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Chemistry 2012

JOHN E. MCMURRY, ROBERT C. FAY, JORDAN FANTINI

Chapter 22

Nuclear Chemistry

Educators


Problem 1

Calculate the mass defect for the formation of an oxygen-16 nucleus in both grams and g/mol, and calculate the binding energy in both MeV/nucleon and kJ/mol. The mass of an $^{16} \mathrm{O}$ atom is 15.99491 $\mathrm{amu}$

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Problem 2

What is the mass change in g/mol for the reaction of sodium metal with chlorine gas $({Cl}_{2})$ to give sodium chloride?
$$2 {Na}(s)+{Cl}_{2}(g) \longrightarrow 2 {NaCl}(s) \quad \Delta E=-820 {kJ}$$

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Problem 3

An alternative pathway for the nuclear fission of 25 ${U}$ produces tellurium-137 and zirconium-97. How much energy in kJ/mol is released in this fission pathway?

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Problem 4

Calculate the amount of energy released in kJ/mol for the fusion reaction of $^{1}\mathrm{H}$ and $^{2} \mathrm{H}$ atoms to yield and $^{3}\mathrm{H}$ atom:
$$^{1} \mathrm{H}+\frac{2}{1} \mathrm{H} \longrightarrow_{2}^{3} \mathrm{He}$$
The atomic masses are $^{1} \mathrm{H}(1.00783 \mathrm{amu}),^{2} \mathrm{H}(2.01410 \mathrm{amu}),$ and $^{3} \mathrm{He}(3.01603 \mathrm{amu})$

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Problem 5

Write a balanced nuclear equation for the reaction of argon-40 with a proton:
$$^{48}_{18} \mathrm{Ar} +^{1}_{1} \mathrm{H} \longrightarrow ?+ ^{1}_{0} \mathrm{N}$$

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Problem 6

Write a balanced nuclear equation for the reaction of uranium-238 with a deuteron $(\frac{2}{1} {H})$ $^{238}_{92} {U}+_{1}^{2} {H} \longrightarrow ?+2 ^{1}_{0} {n}$

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Problem 7

Charcoal found in the Lascaux cave in France, site of many prehistoric cave paintings, was observed in 1950 to decay at a rate of 2.4 disintegrations/min per gram of carbon. What is the age of the charcoal BP if currently living organisms decay at the rate of 15.3 disintegrations/min per gram of carbon? The half-life of $^{14} {C}$ is 5715 years.

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Problem 8

Why does a given nucleus have less mass than the sum of its constituent protons and neutrons?

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Problem 9

In an endothermic chemical reaction, do the products have more mass, less mass, or the same mass as the reactants? Explain.

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Problem 10

What is the wavelength (in nm) of rays whose energy is 1.50 MeV?

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Problem 11

What is the frequency (in Hz) of X rays whose energy is 6.82 keV?

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Problem 12

Calculate the mass defect (in g/mol) for the following nuclei:
(a) $^{52} \mathrm{Fe}$ (atomic mass $=51.94811 \mathrm{amu} )$
(b) $^{92} \mathrm{Mo}$ (atomic mass $=91.90681 \mathrm{amu} )$

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Problem 13

Calculate the mass defect (in g/mol) for the following nuclei:
(a) $^{32}{S}$ (atomic mass $=31.97207 {amu} )$
(b)$^{40}$ Ca (atomic mass $=39.96259 {amu} )$

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Problem 14

Calculate the binding energy (in MeV/nucleon) for the following nuclei:
(a)) $^{58}$ Ni (atomic mass $=57.93535$ amu $)$
b) $^{84} \mathrm{Kr}(\text { atomic mass }=83.91151 \mathrm{amu})$

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Problem 15

Calculate the binding energy (in MeV/nucleon) for the following nuclei:
(a) $^{63} \mathrm{Cu}(\text { atomic mass }=62.93960 \mathrm{amu})$
(b) $^{84} \mathrm{Sr}$ (atomic mass $=83.91343 \mathrm{amu} )$

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Problem 16

What is the energy change $\Delta E(\text { in } \mathrm{kJ} / \mathrm{mol})$ when an $\alpha$ particle is emitted from 1$^{174} \mathrm{Ir}$? The atomic mass of $^{174} \mathrm{Ir}$ is 173.96666 amu, the atomic mass of $^{170} \mathrm{Re}$ is 169.95804 amu, and the atomic mass of a $^{4}4{He}$ atom is 4.00260 amu.
$\frac{174}{77} \mathrm{Ir}$ $\longrightarrow^{170}_{75} \mathrm{Re}+_{2}^{4} \mathrm{He}$ $\Delta E=?$

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Problem 17

Magnesium- 28 is a $\beta$ emitter that decays to aluminum- 28 How much energy is released in kJ/mol? The atomic mass of $^{28} \mathrm{Mg}$ is $27.98388 \mathrm{amu},$ and the atomic mass of $^{28} \mathrm{Al}$ is 27.98191 $\mathrm{amu}$

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Problem 18

What is the mass change in grams accompanying the formation of $\mathrm{NH}_{3}$ from $\mathrm{H}_{2}$ and $\mathrm{N}_{2} ?$
$\mathrm{N}_{2}(g)+3 \mathrm{H}_{2}(g) \longrightarrow 2 \mathrm{NH}_{3}(g) \quad \Delta H^{\circ}=-92.2 \mathrm{kJ}$

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Problem 19

What is the mass change in grams accompanying the formation of $\mathrm{CO}$ and $\mathrm{H}_{2}$ in the water-gas reaction?
$\mathrm{C}(s)+\mathrm{H}_{2} \mathrm{O}(g) \longrightarrow \mathrm{CO}(g)+\mathrm{H}_{2}(g) \quad \Delta H^{\circ}=+131 \mathrm{kJ}$

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Problem 20

A positron has the same mass as an electron $(9.109 \times$ $10^{-31} \mathrm{kg}$ ) but an opposite charge. When the two particles encounter each other, annihilation occurs and only $\gamma$ rays
are produced. How much energy (in kJ/mol) is produced?

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Problem 21

How much energy is released (in kJ) in the fusion reaction of $^{2} \mathrm{H}$ to yield 1 $\mathrm{mol}$ of 3 $\mathrm{He}$ ? The atomic mass of $^{2} \mathrm{H}$ is 2.0141 amu, and the atomic mass of $^{3}$ He is 3.0160 amu.
$2 \} \mathrm{H} \longrightarrow_{2}^{3} \mathrm{He}+\delta \mathrm{n}$

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Problem 22

Give the products of the following nuclear reactions:
(a) $\underset{47}{109} \mathrm{Ag}+\frac{4}{2} \mathrm{He} \longrightarrow ?$
(b) $\frac{10}{5} \mathrm{B}+\frac{4}{2} \mathrm{He} \longrightarrow ?+_{0} \mathrm{n}$

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Problem 23

Balance the following equations for the nuclear fission of
$^{235} \mathrm{U} :$
a) (a) $\frac{235}{92} \mathrm{U} \longrightarrow_{62}^{160} \mathrm{Sm}+_{30}^{72} \mathrm{Zn}+2 \frac{1}{0} \mathrm{n}$
b) $^{235}_{92} \mathrm{U} \longrightarrow \frac{87}{35} \mathrm{Br}+?+2 \stackrel{1}{0} \mathrm{n}$

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Problem 24

Element 109$\left(\frac{266}{109} \mathrm{Mt}\right)$ was prepared in 1982 by bombardment of $^{209} \mathrm{Bi}$ atoms with $^{58} \mathrm{Fe}$ atoms. Identify the other product that must have formed, and write a balanced nuclear equation assuming no other products were formed.

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Problem 25

Molybdenum-99 is formed by neutron bombardment of a naturally occurring isotope. If one neutron is absorbed and no byproducts are formed, what is the starting isotope?

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Problem 26

Californium-246 is formed by bombardment of uranium- 238 atoms. If four neutrons are formed as byproducts, what particle is used for the bombardment?

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Problem 27

Balance the following transmutation reactions:
a) $^{246}_{96} \mathrm{Cm}+^{12} \mathrm{C} \longrightarrow ?$ + 4 $_{0}^{1} \mathrm{n}$
b) $\frac{233}{99} \mathrm{Es}+?$$? \longrightarrow \frac{256}{101} \mathrm{Md}$ $_{0}^{1} \mathrm{n}$
c) $^{250}_{98} \mathrm{Cf}+$ $\frac{11}{5} \mathrm{B}$ $\longrightarrow ?+4 \frac{1}{0} n$

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Problem 28

The electronic systems on the New Horizons spacecraft, launched on January $19,2006,$ and scheduled to reach Pluto on July $14,2015,$ are powered by electricity generated by heat. The heat comes from the radioactive decay of $^{238} \mathrm{Pu}$ in the 11 $\mathrm{kg}$ of $^{238 \mathrm{PuO}_{2}}$ fuel onboard. The generator provided 240 $\mathrm{W}$ when the spacecraft was launched. If the power output is directly proportional to the amount of $^{238 }\mathrm{Pu}$ in the generator, what will the power output be when the spacecraft reaches Pluto? The half life of $^{238 }\mathrm{Pu}$ is 87.7 $\mathrm{y}$ .

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Problem 29

To treat a brain tumor with gamma knife radiosurgery, the patient's head is positioned within a hemispherical dome covered by 201 individual 60 $\mathrm{Co}$ sources directed inward toward the tumor target point. The tumor receives a very high dose of radiation because all the beams converge on it, while any irradiated healthy tissue receives only the radiation of a single beam. For a prescribed dose of 1800 rad, how long should the radiation treatment go if $2.2 \times 10^{11}$ disintegrations of $^{60} \mathrm{Co}$ are required to give a dose of 1.0 rad? Assume that all the sources are directed at the tumor with an activity of 30 $\mathrm{Ci}$ each.

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Problem 30

Thorium-232 decays by a 10 -step series of nuclear reactions, ultimately yielding lead- $208,$ along with 6$\alpha$ particles and 4$\beta$ particles. How much energy (in kJ/mol) is released
during the overall process? The relevant masses are: $^{232} \mathrm{Th}=$ 232.038054 amu $; \quad^{1} 208 \mathrm{Pb}=207.976627 \mathrm{amu}$ clectron $=$ $5.485799 \times 10^{-4}$ amu $;^{4} \mathrm{He}=4.002603 \mathrm{amu}$

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Problem 31

Fraud in science is rare but does happen occasionally. In $1999,$ the creation of three superheavy elements (one new) was aimed when $^{208} \mathrm{Pb}$ was bombarded with $^{\infty} \mathrm{Kr}$ . The claim was subsequently found to be fraudulent and was withdrawn. Identify the isotopes X, Y, and Z that were claimed.

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Problem 32

Calculate the mass defect (in g/mol) and the binding energy (in MeV/nucleon) for the following nuclei. Which of the two is more stable?
(a) $^{50} \mathrm{Cr}$ (atomic mass $=49.94605 \mathrm{amu} )$
(b) $^{64} \mathrm{Zn}$ (atomic mass $=63.92915 \mathrm{amu} )$

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Problem 33

What is the age BP of a bone fragment that shows an average of 2.9 disintegrations/min per gram of carbon in 2005$?$ The carbon in living organisms undergoes an average of 15.3 disintegrations/min per gram, and the half-life of $^{14} \mathrm{C}$ is 5715 years. ( See Section $12.6 . )$

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Problem 34

How much energy (in kJ/mol) is released in the fusion reaction of $^{2} \mathrm{H}$ with $^{3} \mathrm{He} ?$
$$\frac{2}{1} \mathrm{H}+\frac{3}{2} \mathrm{He} \longrightarrow_{2}^{4} \mathrm{He}+_{1}^{1} \mathrm{H}$$
The relevant masses are $^{2} \mathrm{H}(2.0141 \mathrm{amu}),^{3}$ He $(3.0160 \mathrm{amu})$ $^{4} \mathrm{He}\left(4.0026 \text { amu), and }^{\mathrm{i}} \mathrm{H}(1.0078 \mathrm{amu})\right.$

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Problem 35

What is the age of a rock whose $^{40} \mathrm{Ar} /^{40} \mathrm{K}$ ratio is 1.15$?$ The half-life of $^{40} \mathrm{K}$ is $1.28 \times 10^{9}$ years.

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Problem 36

The most abundant isotope of uranium, $238 \mathrm{U},$ does not undergo fission. In a breeder reactor, however, a $^{238} \mathrm{U}$ atom captures a neutron and emits two $\beta$ particles to make a fissionable isotope of plutonium, which can then be used as fuel in a nuclear reactor. Write a balanced nuclear equation.

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Problem 37

It has been estimated that $3.9 \times 10^{23} \mathrm{kJ} / \mathrm{s}$ is radiated into
space by the Sun. What is the rate of the Sun's mass loss in $\mathrm{kg} / \mathrm{s}$ ?

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Problem 38

In a cancer treatment called boron neutron-capture therapy, a drug containing boron-10 is injected into a patient where it selectively binds to cancer cells. Irradiating the affected area with neutrons then induces the following reaction:
$$^{10} \mathrm{B}+^{1} \mathrm{n} \longrightarrow^{4} \mathrm{He}+^{7} \mathrm{Li}+\gamma$$
The radiation kills the cancer cells, leaving the surrounding tissue unharmed. The reactants in this nuclear process have essentially no kinetic energy, but the products have a total kinetic energy of 2.31 MeV. What is the energy of the photon released? Relevant masses are: $^{4} \mathrm{He}(4.002603 \mathrm{amu}), 7 \mathrm{Li}(7.016004 \mathrm{amu})$ $^{10} \mathrm{B}(10.012937 \mathrm{amu})$ $\mathrm{e}^{-}(0.0005486 \mathrm{amu}),$ on $(1.008665 \mathrm{amu})$

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Problem 39

Neptunium- 237 decays by a series of steps to bismuth-209. How many $\alpha$ and $\beta$ particles are produced by this decay process?

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Problem 40

The radioactive isotope $^{100} \mathrm{Tc}$ decays to form the stable isotope $^{100} \mathrm{Tc}$
(a) There are two possible pathways for this decay. Write balanced equations for both.
(b) Only one of the pathways is observed. Calculate the energy released by both pathways, and explain why only one is observed. Relevant masses are: $^{100}$ $\mathrm{Tc}(99.907657 \mathrm{amu})$ $^{100} \mathrm{Mo}(99.90748 \mathrm{amu}), \mathrm{e}^{-}(0.0005486 \mathrm{amu})$

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Problem 41

The radioisotope 226 Ac can decay by any of three different nuclear processes: $\alpha$ emission, $\beta$ emission, or electron
(a) Write a balanced nuclear equation for the decay of $^{226 }$Ac by each decay mode.
(b) For the decay of $^{226} \mathrm{Ac}$ by all processes combined, the first-order rate constant is $k=0.556 \mathrm{d}^{-1} .$ How many days are required for 80.0$\%$ of a sample of $^{226} \mathrm{Ac}$ to decay?

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Problem 42

A small sample of wood from an archeological site in Clovis, New Mexico, was burned in ${O}_{2}$ and the ${CO}_{2}$ produced was bubbled through a solution of ${Ba}({OH})_{2}$ to produce a precipitate of ${BaCO}_{3}$ . When the ${BaCO}_{3}$ was collected by filtration, a 1.000 g sample was found to have a radioactivity of $4.0 \times 10^{-3}$ Bq. The half-life of 14 ${C}$ is $5715 {y},$ and living organisms have a radioactivity due to 14 ${C}$ of 15.3 disintegrations/min per gram of carbon. Assuming that the analysis was carried out in $1960,$ what is the age ${BP}$ of the Clovis site?

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Problem 43

Polonium-210, a naturally occurring radioisotope, is an $\alpha$ emitter, with $t_{1 / 2}=138 \mathrm{d}$ . Assume that a sample of 210 $\mathrm{po}$ with a mass of 0.700 $\mathrm{mg}$ was placed in a 250.0 $\mathrm{mL}$ flask, which was evacuated, sealed, and allowed to sit undisturbed. What would the pressure be inside the flask (in $\mathrm{mm} \mathrm{Hg}$ ) at $20^{\circ} \mathrm{C}$ after 365 days if all the $\alpha$ particles emitted had become helium atoms?

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Problem 44

A blood-volume determination was carried out on a patient by injection with 20.0 mL of blood that had been radioactively labeled with Cr-51 to an activity of 4.10$\mu \mathrm{Ci} / \mathrm{mL}$ . After a brief period to allow for mixing in the body, blood was drawn from the patient for analysis.
Unfortunately, a mixup in the laboratory prevented an immediate analysis, and it was not until 17.0 days later that a measurement on the blood was made. The radiation level was then determined to be 0.00935$\mu \mathrm{Ci} / \mathrm{mL}$ . If $^{51} \mathrm{Cr}$ has $t_{1 / 2}=27.7$ days, what is the volume of blood in the patient?

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Problem 45

Imagine that you have a 0.00750 $\mathrm{M}$ aqueous $\mathrm{MgCl}_{2}$ solution, prepared so that it contains a small amount of radioactive 28 $\mathrm{Mg}$ . The half-life of $^{28} \mathrm{Mg}$ is $20.91 \mathrm{h},$ and the initial activity of the $\mathrm{MgCl}_{2}$ solution is 0.112$\mu \mathrm{Ci} / \mathrm{mL}$ . Assume that 20.00 $\mathrm{mL}$ of this $\mathrm{MgCl}_{2}$ solution is added to 15.00 $\mathrm{mL}$ of 0.01250 $\mathrm{M}$ aqueous $\mathrm{Na}_{2} \mathrm{CO}_{3}$ solution and that the resultant precipitate is then removed by filtration to give a clear filtrate. After a long break to go for a run, you find that the activity of the filtrate measured 2.40 $\mathrm{h}$ after beginning the experiment is 0.029$\mu \mathrm{Ci} / \mathrm{mL}$ . What are themolar concentrations of $\mathrm{Mg}^{2+}$ and $\mathrm{CO}_{3}^{2}$ in the filtrate, and what is the solubility product constant of $\mathrm{MgCO}_{3} ?$

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