What a learner can do afterwards
- Extracts the rotational constant from evenly spaced line spacings and converts it to a bond length
- Converts a vibrational wavenumber into a force constant using the reduced mass
- Identifies the P and R branches of a vibration-rotation band and explains the gap between them
- States which measured quantity limits the precision of the derived bond length
1 · Read
Microwave lines sit evenly spaced by 2B. With B in wavenumbers and c in cm/s, the moment of inertia is I = h / (8 pi squared c B), and I = mu r squared gives the bond length. Carbon monoxide lines sit 3.86 cm^-1 apart, so B = 1.93 cm^-1 and I = 1.45e-46 kg m squared. With mu = 1.14e-26 kg, r = square root of (I / mu) = 113 pm.
Infrared bands report stiffness: at the same mass, a higher wavenumber means a stiffer bond, so triple bonds ring highest. The force constant is k = mu (2 pi c times the wavenumber) squared. Carbon monoxide absorbs at 2143 cm^-1, so k = 1.14e-26 x (2 pi x 3.00e10 x 2143) squared, about 1.86e3 N/m. Heavier isotopes shift the band down as the reduced mass predicts, confirming the assignment.
A vibration-rotation band has two wings around a gap. The R branch above the band origin comes from delta J = +1, the P branch below from delta J = -1. The gap sits where the forbidden delta J = 0 line would land, so number lines outward from it to extract B.
Precision in r is set by how well the line centres are pinned down, since spacing errors feed through B and I into r. Quote only the figures that measurement supports.
Spacing gives the geometry through B and I, the band centre gives the stiffness, and the line measurement sets the precision.
The rewritten readings were written for this dataset.
2 · Watch
Take it off screen
Where it sits
Optional connections
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.