Rotational Energy Level Degeneracy (2024)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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