Final answer:
To find the energy difference between the HOMO and LUMO using Planck's equation, convert the wavelength to meters, apply the equation, and then scale the result up to kilojoules per mole using Avogadro's number. The approximate energy is 2.54 x 10^2 kJ/mol.
Step-by-step explanation:
To calculate the actual energy difference between the HOMO and LUMO for beta-carotene, we follow the relationship that energy (E) can be calculated using the Planck's equation E = hc/λ, where h is Planck's constant (6.626 x 10-34 J·s), c is the speed of light (2.998 x 108 m/s), and λ is the wavelength (in meters).
First, we must convert the wavelength from nanometers to meters: 470 nm is equal to 470 x 10-9 m.
Then apply the Planck's equation:
- E = (6.626 x 10-34 J·s) (2.998 x 108 m/s) / (470 x 10-9 m)
- E = 4.217 x 10-19 J per photon
To find the energy in kJ/mol, we need to multiply by Avogadro's number (6.022 x 1023 mol-1):
- E (kJ/mol) = 4.217 x 10-19 J/photon × 6.022 x 1023 photons/mol × 1 kJ/1000 J
- E (kJ/mol) = ∼ 2.54 x 102 kJ/mol