The half-life for the (first-order) radioactive decay of \({ }^{14} \mathrm{C}\) is \(5730 \mathrm{y}\) (it emits \(\beta\) rays with an energy of \(0.16 \mathrm{MeV}\) ). An archaeological sample contained wood that had only 72 per cent of the \({ }^{14} \mathrm{C}\) found in living trees. What is its age?

First-order decay is a fundamental concept in the study of radioactive elements and their breakdown over time. The defining characteristic of first-order decay is that the rate at which the substance decays is directly proportional to the quantity that remains. This proportionality is key to understanding why the decay process follows an exponential pattern.

In the context of radioactive carbon, as in our exercise with Carbon-14, the important implication of this behavior is that a constant fraction of the remaining Carbon-14 atoms decays per unit of time, regardless of the initial quantity. Thus, whether you start with a gram or a kilogram of Carbon-14, after one half-life, half will have decayed in either case. It's this predictable pattern that allows scientists to use Carbon-14 as a clock to date organic materials.

To improve comprehension for students, it helps to visualize or simulate the decay process, perhaps with a graph showing the exponential decrease of substance over time, reinforcing the concept that as time progresses, the remaining amount of the substance decreases at a rate that becomes gradually slower.

The half-life of a radioactive substance is a critical piece of information in nuclear chemistry. It is defined as the time required for half of the original quantity of a radioactive material to decay. For instance, Carbon-14 has a half-life of approximately 5730 years.

This doesn't mean that exactly half of a sample will have decayed after 5730 years due to the statistical nature of radioactive decay, but it does mean that, on average, a large number of Carbon-14 atoms will take that amount of time to halve. This concept is integral to radiometric dating methods, including the example in our exercise where the age of an archaeological sample is determined.

For students, learning about half-life may involve engaging activities such as 'decay dice' games simulating the decay process, reinforcing the statistical concept behind the half-life. Understanding that despite its unpredictability on a microscopic level, decay follows a highly predictable pattern on a macroscopic scale is essential.

At the heart of the concepts like first-order decay and half-life lies the exponential decay process. The term 'exponential' denotes that the quantity of the substance decreases at a rate proportional to its current value. Mathematically, this relationship is expressed through the natural exponential function, leading to a characteristic steep decline that flattens over time on a graph.

In practical terms, as applied to radioactive decay, this process demonstrates that while a significant portion of the material decays quickly at first, the rate of decay slows down as there becomes less material available to decay. In the exercise concerning Carbon-14, when calculating the age of an archaeological find, we apply this exponential nature of decay to determine how long it's been since the sample had the same level of Carbon-14 as a living tree.

To solidify the students' understanding, animations or visuals illustrating how the curve of remaining material flattens over time can be extremely effective. Explaining that this type of decay pattern is not limited to radioactivity, but can be seen in other natural processes, can help students recognize the broader relevance of the exponential decay process.