Radium-223 decays with a half-life of 11.4 days. How long does it take for a 0.240-mol sample of radium to decay to 1.50 * 10-2 mol?

Radioactive decay is a fundamental process by which an unstable atomic nucleus loses energy by emitting radiation. It occurs naturally in a variety of isotopes, emitting particles such as alpha particles, beta particles, or gamma rays until a more stable configuration is reached. This form of decay follows a predictable pattern described by the half-life, the time required for half of the radioactive substance to disintegrate.

Understanding radioactive decay is crucial in fields such as nuclear medicine, dating of archaeological artifacts, and safety assessments of nuclear materials. The decay process is random for individual atoms but highly predictable for a large number of atoms, hence it's modeled effectively by statistical methods.

The exponential decay formula is used to describe the decrease in the amount of a substance over time due to radioactive decay. The formula is expressed as: \[N(t) = N_0 \times 0.5^{\frac{t}{t_{1/2}}}\], where \(N(t)\) is the amount of substance remaining after time \(t\), \(N_0\) is the initial amount of substance, and \(t_{1/2}\) is the half-life of the substance.

This formula showcases the nature of decay as a continuous and exponentially decreasing process. When plotted on a graph, it produces a characteristic 'decay curve' that halves at each half-life interval, reflecting the nature of the process as one that slows down as the amount of substance decreases.

The half-life period is a measure of the time it takes for half of the radioactive atoms in a given sample to decay. It is a constant for each radioactive isotope, meaning it is not affected by external conditions such as temperature or pressure.

For example, if a substance has a half-life of 11.4 days, it means that after 11.4 days, only half of the original number of radioactive atoms will remain; the other half will have decayed into a different element or a more stable isotope. This information helps in calculating how long it will take for a substance to decay to a certain level and is fundamental for applications such as determining the dosage in medical treatments and estimating the age of materials, among others.

Logarithmic equations are vital when dealing with exponential decay because they allow us to solve for time or the remaining amount of substance. When an equation involves an exponential expression, taking the logarithm of both sides facilitates the extraction of the exponent.

To solve for time in a half-life problem, for instance, we can use a logarithm to 'undo' the exponential part of the decay formula. In doing this, the equation becomes solvable for the unknown variable, often the time elapsed. A logarithm essentially answers the question: 'To what power must we raise a certain base to get a particular number?' In our decay problems, this helps us determine the time period necessary for a substance to reach a specific level.