Half-Life Formula: Components and Applications

Man in lab coat measuring a decayed skull in a white laboratory — The half-life formula is essential in carbon dating, which helps archaeologists determine the age of various remains and artifacts. Adam Gault / Getty Images

In nuclear physics, the concept of half-life plays a crucial role in understanding the decay of radioactive substances. Scientists use the half-life formula in other disciplines to predict the rate of decay, as well as measure the age of ancient artifacts through carbon dating.

What Is Half-life?

In the context of radioactive decay and nuclear physics, half-life describes the time it takes for half of a quantity of a substance undergoing decay to go through transformation. In simpler terms, half-life is the duration it takes for a radioactive substance to lose half of its initial radioactivity.

For example, if you start with a certain amount of a radioactive substance, after one half-life, half of that substance will have decayed, and you will have half of the original amount. After two half-lives, three-quarters will have decayed, and so on.

Radioactive Decay and Isotopes

Half-life is a characteristic property of each radioactive isotope, and it plays a crucial role in understanding the stability and decay of atomic nuclei. You can express the concept mathematically through an exponential decay model, where the rate of decay is proportional to the remaining quantity of the substance.

The half-life of a radioactive isotope — denoted by T_1/2 — varies widely depending on the specific isotope. Each has its own unique half-life. Some isotopes have very short half-lives, measured in seconds or minutes, while others have half-lives that extend over thousands or millions of years.

The concept of half-life is not limited to radioactive decay; other fields like medicine, chemistry and environmental science also measure half-life.

Components of the Half-life Formula

The half-life formula is:

N(t) = N e^–^λt

Here are the formula's different components:

N(t) represents the remaining quantity of the radioactive substance at time t.
N is the initial quantity of the substance at time t = 0. The initial quantity sets the starting point.
e is the base of the natural logarithm.
λ is the decay constant, a measure of the rate of decay for the radioactive isotope.

In the formula, e^–λt is the core exponential decay factor, governing the decrease in quantity over time. As time (t) increases, this factor approaches 0, indicating an exponentially decaying quantity due to radioactive decay.

You can find a half-life calculator online to simplify the process of solving half-life problems.

Applications of the Half-life Formula

Here are a few uses for the half-life formula.

Radioactive decay: Scientists use the half-life formula to describe the decay process of radioactive isotopes. It helps determine the rate of decay and predict how much of a substance will remain after a certain period.
Carbon dating: Carbon dating relies on the half-life of carbon-14 (¹⁴C) to estimate the age of organic materials. By measuring the ratio of carbon-14 to carbon-12 in a sample, scientists can calculate how many half-lives have elapsed since the organism died.
Archaeology and geology: Half-life calculations are essential in dating ancient artifacts and geological samples. For example, scientists can use the decay of uranium to lead to determine the age of rocks.
Medical imaging: Radioactive substances used in medical imaging have known half-lives. Understanding the half-life allows medical professionals to determine the appropriate dosage and timing for imaging procedures.

This article was created in conjunction with AI technology, then was fact-checked and edited by a HowStuffWorks editor.