Final answer:
The wavelength of a photon with an energy of 282 kJ/mol is 423 nm, calculated using the Planck's equation relating energy to the frequency and wavelength of electromagnetic radiation.
Step-by-step explanation:
To calculate the wavelength of a photon when the energy is given, we can use the formula derived from Planck's equation, which relates energy (E) to wavelength (λ) and frequency (ν) of electromagnetic radiation: E = hν and ν = c/λ, where h is Planck's constant (6.626 x 10-34 J·s) and c is the speed of light in a vacuum (3.00 x 108 m/s).
Given the energy of 282 kJ/mol for a beam of light, we first convert this to energy per photon by dividing by Avogadro's number (6.022 x 1023 mol-1):
E (per photon) = 282 x 103 J/mol / 6.022 x 1023 mol-1 = 4.68 x 10-19 J/photon
Next, we use the equation E = hν, and ν = c/λ, to find the wavelength
λ = hc/E
Substituting the values, we get:
λ = (6.626 x 10-34 J·s)(3.00 x 108 m/s) / (4.68 x 10-19 J)
λ = 4.23 x 10-7 meters or 423 nm