Define half-life as it relates to radioactive nuclides and solve half-life problems. Describe the general process by which radioactive dating is used to determine the age of various objects. Calculate the time for a sample to decay. Complete dosage calculations based on nuclide activity.

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The rate of radioactive decay is often characterized by the half-life of a radioisotope. Half-life (left( t_1/2 ight)) is the time required for one half of the nuclei in a sample of radioactive material to decay. After each half-life has passed, one half of the radioactive nuclei will have transformed into a new nuclide (see table below). The rate of decay and the half-life do not depend on the original size of the sample. They also do not depend upon environmental factors such as temperature and pressure.

Table (PageIndex1) Number of Half-Lives PassedFraction RemainingPercentage RemainingMass remaining starting with (80 : extg)
1 1/2 50 (40 : extg)
2 1/4 25 (20 : extg)
3 1/8 12.5 (10 : extg)
4 1/16 6.25 (5.0 : extg)
5 1/32 3.125 (2.5 : extg)

As an example, iodine-131 is a radioisotope with a half-life of 8 days. It decays by beta particle emission into xenon-131.

After eight days have passed, half of the atoms of any sample of iodine-131 will have decayed, and the sample will now be (50\%) iodine-131 and (50\%) xenon-131. After another eight days pass (a total of 16 days or 2 half-lives), the sample will be (25\%) iodine-131 and (75\%) xenon-131. This continues until the entire sample of iodine-131. has completely decayed (see figure below).

Figure (PageIndex1): The half-life of iodine-131 is eight days. Half of a given sample of iodine-131 decays after each eight-day time period elapses.

Half-lives have a very wide range, from billions of years to fractions of a second. Listed below (see table below) are the half-lives of some common and important radioisotopes. Those with half-lives on the scale of hours or days are the ones most suitable for use in medical treatment.

Table (PageIndex2) NuclideHalf-Life (left( t_1/2 ight))Decay Mode
Carbon-14 5730 years (eta^-)
Cobalt-60 5.27 years (eta^-)
Francium-220 27.5 seconds (alpha)
Hydrogen-3 12.26 years (eta^-)
Iodine-131 8.07 days (eta^-)
Nitrogen-16 7.2 seconds (eta^-)
Phosphorus-32 14.3 days (eta^-)
Plutonium-239 24,100 years (alpha)
Potassium-40 (1.28 imes 10^9) years (eta^-) and (cee^-) capture
Radium-226 1600 years (alpha)
Radon-222 3.82 days (alpha)
Strontium-90 28.1 days (eta^-)
Technetium-99 (2.13 imes 10^5) years (eta^-)
Thorium-234 24.1 days (eta^-)
Uranium-235 (7.04 imes 10^8) years (alpha)
Uranium-238 (4.47 imes 10^9) years (alpha)

The following example illustrates how to use the half-life of a sample to determine the amount of radioisotope that remains after a certain period of time has passed.

Decay Series

In many instances, the decay of an unstable radioactive nuclide simply produces another radioactive nuclide. It may take several successive steps to reach a nuclide that is stable. A decay series is a sequence of successive radioactive decays that proceeds until a stable nuclide is reached. The terms reactant and product are generally not used for nuclear reactions. Instead, the terms parent and daugher nuclide are used to to refer to the starting and ending isotopes in a decay process. The figure below shows the decay series for uranium-238.

Figure (PageIndex2): The decay of uranium-238 proceeds along many steps until a stable nuclide, lead-206, is reached. Each decay has its own characteristic half-life.

In the first step, uranium-238 decays by alpha emission to thorium-234 with a half-life of (4.5 imes 10^9) years. This decreases its atomic number by two. The thorium-234 rapidly decays by beta emission to protactinium-234 ((t_1/2 =) 24.1 days). The atomic number increases by one. This continues for many more steps until eventually the series ends with the formation of the stable isotope lead-206.

Artificial Transmutation

As we have seen, transmutation occurs when atoms of one element spontaneously decay and are converted to atoms of another element. Artificial transmutation is the bombardment of stable nuclei with charged or uncharged particles in order to cause a nuclear reaction. The bombarding particles can be protons, neutrons, alpha particles, or larger atoms. Ernest Rutherford performed some of the earliest bombardments, including the bombardment of nitrogen gas with alpha particles to produce the unstable fluorine-18 isotope.

Fluorine-18 quickly decays to the stable nuclide oxygen-17 by releasing a proton.

When beryllium-9 is bombarded with alpha particles, carbon-12 is produced with the release of a neutron.

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Half-life calculations can be based on mass, percent remaining, or dose. Regardless of which one, the concept is still the same. Understanding the radioactivity and half-life of a sample is important for calculating the correct dose for a patient and determining the levels and duration of radioactive emission from a patient after treatment is received.

Frequently, dosages for radioactive isotopes are given the activity in volume. For example, the concentration of (ceI)-137 is given as (50 : mu extCi/mL) (microCurie per milliliter). This relationship can be used to calculate the volume needed for a particular dose. For example, a patient needs (125 : mu extCi) of (ceI)1-51. What volume of a (50 mu extCi) per (10 : extmL) solution should be given?