Isotopes are two or more forms of the same element containing the same atomic number,
meaning the number of protons is the same, but the mass number is different (mass number
is the sum of protons and neutrons in an atom). For example,
12-6C (carbon-12) and 14-6C(carbon-14) are isotopes of carbon. Carbon-12 contains six protons and six electrons, therefore, the atomic number is six. It has six neutrons as well, which is why when both the protons and neutrons are added together it gives twelve, which is its mass number. On the other hand, carbon-14 also has the same atomic number, but has a different mass number. This is because carbon-14 has eight neutrons in its nucleus, but the number of protons and electrons are the same, six. So, carbon-14 has a mass number of 14 because of the eight neutrons and six protons in its nucleus. In general, isotopes have the same atomic number but different mass number because they contain different numbers of neutrons in their nucleus.

What makes an isotope stable or unstable?
A broad way to think about isotope stability is to think that stable isotopes do not undergo
spontaneous nuclear decay. In other words, stable isotopes do not emit radiation. Whereas,
unstable isotopes undergo spontaneous nuclear decay, and emit radiation to become stable.
Atoms with more than 83 protons (this number varies a little, but it is generally accepted that
atoms with atomic numbers greater than 80) cannot reach stability even with their larger
numbers of neutrons trying to stabilize them. All elements beyond bismuth on the periodic
table have a neutron-proton ratio above 1.5 : 1, and therefore are unstable. These unstable
atoms go through some sort of “decay” in order to become stable.

Neutrons are important to maintain the nucleus stability because the nucleus holds the
protons together from getting repulsed. So, having inadequate amounts of neutrons in an
atom’s nucleus contributes to the instability of the nucleus.

The principal factor for determining whether a nucleus is stable is the neutron to proton
ratio. Isotopes of elements with atomic numbers greater than 82 are all unstable. The band
of stability helps determine if the isotopes are stable or unstable.

The graphical representation of the band of stability gives the relative stability of different isotopes of different elements. The band of stability stops at element 83 because there are no known stable isotopes above it. Elements lying outside the band of stability would be too unstable to justify the time and money for an attempt to make it. Another thing that is noticed about the band of stability is that as the number of protons increases, the ratio of neutrons to protons increases. This is because more neutrons are needed to compensate for the increasing proton-proton repulsions. Isotopes occurring above and to the left of the band tend to be beta emitters because they want to lose a neutron and gain a proton, so they are reducing their neutron to proton ratio to become stable. Those lying under the band of stability tend to be alpha emitters because they have too many nucleons, they also change their neutron to proton ratio to become stable

The Odd-even rule
In the odd-even rule, when the numbers of neutrons and protons in the nucleus are both even
numbers, the isotopes tend to be far more stable than when they are both odd. Out of all the 264 stable isotopes, only 5 have odd numbers of both, whereas 157 have even numbers of both, and the rest have a mixed number.