31.6

Radioactive Decay and Activity

The question of which radioactive nucleus in a group of nuclei disintegrates at a given instant is decided like the drawing of numbers in a state lottery; individual disintegrations occur randomly. As time passes, the number N of parent nuclei decreases, as Figure 31-14 shows. This graph of N versus time indicates that the decrease occurs in a smooth fashion, with N approaching zero after enough time has passed. To help describe the graph, it is useful to define the half-life of a radioactive isotope as the time required for one-half of the nuclei present to disintegrate. For example, radium has a half-life of 1600 years, because it takes this amount of time for one-half of a given quantity of this isotope to disintegrate. In another 1600 years, one-half of the remaining radium atoms will have disintegrated, leaving only one-fourth of the original number intact. In Figure 31-14, the number of nuclei present at time is , and the number present at is . The number present at is , and so on. The value of the half-life depends on the nature of the radioactive nucleus. Values ranging from a fraction of a second to billions of years have been found (see Table 31-2).

Figure 31-14 The half-life of a radioactive decay is the time in which one-half of the radioactive nuclei disintegrate.

Table 31-2 Some Half-Lives for Radioactive Decay

Isotope |  | Half-Life
Polonium |  |
Krypton |  | 3.16 min
Radon |  | 3.83 d
Strontium |  | 29.1 yr
Radium |  |
Carbon |  |
Uranium |  |
Indium |  |

The physics of radioactive radon gas in houses. Radon is a naturally occurring radioactive gas produced when radium undergoes decay. There is a nationwide concern about radon as a health hazard because radon in the soil is gaseous and can enter the basement of homes through cracks in the foundation. (It should be noted, however, that the mechanism of indoor radon entry is not well understood and that entry via foundation cracks is likely only part of the story.) Once inside, the concentration of radon can rise markedly, depending on the type of housing construction and the concentration of radon in the surrounding soil. Radon gas decays into daughter nuclei that are also radioactive. The radioactive nuclei can attach to dust and smoke particles that can be inhaled, and they remain in the lungs to release tissue-damaging radiation. Prolonged exposure to high levels of radon can lead to lung cancer. Since radon gas concentrations can be measured with inexpensive monitoring devices, it is recommended that all homes be tested for radon. Example 8 deals with the half-life of radon .

| Example 8 | | | The Radioactive Decay of Radon Gas

Suppose radon atoms are trapped in a basement at the time the basement is sealed against further entry of the gas. The half-life of radon is 3.83 days. How many radon atoms remain after 31 days? Reasoning During each half-life, the number of radon atoms is reduced by a factor of two. Thus, we determine the number of half-lives there are in a period of 31 days and reduce the number of radon atoms by a factor of two for each one.
Solution In a period of 31 days there are (31 days)/(3.83 days) = 8.1 half-lives. In 8 half-lives the number of radon atoms is reduced by a factor of . Ignoring the difference between 8 and 8.1 half-lives, we find that the number of atoms remaining after 31 days is .

The activity of a radioactive sample is the number of disintegrations per second that occur. Each time a disintegration occurs, the number N of radioactive nuclei decreases. As a result, the activity can be obtained by dividing , the change in the number of nuclei, by , the time interval during which the change takes place; the average activity over the time interval is the magnitude of , or . Since the decay of any individual nucleus is completely random, the number of disintegrations per second that occurs in a sample is proportional to the number of radioactive nuclei present, so that

| (31.4)

where is a proportionality constant referred to as the decay constant. The minus sign is present in this equation because each disintegration decreases the number N of nuclei originally present.
The SI unit for activity is the becquerel (Bq); one becquerel equals one disintegration per second. Activity is also measured in terms of a unit called the curie (Ci), in honor of Marie (1867–1934) and Pierre (1859–1906) Curie, the discoverers of radium and polonium. Historically, the curie was chosen as a unit because it is roughly the activity of one gram of pure radium. In terms of becquerels,

The activity of the radium put into the dial of a watch to make it glow in the dark is about , and the activity used in radiation therapy for cancer treatment is approximately a billion times greater, or .
The mathematical expression for the graph of N versus t shown in Figure 31-14 can be obtained from Equation 31.4 with the aid of calculus. The result for the number N of radioactive nuclei present at time t is

| (31.5)

assuming that the number present at is . The exponential e has the value , and many calculators provide the value of . We can relate the half-life of a radioactive nucleus to its decay constant in the following manner. By substituting and into Equation 31.5, we find that . Taking the natural logarithm of both sides of this equation reveals that or

| (31.6)

The following example illustrates the use of Equations 31.5 and 31.6.

| Example 9 | | |

The Activity of Radon As in Example 8, suppose there are radon atoms trapped in a basement. (a) How many radon atoms remain after 31 days? Find the activity (b) just after the basement is sealed against further entry of radon and (c) 31 days later.
Reasoning The number N of radon atoms remaining after a time t is given by , where is the original number of atoms when and is the decay constant. The decay constant is related to the half-life of the radon atoms by . The activity can be obtained from Equation 31.4, .
Solution

a. | The decay constant is

| (31.6)

and the number N of radon atoms remaining after 31 days is

| (31.5)

This value is slightly different from that found in Example 8 because there we ignored the difference between 8.0 and 8.1 half-lives.
b. | The activity can be obtained from Equation 31.4, provided the decay constant is expressed in reciprocal seconds:

| (31.6)

Thus, the number of disintegrations per second is

| (31.4)

The activity is the magnitude of , so initially .
c. | From part (a), the number of radioactive nuclei remaining at the end of 31 days is , and reasoning similar to that in part (b) reveals that .

Check Your Understanding 4

The thallium nucleus is radioactive with a half-life of 3.053 min. At a given instant, the activity of a certain sample of thallium is 2400 Bq. Using the concept of half-life, and without doing any written calculations, determine whether the activity 9 minutes later is (a) a little less than , (b) a little more than , (c) a little less than , (d) a little more than . (The answer is given at the end of the book.)
Background: The half-life of a radioactive sample is the time required for one-half of the nuclei present to disintegrate. The activity is the product of the number of nuclei present and the decay constant.
For similar questions (including calculational counterparts), consult Self-Assessment Test 31.2, which is described at the end of Section 31.9.

Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.