SIF3002 Lecture 19: Rigid Rotor Spectroscopy (Hard)

Questions and Answers

What is the formula for the eigenvalues of a rigid rotor in terms of the rotational quantum number J?

$E_J = \frac{\hbar^2}{2I}J(J+1)$

In spectroscopy, what unit is preferred for reporting the energy levels EJ of a rigid rotor?

cm^-1

What is the formula for the rotational constant B in terms of the moment of inertia I and the rotational quantum number J?

$B = \frac{\hbar}{8\pi^2Ic}J(J+1)$

What are the selection rules for the rotational transition in a rigid rotor?

$∆ J = ±1$ Signup and view all the answers

In the case of absorption of electromagnetic radiation, what is the spacing between adjacent lines in the rotational spectrum of a rigid rotator?

$2B$ Signup and view all the answers

What is the bond length for the molecule C≡O?

1.131 × 10−10 m Signup and view all the answers

In the non-rigid rotor model, how does the new term in the rotational energy equation affect the energy levels?

The energy levels become closer to each other compared with the rigid rotor. Signup and view all the answers

What condition must a molecule satisfy to be microwave active?

Possess a permanent dipole moment Signup and view all the answers

Which classification applies to molecules based on the relative values of principal moments of inertia?

Linear molecules, symmetric tops, spherical tops, asymmetric tops Signup and view all the answers

What effect does rotation have on the bond in a non-rigid rotor model?

Rotation lengthens the bond due to centrifugal force. Signup and view all the answers

Study Notes

Rigid Rotor

The eigenvalues of a rigid rotor are a function of the rotational quantum number J.
The energy levels EJ of a rigid rotor are typically reported in units of cm⁻¹.
The rotational constant B is given by the formula B = ħ / (4πcI), where I is the moment of inertia and ħ is the reduced Planck constant.

Rotational Transition Selection Rules

The selection rules for rotational transitions in a rigid rotor are ΔJ = ±1.

Rotational Spectrum

The spacing between adjacent lines in the rotational spectrum of a rigid rotor is 2B.

Molecular Properties

The bond length for the molecule C≡O is 113 pm.

Non-Rigid Rotor Model

In the non-rigid rotor model, a new term is added to the rotational energy equation, causing the energy levels to be affected.
Rotation causes an increase in the bond length in a non-rigid rotor model.

Microwave Activity

A molecule must have a permanent electric dipole moment to be microwave active.

Moment of Inertia Classification

Molecules can be classified as prolate (Ia > Ib > Ic), oblate (Ia < Ib ≤ Ic), or spherical (Ia = Ib = Ic) based on the relative values of their principal moments of inertia.