Chapter 19

Smoke detectors contain a small amount of americium-241. Its decay product is neptunium-237. Identify the emission from americium-241.

Lead-210 is used to prepare eyes for corneal transplants. Its decay product is bismuth-210. Identify the emission from lead-210.

Write balanced nuclear reactions for the following: (a) Formation of Am-241 through \(\beta\) -emissions. (b) Formation of Kr-81 by K-electron capture. (c) Formation of Ra-223 by \(\alpha\) -emission.

Write balanced nuclear equations for the following: (a) Formation of Mn-52 by positron emission. (b) Formation of Ac-228 by \(\beta\) -emission. (c) Formation of \(\mathrm{Np}-232\) by \(\alpha\) -decay.

Write balanced nuclear equations for (a) the alpha emission resulting in the formation of \(\mathrm{Pa}-233\). (b) the loss of a positron by \(\mathrm{Y}-85\). (c) the fusion of two C-12 nuclei to give sodium-23 and another particle. (d) the fission of Pu-239 to give tin-130, another nucleus, and an excess of two neutrons.

Write balanced nuclear equations for (a) the loss of an alpha particle by Th-230. (b) the loss of a beta particle by lead-210. (c) the fission of U-235 to give Ba-140, another nucleus, and an excess of two neutrons. (d) the K-capture of Ar-37.

An isotope of rutherfordium, \({ }_{10}^{257} \mathrm{Rf}\), is formed by the bombardment of californium- 249 by carbon-12. In the process, neutrons are emitted. The new isotope formed decays rapidly, emitting an alpha particle. (a) How many neutrons are emitted for every Cf-249 bombarded? (b) Write the nuclear symbol for the isotope formed by the decay of Rf-257.

When bismuth-209 is bombarded with nickel-64, one neutron and a new isotope, \(\mathrm{X}\), is formed. The isotope then goes through a series of alpha particle emissions. (a) Write the nuclear symbol for the isotope formed. (b) Write the nuclear symbol for the isotope formed after the third alpha particle emission.

Write balanced nuclear equations for the bombardment of (a) U-238 with a nucleus to produce Fm-249 and five neutrons. (b) Al-26 with an alpha particle to produce P-30. (c) Cu-63 with a nucleus producing \(\mathrm{Zn}-63\) and a neutron. (d) Al-27 with deuterium \(\left({ }_{1}^{2} \mathrm{H}\right)\) to produce an alpha particle and another nucleus.

Write balanced nuclear equations for the bombardment of (a) Fe-54 with an alpha particle to produce another nucleus and two protons. (b) Mo-96 with deuterium \(\left({ }_{1}^{2} \mathrm{H}\right)\) to produce a neutron and another nucleus. (c) Ar-40 with an unknown particle to produce \(\mathrm{K}-43\) and a proton. (d) a nucleus with a neutron to produce a proton and \(\mathrm{P}-31 .\)

Balance the following equations by filling in the blanks. (a) \({ }_{92}^{235} \mathrm{U}+{ }_{0} n \longrightarrow{ }_{54}^{137}=2{ }_{0}^{1} n+\) (b) \({ }_{90}^{232} \mathrm{Th}+{ }_{6}^{12}\) \(\longrightarrow\) \(1 . n+{ }_{96}^{240} \mathrm{Cm}\) (c) \({ }_{2}^{4} \mathrm{He}+{ }_{42}^{96} \mathrm{Mo} \longrightarrow{ }_{43}^{100}\) (d) \(+{ }_{1}^{2} \mathrm{H} \longrightarrow{ }_{84}^{210}+{ }_{0}^{1} n\)

Balance the following nuclear equations by filling in the blanks. (a) Es-249 + neutron \(\longrightarrow 2\) neutrons \(+\) ____\(+\) Gd-161 (b) ______ \(\longrightarrow\) beta particle \(+\mathrm{Co}-59\) (c) \(4 \mathrm{HH} \longrightarrow\)______ \(+2\) positrons (d) \(\mathrm{Mg}-24+\) neutron \(\longrightarrow\) proton \(+\) ________

A source for gamma rays has an activity of 3175 Ci. How many disintegrations are there for this source per minute?

A scintillation counter registers emitted radiation caused by the disintegration of nuclides. If each atom of nuclide emits one count, what is the activity of a sample that registers \(3.00 \times 10^{4}\) disintegrations in five minutes?

A Geiger counter counts \(0.070 \%\) of all particles emitted by a sample. What is the activity that registers \(19.4 \times 10^{3}\) counts in one minute?

Krypton-87 has a rate constant of \(1.5 \times 10^{-4} \mathrm{~s}^{-1}\). What is the activity of a \(2.00-\mathrm{mg}\) sample?

Iodine-131 is used to treat thyroid cancer. It decays by beta emission and has a half-life of \(8.1\) days. (a) Write a balanced nuclear reaction for the decay of iodine-131. (b) What is the activity (in Ci) of a 2.50-mg sample of the isotope?

Lead-210 has a half-life of \(20.4\) years. This isotope decays by beta particle emission. A counter registers \(1.3 \times 10^{4}\) disintegrations in five minutes. How many grams of \(\mathrm{Pb}-210\) are there?

Bromine-82 has a half-life of 36 hours. A sample containing Br-82 was found to have an activity of \(1.2 \times 10^{5}\) disintegrations \(/ \mathrm{min}\). How many grams of Br-82 were present in the sample? Assume that there were no other radioactive nuclides in the sample.

Cobalt-60 is used extensively in medicine as a source of \(\gamma\) -rays. Its half-life is \(5.27\) years. (a) How long will it take a \(\mathrm{Co}-60\) source to decrease to \(18 \%\) of its original activity? (b) What percent of its activity remains after 29 months?

Carbon from a cypress beam obtained from the tomb of an ancient Egyptian king gave \(9.2\) disintegrations/minute of \(C-14\) per gram of carbon. Carbon from living material gives \(15.3\) disintegrations/min of C-14 per gram of carbon. Carbon-14 has a half-life of 5730 years. How old is the beam?

Sandals found in a cave were determined by carbon-14 dating to be \(3.9 \times 10^{2}\) years old. Assuming that carbon from living material gives \(15.3\) disintegrations/min of C-14 per gram of carbon, what is the activity of the C-14 in the sandals in disintegrations/min/g of carbon? ( \(t_{1 / 2}\) of \(\mathrm{C}-14=\) 5730 years)

A sample of a wooden artifact gives \(5.0\) disintegrations \(/ \mathrm{min} / \mathrm{g}\) carbon. The half-life of carbon-14 is 5730 years, and the activity of C-14 in wood just cut down from a tree is \(15.3\) disintegrations \(/ \mathrm{min} / \mathrm{g}\) carbon. How old is the wooden artifact?

A rock from an archaeological dig was found to contain \(0.255 \mathrm{~g}\) of Pb-206 per gram of U-238. Assume that the rock did not contain any Pb-206 at the time of its formation and that U-238 decayed only to Pb-206. How old is the rock? (For \(\mathrm{U}-238, t_{1 / 2}=4.5 \times 10^{9} \mathrm{y}\).)

Plutonium- 239 is used as the energy source for heart pacemakers and space probes. It decays by alpha emission. (a) Calculate \(\Delta m\) in grams when one mole of Pu-239 decays. (b) How much energy (in kilojoules) is given off by the decay of \(2.00 \mathrm{mg}\) of \(\mathrm{Pu}-239 ?\)

Strontium-90 is a dangerous byproduct of atomic testing because it mimics the action of calcium in the body. It decays in two beta emissions to give zirconium- 90 (Nuclear mass \(=89.8824 \mathrm{~g}\) ). (a) Write a balanced nuclear reaction for the overall decay of Sr-90. (b) Calculate \(\Delta m\) in grams when one mole of Sr-90 decays to \(\mathrm{Zr}-90 .\) (c) How much energy (in kilojoules) is given off by the decay of \(6.50 \mathrm{mg}\) of \(\mathrm{Sr}-90 ?\)

Which has the larger binding energy, fluorine- 19 or oxygen-17?

Which has the larger binding energy, \(\mathrm{Mg}-26\) or \(\mathrm{Al}-26 ?\)

Consider the fusion of boron- 10 with an alpha particle. The products of the fusion are carbon-13 and a proton. (a) Write a nuclear reaction for this process. (b) How much energy is released when \(1.00 \mathrm{~g}\) of \(\mathrm{B}-10\) is fused with an \(\alpha\) -particle?

Show by calculation which process produces more energy per gram of material reacting. fission of U-235: \(\quad{ }^{235} \mathrm{U}_{22} \mathrm{U}+{ }_{0} n \longrightarrow{ }_{40}^{94} \mathrm{Zr}+{ }_{58}^{140} \mathrm{Ce}+6_{-1}{\underline{\phantom{xx}}}^{0} e+2{ }_{0}^{1} n\) fusion of deuterium: \(\quad{ }_{1}^{2} \mathrm{H}+{ }_{1}^{2} \mathrm{H} \longrightarrow{ }_{1}^{3} \mathrm{H}+{ }_{1}^{1} \mathrm{H}\) Nuclear masses for Ce-140 and Zr-94 are \(139.8734\) and 93.8841, respectively.

Consider the fission reaction in which U-235 is bombarded by neutrons. The products of the bombardment are rubidium-89, cerium-144, beta particles, and more neutrons. (a) Write a balanced nuclear equation for the bombardment. (b) Calculate \(\Delta E\) when one gram of U-235 undergoes fission. (c) The detonation of TNT, an explosive, evolves \(2.76 \mathrm{~kJ} / \mathrm{g}\). How many kilograms of TNT are required to produce the same amount of energy as one milligram of \(\mathrm{U}-235\) ?

Iodine- 131 is used in the treatment of tumors in the thyroid gland. Its half- life is \(8.1\) days. Suppose that, due to a shipment delay, the I-131 in a hospital's pharmacy is \(2.0\) days old. (a) What percentage of the I-131 has disintegrated? (b) A patient is scheduled to receive \(15.0 \mathrm{mg}\) of \(\mathrm{I}-131 .\) What dosage (in milligrams) should the hospital pharmacist recommend for this patient if the 2 -day-old bottle of \(\mathrm{I}-131\) is used?

An explosion used five tons \((1\) ton \(=2000 \mathrm{lb}\) ) of ammonium nitrate \((\Delta E=-37.0 \mathrm{~kJ} / \mathrm{mol})\). (a) How much energy was released by the explosion? (b) How many grams of TNT \((\Delta E=-2.76 \mathrm{~kJ} / \mathrm{g})\) are needed to release the energy calculated in (a)? (c) How many grams of U-235 are needed to obtain the same amount of energy calculated in (a)? (See the equation in Problem 45.)

The amount of oxygen dissolved in a sample of water can be determined by using thallium metal containing a small amount of the isotope Tl- 204\. When excess thallium is added to oxygen-containing water, the following reaction occurs. $$2 \mathrm{Tl}(s)+\frac{1}{2} \mathrm{O}_{2}(g)+\mathrm{H}_{2} \mathrm{O} \longrightarrow 2 \mathrm{Tl}^{+}(a q)+2 \mathrm{OH}^{-}(a q)$$ After reaction, the activity of a 25.0-mL water sample is 745 counts per minute (cpm), caused by the presence of \(\mathrm{Tl}^{+}-204\) ions. The activity of Tl-204 is \(5.53 \times 10^{5} \mathrm{cpm}\) per gram of thallium metal. Assuming that \(\mathrm{O}_{2}\) is the limiting reactant in the above equation, calculate its concentration in moles per liter.

A 35-mL sample of \(0.050 \mathrm{M} \mathrm{AgNO}_{3}\) is mixed with \(35 \mathrm{~mL}\) of \(0.050 \mathrm{M}\) NaI labeled with I-131. The following reaction occurs. $$\mathrm{Ag}^{+}(a q)+\mathrm{I}^{-}(a q) \longrightarrow \mathrm{AgI}(s)$$ The filtrate is found to have an activity of \(2.50 \times 10^{3}\) counts per minute per milliliter. The \(0.050 \mathrm{M}\) NaI solution had an activity of \(1.25 \times 10^{10}\) counts per minute per milliliter. Calculate \(K_{\text {sp }}\) for AgI.

A \(100.0\) -g sample of water containing tritium, \({ }_{1}^{3} \mathrm{H}\), emits \(2.89 \times 10^{3}\) beta particles per second. Tritium has a half-life of \(12.3\) years. What percentage of all the hydrogen atoms in the water sample is tritium?

Use the half-life of tritium given in Problem 53 to calculate the activity in curies of \(1.00 \mathrm{~mL}\) of \({ }_{1}^{3} \mathrm{H}_{2}\) at STP.

One of the causes of the explosion at Chernobyl may have been the reaction between zirconium, which coated the fuel rods, and steam. $$\mathrm{Zr}(s)+2 \mathrm{H}_{2} \mathrm{O}(g) \longrightarrow \mathrm{ZrO}_{2}(s)+2 \mathrm{H}_{2}(g)$$ If half a metric ton of zirconium reacted, what pressure was exerted by the hydrogen gas produced at \(55^{\circ} \mathrm{C}\) in the containment chamber, which had a volume of \(2.0 \times 10^{4} \mathrm{~L} ?\)

To measure the volume of the blood in an animal's circulatory system, the following experiment was performed. A 5.0-mL sample of an aqueous solution containing \(1.7 \times 10^{5}\) counts per minute (cpm) of tritium was injected into the bloodstream. After an adequate period of time to allow for the complete circulation of the tritium, a 5.0-mL sample of blood was withdrawn and found to have \(1.3 \times 10^{3} \mathrm{cpm}\) on the scintillation counter. Assuming that only a negligible amount of tritium has decayed during the experiment, what is the volume of the animal's circulatory system?

Consider the fission reaction $${ }_{0}^{1} n+{ }_{92}^{235} \mathrm{U} \longrightarrow{ }_{3}{\underline{\phantom{xx}}}_{3}^{89} \mathrm{Rb}+{ }_{55}^{144} \mathrm{Ce}+3_{-1}^{0} e+3{ }_{0}^{1} n$$ How many liters of octane, \(\mathrm{C}_{8} \mathrm{H}_{18}\), the primary component of gasoline, must be burned to \(\mathrm{CO}_{2}(g)\) and \(\mathrm{H}_{2} \mathrm{O}(g)\) to produce as much energy as the fission of one gram of U-235 fuel? Octane has a density of \(0.703 \mathrm{~g} / \mathrm{mL} ;\) its heat of formation is \(-249.9 \mathrm{~kJ} / \mathrm{mol}\).

Radium-226 decays by alpha emission to radon-222. Suppose that \(25.0 \%\) of the energy given off by one gram of radium is converted to electrical energy. What is the minimum mass of lithium that would be needed for the voltaic cell \(\mathrm{Li}\left|\mathrm{Li}^{+} \| \mathrm{Cu}^{2+}\right| \mathrm{Cu}\), at standard conditions, to produce the same amount of electrical work \(\left(\Delta G^{\circ}\right) ?\)

For how many years could all the energy needs of the world be supplied by the fission of \(\mathrm{U}-235\) ? Use the following assumptions: The world has about \(1.0 \times 10^{7}\) metric tons of uranium ore, which are about 0.75\% U-235. The energy consumption of the world is about \(4.0 \times 10^{15} \mathrm{~kJ} / \mathrm{y}\) and does not change with time. The fission of U-235 releases about \(8.0 \times 10^{7} \mathrm{~kJ} / \mathrm{g}\) of \(\mathrm{U}-235\).

When a positron and an electron collide, they annihilate each other and produce two gamma photons, which carry the same amount of energy. What is the wavelength (in nanometers) of these photons?

Classify the following statements as true or false. If false, correct the statement to make it true. (a) The mass number increases in beta emission. (b) A radioactive species with a large rate constant, \(k\), decays very slowly. (c) Fusion gives off less energy per gram of fuel than fission.

Explain how (a) alpha and beta radiation are separated by an electric field. (b) radioactive C-11 can be used as a tracer to study brain disorders. (c) a self-sustaining chain reaction occurs in nuclear fission.

The principle behind the home smoke detector is described on page 516. Americium- 241 is present in such detectors. It has a decay constant of \(1.51 \times 10^{-3} \mathrm{y}^{-1}\). You are urged to check the battery in the detector at least once a year. You are, however, never encouraged to check how much Am-241 remains undecayed. Explain why.

The cleavage of ATP (adenosine triphosphate) to ADP (adenosine diphosphate) and \(\mathrm{H}_{3} \mathrm{PO}_{4}\) may be written as follows: It is interesting to determine which bond (the \(\mathrm{P}-\mathrm{O}\) bond marked a or the \(\mathrm{O}\) - \(\mathrm{P}\) bond marked \(\mathbf{b}\) ) is cleaved by hydrolysis (reaction with water). (a) Outline an experiment (using radioactivity) that can be used to determine where the cleavage results. (b) Describe the results that would lead you to conclude that cleavage results at a, (c) Describe the results that would lead you to conclude that cleavage results at \(\mathbf{b}\). Results show that the cleavage occurs at \(\mathrm{b}\).

An activity of 20 picocuries \(\left(20 \times 10^{-12} \mathrm{Ci}\right)\) of radon- 222 per liter of air in a house constitutes a health hazard to anyone living there. The half-life of radon-222 is \(3.82\) days. Calculate the concentration of radon in air (moles per liter) that corresponds to a 20-picocurie activity level.

It is possible to estimate the activation energy for fusion by calculating the energy required to bring two deuterons close enough to one another to form an alpha particle. This energy can be obtained by using Coulomb's law in the form \(E=8.99 \times 10^{9} q_{1} q_{2} / r\), where \(q_{1}\) and \(q_{2}\) are the charges of the deuterons \(\left(1.60 \times 10^{-19} \mathrm{C}\right), r\) is the radius of the He nucleus, about \(2 \times 10^{-15} \mathrm{~m}\), and \(E\) is the energy in joules. (a) Estimate \(E\) in joules per alpha particle. (b) Using the equation \(E=m v^{2} / 2\), estimate the velocity (meters per second) each deuteron must have if a collision between the two of them is to supply the activation energy for fusion \((m\) is the mass of the deuteron in kilograms).

Consider the reaction $$2{ }_{1}^{2} \mathrm{H} \longrightarrow{ }_{2}^{4} \mathrm{He}$$ (a) Calculate \(\Delta E\) in kilojoules per gram of deuterium fused. (b) How much energy is potentially available from the fusion of all the deuterium in seawater? The percentage of deuterium in water is about \(0.0017 \%\). The total mass of water in the oceans is \(1.3 \times 10^{24} \mathrm{~g}\). (c) What fraction of the deuterium in the oceans would have to be consumed to supply the annual energy requirements of the world \(\left(2.3 \times 10^{17} \mathrm{~kJ}\right) ?\)