Half-Life

Definition of Half-Life

The half-life of a radioactive isotope is the time taken for half of the nuclei in a sample to decay. It is a characteristic property of each radioactive isotope, meaning every isotope has its own fixed half-life that does not change.

For example, if you start with 1000 nuclei of a radioactive isotope, after one half-life only 500 nuclei will remain undecayed. After another half-life, half of those 500 will decay, leaving 250, and so on.

This concept is important because it tells us how quickly a radioactive substance becomes less active over time.

Radioactive Decay Process

Radioactive decay is a random and spontaneous process. This means you cannot predict exactly when a particular nucleus will decay, but you can predict the behaviour of a large number of nuclei statistically.

Decay follows an exponential pattern: the number of undecayed nuclei decreases by half every half-life period. This means the decay rate slows down over time as fewer nuclei remain.

As decay happens, the number of undecayed nuclei reduces, but the half-life remains constant regardless of how many nuclei are left.

Calculations Involving Half-Life

The number of half-lives is the number of times the half-life period has passed during the decay process. It can be calculated as:

$$\text{Number of half-lives} = \frac{\text{Total time elapsed}}{\text{Half-life}}$$

After each half-life, the number of undecayed nuclei halves:

$$\text{Remaining nuclei} = \text{Initial nuclei} \times \left(\frac{1}{2}\right)^{\text{number of half-lives}}$$

For instance, if a radioactive isotope has a half-life of 4 hours and you start with 800 nuclei, after 8 hours (which is 2 half-lives) the number of undecayed nuclei is:

$$800 \times \left(\frac{1}{2}\right)^2 = 800 \times \frac{1}{4} = 200 \text{ nuclei}$$

You can also find the time elapsed if you know how many nuclei remain:

Rearranging the formula:

$$\text{Time elapsed} = \text{Number of half-lives} \times \text{Half-life}$$

where the number of half-lives is found from:

$$\left(\frac{1}{2}\right)^{\text{number of half-lives}} = \frac{\text{Remaining nuclei}}{\text{Initial nuclei}}$$

Decay is often shown on a graph with the number of undecayed nuclei on the vertical axis and time on the horizontal axis. The curve slopes downwards, halving at regular intervals equal to the half-life.

Applications and Implications

Dating archaeological samples: Scientists use half-life to estimate the age of ancient objects by measuring the amount of radioactive isotopes remaining. For example, carbon-14 dating uses the half-life of carbon-14 to date once-living materials up to about 50,000 years old.

Medical uses and safety: Radioactive isotopes with known half-lives are used in medicine for diagnosis and treatment. Knowing the half-life helps doctors choose isotopes that remain active long enough to be useful but decay quickly enough to reduce radiation exposure.

Understanding nuclear stability: Half-life gives insight into how stable a nucleus is. Isotopes with very short half-lives decay quickly and are highly unstable, while those with very long half-lives are more stable.