Radioactivity and Nuclear Physics

(a) Write the complete \({{\rm{\beta }}^{\rm{ - }}}\) decay equation for \({}^{{\rm{90}}}{\rm{Sr}}\) , a major waste product of nuclear reactors. (b) Find the energy released in the decay.

Show that the activity of the \({}^{{\rm{14}}}{\rm{C}}\) in \(1.00\,{\rm{g}}\) of \({}^{{\rm{12}}}{\rm{C}}\) found in living tissue is \(0.250\,{\rm{Bq}}\).

Cow’s milk produced near nuclear reactors can be tested for as little as \(1.00\,{\rm{pCi}}\) of \({}^{{\rm{131}}}{\rm{I}}\) per litre, to check for possible reactor leakage. What mass of\({}^{{\rm{131}}}{\rm{I}}\)  has this activity?

a) Natural potassium contains \({}^{{\rm{40}}}{\rm{K}}\), which has a half-life of \(1.277 \times {10^9}\,{\rm{y}}\). What mass of \({}^{{\rm{40}}}{\rm{K}}\) in a person would have a decay rate of \(4140\,{\rm{Bq}}\)? (b) What is the fraction of \({}^{{\rm{40}}}{\rm{K}}\)  in natural potassium, given that the person has \({\rm{140g}}\) in his body? (These numbers are typical for a \({\rm{70}}\)-kg adult.)

There is more than one isotope of natural uranium. If a researcher isolates \(1.00\,{\rm{mg}}\) of the relatively scarce\({}^{{\rm{235}}}{\rm{U}}\) and finds this mass to have an activity of \(80.0\,{\rm{Bq}}\), what is its half-life in years?

\({}^{{\rm{50}}}{\rm{V}}\) has one of the longest known radioactive half-lives. In a difficult experiment, a researcher found that the activity of \(1.00\,{\rm{kg}}\) of \({}^{{\rm{50}}}{\rm{V}}\)is \(1.75\,{\rm{Bq}}\). What is the half-life in years?

What fraction of the 40 K that was on Earth when it formed \(4.5 \times {10^9}\,{\rm{year}}\) ago is left today?

A 5000-Ci \({}^{60}Co\) source used for cancer therapy is considered too weak to be useful when its activity falls to 3500 Ci. How long after its manufacture does this happen?

World War II aircraft had instruments with glowing radium-painted dials (see figure ). The activity of one such instrument was \(1.0 \times {10^5}\,{\rm{Bq}}\) when new. 

(a) What mass of \(^{{\rm{226}}}{\rm{Ra}}\) was present? 

(b) After some years, the phosphors on the dials deteriorated chemically, but the radium did not escape. What is the activity of this instrument \({\rm{57}}{\rm{.0}}\)years after it was made?

(a) Write the complete \({{\rm{\beta }}^{\rm{ - }}}\] decay equation for \({}^{{\rm{90}}}{\rm{Sr}}\] , a major waste product of nuclear reactors. (b) Find the energy released in the decay.

(a) Calculate the energy released in the \({\rm{\alpha }}\) decay of \({}^{{\rm{238}}}{\rm{U}}\) . (b) What fraction of the mass of a single \({}^{{\rm{238}}}{\rm{U}}\) is destroyed in the decay? The mass of \({}^{{\rm{234}}}{\rm{Th}}\) is \(234.043593\,{\rm{u}}\). (c) Although the fractional mass loss is large for a single nucleus, it is difficult to observe for an entire macroscopic sample of uranium. Why is this?

(a) Write the complete reaction equation for electron capture by\({}^{{\rm{15}}}{\rm{O}}\). (b) Calculate the energy released.

An old campfire is uncovered during an archaeological dig. Its charcoal is found to contain less than \(1/1000\) the normal amount of \({}^{{\rm{14}}}{\rm{C}}\). Estimate the minimum age of the charcoal, noting that \({2^{10}} = 1024\).

(a) Calculate the activity \({\rm{R}}\) in curies of \(1.00\,{\rm{g}}\) of \({}^{{\rm{226}}}{\rm{Ra}}\). (b) Discuss why your answer is not exactly \(1.00\,{\rm{Ci}}\), given that the curie was originally supposed to be exactly the activity of a gram of radium.

The ceramic glaze on a red-orange Fiesta ware plate is \({{\rm{U}}_{\rm{2}}}{{\rm{O}}_{\rm{3}}}\)and contains \({\rm{50}}{\rm{.0}}\)grams of \(^{{\rm{238}}}{\rm{U}}\), but very little \(^{{\rm{235}}}{\rm{U}}\). (a)

1.  What is the activity of the plate? 
2. Calculate the total energy that will be released by the \(^{{\rm{238}}}{\rm{U}}\)decay. 
3. If energy is worth \({\rm{12}}{\rm{.0}}\)cents per\({\rm{kW \times h}}\), what is the monetary value of the energy emitted? (These plates went out of production some 30 years ago, but are still available as collectibles.)

Unreasonable Results

The manufacturer of a smoke alarm decides that the smallest current of \({\rm{\alpha }}\) radiation he can detect is \(1.00\,\mu A\). 

1. Find the activity in curies of an \({\rm{\alpha }}\) emitter that produces a \(1.00\,\mu A\)current of \({\rm{\alpha }}\) particles.  
2. What is unreasonable about this result? 
3. What assumption is responsible?

Construct Your Own Problem

Consider the decay of radioactive substances in the Earth's interior. The energy emitted is converted to thermal energy that reaches the earth's surface and is radiated away into cold dark space. Construct a problem in which you estimate the activity in a cubic meter of earth rock? And then calculate the power generated. Calculate how much power must cross each square meter of the Earth's surface if the power is dissipated at the same rate as it is generated. Among the things to consider are the activity per cubic meter, the energy per decay, and the size of the Earth.

Suppose the range for 5.0 MeVα  ray is known to be 2.0mm in a certain material. Does this mean that every 5.0 MeVα  a ray that strikes this material travels 2.0mm , or does the range have an average value with some statistical fluctuations in the distances traveled? Explain.

What is the difference between γ rays and characteristic x rays? Is either necessarily more energetic than the other? Which can be the most energetic?

Ionizing radiation interacts with matter by scattering from electrons and nuclei in the substance. Based on the law of conservation of momentum and energy, explain why electrons tend to absorb more energy than nuclei in these interactions.

What characteristics of radioactivity show it to be nuclear in origin and not atomic?

What is the source of the energy emitted in radioactive decay? Identify an earlier conservation law, and describe how it was modified to take such processes into account.

If an electric field is substituted for the magnetic field with positive charge instead of the north pole and negative charge instead of the south pole, in which directions will the α , β , and γ rays bend?

Explain how an α particle can have a larger range in air than a β particle with the same energy in lead.

Arrange the following according to their ability to act as radiation shields, with the best first and worst last. Explain your ordering in terms of how radiation loses its energy in matter.

(a) A solid material with low density composed of low-mass atoms.

(b) A gas composed of high-mass atoms.

(c) A gas composed of low-mass atoms.

(d) A solid with high density composed of high-mass atoms.

Often, when people have to work around radioactive materials spills, we see them wearing white coveralls (usually a plastic material). What types of radiation (if any) do you think these suits protect the worker from, and how?

Is it possible for light emitted by a scintillator to be too low in frequency to be used in a photomultiplier tube? Explain.

The weak and strong nuclear forces are basic to the structure of matter. Why we do not experience them directly?

Star Trek fans have often heard the term “antimatter drive.” Describe how you could use a magnetic field to trap antimatter, such as produced by nuclear decay, and later combine it with matter to produce energy. Be specific about the type of antimatter, the need for vacuum storage, and the fraction of matter converted into energy.

Neutrinos are experimentally determined to have an extremely small mass. Huge numbers of neutrinos are created in a supernova at the same time as massive amounts of light are first produced. When the 1987A supernova occurred in the Large Magellanic Cloud, visible primarily in the Southern Hemisphere and some 100,000 light-years away from Earth, neutrinos from the explosion were observed at about the same time as the light from the blast. How could the relative arrival times of neutrinos and light be used to place limits on the mass of neutrinos?

What do the three types of beta decay have in common that is distinctly different from alpha decay?

In a \({\rm{3 \times 1}}{{\rm{0}}^{\rm{9}}}\)-year-old rock that originally contained some\(^{{\rm{238}}}{\rm{U}}\)which has a half-life of \({\rm{4}}{\rm{.5 \times 1}}{{\rm{0}}^{\rm{9}}}\) years, we expect to find some \(^{{\rm{238}}}{\rm{U}}\)remaining in it. Why are \(^{{\rm{226}}}{\rm{Ra}}{{\rm{,}}^{{\rm{222}}}}{\rm{Rn,  and}}{{\rm{ }}^{{\rm{210}}}}{\rm{Po}}\) also found in such a rock, even though they have much shorter half-lives (1600 years, 3.8 days, and 138 days, respectively)?

Does the number of radioactive nuclei in a sample decrease to exactly half its original value in one half-life? Explain in terms of the statistical nature of radioactive decay.

Radioactivity depends on the nucleus and not the atom or its chemical state. Why, then, is one kilogram of uranium more radioactive than one kilogram of uranium hexafluoride?

Explain how a bound system can have less mass than its components. Why is this not observed classically, say for a building made of bricks?

Spontaneous radioactive decay occurs only when the decay products have less mass than the parent, and it tends to produce a daughter that is more stable than the parent. Explain how this is related to the fact that more tightly bound nuclei are more stable. (Consider the binding energy per nucleon.)

To obtain the most precise value of \({\rm{BE}}\) from the equation \({\rm{BE  =  (ZM}}{{\rm{(}}^{\rm{1}}}{\rm{H) + N}}{{\rm{m}}_{\rm{n}}}{\rm{)}}{{\rm{c}}^{\rm{2}}}{\rm{ - m}}{{\rm{(}}^{\rm{A}}}{\rm{X)}}{{\rm{c}}^{\rm{2}}}\) , we should take into account the binding energy of the electrons in the neutral atoms. Will doing this produce a larger or smaller value for\({\rm{BE}}\)? Why is this effect usually negligible?

How does the finite range of the nuclear force relate to the fact that \({\rm{BE/A }}\) is greatest for \({\rm{A }}\)near\({\rm{60}}\).

A physics student caught breaking conservation laws is imprisoned. She leans against the cell wall hoping to tunnel out quantum mechanically. Explain why her chances are negligible. (This is so in any classical situation.)

The energy of \(30.0\,{\rm{eV}}\) is required to ionize a molecule of the gas inside a Geiger tube, thereby producing an ion pair. Suppose a particle of ionizing radiation deposits \(0.500\,{\rm{MeV}}\) of energy in this Geiger tube. What maximum number of ion pairs can it create?

A particle of ionizing radiation creates \({\rm{4000}}\) ion pairs in the gas inside a Geiger tube as it passes through. What minimum energy was deposited, if \(30.0\,{\rm{eV}}\) is required to create each ion pair?

(a) Repeat Exercise \({\rm{31}}{\rm{.2}}\), and convert the energy to joules or calories. (b) If all of this energy is converted to thermal energy in the gas, what is its temperature increase, assuming \(50.0\,{\rm{c}}{{\rm{m}}^{\rm{3}}}\) of ideal gas at \({\rm{0}}{\rm{.250 - }}\)atm pressure? (The small answer is consistent with the fact that the energy is large on a quantum mechanical scale but small on a macroscopic scale.)

Suppose a particle of ionizing radiation deposits \(1.0\,{\rm{MeV}}\) in the gas of a Geiger tube, all of which goes to creating ion pairs. Each ion pair requires \(30.0\,{\rm{eV}}\) of energy. (a) The applied voltage sweeps the ions out of the gas in \(1.00\,{\rm{\mu s}}\) . What is the current? (b) This current is smaller than the actual current since the applied voltage in the Geiger tube accelerates the separated ions, which then create other ion pairs in subsequent collisions. What is the current if this last effect multiplies the number of ion pairs by \({\rm{900}}\)?

Verify that a \(2.3 \times {10^{17}}\,{\rm{km}}\) mass of water at normal density would make a cube \(60\,{\rm{km}}\) on a side, as claimed in Example\({\rm{31}}{\rm{.1}}\). (This mass at nuclear density would make a cube \(1.0\,{\rm{m}}\) on a side.)

Find the length of a side of a cube having a mass of \(1.0\,{\rm{kg}}\) and the density of nuclear matter, taking this to be \(2.3 \times {10^{17}}\,{\rm{kg/}}{{\rm{m}}^{\rm{3}}}\).

What is the radius of an\({\rm{\alpha }}\) particle?

Find the radius of a \(^{{\rm{238}}}{\rm{Pu}}\) nucleus. \(^{{\rm{238}}}{\rm{Pu}}\)is a manufactured nuclide that is used as a power source on some space probes.

(a) Calculate the radius of \(^{{\rm{58}}}{\rm{Ni}}\), one of the most tightly bound stable nuclei. (b) What is the ratio of the radius of \(^{{\rm{58}}}{\rm{Ni}}\) to that of \(^{{\rm{258}}}{\rm{Ha}}\) , one of the largest nuclei ever made? Note that the radius of the largest nucleus is still much smaller than the size of an atom.

The unified atomic mass unit is defined to be\(1\,{\rm{u}} = 1.6605 \times {10^{ - 27}}\,{\rm{kg}}\). Verify that this amount of mass converted to energy yields \(931.5\,{\rm{MeV}}\). Note that you must use four-digit or better values for \({\rm{c}}\) and \({\rm{|}}{{\rm{q}}_{\rm{e}}}{\rm{|}}\).

What is the ratio of the velocity of a \({\rm{\beta }}\) particle to that of an\({\rm{\alpha }}\) particle, if they have the same nonrelativistic kinetic energy?