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 By are presented in a guided tutorial format that provides enhanced interactivity. See Preface for additional details.
The data given for atomic masses in these problems include the mass of the electrons orbiting the nucleus of the electrically neutral atom.
ssm Solution is in the Student Solutions Manual.
This icon represents a biomedical application.
Section 31.1 Nuclear Structure,
Section 31.2 The Strong Nuclear Force and the Stability of the Nucleus

| 1 | ssm By what factor does the nucleon number of a nucleus have to increase in order for the nuclear radius to double?

| 2 | For find (a) the net electrical charge of the nucleus, (b) the number of neutrons, (c) the number of nucleons, (d) the approximate radius of the nucleus, and (e) the nuclear density.

| 3 | In each of the following cases, what element does the symbol X represent and how many neutrons are in the nucleus: (a), (b), (c), (d), and (e)? Use the periodic table on the in side of the back cover as needed.

| 4 | What is the radius of a nucleus of titanium

| 5 | ssm www The largest stable nucleus has a nucleon number of 209, and the smallest has a nucleon number of 1. If each nucleus is assumed to be a sphere, what is the ratio (largest/smallest) of the surface areas of these spheres?

* | 6 | The ratio of the radius of an unknown nucleus to a tritium nucleus is . Both nuclei contain the same number of neutrons. Identify the unknown nucleus in the form . Use the periodic table on the inside of the back cover as needed.

* | 7 | An unknown nucleus contains 70 neutrons and has twice the volume of the nickel nucleus. Identify the unknown nucleus in the form . Use the periodic table on the inside of the back cover as needed.

** | 8 | Conceptual Example 1 provides some useful background for this problem. (a) Determine an approximate value for the density (in kg/m3) of the nucleus. (b) If a BB (radius = 2.3 mm) from an air rifle had a density equal to the nuclear density, what mass would the BB have? (c) Assuming the mass of a supertanker is about kg, how many “supertankers” of mass would this hypothetical BB have?

** | 9 | ssm www Refer to Conceptual Example 1 for a discussion of nuclear densities. A neutron star is composed of neutrons and has a density that is approximately the same as that of a nucleus. What is the radius of a neutron star whose mass is 0.40 times the mass of the sun?
Section 31.3 The Mass Defect of the Nucleus and Nuclear Binding Energy (Note: The atomic mass for hydrogen is 1.007 825 u, including the mass of one electron.)

| 10 | Determine the mass defect (in atomic mass units) for (a) helium, which has an atomic mass of 3.016 030 u, and (b) the isotope of hydrogen known as tritium , which has an atomic mass of 3.016 050 u. (c) On the basis of your answers to parts (a) and (b), state which nucleus requires more energy to disassemble it into its separate and stationary constituent nucleons. Give your reasoning.

| 11 | ssm The earth revolves around the sun, and the two represent a bound system that has a binding energy of . Suppose the earth and sun were completely separated, so that they were infinitely far apart and at rest. What would be the difference between the mass of the separated system and that of the bound system?

| 12 | Find the binding energy (in MeV) for lithium (atomic mass = 7.016 003 u).

| 13 | For lead (atomic mass = 205.974 440 u) obtain (a) the mass defect in atomic mass units, (b) the binding energy (in MeV), and (c) the binding energy per nucleon (in MeV/nucleon).

| 14 | Use the plot of binding energy per nucleon in Figure 31-5 to determine the mass defect for the oxygen nucleus. Express your answer in kilograms.

* | 15 | ssm Two isotopes of a certain element have binding energies that differ by 5.03 MeV. The isotope with the larger binding energy contains one more neutron than the other isotope. Find the difference in atomic mass between the two isotopes.

* | 16 | (a) Energy is required to separate a nucleus into its constituent nucleons, as Figure 31-3 indicates; this energy is the total binding energy of the nucleus. In a similar way one can speak of the energy that binds a single nucleon to the remainder of the nucleus. For example, separating nitrogen into nitrogen and a neutron takes energy equal to the binding energy of the neutron, as shown below:

Find the energy (in MeV) that binds the neutron to the nucleus by considering the mass of (atomic mass = 13.005 738 u) and the mass of (atomic mass = 1.008 665 u), as compared to the mass of (atomic mass = 14.003 074 u). (b) Similarly, one can speak of the energy that binds a single proton to the nucleus:

Following the procedure outlined in part (a), determine the energy (in MeV) that binds the proton (atomic mass = 1.007 825 u) to the nucleus. The atomic mass of carbon is 13.003 335 u. (c) Which nucleon is more tightly bound, the neutron or the proton?
Section 31.4 Radioactivity

| 17 | ssm Write the decay process for , including the chemical symbol and values for Z and A.

| 18 | In the form , identify the daughter nucleus that results when (a) plutonium undergoes decay, (b) sodium undergoes decay, and (c) nitrogen undergoes decay.

| 19 | ssm www Write the decay process for each of the following nuclei, being careful to include Z and A and the proper chemical symbol for each daughter nucleus: (a) and (b) .

| 20 | Find the energy (in MeV) released when decay converts radium (atomic mass = 226.025 40 u) into radon (atomic mass = 222.017 57 u). The atomic mass of an particle is 4.002 603 u.

| 21 | decay occurs for each of the following nuclei. Write the decay process for each, including the chemical symbols and values for Z and A for the daughter nuclei: (a) and (b) .

| 22 | Multiple-Concept Example 7 reviews the concepts needed to solve this problem. When uranium decays, it emits (among other things) a ray that has a wavelength of . Determine the energy (in MeV) of this ray.

| 23 | Find the energy released when lead (atomic mass = 210.988 735 u) undergoes decay to become bismuth (atomic mass = 210.987 255 u).

* | 24 | Review Conceptual Example 5 as background for this problem. The decay of uranium produces thorium (atomic mass = 234.0436 u). In Example 4, the energy released in this decay is determined to be 4.3 MeV. Determine how much of this energy is carried away by the recoiling daughter nucleus and how much by the particle (atomic mass = 4.002 603 u). Assume that the energy of each particle is kinetic energy, and ignore the small amount of energy carried away by the ray that is also emitted. In addition, ignore relativistic effects.

* | 25 | Refer to Interactive Solution 31.25 at to review a model for solving this type of problem. Polonium (atomic mass = 209.982 848 u) undergoes decay. Assuming that all the released energy is in the form of kinetic energy of the particle (atomic mass = 4.002 603 u) and ignoring the recoil of the daughter nucleus (lead , 205.974 440 u), find the speed of the particle. Ignore relativistic effects.

* | 26 | Rado produces a daughter nucleus that is radioactive. The daughter, in turn, produces its own radioactive daughter, and so on. This process continues until lead is reached. What are the total number of particles and the total number of particles that are generated in this series of radioactive decays?

** | 27 | ssm Find the energy (in MeV) released when decay converts sodium (atomic mass = 21.994 434 u) into neon (atomic mass = 21.991 383 u). Notice that the atomic mass for includes the mass of 11 electrons, whereas the atomic mass for includes the mass of only 10 electrons.

** | 28 | Interactive LearningWare 31.1 at reviews the concepts that are involved in this problem. An isotope of beryllium (atomic mass = 7.017 u) emits a ray and recoils with a speed of . Assuming that the beryllium nucleus is stationary to begin with, find the wavelength of the ray.
Section 31.6 Radioactive Decay and Activity

| 29 | The isotope of radium has a decay constant of . What is the half-life (in days) of this isotope?

| 30 | The half-lives in two different samples, A and B, of radioactive nuclei are related according to In a certain period the number of radioactive nuclei in sample A decreases to one-fourth the number present initially. In this same period the number of radioactive nuclei in sample B decreases to a fraction f of the number present initially. Find f

| 31 | ssm How many half-lives are required for the number of radioactive nuclei to decrease to one-millionth of the initial number?

| 32 | Strontium has a half-life of 29.1 yr. It is chemically similar to calcium, enters the body through the food chain, and collects in the bones. Consequently, is a particularly serious health hazard. How long (in years) will it take for 99.9900% of the released in a nuclear reactor accident to disappear?

| 33 | A device used in radiation therapy for cancer contains 0.50 g of cobalt (59.933 819 u). The half-life of is 5.27 yr. Determine the activity of the radioactive material.

| 34 | Iodine is used in diagnostic and therapeutic techniques in the treatment of thyroid disorders. This isotope has a half-life of 8.04 days. What percentage of an initial sample of remains after 30.0 days?

| 35 | ssm The number of radioactive nuclei present at the start of an experiment is . The number present twenty days later is . What is the half-life (in days) of the nuclei?

* | 36 | To make the dial of a watch glow in the dark, kg of radium is used. The half-life of this isotope has a value of yr. How many kilograms of radium disappear while the watch is in use for fifty years?

* | 37 | Refer to Interactive Solution 31.37 at for one approach to solving this problem. To see why one curie of activity was chosen to be Bq, determine the activity (in disintegrations per second) of one gram of radium ().

* | 38 | The isotope (atomic mass = 197.968 u) of gold has a half-life of 2.69 days and is used in cancer therapy. What mass (in grams) of this isotope is required to produce an activity of 315 Ci?

* | 39 | ssm Two radioactive nuclei A and B are present in equal numbers to begin with. Three days later, there are three times as many A nuclei as there are B nuclei. The half-life of species B is 1.50 days. Find the half-life of species A. Section 31.7 Radioactive Dating

| 40 | Multiple-Concept Example 11 reviews most of the concepts that are needed to solve this problem. Material found with a mummy in the arid highlands of southern Peru has a activity per gram of carbon that is 78.5% of the activity present initially. How long ago (in years) did this individual die?

| 41 | ssm The practical limit to ages that can be determined by radio carbon dating is about 41 000 yr. In a 41 000-yr-old sample, what percentage of the original atoms remains?

| 42 | The half-life for the decay of uranium is . Determine the age (in years) of a rock specimen that contains sixty percent of its original number of atoms.

| 43 | The shroud of Turin is a religious artifact known since the Middle Ages. In 1988 its age was measured using the radiocarbon dating technique, which revealed that the shroud could not have been made before 1200 AD. Of the nuclei that were present in the living matter from which the shroud was made, what percentage remained in 1988?

| 44 | Review Conceptual Example 12 before starting to solve this problem. The number of unstable nuclei remaining after a time is N, and the number present initially is N0. Find the ratio for (a) (half-life = 5730 yr), (b) (half-life = 122.2 s; use t = 1.00 h, since otherwise the answer is out of the range of your calculator), and (c) (half-life = 12.33 yr). Verify that your answers are consistent with the reasoning in Conceptual Example 12.

* | 45 | ssm When any radioactive dating method is used, experimental error in the measurement of the sample’s activity leads to error in the estimated age. In an application of the radiocarbon dating technique to certain fossils, an activity of 0.100 Bq per gram of carbon is measured to within an accuracy of . Find the age of the fossils and the maximum error (in years) in the value obtained. Assume that there is no error in the 5730-year half-life of nor in the value of 0.23 Bq per gram of carbon in a living organism.

** | 46 | (a) A sample is being dated by the radiocarbon technique. If the sample were uncontaminated, its activity would be 0.011 Bq per gram of carbon. Find the true age (in years) of the sample. (b) Suppose the sample is contaminated, so that only 98.0% of its carbon is ancient carbon. The remaining 2.0% is fresh carbon, in the sense that the it contains has not had any time to decay. Assuming that the lab technician is unaware of the contamination, what apparent age (in years) would be determined for the sample?

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