Chapter 31

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

For \({ }_{82}^{208} \mathrm{~Pb}\) 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.

In each of the following cases, what element does the symbol \(\mathrm{X}\) represent and how many neutrons are in the nucleus: (a) 195 78 \(\mathrm{X}\), (b) \(\frac{32}{16} \mathrm{X}\), (c) \({ }_{29}^{63} \mathrm{X}\), (d) \({ }_{5}^{11} \mathrm{X}\), and (e) \(\frac{239}{94} X\) ? Use the periodic table on the in side of the back cover as needed.

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?

The ratio \(r_{X} / r_{T}\) of the radius of an unknown nucleus \(A_{X}\) to a tritium nucleus \({ }_{1}^{3} T\) is \(\frac{r_{\mathrm{X}}}{r_{\mathrm{T}}}=1.10 .\) Both nuclei contain the same number of neutrons. Identify the unknown nucleus in the form \(A_{X}\). Use the periodic table on the inside of the back cover as needed.

An unknown nucleus contains 70 neutrons and has twice the volume of the nickel \(60 \mathrm{Ni}\) nucleus. Identify the unknown nucleus in the form \(A_{X}\). Use the periodic table on the inside of the back cover as needed.

Determine the mass defect (in atomic mass units) for (a) helium \({ }_{2}^{3} \mathrm{He},\) which has an atomic mass of \(3.016030 \mathrm{u},\) and \((\mathrm{b})\) the isotope of hydrogen known as tritium \(\frac{3}{1} \mathrm{~T}\) which has an atomic mass of \(3.016050 \mathrm{u}\). (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.

The earth revolves around the sun, and the two represent a bound system that has a binding energy of \(2.6 \times 10^{33} \mathrm{~J}\). 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?

Find the binding energy (in \(\mathrm{MeV}\) ) for lithium \(\frac{7}{3} \mathrm{Li}\) (atomic mass \(=7.016003 \mathrm{u}\) ).

Two isotopes of a certain element have binding energies that differ by \(5.03 \mathrm{MeV}\). 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.

(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 \({ }_{7}^{14} N\) into nitrogen \({ }_{7}^{13} \mathrm{~N}\) and a neutron takes energy equal to the binding energy of the neutron, as shown below: $$ \frac{14}{7} N+\text { Energy } \rightarrow \frac{13}{7} N+\frac{1}{0} n $$ Find the energy (in MeV) that binds the neutron to the \(\frac{14}{7} N\) nucleus by considering the mass of \(13 \mathrm{~N}\) (atomic mass \(=13.005738 \mathrm{u}\) ) and the mass of \(\frac{1}{0^{\mathrm{n}}}\) (atomic mass \(=1.008\) \(665 \mathrm{u}\) ), as compared to the mass of \({ }_{7}^{14} \mathrm{~N}\) ( atomic mass \(=14.003074 \mathrm{u}\) ). (b) Similarly, one can speak of the energy that binds a single proton to the\(\frac{14}{7} \mathrm{~N}\) nucleus: $$ { }_{7}^{14} \mathrm{~N}+\text { Energy } \rightarrow{ }_{6}^{13} \mathrm{C}+{ }_{1}^{1} \mathrm{H} $$ Following the procedure outlined in part (a), determine the energy (in MeV) that binds the proton (atomic mass \(=1.007825 \mathrm{u}\) ) to the \(\frac{14}{7} \mathrm{~N}\) nucleus. The atomic mass of carbon \(\frac{13}{6} \mathrm{C}\) is \(13.003335 \mathrm{u}\) (c) Which nucleon is more tightly bound, the neutron or the proton?

Write the \(\beta^{-}\) decay process for \(\underset{16}^{35} \mathrm{~S}\), including the chemical symbol and values for \(Z\) and \(A\).

Write the \(\beta^{-}\) decay process for \(\frac{35}{16} \mathrm{~S}\), including the chemical symbol and values for \(Z\) and \(A\)

In the form \({ }_{Z}^{A} X\), identify the daughter nucleus that results when (a) plutonium \({ }_{94}^{242} \mathrm{Pu}\) undergoes \(\alpha\) decay, (b) sodium \({ }_{11}^{24}\) Na undergoes \(\beta^{-}\) decay, and (c) nitrogen \(\frac{13}{7} \mathrm{~N}\) undergoes \(\beta^{+}\) decay.

Write the \(\beta^{-}\) decay process for \(\frac{35}{16} \mathrm{~S}\), including the chemical symbol and values for \(Z\) and \(A\)

Find the energy (in \(\mathrm{MeV}\) ) released when \(\alpha\) decay converts radium \({ }_{88}^{226} \mathrm{Ra}\) (atomic mass \(=226.02540 \mathrm{u}\) ) into radon \(\frac{222}{86} \mathrm{Rn}\) (atomic mass \(\left.=222.01757 \mathrm{u}\right)\). The atomic mass of an \alpha particle is \(4.002603 \mathrm{u}\).

Find the energy (in MeV) released when \(\alpha\) decay converts radium 226 Ra (atomic mass \(=226.02540 \mathrm{u}\) ) into radon \(\frac{222}{86} \mathrm{Rn}(\) atomic mass \(=222.01757 \mathrm{u}) .\) The atomic mass of an \(\alpha\) particle is \(4.002603 \mathrm{u}\).

\(\alpha\) 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) 212 84 \(\mathrm{P} \circ\) and (b) \({ }_{92}^{232} \mathrm{U}\)

decay occurs for each of the following nuclei. Write the decay process for each, including the chemical symbols nd values for \(Z\) and \(A\) for the daughter nuclei: (a) \({ }_{84}^{212} \mathrm{P} \circ\) and \((\mathrm{b}){ }_{92}^{232} \mathrm{U}\).

Multiple-Concept Example 7 reviews the concepts needed to solve this problem. When uranium 235 U decays, it emits (among other things) a \(\gamma\) ray that has a wavelength of \(1.14 \times 10^{-11} \mathrm{~m} .\) Determine the energy (in \(\mathrm{MeV}\) ) of this \(\gamma\) ray.

Find the energy released when lead \({ }_{82}^{211} \mathrm{~Pb}\) (atomic mass \(=210.988735 \mathrm{u}\) ) undergoes \(\beta^{-}\) decay to become bismuth \({ }_{83}^{211} \mathrm{Bi}\) (atomic mass \(\left.=210.987255 \mathrm{u}\right)\).

Find the energy released when lead \(\underset{82}^{211} \mathrm{~Pb}\) (atomic mass \(=210.988735 \mathrm{u}\) ) undergoes \(\beta^{-}\) decay to become bismuth \(\underset{83}{211} \mathrm{Bi}\) (atomic mass \(\left.=210.987255 \mathrm{u}\right)\).

Refer to Interactive Solution 31.25 at to review a model for solving this type of problem. Polonium \({ }_{84}^{210} \mathrm{P} \circ(\) atomic mass \(=209.982848 \mathrm{u})\) undergoes \(\alpha\) decay. Assuming that all the released energy is in the form of kinetic energy of the \(\alpha\) particle (atomic mass \(=4.002603 \mathrm{u}\) ) and ignoring the recoil of the daughter nucleus (lead \(826 \mathrm{p}, 205.974440 \mathrm{u}),\) find the speed of the \(\alpha\) particle. Ignore relativistic effects.

Rado \(220 \mathrm{Rn}\) produces a daughter nucleus that is radioactive. The daughter, in turn, produces its own radioactive daughter, and so on. This process continues until lead \(83 \underline{208} \mathrm{~b}\) is reached. What are the total number \(N_{\alpha}\) of \(\alpha\) particles and the total number \(N_{\beta}\) of \(\beta\) particles that are generated in this series of radioactive decays?

Find the energy (in MeV) released when \(\beta^{+}\) decay converts sodium \(\underset{11}{22} \mathrm{Na}\) (atomic mass \(=21.994434 \mathrm{u}\) ) into neon \(\underset{10}{22} \mathrm{Ne}\) (atomic mass \(=21.991383 \mathrm{u}\) ). Notice that the atomic mass for \(22 \mathrm{Na}\) includes the mass of 11 electrons, whereas the atomic mass for 22 Ne includes the mass of only 10 electrons.

Interactive LearningWare 31.1 at reviews the concepts that are involved in this problem. An isotope of beryllium (atomic mass \(=7.017 \mathrm{u}\) ) emits a \(\gamma\) ray and recoils with a speed of \(2.19 \times 10^{4} \mathrm{~m} / \mathrm{s}\). Assuming that the beryllium nucleus is stationary to begin with, find the wavelength of the \(\gamma\) ray.

The isotope \({ }_{88}^{224} \mathrm{Ra}\) of radium has a decay constant of \(2.19 \times 10^{-6} \mathrm{~s}^{-1}\). What is the halflife (in days) of this isotope?

The isotope \({ }_{88}^{224}\) Ra of radium has a decay constant of \(2.19 \times 10^{-6} \mathrm{~s}^{-1}\). What is the halflife (in days) of this isotope?

The half-lives in two different samples, \(A\) and \(B\), of radioactive nuclei are related according to \(T_{1 / 2, \mathrm{~B}}=\frac{1}{2} T_{1 / 2, \mathrm{~A}}\) 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 \(\bar{f}\)

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

A device used in radiation therapy for cancer contains \(0.50 \mathrm{~g}\) of cobalt \(\frac{60}{27} \mathrm{Co}\) \((59.933819 \mathrm{u}) .\) The half-life of \(\underset{27}{60} \mathrm{C} \circ\) is \(5.27 \mathrm{yr}\). Determine the activity of the radioactive material.

Iodine 131 53 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 \(\quad 131\) 53 remains after \(30.0\) days?

Iodine \(\frac{131}{53} \mathrm{I}\) 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 \(\frac{131}{53}\) I remains after 30.0 days?

The number of radioactive nuclei present at the start of an experiment is \(4.60 \times 10^{15}\). The number present twenty days later is \(8.14 \times 10^{14}\). What is the half-life (in days) of the nuclei?

To make the dial of a watch glow in the dark, \(1.000 \times 10^{-9} \mathrm{~kg}\) of radium \(\frac{226}{88} \mathrm{Ra}\) is used. The half-life of this isotope has a value of \(1.60 \times 10^{3}\) yr. How many kilograms of radium disappear while the watch is in use for fifty years?

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 \(3.7 \times 10^{10} \mathrm{~Bq},\) determine the activity (in disintegrations per second) of one gram of radium \(\frac{226}{88} \mathrm{Ra}\left(T_{1 / 2}=1.6 \times 10^{3} \mathrm{yr}\right)\)

The isotope \({ }_{79}^{198} \mathrm{Au}\) (atomic mass \(=197.968 \mathrm{u}\) ) 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 \(315 \mathrm{Ci}\) ?

The isotope \(\frac{198}{79} \mathrm{Au}\) (atomic mass \(=197.968 \mathrm{u}\) ) 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 \(315 \mathrm{Ci} ?\)

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 \(\mathrm{B}\) nuclei. The half-life of species \(\mathrm{B}\) is 1.50 days. Find the half-life of species \(\mathrm{A}\).

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 \({ }_{6}^{14} \mathrm{C}\) atoms remains?

The half-life for the \(\alpha\) decay of uranium \({ }_{92}^{238} \mathrm{U}\) is \(4.47 \times 10^{9}\) yr. Determine the age (in years) of a rock specimen that contains sixty percent of its original number of \(\underset{92}^{238} \mathrm{U}\) atoms

The half-life for the \(\alpha\) decay of uranium \(\frac{238}{92} \mathrm{U}\) is \(4.47 \times 10^{9} \mathrm{yr} .\) Determine the age (in years) of a rock specimen that contains sixty percent of its original number of \(\frac{238}{92} \mathrm{U}\) atoms.

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 \(0.100 \mathrm{~Bq}\) per gram of carbon is measured to within an accuracy of \(\pm 10.0 \%\). 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 \(\frac{14}{6} \mathrm{C}\) nor in the value of 0.23 Bq per gram of carbon in a living organism.

In electrically neutral atoms, how many (a) protons are in the uranium \({ }_{92}^{238} \mathrm{U}\) nucleus, (b) neutrons are in the mercury \({ }_{80}^{202} \mathrm{Hg}\) nucleus, and (c) electrons are in orbit about the niobium \({ }_{41}^{93} \mathrm{Nb}\) nucleus?

Osmium \(\frac{191}{76} \mathrm{Os}\) (atomic mass \(\left.=190.960920 \mathrm{u}\right)\) is converted into Iridium \(\frac{191}{77} \mathrm{Ir}\) (atomic mass \(=190.960584 \mathrm{u}\) ) via \(\beta^{-}\) decay. What is the energy (in \(\mathrm{MeV}\) ) released in this process?

Osmium \({ }_{76}^{191} \mathrm{Os}\) (atomic mass \(=190.960920 \mathrm{u}\) ) is converted into Iridium \(\frac{191}{77} \mathrm{Ir}\) (atomic mass \(=190.960584 \mathrm{u}\) ) via \(\beta^{-}\) decay. What is the energy (in \(\mathrm{MeV}\) ) released in this process?

Complete the following decay processes by stating what the symbol "X" represents \(\left(\mathrm{X}=\alpha, \beta^{-}, \beta^{+}\right.\) or \(\left.\gamma\right)\) a. \(\quad{ }_{82}^{211} \mathrm{~Pb} \rightarrow{ }_{83}^{211} \mathrm{Bi}+\mathrm{X}\) b. \(\quad{ }_{6}^{11} \mathrm{C} \rightarrow{ }_{5}^{11} \mathrm{~B}+\mathrm{X}\) c. \(\quad{ }_{90}^{231} \mathrm{Th}^{*} \rightarrow{ }_{90}^{231} \mathrm{Th}+\mathrm{X}\) d. \(\quad{ }_{84}^{210} \mathrm{P} \circ \rightarrow{ }_{82}^{206} \mathrm{~Pb}+\mathrm{X}\)

The photomultiplier tube in a commercial scintillation counter contains 15 of the special electrodes, or dynodes. Each dynode produces 3 electrons for every electron that strikes it. One photoelectron strikes the first dynode. What is the maximum number of electrons that strike the 15 th dynode?

A sample of ore containing radioactive strontium \({ }_{38}^{90} \mathrm{Sr}\) has an activity of \(6.0 \times 10^{5} \mathrm{~Bq}\) The atomic mass of strontium is \(89.908 \mathrm{u},\) and its half-life is 29.1 yr. How many grams of strontium are in the sample?

Determine the symbol \(A_{Z}^{A} X\) for the parent nucleus whose \(\alpha\) decay produces the same daughter as the \(\beta^{-}\) decay of thallium \(\underset{81}^{208} \mathrm{~T}\).