Half-Life

Grand Canyon image

The Grand Canyon, pictured above, was carved by the rushing waters of the Colorado River over millions of years. The exposed rocks at the bottom of the canyon are almost 2 billion years old. The youngest rocks near the top are about 230 million years old. Therefore, from top to bottom, the rocks provide a continuous record of more than 1.5 billion years of geological history in this region.

Radioactive isotopes can be used to estimate the ages of not only rocks but also of fossils and artifacts made long ago by human beings. Even the age of Earth has been estimated on the basis of radioactive isotopes. The general method is called radioactive dating. To understand how radioactive dating works, you need to understand radioactive decay.

The rate of radioactive decay is different for every radioactive isotope as seen in the table below. Less stable nuclei decay at a faster rate than more stable nuclei. In this lesson, you will learn about the half-lives of radioactive nuclei.

Radioactive Decay
Parent Isotope	Daughter Isotope	Half-life
potassium-40	argon-40	1.3 billion years
uranium-235	lead-207	700 million years
uranium- 234	thorium-230	80,000 years
carbon-14	nitrogen-14	5,700 years

A radioactive isotope has atoms with unstable nuclei. Unstable nuclei naturally decay or break down. They lose energy and particles and become more stable which is what we looked at in the previous lesson. As nuclei decay, they gain or lose protons, so the atoms become different elements. The original, unstable nucleus is called the parent nucleus. After it loses a particle (alpha or beta particle), it forms a daughter nucleus, with a different number of protons. The nucleus of a given radioisotope decays at a constant rate that is unaffected by temperature, pressure, amount of original size, or other conditions outside the nucleus. This rate of decay is called the half-life. The half-life (t_1/2) is the length of time it takes for half of the original amount of the radioisotope to decay into another element.

Parent nucleus To Daugher Nucleus and alpha particle

After each half-life has passed, one-half of the radioactive nucleus will have transformed into a new nucleus. Look at the table below for the breakdown:

Half-lives Passed Chart
Number of Half-lives Passed	Percentage of Radioactive Isotope Remaining	Fraction of Radioactive Isotope Remaining
1	50%	$LaTeX: \frac{1}{2}$
2	25%	$LaTeX: \frac{1}{4}$
3	12.5%	$LaTeX: \frac{1}{8}$
4	6.25%	$LaTeX: \frac{1}{16}$
5	3.125%	$LaTeX: \frac{1}{32}$

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

$LaTeX: ^{131}_{53}$ I $LaTeX: \rightarrow ^{131}_{54}$ Xe + $LaTeX: ^0_{-1}$ e

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), the sample will be 25% iodine-131 and 75% xenon-131. This continues until the entire sample of iodine-131 has completely decayed.

Just to be sure you understand half-life, let's look at another example, Uranium-235. Uranium-235, the parent nucleus, has a half-life of 704 million years. It decays to Lead-207, the daughter nucleus. Look at the chart below for an additional explanation of what happens to the parent nucleus after each half-life in regards to how much is remaining if we start with an 80-gram sample of Uranium-235.

Half-lives
Number of Half-lives	Time Elapsed	Fraction of U-235 remaining	Amount of U-235 remaining
0	0 years	1	80 grams
1	704 million years	$LaTeX: \frac{1}{2}$	40 grams
2	1408 million years	$LaTeX: \frac{1}{4}$	20 grams
3	2112 million years	$LaTeX: \frac{1}{8}$	10 grams

Decay Curves

The decay of radioactive materials can be shown with a graph. The graph below shows the radioactive decay of an imaginary isotope that has a half-life of 1 year.

decay of radioactive substance graph, curve is downward trending
% of radioactive remaining and age of sample in years

Notice how it doesn't take too many half-lives before there is very little parent remaining and most of the isotopes are daughter isotopes. However, remember that isotopes have a wide range of the rate at which they decay.

What can we interpret from a radioactive decay curve? First, you can determine the half-life. This is seen by looking at the red line on the graph above. On the y-axis, you will find the 50% remaining, and then where the graphed line hits on the decay curve determine how much time has elapsed. The time it takes for half of the substance to decay is the half-life. Continuing, you can determine the percent remaining over time. As you can see, the activity decreases by one-half during each succeeding half-life.

Let's answer some questions by looking at a decay curve for iodine-131 as seen below.

Decay Curve for Iodine-131 on graph
downward trending
percentage remaining versus elapsed time in days

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