The particle-on-a-ring is a useful model for the motion of electrons around a conjugated macrocycle such...
The particle-on-a-ring is a useful model for the motion of
electrons around a conjugated macrocycle such as octatetrene, for
example. Treat the molecule as a circular ring of radius 0.480 nm,
with 10 electrons in the conjugated system moving along the
perimeter of the ring. Assume based on the Pauli Exclusion
Principle that in the ground state of the molecule each state is
occupied by two electrons with opposite spins.
(a) Calculate the energy of an electron in the highest occupied
level in joules.
J(b) Calculate the (absolute magnitude of the) angular momentum of
an electron in the highest occupied level in J-s.
J-s(c) Calculate the frequency of radiation in Hz that can induce a
transition between the highest occupied and lowest unoccupied
levels.
Homework Answers
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The particle-on-a-ring is a useful model for the motion of
electrons around a conjugated macrocycle such...
Calculate the wavelength of radiation that would absorb from a molecule of porphyrin at the highest and lowest occupied energy levels. Poryphyrin has 22 pi electrons that can be idealized as particles...
Calculate the wavelength of radiation that would absorb from a
molecule of porphyrin at the highest and lowest occupied energy
levels. Poryphyrin has 22 pi electrons that can be idealized as
particles on a ring with a radius of 0.50nm
g) Assume that the 22 r electrons in free base porphyrin (shown below) may be idealized as particles on a ring of radius 0.50 nm. What wavelength of radiation would it absorb for the excitation from the highest occupied energy...
2. For a conjugated molecule with 18 π electrons. a. Calculate the length of the molecules...
2. For a conjugated molecule with 18 π electrons. a. Calculate the length of the molecules b. Write the value of n for the highest occupied level c. Calculate the longest wavelength transition for this molecule using the 1 dimensional I have an exam coming up soon and I am having trouble understanding these practice questions. particle in a box model.
Electrons in a conjugated molecule can be approximated by the particle-in-a-box model. When an electron in...
Electrons in a conjugated molecule can be approximated by the particle-in-a-box model. When an electron in such a system moves from n1 = 5 to n2 = 7 light with a wavelength of 3568 Angstroms is emitted. Using this information calculate the length L of the unknown molecule in nanometers.
The 22 de localized TT-electrons in the 11 conjugated C-C bonds which make up the center...
The 22 de localized TT-electrons in the 11 conjugated C-C bonds which make up the center ß-carotene molecule behave as if they are in a box 2nm long. Given the usual 2 electrons per energy level that you know from chemistry, the highest energy electron (in the HOMO) is in the n = 11 state. What energy photon is needed to bump it to the n = 12 (LUMO) state?
3. Knowing that the magnesium and aluminum atoms possesses 12 and 13 electrons respectively: (a) Write...
3. Knowing that the magnesium and aluminum atoms possesses 12 and 13 electrons respectively: (a) Write down the electron configuration (nl) of the ground state. Explain your answer. (b) We focus on the electron configuration determined in (a). Considering only the electrons from the outer subshell, add their electronic spins and their orbital angular momenta separately. Label the total spin and the total orbital momentum quantum numbers by s and l, respectively. Write down the possible values spanned by s...
3. Knowing that the magnesium and aluminum atoms possesses 12 and 13 electrons respectively: (a) Write...
3. Knowing that the magnesium and aluminum atoms possesses 12 and 13 electrons respectively: (a) Write down the electron configuration (nl) of the ground state. Explain your answer. (b) We focus on the electron configuration determined in (a). Considering only the electrons from the outer subshell, add their electronic spins and their orbital angular momenta separately. Label the total spin and the total orbital momentum quantum numbers by s and l, respectively. Write down the possible values spanned by s...
7. π electron is an electron which resides in the pi bond(s) of a double bond or a triple bond, o...
7. π electron is an electron which resides in the pi bond(s) of a double bond or a triple bond, or in a conjugated p orbital. The 1,3,5-hexatriene molecule is a conjugated molecule with 6 t electrons. Consider the Tt electrons free to move back and forth along the molecule through the delocalized pi system. Using the particle in a box approximation, treat the carbon chain as a linear one-dimensional "box". Allow each energy level in the box to hold...
1. Use the model of the particle in the ring (rotation in two dimensions) of quantum...
1. Use the model of the particle in the ring (rotation in two dimensions) of quantum mechanics to describe the movement of electrons in the conjugate system of the benzene molecule. Presume that the circumference of the ring is equal to 8.40A. a. ((3 pts) Make a diagram of the energy levels of the electrons Pi in the benzene molecule, clearly identifying the HOMO and the LUMO. b. (3 pts) Calculate the energy of the HOMO and LUMO C. (3...
5. (20 pts total) For a particle-on-a-ring (a simple model for the pi electrons in a...
5. (20 pts total) For a particle-on-a-ring (a simple model for the pi electrons in a cyclic aromatic molecule) the angular momentum operator L has eigenvectors n with eigenvalues Ln-nh, where n = 0, ±1, ±2, , too. L is a linear Hermitian operator representing a physically measurable quantity. The eigenvectors, ψ'm, are normalized. (The variable φ is the polar angle in 2D polar coordinates, with range 0 to 2, though this information should not be necessary.) a) (5 pts)...
4. Estimate the transition frequency for the poryphyrin molecule from m-11 to m 12, assuming that the pi electrons can be modeled as a particle in a ring of radius 440 picometers. (C 7. The...
4. Estimate the transition frequency for the poryphyrin molecule from m-11 to m 12, assuming that the pi electrons can be modeled as a particle in a ring of radius 440 picometers. (C 7. The most probable distance of the electron from the nucleus in a 1ls state hydrogen atom (with wavefunction V1) can be determined by 21. A (A) solving the eigenvalue equation: Rvw rV., finding the maximum in the 1s radial distribution function by differentiation. (C) substituting vi,...
Calculate the wavelength of radiation that would absorb from a
molecule of porphyrin at the highest and lowest occupied energy
levels. Poryphyrin has 22 pi electrons that can be idealized as
particles on a ring with a radius of 0.50nm
g) Assume that the 22 r electrons in free base porphyrin (shown below) may be idealized as particles on a ring of radius 0.50 nm. What wavelength of radiation would it absorb for the excitation from the highest occupied energy...
Electrons in a conjugated molecule can be approximated by the particle-in-a-box model. When an electron in such a system moves from n1 = 5 to n2 = 7 light with a wavelength of 3568 Angstroms is emitted. Using this information calculate the length L of the unknown molecule in nanometers.
The 22 de localized TT-electrons in the 11 conjugated C-C bonds which make up the center ß-carotene molecule behave as if they are in a box 2nm long. Given the usual 2 electrons per energy level that you know from chemistry, the highest energy electron (in the HOMO) is in the n = 11 state. What energy photon is needed to bump it to the n = 12 (LUMO) state?
7. π electron is an electron which resides in the pi bond(s) of a double bond or a triple bond, or in a conjugated p orbital. The 1,3,5-hexatriene molecule is a conjugated molecule with 6 t electrons. Consider the Tt electrons free to move back and forth along the molecule through the delocalized pi system. Using the particle in a box approximation, treat the carbon chain as a linear one-dimensional "box". Allow each energy level in the box to hold...
1. Use the model of the particle in the ring (rotation in two dimensions) of quantum mechanics to describe the movement of electrons in the conjugate system of the benzene molecule. Presume that the circumference of the ring is equal to 8.40A. a. ((3 pts) Make a diagram of the energy levels of the electrons Pi in the benzene molecule, clearly identifying the HOMO and the LUMO. b. (3 pts) Calculate the energy of the HOMO and LUMO C. (3...
5. (20 pts total) For a particle-on-a-ring (a simple model for the pi electrons in a cyclic aromatic molecule) the angular momentum operator L has eigenvectors n with eigenvalues Ln-nh, where n = 0, ±1, ±2, , too. L is a linear Hermitian operator representing a physically measurable quantity. The eigenvectors, ψ'm, are normalized. (The variable φ is the polar angle in 2D polar coordinates, with range 0 to 2, though this information should not be necessary.) a) (5 pts)...
4. Estimate the transition frequency for the poryphyrin molecule from m-11 to m 12, assuming that the pi electrons can be modeled as a particle in a ring of radius 440 picometers. (C 7. The most probable distance of the electron from the nucleus in a 1ls state hydrogen atom (with wavefunction V1) can be determined by 21. A (A) solving the eigenvalue equation: Rvw rV., finding the maximum in the 1s radial distribution function by differentiation. (C) substituting vi,...
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