A 0.0100 -g sample of a radioactive isotope with a half-life of \(1.3 \times 10^{9} \mathrm{yr}\) decays at the rate of \(2.9 \times\) \(10^{4} \mathrm{dpm} .\) Calculate the molar mass of the isotope.

Radioactive isotope decay is a process by which an unstable atomic nucleus loses energy by emitting radiation, such as an alpha particle, beta particle, or gamma ray. This loss of energy results in the conversion of an atom of one type (a radioactive isotope) into an atom of a different element.
Through this process, the radioactive isotope undergoes a transformation over time, changing into more stable forms until it eventually becomes a stable isotope. Different isotopes decay at different rates, which is why understanding the concept of half-life is essential.
When students confront problems involving radioactive decay, they must be able to gauge the decay rate of the isotope in question, which leads directly into the practical uses of such measures for determining the age of materials or the diagnostic use in medicine.

Half-life is a term that describes the time required for one-half of a sample of a radioactive isotope to decay. It effectively measures the rate at which the atoms of the radioactive isotope are transformed into another element or isotope.
In our exercise, the half-life was given as 1.3 billion years, indicating that the isotope decays relatively slowly. If a student were to understand half-life properly, they would know it informs us not only about the time scale of decay but also about the stability of the isotope. A longer half-life would indicate a more stable isotope, and vice versa.

Exercise Connection

By knowing the half-life, students can derive other important values, such as the decay constant, which is crucial for determining the molar mass of the radioactive isotope—as showcased in the exercise.

The decay constant, often symbolized by \(\lambda\), represents the probability of a single atom decaying per unit time and is related to the half-life of the isotope. It's a fixed value for each radioactive isotope and can be calculated using the formula \(\lambda = \frac{\ln(2)}{t_{1/2}}\).
In practical terms, the decay constant allows us to quantify the rate of decay in more precise terms than the half-life. This interpolation between the understanding of half-life and the actual decay occurring in a sample over time provides a bridge for students to grasp more complex decay calculations.

Applying to the Exercise

Understanding the decay constant is critical for the exercise at hand. Among other uses, it enables students to calculate the number of decaying atoms at any moment, which is a stepping stone to determining the sample's molar mass.

Avogadro's number, approximately equal to \(6.022 \times 10^{23}\) entities per mole, is a fundamental constant in chemistry. It denotes the number of constituent particles, usually atoms or molecules, that are contained in one mole of a substance. The mole is a standard unit of measurement for the amount of substance in the International System of Units (SI).
Avogadro's number is indispensable when converting from countable units, like atoms or molecules, to measurable units like grams or moles—a common requirement in chemistry problems.

Link to the Exercise

In the exercise provided, the use of Avogadro's number plays a crucial role in translating the number of atoms in the radioactive sample into moles. This step is pivotal for the subsequent calculation of the molar mass, which brings together all previous concepts from half-life and decay constant to provide the solution.