Why can isotopes have different half-lives?

Isotopes have different half-lives due to variations in their nuclear stability and the nature of their radioactive decay.

Isotopes are atoms of the same element that have different numbers of neutrons in their nucleus. This variation in neutron number affects the stability of the nucleus, which in turn influences the isotope's half-life. The half-life of an isotope is the time it takes for half of the atoms in a sample to decay. It is a measure of the rate at which the isotope undergoes radioactive decay.

The stability of an atomic nucleus is determined by the balance between the forces of attraction (the strong nuclear force) and the forces of repulsion (the electrostatic force) within it. If the balance is tipped, the nucleus becomes unstable and will seek to regain stability through radioactive decay. The nature and rate of this decay process, which can involve the emission of alpha particles, beta particles, or gamma rays, vary from isotope to isotope. This is why different isotopes of the same element can have different half-lives.

For example, the isotope Carbon-14, used in radiocarbon dating, has a half-life of about 5,730 years. This is because its nucleus, with 6 protons and 8 neutrons, is unstable and decays by beta decay. On the other hand, Carbon-12, the most common isotope of carbon, is stable and does not undergo radioactive decay.

In summary, the half-life of an isotope is determined by the stability of its nucleus and the nature of its radioactive decay. Different isotopes have different numbers of neutrons, which affects their nuclear stability and hence their half-lives.