31.7

Radioactive Dating

One important application of radioactivity is the determination of the age of archeological or geological samples. If an object contains radioactive nuclei when it is formed, then the decay of these nuclei marks the passage of time like a clock, half of the nuclei disintegrating during each half-life. If the half-life is known, a measurement of the number of nuclei present today relative to the number present initially can give the age of the sample. According to Equation 31.4, the activity of a sample is proportional to the number of radioactive nuclei, so one way to obtain the age is to compare present activity with initial activity. A more accurate way is to determine the present number of radioactive nuclei with the aid of a mass spectrometer.

The physics of radioactive dating.

The present activity of a sample can be measured, but how is it possible to know what the original activity was, perhaps thousands of years ago? Radioactive dating methods entail certain assumptions that make it possible to estimate the original activity. For instance, the radiocarbon technique utilizes the isotope of carbon, which undergoes decay with a half-life of 5730 yr. This isotope is present in the earth’s atmosphere at an equilibrium concentration of about one atom for every atoms of normal carbon . It is often assumed* that this value has remained constant over the years because is created when cosmic rays interact with the earth’s upper atmosphere, a production method that offsets the loss via decay. Moreover, nearly all living organisms ingest the equilibrium concentration of . However, once an organism dies, metabolism no longer sustains the input of , and decay causes half of the nuclei to disintegrate every 5730 years. Example 10 illustrates how to determine the activity of one gram of carbon in a living organism.

These mummified remains were found in a cave in Mexico in 2002. They are thought to be about 2300 years old. Radioactive dating is one the techniques used to determine the age of such remains. (David Aguilar/Reuters/Landov LLC)

| Example 10 | | |

Activity Per Gram of Carbon in a Living Organism (a) Determine the number of carbon atoms present for every gram of carbon in a living organism. Find (b) the decay constant and (c) the activity of this sample. Reasoning The total number of carbon atoms in one gram of carbon is equal to the corresponding number of moles times Avogadro’s number (see Section 14.1). Since there is only one atom for every atoms of , the number of atoms is equal to the total number of atoms divided by . The decay constant for is , where is the half-life. The activity is equal to the magnitude of , which is equal to the decay constant times the number of atoms present, according to Equation 31.4.
Solution

a. | One gram of carbon (atomic mass = 12 u) is equivalent to 1.0/12 mol. Since Avogadro’s number is atoms/mol and since there is one atom for every atoms of , the number of atoms is

b. | Since the half-life of is 5730 yr , the decay constant is

| (31.6)

c. | Equation 31.4 indicates that , so the magnitude of is .

An organism that lived thousands of years ago presumably had an activity of about 0.23 Bq per gram of carbon. When the organism died, the activity began decreasing. From a sample of the remains, the current activity per gram of carbon can be measured and compared to the value of 0.23 Bq to determine the time that has transpired since death. This procedure is illustrated in Example 11.

A N A L Y Z I N G M U L T I P L E - C O N C E P T P R O B L E M S

| Example 11 | | | The Ice Man

On September 19, 1991, German tourists on a walking trip in the Italian Alps found a Stone-Age traveler, later dubbed the Ice Man, whose body had become trapped in a glacier. Figure 31-15 shows the well-preserved remains, which were dated using the radiocarbon method. Material found with the body had a activity of about 0.121 Bq per gram of carbon. Find the age of the Ice Man’s remains.

Figure 31-15 The two scientists in this photograph are studying the Iceman or “Oetzi,” as he is also called. His frozen remains and various artifacts were discovered in the ice of a glacier in the Italian Alps in 1991. Radiocarbon dating has revealed his age. (AFP/Getty Images News)

Reasoning In the radiocarbon method, the number of radioactive nuclei remaining at a given instant is related to the number present initially, the time that has passed since the Ice Man died, and the decay constant for . Thus, to determine the age of the remains, we will need information about the number of nuclei present when the body was discovered and the number present initially, which can be related to the activity of the material found with the body and the initial activity. To determine the age, we will also need the decay constant, which can be obtained from the half-life of .
Knowns and Unknowns We have the following data:

Description | Symbol | Value | Comment
Explicit Data |  |  |
Activity of material found with body | A | 0.121 Bq | This is the activity per gram of carbon.
Implicit Data |  |  |
Half-life of |  | 5730 yr | The radiocarbon dating method is specified.
Initial activity of material found with body |  | 0.23 Bq | This activity is assumed for one gram of carbon in a living organism.
Unknown Variable |  |  |
Age of Ice Man’s remains | t | ? |

Modeling the Problem

| Step 1 Radioactive Decay
| The number N of radioactive nuclei present at a time t is

| (31.5)

where is the number present initially at and is the decay constant for . Rearranging terms gives

Taking the natural logarithm of both sides of this result, we find that

Solving for t shows that the age of the Ice Man’s remains is given by Equation 1 at the right. To use this result, we need information about the ratio and . We deal with in Step 2 and in Step 3.

| (1)

| Step 2 Activity
| The activity A is the number of disintegrations per second, or , where is the number of disintegrations that occur in the time interval . Noting that according to Equation 31.4, we find for the activity that

Using this expression, we have that

The substitution of this result into Equation 1 is shown at the right. We turn now to Step 3, in order to evaluate the decay constant .

| Step 3 Decay Constant
| The decay constant is related to the half-life according to

| (31.6)

which we substitute into Equation 1, as shown at the right.

Solution Combining the results of each step algebraically, we find that

This result reveals that the age of the Ice Man’s remains is

Note that this solution implies for the activity that

This can be seen by combining the result from Step 2 with Equation 31.5 .
Related Homework: Problems 40 and 52
Radiocarbon dating is not the only radioactive dating method. For example, other methods utilize uranium , potassium , and lead . For such methods to be useful, the half-life of the radioactive species must be neither too short nor too long relative to the age of the sample to be dated, as Conceptual Example 12 discusses.

| Conceptual Example 12 | | | Dating a Bottle of Wine

A bottle of red wine is thought to have been sealed about 5 years ago. The wine contains a number of different kinds of atoms, including carbon, oxygen, and hydrogen. Each of these has a radioactive isotope. The radioactive isotope of carbon is the familiar , with a half-life of 5730 yr. The radioactive isotope of oxygen is and has a half-life of 122.2 s. The radioactive isotope of hydrogen is and is called tritium; its half-life is 12.33 yr. The activity of each of these isotopes is known at the time the bottle was sealed. However, only one of the isotopes is useful for determining the age of the wine accurately. Which is it?
Reasoning and Solution In a dating method that measures the activity of a radioactive isotope, the age of the sample is related to the change in the activity during the time period in question. Here the expected age is about 5 years. This period is only a small fraction of the 5730-yr half-life of . As a result, relatively few of the nuclei would decay during the wine’s life, and the measured activity would change little from its initial value. To obtain an accurate age from such a small change would require prohibitively precise measurements. The isotope is not very useful, either; the difficulty is its relatively short half-life of 122.2 s. During a 5-year period, so many half-lives of 122.2 s would occur that the activity would decrease to a vanishingly small level. It would not be even possible to measure it. The only remaining option is the tritium isotope of hydrogen. The expected age of 5 yr is long enough relative to the half-life of 12.33 yr that a measurable change in activity will occur, but not so long that the activity will have completely vanished for all practical purposes.
Related Homework: Conceptual Question 14, Problem 44

In the radioactive dating technique that uses the carbon isotope, the relatively few atoms can be detected by measuring their activity, as we have seen. It is also possible to use an accelerator mass spectrometer, such as the one in this photograph, to detect these atoms more accurately. (James King Holmes/SPL/Photo Researchers, Inc.)

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