Which one of each pair of nuclides would you expect to have the longer half- life? $$ \begin{array}{l}{\text { a. Cs} - 113 \text { or Cs} -125} \\ {\text { b. Fe}-62 \text { or Fe} -70}\end{array} $$

Radioactive decay is a fundamental process in which an unstable atomic nucleus loses energy by emitting radiation. This spontaneous transformation can result in the change of an element into a different one or a different isotope of the same element. There are several types of radioactive decay, including alpha decay, beta decay, and gamma decay.

The rate at which a radioactive isotope decays is characterized by its half-life, which is the time it takes for half of a given amount of the isotope to decay. This is a constant for any given isotope and can range from fractions of a second to billions of years. In terms of nuclear stability, isotopes with shorter half-lives are considered to be less stable because they decay more rapidly.

The neutron to proton ratio is critical in determining the stability of an isotope. Atoms strive for a balanced ratio between neutrons and protons, which contributes to the stability of the nucleus. The stability of light nuclei is greatest when this ratio is close to 1:1. As atoms increase in size, they require a higher ratio of neutrons to protons to remain stable.

For example, isotopes of smaller atoms, such as carbon or oxygen, have a stable ratio near 1:1, while heavier atoms like cesium will have stable neutron to proton ratios closer to 1.5. This ratio informs us about the likelihood of an isotope to undergo radioactive decay—deviations from the stable range imply a tendency towards decay to achieve stability.

The band of stability refers to the region on a graph where stable nuclei lie when plotting the number of neutrons versus the number of protons. Nuclei within this band have an optimal neutron-to-proton ratio and are less likely to undergo radioactive decay. Elements outside this band are radioactive and will decay into more stable forms.

This band conveys that light elements up to calcium (with proton number 20) tend to have a ratio of 1:1. The slope of the band increases gradually for heavier elements, which means these elements require proportionally more neutrons than protons to be stable. Understanding where a nuclide sits relative to this band can give insights into its stability and predict its half-life, as demonstrated in the solution where Cs-125 and Fe-62 are closer to the band of stability, therefore likely to have longer half-lives.