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# Degrees of Freedom - Definition, Types of Degree of Freedom, Practice Problems & FAQs

Aeroplanes are truly engineering marvels. They overcome highly turbulent and unpredictable currents in the air and complete their flights by undertaking many complex manoeuvres. But have you ever thought of its motion and the directions for a stable flight?

Let us understand the typical movement of an aeroplane.

Consider an aeroplane, if we assume that the centre of mass of this setup is passing through the origin of this cartesian coordinate system. We realise that this plane can move in all three directions of this system or X, Y & Z axes in a straight line. How?

It can move forward and backwards on the X-axis, left and right as well on the Y-axis and up and down on the Z-axis. Do you know what kind of motion is this? It is translational motion.

So, basically, we can say that any particle or object which is undergoing translational motion, it can do so in 3 possible directions of our standard cartesian coordinate system.

But is it the only possible way for the aeroplane to undergo motion? No, there are three more possible movements that this plane can make.

It can roll on the X-axis, It can pitch upwards or downwards on the Y-axis, and can yaw in a circle on the Z-axis. Don’t get scared of these new terms, pitch, roll and yaw are just technical terms used by experts to denote movements.

We can simply relate this to the rotational movement of objects along the coordinate axes.

So, we can conclude that aeroplanes can undergo two types of motion: Translational and Rotational motion, and that too in 6 possible independent ways.

Not only aeroplanes, but any rigid body can also undergo translational and rotational motion. If we look at the movement of gas particles present inside a beaker. They are also constantly moving. In fact, you will be amazed to know that they also exhibit translational and rotational motion while moving. Not only translational and rotational, but they also show vibrational motion at extremely high temperatures.

Well! We can see that any particle whether it is macroscopic or microscopic has complete freedom to

exhibit any sort of motion. These independent ways through which any particle can move is called its “degree of freedom”. It is represented by ‘f’.

Just like the case of the aeroplane, it has 6 independent ways of movement, 3 translational and 3 rotational i.e., overall 6 degrees of freedom.

In a similar way, gas particles also have their individual degree of freedom. Let’s discuss it in detail.

• What is the Degree of Freedom?
• Types of Degree of Freedom
• Degrees of Freedom of a Monoatomic Gas
• Degrees of Freedom of a Diatomic Gas
• Degrees of Freedom of a Triatomic Gas
• Practice Problems

What is the Degree of Freedom?

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

The degrees of freedom refers to the number of ways in which a particle in the gas phase may move, rotate, or vibrate in space. The number of degrees of freedom a particle 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).

## Types of Degrees 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), ability to 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 generally a total degree of freedom in three dimensions of the molecule is determined by: F= 3N.

Let’s discuss the types of degrees of freedom one by one.

### Translational degree of freedom

Translational degrees of freedom arises from a gas particle’s ability to move freely in space. A particle 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 X, Y, and Z-axis. So, the translational motion of the particle of a gas has three degrees of freedom associated with it. This is applicable to all gas particles, whether they are monatomic, diatomic, or polyatomic, as any particle may move freely in all directions in three-dimensional space.

### Rotational degree of freedom

A molecule’s rotational degrees of freedom represents the number of unique ways the molecule may rotate in space about its centre 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 centre 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 mass. This shows that the linear molecule has two rotational degrees of freedom. However, for non-linear molecules, there are three rotational degrees of freedom.

### 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.

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 we get the vibrational degrees of freedom of linear and non-linear molecules.

Vibrational degree of freedom for linear molecules = 3N - 5

Vibrational degree of freedom for linear molecules = 3N - 6

where N is no. of atoms present in a molecule.

## Degrees of Freedom of a 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 axis X, Y and Z.

Examples: Inert gases like helium (He), Neon (Ne), Argon (Ar), etc.

## Degrees of Freedom of a Diatomic Gas

The diatomic molecule can rotate about an 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-axis. So, it has in total of 5 degrees of freedom.

However, at high temperatures, 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.

## Degrees of Freedom of a Triatomic Gas

Polyatomic gases can be linear or non-linear.

Degrees of freedom of a linear triatomic gas

In the case of a linear triatomic molecule, 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 in total 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: CO2, CS2 etc.

## Degrees of freedom of a non-linear triatomic gas

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: H2O, SO2 etc.

 Type of Molecule Translational DOF Rotational DOF Total DOF Monoatomic 3 0 3 Diatomic 3 2 5 Triatomic - Linear 3 2 5 Triatomic - Nonlinear 3 3 6 Polyatomic- Nonlinear 3 3 6

## Practice Problems

Q. Vibrational degrees of freedom are:

A) significant only at high temperatures
B) significant only at low temperatures
C) significant at all temperatures
D) negligible at all temperatures

Solution: Vibrational degrees of freedom are significant only at high temperatures. This is because, for atoms of molecules to vibrate, high energy is required, which is obtained only at higher temperatures.

Q. If vibrational degrees of freedom are active then what will be the degree of freedom of CO2?

A) 2
B) 3
C) 4
D) 9

Solution: If vibrational degrees of freedom are active then the formula for calculating the degree of freedom is 3N where N is atomicity. So, here the molecule is CO2 and its atomicity is 3. So,

Degree of freedom of CO2 = 3N = 3 3 = 9

This molecule has 3 translational, 2 rotational and 4 vibrational degrees of freedom. So, the total number of degrees of freedom is 9.

Q. The degrees of freedom of a polyatomic gas such as ammonia at room temperature is:

A) 2
B) 3
C) 4
D) 6

Solution: The molecular formula of ammonia molecule is NH3 and the structure of ammonia is trigonal pyramidal. Hence this molecule will fall under the category of a polyatomic non-linear molecule and at room temperature, the vibrational degree of freedom is inactive. So, this polyatomic non-linear molecule possesses three degrees of freedom of rotation in addition to three degrees of freedom of translation. Hence, it has six degrees of freedom.

Q. What will be the vibrational degree of freedom of the hydrogen chloride molecule?

A) 1
B) 2
C) 3
D) 4

Solution: The molecular formula of the hydrogen chloride molecule is HCl and the shape of HCl molecule is linear. Hence this molecule will fall under the category of the diatomic linear molecule and the formula for calculating the vibrational degree of freedom in the linear molecule is (3N - 5) , where N is the number of atoms present in a molecule.

So, vibrational degree of freedom of the hydrogen chloride molecule:

= (3N - 5)

= ((32) - 5)

= 1

## Frequently Asked Questions - FAQs

Q. What do you mean by a non-linear molecule?
Non-linear molecules are compounds that have a geometry other than linear geometry i.e, these molecules are not linear, and their atoms are not arranged in a straight line. The shape of these molecules depends on the hybridization of the atomic orbitals of the atoms in the molecule.

### Q. If the vibrational degree of freedom of a linear molecule is 7, then what will be the atomicity of a molecule? Answer: For N atomicity of linear molecules there are total 3N degrees of freedom among them 3 translational, 2 rotational and (3N-5) vibrational degrees of freedom. Given that the vibrational degrees of freedom of a linear molecule is 7 , then we can write,

(3N-5) = 7

So, N comes out to be 4. Hence, if the vibrational degrees of freedom of a linear molecule is 7, then the atomicity of a molecule will be 4.

Q. What will be the atomicity of a gas molecule if the total degree of freedom is 9?