All radioactive decay processes follow first-order kinetics. The half-life of the radioactive isotope tritium \(\left({ }^{3} \mathrm{H}\right.\), or \(\left.\mathrm{T}\right)\) is \(12.3\) years. How much of a \(25.0\)-mg sample of tritium would remain after \(10.9\) years?

Understanding the concept of half-life is essential in the study of radioactive materials. The half-life is the time required for exactly half of a sample of a radioactive isotope to decay. It is a constant value that is characteristic to each isotope. This means that regardless of the amount of the substance you start with, after one half-life, you will have 50% of your original substance left.

In our example with tritium (\( {}^{3}\text{H} \) or T), with a half-life of 12.3 years, it shows that after 12.3 years, any quantity of tritium would be reduced to half of its original amount. Let's say we started with 25.0 mg of tritium; after 12.3 years, we'd expect to have 12.5 mg remaining. If we needed to calculate how much tritium would be left after a time period that is not a multiple of the half-life, then we'd need the decay constant and the first-order decay formula, which will be discussed next.

It's also worth mentioning that half-life calculation doesn't account for chemical reactions or physical processes that the material may undergo, only its inherent radioactivity.

The decay constant, represented as \( k \), is a probability that characterizes the rate of decay of a radioactive isotope. It is directly related to the half-life and is used to describe how quickly an isotope will decay. For first-order kinetics, the relationship between the decay constant and the half-life is given by the formula \( k = \frac{\ln(2)}{t_{1/2}} \). The natural logarithm of 2 (\( \ln(2) \)) is used because decay processes are described by the exponential function.

In our tritium example, we can calculate the decay constant using the provided half-life of 12.3 years. By plugging this into our decay constant formula, we find that \( k \) is a small fraction per year, which indicates that tritium decays slowly. The decay constant gives us a quantitative measure that can be applied across any amount of material and allows us to calculate the remaining quantity of the isotope at any point in time using the first-order decay formula, which brings us to our final concept.

To determine the exact amount of a radioactive element remaining after any given time, not just at its half-life, we employ the first-order decay formula, which is expressed as \( N = N_0 \cdot e^{-kt} \). \( N \) represents the remaining amount of the isotope, \( N_0 \) is the initial amount, \( t \) is the time that has passed, and \( k \) is the decay constant.

So for our tritium with an initial mass of 25.0 mg and wanting to know how much remains after 10.9 years, we substitute these values into our formula to solve for \( N \). Because the exponential term \( e^{-kt} \) shows how much of the substance has decayed over time, it will always be a fraction less than 1 for positive values of \( kt \), reflecting the remaining proportion of the undecayed material.

This formula demonstrates that the amount of substance does not decrease linearly over time but rather at a rate proportional to its current amount—hence 'first-order'. The continuous decay over time creates an exponential decay graph, which is key in understanding radioactivity and the predictable yet non-linear nature of radioactive decay.