Molecular Motion Quantised: The Oscillator and the Rotor
Molecules store energy in vibration and rotation as well as in electrons. Two solvable models, a spring between two masses and a rigid dumbbell, give the level patterns that every molecular spectrum is read against.
What a learner can do afterwards
- Writes the vibrational level pattern as evenly spaced and the rotational pattern as widening with quantum number
- Relates vibrational spacing to force constant and reduced mass, and predicts the shift on deuterium substitution
- Relates rotational spacing to bond length through the moment of inertia
- Explains what zero-point energy is and why a bond cannot be still even at absolute zero
1 · Read
A chemical bond vibrates like a spring between two masses. Stiffer springs and lighter masses oscillate faster, with the frequency set by the square root of stiffness over mass. Quantum rules allow only evenly spaced levels, written E = (v + 1/2) h f. Each step up the ladder costs the same packet.
Swap hydrogen for deuterium in H-Cl and the band moves down. The force constant stays the same, but the reduced mass rises from about 0.97 to 1.89 atomic mass units. Frequency goes as one over the square root of reduced mass, so D-Cl vibrates at about the square root of 0.97 / 1.89, roughly 0.72 times the H-Cl value.
A spinning molecule is a rigid dumbbell with levels E = B J(J + 1). Unlike vibration the gaps widen as J climbs, running 2B, then 4B, then 6B. Measure that spacing to get B, convert B to the moment of inertia I, and solve I = mu r squared for the bond length r.
An even level ladder means vibration. A widening one means rotation, and since each gap grows by 2B, neighbouring rotational lines sit a steady 2B apart. Wider line spacing means larger B, smaller I, and a shorter bond for equal masses. The half quantum at v = 0 is the zero point energy, so a bond never sits still, even at absolute zero.
Vibration climbs an even ladder set by stiffness and mass, while rotation spreads wider with J and reveals the bond length.
The rewritten readings were written for this dataset.
2 · Watch
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.