Frequency to Wavenumber Converter
Common Conversions
| Hz | cm⁻¹ |
|---|---|
| 29980000000 | 1 |
| 299800000000 | 10 |
| 2998000000000 | 100 |
| 14990000000000 | 500 |
| 29980000000000 | 1000 |
| 59960000000000 | 2000 |
| 89940000000000 | 3000 |
| 119900000000000 | 4000 |
| 149900000000000 | 5000 |
| 299800000000000 | 10000 |
| 599600000000000 | 20000 |
| 1499000000000000 | 50000 |
Why this conversion matters in chemistry
Frequency and wavenumber describe the same light, but spectroscopy strongly prefers wavenumber because it's proportional to photon energy. Raw frequency in hertz is what shows up in time-domain measurements, FTIR software outputs, or theoretical calculations — but by the time the result lands in a paper, it's in cm⁻¹. Divide the frequency in Hz by 2.998 × 10¹⁰ cm/s and you have your wavenumber. A C=O stretch at 5.1 × 10¹³ Hz becomes 1700 cm⁻¹, which is the number every organic chemistry textbook quotes. After enough spectra you stop thinking of them as two different scales; you just read one off the other.
Formula
Where the factor comes from
The defining relation is ν̃ ≡ 1/λ, and everything awkward about this conversion follows from the centimeter being welded into the unit symbol. Since ν = c/λ, dividing a frequency by the speed of light returns the reciprocal wavelength — but the speed has to be carried in centimeters per second, 2.997 924 58 × 10¹⁰ cm/s, rather than the meters per second that constant usually wears. Invert it and the multiplier is 3.335 640 95 × 10⁻¹¹ cm⁻¹ per hertz. Read the same relation the other way and one wavenumber is exactly 29.979 245 8 GHz, the form microwave and terahertz work tends to prefer. The speed of light is exact by definition, so the ratio is exact too. Its decimal expansion simply never terminates, because 299 792 458 carries a large odd factor that no power of ten will clear.
Precision and significant figures
Six figures on the multiplier, 3.335 64 × 10⁻¹¹, sit far beyond anything a vibrational spectrum can use; the truncation error is about three parts in 10⁷ and no band on a routine trace is defined to better than a whole wavenumber. Set expectations from the interferometer instead. An FTIR samples its interferogram against an internal helium-neon laser, which makes the wavenumber scale itself remarkably stable — but a spectrum collected at 4 cm⁻¹ resolution cannot support a peak position quoted to two decimals, however confidently the peak-picking routine prints them. Quote band positions to the resolution actually used, and keep extra digits only when the scan was taken at high resolution on a band narrow enough to justify them.
Worked Examples
Mid-IR, right in the fingerprint region. About 30 THz — a clean anchor worth remembering for spectral bookkeeping.
The C–H stretching region. Any aliphatic spectrum has peaks clustered around here.
A generic carbonyl C=O stretch. Exact position shifts with context — aldehydes sit near 1725 cm⁻¹, ketones 1715, esters 1735, amides 1650–1680 — and that sensitivity is exactly what makes the band so diagnostic.
Far-IR, or what physics labs call the terahertz range. Low-energy lattice vibrations and some metal-ligand stretches live out here.
Common mistakes
c left in meters per second
Dividing a frequency in hertz by 2.998 × 10⁸ returns reciprocal meters, not reciprocal centimeters, and the answer lands a hundredfold too large. A C–H stretch emerges near 300 000 instead of 3000 — obvious once seen, easy to miss inside a spreadsheet column. The unit in the symbol dictates the unit in the constant, so carry c as 2.998 × 10¹⁰ cm/s whenever the target axis is cm⁻¹.
Wavenumber called a frequency
Spectroscopy routinely calls a cm⁻¹ value a frequency and sometimes writes it as ν rather than ν̃. It is a reciprocal length. The habit costs nothing in conversation and a great deal in a calculation, where a quantity labeled as a frequency of 1700 fed into E = hν returns a photon energy short by the speed of light — a factor of roughly 3 × 10¹⁰.
Micrometer wavelength inverted without converting
The other common infrared axis is wavelength in micrometers, and ν̃ = 1/λ holds only when λ is expressed in centimeters. A 10 µm band is 10⁻³ cm and therefore 1000 cm⁻¹; inverting the bare 10 gives 0.1, four decades adrift. Any inversion on this axis needs the length taken to centimeters first, whatever prefix it happened to arrive wearing.