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CHEMISTRY: The Molecular Nature of Matter and Change 2016

Martin S. Silberberg, Patricia G. Amateis

Chapter 24

Nuclear Reactions and Their Applications

Educators


Problem 1

How do chemical and nuclear reactions differ in
(a) Magnitude of the energy change?
(b) Effect on rate of increasing temperature?
(c) Effect on rate of higher reactant concentration?
(d) Effect on yield of higher reactant concentration?

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Problem 2

Sulfur has four naturally occurring stable isotopes. The one with the lowest mass number is sulfur-32, which is also the most abundant (95.02%).
(a) What percentage of the $\mathrm{S}$ atoms in a matchhead are $^{32} \mathrm{S}$ ?
(b) The isotopic mass of 31.972070 amu. Is the atomic mass of S larger, smaller, or equal to this mass? Explain.

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Problem 3

What led Marie Curie to draw the following conclusions?
(a) Radioactivity is a property of the element and not the com- pound in which it is found.
(b) A highly radioactive element, aside from uranium, occurs in pitchblende,

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Problem 4

Which of the following processes produce an atom of a different element: (a) $\alpha$ decay; (b) $\beta^{-}$ decay; (c) $\gamma$ emission; (d) $\beta^{7}$ emission; (e) $\mathrm{e}^{-}$ capture? Show how $Z$ and $N$ change, if at all, with each process.

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Problem 5

Why is $\frac{3}{2}$ He stable, but $\frac{2}{2}$ He has never been detected?

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Problem 6

How do the modes of decay differ for a neutron-rich nuclide and a proton-rich nuclide?

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Problem 7

Why might it be difficult to use only a nuclide's $N / Z$ ratio to predict whether it will decay by $\beta^{+}$ emission or by e $^{-}$ capture? What other factor is important?

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Problem 8

Write balanced nuclear equations for the following:
(a) Alpha decay of $^{24} \mathrm{U}$
(b) Electron capture by neptunium-232
(c) Positron emission by $\frac{12}{7 \mathrm{N}}$

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Problem 9

Write balanced nuclear equations for the following:
(a) $\beta^{-}$ decay of sodium-26
(b) $\beta^{-}$ decay of francium-223
(c) Alpha decay of $\frac{212}{83} \mathrm{Bi}$

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Problem 10

Write balanced nuclear equations for the following:
(a) $\beta^{-}$ emission by magnesium- 27
(b) $\beta^{+}$ emission by $\frac{23}{12} \mathrm{Mg}$
(c) Electron capture by 103 $\mathrm{Pd}$

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Problem 11

Write balanced nuclear equations for the following:
(a) $\beta^{-}$ decay of silicon- 32
(b) Alpha decay of polonium-218
(c) Electron capture by 149 $\mathrm{In}$

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Problem 12

Write balanced nuclear equations for the following:
(a) Formation of $\frac{48}{22}$ Ti through positron emission
(b) Formation of silver-10 7 through electron capture
(c) Formation of polonium-206 through $\alpha$ decay

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Problem 13

Write balanced nuclear equations for the following:
(a) Formation of $^{241} \mathrm{Am}$ through $\beta^{-}$ decay
(b) Formation of 28 $\mathrm{Ac}$ through $\beta^{-}$ decay
(c) Formation of 203 $\mathrm{Bi}$ through $\alpha$ decay

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Problem 14

Write balanced nuclear equations for the following:
(a) Formation of $^{186}$ Ir through electron capture
(b) Formation of francium-22 1 through $\alpha$ decay
(c) Formation of iodine- 129 through $\beta^{-}$ decay

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Problem 15

Write balanced nuclear equations for the following:
(a) Formation of $^{52}$ Mn through positron emission
(b) Formation of polonium-2 15 through $\alpha$ decay
(c) Formation of $^{81} \mathrm{Kr}$ through electron capture

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Problem 16

Which nuclide(s) would you predict to be stable? Why?
$$
\text{(a)}^{20} \mathrm{O}
$$
$$
\text{(b)}\frac{59}{2}
$$
$$
\text{(c)}\frac{9}{3} \mathrm{Li}
$$

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Problem 17

Which nuclide(s) would you predict to be stable? Why?
$$
\text{(a)}_{60}^{146} \mathrm{Nd}
$$
$$
\text{(b)}\stackrel{114}{48} \mathrm{Cd}
$$
$$
\text{(c)}\frac{88}{42} \mathrm{Mo}
$$

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Problem 18

Which nuclide(s) would you predict to be stable? Why?
$$
\text{(a)}^{127} \mathrm{I} \quad \text { (b) } \operatorname{tin}-106\quad(\mathrm{c})^{68} \mathrm{As}
$$

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Problem 19

Which nuclide(s) would you predict to be stable? Why?
$$
(\mathrm{a})^{127} \mathrm{I}
$$
$$
\text{(b)}\operatorname{tin}-106
$$
$$
\text{(c)}-32
$$

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Problem 20

What is the most likely mode of decay for each?
$$
\text{(a)}\frac{238}{92} \mathrm{U}
$$
$$
\text{(b)}\frac{48}{24} \mathrm{Cr}
$$
$$
\text{(c)}\frac{50}{25} \mathrm{Mn}
$$

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Problem 21

What is the most likely mode of decay for each?
$$
\text{(a)}111 \mathrm{Ag}
$$
$$
\text{(b)}\stackrel{41}{17} \mathrm{Cl}
$$
$$
\text{(c)}110 \mathrm{Ru}
$$

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Problem 22

What is the most likely mode of decay for each?
$$
\text{(a)}^{15} \mathrm{C} \quad \text { (b) }^{120} \mathrm{Xe}\quad(\mathrm{c})^{224} \mathrm{Th}
$$

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Problem 23

What is the most likely mode of decay for each?
$$
(\mathrm{a})^{106} \mathrm{In} \quad \text { (b) }^{141} \mathrm{Eu}\quad(\mathrm{c})^{241} \mathrm{Am}
$$

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Problem 24

Why is $\frac{52}{24} \mathrm{Cr}$ the most stable isotope of chromium?

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Problem 25

Why is $_{20}^{40} \mathrm{Ca}$ the most stable isotope of calcium?

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Problem 26

277 Np is the parent nuclide of a decay series that starts with \alpha emission, followed by $\beta^{-}$ emission, and then two more $\alpha$ emissions. Write a balanced nuclear equation for each step.

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Problem 27

Why is helium found in deposits of uranium and thorium ores? What kind of radioactive emission produces it?

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Problem 28

In a natural decay series, how many $\alpha$ and $\beta^{-}$ emissions per atom of uranium- 235 result in an atom of lead- 207$?$

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Problem 29

What electronic process is the basis for detecting radioactivity in (a) a scintillation counter; (b) a Geiger-Müller counter?

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Problem 30

What is the reaction order of radioactive decay? Explain.

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Problem 31

After 1 min, three radioactive nuclei remain from an original sample of six. Is it valid to conclude that $t_{1 / 2}$ equals 1 $\min _{12}$ Is this conclusion valid if the original sample contained $6 \times 10^{12}$ nuclei and $3 \times 10^{12}$ remain after 1 min? Explain.

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Problem 32

Radioisotopic dating depends on the constant rate of decay and formation of various nuclides in a sample. How is the proportion of $^{14} \mathrm{C}$ kept relatively constant in living organisms?

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Problem 33

What is the specific activity (in Ci/g) if 1.65 $\mathrm{mg}$ of an isotope emits $1.56 \times 10^{6} \alpha$ particles per second?

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Problem 34

What is the specific activity (in $\mathrm{Ci} / \mathrm{g} )$ if 2.6 $\mathrm{g}$ of an isotope emits $4.13 \times 10^{8} \mathrm{\beta}^{-}$ particles per hour?

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Problem 35

What is the specific activity (in Bq/g) if 8.58$\mu g$ of an isotope emits $7.4 \times 10^{4} \alpha$ particles per minute?

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Problem 36

What is the specific activity (in Bq/g) if 1.07 $\mathrm{kg}$ of an isotope emits $3.77 \times 10^{7} \mathrm{\beta}^{-}$ particles per minute?

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Problem 37

If one-trillionth of the atoms of a radioactive isotope disintegrate each day, what is the decay constant of the process?

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Problem 38

If $2.8 \times 10^{-10 \%}$ of the atoms of a radioactive isotope disintegrate in 1.0 yr, what is the decay constant of the process?

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Problem 39

If $1.00 \times 10^{-12}$ mol of $^{135} \mathrm{Cs}$ emits $1.39 \times 10^{5} \beta^{-}$ particles in
1.00 yr, what is the decay constant?

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Problem 40

If $6.40 \times 10^{-9}$ mol of 176 $\mathrm{W}$ emits $1.07 \times 10^{15} \beta^{+}$ particles in
$1.00 \mathrm{h},$ what is the decay constant?

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Problem 41

The isotope 212 $\mathrm{Bi}$ has a half-life of 1.01 yr. What mass (in mg) of a 2.00 -mg sample will remain after $3.75 \times 10^{3} \mathrm{h}$ ?

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Problem 42

The half-life of radium-226 is $1.60 \times 10^{3}$ yr. How many hours will it take for a $2.50-\mathrm{g}$ sample to decay to the point where 0.185 $\mathrm{g}$ of the isotope remains?

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Problem 43

A rock contains 270$\mu \mathrm{mol}$ of $^{238} \mathrm{U}\left(t_{1 / 2}=4.5 \times 10^{9} \mathrm{yr}\right)$ and 110$\mu \mathrm{mol}$ of $^{206} \mathrm{Pb}$ . Assuming that all the 206 $\mathrm{Pb}$ comes from decay of the $^{238} \mathrm{U},$ estimate the rock's age.

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Problem 44

A fabric remnant from a burial site has a $^{14} \mathrm{C}^{12} \mathrm{C}$ ratio of 0.735 of the original value. How old is the fabric?

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Problem 45

Due to decay of $^{40} \mathrm{K}$ , cow's milk has a specific activity of about $6 \times 10^{-11} \mathrm{mCi}$ per milliter. How many disintegrations of 40 $\mathrm{K}$ nuclei are there per minute in an 8.0 $\mathrm{-oz}$ glass of milk?

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Problem 46

Plutonium-239 $\left(t_{1 / 2}=2.41 \times 10^{4} \text { yr) represents a serious }\right.$ nuclear waste hazard. If seven half-lives are required to reach a tolerable level of radioactivity, how long must $^{23}$ Pu be stored?

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Problem 47

$\mathrm{A}$ rock that contains $3.1 \times 10^{-15} \mathrm{mol}$ of 232 $\mathrm{Th}\left(t_{1 / 2}=\right.$ $1.4 \times 10^{10} \mathrm{yr} )$ has $9.5 \times 10^{4}$ fission tracks, each track representing the fission of one atom of $\mathrm{f}^{232} \mathrm{Th}$ . How old is the rock?

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Problem 48

A volcanic eruption melts a large chunk of rock, and all gases are expelled. After cooling, $40^{\circ}$ Ar accumulates from the ongoing decay of $_{19} \mathrm{K}$ in the rock $\left(t_{1 / 0}=1.25 \times 10^{9} \mathrm{yr}\right) .$ When a piece of rock is analyzed, it is found to contain 1.38 $\mathrm{mmol}$ of $^{40} \mathrm{K}$ and 1.14 $\mathrm{mmol}$ of $^{40} \mathrm{Ar}$ . How long ago did the rock cool?

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Problem 49

Irene and Frederic Joliot-Curie converted 27 13 $\mathrm{Al}$ to $\frac{30}{15} \mathrm{P}$ in 1933. Why was this transmutation significant?

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Problem 50

Early workers mistakenly thought neutron beams were $\gamma$ radiation. Why? What evidence led to the correct conclusion?

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Problem 51

Why must the electrical polarity of the tubes in a linear accelerator be reversed at very short time intervals?

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Problem 52

Why does bombardment with protons usually require higher energies than bombardment with neutrons?

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Problem 53

Determine the missing species in these transmutations, and write a full nuclear equation from the shorthand notation:
$$
\begin{array}{l}{\text { (a) }^{10} \mathrm{B}(\alpha, \mathrm{n})} \\ {\text { (b) } 28 \mathrm{Si}(\mathrm{d},-)^{29} \mathrm{P} \text { ( where d is a deuteron, }^{2} \mathrm{H} )} \\ {\text { (c) }_{-}(\alpha, 2 \mathrm{n})^{244} \mathrm{Cf}}\end{array}
$$

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Problem 54

Name the unidentified species, and write each transmutation process in shorthand notation: (a) gamma irradiation of a nuclide yields a proton, a neutron, and $^{29} \mathrm{Si} ;(\mathrm{b})$ bombardment of 252 $\mathrm{Cf}$ with 10 $\mathrm{B}$ yields five neutrons and a nuclide; (c) bombardment of $^{238} \mathrm{U}$ with a particle yields three neutrons and $^{239} \mathrm{Pu}$ .

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Problem 55

Elements $104,105,$ and 106 have been named rutherfordium (Rf), dubnium (Db), and seaborgium (Sg), respectively. These elements are synthesized from californium- 249 by bombarding with carbon-12, nitrogen-15, and oxygen-18 nuclei, respectively. Four neutrons are formed in each reaction as well. (a) Write balanced nuclear equations for the formation of these elements. (b) Write the equations in shorthand notation.

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Problem 56

The effects on matter of $\gamma$ rays and $\alpha$ particles differ. Explain.

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Problem 57

What is a cation-electron pair, and how does it form?

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Problem 58

Why is ionizing radiation more harmful to children than adults?

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Problem 59

Why is $\cdot$ OH more dangerous than $\mathrm{OH}^{-}$ in an organism?

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Problem 60

$\mathrm{A}$ 135-lb person absorbs $3.3 \times 10^{-7} \mathrm{J}$ of energy from radioactive emissions. (a) How many rads does she receive? (b) How many grays (Gy) does she receive?

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Problem 61

\mathrm{A} 3.6-\mathrm{kg}$ laboratory animal receives a single dose of $8.92 \times 10^{-4} \mathrm{Gy}$ . (a) How many rads did the animal receive? (b) How many joules did the animal absorb?

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Problem 62

A 70 . -kg person exposed to $^{90} \mathrm{Sr}$ absorbs $6.0 \times 10^{5} \beta^{-}$ particles, each with an energy of $8.74 \times 10^{-14} \mathrm{J}$ . (a) How many grays does the person receive? (b) If the RBE is $1.0,$ how many millirems is this? (c) What is the equivalent dose in sieverts (Sv)?

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Problem 63

A laboratory rat weighs 265 $\mathrm{g}$ and absorbs $1.77 \times 10^{10} \mathrm{\beta}^{-}$ particles, each with an energy of $2.20 \times 10^{-13} \mathrm{J}$ . (a) How many rads does the animal receive? (b) What is this dose in Gy? (c) If the $\mathrm{RBE}$ is $0.75,$ what is the equivalent dose in $\mathrm{Sv}$ ?

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Problem 64

If 2.50 $\mathrm{pCi}\left[1 \mathrm{pCi} \text { (picocurie) }=1 \times 10^{-12} \mathrm{Ci}\right]$ of radioactivity from $^{239} \mathrm{Pu}$ is emitted in a 95 -kg human for $65 \mathrm{h},$ and each
disintegration has an energy of $8.25 \times 10^{-13} \mathrm{J}$ , how many grays does the person receive?

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Problem 65

A small region of a cancer patient's brain is exposed for 24.0 $\mathrm{min}$ to 475 $\mathrm{Bq}$ of radioactivity from $^{60} \mathrm{Co}$ for treatment of a tumor. If the brain mass exposed is 1.858 $\mathrm{g}$ and each $\beta^{-}$ particle emitted has an energy of $5.05 \times 10^{-14} \mathrm{J},$ what is the dose in rads?

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Problem 66

What two ways are radioactive tracers used in organisms?

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Problem 67

Why is neutron activation analysis (NAA) useful to art historians and criminologists?

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Problem 68

Positrons cannot penetrate matter more than a few atomic diameters, but positron emission of radiotracers can be monitored in medical diagnosis. Explain

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Problem 69

A steel part is treated to form some iron- $59 .$ Oil used to lubricate the part emits 298$\beta^{-}$ particles (with the energy characteristic of $^{59} \mathrm{Fe}$ ) per minute per milliter of oil. What other information would you need to calculate the rate of removal of the steel from the part during use?

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Problem 70

The oxidation of methanol to formaldehyde can be accomplished by reaction with chromic acid:
$$
\begin{aligned} 6 \mathrm{H}^{+}(a q)+3 \mathrm{CH}_{3} \mathrm{OH}(a q)+& 2 \mathrm{H}_{2} \mathrm{CrO}_{4}(a q) \longrightarrow \\ & 3 \mathrm{CH}_{2} \mathrm{O}(a q)+2 \mathrm{Cr}^{3+}(a q)+8 \mathrm{H}_{2} \mathrm{O}(l) \end{aligned}
$$
The reaction can be studied with the stable isotope tracer 18 $\mathrm{O}$ and mass spectrometry. When a small amount of $\mathrm{CH}_{3}^{18} \mathrm{OH}$ is present in the alcohol reactant, $\mathrm{CH}_{2}^{18} \mathrm{O}$ forms. When a small amount of $\mathrm{H}_{2} \mathrm{Cr}^{18} \mathrm{O}_{4}$ is present, $\mathrm{H}_{2}^{18} \mathrm{O}$ forms. Does chromic acid or methanol supply the $\mathrm{O}$ atom to the aldehyde? Explain.

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Problem 71

Many scientists at first reacted skeptically to Einstein's equation, $E=m c^{2} .$ Why?

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Problem 72

How does a change in mass arise when a nuclide forms from nucleons?

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Problem 73

When a nucleus forms from nucleons, is energy absorbed or released? Why?

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Problem 74

What is the binding energy per nucleon? Why is the binding energy per nucleon, rather than per nuclide, used to comparenuclide stability?

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Problem 75

$\mathrm{A}^{3} \mathrm{H}$ nucleus decays with an energy of 0.01861 $\mathrm{MeV} . \mathrm{Con}$ . vert this energy into (a) electron volts; (b) joules.

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Problem 76

Arsenic-84 decays with an energy of $1.57 \times 10^{-15} \mathrm{kJ}$ per nucleus. Convert this energy into (a) eV;(b) MeV.

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Problem 77

How many joules are released when 1.5 mol of $^{239}$ Pu decays, if each nucleus releases 5.243 $\mathrm{MeV} ?$

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Problem 78

How many MeV are released per nucleus when $3.2 \times 10^{-3}$ mol of chromium - 49 releases $8.11 \times 10^{5} \mathrm{kJ} ?$

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Problem 79

Oxygen-16 is one of the most stable nuclides. The mass of a 16 $\mathrm{O}$ atom is 15.994915 amu. Calculate the binding energy (a) per nucleon in MeV; (b) per atom in MeV; (c) per mole in kJ.

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Problem 80

Lead- 206 is the end product of $^{238} \mathrm{U}$ decay. One $^{206} \mathrm{Pb}$ atom has a mass of 205.974440 amu. Calculate the binding energy (a) per nucleon in MeV; (b) per atom in MeV;(c) per mole in kJ.

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Problem 81

Cobalt- 59 is the only stable isotope of this transition metal. One 5$\% \mathrm{Co}$ atom has a mass of 58.933198 amu. Calculate the binding energy $(\text { a) per nucleon in MeV; (b) per atom in MeV; }$
(c) per mole in kJ.

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Problem 82

Iodine-131 is one of the most important isotopes used in the diagnosis of thyroid cancer. One atom has a mass of 130.906114 amu. Calculate the binding energy (a) per nucleon in MeV; (b) per atom in MeV;(c) per mole in $\mathrm{kJ}$ .

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Problem 83

The $^{80} \mathrm{Br}$ nuclide decays either by $\beta^{-}$ decay or by electron capture. (a) What is the product of each process? (b) Which process releases more energy? (Masses of atoms: $^{80} \mathrm{Br}=79.918528$ amu; $^{80} \mathrm{Kr}=79.916380$ amu; $^{80} \mathrm{Se}=79.916520$ amu; neglect the mass of electrons involved because these are atomic, not nuclear, masses.)

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Problem 84

What is the minimum number of neutrons from each fission event that must be absorbed by other nuclei for a chain reaction to be sustained?

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Problem 85

In what main way is fission different from radioactive decay? Are all fission events in a chain reaction identical? Explain.

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Problem 86

What is the purpose of enrichment in the preparation of fuel rods? How is it accomplished?

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Problem 87

Describe the nature and purpose of these components of a nuclear reactor: (a) control rods; (b) moderator; (c) reflector.

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Problem 88

State an advantage and a disadvantage of heavy-water reactors compared to light-water reactors.

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Problem 89

What are the expected advantages of fusion reactors over fission reactors?

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Problem 90

Why is iron the most abundant element in Earth's core?

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Problem 91

The reaction that will probably power the first commercial fusion reactor is
$$
_{1}^{3} \mathrm{H}+_{1}^{2} \mathrm{H} \longrightarrow_{2}^{4} \mathrm{He}+_{0}^{1} \mathrm{n}
$$
How much energy would be produced per mole of reaction? (Masses of atoms: $^{3} \mathrm{H}=3.01605 \mathrm{amu} ;_{1}^{2} \mathrm{H}=2.0140 \mathrm{amu} ;_{2}^{4}, \mathrm{He}=$ 4.00260 amu; mass of $_{0}^{1} \mathrm{n}=1.008665 \mathrm{amu} . )$

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Problem 92

Some 243 $\mathrm{Am}$ was present when Earth formed, but it all decayed in the next billion years. The first three steps in this decay series are emissions of an $\alpha$ particle, a $\beta^{-}$ particle, and another $\alpha$ particle. What other isotopes were present on the young Earth in a rock that contained some $^{243} \mathrm{Am}$ ?

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Problem 93

Curium-243 undergoes $\alpha$ decay to plutonium- 239 :
$$
^{243} \mathrm{Cm} \longrightarrow^{239} \mathrm{Pu}+\alpha
$$
(a) Find the change in mass, $\Delta m(\text { in } \mathrm{kg})$ . (Masses: $^{243} \mathrm{Cm}=$
243.0614 amu; $239 \mathrm{Pu}=239.0522$ amu; $4 \mathrm{He}=4.0026$ amu; $1 \mathrm{amu}=1.661 \times 10^{-24} \mathrm{g} . )$
(b) Find the energy released in joules.
(c) Find the energy released in kJ/mol of reaction, and comment on the difference between this value and a typical heat of reaction for a chemical change, which is a few hundred $\mathrm{kJ} / \mathrm{mol}$ .

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Problem 94

Plutonium "triggers" for nuclear weapons were manufactured at the Rocky Flats plant in Colorado. An 85 $\mathrm{kg}$ worker inhaled a dust particle containing 1.00$\mu \mathrm{g}$ of 239 $\mathrm{Pu}$ , which resided in his body for 16 $\mathrm{h}\left(t_{1 / 2} \text { of }^{299} \mathrm{Pu}=2.41 \times 10^{4} \text { yr; each disinte- }\right.$gration released 5.15 $\mathrm{MeV}$ ). (a) How many rads did he receive? (b) How many grays?

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Problem 95

Archeologists removed some charcoal from a Native Ameican campfire, burned it in $\mathrm{O}_{2},$ and bubbled the $\mathrm{CO}_{2}$ formed into $\mathrm{Ca}(\mathrm{OH})_{2}$ solution (limewater). The $\mathrm{CaCO}_{3}$ that precipitated was fittered and dried. If 4.58 $\mathrm{g}$ of the $\mathrm{CaCO}_{3}$ had a radioactivity of 3.2 $\mathrm{d} / \mathrm{min}$ , how long ago was the campfire?

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Problem 96

$\mathrm{A}$ 5.4-\mug sample of $^{226 \mathrm{RaCl}_{2}}$ has a radioactivity of $1.5 \times 10^{5} \mathrm{Bq} .$ Calculate $t_{1 / 2}$ of $^{26} \mathrm{Ra}$ .

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Problem 97

How many rads does a $65-$ kg human receive each year from the approximately $10^{-8} \mathrm{g}$ of 14 $\mathrm{C}$ naturally present in her body $\left(t_{1 / 2}=5730 \text { yr; each disintegration releases } 0.156 \mathrm{MeV}\right) ?$

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Problem 98

A sample of AgCl emits $175 \mathrm{nCl} / \mathrm{g},$ A saturated solution prepared from the solid emits $1.25 \times 10^{-2}$ Bq/mL due to radioactive Ag $^{+}$ ions. What is the molar solubility of AgCl?

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Problem 99

The scene below depicts a neutron bombarding $^{255} \mathrm{U}$ :
(a) Is this an example of fission or of fusion? (b) Identify the other nuclide formed. (c) What is the most likely mode of decay of the nuclide with $Z=55 ?$

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Problem 100

What fraction of the $^{235} \mathrm{U}\left(t_{1 / 2}=7.0 \times 10^{8} \mathrm{yr}\right)$ created when Earth was formed would remain after $2.8 \times 10^{9} \mathrm{yr} ?$

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Problem 101

$^{238} \mathrm{U}\left(t_{1 / 2}=4.5 \times 10^{9} \mathrm{yr}\right)$ begins a decay series that ultmately forms $^{206 \mathrm{Pb}}$ . The scene below depicts the relative number of $^{238} \mathrm{U}$ atoms (red) and $^{206} \mathrm{Pb}$ atoms (green) in a mineral. If all the Pb comes from $^{288} \mathrm{U}$ , calculate the age of the sample.

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Problem 102

Technetium-99m is a metastable nuclide used in numerous cancer diagnostic and treatment programs. It is prepared just before use because it decays rapidly through $\gamma$ emission:
$$
99 \mathrm{m} \mathrm{Tc} \longrightarrow^{99} \mathrm{Tc}+\gamma
$$
Use the data below to determine (a) the half-life of $^{9 \mathrm{m}} \mathrm{Tc} ;(\mathrm{b})$ the
percentage of the isotope that is lost if it takes 2.0 $\mathrm{h}$ to prepare and administer the dose.

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Problem 103

How many curies are produced by 1.0 $\mathrm{mol}$ of $^{40} \mathrm{K}\left(t_{1 / 2}=\right.$
$1.25 \times 10^{9} \mathrm{yr} ) ?$ How many becquerels?

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Problem 104

The fraction of a radioactive isotope remaining at time $t$ is $\left(\frac{1}{2}\right)^{t / I_{12}},$ where $t_{1 / 2}$ is the half-life. If the half-life of carbon- 14 is 5730 yr, what fraction of carbon-14 4 in a piece of charcoal remains after (a) 10.0 yr; (b) $10.0 \times 10^{3}$ yr; (c) $10.0 \times 10^{4}$ yr? (d) Why is
radiocarbon dating more reliable for the fraction remaining in part (b) than that in part (a) or in part(c)?

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Problem 105

The isotopic mass of $^{210} \mathrm{Rn}$ is 209.989669 amu. When this nuclide decays by electron capture, it emits 2.368 $\mathrm{MeV} .$ What is the isotopic mass of the resulting nuclide?

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Problem 106

Exactly 0.1 of the radioactive nuclei in a sample decay per hour. Thus, after $n$ hours, the fraction of nuclei remaining is $(0.900)^{n} .$ Find the value of $n$ equal to one half-life.

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Problem 107

In neutron activation analysis (NAA), stable isotopes are bombarded with neutrons. Depending on the isotope and the energy of the neutron, various emissions are observed. What are the products when the following neutron-activated species decay? Write an overall equation in shorthand notation for the reaction starting with the stable isotope before neutron activation.
$$
\text{(a)}\frac{52}{23} \mathrm{V}^{*} \longrightarrow\left[\beta^{-} \text { emission }\right]
$$
$$
\text{(b)}_{29}^{64} \mathrm{Cu}^{*} \longrightarrow\left[\beta^{+} \text { emission }\right]
$$
$$
\text{(c)}_{13}^{28} \mathrm{Al}^{*} \longrightarrow\left[\beta^{-} \text { emission }\right]
$$

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Problem 108

In neutron activation analysis (NAA), stable isotopes are bombarded with neutrons. Depending on the isotope and the energy of the neutron, various emissions are observed. What are the products when the following neutron-activated species decay? Write an overall equation in shorthand notation for the reaction starting with the stable isotope before neutron activation.

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Problem 109

The scene below represents a reaction (with neutrons gray and protons purple) that occurs during the lifetime of a star. (a) Write a balanced nuclear equation for the reaction. (b) If the mass difference is $7.7 \times 10^{-2}$ amu, find the energy (kJ) released.

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Problem 110

What volume of radon will be produced per hour at STP from 1.000 g of $^{226} \mathrm{Ra}\left(t_{1 / 2}=1599 \mathrm{yr} ; 1 \mathrm{yr}=8766 \mathrm{h} ; \text { mass of one }\right.$ 26 $\mathrm{Ra}$ atom $=226.025402 \mathrm{amu} ) ?$

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Problem 111

$^{90} \mathrm{Kr}\left(t_{1 / 2}=32 \mathrm{s}\right)$ is used to study respiration. How soon after being made must a sample be administered to the patient if the activity must be at least 90$\%$ of the original activity?

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Problem 112

Which isotope in each pair is more stable? Why?
$$
\text{(a)}140 \mathrm{Cs} \text { or }^{133} \mathrm{Cs}
$$
$$
\text{(b)}
\begin{array}{l}{79} \\ {35}\end{array} ] \text { or } 38 \mathrm{Br}
$$
$$
\text{(c)}\begin{array}{l}{28} \\ {12}\end{array} \mathrm{Mg} \text { or }_{12}^{24} \mathrm{Mg}
$$
$$
\text{(d)}\frac{14}{7} \mathrm{N} \text { or } \frac{18}{7} \mathrm{N}
$$

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Problem 113

A bone sample containing strontium- 90$\left(t_{1 / 2}=29 \text { yr) emits }\right.$ $7.0 \times 10^{4} \beta^{-}$ particles per month. How long will it take for the emission to decrease to $1.0 \times 10^{4}$ particles per month?

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Problem 114

The $23^{\text { rd }}$ -century starship Enterprise uses a substance called "dilithium crystals" as its fuel.
(a) Assuming that this material is the result of fusion, what is the product of the fusion of two 6 Li nuclei?
(b) How much energy is released per kilogram of dilithium formed? (Mass of one 6 Li atom is 6.015121 amu.)
(c) When four 'H atoms fuse to form 4 He, how many positrons are released?
(d) To determine the energy potential of the fusion processes in parts (b) and (c), compare the changes in mass per kilogram of dilithium and of "He.
(e) Compare the change in mass per kilogram in part (b) to that for the formation of $^{4}$ He by the method used in current fusion reactors (see p. 1088$) .$ (For masses, see Problem $24.91 . )$
(f) Using early $21^{\mathrm{st}}$ -century fusion technology, how much tritium can be produced per kilogram of $^{6} \mathrm{Li}$ in the following reaction: $_{3}^{6} \mathrm{Li}+_{0}^{\mathrm{l}} \mathrm{n} \longrightarrow_{2}^{4} \mathrm{He}+_{1}^{3} \mathrm{H} ?$ When this amount of tritium is fused with deuterium, what is the change in mass? How does this quantity compare with the use of dilithium in part $(\mathrm{b}) ?$

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Problem 115

Uranium and radium are found in many rocky soils throughout the world. Both undergo radioactive decay, and one of the products is radon-2222, the heaviest noble gas $\left(t_{1 / 2}=\right.$
3.82 days). Inhalation of this gas contributes to many lung cancers. According to Environmental Protection Agency recommendations, the level of radioactivity from radon in homes should not
exceed 4.0 $\mathrm{pCi} / \mathrm{L}$ of air.
(a) What is the safe level of radon in Bq/L of air?
(b) A home has a radon measurement of 41.5 $\mathrm{pCi} / \mathrm{L}$ . The owner vents the basement in such a way that no more radon enters the living area. What is the activity of the radon remaining in the room air (in $\mathrm{Bq} / \mathrm{L}$ ) after 9.5 days?
(c) How many more days does it take to reach the EPA recommended level?

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Problem 116

Nuclear disarmament could be accomplished if weapons were not "replenished." The tritium in warheads decays to helium with a half-life of 12.26 yr and must be replaced or the weapon is useless. What fraction of the tritium is lost in 5.50 $\mathrm{yr}$ ?

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Problem 117

A decay series starts with the synthetic isotope 239 $\mathrm{U}$ . The first four steps are emissions of a $\beta^{-}$ particle, another $\beta^{-},$ an $\alpha$ particle, and another $\alpha$ . Write a balanced nuclear equation for each step. Which natural series could start by this sequence?

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Problem 118

How long can a 54 -lb child be exposed to 1.0 $\mathrm{mCi}$ of radiation from 222 $\mathrm{Rn}$ before accumulating 1.0 $\mathrm{mrad}$ if the energy of each disintegration is 5.59 $\mathrm{MeV} ?$

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Problem 119

An earthquake in the area of present-day San Francisco is to be dated by measuring the 14 $\mathrm{C}$ activity $\left(t_{1 / 2}=5730 \text { yr) of }\right.$ parts of a tree uprooted during the event. The tree parts have an activity of $12.9 \mathrm{d} / \mathrm{min} \cdot \mathrm{g} \mathrm{C},$ and a living tree has an activity of 15.3 $\mathrm{d} / \mathrm{min} \cdot \mathrm{g}$ . How long ago did the earthauake occur?

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Problem 120

Were organisms a billion years ago exposed to more or less ionizing radiation than similar organisms today? Explain.

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Problem 121

Tritium $\left(3^{3} \mathrm{H} ; t_{1 / 2}=12.26 \mathrm{yr}\right)$ is continually formed in the upper troposphere by interaction of solar particles with nitrogen. As a result, natural waters contain a small amount of tritium. Two samples of wine are analyzed, one known to be made in 1941 and
another made earlier. The water in the 1941 wine has 2.23 times as much tritium as the water in the other. When was the other wine produced?

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Problem 122

Even though plutonium- 239$\left(t_{1 / 2}=2.41 \times 10^{4} \text { yr) is one of }\right.$ the main fission fuels, it it is still a radiation hazard present in spent uranium fuel from nuclear power plants. How many years does it take for 99$\%$ of the plutonium- 239 in spent fuel to decay?

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Problem 123

Carbon from the remains of an extrinct Australian marsupial, called Diprotodon, has a specific activity of 0.61 $\mathrm{pCi} / \mathrm{g} .$ Modern carbon has a specific activity of 6.89 $\mathrm{pCi} / \mathrm{g}$ . How long ago did the Diprotodon apparently become extinct?

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Problem 124

The reaction that allows for radiocarbon dating is the continual formation of carbon-14 in the unper atmosphere.
$$
\stackrel{14}{7} \mathrm{N}+_{0}^{1} \mathrm{n} \longrightarrow_{6}^{14} \mathrm{C}+_{1}^{1} \mathrm{H}
$$
What is the energy change that is associated with this process
in eV/reaction and in $\mathrm{kJ} / \mathrm{mol}$ reaction? (Masses of atoms: $\frac{14 \mathrm{N}}{7} \mathrm{N}=$ 14.003074 amu; $^{14} \mathrm{C}=14.003241$ amu; $1 \mathrm{H}=1.007825$ amu; mass of $_{0}^{1} \mathrm{n}=1.008665$ amu.)

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Problem 125

What is the nuclear binding energy of a lithium- 7 nucleus in units of kJ/mol and eV/nucleus? (Mass of a lithium- 7 atom $=$ 7.016003 amu.)

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Problem 126

Gadolinium-146 undergoes electron capture. Identify the product, and use Figure 24.2 to find the modes of decay and the two intermediate nuclides in the series:

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Problem 127

Using $21^{\text { st }}$ -century technology, hydrogen fusion requires temperatures around $10^{8} \mathrm{K}$ . But, lower initial temperatures are used if the hydrogen is compressed. In the late $24^{\text { th }}$ century, the starship Leinad uses such methods to fuse hydrogen at $10^{6} \mathrm{K}$ .
(a) What is the kinetic energy of an H atom at $1.00 \times 10^{6} \mathrm{K} ?$
(b) How many H atoms are heated to $1.00 \times 10^{6} \mathrm{K}$ from the energy of one $\mathrm{H}$ and one anti-H atom annihilating each other?
(c) If the heated H atoms of part (b) fuse into 'He atoms (with the loss of two positrons per 'He formed), how much energy (in $\mathrm{J}$ ) is generated?
(d) How much more energy is generated by the fusion in (c) than by the hydrogen-antihydrogen collision in (b)?
(e) Should the captain of the Leinad change the technology and produce $^{3}$ He $\left(\text { mass }=3.01603 \text { amu) instead of }^{4} \mathrm{He} ?\right.$

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Problem 128

A metastable (excited) form of $^{50} \mathrm{Sc}$ changes to its stable form by emitting $\gamma$ radiation with a wavelength of 8.73 $\mathrm{pm} .$ What is the change in mass of 1 mol of the isotope when it undergoes this change?

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Problem 129

A sample of cobalt-60 $\left(t_{1 / 2}=5.27 \text { yr), a powerful } \gamma \text { emit- }\right.$
ter used to treat cancer, was purchased by a hospital on March $1,$ 2012 . The sample must be replaced when its activity reaches $70 . \%$ of the original value. On what date must it be replaced?

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Problem 130

Uranium- 233 decays to thorium-229 by $\alpha$ decay, but the emissions have different energies and products: 83$\%$ emit an $\alpha$ particle with an energy of 4.816 $\mathrm{MeV}$ and give $^{229} \mathrm{Th}$ in its ground state; 15$\%$ emit an $\alpha$ particle of 4.773 MeV and give 229 Th in
excited state I: and 2$\%$ emit a lower energy $\alpha$ particle and give 229 Th in the higher excited state II. Excited state II emits a $\gamma$ ray of 0.060 MeV to reach excited state I. (a) Find the $\gamma$ -ray energy and wavelength that would convert excited state I to the ground state. (b) Find the energy of the $\alpha$ particle that would raise $^{233} \mathrm{U}$ to excited state I.

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Problem 131

Uranium-233 undergoes a slow decay step $\left(t_{1 / 2}=\right.$ $4.5 \times 10^{9}$ yr) followed by a series of fast steps to form the stable isotope $20^{\circ} \mathrm{Pb}$ . Thus, on a time scale of billions of years, 238 $\mathrm{U}$ effectively decays "directly" to $^{206} \mathrm{Pb}$ , and the relative amounts of these isotopes are used to find the age of some rocks. Two students derive equations relating number of half-lives (n) since the rock formed to the amounts of the isotopes:
$$
\left(\frac{1}{2}\right)^{n}=\frac{_{238}^{238} \mathrm{U}}{\frac{806}{82} \mathrm{Pb}}
$$
$$
\left(\frac{1}{2}\right)^{n}=\frac{_{92}^{238} \mathrm{U}}{\frac{928}{92} \mathrm{U}+_{82}^{206} \mathrm{Pb}}
$$

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Problem 132

In the naturally occurring thorium-232 decay series, the steps emit this sequence of particles: $\alpha, \beta^{-}, \beta^{-}, \alpha, \alpha, \alpha, \alpha, \beta^{-}, \beta^{-}$ and $\alpha .$ Write a balanced equation for each step.

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Problem 133

At death, a nobleman in ancient Egypt was mummified and his body contained $1.4 \times 10^{-3} \mathrm{g}$ of $^{40} \mathrm{K}\left(t_{1 / 2}=1.25 \times 10^{9} \mathrm{yr}\right)$ $1.2 \times 10^{-8} \mathrm{g}$ of $^{14} \mathrm{C}\left(t_{1 / 2}=5730 \mathrm{yr}\right),$ and $4.8 \times 10^{-14} \mathrm{g}$ of $^{3} \mathrm{H}\left(t_{1 / 2}=\right.$ 12.26 $\mathrm{yr}$ ). Which nuclide would give the most accurate estimate of the mummy's age? Explain.

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Problem 134

Assuming that many radioactive nuclides can be considered safe after 20 half-lives, how long will it take for each of the following nuclides to be safe: $(a)^{242} \mathrm{Cm}\left(t_{1 / 2}=163 \text { days); }\right.$ (b) $^{214} \mathrm{Po}\left(t_{1 / 2}=1.6 \times 10^{-4} \mathrm{s}\right) ;(\mathrm{c})^{222} \mathrm{Th}\left(t_{1 / 2}=1.39 \times 10^{10} \mathrm{yr}\right) ?$

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Problem 135

An ancient sword has a blade from the early Roman Empire, around 100 AD, but the wooden handle, inlaid wooden decorations, leather ribbon, and leather sheath have different styles. Given the following activities, estimate the age of each part. Which part was made near the time of the blade $\left(t_{1 / 2} \text { of }\right.$

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Problem 136

The starship Voyager, like many other vessels of the newly designed $24^{\text { th }}$ -century fleet, uses antimatter as fuel.
(a) How much energy is released when 1.00 kg each of antimatter and matter annihilate each other?
(b) When the antimatter is atomic antihydrogen, a small amount of it is mixed with excess atomic hydrogen (gathered from interstellar space during flight). The annihilation releases so much heat that the remaining hydrogen nuclei fuse to form 4 $\mathrm{He}$ . If each hydrogen-antihydrogen collision releases enough heat to fuse $1.00 \times 10^{5}$ hydrogen atoms, how much energy (in kJ) is released
per kilogram of antihydrogen?
(c) Which produces more energy per kilogram of antihydrogen, the procedure in part (a) or that in part (b)?

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Problem 137

Use Einstein's equation, the mass in grams of 1 amu, and the relation between electron volts and joules to find the energy equivalent (in MeV) of a mass difference of 1 amu.

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Problem 138

Determine the age of a rock containing 0.065 g of uranium- 238$\left(t_{1 / 2}=4.5 \times 10^{9} \text { yr) and } 0.023 \text { g of lead-206. (Assume that all }\right.$ the lead-206 came from $^{238} \mathrm{U}$ decay.)

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Problem 139

Plutonium-242 decays to uranium- 238 by emission of an $\alpha$ particle with an energy of 4.853 MeV. The 28 $\mathrm{U}$ that forms is unstable and emits a $\gamma$ ray $(\lambda=0.02757 \mathrm{nm}) .$ (a) Write balanced equations for these reactions. (b) What would be the energy of the
$\alpha$ particle if $^{244}$ Pu decayed directly to the more stable $^{288} \mathrm{U} ?$

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Problem 140

Seaborgium- 263 , the first isotope of element 106 synthesized, was produced, along with four neutrons, by bombarding californium- 249 with oxygen-18. The 263 gg then underwent a series of decays starting with three $\alpha$ emissions. Write balanced equations for the synthesis and the three $\alpha$ emissions of $^{263} \mathrm{Sg}$ .

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Problem 141

Some nuclear power plants use plutonium- $239,$ which is produced in breeder reactors. The rate-determining step is the second $\beta^{-}$ emission. How long does it take to make 1.00 $\mathrm{kg}$ of $^{299} \mathrm{Pu}$ if the reaction is complete when the product is $90 . \% 39 \mathrm{Pu}$ ?

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Problem 142

A random-number generator can be used to simulate the probability of a given atom decaying over a given time. For example, the formula $"=$ RANDO" in an Excel spreadsheet returns a random number between 0 and $1 ;$ thus, for one radioactive atom and a time of one half-life, a number less than 0.5 means the atom decays and a number greater than 0.5 means it doesn't. (a) Place the $"=\mathrm{RAND}( )^{\prime \prime}$ in an Excel spreadsheet returns a random number between 0 and $1 ;$ thus, for one radioactive atom and a time of one half-life, a number less than 0.5 means the atom
decays and a number greater than 0.5 means it doesn't.
(a) Place the $"=\mathrm{RAND}(\text { "s }$formula in cells $\mathrm{A} 1$ to $\mathrm{A} 10$ of an Excel spreadsheet. In cell $\mathrm{B} 1,$ place $"=\operatorname{IF}(\mathrm{A} 1<0.5,0,1) . "$ This formula returns 0 if $A 1$ is $<0.5$ (the atom decays) and 1 if $A 1$ is $>0.5$ (the atom does not decay. Place analogous formulas in cells $\mathrm{B} 2$ to $\mathrm{B} 10$ (using the "Fill Down" procedure in Excel). To determine the number of atoms remaining after one half-life, sum cells $\mathrm{B} 1$ to $\mathrm{B} 10$ by placing $"=\operatorname{SUM}(\mathrm{B} 1 : \mathrm{B} 10)^{\prime \prime}$ in cell $\mathrm{B} 12$ . To create a new set of random numbers, click on an empty cell (e.g., B13) and hit "Delete." Perform 10 simulations, each time recording the total number of atoms remaining. Do half of the atoms remain after each half-life? If not, why not?
(b) Increase the number of atoms to 100 by placing suitable formulas in cells $\mathrm{Al}$ to $\mathrm{A} 100, \mathrm{B} 1 \mathrm{to} \mathrm{B} 100$ , and $\mathrm{B} 102 .$ Perform 10 simulations, and record the number of atoms remaining each time. Are these results more realistic for radioactive decay? Explain.

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Problem 143

In the following Excel-based simulation, the fate of 256 atoms is followed over five half-lives. Set up formulas in columns $\mathrm{A}$ and $\mathrm{B},$ as in Problem $24.142,$ and simulate the fate of the sample of 256 atoms over one half-life. Cells $\mathrm{B} 1$ to $\mathrm{B} 256$ should contain 1 or 0. In cell C1, enter “5IF(B150, 0, RAND()).” This returns 0 if the original atom decayed in the previous half-life or a random number between 0 and 1 if it did not. Fill the formula in
C1 down to cell C256. Column D should have formulas similar to those in B, but with modified references, as should columns F, H, and J. Columns E, G, and I should have formulas similar to those in C, but with modified references. In cell B258, enter “5SUM(B1:B256).” This records the number of atoms remaining after the first half-life. Put formulas in cells D258, F258, H258, and J258 to record atoms remaining after subsequent half-lives. (a) Ideally, how many atoms should remain after each half-life? (b) Make a table of the atoms remaining after each half-life in four separate simulations. Compare these outcomes to the ideal outcome. How would you make the results more realistic?

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Problem 144

Representations of three nuclei (with neutrons gray and protons purple) are shown below. Nucleus 1 is stable, but 2 and 3 are not. (a) Write the symbol for each isotope. (b) What is (are) the most likely mode(s) of decay for 2 and 3?

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