Wavenumber to Frequency Converter
Common Conversions
| cm⁻¹ | Hz |
|---|---|
| 1 | 29980000000 |
| 10 | 299800000000 |
| 100 | 2998000000000 |
| 500 | 14990000000000 |
| 1000 | 29980000000000 |
| 2000 | 59960000000000 |
| 3000 | 89940000000000 |
| 4000 | 119900000000000 |
| 5000 | 149900000000000 |
| 10000 | 299800000000000 |
| 20000 | 599600000000000 |
| 50000 | 1499000000000000 |
Why this conversion matters in chemistry
Wavenumber is what spectroscopy reports because it's proportional to energy and reads cleanly in the IR range. Frequency in hertz is what the underlying physics wants — photon energy is E = hν, force-constant calculations are built around vibrational frequencies, quantum-mechanical models return frequencies directly. Multiplying cm⁻¹ by the speed of light in cm/s (2.998 × 10¹⁰ cm/s) converts one to the other. A 1650 cm⁻¹ amide-I C=O stretch becomes 4.95 × 10¹³ Hz, or 49.5 THz. The step shows up every time a molecular dynamics simulation or ab initio calculation outputs vibrational frequencies and you need to compare them against an experimental spectrum.
Formula
Where the factor comes from
Alone in this family, the relation is a straight proportionality rather than a reciprocal. Wavenumber counts wave cycles per centimeter of path; frequency counts them per second of time; the speed of light is the exchange rate between path and time, so ν = c ν̃ with nothing else in the expression. The unit algebra is one prefix step: 299,792,458 m/s × 10² cm/m = 2.99792458 × 10¹⁰ cm/s, which gives 2.99792458 × 10¹⁰ Hz for every cm⁻¹ — 29.9792458 GHz, near enough 30 GHz for mental arithmetic. Since the meter is defined from a fixed value of c and centi is a defined prefix, the factor is exact to every digit. That exactness is what allows a rotational transition to be quoted interchangeably in megahertz and in cm⁻¹ without anyone losing sleep.
Precision and significant figures
Being linear, this conversion neither concentrates nor dilutes precision: the relative uncertainty on the wavenumber is the relative uncertainty on the frequency, at every point in the range. The instrument therefore decides the answer. A survey FTIR run at 4 cm⁻¹ resolution puts a 1000 cm⁻¹ band position within a few parts per thousand, so three significant figures in hertz is already generous. The same exact factor serves the opposite extreme without complaint — microwave and terahertz work measures rotational transitions to sub-kilohertz, something near 10⁻⁸ cm⁻¹, and converts them with no loss at all. Write the digits the measurement earned, and prefer terahertz to strings of zeros: 1000 cm⁻¹ is 29.98 THz.
Worked Examples
Mid-IR fingerprint region. Roughly 30 THz — a clean anchor worth remembering for spectral bookkeeping.
C–H stretching region. Any aliphatic spectrum shows peaks clustered here.
A generic carbonyl C=O stretch. The exact position shifts with chemical context — aldehydes near 1725, ketones 1715, esters 1735, amides 1650–1680 — which is what makes the band so diagnostic.
Far-IR / THz range. Low-energy lattice vibrations and some metal-ligand modes live here.
Common mistakes
Using c in meters per second
The wavenumber is per centimeter, so the speed of light has to be as well: 2.998 × 10¹⁰ cm/s, not 2.998 × 10⁸ m/s. The slip costs a factor of one hundred and hides well, because 1000 cm⁻¹ then emerges as 3.0 × 10¹¹ Hz rather than 3.0 × 10¹³ Hz, and 0.3 THz still reads like a believable spectroscopic frequency.
Ordinary frequency mistaken for angular frequency
ν and ω differ by 2π. Normal-mode analyses and force-constant expressions frequently carry the angular form in rad/s, and 1000 cm⁻¹ is 2.998 × 10¹³ Hz but 1.884 × 10¹⁴ rad/s. Because the force constant goes as the square, substituting one form where the other belongs shifts the result by a factor of about forty. Check which the expression assumes.
Harmonic frequencies compared with observed bands
A quantum-chemistry job returns harmonic wavenumbers, which sit above the observed fundamentals by a few percent because real potentials are anharmonic. Converting an unscaled harmonic value to hertz and setting it against an experimental band compares two different quantities. Apply the scaling appropriate to the method and basis set first, and state in the write-up that you did.