Degree of Freedom

Gases have no or negligible forces of intermolecular attraction between them. So, when they are put in a container, they randomly move in different directions. It can be along the x-axis, y-axis and z-axis. 

The degree of freedom is a number of parameters that determine the state and configuration of an atom in space. All system states are called the system’s phase space, and the degrees of freedom of the system are called the dimensions of the phase space. It can also be represented as the coordinates that perfectly represent both the position and orientation of a particle in space completely. The molecular degree of freedom is the number of ways a gas molecule may rotate, move, or oscillate in space.  

Definition of Degrees of Freedom

The degree of freedom for a dynamic system is the number of directions in which the system can move. It is the number of coordinates that describe the position and configuration of the system.

They are commonly represented by N.

It can also be defined as all independent coordinates required to specify the configuration and position of a thermodynamic system in space.

Irrespective of the molecule type, the maximum number of translational motions a particle possesses is 3.   The rotational motion of a molecule is caused due to the rotation of a molecule about an axis passing through the centre of mass of the molecule. Their number depends upon the structure of the molecule. The vibrational motion of a molecule is caused due to the motion of the molecule about its equilibrium position.

Some hindrances sometimes restrict the motion of the molecules. These are called constraints. 

Degrees of Freedom formula

Considering the situation that there are several gas molecules (A) in a container. Then, the gross total of degrees of freedom is given by N or f = 3A. 

However, if we consider R number of constraints restricting the molecules from moving freely, then the degrees of freedom decrease (due to restriction being an opposing force), and it is now given by,

N or f = 3A – R.

Here, A indicates the number of particles (in this case, gas molecules) in the system and R represents the number of constraints (or independent relations). N or f represents the degree of freedom.

Independent movements can be translation, rotation, vibration, or any combination of these.

So, the degrees of freedom can be of three types:

(i) degrees of freedom of translational motion

(ii)  degrees of freedom of rotational motion

(iii) degrees of freedom of vibrational motion

Degrees of Freedom of Different particles

Three types of degrees of freedom are translation, rotation, and vibration.

For a monatomic gas, degrees of freedom = 3, and all are translational: Molecules of monoatomic gases can move linearly in any direction in space along the coordinate axis, so they can have three independent motions and hence 3 degrees of freedom.

The example includes gases like Argon and Helium.

For a diatomic gas, degrees of freedom = 5, where 3 are translational and 2 are rotational: In diatomic gas molecules, the centre of mass of two atoms is free to move along three coordinate axes. Thus, a diatomic molecule rotates about an axis at right angles to its axis. Therefore, there are 2 degrees of freedom of rotational motion and 3 degrees of freedom of translational motion along the three axes.

The example includes oxygen and nitrogen molecules.

For a non-linear triatomic gas, degrees of freedom = 6, where 3 are translational and 3 are rotational. For a linear triatomic gas, degrees of freedom = 7, where 3 are translational, 3 are rotational, and 1 is vibrational. Triatomic gas molecules have three atoms. 

If all three atoms are aligned along a line, it is a linear molecule. 

But if the three atoms are placed along the vertex of a triangle, then it is a non-linear molecule. 

The molecules of these gases can rotate perpendicular to their axis, passing through the centre of mass in two directions.

On the other hand, the mass’s centre is at the central atom of a triatomic molecule. Thus, it behaves as a diatomic molecule that has 3 degrees of freedom of translation as well as 2 degrees of freedom of rotation. So, in total, there are 5 degrees of freedom.

Polyatomic molecules: The polyatomic molecule can have N (where N is equal to or greater than 2) number of atoms, which can be linear or nonlinear.

As mentioned above, we can also detect degrees of freedom using the minimum number of coordinates required to specify a position. This is done as follows:

1. For a single particle, we need 2 coordinates in a 2-D plane to specify its position and 3 coordinates in a 3-D plane. Thus, its degree of freedom in a 3-D plane is 3.

2. For an object, with a distance `d` between them, consisting of two particles (for example, a diatomic molecule) in a 3-D plane, we can show its degrees of freedom to be 5. 

Let’s say one particle in this body has coordinates (x1, y1, z1) and the other has coordinates (x2, y2, z2) with z2 unknown. 

Application of the formula for distance between two coordinates can be noted as

d=√(x2x1)2 + (y2y1)2 + (z2z1)2 results in one equation with one unknown, in which we can solve for z2. One of x1, x2, y1, y2, z1, or z2 can be unknown.

All possible movements need to be strictly restrained for a body to be in static equilibrium. If a degree of freedom is not restrained, the body is unstable. Then it can freely move around in one or more ways. 

Suppose the restraints are correctly understood and known. In that case, equal constraints and degrees of freedom create a stable system, and the reaction force can be determined using the equilibrium equation. When multiple restraints exceed the number of degrees of freedom, the body is in equilibrium.

Conclusion 

A degree of freedom is considered a useful property as it is not dependent on other variables. It is an essential factor in physics to determine and contribute to a physical system. Gases have no or negligible forces of intermolecular attraction between them. So, when they are put in a container, they randomly move in different directions. The molecular degree of freedom is the number of ways a gas molecule may rotate, move, or oscillate in space. If the degree of freedom is not restrained, the body is unstable.