Chapter 10, Problem 10.49E

Carotenes are molecules with alternating $\text{C}-\text{C}$ and $\text{C}=\text{C}$ bonds in which the electrons are delocalized across the entire alternating bond system. As such, the electrons can be approximated as being a particle-in-a-box. Lycopene is a carotene found in tomatoes and watermelon. Assuming that the alternating carbon-carbon bond system is $2.64\text{nm}$ wide:

(a) What is the energy of the $n=11$ level?

(b) What is the energy of the $n=12$ level?

(c) What is the $\Delta E$ between $n=11$ and $n=12$?

(d) If, by the Bohr frequency condition, $\Delta E=h\upsilon$, what frequency and wavelength of light corresponds to the transition from $n=11$ to $n=12$?