2.6 Halflives and the Rate of Radioactive Decay
 Page ID
 159956
Skills to Develop
 To know how to use halflives to describe the rates of firstorder reactions
Radioactive Decay Rates
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 onehalf the initial value. This period of time is called the halflife of the process, written as t_{1}_{/2}. Thus the halflife 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}.
Figure \(\PageIndex{1}\): The HalfLife of a FirstOrder Reaction. This plot shows the concentration of the reactant in a firstorder reaction as a function of time and identifies a series of halflives, intervals in which the reactant concentration decreases by a factor of 2. In a firstorder reaction, every halflife is the same length of time.
Number of HalfLives  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 halflives of a firstorder reaction is (1/2)^{n} times the initial concentration.
For a firstorder reaction, the concentration of the reactant decreases by a constant with each halflife and is independent of [A].
For a given number of atoms, isotopes with shorter halflives decay more rapidly, undergoing a greater number of radioactive decays per unit time than do isotopes with longer halflives. The halflives of several isotopes are listed in Table 14.6, along with some of their applications.
Radioactive Isotope  HalfLife  Typical Uses 

*The m denotes metastable, where an excited state nucleus decays to the ground state of the same isotope.  
hydrogen3 (tritium)  12.32 yr  biochemical tracer 
carbon11  20.33 min  positron emission tomography (biomedical imaging) 
carbon14  5.70 × 10^{3} yr  dating of artifacts 
sodium24  14.951 h  cardiovascular system tracer 
phosphorus32  14.26 days  biochemical tracer 
potassium40  1.248 × 10^{9} yr  dating of rocks 
iron59  44.495 days  red blood cell lifetime tracer 
cobalt60  5.2712 yr  radiation therapy for cancer 
technetium99m*  6.006 h  biomedical imaging 
iodine131  8.0207 days  thyroid studies tracer 
radium226  1.600 × 10^{3} yr  radiation therapy for cancer 
uranium238  4.468 × 10^{9} yr  dating of rocks and Earth’s crust 
americium241  432.2 yr  smoke detectors 
Note
Radioactive decay is a firstorder 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 halflives have passed?
Solution
Given: mass of radioactive sample of an element, number of halflives
Asked to Solve For: mass of radioactive element after so many halflives
Solve:
All radioactive samples lose half of their mass after each halflife. Thus, one solution is to calculate the mass after each halflife. (This method only works if you are asked to solve for a whole number of halflives). Let the passing of time equal to one halflife 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 halflives, including fractional halflives, then you use the equation: amount remaining = \( \left( \dfrac {1}{2} \right)^n (amount \; at \; start)\), where n= number of halflives. 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 halflives have passed?
 Answer

amount left = \( \left( \dfrac {1}{2} \right)^{4.30} (300. g) = 15.2 g\)
Exercise \(\PageIndex{2}\)
A certain radioactive nuclide has a halflife 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?
 Answer

\( 20.0 \; days \times \dfrac{1 \; halflife}{5.25 \; days} = 3.81 \; halflives\)
amount left = \( \left( \dfrac {1}{2} \right)^{3.81} (100. g) = 7.13 g\)
Radioisotope Dating Techniques
In our earlier discussion, we used the halflife of a firstorder reaction to calculate how long the reaction had been occurring. Because nuclear decay reactions follow firstorder kinetics and have a rate constant that is independent of temperature and the chemical or physical environment, we can perform similar calculations using the halflives 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 carbon14 dating. The carbon14 isotope, created continuously in the upper regions of Earth’s atmosphere, reacts with atmospheric oxygen or ozone to form ^{14}CO_{2}. As a result, the CO_{2} that plants use as a carbon source for synthesizing organic compounds always includes a certain proportion of ^{14}CO_{2} molecules as well as nonradioactive ^{12}CO_{2} and ^{13}CO_{2}. 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 carbon14 nuclei in its tissues decay to nitrogen14 nuclei by a radioactive process known as beta decay, which releases lowenergy electrons (β particles) that can be detected and measured:
\[ \ce{^{14}C \rightarrow ^{14}N + \beta^{−}} \label{21.4.7}\]
The halflife for this reaction is 5700 ± 30 yr.
The ^{14}C/^{12}C 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 ^{14}CO_{2}/^{12}CO_{2} ratio in the atmosphere is constant, which is not strictly correct. Other methods, such as treering 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 ^{14}CO_{2}/^{12}CO_{2} ratio over time.
Figure \(\PageIndex{2}\): Radiocarbon Dating. A plot of the specific activity of ^{14}C versus age for a number of archaeological samples shows an inverse linear relationship between ^{14}C content (a log scale) and age (a linear scale).
Summary
 The halflife of a firstorder reaction is independent of the concentration of the reactants.
 The halflives 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 halflife of a reaction is the time required for the reactant concentration to decrease to onehalf its initial value. Radioactive decay reactions are firstorder reactions.
Modified by Tom Neils (Grand Rapids Community College)