Chemical Science Short Notes: Rotational Spectra!(Download PDF)

Study Notes on Rotational Spectra (Download PDF)

Consider a diatomic molecule in which m1 and m2 are the masses of the two atoms, and r is the equilibrium bond length, rotating about an axis passing through its centre of gravity,

The centre of gravity is defined by the quality of the momenta about it, i.e., 

...........................(1)

The moment of inertia I of a molecule (rotating as a rigid rotor, not subject to centrifugal forces that tend to distort the molecular geometry and change the moments of inertia) is defined as:

..................................(2)

Here, r_i is the difference of the ith particle of mass m_i from the centre of gravity. Since a diatomic molecule has two atoms, so,

Therefore, from Eqs. 1 and 5:

....................................(6)

Hence,

Substituting the above values of r1 and r2 in equation 3. 

---------------------(7)

--------------------(8)

Where J is the rotational quantum number. The energy of a rotating molecule is given by 1/2Iw² . Hence, the quantized rotational energy levels of a rigid diatomic rotor (rotating molecule) are given by

------------------(10)

Using the expression for L from Eq. 9, 

............................(11)

The expression of total rotational energy is given by:

................................(12)

F(J) is called the rotational term. Defining the rotational constant B as:

.............................(13)

If we want to consider centrifugal distortion whose effect on the diatomic rotor is to stretch the bond and hence to increase the moment of inertia, and thereby to reduce the rotational constant and hence bring the energy levels closer than in the rigid-rotor approximation, then the energy level expression (14a) becomes

                              F(J) = B J (J + 1) − D_J J2 (J + 1)² ….............................................. (14c)

Where DJ is the centrifugal distortion constant given by D_j = 4B³/v², where is the molecule's vibrational frequency.

Next, a selection rule is needed to determine the radiative transition between the rotational energy levels. 

..................................(15)

i.e., only those transitions are allowed in which the rotational quantum number changes by unity. The + sign refers to absorption, and the - sign refers to radiation emission. Microwave spectra are usually observed as absorption spectra so that the operative part of the selection rule is ΔJ = +1. For a transition taking place from J to J + 1, the rotational frequency i is given by:

The rotational spectrum of a rigid diatomic molecule consists of a series of lines at 2B, 4B, 6B, 8B, etc. These lines are equally spaced by 2B called frequency separation.

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