Shown below is a molecule known as beta-carotene: This molecule is found in carr

Question

Shown below is a molecule known as beta-carotene:

This molecule is found in carrots and is responsible for their orange color. We will now see why.
This molecule consists of a long straight chain of carbon atoms (black spheres), bonded to hydrogen atoms (white spheres). This molecule can be thought of as a long string, and the electron states correspond to standing waves on that string.

The goal here is to find the different energies corresponding to these standing waves. The energy differences between standing wave states will then determine the possible energies of absorbed photons.
First, we start with some background information. The length of this carbon chain is about L= 18.5 x 10^-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 string. 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 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./2 mv^2 = p^2/2m

4. Find the difference in energy between the 12th and 11th states, deltaE.

5. Set deltaE=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.
To do this assignment, show me the calculations and results for procedures 1-6. Write a brief explanation of why beta-carotine appears orange.

The wavelenght string equation in the book is lambda = 2L/n. i plugged in 11 and 12 for n, (and L the lenght of the molecule), then got the momentums of each electron state, then got the velocities, then plugged that into the kinetic energy, then got the difference in energy btw the 2 states, and divided by Planck to the frequencey, which was around 6 Hz, then speed of light divided by frequency, but the wavelengh is gigantic, when it obviously has to be in the visible range (nanomemters).

Shown below is a molecule known as beta-carotene: This molecule is found in carrots and is responsible for their orange color. We will now see why. This molecule consists of a long straight chain of carbon atoms (black spheres), bonded to hydrogen atoms (white spheres). This molecule can be thought of as a long string, and the electron states correspond to standing waves on that string. The goal here is to find the different energies corresponding to these standing waves. The energy differences between standing wave states will then determine the possible energies of absorbed photons. First, we start with some background information. The length of this carbon chain is about L= 18.5 x 10^-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: Determine the wavelengths of standing waves on the molecule or 'string'. 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 is the same as given in Chapter 12 of the textbook. Once you know lambda , use p = mv = h/lambda , to determine the momentum of the electron. 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./2 mv^2 = p^2/2m Find the difference in energy between the 12th and 11th states, deltaE. Set deltaE=hf to determine the frequency of a photon that can cause an electron to change states. Also calculate the wavelength of the photon. 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. To do this assignment, show me the calculations and results for procedures 1-6. Write a brief explanation of why beta-carotine appears orange. The wavelenght string equation in the book is lambda = 2L/n. i plugged in 11 and 12 for n, (and L the lenght of the molecule), then got the momentums of each electron state, then got the velocities, then plugged that into the kinetic energy, then got the difference in energy btw the 2 states, and divided by Planck to the frequencey, which was around 6 Hz, then speed of light divided by frequency, but the wavelengh is gigantic, when it obviously has to be in the visible range (nanomemters).

Explanation / Answer

1

Related Questions

Navigate

• Browse (All)
• Browse S
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.