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Chapter 21: Problem 72
Radon-222 decays to a stable nucleus by a series of three alpha emissions and two beta emissions. What is the stable nucleus that is formed?
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Putting Concepts Together Potassium ion is present in foods and is an essential nutrient in the human body. One of the naturally occurring isotopes of potassium, potassium- 40 , is radioactive. Potassium-40 has a natural abundance of \(0.0117 \%\) and a half- life \(t_{1 / 2}=1.28 \times 10^{9} \mathrm{yr}\). It undergoes radioactive decay in three ways: \(98.2 \%\) is by electron capture, \(1.35 \%\) is by beta emission, and \(0.49 \%\) is by positron emission. (a) Why should we expect \({ }^{40} \mathrm{~K}\) to be radioactive? (b) Write the nuclear equations for the three modes by which \({ }^{40} \mathrm{~K}\) decays. (c) How many \({ }^{40} \mathrm{~K}^{+}\)ions are present in \(1.00 \mathrm{~g}\) of \(\mathrm{KCl}\) ? (d) How long does it take for \(1.00 \%\) of the \({ }^{40} \mathrm{~K}\) in a sample to undergo radioactive decay? SOLUTION (a) The \({ }^{40} \mathrm{~K}\) nucleus contains 19 protons and 21 neutrons. There are very few stable nuclei with odd numbers of both protons and neutrons (Section 21.2). (b) Electron capture is capture of an inner-shell electron by the nucleus: $$ { }_{19}^{40} \mathrm{~K}+{ }_{-1}^{0} \mathrm{e} \longrightarrow{ }_{18}^{40} \mathrm{Ar} $$ Beta emission is loss of a beta particle \((-1 \mathrm{e})\) ) by the nucleus: $$ { }_{19}^{40} \mathrm{~K} \longrightarrow{ }_{20}^{40} \mathrm{Ca}+{ }_{-1}^{0} \mathrm{e} $$ Positron emission is loss of a positron \(\left(+{ }_{+}^{0} \mathrm{e}\right)\) by the nucleus: $$ { }_{19}^{40} \mathrm{~K} \longrightarrow{ }_{18}^{40} \mathrm{Ar}+{ }_{+1}^{0} \mathrm{e} $$ (c) The total number of \(\mathrm{K}^{+}\)ions in the sample is $$ (1.00 \mathrm{~g} \mathrm{KCl})\left(\frac{1 \mathrm{~mol} \mathrm{KCl}}{74.55 \mathrm{~g} \mathrm{KCl}}\right)\left(\frac{1 \mathrm{~mol} \mathrm{~K}}{1 \mathrm{~mol} \mathrm{KCl}}\right)\left(\frac{6.022 \times 10^{23} \mathrm{~K}^{+}}{1 \mathrm{~mol} \mathrm{~K}^{+}}\right)=8.08 \times 10^{21} \mathrm{~K}^{+} \text {ions } $$ Of these, \(0.0117 \%\) are \({ }^{40} \mathrm{~K}^{+}\)ions: $$ \left(8.08 \times 10^{21} \mathrm{~K}^{+} \text {ions }\right)\left(\frac{0.0117^{40} \mathrm{~K}^{+} \text {ions }}{100^{+} \text {ions }}\right)=9.45 \times 10^{17} \text { potassium-40 ions } $$ (d) The decay constant (the rate constant) for the radioactive decay can be calculated from the half-life, using Equation 21.20: $$ k=\frac{0.693}{t_{1 / 2}}=\frac{0.693}{1.28 \times 10^{9} \mathrm{yr}}=\left(5.41 \times 10^{-10}\right) / \mathrm{yr} $$ The rate equation, Equation \(21.19\), then allows us to calculate the time required: $$ \begin{aligned} \ln \frac{N_{t}}{N_{0}} &=-k t \\ \ln \frac{99}{100} &=-\left[\left(5.41 \times 10^{-10}\right) / \mathrm{yr}\right] t \\ -0.01005 &=-\left[\left(5.41 \times 10^{-10}\right) / \mathrm{yr}\right] t \\ t &=\frac{-0.01005}{\left(-5.41 \times 10^{-10}\right) / \mathrm{yr}}=1.86 \times 10^{7} \mathrm{yr} \end{aligned} $$ That is, it would take \(18.6\) million years for just \(1.00 \%\) of the \({ }^{40} \mathrm{~K}\) in a sample to decay.
What particle is produced during the following decay processes: (a) sodium-24 decays to magnesium-24; (b) mercury-188 decays to gold-188; (c) iodine-122 decays to xenon-122; (d) plutonium-242 decays to uranium-238?
Iodine-131 is a convenient radioisotope to monitor thyroid activity in humans. It is a beta emitter with a half-life of \(8.02\) days. The thyroid is the only gland in the body that uses iodine. A person undergoing a test of thyroid activity drinks a solution of Nal, in which only a small fraction of the iodide is radioactive. (a) Why is NaI a good choice for the source of iodine? (b) If a Geiger counter is placed near the person's thyroid (which is near the neck) right after the sodium iodide solution is taken, what will the data look like as a function of time? (c) A normal thyroid will take up about \(12 \%\) of the ingested iodide in a few hours. How long will it take for the radioactive iodide taken up and held by the thyroid to decay to \(0.01 \%\) of the original amount?
The energy from solar radiation falling on Earth is \(1.07 \times 10^{16} \mathrm{~kJ} / \mathrm{min}\). (a) How much loss of mass from the Sun occurs in one day from just the encrgy falling on Farth? (b) If the energy released in the reaction $$ { }^{235} \mathrm{U}+{ }_{0}^{1} \mathrm{n} \longrightarrow{ }_{56}^{141} \mathrm{Ba}+{ }_{36}^{92} \mathrm{Kr}+3{ }_{0}^{1} \mathrm{n} $$ \(\left({ }^{235} \mathrm{U}\right.\) nuclear mass, \(234.9935 \mathrm{amu} ;{ }^{141} \mathrm{Ba}\) nuclear mass, \(140.8833 \mathrm{amu} ;{ }^{92} \mathrm{Kr}\) nuclear mass, 91.9021 amu) is taken as typical of that occurring in a nuclear reactor, what mass of uranium-235 is required to equal \(0.10 \%\) of the solar energy that falls on Earth in \(1.0\) day?
The table to the right gives the number of protons \((p)\) and neutrons \((n)\) for four isotopes. (a) Write the symbol for each of the isotopes. (b) Which of the isotopes is most likely to be unstable? (c) Which of the isotopes involves a magic number of protons and/or neutrons? (d) Which isotope will yield potassium-39 following positron emission? \begin{equation}\begin{array}{|c|c|c|c|}\hline & {\text { (i) }} & {\text { (ii) }} & {\text { (iii) }} & {\text { (iv) }} \\ \hline p & {19} & {19} & {20} & {20} \\ \hline n & {19} & {21} & {19} & {20} \\ \hline\end{array} \end{equation}
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