Rotational Spectrum of a Rigid Diatomic Rotor

A rigid diatomic rotor refers to a molecule consisting of two atoms that are connected by a rigid bond, allowing the molecule to rotate freely around its center of mass. The rotational spectrum of such a molecule refers to the set of spectral lines observed when the molecule undergoes transitions between different rotational energy levels.

Equally Spaced Lines in the Rotational Spectrum

The rotational spectrum of a rigid diatomic rotor consists of equally spaced lines. The spacing between these lines is related to the rotational constant, denoted as B, which is a characteristic property of the molecule. The rotational constant is defined as the energy difference between two adjacent rotational energy levels.

Explanation of the Correct Answer (Option D)

The correct answer is option D, which states that the spacing between the lines in the rotational spectrum is equal to 2B.

This can be understood by considering the energy levels of a rigid diatomic rotor. The rotational energy levels of a diatomic molecule are given by the expression:

E(J) = BJ(J+1)

Where E(J) is the energy of the Jth rotational level, and J is the rotational quantum number. The rotational constant B determines the energy spacing between adjacent levels.

Now, let's consider the energy difference between two adjacent rotational levels, for example, between J and J+1:

ΔE = E(J+1) - E(J)
= B(J+1)(J+2) - BJ(J+1)
= B(J+1)[(J+2) - J]
= B(J+1)(J+2 - J)
= B(J+1)(2)

From this equation, we can see that the energy difference between adjacent rotational levels is proportional to B multiplied by (J+1)(2). This means that the spacing between the lines in the rotational spectrum is also proportional to B multiplied by (J+1)(2).

Since the spacing between adjacent levels is constant throughout the spectrum, it implies that the value of (J+1)(2) is constant. Therefore, the spacing between the lines in the rotational spectrum is equal to 2B.

Hence, the correct answer is option D, which states that the spacing between the lines in the rotational spectrum of a rigid diatomic rotor is equal to 2B.