First, we start with some background information. The length of this carbon chai
ID: 2257245 • Letter: F
Question
First, we start with some background information. The length of this carbon chain is about L=18.5e-10m, and there are 22 electrons. Due to the exclusion principle, only one electron can be in each state. However, because of electron spin, which has two states, each standing wave can accommodate 2 electrons. Therefore, we need 22/2 = 11 standing wave states to accommodate all these electrons. Energy can be absorbed when an electron jumps into the next highest (i.e. the 12th) state.
Thus, we want to figure out the energies corresponding to the 11th and 12th standing wave states, and set that equal to the absorbed photon energy.
The procedure is:
1. Determine the wavelengths of standing waves on the molecule or %u2018string%u2019. The only rule is that the wave amplitude must be zero at either end. This is exactly the same as for an actual string, so the formula for lambda is the same as given in Chapter 12 of the textbook.
2. Once you know lambda, use p=mv=h/lambda to determine the momentum of the electron.
3. The total energy is the sum of kinetic and potential energies. In this model we assume that, since the molecule has no net charge, the forces due to all other protons and electrons cancel out. Therefore, the electron has kinetic energy only, equal to E=1/2mv^2=p^2/2M
4. Find the difference in energy between the 12th and 11th states, Delta E.
5. Set Delta E=hf to determine the frequency of a photon that can cause an electron to change states. Also calculate the wavelength of the photon.
6. Determine what color of light this wavelength corresponds to. This is the color that the molecule absorbs. The color that we see corresponds to what is not absorbed.
Show me the calculations and results for procedures 1-6. Write a brief explanation of why beta-carotine appears orange.
Explanation / Answer
In analogy with a taut guitar string of length L, a standing wave with zero amplitude at both ends can as a lowest energy state have one-half wavelength spanning L. The next possibility of higher energy is spanning L with one full wavelength (2 half wavelengths), then 3 half wavelengths etc.
So the condition for fitting a standing wave is
L = n* lambda/2, where n= 1,2,3,...
So
lambda = 2L/n, with n=1,2,3,...
Then p = h/lambda= nh/(2L)
E= p^2/(2m) = n^2 h^2 / (8 m L^2)
Delta(E)=E(12)-E(11) = (144-121) h^2/(8mL^2)
= 23 h^2/(8mL^2)
From here you can calculate f with f= Delta(E) / h, lambda from lambda = c/f and look up the color of light that corresponds to this lambda. The complementary color is what is reflected and what we then see.
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