About Lesson

**Translational degree of freedom:-**

- Translation means motion of the body as a whole from one point to another.

- For example:
- Consider the oxygen molecule;it has 2 oxygen atoms which are bonded together.
- The 2 oxygen atoms along with the bond are considered as whole body.
- When the body as a whole is moving from one point to another is known as translational.
- Consider a molecule which is free to move in space and so it will need 3 coordinates(x, y, z) to specify its location.
- Therefore it has 3 degrees of freedom.
- Similarly a molecule which is free to move in a plane which is 2 dimensional and so it needs 2 coordinates to specify its location.
- Therefore it has 2 degrees of freedom.
- Similarly a molecule which is free to move in line it needs 1 coordinate to specify its location.
- Therefore it has 1 degree of freedom.

- Molecules of monoatomic gas have only translational degrees of freedom.This means gases which have only one atom.
- For example:-Helium atom it consists of only one He atom.It will have translationaldegrees of freedom.
- Each translational degree of freedom contributes a term that contains square of some variable of motion.
- The variable of motion means the velocity (v
_{x},v_{y},v_{z}). - The term (
**1/2) mv**will contribute to energy.This is Kinetic energy which is involved with the motion of the molecule from one point to another._{x}^{2}

- The variable of motion means the velocity (v

In thermal equilibrium, the average of each such term is (**1/2) k _{B}T**.

**Rotational Degree of freedom**

- Independent rotations that specify the orientation of a body or system.
- There is rotation of one part of the body with respect to the other part.

- Rotational degree of freedom happens only in diatomic gas.
- Diatomic molecules have rotational degrees of freedom in addition to translational degrees of freedom.
- It is possible in diatomic molecules as 2 atoms are connected together by a bond.So the rotation of one atomw.r.t to other atom.
- In diatomic there is translational in addition to that they have rotational degree of freedom also.
- For example: – Two oxygen atoms joined together by a bond. There are two perpendicular axes.
- There are 2 rotations possible along the two axes.
- They have 3 translational degrees of freedom and also 2 rotational degree of rotation.

- Therefore Rotational degree of freedom contributes a term to the energy that contains square of a rotational variable of motion.
- Rotational variable of motion comes from angular momentum ω.
- Linear velocity is v
_{x},v_{y},v_{z}. Whereas angular velocity is w_{x},w_{y},w_{z}. - E
_{R}(rotational) = (1/2)(I_{1}ω_{1})+(1/2)I_{2}ω_{2}. These are 3 rotationaldegrees of freedom along the 2 perpendicular axes.

- Linear velocity is v
- The total energy contribution due to the degrees of freedom for oxygen molecule.
- There will be 3 translational degree of freedom (1/2)m
_{x}v_{x}^{2},(1/2)m_{y}v_{y}^{2},(1/2)m_{z}v_{z}^{2}) - 2 rotational degree of freedom (1/2)I
_{1}^{2}ω_{1}^{2},(1/2)I_{2}^{2}ω_{2}^{2}

- There will be 3 translational degree of freedom (1/2)m

**Vibrational degree of freedom**

- Some molecules have a mode of vibration,i.e. its atoms oscillate along the inter-atomic axis like a one-dimensional oscillator.
- This vibration is observed in some molecules.
- For example:- CO atoms oscillate along the interatomic axis like a

one-dimensional oscillator.

- Consider two 2 atoms they vibrate along the inter-atomic axis.
- The vibrational energy terms contain square of vibrational variables of motion.
- Total vibrational energy term E
_{v}= (1/2) m (dy/dt)^{2}+ (1/2) ky^{2}where

(1/2) m(dy/dt)^{2}=Kinetic energy and (1/2)ky^{2} =Potential energy and k=force constant one-dimensional oscillator.

- The vibrational degree of freedom contributes 2 terms.