--- title: "Reading Bond Lengths and Force Constants from Spectra" description: "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" canonical: https://lightmysky.com/learn/science/reading-bond-lengths-and-force-constants-from-spectra-mt_5m7J1mFw68 source: https://lightmysky.com/learn/science/reading-bond-lengths-and-force-constants-from-spectra-mt_5m7J1mFw68.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # 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. Subject: Science · Area: Chemistry · Ages 19 to 20 Page: https://lightmysky.com/learn/science/reading-bond-lengths-and-force-constants-from-spectra-mt_5m7J1mFw68 ## Ready when they can - 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 ## Lesson: Turning spectra into bond distances 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. **Example.** 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. **Recap.** Spacing gives the geometry through B and I, the band centre gives the stiffness, and the line measurement sets the precision. ## Practice 8 questions on this page, each with its working shown. ## Needs first - [Selection Rules and Why Some Transitions Never Appear](https://lightmysky.com/learn/science/selection-rules-and-why-some-transitions-never-appear-mt_oJDr81W4fW) ## Optional connections - [Mass Spectra and Infrared: Identifying an Unknown](https://lightmysky.com/learn/science/mass-spectra-and-infrared-identifying-an-unknown-mt_yotOEg9OTn) ## Opens up - [Organometallic Ligands: Carbonyls, Phosphines and Hapticity](https://lightmysky.com/learn/science/organometallic-ligands-carbonyls-phosphines-and-hapticity-mt_JXYSQDYlJ-) - [Electronic Spectra, Franck-Condon and the Origin of Colour](https://lightmysky.com/learn/science/electronic-spectra-franck-condon-and-the-origin-of-colour-mt_y1hKy9nvAY) - [Choosing Between Vibrational Methods: Infrared, Raman and Resonance Raman](https://lightmysky.com/learn/science/choosing-between-vibrational-methods-infrared-raman-and-resonance-raman-mt_YxLAvBIBAR)