Class 11 Physics Kinetic Theory Vibrational degree of freedom

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 Ev = (1/2) m (dy/dt)2+ (1/2) ky2 where

 (1/2) m(dy/dt)2=Kinetic energy and (1/2)ky2 =Potential energy and k=force constant one-dimensional oscillator.

  • The vibrational degree of freedom contributes 2 terms.

(1)

(2)

 

(1)   Rotational motion along two axis perpendicular to line joining two particles (here y and z directions)

(2)  Vibrational motion along line joining the two atoms

Comparison between 3 energy modes

Translational

Rotational

Vibrational

Et=(1/2)mvx2

Er=(1/2)I ω2

Ev = (1/2)m (dy/dt)2+(1/2)ky2

1 squared term is being contributed

1 squared term is being contributed

2 squared term is being contributed

 

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 an average energy equal to (1/2)kBT.

  1. Each translational degree of freedom contributes (1/2) kB
  2. Each rotational degree of freedom contributes (1/2) kB
  3. Each vibrational degree of freedom contributes 2x (1/2)kB

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