Around the year 1900 Ernest Rutherford (a key figure in early investigations into atomic structure) found that his sources of radioactivity were losing their potency. Over time, they began to

emit less and less radiation. He decided to measure this decrease in radiation and discovered a simple law of exponential decay to describe the decrease:

N
       ——  = e-kt
 N0

The e in the formula above is the base of the natural logarithm and is a standard constant of mathematics. It equals 2.7183. If you have a scientific calculator, there is probably a button for this constant. Use this formula to find the fraction of a radionuclide that has not decayed, given the time and the rate constant.

The half-life of a radionuclide (a radioactive isotope) is the amount of time it takes for half of a sample of the material to undergo radioactive decay. On an atomic level, radioactive decay is a random event that can happen at any time. It cannot be hurried up or slowed down. But for a collection of radionuclides there is a way to find out how long it will be before half of the sample has transformed into another material. Through some mathematical work which is not important for this activity, the equation for the half-life of a radionuclide is found to be:

Finally, there is a way to figure out how old something is by using the number of half-lives. For example, carbon-14 (¹⁴ ₆C ) is created in the atmosphere by the action of cosmic neutrons on nitrogen nuclei. Living things take up this carbon-14 and make it part of themselves while they are alive. The half-life of carbon-14 is 5,730 years. It is possible to measure the amount of carbon-14 in an object and compare it to how much carbon-14 there must have been to begin with. This data can be given in the form of a fraction of the original amount of carbon-14. You can use the following relationship to find the number of half-lives which have passed since the animal or plant stopped taking in fresh carbon from the environment (that is, since it died):

     ln(N/N0)
n = ——————————
    -0.693

The ‘ln’ in the formula above is the natural logarithm. Just use the ln button on your calculator. Once you have n simply multiply it by the length of the half-life to find the age of an object. If you know n, just plug it in to find the fraction remaining.

1. Write a balanced nuclear equation for the decay of each of the isotopes in the table given on the first page of this activity.
2. Obtain some graph paper. Choose two isotopes from the list on the previous page. Create two graphs: The x-axis of each graph must be time in years, thousands of years or any convenient unit of time. The y-axis must be the amount of an original isotope remaining at a given time after an arbitrary starting point (i.e., when t = 0 yr, which can be assigned to any actual date). Using the formula relating half-life to the constant ‘k’ and the formula relating k to the fraction of an isotope that remains after a given amount of time create a plot for each isotope. For time inputs use t = 0 and go forward in increments of the half-life of the isotope. For example, for carbon-14 you would have a time data point every 5,730 yrs. Calculate the amount left after each half-life and draw a best-fit curve for each graph. For the starting amount use 6.02 × 10²³ atoms. Look at the next few questions to help you decide how to set up your time scales.
There is a sample set of graphs for Carbon-14 available in Excel format.
3. According to your graphs is there a time when there are no atoms left to decay? Justify your answer.
4. For each isotope that you chose, determine how many half-lives it would take to bring the number of atoms effectively to zero (say to the level of 6 ×10¹⁷ atoms or, as a fraction of a mole: 1 × 10^-6 mol).
5. Which radioisotope in the table would be best to use for dating the charred firewood from a Pleistocene camp? (The Pleistocene is an era of geologic history 1,808,000 to 11,550 years before the present). Why?
6. How many half-lives of that of the nuclide you chose have occurred since the Pleistocene?
7. What fraction of the original material is left? If the fraction is much less than 1% then you may want to change your answer. Radionuclides are often present only in very small amounts and small fractions of small amounts are difficult to detect.
8. Which radioisotope in the table would be best to use for dating the a fossil that is suspected to be from the Precambrian era? (The Precambrian is an era of geologic history 4.5 × 10⁹ years ago to 5.72 × 10⁸ years ago). Why?
9. How many half-lives of the nuclide you chose have occurred since the Precambrian?
10. What fraction of the original material is left? If the fraction is much less than 1% then you may want to change your answer.
11. In 1988, three teams of scientists found that the Shroud of Turin, which was reputed to be the burial cloth of Jesus, contained 91% of the amount of C-14 contained in freshly made cloth of the same material. How old is the Shroud according to the data?
12. A rock containing potassium minerals is analyzed. The sample shows that only 75% of the original ⁴⁰₁₉K is still present. How old is the sample? Explain your reasoning.
13. How much ¹⁴  ₆C would remain after 5 half-lives? How old would a sample be that had gone through five half-lives? Explain your answers.
14. Compare 10 μg samples of carbon-14, potassium-40, rubidium-87, and uranium-238. After one million years, all of them have decayed to some extent. Put them in order from most decayed to least decayed. How much is left of each one?
15. The rate constant for a certain radioactive nuclide is 1.0 × 10^-3 1/hr. What is the half-life of this nuclide in hours?
16. Radioactive copper-64 decays with a half-life of 12.8 days.
    1. What is the value of k in 1/days?
    2. A chemist obtains a fresh sample of ⁶⁴Cu and measures its radioactivity. She then determines that to do an experiment, the radioactivity cannot fall below 25% of the intial measured value. How long does she have before the ⁶⁴Cu is too depleted to work with?
17. The rate constant for a certain radioactive nuclide is 4.7 × 10^-7 1/day. What is the half-life of this nuclide in days? Convert that half-life to years.
18. The first atomic explosion was detonated in the desert north of Alamogordo, New Mexico, on July 16, 1945. What fraction of the strontium-90 (t_½ = 28.8 years) originally produced by that explosion still remains as of July 16, 2006?
19. Iodine-131 is used in the diagnosis and treatment of thyroid disease and has a half-life of 8.1 days. If a patient with thyroid disease consumes a sample of Na¹³¹I containing 10 μg of ¹³¹I, how long will it take for the amount of ¹³¹I to decrease to 0.1 μg?
20. Uranium-238 decays over times into a series of nuclides: ²³⁴Th, ²³⁴Pa, ²³⁴U, ²³⁰Th, ²²⁶Ra, ²²²Rn, ²¹⁸Po, ²¹⁴Pb, ²¹⁴Bi, ²¹⁴Po, ²¹⁰Pb, ²⁰⁶Hg, ²⁰⁶Tl, and finally ²⁰⁶Pb. Write balanced radioactive decay equations for these nuclides. All of the intermediates between ²³⁸U and ²⁰⁶Pb have very short half-lives.
21. A rock contains 0.688 mg of ²⁰⁶Pb for every 1.000 mg of ²³⁸U. Assuming that no lead was originally present, that all the ²⁰⁶Pb formed over the years has remained in the rock, and that the number of nuclides in the intermediate stages of decay between ²³⁸U and ²⁰⁶Pb is negligible, calculate the age of the rock.
22. Assume that the smallest amount of aluminum-26 that can be detected is 1 μg. What is the age-limit on the dating of rocks which started with 100 g of aluminum-26?