Nuclear Reactions and Their Applications

Question: Uranium and radium are found in many rocky soils throughout the world. Both undergo radioactive decay, and one of the products is radon-222, the heaviest noble gas (t=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 p Ci/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 pCi/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 Bq/L) after  9.5 days

(c) How many more days does it take to reach the EPA recommended level?

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 yr?

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

An earthquake in the area of present-day San Francisco is to be dated by measuring the ¹⁴C activity (t = 5730 yr) of parts of a tree uprooted during the event. The tree parts have an activity of 12.9 d/min-g C, and a living tree has an activity of 15.3 d/min C. How long ago did the earthquake occur?

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

Tritium $\left({}^{3}H;t_{1/2}=12.26yr\right)$ 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?

Even though plutonium $-239(t_{t/2}==2.41\times {10}^{4}yr)$is one of the main fission fuels, 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 plutoniumin -239 spent fuel to decay?

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

The reaction that allows for radiocarbon dating is the continual formation of carbon-14 in the upper atmosphere: 

${}_7{}^{14}N{+}_0^{1}n{\to }_6^{14}C{+}_1^{1}H$

What is the energy change associated with this process in eV/reaction and in kJ/mol reaction? (Masses of atoms: ${}_7{}^{14}N=14.003074$amu; ${}_6{}^{14}C=14.003241$ amu;${}_1{}^{1}H=1.007825$  amu;  ${}_0{}^{1}n=1.008665$amu.)

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.)

Using -century technology, hydrogen fusion requires temperatures around 10⁸K . But, lower initial temperatures are used if the hydrogen is compressed. In the late 24^th century, the starship Leinad uses such methods to fuse hydrogen at 10⁶K  . 

(a) What is the kinetic energy of an  atom at $1.00\times {10}^{6}K$? 

(b) How many H atoms are heated to $1.00\times {10}^{6}K$ from the energy of one H and one anti-H atom annihilating each other? 

(c) If these H atoms fuse into ⁴He atoms (with the loss of two positrons per ⁴He  formed), how much energy (in Jr) is generated?

(d) How much more energy is generated by the fusion in than by the hydrogen-antihydrogen collision in ? 

(e) Should the captain of the Leinad change the technology and produce (mass = 3.01603amu)  instead of ⁴He ?

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

A sample of cobalt-60 (t_1/2  = 5.27 yr), a powerful emitter used to treat cancer, was purchased by a hospital on March 1, 2007. The sample must be replaced when its activity reaches 70% of the original value. On what date must it be replaced?

Uranium-233 decays to thorium-229 by decay, but the emissions have different energies and products: 83% emit an  particle with energy of 4.816 MeV and give 229Th in its ground state; 15% emit an  particle of  4.773 MeV and give ²²⁹Th in excited state I; and2%  emit a lower energy  particle and give 229Th in the higher excited state II. Excited state II emits a 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 particle that would convert  ²³³U to excited state II.

Uranium-238  undergoes a slow decay step $(t_{t/2}=4.5\times {10}^{9}yr)$ followed by a series of fast steps to form the stable isotope ²⁰⁶pb. Thus, on a time scale of billions of years, ²³⁸U effectively decays “directly” to ²⁰⁶pb, 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: 

Student 1: ${\left(\frac{1}{2}\right)}^n=\frac{{}_{92}{}^{238}U}{{}_{82}{}^{206}Pb}$

Student 2: ${\left(\frac{1}{2}\right)}^n=\frac{{}_{92}{}^{238}U}{{}_{92}{}^{238}U{+}_{82}^{206}Pb}$

(a) Which equation is correct, and why? 

(b) If a rock contains exactly twice as much ²³⁸U as ²⁰⁶pb, what is its age in years?

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

At death, a nobleman in ancient Egypt was mummified and his body contained $\mathbf{1}\mathbf{.}\mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{3}}\mathit{g}$of ${}^{\mathbf{40}}\mathit{K}\mathbf{ }\left(t_{1/2}=1.25\times {10}^9 yr\right)\mathbf{,}\mathbf{ }\mathbf{1}\mathbf{.}\mathbf{2}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{8}}\mathbf{ }\mathit{g}$ of ${}^{\mathbf{14}}\mathit{C}\mathbf{(}{\mathit{t}}_{\mathbf{1}\mathbf{/}\mathbf{2}}\mathbf{=}\mathbf{ }\mathbf{5730}\mathit{y}\mathit{r}\mathbf{)}$, and $\mathbf{4}\mathbf{.}\mathbf{8}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{14}}\mathit{g}$   of ${}^{\mathbf{3}}\mathit{H}\mathbf{(}{\mathit{t}}_{\mathbf{1}\mathbf{/}\mathbf{2}}\mathbf{=}\mathbf{12}\mathbf{.}\mathbf{26}\mathit{y}\mathit{r}\mathbf{)}$. Which nuclide would give the most accurate estimate of the mummy’s age? Explain

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) ²⁴²cm(t_1/2=163 days); 

(b) ²¹⁴Po_o($t_{\frac{1}{2}}=1.6\times {10}^{-4}\sec$); 

(c) ²³²Th($t_{\frac{1}{2}}=1.39\times {10}^{10}yr$)?

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 $\mathbf{(}{\mathit{t}}_{\mathbf{1}\mathbf{/}\mathbf{2}}\mathbf{ }\mathit{o}\mathit{f}{\mathbf{ }}^{\mathbf{14}}\mathit{C}\mathbf{=}\mathbf{ }\mathbf{5730}\mathit{y}\mathit{r}\mathbf{;}{\mathbf{A}}_{\mathbf{0}}\mathbf{=}\mathbf{15}\mathbf{.}\mathbf{3}\mathit{d}\mathbf{/}\mathit{m}\mathit{i}\mathit{n}\mathbf{.}\mathit{g}\mathbf{)}$?

Part                   (d/min.g)

Handle               10.1

Inlaid wood       13.8

Ribbon               12.1

Sheath                15.0

The star ship Voyager, like many other vessels of the newly designed 24th-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 ⁴He. 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)?

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.

Determine the age of a rock containing 0.065 g of uranium$\mathbf{-}\mathbf{238}\mathbf{(}{\mathit{t}}_{\mathbf{1}\mathbf{/}\mathbf{2}}\mathbf{=}\mathbf{4}\mathbf{.}\mathbf{5}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{9}}\mathit{y}\mathit{r}\mathbf{)}$ and 0.23 g of lead-206 . (Assume all the lead-206 came from U-238 decay.)

Plutonium-242 decays to uranium-238 by emission of an $\alpha$-particle with an energy of 4.853 MeV. The  ²³⁸U that forms unstable and emits a $\gamma$-ray ($\lambda =0.02757 nm$).

(a) Write balanced equations for these reactions.

(b) What would be the energy of the -particle if ²⁴²P decayed directly to the more stable ²³⁸U?

Seaborgium-263  (Sg), the first isotope of element 106 synthesized, was made, together with four neutrons, by bombarding californium-249  with oxygen-18 . It then underwent a series of decays starting with three emissions. Write balanced equations for the synthesis and the three emissions of  ²⁶³Sg.

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