Rotational Spectra of Diatomic Molecules: quick revision, 15 bites

Rotation Is Quantised — That's Why Spectra Appear

Molecules don't spin freely — they spin in steps.

A rotating molecule can only occupy **discrete rotational energy levels**. Transitions between these levels absorb microwave radiation, producing a **line spectrum** rather than a continuous band.

Takeaway. Quantised rotation is the reason rotational spectra consist of sharp, discrete lines.

Rigid Rotor: The Core Model

Two masses, one fixed bond — the simplest spinner.

Takeaway. The rigid rotor model converts a real molecule into a calculable two-mass system.

Rotational Energy Levels of a Rigid Rotor

One formula unlocks every rotational line.

Takeaway. Energy grows as $J(J+1)$, so levels spread further apart as $J$ increases.

Selection Rule: Why Only ΔJ = ±1

Not every jump is allowed — only one step.

For a molecule to absorb a microwave photon, it must have a **permanent dipole moment** and obey the selection rule $\Delta J = \pm 1$.

Takeaway. No permanent dipole → no rotational spectrum, no matter how fast it spins.

Line Spacing Reveals the Bond Length

Equal gaps in the spectrum = one bond length.

Because the gap grows linearly with $J$, the spectrum shows **equally spaced lines** separated by $2B$.

Takeaway. Measure the spacing, get $B$, then get $I$ and the bond length.

Equal Spacing Is the Fingerprint

Equal gaps = rigid rotor signature.

Takeaway. If the spacing is constant, the rigid rotor model fits — and the molecule is not stretching.

Worked Example: HCl Rotational Spectrum

From line spacing to bond length in two steps.

Takeaway. Rotational spectra are a direct ruler for bond length.

Polar vs Non-Polar: The Dipole Trap

N₂ spins — but never shows a spectrum.

Takeaway. Rotation alone is not enough — a permanent dipole is mandatory.

Microwave-Active vs Microwave-Inactive

Same shape, opposite spectra.

Takeaway. Symmetry kills the dipole — and with it, the rotational spectrum.

Reduced Mass: The Hidden Variable

Heavier atoms — smaller steps.

Takeaway. Reduced mass links the spectrum to the actual atoms in the bond.

Remember the Selection Rule

One step up, one step down.

Think **"one J at a time"** — absorption takes $\Delta J = +1$, emission takes $\Delta J = -1$. No skipping levels.

Takeaway. ΔJ = ±1 is the single rule that shapes the entire spectrum.

Exam Pattern: Spacing → Bond Length

Spacing is the question, bond length is the answer.

Takeaway. Most numericals follow this exact five-step chain — memorise the order.

Where Rotational Spectroscopy Is Used

Astronomers read molecules across space.

Takeaway. Rotational spectra are a fingerprint tool for molecules we cannot touch.

Rigid Rotor Is an Approximation

Real bonds stretch — spectra drift.

Takeaway. Rigid rotor works well at low J — deviations signal bond flexibility.

What You Need Before This Topic

Three ideas unlock rotational spectra.

Takeaway. Comfort with quantisation and moments of inertia makes this topic click fast.