Consider the decay series $$\mathrm{A} \longrightarrow \mathrm{B} \longrightarrow \mathrm{C} \longrightarrow \mathrm{D}$$ where \(\mathrm{A}, \mathrm{B},\) and \(\mathrm{C}\) are radioactive isotopes with halflives of \(4.50 \mathrm{~s}, 15.0\) days, and \(1.00 \mathrm{~s},\) respectively, and \(\mathrm{D}\) is nonradioactive. Starting with 1.00 mole of A, and none of \(\mathrm{B}, \mathrm{C},\) or \(\mathrm{D},\) calculate the number of moles of \(\mathrm{A}, \mathrm{B}, \mathrm{C},\) and \(\mathrm{D}\) left after 30 days.

Radioactive isotopes, or radioisotopes, are variants of chemical elements that have unstable nuclei and emit radiation in the form of particles or electromagnetic waves during their decay to become more stable. This natural process is known as radioactive decay, and it is a random but quantifiable event that is a key concept in nuclear physics and chemistry.

Each radioisotope has a characteristic decay pattern involving one or more types of radiation, such as alpha, beta, or gamma rays. For example, in the decay series from the exercise, isotopes A, B, and C are radioisotopes that ultimately decay to form the stable, nonradioactive isotope D. Understanding the properties of these isotopes, such as their half-lives and decay pathways, is critical for predicting their behavior and safely managing their use in various applications, from medicine to energy production.

The half-life of a radioisotope is the time required for half of the atoms in a given sample to decay. It is a measure of the rate at which the isotope decays and it varies widely among different isotopes. For instance, isotope A has a half-life of just 4.5 seconds, whereas isotope B's half-life is much longer at 15.0 days. These differences significantly affect the dynamics of the decay series and the relative amounts of each isotope present after a period.

The half-life of an isotope is a key concept in understanding radioactive decay. It is the period over which half of a given quantity of a radioactive isotope decays into another element or isotope. This half-life can range from fractions of a second to millions of years, depending on the isotope in question.

In the context of the aforementioned problem, halflives of isotopes play a crucial role in determining the number of moles of each isotope present after a certain period. Isotope A, with a half-life of only 4.5 seconds, will quickly decay to isotope B, which has a much longer half-life of 15 days, making it significantly more stable. On the other hand, isotope C has a very short half-life as well (1 second), which means it will rapidly decay to the stable isotope D.

To illustrate, even though we started with 1 mole of isotope A, after 30 days (which is significantly longer than the half-lives of A and C), it is expected that very little to no molecules of A or C would remain. This underscores how half-lives determine the persistence of isotopes over time, which is a fundamental aspect of nuclear chemistry and safety considerations.

The radioactive decay formula is a mathematical representation of the decay process of isotopes. The general equation used is: \(N = N_0 e^{-\text{λ}t}\), where \(N\) is the remaining amount of a substance after time \(t\), \(N_0\) is the initial amount of the substance, \(\lambda\) (lambda) is the decay constant, and \(e\) is the base of natural logarithms.

The decay constant \(\lambda\) is related to the half-life (\(T_{1/2}\)) of the isotope by the formula \(\lambda = \frac{\ln{2}}{T_{1/2}}\). This relation shows how the half-life and the decay constant are inversely proportional; the shorter the half-life, the larger the decay constant, meaning the substance will decay more rapidly. In the exercise, this formula allows us to calculate the exact number of moles left for each isotope after 30 days.

For instance, when applied to isotope B with a known decay constant, we can predict that approximately 0.31 moles of B would remain after 30 days, with the rest decaying into isotope C and then rapidly into D. Understanding the radioactive decay formula aids students not only in solving textbook problems but also in comprehending the fundamental physics driving the stability and transformation of elements.