Quiz on Rotational Energy Level Degeneracy

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Questions and Answers

What is the degeneracy of each rotational energy level (J)?

J(J + 1)

2J + 1 (correct)

J^2

What is the energy difference between two adjacent rotational energy levels?

2hB(J + 1) (correct)

hB(2J + 1)

hB(4J + 2)

hB(J + 1)

What is the moment of inertia (I) for a rigid diatomic molecule?

μr

μr^2

μ/r

μr^2 / (m1 + m2) (correct)

Why is rotation around the bond axis for a rigid diatomic molecule forbidden?

Because the moment of inertia is too small Signup and view all the answers

What is the expression for the energy of a rotational state (εJ)?

hJ(J + 1) / 8π^2I Signup and view all the answers

What is the frequency of the rotational transition J → J + 1?

2B(J + 1) Signup and view all the answers

Why are the energy levels extremely widely spaced for a rigid diatomic molecule?

Because the moment of inertia is very small Signup and view all the answers

What is the expression for the rotational constant (B)?

h / 8π^2I Signup and view all the answers

What is the total number of degrees of freedom for a molecule with N atoms?

3N Signup and view all the answers

How many degrees of freedom are present in a linear molecule's rotational movement?

2 Signup and view all the answers

What type of energy is described by the equation 𝐸𝑡𝑟𝑎𝑛𝑠. = ℎ²/8𝑚𝑎²?

Translational energy Signup and view all the answers

What is the number of vibrational degrees of freedom for a nonlinear molecule?

3N - 6 Signup and view all the answers

What is the significance of the equation 𝐾.𝐸 𝑏𝑦 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦 = 3/2 𝐾 𝑇 / 𝐵?

Relates thermal energy to temperature Signup and view all the answers

How many degrees of freedom are present in a molecule's translational movement?

3 Signup and view all the answers

What type of movement is described by the equation 𝐸𝑡𝑟𝑎𝑛𝑠. = ℎ²(n_x² + n_y² + n_z²) / 8𝑚𝑎²?

Translational energy in three dimensions Signup and view all the answers

How many degrees of freedom are present in a linear molecule's vibrational movement?

3N - 5 Signup and view all the answers

What is the reason for the lack of interaction between linear molecules and electromagnetic radiation during rotation around the bond axis?

There is no change in the dipole moment component during rotation Signup and view all the answers

What is the selection rule for allowed transitions in rotational spectroscopy?

ΔJ = ±1 Signup and view all the answers

What is the effect of isotopic substitution on a molecule's rotation constant (B)?

Decrease in B Signup and view all the answers

What is the condition for a molecule to interact with electromagnetic radiation during rotation?

The molecule must possess a permanent dipole moment Signup and view all the answers

What is the mathematical expression for the rotational energy level with maximum population?

𝐸𝐽𝑚𝑎𝑥. = 𝐾𝐵 𝑇 / 2ℎ𝐶𝐵 Signup and view all the answers

Why do transitions from J=0 to J=1 not occur under normal spectroscopic conditions?

The molecule requires a great deal of energy to be raised from J=0 to J=1 Signup and view all the answers

What is the result of the selection rule ΔJ = ±1 in rotational spectroscopy?

Only transitions with ΔJ = ±1 are allowed Signup and view all the answers

What happens to the internuclear distance in a molecule when an atom is replaced by its isotope?

The internuclear distance remains unchanged Signup and view all the answers

Study Notes

Rotational Energy Levels of a Rigid Diatomic Molecule

The degeneracy of state (gJ) for each rotational energy level (J) is given by gJ = 2J + 1.
The energy levels are given by εJ = ℎJ(J+1) / 8π2IC.
Rotation around the bond axis for a rigid diatomic molecule is forbidden.

Moment of Inertia (I) and Rotation Constant (B)

The moment of inertia (I) is very small (~ 0) around the bond.
I = μr2, where μ = (m1m2) / (m1 + m2) and r is the internuclear distance.
The rotation constant (B) is given by B = ℎ / 8π2IC.

Energy Levels and Transitions

The energy level spacing is extremely wide due to the small moment of inertia.
The allowed transitions obey the selection rule ΔJ = ±1.
Only transitions in which J changes by one unit are spectroscopically allowed.

Degrees of Freedom in a Molecule

A molecule has a total of 3N degrees of freedom, where N is the number of atoms.
The degrees of freedom can be broken down into three categories: translational, rotational, and vibrational.
Translational and rotational movements have 3 degrees of freedom each, while vibrational movements have 3N - 6 (non-linear) or 3N - 5 (linear) degrees of freedom.

Rotational Energy Level with Max. Population

The maximum population of a rotational energy level is given by Jmax = √(KBT / 2ℎB) - 1/2.

Selection Rule and Dipole Moment

The selection rule dictates that the rotational quantum number has to change by unity during emission or absorption.
The molecule must possess a permanent dipole moment for rotational spectroscopy.

The Effect of Isotopic Substitution

Isotopic substitution changes the atomic and molecular mass, resulting in changes to the moment of inertia (I) and the rotation constant (B).
The internuclear distance remains relatively unchanged.

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Description

This quiz is about the formula for the degeneracy of rotational energy levels in quantum mechanics, where gJ is the degeneracy of the Jth rotational energy level.

Rotational Energy Level Degeneracy

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