Given that the energy of a mole of photons is 2.83 × 10^7 J, what is the wavelength in nanometers of the light? (Use h = 6.626 × 10^{-34} J·s, c = 3.00 × 10^8 m/s, and Avogadro's number = 6.022 × 10^{23} mol^{-1})

• A. 167 nm
• B. 850 nm
• C. 212 nm
• D. 423 nm

Answer: (D)

Verified step by step guidance

1. Identify the given information: energy per mole of photons \(E_{\text{mole}} = 2.83 \times 10^{7}\) J, Planck's constant \(h = 6.626 \times 10^{-34}\) J·s, speed of light \(c = 3.00 \times 10^{8}\) m/s, and Avogadro's number \(N_A = 6.022 \times 10^{23}\) mol\(^{-1}\).
2. Calculate the energy of a single photon by dividing the energy per mole by Avogadro's number: \(E_{\text{photon}} = \frac{E_{\text{mole}}}{N_A}\).
3. Use the relationship between the energy of a photon and its wavelength: \(E_{\text{photon}} = \frac{h c}{\lambda}\), where \(\lambda\) is the wavelength in meters.
4. Rearrange the formula to solve for the wavelength: \(\lambda = \frac{h c}{E_{\text{photon}}}\).
5. Convert the wavelength from meters to nanometers by multiplying by \$10^{9}\(, since \)1\( nm = \)10^{-9}$ m.