Problems & Exercises

A neutron generator uses an $\alpha$ source, such as radium, to bombard beryllium, inducing the reaction ${}^4\text{He}+{}^9\text{Be}\to {}^{\text{12}}\text{C}+n$. Such neutron sources are called RaBe sources, or PuBe sources if they use plutonium to get the $\alpha$ s. Calculate the energy output of the reaction in MeV.

The purpose of producing ${}^{\text{99}}\text{Mo}$ (usually by neutron activation of natural molybdenum, as in the preceding problem) is to produce ${}^{\text{99m}}\text{Tc.}$ Using the rules, verify that the ${\beta }^{-}$ decay of ${}^{\text{99}}\text{Mo}$ produces ${}^{\text{99m}}\text{Tc}$. (Most ${}^{\text{99m}}\text{Tc}$ nuclei produced in this decay are left in a metastable excited state denoted ${}^{\text{99m}}\text{Tc}$.)

Table 32.1 indicates that 7.50 mCi of ${}^{\text{99m}}\text{Tc}$ is used in a brain scan. What is the mass of technetium?

(a) Neutron activation of sodium, which is 100%${}^{\text{23}}\text{Na}$, produces ${}^{\text{24}}\text{Na}$, which is used in some heart scans, as seen in Table 32.1. The equation for the reaction is ${}^{\text{23}}\text{Na}+n\to {}^{\text{24}}\text{Na}+\gamma$. Find its energy output, given the mass of ${}^{\text{24}}\text{Na}$ is 23.990962 u.

(b) What mass of ${}^{\text{24}}\text{Na}$ produces the needed 5.0-mCi activity, given its half-life is 15.0 h?

What is the dose in mSv for: (a) a 0.1 Gy x-ray? (b) 2.5 mGy of neutron exposure to the eye? (c) 1.5 mGy of $\alpha$ exposure?

How many Gy of exposure is needed to give a cancerous tumor a dose of 40 Sv if it is exposed to $\alpha$ activity?

One half the $\gamma$ rays from ${}^{\text{99m}}\text{Tc}$ are absorbed by a 0.170-mm-thick lead shielding. Half of the $\gamma$ rays that pass through the first layer of lead are absorbed in a second layer of equal thickness. What thickness of lead will absorb all but one in 1000 of these $\gamma$ rays?

In the 1980s, the term picowave was used to describe food irradiation in order to overcome public resistance by playing on the well-known safety of microwave radiation. Find the energy in MeV of a photon having a wavelength of a picometer.

A beam of 168-MeV nitrogen nuclei is used for cancer therapy. If this beam is directed onto a 0.200-kg tumor and gives it a 2.00-Sv dose, how many nitrogen nuclei were stopped? (Use an RBE of 20 for heavy ions.)

Calculate the dose in Sv to the chest of a patient given an x-ray under the following conditions. The x-ray beam intensity is $\text{1.50 W}{\text{/m}}^2$, the area of the chest exposed is $\text{0.0750}{\text{m}}^2$, 35.0% of the x-rays are absorbed in 20.0 kg of tissue, and the exposure time is 0.250 s.

What is the mass of ${}^{60}\text{Co}$ in a cancer therapy transillumination unit containing 5.00 kCi of ${}^{60}\text{Co}$?

Naturally occurring ${}^{\text{40}}\text{K}$ is listed as responsible for 16 mrem/y of background radiation. Calculate the mass of ${}^{\text{40}}\text{K}$ that must be inside the 55-kg body of a woman to produce this dose. Each ${}^{\text{40}}\text{K}$ decay emits a 1.32-MeV $\beta$, and 50% of the energy is absorbed inside the body.

The annual radiation dose from ${}^{14}\text{C}$ in our bodies is 0.01 mSv/y. Each ${}^{14}\text{C}$ decay emits a ${\beta }^{–}$ averaging 0.0750 MeV. Taking the fraction of ${}^{14}\text{C}$ to be $1.3\times {10}^{\mathrm{–12}}\text{N}$ of normal ${}^{12}\text{C}$, and assuming the body is 13% carbon, estimate the fraction of the decay energy absorbed. (The rest escapes, exposing those close to you.)

Verify that the total number of nucleons, total charge, and electron family number are conserved for each of the fusion reactions in the proton-proton cycle in

and

(List the value of each of the conserved quantities before and after each of the reactions.)

Show that the total energy released in the proton-proton cycle is 26.7 MeV, considering the overall effect in ${}^1\text{H}+{}^1\text{H}\to {}^2\text{H}+e^{+}+v_{\text{e}}$, ${}^1\text{H}+{}^2\text{H}\to {}^3\text{He}+\gamma$, and ${}^3\text{He}+{}^3\text{He}\to {}^4\text{He}+{}^1\text{H}+{}^1\text{H}$ and being certain to include the annihilation energy.

The energy produced by the fusion of a 1.00-kg mixture of deuterium and tritium was found in Example Calculating Energy and Power from Fusion. Approximately how many kilograms would be required to supply the annual energy use in the United States?

Two fusion reactions mentioned in the text are

$n+{}^3\text{He}\to {}^4\text{He}+\gamma$

and

$n+{}^1\text{H}\to {}^2\text{H}+\gamma$.

Both reactions release energy, but the second also creates more fuel. Confirm that the energies produced in the reactions are 20.58 and 2.22 MeV, respectively. Comment on which product nuclide is most tightly bound, ${}^4\text{He}$ or ${}^2\text{H}$.

How many kilograms of water are needed to obtain the 198.8 mol of deuterium, assuming that deuterium is 0.01500% (by number) of natural hydrogen?

Another set of reactions that result in the fusing of hydrogen into helium in the Sun and especially in hotter stars is called the carbon cycle. It is
$\begin{array}{lll}{}^{\text{12}}\text{C}+{}^1\text{H}& \to & {}^{\text{13}}\text{N}+\gamma ,\\ {}^{\text{13}}\text{N}& \to & {}^{\text{13}}\text{C}+e^{+}+v_e,\\ {}^{\text{13}}\text{C}+{}^1\text{H}& \to & {}^{\text{14}}\text{N}+\gamma ,\\ {}^{\text{14}}\text{N}+{}^1\text{H}& \to & {}^{\text{15}}\text{O}+\gamma ,\\ {}^{\text{15}}\text{O}& \to & {}^{\text{15}}\text{N}+e^{+}+v_e,\\ {}^{\text{15}}\text{N}+{}^1\text{H}& \to & {}^{\text{12}}\text{C}+{}^4\text{He.}\end{array}$
Write down the overall effect of the carbon cycle (as was done for the proton-proton cycle in $2e^{-}+4{}^1\text{H}\to {}^4\text{He}+{2v}_e+\mathrm{6\gamma }$). Note the number of protons ( ${}^1\text{H}$) required and assume that the positrons ( $e^{+}$) annihilate electrons to form more $\gamma$ rays.

Verify that the total number of nucleons, total charge, and electron family number are conserved for each of the fusion reactions in the carbon cycle given in the above problem. (List the value of each of the conserved quantities before and after each of the reactions.)

Integrated Concepts

Find the amount of energy given to the ${}^4\text{He}$ nucleus and to the $\gamma$ ray in the reaction $n{+}^3\text{He}{\to }^4\text{He}+\gamma$, using the conservation of momentum principle and taking the reactants to be initially at rest. This should confirm the contention that most of the energy goes to the $\gamma$ ray.

Integrated Concepts

(a) Estimate the years that the deuterium fuel in the oceans could supply the energy needs of the world. Assume world energy consumption to be ten times that of the United States which is $8\times {\text{10}}^{\text{19}}$ J/y and that the deuterium in the oceans could be converted to energy with an efficiency of 32%. You must estimate or look up the amount of water in the oceans and take the deuterium content to be 0.015% of natural hydrogen to find the mass of deuterium available. Note that approximate energy yield of deuterium is $3\text{.}\text{37}\times {\text{10}}^{\text{14}}$ J/kg.

(b) Comment on how much time this is by any human measure. (It is not an unreasonable result, only an impressive one.)

(a) Calculate the energy released in the neutron-induced fission (similar to the spontaneous fission in Example 32.3)

$$n+{}^{\text{238}}\text{U}\to {}^{\text{96}}\text{Sr}+{}^{\text{140}}\text{Xe}+3\mathrm{n,}$$

given $m({}^{\text{96}}\text{Sr})=\text{95.921750 u}$ and $m({}^{\text{140}}\text{Xe})=\text{139.92164}$. (b) This result is about 6 MeV greater than the result for spontaneous fission. Why? (c) Confirm that the total number of nucleons and total charge are conserved in this reaction.

(a) Calculate the energy released in the neutron-induced fission reaction

$$n+{}^{\text{239}}\text{Pu}\to {}^{\text{96}}\text{Sr}+{}^{\text{140}}\text{Ba}+4n,$$

given $m({}^{\text{96}}\text{Sr})=\text{95}\text{.}\text{921750 u}$ and $m({}^{\text{140}}\text{Ba})=\text{139}\text{.}\text{910581 u}$.

(b) Confirm that the total number of nucleons and total charge are conserved in this reaction.

Breeding plutonium produces energy even before any plutonium is fissioned. (The primary purpose of the four nuclear reactors at Chernobyl was breeding plutonium for weapons. Electrical power was a by-product used by the civilian population.) Calculate the energy produced in each of the reactions listed for plutonium breeding just following Example 32.4. The pertinent masses are $m({}^{\text{239}}\text{U})=\text{239.054289 u}$, $m({}^{\text{239}}\text{Np})=\text{239.052932 u}$, and $m({}^{\text{239}}\text{Pu})=\text{239.052157 u}$.

The electrical power output of a large nuclear reactor facility is 900 MW. It has a 35.0% efficiency in converting nuclear power to electrical.

(a) What is the thermal nuclear power output in megawatts?

(b) How many ${}^{\text{235}}\text{U}$ nuclei fission each second, assuming the average fission produces 200 MeV?

(c) What mass of ${}^{\text{235}}\text{U}$ is fissioned in one year of full-power operation?

Find the mass converted into energy by a 12.0-kT bomb.

Fusion bombs use neutrons from their fission trigger to create tritium fuel in the reaction $n{+}^6\text{Li}{\to }^3\text{H}{+}^4\text{He}$. What is the energy released by this reaction in MeV?

A radiation-enhanced nuclear weapon (or neutron bomb) can have a smaller total yield and still produce more prompt radiation than a conventional nuclear bomb. This allows the use of neutron bombs to kill nearby advancing enemy forces with radiation without blowing up your own forces with the blast. For a 0.500-kT radiation-enhanced weapon and a 1.00-kT conventional nuclear bomb: (a) Compare the blast yields. (b) Compare the prompt radiation yields.

Assume one-fourth of the yield of a typical 320-kT strategic bomb comes from fission reactions averaging 200 MeV and the remainder from fusion reactions averaging 20 MeV.

(a) Calculate the number of fissions and the approximate mass of uranium and plutonium fissioned, taking the average atomic mass to be 238.

(b) Find the number of fusions and calculate the approximate mass of fusion fuel, assuming an average total atomic mass of the two nuclei in each reaction to be 5.

(c) Considering the masses found, does it seem reasonable that some missiles could carry 10 warheads? Discuss, noting that the nuclear fuel is only a part of the mass of a warhead.

It is estimated that weapons tests in the atmosphere have deposited approximately 9 MCi of ${}^{\text{90}}\text{Sr}$ on the surface of the earth. Find the mass of this amount of ${}^{\text{90}}\text{Sr}$.

Integrated Concepts

One scheme to put nuclear weapons to nonmilitary use is to explode them underground in a geologically stable region and extract the geothermal energy for electricity production. There was a total yield of about 4,000 MT in the combined arsenals in 2006. If 1.00 MT per day could be converted to electricity with an efficiency of 10.0%:

(a) What would the average electrical power output be?

(b) How many years would the arsenal last at this rate?

Critical Thinking A spherical target in a body is being targeted with ${\beta }^{+}$ radiation. The target is centered at $\text{(0}\text{.00, 2}\text{.00, 3}\text{.00)}$ on a grid that is in centimeters. The target has a radius of 0.500 centimeters. (a) What two points on the x-axis are at the extremes of where the positrons annihilate the electrons? (b) What two points on the y-axis are at the extremes of where the positrons annihilate the electrons? (c) What can be said about the directions of the pair of gamma rays that result from the annihilations? (d) Can the use of ${\beta }^{+}$ give information in three dimensions?