For a particle in a ring, the energy levels can be expressed as En= [(h)2/ (2m)]...
For a particle in a ring, the energy levels can be expressed as
En= [(h)2/ (2m)] [(n)/(L)]2
where n= 0, ±1, ±2, …, L = 2πr is the circumference of the ring, and r is the radius of the ring. If r=4.0 x 10-10 m, find the energy and the corresponding wavelength (λ) for the n = 0 to n=1 transition.
Homework Answers
The energy and the corresponding wavelength for the n = 0 to n = 1 transition are 3.82*10—30 J and 52054.4 m respectively.
The detail calculation is shown below.
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For a particle in a ring, the energy levels can be expressed
as
En= [(h)2/ (2m)]...
The Schrodinger equation for a particle in a ring can be expressed as H cap psi(phi)...
The Schrodinger equation for a particle in a ring can be expressed as H cap psi(phi) = E psi(phi), where H cap = -h^2 partial^2/21 partial phi^2; psi(phi) = 1/squareroot 2 pi e^-ik phi, k is an integer What is the potential energy? What is the kinetic energy? Write the Schrodinger equation Derive the eigenvalue Show that psi(phi) = psi(phi + 2 pi) Derive an expression for the probability density for a particle in a ring?
CHEM20020 Structure and Properties Question 8. [time = 10 minutes) Benzene (C6H) has a cyclic structure...
CHEM20020 Structure and Properties Question 8. [time = 10 minutes) Benzene (C6H) has a cyclic structure with Dh symmetry. Using a particle-on-a-ring model calculate the energy and wavelength for the lowest electronic trans elength for the lowest electronic transition of the benzene molecule. Clearly explain (using a diagram) which energy levers are sing a diagram) which energy levels are involved in the transition. Make the following assumptions: 1. The r electrons can be approximated as moving on a 2-D ring....
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...
The energy levels for a particle in a 1-D box of dimension (L) is provided by...
The energy levels for a particle in a 1-D box of dimension (L) is provided by the following expression: n h2 E, 2mL2 where m is the mass of the particle, h-bar = h/2 andn is the quantum number. Evaluate the energy associated with the first level (n 1) for an argon atom in a 10 A 1-D box. Select one: a. 8.27 x 10-25 kJ/mol b. 4.98 x 10 1 kJ/mol c. 8.27 x 1028 kJ/mol d. 4.98 x...
Consider a simple single quantum particle with the energy levels of the harmonic oscillator En =...
Consider a simple single quantum particle with the energy levels of the harmonic oscillator En = (n + 1/2)ℏω. This particle is in thermal contact with a reservoir with temperature T. a) Calculate the partition function of this particle. b) Calculate the internal energy of the particle as a function of temperature. Deduce and interpret the state of this energy at low and high temperatures. c) Calculate the specific temperature of this particle at constant pressure.
Figure 8.3 gives the energy and degeneracy of the first 5 levels for a particle in...
Figure 8.3 gives the energy and degeneracy of the
first 5 levels for a particle in a cubic box. Find the energy and
degeneracy of the next 3 levels (that is the 6th, 7th and
8th).
m? Degeneracy 4E.. 12 None 3 SE 93 2E0 6 Eo. None Figure 8.3 An energy-level di- agram for a particle confined to a cubic box. The ground-state energy is Ep = 37'h/2m/?. and ?? ni + n + n. Note that most of...
need # 4 or 5 o Vibrational spectroscopy of the NO molecule (with absorption at 1878 cm isotope masses of No14 and 0-16, respectively) reveals Assuming that this transition represents the energy s...
need # 4 or 5
o Vibrational spectroscopy of the NO molecule (with absorption at 1878 cm isotope masses of No14 and 0-16, respectively) reveals Assuming that this transition represents the energy spacing between vibrational energy levels, calculate the force constant of the bond Assuming that the "N"O molecule has a bond with the same force constant as in part a, predict the position (in cm) of the absorbance peak for this molecule. 1. a. b 2. Normalize the first...
For a particle on a ring, the wavefunctions are: (2T)1/2 where m' # 0, 1, £2,...
For a particle on a ring, the wavefunctions are: (2T)1/2 where m' # 0, 1, £2, The Hamiltonian (total energy) operator for the particle on a ring is: h2 d2 Apply the Hamiltonian to the particle on a ring function to find an expression for total energy. (2 pts)
7) Consider the transition from the energy levels n=4 to n = 2 a) what is...
7) Consider the transition from the energy levels n=4 to n = 2 a) what is the frequency and wavelength associated with this transition? 9) What type of electron orbital (i.e. s, p, d, or f) is designated by 2, 1 a) n 1, m= -1? 1, 1- 0, m/ b) n 0? c) n 5, 1 2, m/= 2?
Question # 1: Find the unit of energy in the energy expression of a free particle...
Question # 1: Find the unit of energy in the energy expression of a free particle in 1-D box: Question # 2: A proton in a box is in a state n = 5 falls to a state n = 4 and loose energy with a wavelength of 2000 nm, what is the length of the box? (answer: 4 x 10 m) Question # 3: a. Consider an electron confined to move in an atom in one dimension over a...
The Schrodinger equation for a particle in a ring can be expressed as H cap psi(phi) = E psi(phi), where H cap = -h^2 partial^2/21 partial phi^2; psi(phi) = 1/squareroot 2 pi e^-ik phi, k is an integer What is the potential energy? What is the kinetic energy? Write the Schrodinger equation Derive the eigenvalue Show that psi(phi) = psi(phi + 2 pi) Derive an expression for the probability density for a particle in a ring?
CHEM20020 Structure and Properties Question 8. [time = 10 minutes) Benzene (C6H) has a cyclic structure with Dh symmetry. Using a particle-on-a-ring model calculate the energy and wavelength for the lowest electronic trans elength for the lowest electronic transition of the benzene molecule. Clearly explain (using a diagram) which energy levers are sing a diagram) which energy levels are involved in the transition. Make the following assumptions: 1. The r electrons can be approximated as moving on a 2-D ring....
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...
The energy levels for a particle in a 1-D box of dimension (L) is provided by the following expression: n h2 E, 2mL2 where m is the mass of the particle, h-bar = h/2 andn is the quantum number. Evaluate the energy associated with the first level (n 1) for an argon atom in a 10 A 1-D box. Select one: a. 8.27 x 10-25 kJ/mol b. 4.98 x 10 1 kJ/mol c. 8.27 x 1028 kJ/mol d. 4.98 x...
Figure 8.3 gives the energy and degeneracy of the
first 5 levels for a particle in a cubic box. Find the energy and
degeneracy of the next 3 levels (that is the 6th, 7th and
8th).
m? Degeneracy 4E.. 12 None 3 SE 93 2E0 6 Eo. None Figure 8.3 An energy-level di- agram for a particle confined to a cubic box. The ground-state energy is Ep = 37'h/2m/?. and ?? ni + n + n. Note that most of...
need # 4 or 5
o Vibrational spectroscopy of the NO molecule (with absorption at 1878 cm isotope masses of No14 and 0-16, respectively) reveals Assuming that this transition represents the energy spacing between vibrational energy levels, calculate the force constant of the bond Assuming that the "N"O molecule has a bond with the same force constant as in part a, predict the position (in cm) of the absorbance peak for this molecule. 1. a. b 2. Normalize the first...
For a particle on a ring, the wavefunctions are: (2T)1/2 where m' # 0, 1, £2, The Hamiltonian (total energy) operator for the particle on a ring is: h2 d2 Apply the Hamiltonian to the particle on a ring function to find an expression for total energy. (2 pts)
7) Consider the transition from the energy levels n=4 to n = 2 a) what is the frequency and wavelength associated with this transition? 9) What type of electron orbital (i.e. s, p, d, or f) is designated by 2, 1 a) n 1, m= -1? 1, 1- 0, m/ b) n 0? c) n 5, 1 2, m/= 2?
Question # 1: Find the unit of energy in the energy expression of a free particle in 1-D box: Question # 2: A proton in a box is in a state n = 5 falls to a state n = 4 and loose energy with a wavelength of 2000 nm, what is the length of the box? (answer: 4 x 10 m) Question # 3: a. Consider an electron confined to move in an atom in one dimension over a...
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