Calculate that half-life of cesium-135 if seven-eighths of a sample decays in \(6 \times 10^{6} \mathrm{y}\) .

Exponential decay is a process where the quantity of a substance decreases over time at a rate proportional to its current value. In the context of nuclear chemistry, this relates to how unstable atoms lose particles from their nucleus, leading to a reduction in the number of radioactive atoms over time.

Mathematically, the exponential decay model is represented by the formula \(N = N_0 \times e^{-rt}\), where:

• \(N\) is the amount of the substance at time \(t\),
• \(N_0\) is the initial quantity of the substance,
• \(r\) is the decay constant, which represents the rate of decay,
• \(e\) is the base of the natural logarithm, approximately equal to 2.71828.

What's crucial to grasp is that this rate of change is constant in percentage, not in absolute terms. This means that no matter how much substance you have, a fixed percentage will decay over a standard interval—a defining characteristic of exponential processes.

The decay constant, denoted by \(r\), is a paramount concept in understanding exponential decay. It's a measure of how quickly a radioactive substance undergoes decay. The larger the decay constant, the faster the substance will decay.

Calculating \(r\) involves rearranging the exponential decay formula to solve for \(r\), using the known initial and remaining amounts of a substance and the time elapsed. From the exercise, using the provided data points and solving for \(r\) involves logarithmic functions, reflecting the inverse of the exponential function used to model decay.

Knowing \(r\) allows us to determine the half-life, which is the time taken for half of the radioactive material to decay. The relationship between half-life (\(T_{1/2}\)) and the decay constant is given by the formula \(T_{1/2} = \frac{0.693}{r}\), where 0.693 is the natural logarithm of 2, representing the fraction of the substance that decays after one half-life.

Radioactive decay stands as a natural process by which an unstable atomic nucleus loses energy by emitting radiation in the form of particles or electromagnetic waves. There are different types of decay, such as alpha, beta, and gamma decay, each with a unique set of characteristics and emissions.

The rate of radioactive decay is unpredictable for individual atoms; however, for a large number of atoms, the decay rate can be statistically predicted, hence the use of exponential decay models. The half-life is a key concept within this framework and refers to the duration required for half of the radioactive substance to transform or decay. It is noteworthy that half-life is independent of the initial amount of the material and only depends on the decay constant, indicative of the specific type of decay the nuclei undergo.

Nuclear chemistry is the sub-discipline of chemistry that deals with the chemical and physical properties of elements as influenced by changes in the structure of the atomic nucleus. It mainly involves the study of radioactive elements, their behavior, transformation, and the interactions and processes that drive changes in the nucleus—like nuclear fission, fusion, and radioactive decay.

Nuclear chemistry is essential in various applications, including energy production in nuclear reactors, radiometric dating in archaeology, and nuclear medicine in medical treatments. A sound understanding of concepts like radioactive decay patterns, half-life, and decay constants is foundational in this field, as these allow us to predict the behavior of radioactive substances over time and manage their use in diverse real-world applications.