Chapter 23

In Chapter \(2,\) the law of conservation of mass was introduced as an important principle in chemistry. The discovery of nuclear reactions forced scientists to modify this law. Explain why, and give an example illustrating that mass is not conserved in a nuclear reaction.

Outline how nuclear reactions are carried out in the laboratory. Describe the artificial nuclear reactions used to make an element with an atomic number greater than 92.

Explain how carbon- 14 is used to estimate the ages of archeological artifacts. What are the limitations for use of this technique?

Describe how the concept of half-life for nuclear decay is used.

What is a radioactive decay series? Explain why radium and polonium are found in uranium ores.

The interaction of radiation with matter has both positive and negative consequences. Discuss briefly the hazards of radiation and the way that radiation can be used in medicine.

Define the terms curie, rad, and rem.

Complete the following nuclear equations. Write the mass number and atomic number for the remaining particle, as well as its symbol. (a) \(^{54}_{26} \mathrm{Fe}+_{2}^{4} \mathrm{He} \longrightarrow 2_{0}^{1} \mathrm{n}+?\) (b) \(_{13}^{27} \mathrm{Al}+_{2}^{4} \mathrm{He} \longrightarrow_{15}^{30} \mathrm{P}+?\) (c) \(^{32}_{16} \mathrm{S}+\frac{1}{0} \mathrm{n} \longrightarrow_{1}^{1} \mathrm{H}+?\) (d) \(_{42}^{96} \mathrm{Mo}+_{1}^{2} \mathrm{H} \longrightarrow_{0}^{1} \mathrm{n}+?\) (e) \(_{42}^{98} \mathrm{Mo}+_{0}^{1} \mathrm{n} \longrightarrow_{43}^{99} \mathrm{Tc}+?\) (f) \(^{18}_{9} \mathrm{F} \longrightarrow^{18}_{8} \mathrm{Q}+?\)

Complete the following nuclear equations. Write the mass number, atomic number, and symbol for the remaining particle. (a) \(^{9}_{4} \mathrm{Be}+? \longrightarrow_{3}^{6} \mathrm{Li}+_{2}^{4} \mathrm{He}\) (b) \(?+_{0}^{1} n \longrightarrow_{11}^{24} \mathrm{Na}+_{2}^{4} \mathrm{He}\) (c) \(_{20}^{40} \mathrm{Ca}+? \longrightarrow_{19}^{40} \mathrm{K}+_{1}^{1} \mathrm{H}\) (d) \(^{241}_{95} \mathrm{Am}+_{2}^{4} \mathrm{He} \longrightarrow_{97}^{243} \mathrm{Bk}+?\) (e) \(^{246}_{96} \mathrm{Cm}+_{6}^{12} \mathrm{C} \longrightarrow 4_{0}^{1} \mathrm{n}+?\) (f) \(^{238}_{92} \mathrm{d}+? \longrightarrow_{100}^{249} \mathrm{Fm}+5_{0}^{1} \mathrm{n}\)

Complete the following nuclear equations. Write the mass number, atomic number, and symbol for the remaining particle. (a) \(^{111}_{47} \mathrm{Ag} \longrightarrow_{48}^{111} \mathrm{Cd}+?\) (b) \(_{36}^{87} \mathrm{Kr} \longrightarrow_{-1}^{0} \beta+?\) (c) \(^{231}_{91} \mathrm{Pa} \longrightarrow_{89}^{227} \mathrm{Ac}+?\) (d) \(^{230}_{90} \mathrm{Th} \longrightarrow_{2}^{4} \mathrm{He}+?\) (e) \(_{35}^{82} \mathrm{Br} \longrightarrow_{36}^{82} \mathrm{Kr}+?\) (f) \(? \longrightarrow_{12}^{24} \mathrm{Mg}+_{-1}^{0} \beta\)

Complete the following nuclear equations. Write the mass number, atomic number, and symbol for the remaining particle. (a) \(_{10}^{19} \mathrm{Ne} \longrightarrow_{+1}^{0} \beta+?\) (b) \(_{26}^{59} \mathrm{Fe} \longrightarrow_{-1}^{0} \beta+?\) (c) \(_{19}^{40} \mathrm{K} \longrightarrow_{-1}^{0} \beta+?\) (d) \(^{37}_{18}\mathrm{Ar}+_{-1}^{0} \mathrm{e}(\text {electron capture}) \longrightarrow?\) (e) \(^{55}_{26} \mathrm{Fe}+{0}_{-1} \mathrm{e}(\text {electron capture}) \longrightarrow ?\) (f) \(^{26}_{13} \mathrm{Al} \longrightarrow_{12}^{25} \mathrm{Mg}+?\)

The uranium-235 radioactive decay series, beginning with \(_{92}^{235} \mathrm{U}\) and ending with \(_{82}^{207} \mathrm{Pb},\) occurs in the following sequence: \(\alpha, \beta, \alpha, \beta, \alpha, \alpha, \alpha, \alpha, \beta, \beta, \alpha .\) Write an equation for each step in this series.

The thorium-232 radioactive decay series, beginning with \(^{232}_{90}\)Th and ending with \(^{208}_{82}\)Pb, occurs in the following sequence: \(\alpha, \beta, \beta, \alpha, \alpha, \alpha, \alpha, \beta, \beta, \alpha .\) Write an equation for each step in this series.

What particle is emitted in the following nuclear reactions? Write an equation for each reaction. (a) Gold-198 decays to mercury-198. (b) Radon-222 decays to polonium-218. (c) Cesium-137 decays to barium-137. (d) Indium-110 decays to cadmium-110.

What is the product of the following nuclear decay processes? Write an equation for each process. (a) Gallium-67 decays by electron capture. (b) Potassium- 38 decays with positron emission. (c) Technetium-99m decays with \(\gamma\) emission. (d) Manganese-56 decays by \(\beta\) emission.

(a) Which of the following nuclei decay by \(-1 \beta\) decay? $$^{3} \mathbf{H} \quad^{16} \mathbf{O} \quad^{20} \mathbf{F} \quad^{13} \mathbf{N}$$ (b) Which of the following nuclei decays by \(+1 \beta\) decay? $$^{238} \mathbf{U} \quad^{19} \mathbf{F} \quad^{22} \mathbf{N a} \quad^{24} \mathbf{N a}$$

(a) Which of the following nuclei decay by \(_{-1}^{0} \beta\) decay? $$^{1} \mathrm{H} \quad^{23} \mathrm{Mg} \quad^{32} \mathrm{P} \quad^{20} \mathrm{Ne}$$ (b) Which of the following nuclei decay by \(^{0}_{+1} \beta\) decay? $$^{235} \mathrm{U} \quad^{35} \mathrm{Cl} \quad^{38} \mathrm{K} \quad^{24} \mathrm{Na}$$

Calculate the binding energy per nucleon for calcium-40, and compare your result with the value for calcium-40 in Figure \(23.4 .\) Masses needed for this calculation are $$_{1}^{1} \mathrm{H}=1.00783,_{0}^{1} \mathrm{n}=1.00867, \text { and }_{20}^{40} \mathrm{Ca}=39.96259.$$

Calculate the binding energy per mole of nucleons for \(_{8}^{16} \mathrm{O} .\) Masses needed for this calculations are \(_{1}^{1} \mathrm{H}=1.00783\) \(_{0}^{1} \mathrm{n}=1.00867,\) and \(^{16}_{8} \mathrm{O}=15.99492.\)

Copper acetate containing \(^{64} \mathrm{Cu}\) is used to study brain tumors. This isotope has a half-life of 12.7 h. If you begin with \(25.0 \mu \mathrm{g}\) of \(^{64} \mathrm{Cu},\) what mass in micrograms remains after \(64 \mathrm{h} ?\)

Gold-198 is used in the diagnosis of liver problems. The half-life of \(^{198} \mathrm{Au}\) is 2.69 days. If you begin with \(2.8 \mu \mathrm{g}\) of this gold isotope, what mass remains after 10.8 days?

Iodine-131 is used to treat thyroid cancer. (a) The isotope decays by \(\beta\) particle emission. Write a balanced equation for this process. (b) Iodine-131 has a half-life of 8.04 days. If you begin with \(2.4 \mu \mathrm{g}\) of radioactive \(^{131} \mathrm{I},\) what mass remains after 40.2 days?

Phosphorus- 32 is used in the form of \(\mathrm{Na}_{2} \mathrm{HPO}_{4}\) in the treatment of chronic myeloid leukemia, among other things. (a) The isotope decays by \(\beta\) particle emission. Write a balanced equation for this process. (b) The half-life of \(^{32} \mathrm{P}\) is 14.3 days. If you begin with \(4.8 \mu \mathrm{g}\) of radioactive \(^{32} \mathrm{P}\) in the form of \(\mathrm{Na}_{2} \mathrm{HPO}_{4},\) what mass remains after 28.6 days (about one month)?

Gallium-67 \(\left(t_{1 / 2}=78.25 \mathrm{h}\right)\) is used in the medical diagnosis of certain kinds of tumors. If you ingest a compound containing 0.015 mg of this isotope, what mass (in milligrams) remains in your body after 13 days? (Assume none is excreted.)

Iodine- \(131(t_{1 / 2}=8.04 \text { days), a } \beta\) emitter, is used to treat. thyroid cancer. (a) Write an equation for the decomposition of \(^{131}\) I. (b) If you ingest a sample of NaI containing \(^{131}\) I, how much time is required for the activity to decrease to \(35.0 \%\) of its original value?

Radon has been the focus of much attention recently because it is often found in homes. Radon-222 emits \(\alpha\) particles and has a half-life of 3.82 days. (a) Write a balanced equation to show this process. (b) How long does it take for a sample of \(^{222} \mathrm{Rn}\) to decrease to \(20.0 \%\) of its original activity?

A sample of wood from a Thracian chariot found in an excavation in Bulgaria has a \(^{14} \mathrm{C}\) activity of \(11.2 \mathrm{dpm} / \mathrm{g}\) Estimate the age of the chariot and the year it was made \((t_{1 / 2} \text { for }^{14} \mathrm{C} \text { is } 5.73 \times 10^{3} \text { years, and the activity of }^{14} \mathrm{C}\) in. living material is \(14.0 \mathrm{dpm} / \mathrm{g}).\)

A piece of charred bone found in the ruins of a Native American village has a \(^{14} \mathrm{C}:^{12} \mathrm{C}\) ratio that is \(72 \%\) of the radio found in living organisms. Calculate the age of the bone fragment.

Strontium-90 is a hazardous radioactive isotope that resulted from atmospheric nuclear testing. A sample of strontium carbonate containing \(^{90} \mathrm{Sr}\) is found to have an activity of \(1.0 \times 10^{3} \mathrm{dpm} .\) One year later the activity of this sample is 975 dpm. (a) Calculate the half-life of strontium-90 from this information. (b) How long will it take for the activity of this sample to drop to \(1.0 \%\) of the initial value?

Radioactive cobalt-60 is used extensively in nuclear medicine as a \(\gamma\) -ray source. It is made by a neutron capture reaction from cobalt-59, and it is a \(\beta\) emitter; \(\beta\) emission is accompanied by strong \(\gamma\) radiation. The half-life of cobalt-60 is 5.27 years. (a) How long will it take for a cobalt-60 source to decrease to one eighth of its original activity? (b) What fraction of the activity of a cobalt-60 source remains after 1.0 year?

Scandium occurs in nature as a single isotope, scandium-45. Neutron irradiation produces scandium-46, a \(\beta\) emitter with a half-life of 83.8 days. If the initial activity is \(7.0 \times 10^{4} \mathrm{dpm},\) draw a graph showing disintegrations per minute as a function of time during a period of one year.

Phosphorus occurs in nature as a single isotope, phosphorus- \(31 .\) Neutron irradiation of phosphorus- \(31\) produces phosphorus- \(32,\) a \(\beta\) emitter with a half-life of 14.28 days. Assume you have a sample containing phosphorus- 32 that has a rate of decay of \(3.2 \times 10^{6} \mathrm{dpm.}\) Draw a graph showing disintegrations per minute as a function of time during a period of one year.

Sodium-23 (in a sample of NaCl) is subjected to neutron bombardment in a nuclear reactor to produce \(^{24}\) Na. When removed from the reactor, the sample is radioactive, with \(\beta\) activity of \(2.54 \times 10^{4} \mathrm{dpm} .\) The decrease in radioactivity over time was studied, producing the following data: $$\begin{array}{lc}\hline \text { Activity }(\mathrm{dpm}) & \text { Time }(\mathrm{h}) \\\\\hline 2.54 \times 10^{4} & 0 \\ 2.42 \times 10^{4} & 1 \\\2.31 \times 10^{4} & 2 \\\2.00 \times 10^{4} & 5 \\\1.60 \times 10^{4} & 10 \\\1.01 \times 10^{4} & 20 \\\\\hline\end{array}$$ (a) Write equations for the neutron capture reaction and for the reaction in which the product of this reaction decays by \(\beta\) emission. (b) Determine the half-life of sodium- 24.

The isotope of polonium that was most likely isolated by Marie Curie in her pioneering studies is polonium-210. A sample of this element was prepared in a nuclear reaction. Initially its activity ( \(\alpha\) emission) was \(7840 \mathrm{dpm}\). Measuring radioactivity over time produced the following data: $$\begin{array}{ll}\hline \text { Activity }(\mathrm{dpm}) & \text { Time }(\text { days }) \\\\\hline 7840 & 0 \\\7570 & 7 \\\7300 & 14 \\\5920 & 56 \\\5470 & 72 \\\\\hline\end{array}$$ Determine the half-life of polonium-210.

There are two isotopes of americium, both with half-lives sufficiently long to allow the handling of large quantities. Americium-241, with a half-life of 432 years, is an \(\alpha\) emitter. It is used in smoke detectors. The isotope is formed from \(^{239} \mathrm{Pu}\) by absorption of two neutrons followed by emission of a \(\beta\) particle. Write a balanced equation for this process.

Americium-240 is made by bombarding plutonium-239 with \(\alpha\) particles. In addition to \(^{240} \mathrm{Am}\), the products are a proton and two neutrons. Write a balanced equation for this process.

To synthesize the heavier transuranium elements, a nucleus must be bombarded with a relatively large particle. If you know the products are californium- 246 and four neutrons, with what particle would you bombard uranium-238 atoms?

Element \(^{287} 114\) was made by firing a beam of \(^{48} \mathrm{Ca}\) ions at \(^{242}\mathrm{Pu.}\) Three neutrons were ejected in the reaction. Write a balanced nuclear equation for the synthesis of this super-heavy element.

Element \(^{287} 114\) decayed by \(\alpha\) emission with a half-life of about \(5 s\). Write an equation for this process.

Deuterium nuclei \(\left(_{1}^{2} \mathrm{H}\right)\) are particularly effective as bombarding particles to carry out nuclear reactions. Complete the following equations: (a) \(_{48}^{114} \mathrm{Cd}+_{1}^{2} \mathrm{H} \longrightarrow ?+_{1}^{1} \mathrm{H}\) (b) \(_{3}^{6} \mathrm{Li}+_{1}^{2} \mathrm{H} \longrightarrow ?+_{0}^{1} \mathrm{n}\) (c) \(_{20}^{40} \mathrm{Ca}+_{1}^{2} \mathrm{H} \longrightarrow_{19}^{38} \mathrm{K}+?\) (d) \(?+_{1}^{2} \mathrm{H} \longrightarrow_{30}^{65} \mathrm{Zn}+\gamma\)

Some of the reactions explored by Rutherford and others are listed below. Identify the unknown species in each reaction. (a) \(^{14}_{7} \mathrm{p}+_{2}^{4} \mathrm{He} \longrightarrow_{8}^{17} \mathrm{O}+?\) (b) \(_{4}^{9} \mathrm{Be}+_{2}^{4} \mathrm{He} \longrightarrow ?+_{0}^{1} \mathrm{n}\) (c) \(?+_{2}^{4} \mathrm{He} \longrightarrow_{15}^{30} \mathrm{P}+_{0}^{1} \mathrm{n}\) (d) \(^{239}_{94} \mathrm{Pu}+_{2}^{4} \mathrm{He} \longrightarrow ?+_{0}^{1} \mathrm{n}\)

Boron is an effective absorber of neutrons. When boron-10 adds a neutron, an \(\alpha\) particle is emitted. Write an equation for this nuclear reaction.

Tritium, \(^{3}_{1} \mathrm{H},\) is one of the nuclei used in fusion reactions. This isotope is radioactive, with a half-life of 12.3 years. Like carbon-14, tritium is formed in the upper atmosphere from cosmic radiation, and it is found in trace amounts on earth. To obtain the amounts required for a fusion reaction, however, it must be made via a nuclear reaction. The reaction of \(_{3}^{6}\) Li with a neutron produces tritium and an \(\alpha\) particle. Write an equation for this nuclear reaction.

A technique to date geological samples uses rubidium-\(87,\) a long-lived radioactive isotope of rubidium \(\left(t_{1 / 2}=\right.\) \(4.8 \times 10^{10}\) years). Rubidium-87 decays by \(\beta\) emission to strontium-87. If the rubidium-87 is part of a rock or mineral, then strontium-87 will remain trapped within the crystalline structure of the rock. The age of the rock dates back to the time when the rock solidified. Chemical analysis of the rock gives the amounts of \(^{87} \mathrm{Rb}\) and \(^{87}\) Sr. From these data, the fraction of \(^{87} \mathrm{Rb}\) that remains can be calculated. Analysis of a stony meteorite determined that \(1.8 \mathrm{mmol}\) of \(^{87} \mathrm{Rb}\) and \(1.6 \mathrm{mmol}\) of \(^{87} \mathrm{Sr}\) were present. Estimate the age of the meteorite. (Hint: The amount of \(^{87} \mathrm{Rb}\) at \(t_{0}\) is moles \(^{87} \mathrm{Rb}+\) moles \(^{87} \mathrm{Sr} .\))

The oldest-known fossil found in South Africa has been dated based on the decay of Rb-87. $$^{87} \mathrm{Rb} \longrightarrow^{87} \mathrm{Sr}+_{-1}^{0} \beta \quad t_{1 / 2}=4.8 \times 10^{10} \text { years }$$ If the ratio of the present quantity of \(^{87} \mathrm{Rb}\) to the original quantity is \(0.951,\) calculate the age of the fossil.

The age of minerals can sometimes be determined by measuring the amounts of \(^{206} \mathrm{Pb}\) and \(^{238} \mathrm{U}\) in a sample. This determination assumes that all of the \(^{206} \mathrm{Pb}\) in the sample comes from the decay of \(^{238} \mathrm{U}\). The date obtained identifies when the rock solidified. Assume that the ratio of \(^{206} \mathrm{Pb}\) to \(^{238} \mathrm{U}\) in an igneous rock sample is \(0.33 .\) Calculate the age of the rock. \((t_{1 / 2} \text { for }^{238} \mathrm{U} \text { is } 4.5 \times 10^{9}\) years.).

In June \(1972,\) natural fission reactors, which operated billions of years ago, were discovered in Oklo, Gabon. At present, natural uranium contains \(0.72 \%^{235} \mathrm{U} .\) How many years ago did natural uranium contain \(3.0 \%^{235} \mathrm{U},\) the amount needed to sustain a natural reactor? \(\left(t_{1} / 2 \text { for }^{235} \mathrm{U}\right.\) is \(7.04 \times 10^{8}\) years.)

If a shortage in worldwide supplies of fissionable uranium arose, it would be possible to use other fissionable nuclei. Plutonium, one such fuel, can be made in "breeder" reactors that manufacture more fuel than they consume. The sequence of reactions by which plutonium is made is as follows: (a) \(\mathrm{A}^{238} \mathrm{U}\) nucleus undergoes an \((\mathrm{n}, \gamma)\) to produce \(^{239} \mathrm{U}\) (b) \(^{239} \mathrm{U}\) decays by \(\beta\) emission \(\left(t_{1 / 2}=23.5 \mathrm{min}\right)\) to give an isotope of neptunium. (c) This neptunium isotope decays by \(\beta\) emission to give a plutonium isotope. (d) The plutonium isotope is fissionable. On collision of one of these plutonium isotopes with a neutron, fission occurs, with at least two neutrons and two other nuclei as products. Write an equation for each of the nuclear reactions.

When a neutron is captured by an atomic nucleus, energy is released as \(\gamma\) radiation. This energy can be calculated based on the change in mass in converting reactants to products. For the nuclear reaction \(_{3}^{6} \mathrm{Li}+_{0}^{1} \mathrm{n} \longrightarrow_{3}^{7} \mathrm{Li}+\gamma:\) (a) Calculate the energy evolved in this reaction (per atom). Masses needed for this calculation are \(_{3}^{6} \mathrm{Li}=\) \(6.01512,_{0}^{1} n=1.00867,\) and \(_{3}^{7} \mathrm{Li}=7.01600\). (b) Use the answer in part (a) to calculate the wavelength of the \(\gamma\) -rays emitted in the reaction.

To measure the volume of the blood system of an animal, the following experiment was done. A 1.0 -mL sample of an aqueous solution containing tritium, with an activity of \(2.0 \times 10^{6} \mathrm{dps},\) was injected into the animal's bloodstream. After time was allowed for complete circulatory mixing, a 1.0-mL blood sample was withdrawn and found to have an activity of \(1.5 \times 10^{4} \mathrm{dps} .\) What was the volume of the circulatory system? (The half-life of tritium is 12.3 years, so this experiment assumes that only a negligible amount of tritium has decayed in the time of the experiment.)