Determine the wavelength of a single photon, given that 6.458 moles of photons with identical wavelengths have a total energy of 1055.6 kJ.

To determine the wavelength of a single photon, given the total energy and the number of moles of photons, we can proceed step by step using Planck's equation and other relevant formulas.

### Steps:

1. **Energy of one mole of photons**:
   - Given: 6.458 moles of photons have a total energy of \( 1055.6 \, \text{kJ} \).
   - To find the energy of one mole of photons:
   \[
   E_{\text{one mole}} = \frac{1055.6 \, \text{kJ}}{6.458 \, \text{mol}} = 163.48 \, \text{kJ/mol}
   \]

2. **Convert energy to joules**:
   Since \( 1 \, \text{kJ} = 1000 \, \text{J} \), the energy per mole of photons in joules is:
   \[
   E_{\text{one mole}} = 163.48 \, \text{kJ/mol} \times 1000 \, \frac{\text{J}}{\text{kJ}} = 163480 \, \text{J/mol}
   \]

3. **Energy of a single photon**:
   Since one mole of photons contains Avogadro's number of photons, \( N_A = 6.022 \times 10^{23} \, \text{photons/mol} \), the energy of a single photon is:
   \[
   E_{\text{photon}} = \frac{163480 \, \text{J/mol}}{6.022 \times 10^{23} \, \text{photons/mol}} = 2.715 \times 10^{-19} \, \text{J}
   \]

4. **Wavelength of the photon**:
   We can use Planck's equation to relate the energy of a photon to its wavelength:
   \[
   E = \frac{hc}{\lambda}
   \]
   where:
   - \( E \) is the energy of the photon,
   - \( h \) is Planck's constant, \( h = 6.626 \times 10^{-34} \, \text{J·s} \),
   - \( c \) is the speed of light, \( c = 3.00 \times 10^8 \, \text{m/s} \),
   - \( \lambda \) is the wavelength of the photon.

   Rearranging for \( \lambda \):
   \[
   \lambda = \frac{hc}{E}
   \]

   Substituting the values:
   \[
   \lambda = \frac{(6.626 \times 10^{-34} \, \text{J·s}) \times (3.00 \times 10^8 \, \text{m/s})}{2.715 \times 10^{-19} \, \text{J}} = 7.32 \times 10^{-7} \, \text{m}
   \]

5. **Convert to nanometers**:
   Since \( 1 \, \text{m} = 10^9 \, \text{nm} \):
   \[
   \lambda = 7.32 \times 10^{-7} \, \text{m} \times 10^9 \, \frac{\text{nm}}{\text{m}} = 732 \, \text{nm}
   \]

### Final Answer:
The wavelength of a single photon is \( 732 \, \text{nm} \).

Would you like further details or explanations on any part of this? Here are some related questions for further practice:

1. What is the energy of a photon with a wavelength of 500 nm?
2. How does wavelength affect the energy of a photon?
3. What is the frequency of a photon with the wavelength of 732 nm?
4. How does the Planck constant relate to the energy of photons?
5. Calculate the energy of a single photon with a frequency of \( 4 \times 10^{14} \, \text{Hz} \).

**Tip**: Always check units when working with physical constants to ensure that your calculations are consistent!

This guide explains how to calculate the wavelength of a photon, given 6.458 moles of photons with a total energy of 1055.6 kJ. Learn key concepts like Planck's equation and energy-wavelength relationships, including detailed step-by-step solutions.

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