In the previous blog we introduced spectroscopy and its different techniques. In this blog we will discuss Rotational Spectroscopy (also known as Microwave Spectroscopy) in detail.

Rotational spectroscopy measures a high-resolution spectrum where the spectral pattern is determined by the three-dimensional structure of the molecule. The quantized energy levels for the spectroscopy come from the overall rotational motion of the molecule. The rotational kinetic energy is determined by the three moments-of-inertia in the principal axis system. Any changes in the mass distribution will produce a different energy level structure and spectroscopic transition frequencies. Therefore, structural isomers have distinct rotational spectra but enantiomers, which have the same set of bond lengths and bond angles, have identical rotational spectra. In order for the rotational motion of the molecule to couple with light, it is necessary for the molecule to have a permanent dipole moment. Molecule must have dipole moment (Most heteronuclear molecules possess a permanent dipole moment e.g. HCl, NO, CO, water)

**Theory**

When a gas molecule is irradiated with microwave radiation, a photon can be absorbed through the interaction of the photon’s electronic field with the electrons in the molecules. For the microwave region this energy absorption is in the range needed to cause transitions between rotational states of the molecule. However, only molecules with a permanent dipole that changes upon rotation can be investigated using microwave spectroscopy. This is due to the fact that there must be a charge difference across the molecule for the oscillating electric field of the photon to impart a torque upon the molecule around an axis that is perpendicular to this dipole and that passes through the molecules center of mass.

**Rotational Symmetries**

To analyze molecules for rotational spectroscopy, we can break molecules down into 5 categories based on their shapes and their moments of inertia around their 3 orthogonal rotational axes:

Diatomic Molecules

Linear Molecules

Spherical Tops

Symmetrical Tops

Asymmetrical Tops

Diatomic Molecules :

The rotations of a diatomic molecule can be modeled as a rigid rotor. This rigid rotor model has two masses attached to each other with a fixed distance between the two masses.

It has an inertia (I) that is equal to the square of the fixed distance between the two masses multiplied by the reduced mass of the rigid rotor.

Ie=μr^{2}

μ=m_{1*}m_{2}/m_{1}+m_{2}

Using quantum mechanical calculations it can be shown that the energy levels of the rigid rotator depend on the inertia of the rigid rotator and the quantum rotational number J.

E_{J}=B_{e} J(J+1)

B_{e} =h/8π^{2}cI_{e}_{}

However, this rigid rotor model fails to take into account that bonds do not act like a rod with a fixed distance, but like a spring. This means that as the angular velocity of the molecule increases so does the distance between the atoms. This leads us to the **nonrigid rotor model** in which a centrifugal distortion term (DeDe) is added to the energy equation to account for this stretching during rotation.

E_{J} (cm^{-1}) = B_{e} J(J+1)–D_{e} J2(J+1)

### Linear Molecules :

Linear molecules behave in the same way as diatomic molecules when it comes to rotations. For this reason they can be modeled as a non-rigid rotor just like diatomic molecules. This means that linear molecule have the same equation for their rotational energy levels. The only difference is there are now more masses along the rotor. This means that the inertia is now the sum of the distance between each mass and the center of mass of the rotor multiplied by the square of the distance between them.

Spherical Tops :

Spherical tops are molecules in which all three orthogonal rotations have equal inertia and they are highly symmetrical. This means that the molecule has no dipole and for this reason spherical tops do not give a microwave rotational spectrum.

Molecular example of Spherical top: Methane (CH_{4})

### Symmetrical Tops :

Symmetrical tops are molecules with two rotational axes that have the same inertia and one unique rotational axis with a different inertia. Symmetrical tops can be divided into two categories based on the relationship between the inertia of the unique axis and the inertia of the two axes with equivalent inertia. If the unique rotational axis has a greater inertia than the degenerate axes the molecule is called an oblate symmetrical top. If the unique rotational axis has a lower inertia than the degenerate axes the molecule is called a prolate symmetrical top. For simplification think of these two categories as either frisbees for oblate tops or footballs for prolate tops.

### Asymmetrical Tops :

Asymmetrical tops have three orthogonal rotational axes that all have different moments of inertia and most molecules fall into this category. Unlike linear molecules and symmetric tops these types of molecules do not have a simplified energy equation to determine the energy levels of the rotations. These types of molecules do not follow a specific pattern and usually have very complex microwave spectra.

Molecular example of asymmetrical tops: acetone. (CH_{3}-CO-CH_{3})

**The intensities of spectral lines**

The line intensity depends on the population density of the molecules in a rotational level. More population means there are more molecules undergoing transition from that level to upper or lower level and this transition will show as a line or peak in the spectrum.

There are two factors deciding the population of the levels

**Boltzmann distribution**

The rotational energy of the lowest energy level is zero as J=0, if we have N** _{0}** molecules in this state, then the number in higher state is given by:

N** _{J}**/N

**= exp (-E**

_{0}_{J}/kT) = exp{-BhcJ (J+1)/kT}

And this value, for J=0 : T=300K comes almost equal to one

**The figure shows the rapid decrease in N_{J}/N_{0} with increasing J and with larger B**

**Degeneracy of energy states**

Degeneracy is the existence of two or more energy states which have exactly the same energy.Each energy level is (2J+1) fold degenerate

Overall result from both these effects,

Population is proportional to (2J+1) exp (-E_{J}/kT)

Applications of Rotational Spectroscopy

- We can calculate exact bond length of a chemical bond

Let’s consider the example of Carbon Monoxide (CO). The first rotational line of CO was observed at 3.84235 cm^{-1}, which means B = 1.921 cm^{-1 }^{}

^{ }And we know I = h/8π^{2}Bc , so Value of I (Moment of Inertia) can be calculated “

I = 14.5695 *10^{-47} kg m^{2}

Reduced mass for Carbon (12.0) and Oxygen (15.9994) is

μ=m_{1*}m_{2}/m_{1}+m_{2}

μ = 11.383 * 10^{-27 }kg

Now, the bond length can be calculated as

r^{2 }= I/μ

r = 0.1131 nm (nano-meter)

2.We can calculate exact atomic weight of an Isotope.

For this we will take the example of Carbon Monoxide again but in this case we will take two different carbons. One is ^{12}C^{16}O and another is ^{13}C^{16}O and their B values are B = 1.921 cm^{-1} and B^{` }= 1.836 cm^{-1} respectively. One important point to be noted is that the bond length will be same for both ^{12}C^{16}O and ^{13}C^{16}O.

If we divide B by B^{` }then that result comes equal to 1.046

We know the mass of Carbon-12 is 12 and mass of oxygen-16 is 15.9994

The value of Carbon -13 is found to be 13.0007.

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