Degrees of Freedom

Degrees of Freedom is defined as an independent physical parameter in the explanation of the state of a physical system. Degrees of freedom is the total number of ways a molecule in a gas can move, vibrate, or rotate in space. Number of degrees of freedom of a molecule plays a role in determining the values of different thermodynamic variables using the equipartition theorem.

Degrees of freedom have three types which are translational, rotational, and vibrational. The number of degrees of freedom of all types which are possessed by the molecule depends on both the number of atoms present in the molecule as well as the geometry of the molecule.

Types of Degrees of freedom

Gaseous molecule contains a specific number of degrees of freedom like the ability to translate (motion of its centre of mass through space), ability to rotate around its centre of mass, or ability to vibrate (as bond length and angle changes). Several physical and chemical properties depend on the energy related with all modes of motion.

When a molecule contains n number of free particles then formula for degrees of freedom in 3D is given as F= 3n

Here, F = Degrees of freedom

Degrees of freedom are divided into three parts which are as follows

1. Translation Degrees of freedom
2. Rotational Degrees of freedom
3. Vibrational Degrees of freedom

Translation Degrees of freedom

Translational degrees of freedom occur from a gas molecule which has the ability to move independently in space. A molecule moves in the directions of x, y, and z of a Cartesian Coordinate. If the centre of mass of a particle moves from its starting point to a new point then the particle is said to have a translational motion along the x-axis, y-axis and z-axis. Hence the translational motion of molecules of a gas has three degrees of freedom which are associated with it. This is suitable for all gas molecules (monatomic, diatomic, or polyatomic), because any molecule can move freely in each direction in 3D space.

Rotational Degrees of freedom

The rotational degrees of freedom of a molecule depict the number of distinct (unique) ways that the molecule can rotate in space about its centre of mass when the orientation of the molecule changes. A monatomic gas molecule, like an inert gas, has no rotational degrees of freedom, since the centre of mass sits directly on the atom and no rotation producing a change is possible. The rotational degrees of freedom is two for linear molecules whereas nonlinear molecules have three rotational degrees of freedom.

Vibrational Degrees of freedom

The atoms of a molecule also vibrate and the vibration of the atoms of a molecule moderately change the inter nuclear gap between atoms of the molecule. The number of vibrational degrees of freedom of a molecule is dictated by studying the number of unique methods the atoms inside the molecule can move relative to one another like in bond stretches.

As we have already known, the atoms possess only a translational degree of freedom. Vibrational degree of freedom is one for diatomic molecules. The bonds of the molecules during the vibrational motion act like a spring and the molecule shows simple harmonic motion.

Degree of freedom            Monatomic             Linear                 Non-linear

Translational                                3                          3                               3

Rotational                                    0                          2                                3

Vibrational                                   0                       3n-5                           3n-6

Total                                            3                         3n                               3n

Equipartition Law of Energy

According to the Law of Equipartition of Energy total energy in thermal equilibrium for a dynamic system is split equally among the degrees of freedom.

Kinetic energy for a single molecule along

x-axis 1/2 mv²x

y-axis 1/2 mv²y

z-axis 1/2 mv²z

At thermal equilibrium average kinetic energy for a single molecule of gas along

x-axis 1/2 mv²x

y-axis 1/2 mv²y

z-axis 1/2 mv²z

Average kinetic energy is

1/2 mv²_rms=32 K_bT

Here,

v_rms= Root mean square

K_b=Boltzmann constant

T = Temperature

Formula for degrees of freedom

The average kinetic energy per degree of freedom for monatomic gas is

KE_X=1/2 K_bT

Or

KE_X=3/2 RT

The average kinetic energy per degree of freedom for triatomic gas is

Translational degrees of freedom = 6

Therefore,

KEx=3RT

Degree of freedom of diatomic molecule

Diatomic molecule will rotate about all axes perpendicularly to its own axis. Therefore, diatomic molecules have 2 (two) rotational degrees of freedom and also has 3 (three) translational degrees of freedom along three axes. Diatomic molecule has 1 (one) vibrational degree of freedom. Hence a diatomic molecule contains six degrees of freedom at high temperature. (at room temperature, the degree of freedom of the diatomic molecule is five).

Degree of freedom of triatomic molecule

The centre of mass triatomic molecule lies at the centre of the atom. Therefore, it behaves like a diatomic molecule with three translational degrees of freedom and two rotational degrees of freedom hence it has five degrees of freedom at room temperature.

Conclusion

The degrees of freedom in a statistical calculation describe the number of variables that can vary in a computation. The degrees of freedom can be determined to verify that chi-square tests, t-tests, and even more advanced f-tests are statistically valid. These tests are frequently used to compare observed data with data that would be expected to be produced if a particular hypothesis were true.For example, imagine a drug trial is undertaken on a group of patients, and it is expected that the patients who received the medicine will have higher heart rates than those who did not. The test findings may then be examined to see if the difference in heart rates is substantial, and degrees of freedom could be factored into the equation.