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: Convert energy per mole to energy per photon

The energy per mole \(E_{mol}=4.84\times 10^{2}\text{ kJ/mol}=4.84\times 10^{5}\text{ J/mol}\).
Using \(E = \frac{E_{mol}}{N_{A}}\), where \(N_{A}=6.022\times 10^{23}\text{ mol}^{-1}\).
\(E=\frac{4.84\times 10^{5}\text{ J/mol}}{6.022\times 10^{23}\text{ mol}^{-1}}\)
\(E = 8.04\times 10^{-19}\text{ J}\)

Step2: Use the formula \(E=\frac{hc}{\lambda}\) to find \(\lambda\)

We know that \(h = 6.626\times 10^{-34}\text{ Js}\) and \(c=3\times 10^{8}\text{ m/s}\).
From \(E=\frac{hc}{\lambda}\), we can solve for \(\lambda\): \(\lambda=\frac{hc}{E}\)
Substitute the values: \(\lambda=\frac{6.626\times 10^{-34}\text{ Js}\times3\times 10^{8}\text{ m/s}}{8.04\times 10^{-19}\text{ J}}\)
\(\lambda=\frac{1.9878\times 10^{-25}\text{ Jm}}{8.04\times 10^{-19}\text{ J}}\)
\(\lambda = 2.47\times 10^{-7}\text{ m}\)

Step3: Convert meters to nanometers

Since \(1\text{ m}=10^{9}\text{ nm}\), then \(\lambda=(2.47\times 10^{-7}\text{ m})\times10^{9}\text{ nm/m}\)
\(\lambda = 247\text{ nm}\)

Answer:

247