Elements are defined by the number of protons in an atom's nucleus. So, for example, an atom with 6 protons must be carbon and an atom with 92 protons must be uranium.
In addition to protons, the atoms of every element (except the simplest form of hydrogen) also contain neutrons.
Isotopes occur when an element's atoms exist with different numbers of neutrons.
As a result of their having different numbers of neutrons, an element's isotopes differ in mass. Atomic mass has very little bearing on chemical reactions; therefore the chemical reactions of an element's different isotopes are almost identical.
The physical properties of atoms, however, do depend on mass. This enables isotopes to be separated from one another by methods such as diffusion and fractional distillation.
The stability of each atom's nucleus depends on the ratio of protons to neutrons. Many isotopes have a ratio of protons to neutrons that renders them unstable and, as a result, they are radioactive.
Consider carbon, which exists naturally with 6, 7 or 8 neutrons. These carbon isotopes have atomic masses of 12, 13 and 14. (atomic mass = mass protons + mass neutrons). The isotopes are called carbon-12, carbon-13 and carbon-14. Alternatively, they may be written 12C, 13C and 14C. Carbon-12 and carbon-13 are stable. Carbon-14 is unstable, decaying with a half-life of about 5,700 years. It is produced in earth's atmosphere by cosmic ray bombardment of nitrogen-14.
Uranium has three naturally occurring isotopes. These are uranium-234, uranium-235, and uranium-238. Since each atom of uranium has 92 protons, the isotopes must have 142, 143 and 146 neutrons respectively. Uranium has no stable isotopes. It is present in the earth's crust because natural rates of decay - particularly for 235U and 238U are extremely slow.
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