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Isotopes are atoms of the same element that have different numbers of neutrons in their nuclei while sharing the same number of protons. The number of protons determines the element's identity, while the number of neutrons affects the isotope's atomic mass.

Atoms are composed of a dense central nucleus containing protons and neutrons, surrounded by orbiting electrons. Protons have a positive charge, while neutrons are neutral (no charge). Since protons are positively charged, they repel each other due to electromagnetic forces. Neutrons, on the other hand, help to stabilize the nucleus by adding a strong nuclear force that counteracts the electrostatic repulsion between protons.

Isotopes arise because the number of neutrons in the nucleus can vary without significantly affecting the atom's chemical behavior. Elements are defined by their number of protons, which determines their atomic number. For example, carbon always has six protons, so any atom with six protons is carbon. However, carbon isotopes can have different numbers of neutrons. Carbon-12 (written as 12C) has six protons and six neutrons, carbon-13 (13C) has six protons and seven neutrons, and carbon-14 (14C) has six protons and eight neutrons.

The differing number of neutrons among isotopes affects their atomic mass. Atomic mass is calculated by summing the masses of protons and neutrons in the nucleus. Since neutrons and protons have slightly different masses, isotopes of the same element have different atomic masses. For example, carbon-12 has an atomic mass of approximately 12 atomic mass units (amu), carbon-13 has an atomic mass of approximately 13 amu, and carbon-14 has an atomic mass of approximately 14 amu.

Isotopes can exhibit different physical properties due to their varying atomic masses. For instance, carbon-12 and carbon-13 are stable isotopes, while carbon-14 is radioactive and undergoes radioactive decay over time. Isotopes with unstable nuclei can be used for radiometric dating or medical imaging purposes.

In summary, isotopes are variants of the same element with different numbers of neutrons. While the number of protons determines the element's identity, the number of neutrons affects the isotope's atomic mass and may give rise to distinct physical properties or behaviors.

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