Lesson 5 · 30 min

Infrared spectroscopy: molecular vibrations

Bonds are springs and infrared spectra record their vibrations. Why does CO₂ absorb infrared while N₂ does not? Why are symmetric stretches invisible? And how do group frequencies reveal what a molecule contains?

In this lesson

Link a vibration’s wavenumber to bond stiffness and atomic masses with the harmonic model ν̃ = (1/2πc)√(k/μ)
Count a molecule’s vibrations (3N − 6 or 3N − 5) and describe stretches and bends
Apply the selection rule — only vibrations that change the dipole moment absorb infrared — and explain IR-silent symmetric stretches and the complementary Raman spectrum
Use group-frequency regions and the fingerprint region to identify functional groups and carbonate, sulfate and nitrate

N₂ and O₂ make up 99 % of the air and are completely transparent to the infrared radiation the Earth gives off; CO₂, at about 0.04 %, and a little water vapour absorb a great deal of it — that is why there is a greenhouse effect. The same physics lets chemists “see” the skeleton of a molecule: an infrared spectrum records how a molecule vibrates. But why does N₂ not absorb while CO₂ does? Neither has a permanent dipole moment.

Predict first

Which of these gases absorbs infrared light?

Pick an answer and lock it in before you look. A wrong guess teaches more than no guess.

All of them: N₂, O₂, CO₂, Ar Only CO₂ N₂ and O₂, but not CO₂ (it has no dipole) None of them

Bonds are springs

Treat the bond between two atoms as a spring: the stiffer the bond (the larger its force constant k) and the lighter the atoms (the smaller the reduced mass μ), the faster it vibrates. Infrared spectra are plotted in wavenumber ν̃ (waves per centimetre, cm⁻¹), which is proportional to frequency and energy; wavelength follows from λ/µm = 10⁴/(ν̃/cm⁻¹). The mid-infrared runs from about 4000 to 400 cm⁻¹ (2.5–25 µm).

\tilde\nu = \frac{1}{2\pi c}\sqrt{\frac{k}{\mu}},\qquad \mu = \frac{m_1 m_2}{m_1 + m_2}

HCl: ωₑ = 2991 cm⁻¹ corresponds to k ≈ 516 N/m; anharmonicity puts the observed band at 2886 cm⁻¹.

$\tilde\nu$: vibrational wavenumber, cm⁻¹
$c$: speed of light (in cm/s)
$k$: force constant of the bond, N/m
$\mu$: reduced mass

A molecule of N atoms has 3N degrees of freedom; subtract 3 translations and 3 rotations (only 2 for a linear molecule) and 3N − 6 (3N − 5 if linear) normal vibrations remain: 3 for H₂O, 4 for CO₂ (its bend is doubly degenerate), 9 for CH₄. Vibrations are stretches (symmetric or asymmetric), which change bond lengths, and bends, which change angles.

Selection rule: a vibration absorbs infrared light only if it changes the dipole moment; it appears in the Raman spectrum only if it changes the polarisability. For molecules with a centre of symmetry, such as CO₂, no vibration is both IR and Raman active (the rule of mutual exclusion), so the two spectra complement each other. Highly symmetric ions behave the same way: the symmetric stretches ν₁ of CO₃²⁻, NO₃⁻ and SO₄²⁻ are invisible in the infrared yet are the strongest lines of their Raman spectra.

Normal modes · the dipole rule · group frequencies

Infrared spectra: molecules in motion

IR inactive · For three-dimensional species the IR activity comes from the symmetry tables (the arrows only sketch the motion).

Normal modes (9 = 3N − 6)

Show organic group regions

Compare the oxo-anions: SO₄²⁻ absorbs only at 1105 and 611 cm⁻¹ (two triply degenerate modes), while CO₃²⁻ and NO₃⁻ each show three bands, the strongest at 1415 or 1390 cm⁻¹. Then choose H₂O, NH₃ and CH₄: their X–H stretches all lie above 3000 cm⁻¹. Turn on the organic group regions and match them with the O–H, N–H and C–H bands.

Fundamental vibrations of three oxo-anions (cm⁻¹): IR = infrared active, R = Raman active
Ion	ν₁ sym. stretch	ν₂	ν₃ asym. stretch	ν₄
$\ce{CO3^2-}$	1063 (R)	879 (IR)	1415 (IR, R)	680 (IR, R)
$\ce{NO3-}$	1050 (R)	831 (IR)	1390 (IR, R)	720 (IR, R)
$\ce{SO4^2-}$	983 (R)	450 (R)	1105 (IR, R)	611 (IR, R)

Source: K. Nakamoto, Infrared and Raman Spectra of Inorganic and Coordination Compounds, Part A, 6th ed. (2009), ch. 2.

Although a vibration involves the whole molecule, X–H, C=O and C≡N vibrations stay largely confined to one bond and appear at similar wavenumbers in every molecule — group frequencies: O–H 3200–3600 cm⁻¹ (broad and rounded when hydrogen-bonded), N–H 3300–3500, C–H 2850–3100 (above 3000 for alkenes and aromatics), C≡N and C≡C 2100–2260, C=O 1670–1780 (ketones about 1715), C=C 1600–1680 and C–O 1000–1300. Below 1500 cm⁻¹ lies the fingerprint region, vibrations of the whole skeleton that differ for every compound and are matched against reference spectra.

Worked example

Force constants of C=O and C–O

A ketone’s C=O stretch is at 1715 cm⁻¹ and an alcohol’s C–O stretch near 1100 cm⁻¹. Treat each as an isolated diatomic oscillator (m_C = 12.000 u, m_O = 15.995 u) and estimate both force constants.

Concept check

The infrared spectrum of calcite (solid calcium carbonate) shows nothing at 1086 cm⁻¹ — yet that is the strongest line in its Raman spectrum (the same vibration of free CO₃²⁻ in solution is at 1063 cm⁻¹). Why?

1086 cm⁻¹ is outside an infrared instrument’s range It is the symmetric stretch of CO₃²⁻: all three oxygens move out together, the dipole stays zero, so it is IR inactive Carbonate does not vibrate in a solid Ca²⁺ absorbs light of that wavenumber

Try the calculation

H³⁵Cl absorbs at 2886 cm⁻¹. With the same force constant, where does D³⁵Cl absorb? (Atomic masses: H 1.0078 u, D 2.0141 u, ³⁵Cl 34.969 u.)

Key takeaways

ν̃ = (1/2πc)√(k/μ): stiffer bonds and lighter atoms vibrate at higher wavenumbers; isotopic substitution shifts bands as √μ.
An N-atom molecule has 3N − 6 (linear 3N − 5) normal vibrations: stretches and bends.
Only vibrations that change the dipole moment absorb infrared; those that change the polarisability appear in the Raman spectrum; with a centre of symmetry the two are mutually exclusive.
Group frequencies (O–H, N–H, C–H, C=O …) identify functional groups; the fingerprint region below 1500 cm⁻¹ identifies the whole molecule.

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Data and references

T. Shimanouchi, Tables of Molecular Vibrational Frequencies, Consolidated Volume I, NSRDS-NBS 39 (1972), via the NIST Chemistry WebBook.
K. Nakamoto, Infrared and Raman Spectra of Inorganic and Coordination Compounds, Part A, 6th ed., Wiley (2009).
S. Gunasekaran, G. Anbalagan, S. Pandi, “Raman and infrared spectra of carbonates of calcite structure”, J. Raman Spectrosc. 37, 892 (2006) (calcite ν₁ near 1086 cm⁻¹, Raman only).
K. P. Huber, G. Herzberg, Constants of Diatomic Molecules (1979), via the NIST Chemistry WebBook (HCl ωₑ, ωₑxₑ).
D. L. Pavia, G. M. Lampman, G. S. Kriz, J. R. Vyvyan, Introduction to Spectroscopy, 5th ed., Cengage (2015), ch. 2.
P. Atkins, J. de Paula, J. Keeler, Atkins’ Physical Chemistry, 11th ed., OUP (2018), topics 11C–11D.