A sample of radioactive \(\frac{35}{16} \mathrm{S}\) disintegrates at a rate of \(1.00 \times 10^{3}\) atoms \(\min ^{-1} .\) The half-life of \(_{16}^{35} \mathrm{S}\) is \(87.9 \mathrm{d}\) How long will it take for the activity of this sample to decrease to the point of producing (a) \(253 ;\) (b) \(104 ;\) and (c) 52 dis \(\min ^{-1} ?\)

The half-life of a radioactive isotope is the time required for half of the unstable nuclei in a sample to undergo radioactive decay. It's a constant value that is characteristic of each radioactive element. For students tackling problems involving half-life, it's crucial to understand that this period of time doesn’t vary depending on the amount of the substance you have. Whether it's a gram or a ton, the half-life remains unchanged.

In the given exercise, the radioactive isotope sulfur-35 has a half-life of 87.9 days. This means that every 87.9 days, the number of sulfur-35 atoms decreases by half due to decay. This provides a predictable pattern which can be utilized to determine how the activity — the rate at which the atoms decay — will change over time.

Conversion for Consistency in Units

A common stumbling block in solving radioactive decay problems is unit conversion. The initial step often involves ensuring that the units for the decay rate and half-life are consistent. In this case, the activity is measured in atoms per minute while the provided half-life is in days; therefore, converting the half-life from days to minutes is essential for accurate calculations.

The radioactive disintegration rate, often referred to as the activity of a radioactive sample, represents the number of atoms decaying per unit of time. It's measured in disintegrations per minute (dpm) or disintegrations per second, also known as becquerels (Bq) in the International System of Units.

In practice, the disintegration rate decreases over time as there are fewer and fewer undecayed atoms available in the sample. It's a direct measure of the quantity of the radioactive substance and its propensity to decay. In the original exercise, starting with a disintegration rate of 1.00 x 10^3 atoms per minute, we are tasked with finding out the time it takes for this rate to decrease to 253, 104, and 52 dpm respectively.

Applying the Concept to Calculations

When performing these calculations, it's important to start with the understanding that the activity of the sample is directly proportional to the number of undecayed atoms present. As the number of these atoms decreases due to decay, so does the activity. By using the half-life formula and incorporating the initial activity, students can calculate how much time passes before the activity falls to any given value.

Radioactive decay follows a pattern known as exponential decay. This means that the quantity of a radioactive substance decreases at a rate proportional to its current amount. Graphically, exponential decay is represented by a curve that starts off steep and gradually flattens out as time progresses. One of the fundamental qualities of exponential decay is that, regardless of the starting amount, the percentage decrease over each time period is constant.

Understanding the mathematical model of exponential decay is key for students to correctly solve radioactive decay problems. The formula that describes this process usually involves the initial amount (N0), the remaining amount (N), the decay constant (λ), and the time elapsed (t). However, when dealing with half-life, the decay constant is substituted with the half-life period to simplify calculations.

Application in Exercises

In the exercise provided, knowing that decay is exponential is crucial for utilizing the correct logarithmic relationship to find the time it takes for the activity to drop to specified levels. By relating the current activity to the initial activity and the half-life, students can identify how the activity changes over time in a process that is fundamentally exponential in nature.