Rafioactivity Problems (Lecture 28A – PHY 315)

Rafioactivity Problems (Lecture 28A – PHY 315)
29.3 Radioactivity
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.

Identify the daughter nuclide when 40
19 K decays via b decay.
232
90Th decays via α decay. Write out the reaction and identify the daughter nuclide.
Write out the reaction and identify the daughter nuclide when 22
11 Na decays by electron capture.
Write out the reaction and identify the daughter nuclide when 22
11 Na decays by emitting a positron.
226
222
4
226
Radium-226 decays as 88 Ra ® 86 Rn+ 2 He . If the 88 Ra nucleus is at rest before the decay and
the 222
86 Rn nucleus is in its ground state, estimate the kinetic energy of the alpha particle. (Assume that
the 222
86 Rn nucleus takes away an insignificant fraction of the kinetic energy.)
Calculate the kinetic energy of the alpha particle in Problem 25. This time, do not assume that
the 222
86 Rn nucleus is at rest after the reaction. Start by figuring out the ratio of the kinetic energies of the
alpha particle and the 222
86 Rn nucleus.
31
Which decay mode would you expect for radioactive 14
Si : α, β, or β+? Explain. [Hint: Look at the
neutron-to-proton ratio.]
Show that the spontaneous alpha decay of 19Ο is not possible.

Calculate the maximum kinetic energy of the beta particle when 40
19 K decays via β decay.

Calculate the energy of the antineutrino when 90
38 Sr decays via β decay if the beta particle has a kinetic
energy of 435 keV.
+
An isotope of sodium, 22
11 Na , decays by β emission. Estimate the maximum possible kinetic energy of
the positron by assuming that the kinetic energy of the daughter nucleus and the total energy of the
neutrino emitted are both zero. [Hint: Remember to keep track of the electron masses.]
The nucleus in a 127 N atom captures one of the atom’s electrons, changing the nucleus to 126 C and
emitting a neutrino. What is the total energy of the emitted neutrino? [Hint:You can use the classical
expression for the kinetic energy of the 126 C atom and the extremely relativistic expression for the
kinetic energy of the neutrino.]
29.4 Radioactive Decay Rates and Half-Lives
33. A certain radioactive nuclide has a half-life of 200.0 s. A sample containing just this one radioactive
nuclide has an initial activity of 80,000.0 s-1. (a) What is the activity 600.0 s later? (b) How many
nuclei were there initially? (c) What is the probability per second that any one of the nuclei decays?
34. Calculate the activity of 1.0 g of radium-226 in Ci.
35. What is the activity in Bq of 1.0 kg of 238U?
36. The half-life of I-131 is 8.0 days. A sample containing I-131 has an activity of 6.4 108 Bq. How
many days later will the sample have an activity of 2.5 106 Bq?
37. Some bones discovered in a crypt in Guatemala are carbon dated. The 14C activity of the bones is
measured to be 0.242 Bq per gram of carbon. Approximately how old are the bones?
38. In this problem, you will verify the statement (in Section 29.4) that the 14C activity in a living sample is
0.25 Bq per gram of carbon. (a) What is the decay constant l for 14C? (b) How many 14C atoms are in
1.00 g of carbon? One mole of carbon atoms has a mass of 12.011 g, and the relative abundance of 14C
is 1.3 10-12. (c) Using your results from parts (a) and (b), calculate the 14C activity per gram of carbon
in a living sample.
39. Carbon-14 dating is used to date a bone found at an archaeological excavation. If the ratio of C-14 to
C-12 atoms is 3.25 10-13, how old is the bone? [Hint: Note that this ratio is
40.
41.
42.
43.
1
4
the ratio of 1.3 10-12
that is found in a living sample.]
A sample of radioactive 214
83 Bi , which has a half-life of 19.9 min, has an activity of 0.058 Ci. What is its
activity 1.0 h later?
The activity of a sample containing radioactive 108Ag is 6.4 104 Bq. Exactly 12 min later, the activity
is 2.0 103 Bq. Calculate the half-life of 108Ag.
A radioactive sample has equal numbers of 15Ο and 19Ο nuclei. Use the half-lives found in Appendix B
to determine how long it will take before there are twice as many 15Ο nuclei as 19Ο. What percent of
the 19Ο nuclei have decayed during this time?
t /T
Show mathematically that 2- t / T1/ 2 = (12 ) 1/ 2 = e- t / τ if and only if T1/2 = τ ln 2. [Hint: Take the natural
logarithm of each side.]
44. The Physics at Home in Section 29.4 suggests tossing coins as a model of radioactive decay. An
improved version is to toss a large number of dice instead of coins: each die that comes up a “one”
represents a nucleus that has decayed. Suppose that N dice are tossed. (a) What is the average number
of dice you expect to decay on one toss? (b) What is the average number of dice you expect to remain
undecayed after three tosses? (c) What is the average number of dice you expect to remain undecayed
after four tosses? (d) What is the half-life in numbers of tosses?