Degree of Freedom

What is a Degree of Freedom?

The number of independent ways in which a molecule of gas can move is called the degree of freedom. 

It is an independent physical parameter in the formal description of the state of a physical system. The degrees of freedom refers to the number of ways a molecule in the gas phase may move, rotate, or vibrate in space. The number of degrees of freedom a molecule possesses plays a role in estimating the values of various thermodynamic variables using the equipartition theorem.

There are three types of degrees of freedom, such as translational, rotational, and vibrational. The number of degrees of freedom of each type possessed by a molecule depends on both the number of atoms in the molecule and the geometry of the molecule, with geometry referring to the way in which the atoms are arranged in space.

Table of Contents

• Types of degree of freedom
• Equipartition law of energy
• Degree of freedom of monoatomic gas
• Degree of freedom of diatomic molecules
• Degree of freedom of triatomic molecules
• Frequently Asked Questions

Types of degree of freedom

A gaseous molecule has a certain number of degrees of freedom, such as the ability to translate (the motion of its centre of mass through space), rotate around its centre of mass, or vibrate (as its bond lengths and angles change). Many physical and chemical properties depend on the energy associated with each of these modes of motion. If a molecule has N number of independent particles, then total degree of freedom in three dimensions of the molecule is determined by: F= 3N

(a) Translational degree of freedom

Translational degrees of freedom arise from the ability of gas molecules to move freely in space. A molecule may move in the x, y, and z directions of a Cartesian coordinate system. When the centre of mass of a particle moves from its initial position to a new position, we say that the particle is having a translational motion along the x-axis, y-axis and z-axis. So, the translational motion of the molecule of gas has three degrees of freedom associated with it. This is applicable for all gas molecules, whether they are monatomic, diatomic, or polyatomic, as any molecule may move freely in all directions in three-dimensional space.

.        

(b) Rotational degree of freedom

A molecule’s rotational degrees of freedom represent the number of unique ways the molecule may rotate in space about its center of mass with a change in the molecule’s orientation. A monatomic gaseous molecule such as a noble gas possesses no rotational degrees of freedom, as the center of mass sits directly on the atom and no rotation which creates change is possible. In the below image a diatomic molecule lying along the Y-axis can undergo rotation about the mutually perpendicular X-axis and Z-axis passing through its centre of gravity, This shows that the linear molecule has two rotational degrees of freedom. However, non-linear molecules have three rotational degrees of freedom.  

(c) Vibrational degree of freedom

The atoms of a molecule can also vibrate and these vibrations of the atoms of a molecule slightly change the internuclear distances between the atoms of the molecule. The number of vibrational degrees of freedom (or vibrational modes) of a molecule is determined by examining the number of unique ways the atoms within the molecule may move relative to one another, such as in bond stretches or bends. 

As already mentioned, atoms possess only a translational degree of freedom. A diatomic molecule has only one vibrational degree of freedom During the vibrational motion the bonds of the molecules behave like a spring and the molecule exhibits simple harmonic motion.

A polyatomic molecule containing  N atoms has 3N degrees of freedom. If we subtract the translational and rotational degree of freedom from the total degree of freedom please find the total number of vibrational degrees of freedom of linear and nonlinear molecules. 

Equipartition law of energy

For a system in equilibrium, there is an average energy of ½ kT or ½ RT per molecule associated with each degree of freedom. (where k = Boltzmann constant and T is the temperature of the system). This energy associated with each degree of freedom is in the form of kinetic energy and potential energy. 

• One translational degree of freedom = ½ kT or ½ RT
• One rotational degree of freedom= ½ kT or ½ RT
• One vibrational degree of freedom=  kT or RT

Note:  As regards the vibrational motion, two atoms oscillate against each other therefore both potential and kinetic energy the energy of vibration involve two degrees of freedom, so that vibrational motion in a molecule is associated with energy= 2 x  ½ kT =  kT

Total energy E= E_tr + E_rot + E_vib + E_elc

Points to be noted: 

• At room temperatures, the degrees of freedom need not include the vibrational modes. For molecules to vibrate in their normal modes they require much higher energies which is not possible at room temperature.

Energy contribution for linear molecules

Energy contribution for non-linear molecules

Degree of freedom of monoatomic gas

• Since a monatomic molecule consists of only a single atom of point mass it has three degrees of freedom of translatory motion along the three coordinate axes x, y and z. 
• Examples: Molecules of Inert gases like helium(He), Neon(Ne), Argon(Ar), etc.

Degree of freedom of diatomic molecule

• The diatomic molecule can rotate about any axis at right angles to its own axis. Hence it has two rotational degrees of freedom, in addition, it has three translational degrees of freedom along the three axes. A diatomic molecule shows one vibrational degree of freedom. So, a diatomic molecule has a total of six degrees of freedom at high temperatures. 
• At room temperature the total degree of freedom of a diatomic molecule is five because vibrational motion is not contributed. Examples: molecules of O₂, N₂, CO, Cl₂, etc.

Degree of freedom of triatomic molecule

• In the case of a triatomic molecule of linear type, the centre of mass lies at the central atom. 
• It, therefore, behaves like a diatomic molecule with three degrees of freedom of translation and two degrees of freedom of rotation, it has five degrees of freedom as shown at room temperature.
• At high temperatures, It shows four vibrational degrees of freedom. Hence, it shows a total of nine degrees of freedom. Examples: molecules of CO₂, CS₂, etc.
• At room temperature a triatomic nonlinear molecule possesses three degrees of freedom of rotation in addition to three degrees of freedom of translation. Hence it has six degrees of freedom. 
• At high temperatures, it shows a total of nine degrees of freedom. Examples : molecules of H₂O, SO₂, etc.