# 2.6 Half-lives and the Rate of Radioactive Decay

Skills to Develop

• To know how to use half-lives to describe the rates of first-order reactions

Radioactivity, or radioactive decay, is the emission of a particle or a photon that results from the spontaneous decomposition of the unstable nucleus of an atom. The rate of radioactive decay is an intrinsic property of each radioactive isotope that is independent of the chemical and physical form of the radioactive isotope. The rate is also independent of temperature. Because there are so many unstable nuclei that decay, we need a method to describe and compare the rates at which these nuclei decay. One approach to describing reaction rates is based on the time required for the number of unstable nuclei to decrease to one-half the initial value. This period of time is called the half-life of the process, written as t1/2. Thus the half-life of a nuclear decay process is the time required for the number of unstable nuclei to decrease from [A]0 to 1/2[A]0.

For all nuclear decay processes (which follow first-order reaction kinetics), each successive half-life is the same length of time, as shown in Figure $$\PageIndex{1}$$, and is independent of [A].

Figure $$\PageIndex{1}$$: The Half-Life of a First-Order Reaction. This plot shows the concentration of the reactant in a first-order reaction as a function of time and identifies a series of half-lives, intervals in which the reactant concentration decreases by a factor of 2. In a first-order reaction, every half-life is the same length of time.

Number of Half-Lives Percentage of Reactant Remaining
1 $$\dfrac{100\%}{2}=50\%$$ $$\dfrac{1}{2}(100\%)=50\%$$
2 $$\dfrac{50\%}{2}=25\%$$ $$\dfrac{1}{2}\left(\dfrac{1}{2}\right)(100\%)=25\%$$
3 $$\dfrac{25\%}{2}=12.5\%$$ $$\dfrac{1}{2}\left(\dfrac{1}{2}\right )\left (\dfrac{1}{2}\right)(100\%)=12.5\%$$
n $$\dfrac{100\%}{2^n}$$ $$\left(\dfrac{1}{2}\right)^n(100\%)=\left(\dfrac{1}{2}\right)^n\%$$

As you can see from this table, the amount of reactant left after n half-lives of a first-order reaction is (1/2)n times the initial concentration.

For a first-order reaction, the concentration of the reactant decreases by a constant with each half-life and is independent of [A].

For a given number of atoms, isotopes with shorter half-lives decay more rapidly, undergoing a greater number of radioactive decays per unit time than do isotopes with longer half-lives. The half-lives of several isotopes are listed in Table 14.6, along with some of their applications.

Table $$\PageIndex{2}$$: Half-Lives and Applications of Some Radioactive Isotopes
*The m denotes metastable, where an excited state nucleus decays to the ground state of the same isotope.
hydrogen-3 (tritium) 12.32 yr biochemical tracer
carbon-11 20.33 min positron emission tomography (biomedical imaging)
carbon-14 5.70 × 103 yr dating of artifacts
sodium-24 14.951 h cardiovascular system tracer
phosphorus-32 14.26 days biochemical tracer
potassium-40 1.248 × 109 yr dating of rocks
iron-59 44.495 days red blood cell lifetime tracer
cobalt-60 5.2712 yr radiation therapy for cancer
technetium-99m* 6.006 h biomedical imaging
iodine-131 8.0207 days thyroid studies tracer
uranium-238 4.468 × 109 yr dating of rocks and Earth’s crust
americium-241 432.2 yr smoke detectors

Note

Radioactive decay is a first-order process.

Example $$\PageIndex{1}$$

If you have a 120 gram sample of a radioactive element, how many grams of that element will be left after 3 half-lives have passed?

Solution

Given: mass of radioactive sample of an element, number of half-lives

Asked to Solve For: mass of radioactive element after so many half-lives

Solve:

All radioactive samples lose half of their mass after each half-life. Thus, one solution is to calculate the mass after each half-life. (This method only works if you are asked to solve for a whole number of half-lives). Let the passing of time equal to one half-life be represented by and arrow, →. Then the solution is:

120 g → 60 g → 30 g → 15 g

If you want to solve for any number of half-lives, including fractional half-lives, then you use the equation: amount remaining = $$\left( \dfrac {1}{2} \right)^n (amount \; at \; start)$$, where n= number of half-lives. Then the solution is:

amount remaining = $$\left( \dfrac {1}{2} \right)^3 (120 g) = 15 g$$

Exercise $$\PageIndex{1}$$

If you have a 300. gram sample of a radioactive element, how many grams of that element will be left after 4.30 half-lives have passed?

amount left = $$\left( \dfrac {1}{2} \right)^{4.30} (300. g) = 15.2 g$$

Exercise $$\PageIndex{2}$$

A certain radioactive nuclide has a half-life of 5.25 days. If you start with 100. grams of this nuclide, how many grams of the nuclide will be left after 20.0 days?

$$20.0 \; days \times \dfrac{1 \; half-life}{5.25 \; days} = 3.81 \; half-lives$$

amount left = $$\left( \dfrac {1}{2} \right)^{3.81} (100. g) = 7.13 g$$

In our earlier discussion, we used the half-life of a first-order reaction to calculate how long the reaction had been occurring. Because nuclear decay reactions follow first-order kinetics and have a rate constant that is independent of temperature and the chemical or physical environment, we can perform similar calculations using the half-lives of isotopes to estimate the ages of geological and archaeological artifacts. The techniques that have been developed for this application are known as radioisotope dating techniques.

The most common method for measuring the age of ancient objects is carbon-14 dating. The carbon-14 isotope, created continuously in the upper regions of Earth’s atmosphere, reacts with atmospheric oxygen or ozone to form 14CO2. As a result, the CO2 that plants use as a carbon source for synthesizing organic compounds always includes a certain proportion of 14CO2 molecules as well as nonradioactive 12CO2 and 13CO2. Any animal that eats a plant ingests a mixture of organic compounds that contains approximately the same proportions of carbon isotopes as those in the atmosphere. When the animal or plant dies, the carbon-14 nuclei in its tissues decay to nitrogen-14 nuclei by a radioactive process known as beta decay, which releases low-energy electrons (β particles) that can be detected and measured:

$\ce{^{14}C \rightarrow ^{14}N + \beta^{−}} \label{21.4.7}$

The half-life for this reaction is 5700 ± 30 yr.

The 14C/12C ratio in living organisms is 1.3 × 10−12, with a decay rate of 15 dpm/g of carbon (Figure $$\PageIndex{2}$$). Comparing the disintegrations per minute per gram of carbon from an archaeological sample with those from a recently living sample enables scientists to estimate the age of the artifact, as illustrated in Example 11.Using this method implicitly assumes that the 14CO2/12CO2 ratio in the atmosphere is constant, which is not strictly correct. Other methods, such as tree-ring dating, have been used to calibrate the dates obtained by radiocarbon dating, and all radiocarbon dates reported are now corrected for minor changes in the 14CO2/12CO2 ratio over time.

Figure $$\PageIndex{2}$$: Radiocarbon Dating. A plot of the specific activity of 14C versus age for a number of archaeological samples shows an inverse linear relationship between 14C content (a log scale) and age (a linear scale).

## Summary

• The half-life of a first-order reaction is independent of the concentration of the reactants.
• The half-lives of radioactive isotopes can be used to date objects.

The rate of decay, or activity, of a sample of a radioactive substance is the rate of decrease in the number of radioactive nuclei per unit time. The half-life of a reaction is the time required for the reactant concentration to decrease to one-half its initial value. Radioactive decay reactions are first-order reactions.

Modified by Tom Neils (Grand Rapids Community College)