Problems

Note to Instructors: Most of the homework problems in this chapter are available for assignment via an online homework management program such as WileyPLUS or WebAssign, and those marked with the icon are presented in a guided tutorial format that provides enhanced interactivity. See Preface for additional details.
In working these problems, ignore relativistic effects.
ssm Solution is in the Student Solutions Manual.
This icon represents a biomedical application.
Section 30.1 Rutherford Scattering and the Nuclear Atom

| 1 | ssm The nucleus of the hydrogen atom has a radius of about . The electron is normally at a distance of about from the nucleus. Assuming the hydrogen atom is a sphere with a radius of find (a) the volume of the atom, (b) the volume of the nucleus, and (c) the percentage of the volume of the atom that is occupied by the nucleus.

| 2 | In a Rutherford scattering experiment a target nucleus has a diameter of . The incoming particle has a mass of . What is the kinetic energy of an particle that has a de Broglie wavelength equal to the diameter of the target nucleus? Ignore relativistic effects.

| 3 | The nucleus of a hydrogen atom is a single proton, which has a radius of about . The single electron in a hydrogen atom normally orbits the nucleus at a distance of . What is the ratio of the density of the hydrogen nucleus to the density of the complete hydrogen atom?

| 4 | Review Conceptual Example 1 and use the information therein as an aid in working this problem. Suppose you’re building a scale model of the hydrogen atom, and the nucleus is represented by a ball of radius 3.2 cm (somewhat smaller than a baseball). How many miles away should the electron be placed?

* | 5 | ssm The nucleus of a copper atom contains 29 protons and has a radius of . How much work (in electron volts) is done by the electric force as a proton is brought from infinity, where it is at rest, to the “surface” of a copper nucleus?

* | 6 | There are Z protons in the nucleus of an atom, where Z is the atomic number of the element. An particle carries a charge of +2e. In a scattering experiment, an particle, heading directly toward a nucleus in a metal foil, will come to a halt when all the particle’s kinetic energy is converted to electric potential energy. In such a situation, how close will an particle with a kinetic energy of come to a gold nucleus (Z = 79)?
Section 30.2 Line Spectra
Section 30.3 The Bohr Model of the Hydrogen Atom

| 7 | ssm www Concept Simulation 30.1 at reviews the concepts on which the solution to this problem depends. The electron in a hydrogen atom is in the first excited state, when the electron acquires an additional 2.86 eV of energy. What is the quantum number n of the state into which the electron moves?

| 8 | Using the Bohr model, determine the ratio of the energy of the nth orbit of a triply ionized beryllium atom (Be3+, Z = 4) to the energy of the nth orbit of a hydrogen atom (H).

| 9 | A singly ionized helium atom (He+) has only one electron in orbit about the nucleus. What is the radius of the ion when it is in the second excited state?

| 10 | (a) What is the minimum energy (in electron volts) that is required to remove the electron from the ground state of a singly ionized helium atom (He+, Z = 2)? (b) What is the ionization energy for He+?

| 11 | ssm Find the energy (in joules) of the photon that is emitted when the electron in a hydrogen atom undergoes a transition from the n = 7 energy level to produce a line in the Paschen series.

| 12 | A hydrogen atom is in the ground state. It absorbs energy and makes a transition to the n = 3 excited state. The atom returns to the ground state by emitting two photons. What are their wavelengths?

| 13 | Consider the Bohr energy expression (Equation 30.13) as it applies to singly ionized helium He+ (Z = 2) and doubly ionized lithium Li2+ (Z = 3). This expression predicts equal electron energies for these two species for certain values of the quantum number n (the quantum number is different for each species). For quantum numbers less than or equal to 9, what are the lowest three energies (in electron volts) for which the helium energy level is equal to the lithium energy level?

| 14 | In the hydrogen atom the radius of orbit B is sixteen times greater than the radius of orbit A. The total energy of the electron in orbit A is −3.40 eV. What is the total energy of the electron in orbit B?

* | 15 | ssm A wavelength of 410.2 nm is emitted by the hydrogen atoms in a high-voltage discharge tube. What are the initial and final values of the quantum number n for the energy level transition that produces this wavelength?

* | 16 | The energy of the n = 2 Bohr orbit is −30.6 eV for an unidentified ionized atom in which only one electron moves about the nucleus. What is the radius of the n = 5 orbit for this species?

* | 17 | ssm For atomic hydrogen, the Paschen series of lines occurs when nf = 3, whereas the Brackett series occurs when nf = 4 in Equation 30.14. Using this equation, show that the ranges of wave lengths in these two series overlap.

* | 18 | Interactive LearningWare 30.1 at reviews the concepts that play roles in this problem. A hydrogen atom emits a photon that has momentum with a magnitude of . This photon is emitted because the electron in the atom falls from a higher energy level into the n = 1 level. What is the quantum number of the level from which the electron falls? Use a value of for Planck’s constant.

** | 19 | A diffraction grating is used in the first order to separate the wavelengths in the Balmer series of atomic hydrogen. (Section 27.7 discusses diffraction gratings.) The grating and an observation screen (see Figure 27-32) are separated by a distance of 81.0 cm. You may assume that is small, so when radian measure is used for . How many lines per centimeter should the grating have so that the longest and the next-to-the-longest wavelengths in the series will be separated by 3.00 cm on the screen?

** | 20 | A certain species of ionized atoms produces an emission line spectrum according to the Bohr model, but the number of protons Z in the nucleus is unknown. A group of lines in the spectrum forms a series in which the shortest wavelength is 22.79 nm and the longest wavelength is 41.02 nm. Find the next-to-the-longest wavelength in the series of lines.
Section 30.5 The Quantum Mechanical Picture of the Hydrogen Atom

| 21 | The orbital quantum number for the electron in a hydrogen atom is . What is the smallest possible value (algebraically) for the total energy of this electron? Give your answer in electron volts.

| 22 | A hydrogen atom is in its second excited state. Determine, according to quantum mechanics, (a) the total energy (in eV) of the atom, (b) the magnitude of the maximum angular momentum the electron can have in this state, and (c) the maximum value that the z component Lz of the angular momentum can have.

| 23 | ssm www The principal quantum number for an electron in an atom is n = 6, and the magnetic quantum number is . What possible values for the orbital quantum number could this electron have?

| 24 | The maximum value for the magnetic quantum number in state A is , while in state B it is . What is the ratio LA/LB of the magnitudes of the orbital angular momenta of an electron in these two states?

* | 25 | Interactive Solution 30.25 at offers one approach to problems of this type. For an electron in a hydrogen atom, the z component of the angular momentum has a maximum value of . Find the three smallest possible values (algebraically) for the total energy (in electron volts) that this atom could have.

* | 26 | Review Conceptual Example 6 as background for this problem. For the hydrogen atom, the Bohr model and quantum mechanics both give the same value for the energy of the nth state. However, they do not give the same value for the orbital angular momentum L. (a) For n = 1, determine the values of L [in units of ] predicted by the Bohr model and quantum mechanics. (b) Repeat part (a) for n = 3, noting that quantum mechanics permits more than one value of when the electron is in the n = 3 state.

** | 27 | ssm www An electron is in the n = 5 state. What is the smallest possible value for the angle between the z component of the orbital angular momentum and the orbital angular momentum? Section 30.6 The Pauli Exclusion Principle and the Periodic Table of the Elements

| 28 | Two of the three electrons in a lithium atom have quantum numbers of n = 1, , , and n = 1, , , . What quantum numbers can the third electron have if the atom is in (a) its ground state and (b) its first excited state?

| 29 | In the style shown in Table 30-3, write down the ground-state electronic configuration for arsenic As (Z = 33). Refer to Figure 30-17 for the order in which the subshells fill.

| 30 | Figure 30-17 was constructed using the Pauli exclusion principle and indicates that the n = 1 shell holds 2 electrons, the n = 2 shell holds 8 electrons, and the n = 3 shell holds 18 electrons. These numbers can be obtained by adding the numbers given in the figure for the subshells contained within a given shell. How many electrons can be put into the n = 5 shell, which is only partly shown in the figure?

| 31 | Write down the fourteen sets of the four quantum numbers that correspond to the electrons in a completely filled 4f subshell.

* | 32 | What is the atom with the smallest atomic number that contains the same number of electrons in its s subshells as it does in its d subshell? Refer to Figure 30-17 for the order in which the subshells fill. Section 30.7 X-Rays

| 33 | ssm Molybdenum has an atomic number of Z = 42. Using the Bohr model, estimate the wavelength of the X-ray.

| 34 | Interactive LearningWare 30.2 at reviews the concepts that are pertinent to this problem. By using the Bohr model, decide which element is likely to emit a X-ray with a wavelength of .

| 35 | Interactive Solution 30.35 at provides one model for solving problems such as this one. An X-ray tube is being operated at a potential difference of 52.0 kV. What is the Bremsstrahlung wavelength that corresponds to 35.0% of the kinetic energy with which an electron collides with the metal target in the tube?

| 36 | What is the minimum potential difference that must be applied to an X-ray tube to knock a K-shell electron completely out of an atom in a copper (Z = 29) target? Use the Bohr model as needed.

* | 37 | ssm An X-ray tube contains a silver (Z = 47) target. The high voltage in this tube is increased from zero. Using the Bohr model, find the value of the voltage at which the X-ray just appears in the X-ray spectrum.

* | 38 | Multiple-Concept Example 9 reviews the concepts that are important in this problem. An electron, traveling at a speed of , strikes the target of an X-ray tube. Upon impact, the electron decelerates to one-quarter of its original speed, emitting an X-ray in the process. What is the wavelength of the X-ray photon?
Section 30.8 The Laser

| 39 | ssm www A laser is used in eye surgery to weld a detached retina back into place. The wavelength of the laser beam is 514 nm, and the power is 1.5 W. During surgery, the laser beam is turned on for 0.050 s. During this time, how many photons are emitted by the laser?

| 40 | The dye laser used in the treatment of the port-wine stain in Figure 30-30 (see Section 30.9) has a wavelength of 585 nm. A carbon dioxide laser produces a wavelength of . What is the minimum number of photons that the carbon dioxide laser must produce to deliver at least as much or more energy to a target as does a single photon from the dye laser?

| 41 | A pulsed laser emits light in a series of short pulses, each having a duration of 25.0 ms. The average power of each pulse is 5.00 mW, and the wavelength of the light is 633 nm. Find (a) the energy of each pulse and (b) the number of photons in each pulse.

| 42 | A laser peripheral iridotomy is a procedure for treating an eye condition known as narrow-angle glaucoma, in which pressure buildup in the eye can lead to loss of vision. A neodymium YAG laser (wavelength = 1064 nm) is used in the procedure to punch a tiny hole in the peripheral iris, thereby relieving the pressure buildup. In one application the laser delivers of energy to the iris in creating the hole. How many photons does the laser deliver?

* | 43 | Fusion is the process by which the sun produces energy. One experimental technique for creating controlled fusion utilizes a solid-state laser that emits a wavelength of 1060 nm and can produce a power of for a pulse duration of . In contrast, the helium/neon laser used at the checkout counter in a bar-code scanner emits a wavelength of 633 nm and produces a power of about . How long (in days) would the helium/neon laser have to operate to produce the same number of photons that the solid-state laser produces in ?

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