Ch.2 Quantum Mechanics And Spectroscopy Verified Test Bank

Chapter 2

Problem 2.1: Calculate the wavelength and energy of photons necessary to promote an electron from the n = 3 state to the n = 4, 5, 6, and 7 states in hydrogen. Label the region of the electromagnetic spectrum for each of the emission lines.

Problem 2.2: Can an electron in a hydrogen atom moving from one energy level to a lower energy level cause a transition in the microwave region of the spectrum? Investigate only transitions between adjacent states of principal quantum number n.

Problem 2.3: This problem makes use of the four- level system and associated emission lines from Figure 2.5. Assume there is an additional energy level E₅ that has an associated emission to E₄ with a wavelength of 710 nm. Sketch the energy level diagram for this five- level system, clearly labeling the energies of all E_excited to E₁ transitions.

Problem 2.4:

(a) Calculate the reduced mass for NO and HCl.

(b) Would the reduced mass of H- ³⁵Cl be different than H- ³⁷Cl? If so, might this lead to a difference in infrared absorption? Support your response with a calculation.

(c) For Example 2.3, would you expect to measure infrared absorption for O₂?

Problem 2.5: The spring constant between atoms in HCl is around 516 N/ m. Calculate the angular frequency and vibrational frequency of HCl molecule. Determine the energy difference between the n = 0 and n = 1 states for this molecule in joules. Is this an allowed transition? What photon wavelength, in units of microns and wave numbers, would be associated with absorption from n = 0 to n = 1? Label the region of the electromagnetic spectrum.

Problem 2.6: The absorption from n = 0 to n = 1 occurs at 2170 cm^-1 in ¹²C– ¹⁶O.

(a) Calculate the spring constant between atoms in this molecule.

(b) State why this is an allowed transition.

Problem 2.7: The frequency spacing between rotational levels is 511 MHz in HBr. Calculate the difference in energy of the J = 0 and J = 1 states. Calculate the bond length for HBr.

Problem 2.8: Rotational absorption is observed at 0.13 cm for the CO molecule. The transition is associated with the J=1 to J=2 energy levels. Use this information to estimate the rotational inertia for CO.

Problem 2.9: Suppose two different states (with no degeneracy so that g₁ = g₂ = 1) have energies E₂ = 2 × 10^–22 J and E₁ = 0.5 × 10^–22 J. At what temperature will N₂ have a population will N₂/ N₁ = 100? Suppose now that the degeneracies are g₁ = 3, g₂ = 2. Is it possible to find a temperature where the population in N₂ is larger than N₁?

Problem 2.10: Suppose an electron is trapped in one dimension to a length of 2 nm, the length of some polymer.

(a) What is the wavelength absorbed by such a system from the ground state to the first excited state? Hint: This can be modeled as a particle in a one- dimensional box.

(b) How does the length of the polymer (the length of the box) affect the absorption wavelength from the ground state to the first excited state? In order to answer this question, use a spreadsheet to calculate the wavelengths associated with this transition for one- dimensional boxes that are from 1 to 200 nm in size at intervals of 10 nm. Make a plot of wavelength of absorption versus polymer length.

Problem 2.11: Carbon- 13 (¹³C) has a gyromagnetic ratio of 6.73 × 10⁷ T^{– 1}, s^{– 1}. Repeat Example 2.8 for a carbon-13 nucleus.

Problem 2.12: Carbon- 13 (¹³C) has a gyromagnetic ratio of 6.73 × 10⁷ T^{– 1}, s^{– 1}. Repeat Example 2.9 for a carbon-13 nucleus.

Problem 2.13: You are considering purchasing a new NMR instrument so that you can do ¹H NMR. Suppose one instrument would provide a magnetic field of 10 T and another would provide a field of 3 T. For a given molecule, would these two instruments have two different absorption wavelengths between spin states? If so, calculate the two different wavelengths that would cause absorption, ignoring the internal magnetic field of the molecule.

EXERCISE 2.1: The spring constant associated with NO is 1530 N/ m. Calculate the frequency of oscillation for this molecule.

EXERCISE 2.2: The spring constant of the diatomic molecule NO is 1530 N/ m.

(a) Calculate the energy level difference from n = 0 to n = 1

(b) Calculate the energy level difference from n = 1 to n = 2.

EXERCISE 2.3: The spring constant associated with CO is 1860 N/ m. Calculate the frequency of oscillation for this molecule.

EXERCISE 2.4: The molecular vibration of HF is well described with a spring constant of 970 N/m. What is the wavelength of photon absorbed from the n = 1 to n = 2 states?

EXERCISE 2.5: Using spreadsheet software, calculate the energies for vibrational states from n=0 to n=10 for HI. HI has a spring constant of 320 N/ m. Using your spreadsheet, calculate the energy level differences between adjacent levels up to n=10.

EXERCISE 2.6: It can be quite useful to move beyond the simple harmonic oscillator model of diatomic molecules. Using the Morse potential, a more accurate model for the molecular potential that takes into account the asymmetric molecular potential, the energy levels are:

where D_e is the bond energy (the energy from the minimum of the potential the energy at which the bond is broken). Note that the first term in the equation is what we found using the simple harmonic oscillator potential. The second term is often referred to as the anharmonic correction.

The dissociation energy for CO is 11.2 eV (1.79 × 10^–18 J) the force constant is 1860 N/ m. Calculate the energy difference between the ground state and the first excited state two ways, with the energy equation for the simply harmonic oscillator and with the anharmonic correction. How large is the correction associated with the anharmonic potential?

EXERCISE 2.7: For two cases, using the energy levels from the harmonic oscillator and then with the full Morse potential energy state, use spreadsheet software to produce a plot of energy level difference for vibration as a function of quantum number n. Describe the relationship and discuss the role of the anharmonic correction. Use values for HCl for your calculation (k = 80 N/ m, disassociation energy = 7.0 × 10^{– 19} J).

EXERCISE 2.8: The disassociation energy for NO is 7.0 eV (1.12 × 10^{– 18} J) and the effective spring constant is 1530 N/ m. Calculate the energy level difference between the n = 5 and n = 6 states and the associated absorption wavelength using the harmonic oscillator model and the Morse poten­tial model (see Exercise 2.7).

EXERCISE 2.9: A gas sample is thought to be either LiBr or LiI. For both molecules calculate the difference in energy due to rotation alone between J = 0 and J = 1. Then compare your answer to the energy due to rotation alone between J = 1 and J = 2 states. Are these energies above or below the thermal energy of 0.04 eV?

EXERCISE 2.10: We have now plotted energy level diagrams for several systems (hydrogen atom, vibrating diatomic molecules, and rotating diatomic molecules) in this chapter. Make such an energy level sketch of the particle in a box system described in Problem 2.10.

EXERCISE 2.11: At what temperature might you expect to have populated an excited state E₂ to about 10% of N₁ from a ground state E₁ when the energy level difference is 5 × 10^–20 J? What wavelength photon would be associated with the absorption between these two states? Degeneracies can be assumed to be one.

EXERCISE 2.12: Produce a spreadsheet tool that can be used to investigate the relationship between temperature, energy level difference, and populations in N₁ and N₂. Your tool should allow you to input the temperature and then plot the ratio of the populations for a range of energy level differences. What other relationships can you explore with your tool?

EXERCISE 2.13: You are doing proton NMR and would like to investigate the role of the magnetic field on the ratio of populations N₂/N₁ between two spin states in proton NMR at 300K. Ignore any particular local magnetic field contributions for the molecule being investigated. Use a spreadsheet tool to produce a plot of N₁/ N₂ as a function of magnetic field. Use your tool to determine the B field at which N₁ = 1.02 N₂. The largest NMR magnets can achieve a B field of around 23 T. Comment on the viability of getting to your B field.