User:David-i98/Vibrotspec draft

Vibration-rotation spectroscopy allows information about the structure of small molecules to be determined based on transitions between between different vibrational and rotational energy levels.

Vibrational energy levels
For an initial approximation, the vibrational energy levels of a diatomic molecule are given by:


 * $$E_{\nu}=\left(\nu + {1\over 2}\right)\hbar\omega$$

where:

$$\nu$$ is the vibrational energy level,

$$E_{\nu}$$ is the energy of that vibrational level, and

$$\omega$$ is given by:


 * $$\omega = \left( \frac{k}{\mu} \right)^{1/2}$$

where:

k is the force constant for the bond in question, and

$$\mu$$ is the reduced mass.

Rotational energy levels
The rotational energy levels of a diatomic molecule are given by:


 * $$E_{J}=J(J+1)B$$

where:

J is the energy level being considered,

B is the molecule's rotational constant, given by:


 * $$B=\frac{\hbar^2}{2I}$$

where:

I is the moment of inertia, given by:


 * $$I=\mu R^2$$

where:

$$\mu$$ is the reduced mass, and

R is the bond length.

Constructing a vibration-rotation energy level diagram
It can be shown that the difference between vibrational energy levels is significantly more than the difference between rotational energy levels.

For example, in the case of H-37Cl, we have:

$$\mu = \left(\frac{1 \times 37}{1+37}\right)$$ = 0.974 u = 1.62 × 10-27 kg

k = 520 Nm-1

Thus, $$\omega$$= 5.67 × 1014 Hz

The bond length of HCl, R, is 1.28 × 10-10m.

Thus, I = 2.650 × 10-47

and B =