A microwave spectrum of a diatomic molecule shows equally spaced absorption lines separated by 20.0 GHz. A student reports the rotational constant B as 20.0 GHz. What is wrong?
ANothing — the line spacing directly equals B
BThe spacing equals 2B, so B = 10.0 GHz; the student failed to divide by two
CThe spacing equals B/2, so B = 40.0 GHz; the student should have multiplied by two
DB cannot be determined from line spacing alone — you also need the transition frequencies
For a rigid diatomic rotor, the transition frequency from J to J+1 is ν = 2B(J+1). The first line (J=0→1) is at 2B, the second at 4B, so adjacent lines are separated by 2B — not B. Dividing the observed spacing by 2 gives B. This is the most common numerical error in rotational spectroscopy problems.
Question 2 Multiple Choice
Why does N₂ show no microwave absorption spectrum despite being a rotating diatomic molecule with well-defined rotational energy levels?
AN₂ rotational energy levels are too closely spaced for microwave radiation to resolve
BN₂'s rotational constant B is zero because both atoms have equal mass
CN₂ is homonuclear and has no permanent dipole moment, so microwave radiation has no oscillating electric field component to couple to rotational transitions
DN₂ absorbs in the infrared rather than the microwave region
The selection rule requires a permanent dipole moment: microwave radiation interacts with a molecule's rotating electric dipole. Homonuclear diatomics have identical atoms, so there is no charge separation and no dipole. The oscillating electric field of the microwave has nothing to 'grip.' N₂ has well-defined rotational levels but is microwave-inactive. HCl, CO, and other heteronuclear diatomics are microwave-active.
Question 3 True / False
For a rigid diatomic rotor, the rotational constant B can be extracted directly from the spacing between adjacent lines in the microwave absorption spectrum.
TTrue
FFalse
Answer: True
Adjacent lines are separated by 2B (since ν = 2B(J+1) and consecutive J values differ by 1). Measuring any adjacent pair and dividing by 2 gives B immediately. From B = h/(8π²Ic), you can extract the moment of inertia I, and from I = μr² you get the bond length r. This chain makes microwave spectroscopy the most precise structural technique for simple diatomics.
Question 4 True / False
At high rotational quantum numbers J, centrifugal distortion causes the spacing between adjacent microwave absorption lines to increase.
TTrue
FFalse
Answer: False
Centrifugal distortion causes the spacing to decrease at high J. As the molecule spins faster at higher J, centrifugal force stretches the bond, increasing the moment of inertia and decreasing the effective rotational constant. The corrected energy expression adds −D_J[J(J+1)]², which progressively reduces the transition frequencies at high J, compressing the line spacing.
Question 5 Short Answer
Explain how a microwave spectrum of a diatomic molecule yields the bond length. What measurements and calculations are required?
Think about your answer, then reveal below.
Model answer: Measure the spacing between any two adjacent absorption lines in the spectrum; this spacing equals 2B. Divide by 2 to get the rotational constant B. Use B = h/(8π²Ic) to calculate the moment of inertia I. For a diatomic, I = μr², where μ = m₁m₂/(m₁+m₂) is the reduced mass (known from atomic masses). Solving for r gives the bond length.
The directness of this chain — spectrum → B → I → r — is what makes rotational spectroscopy uniquely powerful for structural determination. Unlike vibrational or electronic spectroscopy, the bond length drops out with minimal modeling assumptions, giving precision to five or six significant figures.