dof (f
_{v}) = 3N - f_{t }- f_{r}

In a monoatomic species, rotational and vibrational modes of motion are absent. Hence three degrees of freedom corresponds to three translational motion along three different axes.

For a diatomic molecule, total degree of freedom = 3 × 2 = 6

__Break up of the Degree of Freedom: __**(i) **3 translation degrees of freedom representing translation motion of center of mass in three independent directions.**(ii)** Two possible axes of rotation, hence two rotational degrees of freedom.**(iii)** One vibration degree of freedom.

Vibrational DOF is active only at high temperatures.

**Q. Find the total degree of freedom and break up as translational, rotational, or vibrational DOFs in following cases.****(i)** O = C = O**(ii)****(iii)** He**(iv)** NH_{3}**Solution.****(i)** CO_{2}: Total dof = 3 x 3 = 9

Translational = 3

Rotational = 2 (∵ linear molecule)

Vibrational = 4

**(ii)** SO_{2}: Total dof = 3 x 3 = 9

Translational = 3

Rotational = 3 (∵ Bent molecule)

Vibrational = 3

**(iii) **He: Total dof = 3 (∵ Monoatomic molecule)

Translational = 3

**(iv) **NH_{3}: Total dof = 3 x 4 = 12

Translational = 3

Rotational = 3 (∵ non linear molecule)

Vibrational = 6

Energy equal to ½ kT is associated with each translational and rotational degree of freedom per ideal gas molecule.

Energy equal to kT is associated with each vibrational degree of freedom per ideal gas molecule.

Where k = R/N_{A} Boltzmann constant

∴ For n moles,

On ignoring vibrational degree of freedom

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