What is the energy of one mole of photons?

The energy \( W_{\text p} \) of a single photon is given by the following quantum hypothesis:

Here \( c ~=~ 3 \cdot 10^8 \, \frac{\mathrm m}{\mathrm s} \) is the speed of light and \( h ~=~ 6.6 \,\cdot\, 10^{-34} \, \mathrm{Js} \) is the Planck's constant. The energy \(W_{\text p}\) of a photon depends only on the wavelength \( \lambda \) of the light.

The energy of one mole of photons, let us call it \(W_{\text{mol}}\), is the energy \(W_{\text p}\) of a single photon multiplied by the number of photons per mole. The Avogadro constant \(N_{\text A} = 6 \cdot 10^{23} \, \frac{1}{\mathrm{mol}} \) provides us with the number of photons per mole. Therefore, the photon energy per mole is given by:

Substitute equation 1 into 2:

So if you insert a concrete wavelength \(\lambda\) into Eq. 3, you get the energy of \( 6 \cdot 10^{23} \) photons, which just form one mole.

You want to know what is the photon energy per mole \( W_{\text{mol}} \) of a red light that has wavelength \( \lambda = 780 \, \mathrm{nm} \). This corresponds to \( \lambda = 780 \cdot 10^{-9} \, \mathrm{m} \). Insert the wavelength into Eq. 3:

Example: Photon energy of one mole

Formula anchor

You can express the energy \( W_{\text p} \) of a photon with the light frequency \(f\):

Thus, the photon energy per mole can also be calculated in the following way: