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The atomic number is a fundamental feature that defines each element in the periodic table. It represents the number of protons found in the nucleus of an atom. Given that an element's identity is determined by its proton count, all isotopes of the same element share the same atomic number.

For example, carbon has an atomic number of 6, meaning every isotope of carbon - whether it's Carbon-12, Carbon-13, or the radioactive Carbon-14 - contains exactly 6 protons. The atomic number is essential in chemistry because it dictates the element's placement in the periodic table and its chemical behavior.

Isotopes of the same element vary specifically in the number of neutrons in their atomic nuclei. The neutron count contributes to the atomic mass of an isotope, which is expressed as an integer for most stable elements and commonly known as the mass number.

To calculate the number of neutrons in an isotope, subtract the atomic number from the mass number. For instance, the isotope nitrogen-15 has an atomic number of 7 (protons) and a mass number of 15, resulting in a neutron count of 8. This variation in neutron count among isotopes causes differences in mass and other physical properties but does not affect the chemical properties.

While isotopes of the same element do have identical chemical properties due to having the same number of electrons in their neutral state, their physical properties can differ dramatically. This is due to the difference in their mass, which results from varying neutron counts.

Physical properties such as density, rate of diffusion, and nuclear stability can all vary among isotopes. For example, heavy water (D₂O), which contains the isotope deuterium (hydrogen with one neutron), has a slightly higher boiling point than regular water (H₂O). Isotopic variation can be incredibly useful in certain applications, like tracing chemical pathways or dating archaeological finds.

Radioactive isotopes, or radioisotopes, are isotopes that have unstable nuclei and decay over time, emitting radiation in the form of alpha particles, beta particles, or gamma rays. This property of radioisotopes can make them invaluable in many fields, including medicine, where they're used for imaging and treatment, and in archeology, where carbon-14 dating is a tool for determining the age of artifacts.

Radioactive isotopes differ from stable isotopes not in their chemical nature but in their nuclear stability. The decay of a radioactive isotope can lead to the formation of a different element altogether as it loses protons during alpha and beta decays. Understanding radioactive isotopes and their half-lives - the time it takes for half of a sample to decay - is crucial when handling or utilizing them for scientific or practical purposes.