Chapter 42

A \({ }^{7} \mathrm{Li}\) nucleus with a kinetic energy of \(3.00 \mathrm{MeV}\) is sent toward a \({ }^{232} \mathrm{Th}\) nucleus. What is the least center-to- center separation between the two nuclei, assuming that the (more massive) \({ }^{232} \mathrm{Th}\) nucleus does not move?

In a Rutherford scattering experiment, assume that an incident alpha particle (radius \(1.80 \mathrm{fm}\) ) is headed directly toward a target gold nucleus (radius \(6.23 \mathrm{fm}\) ). What energy must the alpha particle have to just barely "touch" the gold nucleus?

The nuclide \({ }^{14} \mathrm{C}\) contains (a) how many protons and (b) how many neutrons?

A neutron star is a stellar object whose density is about that of nuclear matter, \(2 \times 10^{17} \mathrm{~kg} / \mathrm{m}^{3} .\) Suppose that the Sun were to collapse and become such a star without losing any of its present mass. What would be its radius?

What is the binding energy per nucleon of the americium isotope \({ }_{95}^{244} \mathrm{Am} ?\) Here are some atomic masses and the neutron mass. $$\begin{array}{lr}{\underline{\phantom{xx}}}_{95}^{244} \mathrm{Am} & 244.064279 \mathrm{u} \\\\\mathrm{n} & 1.008665 \mathrm{u}\end{array} { }^{1} \mathrm{H} \quad 1.007825 \mathrm{u}$$

What is the binding energy per nucleon of the europium isotope \(\frac{152}{63} \mathrm{Eu}\) ? Here are some atomic masses and the neutron mass. $$\begin{array}{lr}\frac{152}{63} \mathrm{Eu} & 151.921742 \mathrm{u} \\\\\mathrm{n} & 1.008665 \mathrm{u}\end{array} { }^{1} \mathrm{H} \quad 1.007825 \mathrm{u}$$

Because the neutron has no charge, its mass must be found in some way other than by using a mass spectrometer. When a neutron and a proton meet (assume both to be almost stationary), they combine and form a deuteron, emitting a gamma ray whose energy is \(2.2233 \mathrm{MeV}\). The masses of the proton and the deuteron are 1.007276467 u and \(2.013553212 \mathrm{u},\) respectively. Find the mass of the neutron from these data.

What is the binding energy per nucleon of the rutherfordium isotope \({ }_{104}^{259} \mathrm{Rf}\) ? Here are some atomic masses and the neutron mass. $$\begin{array}{lr}\frac{259}{104} \mathrm{Rf} & 259.10563 \mathrm{u} \\\\\mathrm{n} & 1.008665 \mathrm{u}\end{array} { }^{1} \mathrm{H} \quad 1.007825 \mathrm{u}$$

A periodic table might list the average atomic mass of magnesium as being \(24.312 \mathrm{u},\) which is the result of weighting the atomic masses of the magnesium isotopes according to their natural abundances on Earth. The three isotopes and their masses are \({ }^{24} \mathrm{Mg}(23.98504 \mathrm{u}),{ }^{25} \mathrm{Mg}(24.98584 \mathrm{u}),\) and \({ }^{26} \mathrm{Mg}(25.98259 \mathrm{u})\) The natural abundance of \({ }^{24} \mathrm{Mg}\) is \(78.99 \%\) by mass (that is, \(78.99 \%\) of the mass of a naturally occurring sample of magnesium is due to the presence of \({ }^{24} \mathrm{Mg}\) ). What is the abundance of (a) \({ }^{25} \mathrm{Mg}\) and (b) \({ }^{26} \mathrm{Mg} ?\)

(a) Show that the total binding energy \(E_{\mathrm{bc}}\) of a given nuclide is $$E_{\mathrm{be}}=Z \Delta_{\mathrm{H}}+N \Delta_{\mathrm{n}}-\Delta$$ where \(\Delta_{\mathrm{H}}\) is the mass excess of \({ }^{1} \mathrm{H}, \Delta_{\mathrm{n}}\) is the mass excess of a neutron, and \(\Delta\) is the mass excess of the given nuclide. (b) Using this method, calculate the binding energy per nucleon for \({ }^{197}\) Au. Compare your result with the value listed in Table \(42-1 .\) The needed mass excesses, rounded to three significant figures, are \(\Delta_{\mathrm{H}}=+7.29 \mathrm{MeV}\) \(\Delta_{n}=+8.07 \mathrm{MeV},\) and \(\Delta_{197}=-31.2 \mathrm{MeV} .\) Note the economy of calculation that results when mass excesses are used in place of the actual masses.

Go An \(\alpha\) particle ('He nucleus) is to be taken apart in the following steps. Give the energy (work) required for each step: (a) remove a proton, (b) remove a neutron, and (c) separate the remaining proton and neutron. For an \(\alpha\) particle, what are (d) the total binding energy and (e) the binding energy per nucleon? (f) Does either match an answer to (a), (b), or (c)? Here are some atomic masses and the neutron mass. $$ \begin{array}{llll} { }^{4} \mathrm{He} & 4.00260 \mathrm{u} & { }^{2} \mathrm{H} & 2.01410 \mathrm{u} \\ { }^{3} \mathrm{H} & 3.01605 \mathrm{u} & { }^{1} \mathrm{H} & 1.00783 \mathrm{u} \\ \mathrm{n} & 1.00867 \mathrm{u} & & \end{array} $$

A penny has a mass of 3.0 g. Calculate the energy that would be required to separate all the neutrons and protons in this coin from one another. For simplicity, assume that the penny is made entirely of \({ }^{63} \mathrm{Cu}\) atoms (of mass \(62.92960 \mathrm{u}\) ). The masses of the proton-plus-electron and the neutron are \(1.00783 \mathrm{u}\) and \(1.00866 \mathrm{u}\), respectively.

Cancer cells are more vulnerable to \(\mathrm{x}\) and gamma radiation than are healthy cells. In the past, the standard source for radiation therapy was radioactive \({ }^{60} \mathrm{Co},\) which decays, with a half-life of \(5.27 \mathrm{y},\) into an excited nuclear state of \({ }^{60} \mathrm{Ni}\). That nickel isotope then immediately emits two gamma-ray photons, each with an approximate energy of \(1.2 \mathrm{MeV}\). How many radioactive \({ }^{60} \mathrm{Co}\) nuclei are present in a \(6000 \mathrm{Ci}\) source of the type used in hospitals? (Energetic particles from linear accelerators are now used in radiation therapy.)

The half-life of a radioactive isotope is 140 d. How many days would it take for the decay rate of a sample of this isotope to fall to one-fourth of its initial value?

A radioactive nuclide has a half-life of \(30.0 \mathrm{y}\). What fraction of an initially pure sample of this nuclide will remain undecayed at the end of (a) \(60.0 \mathrm{y}\) and (b) \(90.0 \mathrm{y}\) ?

The plutonium isotope \({ }^{239} \mathrm{Pu}\) is produced as a by-product in nuclear reactors and hence is accumulating in our environment. It is radioactive, decaying with a half-life of \(2.41 \times 10^{4} \mathrm{y}\). (a) How many nuclei of Pu constitute a chemically lethal dose of \(2.00 \mathrm{mg} ?\) (b) What is the decay rate of this amount?

A radioactive isotope of mercury, \({ }^{197} \mathrm{Hg}\), decays to gold, \({ }^{197} \mathrm{Au},\) with a disintegration constant of \(0.0108 \mathrm{~h}^{-1} .\) (a) Calculate the half-life of the \({ }^{197} \mathrm{Hg}\). What fraction of a sample will remain at the end of (b) three half-lives and (c) 10.0 days?

The half-life of a particular radioactive isotope is \(6.5 \mathrm{~h}\). If there are initially \(48 \times 10^{19}\) atoms of this isotope, how many remain at the end of \(26 \mathrm{~h}\) ?

Consider an initially pure \(3.4 \mathrm{~g}\) sample of \({ }^{67} \mathrm{Ga},\) an isotope that has a half-life of \(78 \mathrm{~h}\). (a) What is its initial decay rate? (b) What is its decay rate \(48 \mathrm{~h}\) later?

When aboveground nuclear tests were conducted, the explosions shot radioactive dust into the upper atmosphere. Global air circulations then spread the dust worldwide before it settled out on ground and water. One such test was conducted in October 1976 . What fraction of the \({ }^{90} \mathrm{Sr}\) produced by that explosion still existed in October 2006 ? The half-life of \({ }^{90} \mathrm{Sr}\) is \(29 \mathrm{y}\).

Calculate the mass of a sample of (initially pure) \({ }^{40} \mathrm{~K}\) that has an initial decay rate of \(1.70 \times 10^{5}\) disintegrations/s. The isotope has a half-life of \(1.28 \times 10^{9} \mathrm{y}\).

A certain radionuclide is being manufactured in a cyclotron at a constant rate \(R\). It is also decaying with disintegration constant \(\lambda\). Assume that the production process has been going on for a time that is much longer than the half-life of the radionuclide. (a) Show that the number of radioactive nuclei present after such time remains constant and is given by \(N=R / \lambda .\) (b) Now show that this result holds no matter how many radioactive nuclei were present initially. The nuclide is said to be in secular equilibrium with its source; in this state its decay rate is just equal to its production rate.

Plutonium isotope \({ }^{239} \mathrm{Pu}\) decays by alpha decay with a halflife of \(24100 \mathrm{y}\). How many milligrams of helium are produced by an initially pure \(12.0 \mathrm{~g}\) sample of \({ }^{239} \mathrm{Pu}\) at the end of \(20000 \mathrm{y} ?\) (Consider only the helium produced directly by the plutonium and not by any by-products of the decay process.)

The radionuclide \({ }^{64} \mathrm{Cu}\) has a half-life of \(12.7 \mathrm{~h}\). If a sample contains \(5.50 \mathrm{~g}\) of initially pure \({ }^{64} \mathrm{Cu}\) at \(t=0,\) how much of it will decay between \(t=14.0 \mathrm{~h}\) and \(t=16.0 \mathrm{~h} ?\)

The radionuclide \({ }^{56} \mathrm{Mn}\) has a half-life of \(2.58 \mathrm{~h}\) and is produced in a cyclotron by bombarding a manganese target with deuterons. The target contains only the stable manganese isotope \({ }^{55} \mathrm{Mn},\) and the manganese \(-\) deuteron reaction that produces \({ }^{56} \mathrm{Mn}\) is $${ }^{55} \mathrm{Mn}+\mathrm{d} \rightarrow{ }^{56} \mathrm{Mn}+\mathrm{p}$$ If the bombardment lasts much longer than the half-life of \({ }^{56} \mathrm{Mn}\), the activity of the \({ }^{56} \mathrm{Mn}\) produced in the target reaches a final value of \(8.88 \times 10^{10}\) Bq. (a) At what rate is \({ }^{56}\) Mn being produced? (b) How many \({ }^{56}\) Mn nuclei are then in the target? (c) What is their total mass?

A source contains two phosphorus radionuclides, \({ }^{32} \mathrm{P}\left(T_{1 / 2}=\right.\) \(14.3 \mathrm{~d}\) ) and \({ }^{33} \mathrm{P}\left(T_{1 / 2}=25.3 \mathrm{~d}\right.\) ). Initially, \(10.0 \%\) of the decays come from \({ }^{33} \mathrm{P}\). How long must one wait until \(90.0 \%\) do so?

A \(1.00 \mathrm{~g}\) sample of samarium emits alpha particles at a rate of 120 particles/s. The responsible isotope is \({ }^{147} \mathrm{Sm},\) whose natural abundance in bulk samarium is \(15.0 \%\). Calculate the half-life.

A radioactive sample intended for irradiation of a hospital patient is prepared at a nearby laboratory. The sample has a half-life of \(83.61 \mathrm{~h}\). What should its initial activity be if its activity is to be \(7.4 \times 10^{8} \mathrm{~Bq}\) when it is used to irradiate the patient \(24 \mathrm{~h}\) later?

In \(1992,\) Swiss police arrested two men who were attempting to smuggle osmium out of Eastern Europe for a clandestine sale. However, by error, the smugglers had picked up \({ }^{137} \mathrm{Cs}\). Reportedly, each smuggler was carrying a \(1.0 \mathrm{~g}\) sample of \({ }^{137} \mathrm{Cs}\) in a pocket! In (a) bequerels and (b) curies, what was the activity of each sample? The isotope \({ }^{137} \mathrm{Cs}\) has a half-life of \(30.2 \mathrm{y}\). (The activities of radioisotopes commonly used in hospitals range up to a few millicuries.)

The radioactive nuclide \({ }^{99}\) Tc can be injected into a patient's bloodstream in order to monitor the blood flow, measure the blood volume, or find a tumor, among other goals. The nuclide is produced in a hospital by a "cow" containing \({ }^{99} \mathrm{Mo},\) a radioactive nuclide that decays to \({ }^{99} \mathrm{Tc}\) with a half-life of \(67 \mathrm{~h}\). Once a day, the cow is "milked" for its \({ }^{99} \mathrm{Tc},\) which is produced in an excited state by the \({ }^{99} \mathrm{Mo} ;\) the \({ }^{99} \mathrm{Tc}\) de- excites to its lowest energy state by emitting a gamma-ray photon, which is recorded by detectors placed around the patient. The de-excitation has a half- life of \(6.0 \mathrm{~h}\). (a) By what process does \({ }^{99}\) Mo decay to \({ }^{99} \mathrm{Tc} ?\) (b) If a patient is injected with an \(8.2 \times 10^{7}\) Bq sample of \({ }^{99} \mathrm{Tc}\), how many gamma-ray photons are initially produced within the patient each second? (c) If the emission rate of gamma-ray photons from a small tumor that has collected \({ }^{99} \mathrm{Tc}\) is 38 per second at a certain time, how many excited-state \({ }^{99} \mathrm{Tc}\) are located in the tumor at that time?

How much energy is released when a \({ }^{238} \mathrm{U}\) nucleus decays by emitting (a) an alpha particle and (b) a sequence of neutron, proton, neutron, proton? (c) Convince yourself both by reasoned argument and by direct calculation that the difference between these two numbers is just the total binding energy of the alpha particle. (d) Find that binding energy. Some needed atomic and particle masses are $$ \begin{array}{llll} { }^{238} \mathrm{U} & 238.05079 \mathrm{u} & { }^{234} \mathrm{Th} & 234.04363 \mathrm{u} \\ { }^{237} \mathrm{U} & 237.04873 \mathrm{u} & { }^{4} \mathrm{He} & 4.00260 \mathrm{u} \\ { }^{236} \mathrm{~Pa} & 236.04891 \mathrm{u} & { }^{1} \mathrm{H} & 1.00783 \mathrm{u} \\ { }^{235} \mathrm{~Pa} & 235.04544 \mathrm{u} & \mathrm{n} & 1.00866 \mathrm{u} \end{array} $$

Generally, more massive nuclides tend to be more unstable to alpha decay. For example, the most stable isotope of uranium, \({ }^{238} \mathrm{U},\) has an alpha decay half-life of \(4.5 \times 10^{9} \mathrm{y} .\) The most stable isotope of plutonium is \({ }^{244} \mathrm{Pu}\) with an \(8.0 \times 10^{7} \mathrm{y}\) half-life, and for curium we have \({ }^{248} \mathrm{Cm}\) and \(3.4 \times 10^{5} \mathrm{y}\). When half of an original sample of \({ }^{238} \mathrm{U}\) has decayed, what fraction of the original sample of (a) plutonium and (b) curium is left?

Large radionuclides emit an alpha particle rather than other combinations of nucleons because the alpha particle has such a stable, tightly bound structure. To confirm this statement, calculate the disintegration energies for these hypothetical decay processes and discuss the meaning of your findings: (a) \({ }^{235} \mathrm{U} \rightarrow{ }^{232} \mathrm{Th}+{ }^{3} \mathrm{He},\) (b) \({ }^{235} \mathrm{U} \rightarrow{ }^{231} \mathrm{Th}+{ }^{4} \mathrm{He}\) (c) \({ }^{235} \mathrm{U} \rightarrow{ }^{230} \mathrm{Th}+{ }^{5} \mathrm{He}\) The needed atomic masses are $$ \begin{array}{llll} { }^{232} \mathrm{Th} & 232.0381 \mathrm{u} & { }^{3} \mathrm{He} & 3.0160 \mathrm{u} \\ { }^{231} \mathrm{Th} & 231.0363 \mathrm{u} & { }^{4} \mathrm{He} & 4.0026 \mathrm{u} \\ { }^{230} \mathrm{Th} & 230.0331 \mathrm{u} & { }^{5} \mathrm{He} & 5.0122 \mathrm{u} \\ { }^{235} \mathrm{U} & 235.0429 \mathrm{u} & & \end{array} $$

Under certain rare circumstances, a nucleus can decay by emitting a particle more massive than an alpha particle. Consider the decays $${ }^{223} \mathrm{Ra} \rightarrow{ }^{209} \mathrm{~Pb}+{ }^{14} \mathrm{C} \quad \text { and } \quad{ }^{223} \mathrm{Ra} \rightarrow{ }^{219} \mathrm{Rn}+{ }^{4} \mathrm{He}$$ Calculate the \(Q\) value for the (a) first and (b) second decay and determine that both are energetically possible. (c) The Coulomb barrier height for alpha-particle emission is \(30.0 \mathrm{MeV}\). What is the barrier height for \({ }^{14} \mathrm{C}\) emission? (Be careful about the nuclear radii.) The needed atomic masses are $$ \begin{aligned} &\begin{array}{llll} { }^{223} \mathrm{Ra} & 223.01850 \mathrm{u} & { }^{14} \mathrm{C} & 14.00324 \mathrm{u} \end{array}\\\ &{ }^{209} \mathrm{~Pb} \quad 208.98107 \mathrm{u} \quad{ }^{4} \mathrm{He} \quad 4.00260 \mathrm{u}\\\ &{ }^{219} \mathrm{Rn} \quad 219.00948 \mathrm{u} \end{aligned} $$

The cesium isotope \({ }^{137} \mathrm{Cs}\) is present in the fallout from aboveground detonations of nuclear bombs. Because it decays with a slow \((30.2 \mathrm{y})\) half-life into \({ }^{137} \mathrm{Ba},\) releasing considerable energy in the process, it is of environmental concern. The atomic masses of the Cs and \(\mathrm{Ba}\) are 136.9071 and \(136.9058 \mathrm{u},\) respectively; calculate the total energy released in such a decay.

Some radionuclides decay by capturing one of their own atomic electrons, a \(K\) -shell electron, say. An example is $${ }^{49} \mathrm{~V}+\mathrm{e}^{-} \rightarrow{ }^{49} \mathrm{Ti}+\nu, \quad T_{1 / 2}=331 \mathrm{~d}$$ Show that the disintegration energy \(Q\) for this process is given by $$Q=\left(m_{\mathrm{V}}-m_{\mathrm{Ti}}\right) c^{2}-E_{K}$$ where \(m_{\mathrm{v}}\) and \(m_{\mathrm{Ti}}\) are the atomic masses of \({ }^{49} \mathrm{~V}\) and \({ }^{49} \mathrm{Ti},\) respectively, and \(E_{K}\) is the binding energy of the vanadium \(K\) -shell electron. (Hint: Put \(\mathbf{m}_{\mathrm{V}}\) and \(\mathbf{m}_{\mathrm{Ti}}\) as the corresponding nuclear masses and then add in enough electrons to use the atomic masses.)

The radionuclide \({ }^{11} \mathrm{C}\) decays according to $${ }^{11} \mathrm{C} \rightarrow{ }^{11} \mathrm{~B}+\mathrm{e}^{+}+\nu, \quad T_{1 / 2}=20.3 \mathrm{~min}$$ The maximum energy of the emitted positrons is \(0.960 \mathrm{MeV}\). (a) Show that the disintegration energy \(Q\) for this process is given by $$Q=\left(m_{\mathrm{C}}-m_{\mathrm{B}}-2 m_{\mathrm{e}}\right) c^{2}$$ where \(m_{\mathrm{C}}\) and \(m_{\mathrm{B}}\) are the atomic masses of \({ }^{11} \mathrm{C}\) and \({ }^{11} \mathrm{~B}\), respectively, and \(m_{e}\) is the mass of a positron. (b) Given the mass values \(m_{\mathrm{C}}=11.011434 \mathrm{u}, m_{\mathrm{B}}=11.009305 \mathrm{u},\) and \(m_{\mathrm{e}}=\) \(0.0005486 \mathrm{u},\) calculate \(Q\) and compare it with the maximum energy of the emitted positron given above. (Hint: Let \(\mathbf{m}_{\mathrm{C}}\) and \(\mathbf{m}_{\mathrm{B}}\) be the nuclear masses and then add in enough electrons to use the atomic masses.)

A \(5.00 \mathrm{~g}\) charcoal sample from an ancient fire pit has a \({ }^{14} \mathrm{C}\) activity of 63.0 disintegrations/min. A living tree has a \({ }^{14} \mathrm{C}\) activity of 15.3 disintegrations/min per \(1.00 \mathrm{~g}\). The half-life of \({ }^{14} \mathrm{C}\) is \(5730 \mathrm{y}\). How old is the charcoal sample?

The isotope \({ }^{238} \mathrm{U}\) decays to \({ }^{206} \mathrm{~Pb}\) with a half-life of \(4.47 \times 10^{9} \mathrm{y}\). Although the decay occurs in many individual steps, the first step has by far the longest half-life; therefore, one can often consider the decay to go directly to lead. That is, $${ }^{238} \mathrm{U} \rightarrow{ }^{206} \mathrm{~Pb}+\text { various decay products. }$$ A rock is found to contain \(4.20 \mathrm{mg}\) of \({ }^{238} \mathrm{U}\) and \(2.135 \mathrm{mg}\) of \({ }^{206} \mathrm{~Pb}\). Assume that the rock contained no lead at formation, so all the lead now present arose from the decay of uranium. How many atoms of (a) \({ }^{238} \mathrm{U}\) and (b) \({ }^{206} \mathrm{~Pb}\) does the rock now contain? (c) How many atoms of \({ }^{238} \mathrm{U}\) did the rock contain at formation? (d) What is the age of the rock?

The isotope \({ }^{40} \mathrm{~K}\) can decay to either \({ }^{40} \mathrm{Ca}\) or \({ }^{40} \mathrm{Ar} ;\) assume both decays have a half-life of \(1.26 \times 10^{9} \mathrm{y}\). The ratio of the Ca produced to the Ar produced is \(8.54 / 1=8.54 .\) A sample originally had only \({ }^{40} \mathrm{~K}\). It now has equal amounts of \({ }^{40} \mathrm{~K}\) and \({ }^{40} \mathrm{Ar}\); that is, the ratio of \(\mathrm{K}\) to \(\mathrm{Ar}\) is \(1 / 1=1 .\) How old is the sample? (Hint: Work this like other radioactive-dating problems, except that this decay has two products.)

The nuclide \({ }^{198} \mathrm{Au},\) with a half-life of \(2.70 \mathrm{~d},\) is used in cancer therapy. What mass of this nuclide is required to produce an activity of \(250 \mathrm{Ci} ?\)

A radiation detector records 8700 counts in 1.00 min. Assuming that the detector records all decays, what is the activity of the radiation source in (a) becquerels and (b) curies?

An organic sample of mass \(4.00 \mathrm{~kg}\) absorbs \(2.00 \mathrm{~mJ}\) via slow neutron radiation \((\mathrm{RBE}=5) .\) What is the dose equivalent (mSv)?

A \(75 \mathrm{~kg}\) person receives a whole-body radiation dose of \(2.4 \times 10^{-4} \mathrm{~Gy},\) delivered by alpha particles for which the \(\mathrm{RBE}\) factor is 12. Calculate (a) the absorbed energy in joules and the dose equivalent in (b) sieverts and (c) rem.

An \(85 \mathrm{~kg}\) worker at a breeder reactor plant accidentally ingests \(2.5 \mathrm{mg}\) of \({ }^{239} \mathrm{Pu}\) dust. This isotope has a half- life of \(24100 \mathrm{y}\), decaying by alpha decay. The energy of the emitted alpha particles is \(5.2 \mathrm{MeV},\) with an \(\mathrm{RBE}\) factor of \(13 .\) Assume that the plutonium resides in the worker's body for \(12 \mathrm{~h}\) (it is eliminated naturally by the digestive system rather than being absorbed by any of the internal organs) and that \(95 \%\) of the emitted alpha particles are stopped within the body. Calculate (a) the number of plutonium atoms ingested, (b) the number that decay during the \(12 \mathrm{~h},\) (c) the energy absorbed by the body, (d) the resulting physical dose in grays, and (e) the dose equivalent in sieverts.

In a certain rock, the ratio of lead atoms to uranium atoms is \(0.300 .\) Assume that uranium has a half-life of \(4.47 \times 10^{9} \mathrm{y}\) and that the rock had no lead atoms when it formed. How old is the rock?

A typical chest \(x\) -ray radiation dose is \(250 \mu \mathrm{Sv},\) delivered by x rays with an RBE factor of \(0.85 .\) Assuming that the mass of the exposed tissue is one-half the patient's mass of \(88 \mathrm{~kg}\), calculate the energy absorbed in joules.

How many years are needed to reduce the activity of \({ }^{14} \mathrm{C}\) to 0.020 of its original activity? The half-life of \({ }^{14} \mathrm{C}\) is \(5730 \mathrm{y}\).

Radioactive element \(A A\) can decay to either element \(B B\) or element \(C C\). The decay depends on chance, but the ratio of the resulting number of \(B B\) atoms to the resulting number of \(C C\) atoms is always \(2 / 1 .\) The decay has a half-life of 8.00 days. We start with a sample of pure \(A A .\) How long must we wait until the number of \(C C\) atoms is 1.50 times the number of \(A A\) atoms?

A radium source contains \(1.00 \mathrm{mg}\) of \({ }^{226} \mathrm{Ra},\) which decays with a half-life of \(1600 \mathrm{y}\) to produce \({ }^{222} \mathrm{Rn},\) a noble gas. This radon isotope in turn decays by alpha emission with a half-life of \(3.82 \mathrm{~d}\). If this process continues for a time much longer than the half-life of \({ }^{222} \mathrm{Rn}\), the \({ }^{222}\) Rn decay rate reaches a limiting value that matches the rate at which \({ }^{222} \mathrm{Rn}\) is being produced, which is approximately constant because of the relatively long half-life of \({ }^{226} \mathrm{Ra}\). For the source under this limiting condition, what are (a) the activity of \({ }^{226} \mathrm{Ra},\) (b) the activity of \({ }^{222} \mathrm{Rn},\) and \((\mathrm{c})\) the total mass of \({ }^{222} \mathrm{Rn} ?\)