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Radioactive Half-life

Here’s a link to an article about isotopes that might be interesting when this topic is coverd:


See also: PhET Radioactive Dating Game

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. The rate of production of radioactive particles depends only on the number of radioactive atoms present. The number of atoms present in a sample decreases according to an exponetial decay and therefore the rate of decay decreases according to the same law.

——  = e-kt                                       Equation 1
N = no. of atoms left, N0 = no. atoms you started with, e ≅ 2.7183 (a mathematical constant),
k = a rate constant specific to a substance, and t = time
N/N0 is the fraction of the material that is left after some amount of time has passed (for ex., 0.50 or 50%).

The e in the formula above is the base of the natural logarithm and is a standard constant of mathematics. It is approximately 2.7183. If you have a scientific calculator, there is probably a button for this constant and for the logarithm (LN) which is based on it. Use this formula to find the fraction of a radionuclide that has not decayed (that is, the fraction remaining, N/N0), given the time and the rate constant. With suitable mathematical manipulations it is possible to use this same formula to calculate the elapsed time using the fraction remaining (ln(N/N0) = –kt). The fraction remaining may also be expressed in terms of relative rates of decay. The rate of decay depends on the number of atoms so as the number of atoms decreases, so does the rate of decay.

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 a large collection of radionuclides will follow a strict statistical law consistent with the random decays that gives an overall constant rate. The way this rate changes over time can best be understood using Equation 1 but it can also be expressed in terms of half-lives. Using Equation 1 it is possible to define the period of time needed for a sample to reach the point where the rate has decreased to half of its original value (which is the moment when half of the original sample remains or N/N0 = 0.5). The time needed to reach this point is called the half-life. The rate constant that governs the decay of a particular radionuclide can be used to calculate the length of a half-life.

t½ = ln(2)/k                                  Equation 2
where t½ is the half-life and k is the rate constant from the previous equation

The primary way to calculate the time elapsed given a certain fraction remaining is to use Equation 1. There is also a way to figure out how old something is by using the number of half-lives. At one half-life, N/N0 = 0.5 and n (the number of half-lives) equals one. At this time one half-life has elapsed. When the fraction remaining is 1/4 then n = 2 and the total elapsed time equals two half lives. The number of half-lives can be any number and once you have n you can simply multiply it by the length of the half-life to find the age of an object using this formula: ttotal = nt½. Equation 3 is the fraction remaining in terms of the number of half-lives.

N/N0 = (1/2)n   or   n = ——————————            Equation 3
n = the number of half-lives

An isotope important to the science of archaeology is Carbon-14 (14 6C). It is created in the atmosphere by the action of cosmic rays which enables the neutron bombardment of 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 5,715 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. Using this information it is possible to calculate the amount of time that has elapsed since a living thing died, such as a piece of wood carved into a cup.

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Nuclide Half-life k Decay Mode
14  6C 5,715 yr 1.213 × 10-4 1/yr β-
4019K 1.248 × 109 yr 5.553 × 10-10 1/yr β-
8737Rb 4.97 × 1010 yr 1.394 × 10-11 1/yr β-
23892U 4.468 × 109 yr 1.551 × 10-10 1/yr α
     Refresher on Logarithms
ln(ex) = x
eln(M) = M
     ln(M·N) = ln(M) + ln(N)
ln(M/N) = ln(M) – ln(N)
ln(Mx) = x·ln(M)
     (ab· ac = ab + c)
(ab ÷ ac = ab – c)
((ab)c = ab·c)

Answer the following questions of perform the calculations required, showing work.

  1. Write a balanced nuclear equation for the decay of each of the isotopes in the table given above.
  2. When using the decay rates of naturally occurring radioactive isotopes to measure the age of objects there are some natual limits. These limits depend on the isotope being used.
    1. What fraction of any isotope remains after nine half-lives?
    2. If lab techniques are unable to measure the amount of an isotope that remains after it reaches this level of decay, then approximately what is the oldest age that can be established using carbon-14?
    3. What is the oldest object whose age can be established by measurements of the isotope potassium-40? Assume that it can be detected down to the amount that would remain after 9 half-lives.
    4. Although uranium-238 has a much longer half-life than carbon-14 it is not used to find the ages of fossils that come from the Mesozoic era (the age of dinosaurs). It is only used to find the ages of rocks. Explain.
  3. 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 without doing any calculations. Explain your reasoning.
  4. The risks of exposure to a radioactive material depends on two things: how many atoms of the material there are and how rapidly it decays. Based on the data in the table at the bottom of the first page of this packet, place the isotopes in order from least risk to greatest risk for exposure to a sample with the same number of atoms. Explain your reasoning.
  5. How much 14  6C would remain after 5 half-lives? How old would a sample be that had gone through five half-lives?

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  1. Use algebra to show the steps necessary to find the relationship between the half-life and the rate constant starting from the basic decay equation (N/N0 = e–kt). Consider that when one half-life has elapsed exactly one half of the original amount of a radioactive isotope remains in its original form and that fractions such as 1/2 can be written using exponents: 1/2 = 2–1.
  2. 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?
  3. 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 carbon-14 contained in freshly made cloth of the same material. How old is the Shroud according to the data? What year do you suppose it was actually made?
  4. A wooden artifact from a site in Europe has a 14C activity of 39.0 counts per minute. By comparison, the expected rate of decay for an object at the present time is 58.2 counts per minute. Since the rate of decay decreases in exactly the same way as the number of atoms, a ratio of decay rates can be used in the same way as a ratio of the number of atoms. What is the age of the wooden artifact?
  5. 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?
  6. 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? (There are two ways to do this calculation; show work for both).
    3. How long will it be until the amount of 64Cu remaining is only 9%?
  7. The first atomic explosion was detonated in the desert north of Alamogordo, New Mexico, on July 16, 1945. What fraction of the strontium-90 (t½ = 28.8 years) originally produced by that explosion still remains as of December 31, 2018?
  8. Iodine-131 is used in the diagnosis and treatment of thyroid disease and has a half-life of 8.02 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?
  9. Radioactive dating is accomplished by establishing the ratio by numbers of atoms between the initial nucleus and the final nucleus to which it decays. 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. (The exact mass of 206Pb is 205.974449 g/mol and the exact mass of 238U is 238.050784 g/mol).
See also the simpler activity I wrote to teach the idea of half-life in a way that avoids using logarithms: Half-lives Basics
Last updated: Dec 05, 2018       Home