Use the particle-in-a-box model to estimate the wavelength of the lowest energy transition in $\beta$ carotene, whose structure is shown in Figure 10.36. Assume that the average carbon-carbon distance is 0.15 nm.

Question

Quizplus · Accepted Answer

In the particle-in-a-box model, the energy levels of a molecule are given by the equation:

E = (n^2 * h^2) / (8 * m * L^2)

where 
n = quantum number (1, 2, 3, ...)
h = Planck's constant 
m = mass of the particle 
L = length of the box (in this case, the length of the carotene molecule)

The lowest energy transition corresponds to the transition between the ground state (n=1) and the first excited state (n=2). The energy of this transition is given by:

ΔE = E2 - E1 = (4 - 1) * (h^2 / 8 * m * L^2)

We can estimate the length L of the carotene molecule as follows:
- The molecule has 11 double bonds, which are separated by single bonds.
- The average carbon-carbon distance is given as 0.15 nm.
- We can assume that each double bond contributes a length of about 0.12 nm (based on a typical length of a double bond, which is around 1.33 Å or 0.133 nm).

Therefore, the length of the molecule can be estimated as:

L = 11 * 0.12 nm + 10 * 0.15 nm = 2.07 nm

Using this value, we can calculate the wavelength of the lowest energy transition:

ΔE = hc/λ

λ = hc/ΔE = 694.7 nm

Therefore, the best choice is D: 690 nm.

Use the Particle-In-A-Box Model to Estimate the Wavelength of the Lowest