21.E: Exercises

21.1: Radioactivity

Q21.1.1

Why are many radioactive substances warm to the touch? Why do many radioactive substances glow?

Q21.1.2

Describe the differences between nonionizing and ionizing radiation in terms of the intensity of energy emitted and the effect each has on an atom or molecule after collision. Which nuclear decay reactions are more likely to produce ionizing radiation? nonionizing radiation?

Q21.1.3

Would you expect nonionizing or ionizing radiation to be more effective at treating cancer? Why?

S21.1.3

Ionizing radiation is higher in energy and causes greater tissue damage, so it is more likely to destroy cancerous cells.

Q21.1.4

Historically, concrete shelters have been used to protect people from nuclear blasts. Comment on the effectiveness of such shelters.

Q21.1.5

Gamma rays are a very high-energy radiation, yet α particles inflict more damage on biological tissue. Why?

Q21.1.6

List the three primary sources of naturally occurring radiation. Explain the factors that influence the dose that one receives throughout the year. Which is the largest contributor to overall exposure? Which is the most hazardous?

 

S21.1.6

Three primary naturally occurring radiations are radium, uranium and thorium, each all having long half lives. Inhalation of air, ingestion of food and water,terrestrial radation from the ground and cosmic radiation from space are all factors tat influence the does that a person receives throughout the year. Inhalation of the air is the largest contributor to exposure. Radiation can damage DNA or kill cells. When radiation is exposed to your body, it will collide with atoms and this will change and damage your DNA.

Q21.1.7

Because radon is a noble gas, it is inert and generally unreactive. Despite this, exposure to even low concentrations of radon in air is quite dangerous. Describe the physical consequences of exposure to radon gas. Why are people who smoke more susceptible to these effects?

Q21.1.8

Most medical imaging uses isotopes that have extremely short half-lives. These isotopes usually undergo only one kind of nuclear decay reaction. Which kind of decay reaction is usually used? Why? Why would a short half-life be preferred in these cases?

 

Beta decay. Alfa decay can be easily stopped by paper, which means it can not be used to see inside people's body. Also, Gamma rays are really dangerous for human, that even a short period of time exploding to it will have negative effect on human body. Thus, Beta decay is the perfect choice. It can be used to see through human's body and stopped by aluminum or some other metals.

 

Since all these radioactive decays are harmful for human body, if the half time of these reactions are short, the time exploding to these reactions will be short too.

Q21.1.9

Which would you prefer: one exposure of 100 rem, or 10 exposures of 10 rem each? Explain your rationale.

S21.1.9

Ten exposures of 10 rem are less likely to cause major damage.

Q21.1.10

A 2.14 kg sample of rock contains 0.0985 g of uranium. How much energy is emitted over 25 yr if 99.27% of the uranium is ²³⁸U, which has a half-life of 4.46 × 10⁹ yr, if each decay event is accompanied by the release of 4.039 MeV? If a 180 lb individual absorbs all of the emitted radiation, how much radiation has been absorbed in rads?

Q21.1.11

There is a story about a “radioactive boy scout” who attempted to convert thorium-232, which he isolated from about 1000 gas lantern mantles, to uranium-233 by bombarding the thorium with neutrons. The neutrons were generated via bombarding an aluminum target with α particles from the decay of americium-241, which was isolated from 100 smoke detectors. Write balanced nuclear reactions for these processes. The “radioactive boy scout” spent approximately 2 h/day with his experiment for 2 yr. Assuming that the alpha emission of americium has an energy of 5.24 MeV/particle and that the americium-241 was undergoing 3.5 × 10⁶ decays/s, what was the exposure of the 60.0 kg scout in rads? The intrepid scientist apparently showed no ill effects from this exposure. Why?

241/95 Am---> 4/2 He + 237/93Np---> 4/2He + 233/91Pa----> 1/0n+ 232/91Th---> 1/1 H + 233/92 U

By adding alpha particles to the products side of the reaction, he was able to reduce the mass number by 4 and the atomic number by 2 to get the products he wanted. Bombardment with neutrons and 1 H was required to lower to the mass number to get Th and then raise both the mass number and the atomic number to yield Uranium.

2 hours*365*2= 1460 hours of exposure.*60min/1hr*60s/1min= 5.26*10^6s of exposure

1MeV= (1.6022*10^-13 Joules) * (5.24 MeV/particle)*2 particles= 1.679*10^-12 Joules.

(1.679*10^-12 Joules) * (1 amu/ 1.4924*10^-10 Joules)= 2.51*10^-13 amu

E=mc^2

E=(2.51*10^-13 amu)(1.66*10^-22kg/amu)(2.9998*10^8m/s)^2= (3.75*10^-18 kgm^2/s)*(3.5*10^6 decays/s)= 1.31*10^-11 joules of exposure per second.

The scientist showed no ill effects from this exposure because if we multiple the energy in joules of exposure per second, 1.31*10^-11, by the total amount of seconds of exposure, 5.26*10^6s, we find that he was only exposed to 6.9*10^-5 joules of radiation throughout the span of two years. This is a very small amount of radiation for such a long span of time.

In order to plug in the values for this equation, we must convert the given MeV to Joules with the known conversion rate. Similarly, we must convert Joules to amu with another known conversion rate. Then we can plug in the values and multiply by c^2 but we must not forget to multiple the amu by the conversion rate to kg in order to yield Joules. After all of this is done, we multiple the amount of Joules of exposure per second by the total amount of exposure in seconds in order to find out the total amount of exposure over the two year span.

21.2: Patterns of Nuclear Stability

Q21.2.1

How do chemical reactions compare with nuclear reactions with respect to mass changes? Does either type of reaction violate the law of conservation of mass? Explain your answers.

Q21.2.2

Why is the amount of energy released by a nuclear reaction so much greater than the amount of energy released by a chemical reaction?

Q21.2.3

Explain why the mass of an atom is less than the sum of the masses of its component particles.

Q21.2.4

The stability of a nucleus can be described using two values. What are they, and how do they differ from each other?

Q21.2.5

In the days before true chemistry, ancient scholars (alchemists) attempted to find the philosopher’s stone, a material that would enable them to turn lead into gold. Is the conversion of Pb → Au energetically favorable? Explain why or why not.

Q21.2.6

Describe the energy barrier to nuclear fusion reactions and explain how it can be overcome.

Q21.2.7

Imagine that the universe is dying, the stars have burned out, and all the elements have undergone fusion or radioactive decay. What would be the most abundant element in this future universe? Why?

Q21.2.8

Numerous elements can undergo fission, but only a few can be used as fuels in a reactor. What aspect of nuclear fission allows a nuclear chain reaction to occur?

Q21.2.9

How are transmutation reactions and fusion reactions related? Describe the main impediment to fusion reactions and suggest one or two ways to surmount this difficulty.

Q21.2.10

Using the information provided in Chapter 33, complete each reaction and calculate the amount of energy released from each in kilojoules.

1. ²³⁸Pa → ? + β⁻
2. ²¹⁶Fr → ? + α
3. ¹⁹⁹Bi → ? + β⁺

Q21.2.22

Using the information provided in Chapter 33, complete each reaction and calculate the amount of energy released from each in kilojoules.

1. ¹⁹⁴Tl → ? + β⁺
2. ¹⁷¹Pt → ? + α
3. ²¹⁴Pb → ? + β⁻

Q21.2.23

Using the information provided in Chapter 33, complete each reaction and calculate the amount of energy released from each in kilojoules per mole.

1. \(_{91}^{234}\textrm{Pa}\rightarrow \,?+\,_{-1}^0\beta\)
2. \(_{88}^{226}\textrm{Ra}\rightarrow \,?+\,_2^4\alpha\)

Q21.2.24

Using the information provided in Chapter 33, complete each reaction and then calculate the amount of energy released from each in kilojoules per mole.

1. \(_{27}^{60}\textrm{Co}\rightarrow\,?+\,_{-1}^0\beta\) (The mass of cobalt-60 is 59.933817 amu.)
2. technicium-94 (mass = 93.909657 amu) undergoing fission to produce chromium-52 and potassium-40

Q21.2.25

Using the information provided in Chapter 33, predict whether each reaction is favorable and the amount of energy released or required in megaelectronvolts and kilojoules per mole.

1. the beta decay of bismuth-208 (mass = 207.979742 amu)
2. the formation of lead-206 by alpha decay

Q21.2.26

Using the information provided, predict whether each reaction is favorable and the amount of energy released or required in megaelectronvolts and kilojoules per mole.

1. alpha decay of oxygen-16
2. alpha decay to produce chromium-52

Q21.2.27

Calculate the total nuclear binding energy (in megaelectronvolts) and the binding energy per nucleon for ⁸⁷Sr if the measured mass of ⁸⁷Sr is 86.908877 amu.

1. the calculated mass
2. the mass defect
3. the nuclear binding energy
4. the nuclear binding energy per nucleon

Q21.2.30

The experimentally determined mass of ²⁹S is 28.996610 amu. Find each of the following.

1. the calculated mass
2. the mass defect
3. the nuclear binding energy
4. the nuclear binding energy per nucleon

Q21.2.31

Calculate the amount of energy that is released by the neutron-induced fission of ²³⁵U to give ¹⁴¹Ba, ⁹²Kr (mass = 91.926156 amu), and three neutrons. Report your answer in electronvolts per atom and kilojoules per mole.

Q21.2.31

Calculate the amount of energy that is released by the neutron-induced fission of ²³⁵U to give ⁹⁰Sr, ¹⁴³Xe, and three neutrons. Report your answer in electronvolts per atom and kilojoules per mole.

Q21.2.33

Calculate the amount of energy released or required by the fusion of helium-4 to produce the unstable beryllium-8 (mass = 8.00530510 amu). Report your answer in kilojoules per mole. Do you expect this to be a spontaneous reaction?

Q21.2.34

Calculate the amount of energy released by the fusion of ⁶Li and deuterium to give two helium-4 nuclei. Express your answer in electronvolts per atom and kilojoules per mole.

Q21.2.35

How much energy is released by the fusion of two deuterium nuclei to give one tritium nucleus and one proton? How does this amount compare with the energy released by the fusion of a deuterium nucleus and a tritium nucleus, which is accompanied by ejection of a neutron? Express your answer in megaelectronvolts and kilojoules per mole. Pound for pound, which is a better choice for a fusion reactor fuel mixture?

Numerical Answers

1

1. \(_{91}^{238}\textrm{Pa}\rightarrow\,_{92}^{238}\textrm{U}+\,_{-1}^0\beta\); −5.540 × 10⁻¹⁶ kJ
2. \(_{87}^{216}\textrm{Fr}\rightarrow\,_{85}^{212}\textrm{At}+\,_{2}^4\alpha\); −1.470 × 10⁻¹⁵ kJ
3. \(_{83}^{199}\textrm{Bi}\rightarrow\,_{82}^{199}\textrm{Pb}+\,_{+1}^0\beta\); −5.458 × 10⁻¹⁶ kJ

3.  

1. \(_{91}^{234}\textrm{Pa}\rightarrow\,_{92}^{234}\textrm{U}+\,_{-1}^0\beta\); 2.118 × 10⁸ kJ/mol
2. \(_{88}^{226}\textrm{Ra}\rightarrow\,_{86}^{222}\textrm{Rn}+\,_{2}^4\alpha\); 4.700 × 10⁸ kJ/mol

5.  

1. The beta decay of bismuth-208 to polonium is endothermic (ΔE = 1.400 MeV/atom, 1.352 × 10⁸ kJ/mol).
2. The formation of lead-206 by alpha decay of polonium-210 is exothermic (ΔE = −5.405 MeV/atom, −5.218 × 10⁸ kJ/mol).

7. 757 MeV/atom, 8.70 MeV/nucleon

9.  

1. 53.438245 amu
2. 0.496955 amu
3. 463 MeV/atom
4. 8.74 MeV/nucleon

11. −173 MeV/atom; 1.67 × 10¹⁰ kJ/mol

13. ΔE = + 9.0 × 10⁶ kJ/mol beryllium-8; no

15. D–D fusion: ΔE = −4.03 MeV/tritium nucleus formed = −3.89 × 10⁸ kJ/mol tritium; D–T fusion: ΔE = −17.6 MeV/tritium nucleus = −1.70 × 10⁹ kJ/mol; D–T fusion

21.3: Nuclear Transmutations

Q21.3.1

Why do scientists believe that hydrogen is the building block of all other elements? Why do scientists believe that helium-4 is the building block of the heavier elements?

Q21.3.2

How does a star produce such enormous amounts of heat and light? How are elements heavier than Ni formed?

Q21.3.3

Propose an explanation for the observation that elements with even atomic numbers are more abundant than elements with odd atomic numbers.

S21.3.3

The raw material for all elements with Z > 2 is helium (Z = 2), and fusion of helium nuclei will always produce nuclei with an even number of protons.

Q21.3.4

During the lifetime of a star, different reactions that form different elements are used to power the fusion furnace that keeps a star “lit.” Explain the different reactions that dominate in the different stages of a star’s life cycle and their effect on the temperature of the star.

Q21.3.5

A line in a popular song from the 1960s by Joni Mitchell stated, “We are stardust….” Does this statement have any merit or is it just poetic? Justify your answer.

Q21.3.6

If the laws of physics were different and the primary element in the universe were boron-11 (Z = 5), what would be the next four most abundant elements? Propose nuclear reactions for their formation.

Q21.3.7

Write a balanced nuclear reaction for the formation of each isotope.

1. ²⁷Al from two ¹²C nuclei
2. ⁹Be from two ⁴He nuclei

Q21.3.8

At the end of a star’s life cycle, it can collapse, resulting in a supernova explosion that leads to the formation of heavy elements by multiple neutron-capture events. Write a balanced nuclear reaction for the formation of each isotope during such an explosion.

1. ¹⁰⁶Pd from nickel-58
2. selenium-79 from iron-56

Q21.3.9

When a star reaches middle age, helium-4 is converted to short-lived beryllium-8 (mass = 8.00530510 amu), which reacts with another helium-4 to produce carbon-12. How much energy is released in each reaction (in megaelectronvolts)? How many atoms of helium must be “burned” in this process to produce the same amount of energy obtained from the fusion of 1 mol of hydrogen atoms to give deuterium?

21.4: Rates of Radioactive Decay

Q21.4.1

What do chemists mean by the half-life of a reaction?

Q21.4.2

If a sample of one isotope undergoes more disintegrations per second than the same number of atoms of another isotope, how do their half-lives compare?

Q21.4.3

Half-lives for the reaction A + B → C were calculated at three values of [A]₀, and [B] was the same in all cases. The data are listed in the following table:

[A]0 (M)	t½ (s)
0.50	420
0.75	280
1.0	210

Does this reaction follow first-order kinetics? On what do you base your answer?

S21.4.3

1. No; the reaction is second order in A because the half-life decreases with increasing reactant concentration according to t_1/2 = 1/k[A₀].

Q21.4.4

Ethyl-2-nitrobenzoate (NO₂C₆H₄CO₂C₂H₅) hydrolyzes under basic conditions. A plot of [NO₂C₆H₄CO₂C₂H₅] versus t was used to calculate t½, with the following results:

[NO2C6H4CO2C2H5] (M/cm^3)	t½ (s)
0.050	240
0.040	300
0.030	400

Is this a first-order reaction? Explain your reasoning.

Q21.4.5

Azomethane (CH₃N₂CH₃) decomposes at 600 K to C₂H₆ and N₂. The decomposition is first order in azomethane. Calculate t½ from the data in the following table:

Time (s)	\(P_{\large{\mathrm{CH_3N_2CH_3}}}\) (atm)
0	8.2 × 10^−2
2000	3.99 × 10^−2
4000	1.94 × 10^−2

How long will it take for the decomposition to be 99.9% complete?

S21.4.5

t_1/2 = 1.92 × 10³ s or 1920 s; 19100 s or 5.32 hrs.

Q21.4.6

The first-order decomposition of hydrogen peroxide has a half-life of 10.7 h at 20°C. What is the rate constant (expressed in s⁻¹) for this reaction? If you started with a solution that was 7.5 × 10⁻³ M H₂O₂, what would be the initial rate of decomposition (M/s)? What would be the concentration of H₂O₂ after 3.3 h?

Application Problems

1. Until the 1940s, uranium glazes were popular on ceramic dishware. One brand, Fiestaware, had bright orange glazes that could contain up to 20% uranium by mass. Although this practice is less common today due to the negative association of radiation, it is still possible to buy Depression-era glassware that is quite radioactive. Aqueous solutions in contact with this “hot” glassware can reach uranium concentrations up to 10 ppm by mass. If 1.0 g of uranium undergoes 1.11 × 10⁷ decays/s, each to an α particle with an energy of 4.03 MeV, what would be your exposure in rem and rad if you drank 1.0 L of water that had been sitting for an extended time in a Fiestaware pitcher? Assume that the water and contaminants are excreted only after 18 h and that you weigh 70.0 kg.
2. Neutrography is a technique used to take the picture of an object using a beam of neutrons. How does the penetrating power of a neutron compare with alpha, beta, and gamma radiation? Do you expect similar penetration for protons? How would the biological damage of each particle compare with the other types of radiation? (Recall that a neutron’s mass is approximately 2000 times the mass of an electron.)
3. Spent fuel elements in a nuclear reactor contain radioactive fission products in addition to heavy metals. The conversion of nuclear fuel in a reactor is shown here:

Neglecting the fission products, write balanced nuclear reactions for the conversion of the original fuel to each product.

4. The first atomic bomb used ²³⁵U as a fissile material, but there were immense difficulties in obtaining sufficient quantities of pure ²³⁵U. A second fissile element, plutonium, was discovered in 1940, and it rapidly became important as a nuclear fuel. This element was produced by irradiating ²³⁸U with neutrons in a nuclear reactor. Complete the series that produced plutonium, all isotopes of which are fissile:

\(_{92}^{238}\textrm U+\,_0^1\textrm n\rightarrow\,\textrm U\rightarrow\,\textrm{Np}\rightarrow\,\textrm{Pu}\)

5. Boron neutron capture therapy is a potential treatment for many diseases. As the name implies, when boron-10, one of the naturally occurring isotopes of boron, is bombarded with neutrons, it absorbs a neutron and emits an α particle. Write a balanced nuclear reaction for this reaction. One advantage of this process is that neutrons cause little damage on their own, but when they are absorbed by boron-10, they can cause localized emission of alpha radiation. Comment on the utility of this treatment and its potential difficulties.

6. An airline pilot typically flies approximately 80 h per month. If 75% of that time is spent at an altitude of about 30,000 ft, how much radiation is that pilot receiving in one month? over a 30 yr career? Is the pilot receiving toxic doses of radiation?

7. At a breeder reactor plant, a 72 kg employee accidentally inhaled 2.8 mg of ²³⁹Pu dust. The isotope decays by alpha decay and has a half-life of 24,100 yr. The energy of the emitted α particles is 5.2 MeV, and the dust stays in the employee’s body for 18 h.

1. How many plutonium atoms are inhaled?
2. What is the energy absorbed by the body?
3. What is the physical dose in rads?
4. What is the dose in rems? Will the dosage be fatal?

8. For many years, the standard source for radiation therapy in the treatment of cancer was radioactive ⁶⁰Co, which undergoes beta decay to ⁶⁰Ni and emits two γ rays, each with an energy of 1.2 MeV. Show the sequence of nuclear reactions. If the half-life for beta decay is 5.27 yr, how many ⁶⁰Co nuclei are present in a typical source undergoing 6000 dps that is used by hospitals? The mass of ⁶⁰Co is 59.93 amu.

9. It is possible to use radioactive materials as heat sources to produce electricity. These radioisotope thermoelectric generators (RTGs) have been used in spacecraft and many other applications. Certain Cold War–era Russian-made RTGs used a 5.0 kg strontium-90 source. One mole of strontium-90 releases β particles with an energy of 0.545 MeV and undergoes 2.7 × 10¹³ decays/s. How many watts of power are available from this RTG (1 watt = 1 J/s)?

10. Potassium consists of three isotopes (potassium-39, potassium-40, and potassium-41). Potassium-40 is the least abundant, and it is radioactive, decaying to argon-40, a stable, nonradioactive isotope, by the emission of a β particle with a half-life of precisely 1.25 × 10⁹ yr. Thus the ratio of potassium-40 to argon-40 in any potassium-40–containing material can be used to date the sample. In 1952, fragments of an early hominid, Meganthropus, were discovered near Modjokerto in Java. The bone fragments were lying on volcanic rock that was believed to be the same age as the bones. Potassium–argon dating on samples of the volcanic material showed that the argon-40-to-potassium-40 molar ratio was 0.00281:1. How old were the rock fragments? Could the bones also be the same age? Could radiocarbon dating have been used to date the fragments?

Answers

1. 6.6 × 10⁻³ rad; 0.13 rem

\(_{92}^{235}\textrm U+\,_0^1\textrm n\rightarrow\,_{92}^{236}\textrm U+\,\gamma\)

\(_{92}^{238}\textrm U+\,_0^1\textrm n\rightarrow\,_{92}^{239}\textrm U+\,\gamma\)

\(_{92}^{239}\textrm U\rightarrow\,_{93}^{239}\textrm{Np}+\,_{-1}^0\beta\)

\(_{93}^{239}\textrm{Np}\rightarrow\,_{94}^{239}\textrm{Pu}+\,_{-1}^0\beta\)

\(_{94}^{239}\textrm{Pu}\rightarrow\,_{95}^{239}\textrm{Am}+\,_{-1}^0\beta\)

\(_{95}^{239}\textrm{Am}\rightarrow\,_{96}^{239}\textrm{Cm}+\,_{-1}^0\beta\)

1. 7.1 × 10¹⁸ atoms of Pu
2. 0.35 J
3. 0.49 rad
4. 9.8 rem; this dose is unlikely to be fatal.

9. 130 W