Reading Bond Lengths and Force Constants from Spectra
Rotational and vibrational spectra are measured in wavenumbers and converted into molecular quantities. Line spacings give the moment of inertia and so the bond length; the band centre gives the force constant.
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
به انگلیسیTake 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.