Question

The spacing between successive rotational energy levels of a diatomic molecule XY and its heavier isotopic analogue X'Y' varies with the rotational quantum number, J, as

• A.

  

• B.

  

• C.

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• D.

  

Answer 1

The Correct Option is D

To determine how the spacing between successive rotational energy levels of a diatomic molecule XY and its heavier isotopic analogue X'Y' varies with the rotational quantum number \( J \), we need to understand the formula for the rotational energy levels of a diatomic molecule.

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

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

where:

• \( E_J \) is the rotational energy level.
• \( J \) is the rotational quantum number.
• \( \hbar \) is the reduced Planck's constant.
• \( I \) is the moment of inertia of the molecule.

The moment of inertia \( I \) is defined as:

\(I = \mu r^2\)

where:

• \( \mu \) is the reduced mass of the diatomic molecule.
• \( r \) is the internuclear distance.

The spacing between successive rotational levels \( \Delta E \) is:

\(\Delta E = E_{J+1} - E_J = \frac{(J+1)(J + 2) \hbar^2}{2I} - \frac{J(J + 1) \hbar^2}{2I}\)

Upon simplifying, this becomes:

\(\Delta E = \frac{(2J + 3)\hbar^2}{2I}\)

For isotopes, the moment of inertia changes because the reduced mass \( \mu \) depends on the masses of the atoms. For a heavier isotope, \( I' > I \), which means \( \Delta E' < \Delta E \).

Now we can evaluate which option corresponds to this logic and behavior.

The correct answer is the one which acknowledges the decrease in spacing due to an increase in the moment of inertia for the heavier isotopic analogue. Hence, the correct option is the above image, which represents the decrease in rotational level spacing.

Therefore, the spacing decreases as the isotopic mass increases, demonstrating how they vary with the rotational quantum number, \( J \).