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MCAT Problems 7
27.6 Spectroscopy and Early Models of the Atom;
27.7 The Bohr Model of the Hydrogen Atom; Atomic Energy Levels
31. What is the orbital radius of the electron in the n = 3 state of hydrogen?
32. Find the energy for a hydrogen atom in the stationary state n = 4.
33. (a) What is the difference in radius between the n = 1 state and the n = 2 state for hydrogen? (b) What
is the difference in radius between the n = 100 state and the n = 101 state for hydrogen? How do the
neighboring orbital separations compare for large and small n values?
34. Find the Bohr radius of doubly ionized lithium (Li2+).
35. Find the energy in eV required to remove the remaining electron from a doubly ionized lithium (Li 2+)
atom.
36. How much energy must be supplied to a hydrogen atom to cause a transition from the ground state to
the n = 4 state?
37. A hydrogen atom in its ground state absorbs a photon of energy 12.1 eV. To what energy level is the
atom excited?
38. The Bohr theory of the hydrogen atom neglects gravitational forces between the electron and the
proton. Make a calculation to justify this omission. [Hint: Find the ratio of the gravitational and
electrostatic forces acting on the electron due to the proton.]
39. Use the Bohr theory to find the energy necessary to remove the electron from a hydrogen atom initially
in its ground state.
40. How much energy is required to ionize a hydrogen atom initially in the n = 2 state?
41. What is the smallest energy photon that can be absorbed by a hydrogen atom in its ground state?
42. Find the wavelength of the radiation emitted when a hydrogen atom makes a transition from the n = 6
to the n = 3 state.
46. By directly substituting the values of the fundamental constants, show that the ground state energy for
hydrogen in the Bohr model E1 = –mek2e4/(2 h 2 ) has the numerical value –13.6 eV.
47. Calculate, according to the Bohr model, the speed of the electron in the ground state of the hydrogen
atom.
50. A particle collides with a hydrogen atom in the n = 2 state, transferring 15.0 eV of energy to the atom.
As a result, the electron breaks away from the hydrogen nucleus. What is the kinetic energy of the
electron when it is far from the nucleus?
51. A hydrogen atom has an electron in the n = 5 level. (a) If the electron returns to the ground state by
emitting radiation, what is the minimum number of photons that can be emitted? (b) What is the
maximum number that might be emitted?
52. The Paschen series in the hydrogen emission spectrum is formed by electron transitions from ni > 3 to
nf = 3. (a) What is the longest wavelength in the Paschen series? (b) What is the wavelength of the
series limit (the lower bound of the wavelengths in the series)? (c) In what part or parts of the EM
spectrum is the Paschen series found (IR, visible, UV, etc.)?
53. A fluorescent solid absorbs a photon of ultraviolet light of wavelength 320 nm. If the solid dissipates
0.500 eV of the energy and emits the rest in a single photon, what is the wavelength of the emitted
light?
28.2 Matter Waves
1.
What is the de Broglie wavelength of a basketball of mass 0.50 kg when it is moving at 10 m/s? Why
don’t we see diffraction effects when a basketball passes through the circular aperture of the hoop?
2. A fly with a mass of 1.0 –4 kg crawls across a table at a speed of 2 mm/s. Compute the de Broglie
wavelength of the fly and compare it to the size of a proton (about 1 fm, 1 fm = 10 –15 m).
3. An 81-kg student who has just studied matter waves is concerned that he may be diffracted as he walks
through a doorway that is 81 cm across and 12 cm thick. (a) If the wavelength of the student must be
about the same size as the doorway to exhibit diffraction, what is the fastest the student can walk
through the doorway to exhibit diffraction? (b) At this speed, how long would it take the student to
walk through the doorway?
4. What are the de Broglie wavelengths of electrons with the following values of kinetic energy? (a) 1.0
eV; (b) 1.0 keV.
5. What is the ratio of the wavelength of a 0.100-keV photon to the wavelength of a 0.100-keV electron?
6. What is the magnitude of the momentum of an electron with a de Broglie wavelength of 0.40 nm?
7. What is the de Broglie wavelength of an electron moving at speed 53 c?
8. The distance between atoms in a crystal of NaCl is 0.28 nm. The crystal is being studied in a neutron
diffraction experiment. At what speed must the neutrons be moving so that their de Broglie wavelength
is 0.28 nm?
9. An x-ray diffraction experiment using 16-keV x-rays is repeated using electrons instead of x-rays.
What should the kinetic energy of the electrons be in order to produce the same diffraction pattern as
the x-rays (using the same crystal)?
10. Neutron diffraction by a crystal can be used to make a velocity selector for neutrons. Suppose the
spacing between the relevant planes in the crystal is d = 0.20 nm. A beam of neutrons is incident at an
angle θ = 10.0° with respect to the planes. The incident neutrons have speeds ranging from 0 to 2.0
4 m/s. (a) What wavelength(s) are strongly reflected from these planes? [ Hint: Bragg’s law, Eq.
(25-15), applies to neutron diffraction as well as to x-ray diffraction.] (b) For each of the
wavelength(s), at what angle with respect to the incident beam do those neutrons emerge from the
crystal?
11. A nickel crystal is used as a diffraction grating for x-rays. Then the same crystal is used to diffract
electrons. If the two diffraction patterns are identical, and the energy of each x-ray photon is E = 20.0
keV, what is the kinetic energy of each electron?