2. Law of equipartition of energy

Law of Equipartition of energy: Degrees of Freedom

Degrees of Freedom can be defined as independent displacements or rotations that specify the orientation of a body or system. A molecule free to move in space needs three coordinates to specify its location.

If it is constrained to move in a plane it needs to. If constrained to move along a line, it needs just one coordinate to locate it.

For example:-Consider a room and if we tie a thick rope from one wall to another.

Take a ball which is moving straight on the rope. The ball has only 1 degree of freedom. It can move only in one particular dimension. Consider if the ball is on the floor which is two-dimensional, then the ball can move along 2 directions. The ball has 2 degrees of freedom.

Consider if we throw the ball in space which is 3 dimensional. Then the ball can move in 3 dimensions. Therefore degree of freedom tells us in how many ways a body can move or rotate or vibrate.

Categories of Degrees of Freedom

1. Translational degrees of freedom.

2. Rotational degrees of freedom.

3. Vibrational degrees of freedom.

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 the 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, and 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 needs 1 coordinate to specify its location. Therefore it has 1 degree of freedom.
• Molecules of monatomic 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 translational degrees of freedom.
• Each translational degree of freedom contributes a term that contains a square of some variable of motion.

The variable of motion means the velocity (v_x, v_y,v_z ) 

The term   will contribute to energy. This is Kinetic energy which is involved with the motion of the molecule from one point to another.

In thermal equilibrium, the average of each such term is  

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 atom w.r.t. to another atom.

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 degrees of rotation.

Therefore Rotational degree of freedom contributes a term to the energy that contains the 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 (ω_x, ω_y, ω_z).

These are 3 rotational degrees of freedom along the 2 perpendicular axes.

The total energy contribution due to the degrees of freedom for oxygen molecules.  There will be 3 translational degree of freedom and two rotational degree of freedom

Kinetic energy contribution from translational motion is given by 

Kinetic energy contribution from rotational motion is given by 

Vibrational degree of freedom

Some molecules have a mode of vibration, i.e. 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 vibrating along the inter-atomic axis. The vibrational energy terms contain squares of vibrational variables of motion.

Total vibrational energy term

Where, = Kinetic energy and   = Potential energy and k = force constant one-dimensional oscillator.

The vibrational degree of freedom contributes 2 terms.

(1) In the first figure rotational motion along two axes perpendicular to line joining two particles (here y and z directions) is shown.

(2) In the second figure vibrational motion along line joining the two atoms is shown

Comparison between 3 energy modes

Law of Equipartition of energy

According to this law, in equilibrium, the total energy is equally distributed in all possible energy modes, with each mode having average energy equal to  .

1. Each translational degree of freedom contributes  
2. Each rotational degree of freedom contributes  
3. Each vibrational degree of freedom contributes  

Specific Heat Capacity for monoatomic gases Monoatomic gases will only have a translational degree of freedom. Maximum they can have is three translational degrees of freedom. Each degree of freedom will contribute  . Therefore 3 degrees of freedom will contribute

• By using law of equipartition of energy, the total internal energy of 1 mole of gas  .
• Specific heat capacity at constant volume  .
• For an ideal gas C_P - C_V=R , By using above equation
• Ratio of specific heats γ=CP/CV=(5/3).

Specific Heat of Diatomic gases (rigid)

A rigid diatomic gas means they will have translational as well as rotational degrees of freedom but not vibrational. They are rigid oscillators.

A rigid diatomic molecule will have 3 translational degrees of freedom and 2 rotational degrees of freedom. Total 5 degrees of freedom.

• By law of equipartition of energy, each degree of freedom will contribute .
• Therefore 5 degree of freedom will contribute Therefore the total internal energy of 1 mole of gas
• Specific heat capacity at constant volume .
• Specific heat capacity at constant pressure of a rigid diatomic is given as .

• Ratio of specific heats γ=C_p/C_v= (7/5).

Specific Heat of Diatomic gases (non-rigid)

A non-rigid diatomic gas has translational, rotational as well as vibrational degrees of freedom.

There will be 3 translational degrees of freedom and 2 rotational degrees of freedom and 1 vibrational degree of freedom.

• Total contribution by translational =  , rotational  and vibrational =k_BT.
• Total contribution from 6 degree of freedom =  
• Total Internal energy for 1 mole Specific heat at constant volume Cv=dU/dT = (7/2) R.

Specific heat at constant pressure C_p=C_v+R= (9/2) R.

Ratio of specific heat γ= C_p/C_v = (9/7)

There are two independent axes of rotation (1) and (2) normal to the axis joining the two oxygen molecule. It has 3 translational and 2 rotational degrees of freedom

Specific Heat Capacity for polyatomic gases

Polyatomic gases will have 3 translational degree of freedom, 3 rotational degrees of freedom and ‘f’ number of vibrational modes.

• Total internal energy of 1 mole of gas = ((3/2) + (3/2) +f) RT = (3 + f) RT.
• Specific heat at constant volume C_v=dU/dT = (3 + f) R
• Specific heat at constant pressure C_p=C_V+R=(4 + f) R
• Ratio of specific heat γ= C_p/C_V = (4 + f)/(3 + f)

Specific Heat Capacity for solids

Consider there are N atoms in a solid. Each atom can oscillate about its mean position. Therefore vibrational degree of freedom = k_BT

• In one-dimensional average energy=k_BT, in three-dimensional average energy =3k_BT
• Therefore total internal energy (U) of 1 mole of solid = 3K_BTxN_A = 3RT

• At constant pressure, ΔQ = ΔU + PΔV change in volume is very less in solids .Therefore ΔV = 0. So we have finally ΔQ = ΔU for solids.
• Specific heat at constant volume    

• Specific heat at constant pressure  as ΔQ = ΔU, Therefore CV=dU/dT=3R
• Therefore CP = CV = 3R

Specific Heat Capacity of water

Consider water as solid, so it will have ‘N’ number of atoms. Therefore for each atom average energy =3kBT

• Number of molecules in H2O= 3 atoms.
• Total internal energy  U=3k_BT×3×N_A=9RT
• CV = CP = 9R.

Conclusion on Specific heat

• According to classical mechanics, the specific heat which is calculated based on the degree of freedom should be independent of temperature.
• However  T→0,degree of freedom becomes inefficient.
• This shows classical mechanics is not enough; as a result quantum mechanics came into play.
• According to quantum mechanics minimum non-zero energy is required for a degree of freedom to come into play.
• Specific heats of all substances approach zero as T→0.

Mean free path

Mean free path is the average distance between the two successive collisions.

Inside the gas there are several molecules which are randomly moving and colliding with each other. The distance which a particular gas molecule travels without colliding is known as the mean free path.

Expression for mean free path

Consider each molecule of gas is a sphere of diameter (d).The average speed of each molecule is <v>.

Suppose the molecule suffers collision with any other molecule within the distance (d). Any molecule which comes within the distance range of its diameter will have a collision with that molecule.

The volume within which a molecule suffers collision =<v>Δtπd².

Let number of molecules per unit volume =n

Therefore the total number of collisions in time Δt =<v>Δtπd²(n)

Rate of collision  = <v>πd²n.

Suppose time between collision  T= 1/(<v>nπd²) 

Average distance between collision = T<v>= 1/(πd²n).

1/(πd²n), this value was modified and a factor was introduced.

Mean free path (l) = .

Conclusion: - Mean free path depends inversely on:

1. Number density (number of molecules per unit volume)
2. Size of the molecule.

The volume swept by a molecule in time Δtin which any molecule will collide with it.