||

1. INT What initial speed must an alpha particle have to just touch the surface of a ${}^{197}\text{Au}$ gold nucleus before being turned back? Assume the nucleus stays at rest.

   Hint: The alpha particle is not a point particle.

2. What is the initial energy (in MeV) of the alpha particle?

||| INT Particle accelerators fire protons at target nuclei so that investigators can study the nuclear reactions that occur. In one experiment, the proton needs to have 20 MeV of kinetic energy as it impacts a ${}^{207}\text{Pb}$ nucleus. With what initial kinetic energy (in MeV) must the proton be fired toward the lead target? Assume the nucleus stays at rest.

Hint: The proton is not a point particle.

|| INT Stars are powered by nuclear reactions that fuse hydrogen into helium. The fate of many stars, once most of the hydrogen is used up, is to collapse, under gravitational pull, into a neutron star. The force of gravity becomes so large that protons and electrons are fused into neutrons in the reaction ${\text{p}}^{+}+{\text{e}}^{-}\to \text{n}+\nu$. The entire star is then a tightly packed ball of neutrons with the density of nuclear matter.

1. Suppose the sun collapses into a neutron star. What will its radius be? Give your answer in km.

2. The sun’s rotation period is now 27 days. What will its rotation period be after it collapses?

Rapidly rotating neutron stars emit pulses of radio waves at the rotation frequency and are known as pulsars.

|| The element gallium has two stable isotopes: ${}^{69}\text{Ga}$ with an atomic mass of 68.92 u and ${}^{71}\text{Ga}$ with an atomic mass of 70.92 u. A periodic table shows that the chemical atomic mass of gallium is 69.72 u. What is the percent abundance of ${}^{69}\text{Ga}?$

|| You learned in Chapter 41 that the binding energy of the electron in a hydrogen atom is 13.6 eV.

1. By how much does the mass decrease when a hydrogen atom is formed from a proton and an electron? Give your answer both in atomic mass units and as a percentage of the mass of the hydrogen atom.

2. By how much does the mass decrease when a helium nucleus is formed from two protons and two neutrons? Give your answer both in atomic mass units and as a percentage of the mass of the helium nucleus.

3. Compare your answers to parts a and b. Why do you hear it said that mass is “lost” in nuclear reactions but not in chemical reactions?

|| Use the graph of binding energy to estimate the total energy released if three ${}^4\text{He}$ nuclei fuse together to form a ${}^{12}\text{C}$ nucleus.

|| Use the graph of binding energy to estimate the total energy released if a nucleus with mass number 240 fissions into two nuclei with mass number 120.

|| Could a ${}^{56}\text{Fe}$ nucleus fission into two ${}^{28}\text{Al}$ nuclei? Your answer, which should include some calculations, should be based on the curve of binding energy.

|| INT What energy (in MeV) alpha particle has a de Broglie wavelength equal to the diameter of a ${}^{238}\text{U}$ nucleus?

|| The activity of a sample of the cesium isotope ${}^{137}\text{Cs}$, with a half-life of 30 years, is $2.0\times {10}^8\text{ Bq}$. Many years later, after the sample has fully decayed, how many beta particles will have been emitted?

|| What is the age in years of a bone in which the ${}^{14}\text{C}{/}^{12}\text{C}$ ratio is measured to be $1.65\times {10}^{-13}?$

|| A 1 Ci source of radiation is a significant source. ${}^{238}\text{U}$ is an alpha emitter. What mass of ${}^{238}\text{U}$ has an activity of 1 Ci?

| ${}^{137}\text{Cs}$ is a common product of nuclear fission. Suppose an accident spills 550 mCi of ${}^{137}\text{Cs}$ in a lab room.

1. What mass of ${}^{137}\text{Cs}$ is spilled?

2. If the spill is not cleaned up, how long will it take until the radiation level drops to an acceptable level, for a room this size, of 25 mCi?

|| A 115 mCi radioactive tracer is made in a nuclear reactor. When it is delivered to a hospital 16 hours later its activity is 95 mCi. The lowest usable level of activity is 10 mCi.

1. What is the tracer’s half-life?

2. For how long after delivery is the sample usable?

|| Beta-plus decay is ${}^A\text{X}{}_Z{\to }^A\text{Y}{}_{Z-1}+\text{e}{}^{+}+\nu$.

1. Determine the mass threshold for beta-plus decay. That is, what is the minimum atomic mass $m_{\text{X}}$ for which this decay is energetically possible? Your answer will be in terms of the atomic mass $m_{\text{Y}}$ and the electron mass $m_{\text{e}}$.

2. Can ${}^{13}\text{N}$ undergo beta-plus decay into ${}^{13}\text{C}?$ If so, how much energy is released in the decay?

||| INT The radium isotope ${}^{223}\text{Ra}$, an alpha emitter, has a half-life of 11.43 days. You happen to have a 1.0 g cube of ${}^{223}\text{Ra}$, so you decide to use it to boil water for tea. You fill a well-insulated container with 100 mL of water at $18\text{°}\text{C}$ and drop in the cube of radium. How long will it take the water to boil?

|| How many half-lives must elapse until (a) 90% and (b) 99% of a radioactive sample of atoms has decayed?

|| A sample contains radioactive atoms of two types, A and B. Initially there are five times as many A atoms as there are B atoms. Two hours later, the numbers of the two atoms are equal. The half-life of A is 0.50 hour. What is the half-life of B?

| The half-life of the uranium isotope ${}^{235}\text{U}$ is 700 million years. The earth is approximately 4.5 billion years old. How much more ${}^{235}\text{U}$ was there when the earth formed than there is today? Give your answer as the then-to-now ratio.

|| Radioactive isotopes often occur together in mixtures. Suppose a 100 g sample contains ${}^{131}\text{Ba}$, with a half-life of 12 days, and ${}^{47}\text{Ca}$, with a half-life of 4.5 days. If there are initially twice as many calcium atoms as there are barium atoms, what will be the ratio of calcium atoms to barium atoms 2.5 weeks later?

|| BIO There is evidence that low-energy x rays have an RBE slightly greater than 1. Suppose that 10 keV photons with an RBE of 1.2 are used to make a chest x ray. A 60 kg person receives a 0.30 mSv dose from a chest x ray that exposes 25% of the patient’s body. How many x ray photons are absorbed in the patient’s body?

|| BIO The uranium isotope ${}^{238}\text{U}$ is naturally present at low levels in many soils. One of the nuclei in the decay series of ${}^{238}\text{U}$ is the radon isotope ${}^{222}\text{Rn}$, which decays by emitting a 5.50 MeV alpha particle with $t_{1\text{/}2}=3.82\text{days}$. Radon is a gas, and it tends to seep from soil into basements. The Environmental Protection Agency recommends that homeowners take steps to remove radon, by pumping in fresh air, if the radon activity exceeds 4 pCi per liter of air.

1. How many ${}^{222}\text{Rn}$ atoms are there in $1{\text{m}}^3$ of air if the activity is $4\text{pCi}\text{/}\text{L}?$

2. The range of alpha particles in air is $\approx 3\text{cm}$. Suppose we model a person as a 180-cm-tall, 25-cm-diameter cylinder with a mass of 65 kg. Only decays within 3 cm of the cylinder can cause exposure, and only $\approx 50\text{\%}$ of the decays direct the alpha particle toward the person. Determine the dose in mSv per year for a person who spends the entire year in a room where the activity is $4\text{pCi}\text{/}\text{L}$.

3. Does the EPA recommendation seem appropriate? Why?

|| BIO The plutonium isotope ${}^{239}\text{Pu}$ has a half-life of 24,000 years and decays by the emission of a 5.2 MeV alpha particle. Plutonium is not especially dangerous if handled because the activity is low and the alpha radiation doesn’t penetrate the skin. However, there are serious health concerns if even the tiniest particles of plutonium are inhaled and lodge deep in the lungs. This could happen following any kind of fire or explosion that disperses plutonium as dust.

1. Soot particles are roughly $1\mu \text{m}$ in diameter. What is the activity of a $1.0\text{-}\mu \text{m-diameter}$ particle of ${}^{239}\text{Pu}?$ The density of plutonium is $19\text{,}800{\text{kg/m}}^3$.

2. How many decays occur in 1.0 yr?

||| BIO The rate at which a radioactive tracer is lost from a patient’s body is the rate at which the isotope decays plus the rate at which the element is excreted from the body. Medical experiments have shown that stable isotopes of a particular element are excreted with a 6.0 day half-life. A radioactive isotope of the same element has a half-life of 9.0 days. What is the effective half-life of the isotope, in days, in a patient’s body?

|| All the very heavy atoms found in the earth were created long ago by nuclear fusion reactions in a supernova, an exploding star. The debris spewed out by the supernova later coalesced into the gases from which the sun and the planets of our solar system were formed. Nuclear physics suggests that the uranium isotopes ${}^{235}\text{U}$ and ${}^{238}\text{U}$ should have been created in roughly equal numbers. Today, 99.28% of uranium is ${}^{238}\text{U}$ and only 0.72% is ${}^{235}\text{U}$. How long ago did the supernova occur?

||| The technique known as potassium-argon dating is used to date old lava flows. The potassium isotope ${}^{40}\text{K}$ has a 1.28-billion-year half-life and is naturally present at very low levels. ${}^{40}\text{K}$ decays by two routes: 89% undergo beta-minus decay into ${}^{40}\text{Ca}$ while 11% undergo electron capture to become ${}^{40}\text{Ar}$. Argon is a gas, and there is no argon in flowing lava because the gas escapes. Once the lava solidifies, any argon produced in the decay of ${}^{40}\text{K}$ is trapped inside and cannot escape. A geologist brings you a piece of solidified lava in which you find the ${}^{40}\text{Ar}{/}^{40}\text{K}$ ratio to be 0.013. What is the age of the rock?

||| Alpha decay occurs when an alpha particle tunnels through the Coulomb barrier. FIGURE CP42.65 shows a simple one dimensional model of the potential-energy well of an alpha particle in a nucleus with $A\approx 235$. The 15 fm width of this one-dimensional potential-energy well is the diameter of the nucleus. Further, to keep the model simple, the Coulomb barrier has been modeled as a 20-fm-wide, 30-MeV-high rectangular potential-energy barrier. The goal of this problem is to calculate the half-life of an alpha particle in the energy level $E=5.0\text{MeV}$.

1. What is the kinetic energy of the alpha particle while inside the nucleus? What is its kinetic energy after it escapes from the nucleus?

2. Consider the alpha particle within the nucleus to be a point particle bouncing back and forth with the kinetic energy you found in part a. What is the particle’s collision rate, the number of times per second it collides with a wall of the potential?

3. What is the tunneling probability $P_{\text{tunnel}}?$

4. $P_{\text{tunnel}}$ is the probability that on any one collision with a wall the alpha particle tunnels through instead of reflecting. The probability of not tunneling is $1-P_{\text{tunnel}}$. Hence the probability that the alpha particle is still inside the nucleus after $N$ collisions is $(1-P_{\text{tunnel}}{)}^N\approx 1-N{P}_{\text{tunnel}}$, where we’ve used the binomial approximation because $P_{\text{tunnel}}\ll 1$. The half-life is the time at which half the nuclei have not yet decayed. Use this to determine (in years) the half-life of the nucleus.

FIGURE CP42.65

||| It might seem strange that in beta decay the positive proton, which is repelled by the positive nucleus, remains in the nucleus while the negative electron, which is attracted to the nucleus, is ejected. To understand beta decay, let’s analyze the decay of a free neutron that is at rest in the laboratory. We’ll ignore the antineutrino and consider the decay $\text{n}\to {\text{p}}^{+}+{\text{e}}^{-}$. The analysis requires the use of relativistic energy and momentum, from Chapter 36.

1. What is the total kinetic energy, in MeV, of the proton and electron?

2. Write the equation that expresses the conservation of relativistic energy for this decay. Your equation will be in terms of the three masses $m_{\text{n}}\text{,}{m}_{\text{p}}$, and $m_{\text{e}}$ and the relativistic factors ${\gamma }_{\text{p}}$ and ${\gamma }_{\text{e}}$.

3. Write the equation that expresses the conservation of relativistic momentum for this decay. Let $v$ represent speed, rather than velocity, then write any minus signs explicitly.

4. You have two simultaneous equations in the two unknowns $v_{\text{p}}$ and $v_{\text{e}}$. To help in solving these, first prove that $\gamma v={({\gamma }^2-1)}^{1\text{/}2}c$.

5. Solve for $v_{\text{p}}$ and $v_{\text{e}}$. (It’s easiest to solve for ${\gamma }_{\text{p}}$ and ${\gamma }_{\text{e}}$, then find $v$ from $\gamma$.) First get an algebraic expression for each, in terms of the masses. Then evaluate each, giving $v$ as a fraction of $c$.

6. Calculate the kinetic energy in MeV of the proton and the electron. Verify that their sum matches your answer to part a.

7. Now explain why the electron is ejected in beta decay while the proton remains in the nucleus.