Given that the energy of a mole of photons is 1.91 \times 10^6 \text{ J}, what is the wavelength in nanometers of the light? (Planck's constant h = 6.626 \times 10^{-34} \text{ J} \cdot \text{s}, speed of light c = 3.00 \times 10^8 \text{ m/s}, Avogadro's number N_A = 6.022 \times 10^{23} \text{ mol}^{-1})

• A. 1.04 \times 10^3 nm
• B. 627 nm
• C. 104 nm
• D. 312 nm

Answer: (B)

Verified step by step guidance

1. Identify the given information: energy per mole of photons \(E_{\text{mole}} = 1.91 \times 10^{6} \text{ J}\), Planck's constant \(h = 6.626 \times 10^{-34} \text{ J} \cdot \text{s}\), speed of light \(c = 3.00 \times 10^{8} \text{ m/s}\), and Avogadro's number \(N_A = 6.022 \times 10^{23} \text{ mol}^{-1}\).
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: \(E_{\text{photon}} = \frac{h \cdot c}{\lambda}\), where \(\lambda\) is the wavelength in meters.
4. Rearrange the formula to solve for the wavelength: \(\lambda = \frac{h \cdot c}{E_{\text{photon}}}\).
5. Convert the wavelength from meters to nanometers by multiplying by \$10^{9}\(, since \)1 \text{ m} = 10^{9} \text{ nm}$.