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The atomic number is a fundamental characteristic of an element. It tells us the number of protons present in the nucleus of an atom, which unequivocally determines the type of element. For instance, all atoms with 6 protons are classified as carbon atoms. This crucial number is not only fundamental in defining the element itself, but it's also used to organize elements in the periodic table.

Understanding the atomic number is essential when studying isotopes. Despite the isotopes having different numbers of neutrons, the atomic number remains the same across all isotopes of a given element. This is because isotopes are variants of the same element and thus have the same number of protons. So, for the carbon isotopes mentioned in the exercise, whether it is Carbon-12, Carbon-13, or Carbon-14, each atom still contains 6 protons.

While the atomic number is about proton count, the mass number dives into the collective weight of an atom. It is the sum of protons and neutrons in the nucleus. Since protons and neutrons make up most of an atom's mass, the mass number effectively measures how 'heavy' an atom is.

Why does the Mass Number Matter for Isotopes?

Let's consider our carbon example: the mass number of Carbon-12 is 12 (6 protons + 6 neutrons), Carbon-13 is 13 (6 protons + 7 neutrons), and Carbon-14 is 14 (6 protons + 8 neutrons). Although they are all carbon, they differ in mass due to the variation in the number of neutrons. This difference in mass number between isotopes influences their stability and can impact their behavior in natural processes, including the ones leading to radioactive decay.

Radioactive decay is a spontaneous process by which unstable atomic nuclei lose energy by emitting radiation. This can occur in various forms, including the emission of alpha particles, beta particles, or gamma rays. Isotopes that undergo this process are known as radioactive isotopes or radioisotopes.

Among the carbon isotopes, Carbon-14 is a famous example of a radioisotope. It is used in radiocarbon dating because it decays over time, allowing scientists to estimate the age of carbon-containing materials. The rate of decay, also known as a half-life, is unique to each radioisotope. For Carbon-14, the half-life is approximately 5,730 years, meaning it takes this length of time for half the atoms in a sample to decay. This contrast in stability and half-life among isotopes can lead to practical applications in medicine, archaeology, and several other fields.