Relationship between Degree of Freedom and Thermodynamics

Thermodynamics is the branch of science that deals with the quantitative relationship between heat and other forms of energies. The word thermodynamics means the flow of heat. 

Thermodynamics is fundamental and of great importance in physical chemistry. It helps us predict whether a given chemical reaction process is feasible under given conditions of temperature, pressure, and concentration. 

Terminologies in Thermodynamics

There are a few terms related to the field of thermodynamics.

• System: It refers to the part of the universe in which we carry out observations.
• Surroundings: The part of the universe other than the system is known as surroundings.
• Boundary: The wall that separates the system from the surroundings is called the boundary.
• Thermal Equilibrium: If there is no flow of heat from one side of the system to another, the system is said to be in thermal equilibrium.

Degree of Freedom (F)

We can generally describe the motion of atoms and molecules in terms of the degrees of freedom they possess. We can define the degrees of freedom of a molecule as the independent number of parameters required to completely describe the molecule’s state. 

It represents the number of coordinates necessary to specify the position of all the atoms in the molecule. Generally, we use x,y, and z to determine the degree of freedom.

When a gaseous molecule is heated, its energy may bring about three kinds of motion in it. These motions are the translational motion, the rotational motion, and the vibrational motion. We can express it by saying that the molecule possesses translational, rotational, and vibrational degrees of freedom. 

A molecule containing an ‘n’ number of atoms will have 3n degrees of freedom.

Translation Degree of Freedom

Translational motion is how a molecule shifts from one point in space to another. During the molecule’s motion, the centre of mass shifts from one point to another. 

Three coordinates are required to specify the centre of mass of the molecule. Hence, a molecule has three translational degrees of freedom.

Rotational Degree of Freedom

The motion of a molecule around an axis is called rotational motion. There are only two rotational degrees of freedom for any diatomic molecule or any linear polyatomic molecule (carbon dioxide). We do not count the rotation about the molecular axis.

For a non-linear molecule (water molecule), there are three rotational degrees of freedom corresponding to rotation about all the three Cartesian axes.

Vibrational Degree of Freedom

Molecules vibrate along their axis and undergo twisting, wagging, to-fro, back-forth, etc. We can use these vibrating motions to describe the degrees of freedom of a molecule. For a molecule having N number of atoms, the vibrational degree of freedom is given by:

• Linear Molecule (3N-5)
• Non-Linear Molecule (3N-6)

Degrees of Freedom of Gases with One or N Number of Molecules

Equipartition Theorem

According to the law of equipartition of energy, the total energy of a molecule is divided equally among the various degrees of freedom of the molecule. The distribution of kinetic energy along the x,y and z directions is given by:

E = E(x) + E(y) + E(z)

Since the motion of gas molecules is random and the motion of along three cartesian axes is equally probable so, we can conclude that:

E = E(x) = E(y) = E(z) = kT = RT

Where,

k is the Boltzmann constant. 

R is the gas constant.

T is the temperature.

Translational degree of freedom contributes RT per molecule, rotational degree of freedom contributes RT per molecule, and vibrational degree of freedom contributes RT per molecule.

Relationship between Degree of Freedom and Thermodynamics

The internal energy (U) of a thermodynamic system is the total energy within the substance. We can express it in terms of specific heat capacity at constant volume (Cv) and temperature.

U = Cv . T

Specific heat capacity at constant volume, Cv = F.

Where,

F is the degree of freedom. 

R is the gas constant = 8.314 J/(K mol)

Conclusion

Thermodynamics is the branch of science that deals with the quantitative relationship between heat and other energy forms. A molecule’s degree of freedom is the independent number of parameters required to completely describe the molecule’s state. 

The degree of freedom relates to the molecule’s motion when heated. Therefore, there are three degrees of freedom: translational, rotational, and vibrational.