Question

what is the wavelength, in nanometers, of light with an energy content of 4.84×10² kj mol⁻¹?
hint: the unit mol⁻¹ should be interpreted as \per mole of photons\.

enter your answer with three significant figures.
enter scientific notation as 1.23e4.
do not include units.

λ = number nm

Explanation:

Step1: Calculate energy per photon

The energy of one mole of photons \(E = 4.84\times10^{2}\text{ kJ/mol}=4.84\times 10^{5}\text{ J/mol}\).
Using \(E = N_{A}h
u\) (where \(N_{A}=6.022\times 10^{23}\text{ mol}^{-1}\), \(h = 6.626\times10^{-34}\text{ J s}\)), the energy per photon \(e=\frac{E}{N_{A}}\).

$$e=\frac{4.84\times 10^{5}\text{ J/mol}}{6.022\times 10^{23}\text{ mol}^{-1}}$$

$$e\approx8.04\times 10^{-19}\text{ J}$$

Step2: Relate energy and wavelength

We know that \(e = h
u\) and \(
u=\frac{c}{\lambda}\) (where \(c = 3\times 10^{8}\text{ m/s}\)), so \(e=\frac{hc}{\lambda}\).
Rearranging for \(\lambda\), we get \(\lambda=\frac{hc}{e}\).
Substitute \(h = 6.626\times10^{-34}\text{ J s}\), \(c = 3\times 10^{8}\text{ m/s}\) and \(e = 8.04\times 10^{-19}\text{ J}\)

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Convert meters to nanometers (\(1\text{ m}=10^{9}\text{ nm}\))
\(\lambda=2.47\times 10^{-7}\times10^{9}\text{ nm}=247\text{ nm}\)

Answer:

247