Radon-220 undergoes alpha decay with a half-life of \(55.6 \mathrm{~s}\). Assume there are 16,000 atoms present initially and make a table showing how many atoms will be present at \(0 \mathrm{~s}, 55.6 \mathrm{~s}, 111.2 \mathrm{~s}\), \(166.8 \mathrm{~s}, 222.4 \mathrm{~s},\) and \(278.0 \mathrm{~s}\) (all multiples of the half-life). Now calculate how many atoms will be present at \(50 \mathrm{~s}, 100 \mathrm{~s},\) and \(200 \mathrm{~s}\) (not multiples of the half-life). Make a graph with number of atoms present on the \(y\) -axis and total time on the \(x\) -axis.

Let's start by exploring the term 'half-life,' which is crucial when discussing alpha decay and other types of radioactive processes. In essence, the half-life is a measure of time required for half of the radioactive atoms in a sample to decay. This is not just a theoretical construct; it's a practical tool used by scientists to predict how long a radioactive substance will remain active. It's akin to a timer ticking away, with each 'tick' representing a half-life interval. At each tick, the quantity of undecayed atoms is cut in half.

For example, for Radon-220, with a half-life of 55.6 seconds, if we begin with 16,000 atoms, after 55.6 seconds, we would expect to have 8,000 atoms remaining. After another 55.6 seconds (a total of 111.2 seconds), this number would be halved again to 4,000, and so forth. This halving continues, providing a clear and predictable pattern of decay. This concept is inherently tied to the principle of exponential decay, forming the foundation of how we understand and calculate the persistence of radioactive substances over time.

Radioactive decay, more broadly, is the process by which an unstable atomic nucleus loses energy by emitting radiation. There are several types of radioactive decay, and alpha decay is one among them. In alpha decay specifically, the nucleus emits an alpha particle, which consists of two protons and two neutrons, effectively decreasing the atomic number by two and the mass number by four.

During this process, one element transforms into another, and Radon-220 is no exception. Because radioactive decay is random but statistically predictable, we can't tell when a specific atom will decay, but we can predict how many atoms out of a larger group will decay over a certain period—hence the relevance of half-life in providing a statistical understanding of decay. This predictability is essential for various practical applications, such as medical treatments using radioactive isotopes, and for understanding natural processes in environmental science and geology.

To delve into the mathematical heart of decay processes, we address exponential decay. Exponentially decaying systems reduce by a consistent percentage over a predefined interval—a key characteristic that distinguishes this pattern from other types of decline. The formula for exponential decay in the context of radioactive substances is expressed as \( N = N_0 (1/2)^{\frac{t}{T}} \), where \( N \) represents the number of remaining atoms after time \( t \), \( N_0 \) is the initial number of atoms, and \( T \) represents the half-life.

In practical terms, if we want to calculate the number of Radon-220 atoms remaining at a time period that isn't a multiple of its half-life, this formula is the tool we'd use. Diagrammatically, if we plot these values on a graph over time, the curve we obtain is not linear but rather takes the shape of a declining slope, rapidly dropping off at first and then gradually flattening as time progresses. This is indicative of exponential decay, illustrating that the quantity's rate of decrease is proportional to its current value.