For each pair of isotopes listed, predict which one is less stable: (a) \({ }_{3}^{6} \mathrm{Li}\) or \({ }_{3}^{9} \mathrm{Li}\), (b) \({ }_{11}^{23} \mathrm{Na}\) or \({ }_{11}^{25} \mathrm{Na}\) (c) \({ }_{20}^{48} \mathrm{Ca}\) or \({ }_{21}^{48} \mathrm{Sc}\).

The neutron to proton ratio, often abbreviated as n/p ratio, is a critical factor in determining the stability of an isotope. An isotope consists of atoms with the same number of protons but different numbers of neutrons. A balanced n/p ratio is pivotal because it influences the forces at play within an atomic nucleus.

For lighter elements (up to calcium, with an atomic number of 20), isotopes with a n/p ratio close to 1 tend to be more stable. This is because the nuclear forces that bind protons and neutrons together in the nucleus are most effective when the numbers are balanced. However, as elements become heavier, they require a higher proportion of neutrons to maintain nuclear stability. This need arises due to the greater repulsion between the increased number of protons in heavier elements. Excessive or insufficient neutrons in relation to protons can lead to instability, making the isotope susceptible to radioactive decay.

Nuclear stability is the propensity of an atomic nucleus to remain unchanged or undecayed over time. A stable nucleus manages to balance the repulsive electrostatic forces between the positively charged protons and the attractive nuclear forces that hold the nucleus together.

Several factors contribute to nuclear stability, including the neutron to proton ratio discussed earlier. Other factors include the energy levels within the nucleus and the existence of magic numbers – specific numbers of nucleons (protons and neutrons) that render a nucleus particularly stable.

What makes a nucleus unstable?

If the forces within the nucleus are imbalanced, if there are too many or too few neutrons, or if the nucleons are not in an 'optimal' arrangement, instability occurs. Unstable nuclei will seek out stability through various forms of radioactive decay, such as alpha emission, beta decay, or gamma radiation, which results in the adjustment of the number of protons, neutrons, or the energy state within the nucleus.

Isotopes can broadly be classified as stable or unstable based on their ability, or lack thereof, to maintain an unchanged nucleus over time. Stable isotopes do not undergo radioactive decay and exist in nature without changing for exceedingly long periods. These isotopes serve as valuable tools in various scientific fields, such as archaeology for carbon dating and medicine for diagnostics.

On the other hand, unstable isotopes are those that do not have the balance required for longevity and spontaneously undergo radioactive decay in an attempt to achieve stability. This process can emit radiation and transmute the isotope into a different element or isotope altogether. The rate at which this decay happens is characteristic of each unstable isotope and is quantified by its half-life – the time taken for half of the unstable isotopes in a sample to decay. Understanding the distinction between stable and unstable isotopes, along with their decay patterns, is invaluable for applications ranging from nuclear energy production to medical treatments with radioisotopes.