What is the half-life for the first-order decay of phosphorus-32? $$\left(\frac{14}{6} \mathrm{C} \longrightarrow_{7}^{14} \mathrm{N}+\mathrm{e}^{-}\right)$$ The rate constant for the decay is \(4.85 \times 10^{-2}\) day \(^{-1}\).

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation in the form of particles or electromagnetic waves. This transformation into a more stable state occurs spontaneously and is a natural phenomenon found in many elements, such as uranium, radium, and phosphorus-32, among others.

At the heart of radioactive decay is the nucleus of the atom, which contains protons and neutrons. An imbalance in the number of these particles gives rise to a form of instability that the atom seeks to correct by undergoing decay. There are different types of radioactive decay, such as alpha decay, beta decay (as in our example with phosphorus-32), and gamma decay, each defined by the type of radiation emitted during the process.

The rate at which a particular isotope decays is characterized by its half-life—the time it takes for half of a given sample to decay. Understanding decay patterns is crucial not only in fields like nuclear physics and geology but also in medical applications, such as in the treatment and diagnosis of diseases using radioisotopes.

The calculation of the half-life in radioactive decay is crucial to various scientific and medical fields because it helps determine how long a radioactive substance remains active. For first-order decay, which is common among many radioisotopes, the half-life (\( t_{1/2} \) ) is inversely related to the decay rate constant (\( k \) ). This relationship is mathematically represented by the formula \( t_{1/2} = \frac{0.693}{k} \).

To calculate the half-life, one must first understand what the rate constant signifies. It's a measure of the speed of the decay process: a high rate constant means the substance decays quickly, while a low rate constant indicates slower decay. When using the formula, the 0.693 factor is derived from the natural logarithm of 2, since the half-life is based on the concept of 'halving' the amount of substance over time.

By rearranging the formula, one can see that the half-life is directly proportional to this logarithmic factor and inversely proportional to the rate constant. This allows for the estimation of how long it takes for half the nuclei in a radioactive sample to decay, which can be vital information when dealing with radioactive materials in various contexts.

In the context of radioactive decay, the rate constant (\( k \) ) is a numerical value that expresses the speed at which the decay process occurs. It is specific to each radioactive isotope and can vary widely among different substances. This value is crucial for calculating the half-life as it determines how rapidly an isotope will decay over time.

For first-order decay reactions, where the rate of decay depends linearly on the amount of the undecayed substance remaining, the rate constant is found in the exponent of the decay's mathematical expression. It is often given in units like day\textsuperscript{-1}, which indicates how much of the substance decays each day.

As demonstrated in the exercise with phosphorus-32, with a rate constant \( k = 4.85 \times 10^{-2} \) day\textsuperscript{-1}, knowing \( k \) allows us to calculate the half-life of the substance with the formula provided. Thus, the rate constant is a cornerstone for understanding the kinetics of radioactive decay and facilitates the practical use of radioactive isotopes in a variety of applications from nuclear power generation to medical imaging.