Explain the concept of half-life.

Radioactive decay is a fundamental concept in nuclear physics, where unstable atomic nuclei lose energy by emitting radiation. It's an inherently random process, with each unstable isotope, or 'radioisotope', having a specific probability of decay over time. This probability is not affected by external factors such as temperature or pressure.

In practice, unstable isotopes—those that have an imbalance in their proton-to-neutron ratio—reach stability by transforming into different isotopes or elements through processes such as alpha, beta, and gamma decay. For instance, Uranium-238 decays to form Thorium-234, releasing alpha particles during this transformation. The rate at which these processes happen is characterized by the decay constant ( lambda ), which is a unique value for each isotope.

Radiometric dating is a technique used to determine the age of ancient materials, such as archaeological specimens and geological formations. It relies on the known half-life of isotopes contained within these materials. By measuring the amount of the original radioactive isotope and its decay products, scientists can calculate the time that has elapsed since the material was formed.

For example, Carbon-14 dating is a common method for dating organic materials up to about 60,000 years old. As living organisms take in Carbon-14 naturally, its ratio to Carbon-12 remains constant. Upon death, Carbon-14 uptake stops, and its slow decay begins. Over time, the ratio of Carbon-14 to Carbon-12 decreases, and analyzing this ratio provides an estimate of when the organism stopped taking in carbon—essentially, the time of its death.

The decay pattern refers to the predictable way in which radioactive isotopes degrade over time. Unlike other changeable patterns, the radioactive decay pattern follows a specific pathway: the exponential decay curve. This curve illustrates that the rate at which the isotope decays is proportional to the amount of the substance present at any given time.

As such, if we were to graph the remaining amount of a radioisotope against time, we would see a steady decline that halves with each successive half-life period. This is a crucial concept for students to grasp as it emphasizes that the radioactive substance doesn't diminish at a linear rate but rather decreases rapidly at first and then more gradually over successive periods.

Isotope stability is intimately connected to the concept of half-life, as it describes the resistance of an isotope to change or decay. Stable isotopes are those which do not undergo radioactive decay over time and remain constant. In contrast, unstable isotopes spontaneously decay, seeking a more balanced nuclear composition.

The journey to stability can occur in a single step or through a series of decay events, resulting in a chain of intermediate isotopes with their half-lives until a stable isotope is reached. For example, Uranium-238 undergoes a series of decay steps, known as a decay chain, to eventually become the stable isotope Lead-206. These chains are key to understanding the natural processes that shape our planet's geological and biological history.