Degree of Freedom – Complete Explanation for JEE, NEET & CBSE

Degree of Freedom – Complete Explanation

Degree of Freedom (DoF) is one of the most important concepts in Thermodynamics and Kinetic Theory of Gases. It plays a crucial role in understanding internal energy, heat capacity, and gas behavior in JEE, NEET, and CBSE exams.

Definition:

Degree of freedom is the number of independent ways in which a molecule
can store energy.

1. Degrees of Freedom in Space (Translational)

In three-dimensional space, any molecule can move independently along x, y, and z directions.

Motion along x-axis → 1 degree
Motion along y-axis → 1 degree
Motion along z-axis → 1 degree

Total translational degrees of freedom = 3 (for all gases)

2. Rotational Degrees of Freedom

Type of Molecule	Rotational Degrees of Freedom
Monoatomic (He, Ne)	0
Diatomic (H₂, N₂)	2
Linear polyatomic (CO₂)	2
Non-linear polyatomic (H₂O, NH₃)	3

3. Vibrational Degrees of Freedom

Each vibrational mode contributes two degrees of freedom: one kinetic and one potential.

Molecule Type	Vibrational Degrees of Freedom
Monoatomic	0
Diatomic	2
Linear polyatomic	3N − 5
Non-linear polyatomic	3N − 6

N = number of atoms in the molecule

4. Total Degrees of Freedom

f = + Rotational + Vibrational

Examples

Monoatomic gas (He):

Translational = 3
Rotational = 0
Vibrational = 0

f = 3

Diatomic gas (N₂) at room temperature:

Translational = 3
Rotational = 2
Vibrational ≈ 0

f = 5

5. Equipartition of Energy

According to the equipartition theorem, each degree of freedom contributes energy equal to:

Energy per molecule = ½ kT

Energy per mole = ½ RT

6. Relation with Heat Capacity

C_V = (f / 2) R

C_P = C_V + R = (f + 2)/2 · R

γ = C_P / C_V = (f + 2) / f

Gas	Degrees of Freedom (f)	γ
Monoatomic	3	1.67
Diatomic	5	1.40
Polyatomic (non-linear)	6	1.33

7. Temperature Dependence

Temperature	Active Degrees of Freedom
Low temperature	Only translational
Room temperature	Translational + rotational
High temperature	Translational + rotational + vibrational

Exam Tip:

Vibrational degrees of freedom are usually inactive at room temperature
and become active only at high temperatures.

8. One-Line Exam Summary

Degree of freedom is the number of independent ways in which a molecule can store energy, determining its internal energy and heat capacity.

Degree of Freedom Quiz

Test Your Knowledge: Degrees of Freedom