Page 513 - Physics Coursebook 2015 (A level)

Chapter 31: Nuclear physics
QUESTIONS
15 The isotope nitrogen-13 has a half-life of 10 min. A sample initially contains 8.0 × 1010 undecayed nuclei.
a Write down an equation to show how the number undecayed, N, depends on time, t.
b Determine how many nuclei will remain after 10 min, and after 20 min.
c Determine how many nuclei will decay during the first 30 min.
16 A sample of an isotope for which λ = 0.10 s−1 contains 5.0 × 109 undecayed nuclei at the start of an experiment. Determine:
a the number of undecayed nuclei after 50 s
b its activity after 50 s.
17 The value of λ for protactinium-234 is
9.63 × 10−3 s−1. Table 31.5 shows the number of undecayed nuclei, N, in a sample.
Copy and complete Table 31.5. Draw a graph of N against t, and use it to find the half-life t1/2 of protactinium-234.
t/s N
Table 31.5 Data for Question 17.
Decay constant and half-life
A radioactive isotope that decays rapidly has a short half-life t1⁄2. Its decay constant must be large, since the probability per unit time of an individual nucleus decaying must be high. What is the connection between the decay constant and the half-life?
In a time equal to one half-life t1⁄2, the number of undecayed nuclei is halved. Hence the equation:
N = N0 e(−λt) becomes:
N = e(−λt ) = 1 N1⁄22
0 Therefore:
e(λt1⁄2) = 2
λt1⁄2 = ln2≈0.693
(remember if ex = y, then x = lny).
The half-life of an isotope and the decay constant are
inversely proportional to each other. That is: λ = 0.693
Thus if we know either t1⁄2 or λ, we can calculate the other. For a nuclide with a very long half-life, we might not wish to sit around waiting to measure the half-life; it is easier to determine λ by measuring the activity (and using A = λN), and use that to determine t1⁄2.
Note that the units of λ and t1⁄2 must be compatible; for example, λ in s−1 and t1/2 in s.
QUESTIONS
18
Figure 31.13 shows the decay of a radioactive isotope of caesium, 134Cs. Use the graph to
0
20
40
60
80
100
120
140
400
330
t1⁄2
19
20
21
and hence find the decay constant in year−1.
400
300
200
100
002468 Time / years
Figure 31.13 Decay graph for a radioactive isotope of caesium – see Question 18.
The decay constant of a particular isotope is known to be 3.0 × 10−4 s−1. Determine how long it will take for the activity of a sample of this substance to decrease to one-eighth of its initial value.
The isotope 167N decays with a half-life of 7.4 s.
a Calculate the decay constant for this nuclide.
b A sample of 176N initially contains 5000 nuclei. Determine how many will remain after a time of:
i 14.8s ii 20.0s.
A sample contains an isotope of half-life t1/2.
a Show that the fraction f of nuclei in the sample which remain undecayed after a time t is given by the equation:
f=(12)nwhenn=t t1/2
b Calculate the fraction f after each of the following times:
i t1/2 ii 2t1/2 iii 2.5t1/2 iv 8.3t1/2
55
determine the half-life of this nuclide in years,
501
Activity / Bq