Question

If the energy of 1.00 mole of photons is 441 kj, what is the wavelength of the light?

Answer 1

1 mol of photons contained a number of photons equal to Avogadro number:
$N=N_A = 6.022 \cdot 10^{23}$
The total energy of the mole of photons is $E=441 kJ=4.41 \cdot 10^5 J$, so the energy of a single photon is the total energy divided by the number of photons:
$E_1 = \frac{E}{N} = \frac{4.41 \cdot 10^5 J}{6.022 \cdot 10^{23}} =7.32 \cdot 10^{-19}J$

The energy of a single photon is related to its frequency f:
$E_1 = hf$
where h is the Planck constant. From this formula, we find the frequency of the photons in the problem:
$f= \frac{E_1}{h}= \frac{7.32 \cdot 10^{-9} J}{6.6 \cdot 10^{-34}Js} =1.1 \cdot 10^{15} Hz$

And from the frequency we can finally calculate the wavelength $\lambda$, using the relationship between wavelength, frequency and speed of light (c) for photons:
$\lambda= \frac{c}{f}= \frac{3 \cdot 10^8 m/s}{1.1 \cdot 10^{15} Hz} =2.73 \cdot 10^{-7} m = 273 nm$