Planck–Einstein Relations for Photons
A photon’s wavelength \( \lambda \), frequency \( \nu \) and energy \(E\) are tied together by
\[
\boxed{\,E = h\nu = \dfrac{hc}{\lambda}\,}, \qquad
c = 299\,792\,458\ \mathrm{m\,s^{-1}},\quad
h = 6.62607015\times 10^{-34}\ \mathrm{J\,s}.
\]
From this, knowing any one of \( \lambda \), \( \nu \), or \(E\) determines the other two:
\( \lambda = \dfrac{hc}{E} = \dfrac{c}{\nu}\) and \( \nu = \dfrac{c}{\lambda} = \dfrac{E}{h}\).
Per-photon and per-mole energies
The calculator reports energy both per photon and per mole. Using Avogadro’s constant
\( N_{\mathrm A} = 6.02214076\times 10^{23}\ \mathrm{mol^{-1}} \),
\[
E_{\text{molar}} = N_{\mathrm A}\,E_{\text{photon}}.
\]
In electronvolts, \(1\ \mathrm{eV} = 1.602176634\times 10^{-19}\ \mathrm{J}\).
Wavenumber (spectroscopy)
Infrared and Raman spectroscopy often use the wavenumber \( \tilde{\nu} \) (in \( \mathrm{cm^{-1}} \)):
\[
\tilde{\nu} = \frac{1}{\lambda\ (\mathrm{cm})}, \qquad
E_{\text{photon}} = hc\,\tilde{\nu}\,(100\ \mathrm{m^{-1}\,cm}).
\]
Handy conversions (exact or to standard precision):
- \(hc = 1.98644586\times10^{-25}\ \mathrm{J\,m} \approx 1240\ \mathrm{eV\,nm}\).
- \(N_{\mathrm A}hc = 0.11962656\ \mathrm{J\,m\,mol^{-1}}
= 11.962656\ \mathrm{J\,cm\,mol^{-1}}\).
- \(\displaystyle E_{\text{molar}}(\mathrm{kJ\,mol^{-1}}) \approx 0.01196266 \times \tilde{\nu}\ (\mathrm{cm^{-1}}).\)
- \(\displaystyle E(\mathrm{eV}) \approx \dfrac{1240}{\lambda(\mathrm{nm})}.\)
Units used in the tool
- \(\lambda\): m, μm, nm, Å (\(\text{Å}=10^{-10}\ \mathrm{m}\)).
- \(\nu\): Hz, MHz, GHz, THz.
- \(E\): J and eV per photon; J·mol⁻¹ and kJ·mol⁻¹ per mole.
- \(\tilde{\nu}\): cm⁻¹.
How to use the calculator
- Select the known quantity (wavelength, frequency, energy per photon, energy per mole, or wavenumber) and its unit.
- Enter a single value and press Calculate.
- The tool converts the input to SI, applies \(E=h\nu=\dfrac{hc}{\lambda}\), and reports all related quantities:
\( \lambda \), \( \nu \), \( \tilde{\nu} \), \(E\) per photon, and \(E\) per mole.
Note on media: The relations above assume vacuum. In a material of refractive index \(n\),
the phase speed is \(v=c/n\). The frequency remains the same while the wavelength becomes
\( \lambda_{\text{medium}}=\lambda_{\text{vac}}/n \).
