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Half-life is a crucial concept in understanding radioactive decay. It represents the time required for half of the radioactive atoms in a sample to decay. This means if you start with 100 atoms, after one half-life, you'll have 50 atoms remaining. For instance, if isotope \(X\) has a half-life of 4 days, this implies that in 4 days, half of its atoms will have decayed. Similarly, isotope \(Y\) with a 16-day half-life indicates a slower decay process and requires more time for the same reduction. Understanding half-life helps predict how quickly a substance will lose its radioactivity, which is essential for applications that range from medical treatments to archaeological dating.

The decay constant, represented by \(\lambda\), provides a measure of the probability of decay per unit time. It is directly related to the half-life through the formula \(\lambda = \frac{\ln(2)}{T_{1/2}}\), where \(T_{1/2}\) is the half-life. For example, isotope \(X\) with a half-life of 4 days will have a decay constant \(\lambda_x = \frac{\ln(2)}{4}\). Meanwhile, isotope \(Y\) with a 16-day half-life will have a smaller decay constant \(\lambda_y = \frac{\ln(2)}{16}\). A larger decay constant implies a faster rate of decay, indicating that the substance is more radioactive and will lose its radioactivity more rapidly.

Radioactive isotopes, often referred to as radioisotopes, are variants of elements that emit radiation as they decay over time. Each isotope has its unique half-life and decay constant, determining how quickly it will decay. Isotopes like \(X\) and \(Y\) used in the exercise emit radiation at different rates due to their distinct half-life values.

• These isotopes are used in a wide range of applications, including medical imaging and treatment, scientific research, and power generation.
• They help track substance movement in biological systems or dating of archaeological artifacts or geological samples.

Understanding the behavior of such isotopes allows for safe and effective use in various fields.

The activity of a radioactive sample refers to the rate at which a sample decays per unit time. It is often measured in units such as curies or becquerels. Activity \(A\) is determined by the expression \(A = \lambda N\), where \(N\) is the number of radioactive atoms, and \(\lambda\) is the decay constant. For the mixture of isotopes \(X\) and \(Y\), the given activity initially is 1 curie. After 8 days, it reduces to 0.5 curie, demonstrating a 50% decrease. This reduction illustrates how combined isotopes decay over time, allowing determination of the initial ratio of atoms present, as explored in the problem. Activity offers insight into how radioactive substances evolve, which is vital for both scientific study and practical applications.