How do I convert reciprocal centimeters to nanometers? cm^-1 to nm? Example: 2100 cm^-1 = 476 nm, but HOW do I get there?
The Correct Answer and Explanation is:
To convert from reciprocal centimeters (cm⁻¹) to nanometers (nm), you are essentially converting a wavenumber (a measure of frequency, used in spectroscopy) into a wavelength (the physical length of a wave). Here’s how it works:
✅ Correct Conversion Formula:
λ(nm)=107ν~(cm−1)\lambda(\text{nm}) = \frac{10^7}{\tilde{\nu}(\text{cm}^{-1})}λ(nm)=ν~(cm−1)107
Where:
- λ\lambdaλ is the wavelength in nanometers (nm)
- ν~\tilde{\nu}ν~ is the wavenumber in cm⁻¹
✏️ Step-by-Step Example:
Convert 2100 cm⁻¹ to nm:λ=1072100=4761.9 nm≈476 nm\lambda = \frac{10^7}{2100} = 4761.9 \, \text{nm} \approx 476 \, \text{nm}λ=2100107=4761.9nm≈476nm
📘 Textbook-Style Explanation:
In spectroscopy, energy is often described using wavenumber, which is the number of wave cycles per centimeter (cm⁻¹). This is an inverse unit — the larger the wavenumber, the shorter the wavelength. To convert this to a wavelength in nanometers, we use the fundamental relationship between wavenumber and wavelength:ν~=1λ(cm)\tilde{\nu} = \frac{1}{\lambda(\text{cm})}ν~=λ(cm)1
Rearranging gives:λ(cm)=1ν~\lambda(\text{cm}) = \frac{1}{\tilde{\nu}}λ(cm)=ν~1
But we want the wavelength in nanometers, not centimeters. Since:1 cm=107 nm1 \, \text{cm} = 10^7 \, \text{nm}1cm=107nm
We multiply both sides by 10710^7107 to get:λ(nm)=107ν~(cm−1)\lambda(\text{nm}) = \frac{10^7}{\tilde{\nu}(\text{cm}^{-1})}λ(nm)=ν~(cm−1)107
This is the key formula used in infrared (IR) and UV/Vis spectroscopy. It allows scientists to switch between energy values (in cm⁻¹) and physical wavelength (in nm), which is essential for analyzing and interpreting spectra.
For example, 2100 cm⁻¹ corresponds to a wavelength of approximately 476 nm, placing it in the visible to near-infrared region of the spectrum.
