Given specimens neon-24 \(\left(t_{1 / 2}=3.38 \mathrm{min}\right)\) and bismuth- \(211\left(t_{1 / 2}=2.14 \mathrm{min}\right)\) of equal mass, which one would have greater activity and why?

The half-life of a radioactive substance is a fundamental concept that helps us understand the rate at which radioactive decay occurs. Essentially, it is the time taken for half of the radioactive nuclei in a sample to transform into other elements or isotopes through the process of decay. Half-life is a statistical measure and it remains constant regardless of the initial amount of the substance.

For example, if we have a sample of a radioactive isotope with a half-life of 4 years, after 4 years, only half of the original radioactive atoms would remain unchanged. After another 4 years (which makes a total of 8 years), only a quarter of the original number would remain, and so on. This exponential decay continues until the material has fully transformed. It's important to note that the half-life is an inherent property of each radioactive isotope, thus can vary greatly—from fractions of a second to millions of years—depending on the isotope in question.

Radioactive activity is a quantitative measurement of the rate at which a radioactive substance decays. It is typically measured in units of becquerels (Bq) or curies (Ci), with one becquerel being defined as one decay per second. The activity of a substance directly depends on the number of radioactive nuclei present and their half-life.

Mathematically, the relationship between activity (\textbf{A}), the number of radioactive nuclei (\textbf{N}), and the decay constant (\textbf{λ}), which is related to the half-life, is expressed by the equation \( A = λN \). Because \( λ \) is inversely proportional to \( t_{1/2} \) (the half-life), the activity increases as the half-life decreases. This means that shorter half-lives result in higher activity, assuming the same number of radioactive nuclei. This concept explains why a substance with a shorter half-life will be more active and thus, decay more rapidly.

Radioactive nuclei are the unstable atoms that constitute a radioactive substance. These nuclei have an imbalance of protons and neutrons which makes them inherently unstable. To achieve stability, these nuclei undergo radioactive decay, where they emit particles such as alpha particles, beta particles, or gamma rays, and transform into different nuclei.

It's the number of these unstable nuclei that determines the overall activity of the radioactive sample. The decay process is random for individual nuclei, but large populations of radioactive atoms decay at a rate determined by their half-life. In practice, the decay rate of a radioactive sample is predictable and allows us to use radioactive materials for a variety of applications, from medical treatments to dating archaeological finds.