ROTATIONAL PARTITION FUNCTION

The rotational partition function can be calculated from the moment of inertia of the molecule which can be obtained from microwave spectra. An expression for rotational partition function of a diatomic molecule can be deduced in terms of moment of inertia assuming that the molecule is rigid rotor.

According to wave mechanics the rotational energy of a diatomic molecule is given by

Erot =   h²     J (J+1)........................ (1)

          8π²l

I = moment of inertia = μr² ; J = rotational quantum number

According to definitions the rotational partition function is given by 

qrot = ∑ ɡrot e⁻ ᵉʳᵒᵗ/kT.................(2)

For each rotational level the degeneracy is given by grot = 2J+1. Substituting the values of grot and ∈rot in equation (2) we get,

qrot = ∑ ʲⱼ⁼₌₀∞ (2J +1) e⁻   h²J(J+1)........(3)

                                               8π²lkT

If I is sufficiently large, the rotational energy levels are so close to be continuous. This is the case for all diatomic molecules, except those like H₂, D₂ etc. Summation can be replaced by integration.

qrot = ∫₀∞ 2(J+1) e⁻  h²J(J+1) ; dJ .......(4)

                                    8π²l kT

put h²/8π²lk = θrot ; qrot = ∫₀∞, 2(J+1)

e⁻θrot J(J+1)T      dJ.................... (5)

               χ

Let χ = θrot  J (J+1)

               T

dχ =  θrot  (2J + 1) dJ

             T

dJ =   T   .        1    . dχ

       θrot     (2J+1)

qrot = ∫₀∞ 2(J+1) e⁻ˣ .   T     .      1        .dχ

                                       θrot    (2J+1)

=  T   ∫₀∞ e⁻ˣ dχ = −   T   [e⁻ˣ]₀∞ =   T  

θrot                          θrot                 θrot

[e−∞ − e−⁰ ] =     T    ....................(6)

                         θrot

qrot =    8π² lkT     ......................(7)

                h²

The quantity h²/8π² l k has the dimensions of temperature and is called characteristic rotation temperature which can be written as,

θᵣₒₜ =        T.       .......................... (8)

             qrot

The value of θrot T. gives an idea as to how the energy levels are to KT. The assumption of replacement of summation by integration used in evaluating qrot is valid when θrot is far less than T.

Equation (7) can be applied to rotational degrees of freedom in which identical configuration occur only after a rotation of 2π. The number of distinguishable configuration possible for a given molecule during a rotation gives symmetry number σ. For example for homonuclear diatomic molecules, σ = 2, whereas for heteronuclear diatomic molecules, σ = 1. Hence equation (7) should be divided by symmetry number.

      qrot =    8π² l kT     ..................(9)

                   σh²

σ - symmetry number.