Content check — College Physics (AP), Chapter 32: Medical Applications of Nuclear Physics

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Chapter 32: Medical Applications of Nuclear Physics

38 problems · 1 figure

A neutron generator uses an

α

source, such as radium, to bombard beryllium, inducing the reaction

^{4} He +

^{9} Be \to

^{12} C +

n

. Such neutron sources are called RaBe sources, or PuBe sources if they use plutonium to get the

α

s. Calculate the energy output of the reaction in MeV.

Answer

5.701 MeV

32-3 §32.1 source ↗

The purpose of producing

^{99} Mo

(usually by neutron activation of natural molybdenum, as in the preceding problem) is to produce

^{99m} Tc.

Using the rules, verify that the

β^{-}

decay of

^{99} Mo

produces

^{99m} Tc

. (Most

^{99m} Tc

nuclei produced in this decay are left in a metastable excited state denoted

^{99m} Tc

Answer

_{42}^{99} {Mo}_{57} \to

_{43}^{99} {Tc}_{56} +

β^{-} +

{\bar{v}}_{e}

32-5 §32.1 source ↗

Table 32.1 indicates that 7.50 mCi of

^{99m} Tc

is used in a brain scan. What is the mass of technetium?

Table 32.1 Diagnostic Uses of Radiopharmaceuticals in the book ↗

Procedure, isotope	Typical activity (mCi), where $1 mCi = 3.7 \times 10^{7} Bq$
*Brain scan*
$^{99m} Tc$	7.5
$^{113m} In$	7.5
$^{11} C (PET)$	20
$^{13} N (PET)$	20
$^{15} O (PET)$	50
$^{18} F (PET)$	10
*Lung scan*
$^{99m} Tc$	2
$^{133} Xe$	7.5
*Cardiovascular blood pool*
$^{131} I$	0.2
$^{99m} Tc$	2
*Cardiovascular arterial flow*
$^{201} Tl$	3
$^{24} Na$	7.5
*Thyroid scan*
$^{131} I$	0.05
$^{123} I$	0.07
*Liver scan*
$^{198} Au$ (colloid)	0.1
$^{99m} Tc$ (colloid)	2
*Bone scan*
$^{85} Sr$	0.1
$^{99m} Tc$	10
*Kidney scan*
$^{197} Hg$	0.1
$^{99m} Tc$	1.5

Answer

1 . 43 \times 10^{- 9} g

32-7 §32.1 source ↗

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

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

Table 32.1 Diagnostic Uses of Radiopharmaceuticals in the book ↗

Procedure, isotope	Typical activity (mCi), where $1 mCi = 3.7 \times 10^{7} Bq$
*Brain scan*
$^{99m} Tc$	7.5
$^{113m} In$	7.5
$^{11} C (PET)$	20
$^{13} N (PET)$	20
$^{15} O (PET)$	50
$^{18} F (PET)$	10
*Lung scan*
$^{99m} Tc$	2
$^{133} Xe$	7.5
*Cardiovascular blood pool*
$^{131} I$	0.2
$^{99m} Tc$	2
*Cardiovascular arterial flow*
$^{201} Tl$	3
$^{24} Na$	7.5
*Thyroid scan*
$^{131} I$	0.05
$^{123} I$	0.07
*Liver scan*
$^{198} Au$ (colloid)	0.1
$^{99m} Tc$ (colloid)	2
*Bone scan*
$^{85} Sr$	0.1
$^{99m} Tc$	10
*Kidney scan*
$^{197} Hg$	0.1
$^{99m} Tc$	1.5

Answer

(a) 6.958 MeV

(b) $5 . 7 \times 10^{- 10} g$

32-8 §32.2 source ↗

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

α

exposure?

Answer

(a) 100 mSv

(b) 80 mSv

32-10 §32.2 source ↗

How many Gy of exposure is needed to give a cancerous tumor a dose of 40 Sv if it is exposed to

α

activity?

Answer

~2 Gy

32-12 §32.2 source ↗

One half the

γ

rays from

^{99m} Tc

are absorbed by a 0.170-mm-thick lead shielding. Half of the

γ

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

γ

rays?

Answer

1.69 mm

32-14 §32.2 source ↗

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.

Answer

1.24 MeV

32-16 §32.3 source ↗

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.)

Answer

7.44 \times 10^{8}

32-18 §32.3 source ↗

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

1.50 W {/m}^{2}

, the area of the chest exposed is

0.0750 m^{2}

, 35.0% of the x-rays are absorbed in 20.0 kg of tissue, and the exposure time is 0.250 s.

Answer

4.92 \times 10^{–4} Sv

32-20 §32.3 source ↗

What is the mass of

^{60} Co

in a cancer therapy transillumination unit containing 5.00 kCi of

^{60} Co

Answer

4.43 g

32-22 §32.3 source ↗

Naturally occurring

^{40} K

is listed as responsible for 16 mrem/y of background radiation. Calculate the mass of

^{40} K

that must be inside the 55-kg body of a woman to produce this dose. Each

^{40} K

decay emits a 1.32-MeV

β

, and 50% of the energy is absorbed inside the body.

Answer

0.010 g

32-24 §32.3 source ↗

The annual radiation dose from

^{14} C

in our bodies is 0.01 mSv/y. Each

^{14} C

decay emits a

β^{-}

averaging 0.0750 MeV. Taking the fraction of

^{14} C

to be

1.3 \times 10^{–12} N

of normal

^{12} C

, and assuming the body is 13% carbon, estimate the fraction of the decay energy absorbed. (The rest escapes, exposing those close to you.)

Answer

95%

32-26 §32.5 source ↗

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

^{1} H +^{1} H \to^{2} H + e^{+} + v_{e},

^{1} H +^{2} H \to^{3} He + γ,

and

^{3} He +^{3} He \to^{4} He +^{1} H +^{1} H .

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

Answer

(a) $A =1+1=2$ , $Z =1+1=1+1$ , $efn = 0 = - 1 + 1$

(b) $A =1+2=3$ , $Z =1+1=2$ , $efn=0=0$

32-28 §32.5 source ↗

Show that the total energy released in the proton-proton cycle is 26.7 MeV, considering the overall effect in

^{1} H +

^{1} H \to

^{2} H +

e^{+} +

v_{e}

^{1} H +^{2} H \to^{3} He + γ

, and

^{3} He +

^{3} He \to

^{4} He +

^{1} H +

^{1} H

and being certain to include the annihilation energy.

Answer

\begin{array}{l} E & = & m_{i} - m_{f} c^{2} \\ = & 4 m^{1} H - m^{4} He c^{2} \\ = & 4 (1.007825) - 4 . 002603 (931.5 MeV) \\ = & 26.73 MeV \end{array}

32-30 §32.5 source ↗

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?

Answer

3.12 \times 10^{5} kg

(about 200 tons)

32-32 §32.5 source ↗

Two fusion reactions mentioned in the text are

$n +^{3} He \to^{4} He + γ$

and

$n +^{1} H \to^{2} H + γ$ .

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, $^{4} He$ or $^{2} H$ .

Answer

$\begin{array}{l} E & = & m_{i} - m_{f} c^{2} \\ E_{1} & = & 1.008665 + 3.016030 - 4.002603 (931.5 MeV) \\ = & 20.58 MeV \\ E_{2} & = & 1 . 008665 + 1 . 007825 - 2 . 014102 (931.5 MeV) \\ = & 2.224 MeV \end{array}$

$^{4} He is more tightly bound, since this reaction gives off more energy per nucleon.$

32-34 §32.5 source ↗

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?

Answer

1 . 19 \times 10^{4} kg

32-36 §32.5 source ↗

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}{l} ^{12} C +^{1} H & \to & ^{13} N + γ, \\ ^{13} N & \to & ^{13} C + e^{+} + v_{e}, \\ ^{13} C +^{1} H & \to & ^{14} N + γ, \\ ^{14} N +^{1} H & \to & ^{15} O + γ, \\ ^{15} O & \to & ^{15} N + e^{+} + v_{e}, \\ ^{15} N +^{1} H & \to & ^{12} C +^{4} He. \end{array}

Write down the overall effect of the carbon cycle (as was done for the proton-proton cycle in

2 e^{-} +

4^{1} H \to

^{4} He +

{2 v}_{e} +

6γ

). Note the number of protons (

^{1} H

) required and assume that the positrons (

e^{+}

) annihilate electrons to form more

γ

rays.

Answer

2 e^{-} +

4^{1} H \to

^{4} He +

7γ +

{2 v}_{e}

32-38 §32.5 source ↗

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.)

Answer

(a) $A =12+1=13$ , $Z =6+1=7$ , $efn = 0 = 0$

(b) $A =13=13$ , $Z =7=6+1$ , $efn = 0 = - 1 + 1$

(d) $A =14 + 1=15$ , $Z =7+1=8$ , $efn = 0 = 0$

(e) $A =1 5 =15$ , $Z =8=7+1$ , $efn = 0 = - 1 + 1$

(f) $A =15 + 1=12 + 4$ , $Z =7+1=6 + 2$ , $efn = 0 = 0$

32-40 §32.5 source ↗

Integrated Concepts

Find the amount of energy given to the $^{4} He$ nucleus and to the $γ$ ray in the reaction $n +^{3} He \to^{4} He + γ$ , 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 $γ$ ray.

Answer

$E_{γ} = 20.6 MeV$

$E_{^{4} He} =$ $5.68 \times 10^{-2} MeV$

32-42 §32.5 source ↗

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 $8 \times 10^{19}$ 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 $3 . 37 \times 10^{14}$ J/kg.

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

Answer

(a) $3 \times 10^{9} y$

(b) This is approximately half the lifetime of the Earth.

32-43 §32.6 source ↗

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

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

given $m (^{96} Sr) =$ $95.921750 u$ and $m (^{140} Xe) =$ $139.92164$ . (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.

Example 32.3 Calculating Energy Released by Fission in the book ↗

Answer

(a) 177.1 MeV

(b) Because the gain of an external neutron yields about 6 MeV, which is the average $BE/ A$ for heavy nuclei.

32-45 §32.6 source ↗

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

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

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

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

Answer

(a) 180.6 MeV

(b) $A =$ $1 +$ $239 =$ $96 +$ $140 +$ $1 +$ $1 +$ $1 +$ $1, Z =$ $94 =$ $38 +$ $56, efn =$ $0 =$ $0$

32-47 §32.6 source ↗

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 (^{239} U) =

239.054289 u

m (^{239} Np) =

239.052932 u

, and

m (^{239} Pu) =

239.052157 u

Example 32.4 Calculating Energy from a Kilogram of Fissionable Fuel in the book ↗

Answer

$^{238} U +$ $n \to$ $^{239} U +$ $γ$ 4.81 MeV

$^{239} U \to$ $^{239} Np +$ $β^{-} +$ $v_{e}$ 0.753 MeV

$239 Np \to$ $239 Pu +$ $β^{-} +$ $v_{e}$ 0.211 MeV

32-49 §32.6 source ↗

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 $^{235} U$ nuclei fission each second, assuming the average fission produces 200 MeV?

Answer

(a) $2 . 57 \times 10^{3} MW$

(b) $8.03 \times 10^{19} fission/s$

32-51 §32.7 source ↗

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

Answer

0.56 g

32-53 §32.7 source ↗

Fusion bombs use neutrons from their fission trigger to create tritium fuel in the reaction

n +^{6} Li \to^{3} H +^{4} He

. What is the energy released by this reaction in MeV?

Answer

4.781 MeV

32-55 §32.7 source ↗

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.

Answer

(a) Blast yields $2.1 \times 10^{12} J$ to $8.4 \times 10^{11} J$ , or 2.5 to 1, conventional to radiation enhanced.

(b) Prompt radiation yields $6 . 3 \times 10^{11} J$ to $2.1 \times 10^{11} J$ , or 3 to 1, radiation enhanced to conventional.

32-57 §32.7 source ↗

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.

Answer

(a) $1 . 1 \times 10^{25} fissions$ , 4.4 kg

(b) $3.2 \times 10^{26} fusions$ , 2.7 kg

(c) The nuclear fuel totals only 6 kg, so it is quite reasonable that some missiles carry 10 overheads. The mass of the fuel would only be 60 kg and therefore the mass of the 10 warheads, weighing about 10 times the nuclear fuel, would be only 1500 lbs. If the fuel for the missiles weighs 5 times the total weight of the warheads, the missile would weigh about 9000 lbs or 4.5 tons. This is not an unreasonable weight for a missile.

32-59 §32.7 source ↗

It is estimated that weapons tests in the atmosphere have deposited approximately 9 MCi of

^{90} Sr

on the surface of the earth. Find the mass of this amount of

^{90} Sr

Answer

7 \times 10^{4} g

32-61 §32.7 source ↗

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?

Answer

(a) $4 . 86 \times 10^{9} W$

(b) 11.0 y

32-62 §32.7 source ↗

Critical Thinking A spherical target in a body is being targeted with

β^{+}

radiation. The target is centered at

(0 .00, 2 .00, 3 .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

β^{+}

give information in three dimensions?

Answer

(a) $(- 0.500, 2.00, 3.00)$ and $(0.500, 2.00, 3.00)$

(b) $(0.000, 1.50, 3.00)$ and $(0.000, 2.50, 3.00)$

(d) Yes.

AP Test Prep 32.1 §32.2 source ↗

A patient receives A rad of radiation as part of her treatment and absorbs E J of energy. The RBE of the radiation particles is R. If the RBE is increased to 1.5R, what will be the energy absorbed by the patient?

a.
1.5E J
b.
E J
c.
0.75E J
d.
0.67E J

Answer

B. E J

AP Test Prep 32.3 §32.5 source ↗

This figure shows a graph of the potential energy between two light nuclei as a function of the distance between them. Fusion can occur between the nuclei if the distance is

a.
large so that kinetic energy is low.
b.
large so that potential energy is low.
c.
small so that nuclear attractive force can overcome Coulomb’s repulsion.
d.
small so that nuclear attractive force cannot overcome Coulomb’s repulsion.

The figure is a graph with an arrow pointing up for the y-, vertical axis and labeled PEtot. The x-, horizontal axis is labeled with arrow r. A red line indicating the plot starts horizontally below the x-axis and then quickly rises toward the horizontal line marked 0 and is labeled Attractive nuclear below the 0 line. There is a light nuclei dot on the slope of the line above the x-axis with an arrow pointing down to the left along the slope and is labeled Pulled together. After the plot reaches a maximum, the red line of the plot moves down and at less of slope. This portion of the plot is labeled Repulsive Coulomb. On this portion is another light nuclei dot labeled Repelled with a line pointing down and to the right along the slope of the plot. — Figure 32.31 in the book ↗

Answer

C. small so that nuclear attractive force can overcome Coulomb’s repulsion.

AP Test Prep 32.5 §32.5 source ↗

When deuterium and tritium nuclei fuse to produce helium, what else is produced?

a.
positron
b.
proton
c.
α-particle
d.
neutron

Answer

D. neutron

AP Test Prep 32.7 §32.6 source ↗

Which of the following statements about nuclear fission is true?

a.
No new elements can be produced in a fission reaction.
b.
Energy released in fission reactions is generally less than that from fusion reactions.
c.
In a fission reaction, two light nuclei are combined into a heavier one.
d.
Fission reactions can be explained on the basis of the conservation of mass-energy.

Answer

D. Fission reactions can be explained on the basis of the conservation of mass-energy.

AP Test Prep 32.9 §32.6 source ↗

In a neutron-induced fission reaction of ²³⁹Pu, which of the following is produced along with ⁹⁶Sr and four neutrons?

a.
$_{56}^{139} Ba$
b.
$_{56}^{140} Ba$
c.
$_{54}^{139} Xe$
d.
$_{54}^{140} Xe$

Answer

_{56}^{140} Ba

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Problems and answers are adapted from College Physics for AP® Courses 2e by Gregg Wolfe, Erika Gasper, John Stoke, Julie Kretchman, David Anderson and others, OpenStax, and are used under the CC BY-NC-SA 4.0 licence. Access for free at https://openstax.org/books/college-physics-ap-courses-2e/pages/1-connection-for-ap-r-courses. Illustrations: Copyright Rice University, OpenStax, under CC BY-NC-SA 4.0 license. This adaptation is released under the same licence, is non-commercial, and is neither affiliated with nor endorsed by OpenStax or Rice University.