None 4. The harmonic oscillator only works for parabolic potentials, but true potentials are of course anharmonic.Consider the anharmonic potential that gives the energy levels with the following expression:E₁ =159.12(v+0.5)-0.725(v+0.5)- 0.0011(v+0.5)³Determine the vibrational quantum number Vmax at which the vibrational level spacing vanishes, andestimate the dissociation energy Do. The known dissociation energy is Do=0.73 eV. How well does thisanharmonic potential represent the true potential?

Answered: Image /qna-images/answer/d9bbb686-3acd-48f2-8d20-5173920c92d3.jpg

Answered: 4. The harmonic oscillator only works…

Chemistry

10th Edition

ISBN:9781305957404

Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste

Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste

Chapter1: Chemical Foundations

Section: Chapter Questions

Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...

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We can use the classical harmonic oscillator to think about molecular bonds. The HCI molecule has a force constant k = 481 N/m. For the mass, use the reduced mass, which is defined as µ = (m₁m₂)/(m₁+m₂). a) Plot the potential energy of HCl from -1 to 1 Å. What happens to the curvature of the potential as the force constant is varied? What does this mean physically? b) Plot position as a function of time for a total energy of 6 x 10-20 J. What is the period of the motion? How does the period change as the force constant is varied? Explain why this makes sense physically.
The Hydrogen molecule vibrates at a frequency of 1.23 x 10¹4 Hz for v = 0 to v = 1 transition. Calculate a.) The force constant of the H-H bond b.) The energy separation between v = 1 and v = 2, assuming that the hydrogen molecule behave as an ideal harmonic oscillator.
3. Hydrogen atoms (ignore spin) are first at a basic state. At the time t > 0, atoms experience the electric field Ae-bt with A> 0 and b>0, and is a unit vector in the z direction. Determine the probability of transition to the first excitation state at the time t → ∞.
As noted in lecture, the rigid rotor model can be improved by recognizing that in a realistic anharmonic potential, the bond length increases with the vibrational quantum number v. Thus, the rotational constant depends on v, and it can be shown that By = Be – ae(v +). For 'H®Br, B = 8.473 cm1 and a = 0.226 cm². Use this information to calculate the bond length for HBr a) as a rigid rotor, and b) as a nonrigid rotor in the ground vibrational state. Find a literature value for this bond length (cite your source) and compare your answers. Under what conditions would you expect the nonrigid rotor to be a significantly better model?
Assuming the vibrations of a ¹4N2 molecule are equivalent to those of a harmonic oscillator with a force constant kr = 2293.8 N/m, what is the zero-point energy of vibration of this molecule? Use m(14N) = 14.0031 mu.
Calculate the proportion of I2 molecules in their basal state, first and second excited state at 25 °C. The wave number is 214.6 cm−1 Hints: Equidistant levels, relation wave numbers and energy y (kT/hc) = 207.226 cm−1 2) At what temperature does the level ν = 1 of I2 have i) half of the population of the basal state, ii) the same population as the basic state? I need to understand and solve the second question
Suppose that the internuclear distance may be written R = Re + x where Re is the equilibrium bond length. Also suppose that the potential well is symmetrical and confines the oscillator to small displacements. Deduce expressions for 1/⟨R⟩2, 1/⟨R2⟩, and ⟨1/R2⟩ to the lowest non-zero power of ⟨x2⟩/Re2 and confirm that the values are not the same.
2. What are the term symbols for the microstates possible for the 1s 2s2p' electronic configuration of boron?
Light-induced degradation of molecules, also called photobleaching, is a serious problem in applications that require very high intensities. A molecule of a fluorescent dye commonly used to label biopolymers can withstand about 106 excitations by photons before light-induced reactions destroy its π system and the molecule no longer fluoresces. For how long will a single dye molecule fluoresce while being excited by 1.0 mW of 488 nm radiation from a continuous-wave laser? You may assume that the dye has an absorption spectrum that peaks at 488 nm and that every photon delivered by the laser is absorbed by the molecule.
Calculate the proportion of I2 molecules in their basal state, first and second excited state at 25 °C. The wave number is 214.6 cm−1 Hints: Equidistant levels, relation wave numbers and energy y (kT/hc) = 207.226 cm−1 a) At what temperature does the level ν = 1 of I2 have i) half of the population of the basal state, ii) the same population as the basic state?
The Morse potential energy is very useful as a simple representation of the actual molecular potential energy. When 85Rb1H was studied, it was found that ˜ν = 936.8 cm−1 and x_eν˜ = 14.15 cm−1. Plot the potential energy curve from 50 pm to 800 pm aroundRe = 236.7 pm. Then go on to explore how the rotation of a molecule may weaken its bond by allowing for the kinetic energy of rotation of a molecule and plotting V∗ = V + hcBJ˜ (J + 1) with B˜. Plot these curves on the same diagram for J = 40, 80, and 100, and observe how the dissociation energy is affected by the rotations. Hints: Taking B˜ = 3.020 cm−1 as the equilibrium bond length will greatly simplify the calculation. The mass of 85Rb is 84.9118mu
A diatomic Hydrogen (H2) molecule with a potential between its two Hydrogen atoms can be modelled using the Morse potential: V(x) = V0 e−αx (e−αx - 2), where α = 19.3 nm−1 and V0 = 4.52 eV. Assuming the H2 molecule has no rotational energy. Determine the minimum value of the Morse potential, V . (State in eV, to 3 s.f.)

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