Questions: The Rigid Rotor Model of Molecular Rotation

Question 1

A student argues: 'Because the rotational energy levels of a diatomic are not equally spaced, the absorption lines in its microwave spectrum must also be unevenly spaced.' Is this reasoning correct?

• A. Yes — unequal level spacing always produces unequal spectral line spacing
• B. No — the energy levels are actually equally spaced in the rigid rotor
• C. No — the energy levels are unequally spaced, but the spectral lines are equally spaced (separated by 2B) because the transition energies form an arithmetic sequence
• D. Partially — lines are equally spaced only for low values of J

Answer: (C)

The energy levels E_J = BJ(J+1) are not equally spaced — the gap between level J and J+1 is 2B(J+1), which grows with J. However, the microwave selection rule requires ΔJ = +1, so the observed transition frequencies are 2B, 4B, 6B, 8B, … These form a perfectly even progression, each line separated from the next by 2B. The spectral lines ARE equally spaced even though the energy levels are not. This is why measuring the line spacing directly gives you 2B.

Question 2

You measure the microwave spectrum of H₂ and D₂ (deuterium). The bond length of D₂ is essentially the same as H₂. How does the rotational constant B of D₂ compare to that of H₂?

• A. B is the same for both, since bond length determines I and the bond length is unchanged
• B. B is larger for D₂ because heavier atoms rotate faster
• C. B is smaller for D₂ because the larger reduced mass increases the moment of inertia, which decreases B
• D. B is larger for D₂ because the heavier nuclei require higher energy to rotate

Answer: (C)

The rotational constant B = ℏ/(4πcI) and the moment of inertia I = μr², where μ is the reduced mass. For D₂, each deuterium atom has roughly twice the mass of hydrogen, so the reduced mass μ ≈ doubles. Since r is unchanged, I doubles. Because B is inversely proportional to I, B roughly halves for D₂. This demonstrates why B depends on reduced mass — not just bond length. A student who only thinks about bond length will incorrectly predict no change.

Question 3

The degeneracy of rotational level J is 2J+1, meaning that J = 3 has seven distinct quantum states all at the same energy.

• T. True
• F. False

Answer: True

The magnetic quantum number M_J can take integer values from −J to +J, giving 2J+1 possible values. For J = 3, M_J ∈ {−3, −2, −1, 0, 1, 2, 3}, which is 7 states. In the absence of an external field, all have the same energy. This degeneracy matters for spectroscopy: higher-J levels contain more states, so more molecules can populate them (weighted by the Boltzmann factor), which affects the relative intensities of spectral lines and produces the characteristic intensity envelope seen in real microwave spectra.

Question 4

The spacing between adjacent rotational energy levels decreases as the quantum number J increases.

• T. True
• F. False

Answer: False

The energy gap between level J and level J+1 is E_{J+1} − E_J = 2B(J+1), which increases linearly with J. So the higher you go in J, the larger the energy gap between adjacent levels. This is the opposite of, say, a particle in a box or the harmonic oscillator (where levels are equally spaced). The increasing spacing is a direct consequence of the J(J+1) dependence of rotational energies, which itself comes from the quantization of angular momentum.

Question 5

A microwave spectrum of CO shows equally spaced absorption lines. Describe the steps you would take to extract the C–O bond length from this spectrum.

Model answer: Measure the spacing between adjacent lines; this equals 2B. Divide by 2 to get B in cm⁻¹. Use B = ℏ/(4πcI) to compute I = ℏ/(4πcB). Use I = μr² where μ = m_C m_O/(m_C + m_O) is the reduced mass of the CO molecule. Solve for r = √(I/μ).

The procedure works because every observable in a microwave spectrum is a direct consequence of the rigid rotor model. The line spacing encodes 2B; B encodes I (the moment of inertia); I encodes the bond geometry through I = μr². The reduced mass uses atomic masses from the periodic table, which are precisely known, so the only remaining unknown is r. This is why microwave spectroscopy is one of the most accurate methods for measuring bond lengths.