# What is the energy of one mole of photons?

Level 2 (without higher mathematics)

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

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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:

Photon energy per mole is the product of the Avogardo constant with the energy of a photon
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Substitute equation 1 into 2:

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

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

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