KAHIBARO
Table of Contents
Same element, different mass
An isotope is a form of an element that has the same number of protons but a different number of neutrons. Since the number of protons determines the element, all isotopes of a given element are the same element. What changes from one isotope to another is the mass number.
If the atomic number is written as $Z$, and the number of neutrons is written as $N$, then the mass number is
$$
A = Z + N
$$
For isotopes of the same element, $Z$ stays the same, but $N$ changes, so $A$ changes too.
Important idea: Isotopes have the same atomic number $Z$ but different neutron number $N$.
$$
A = Z + N
$$
Same $Z$ means same element. Different $N$ means different isotope.
How isotopes are written
A nucleus is often written in the form
$$
{}^{A}_{Z}X
$$
Here, $X$ is the chemical symbol, $Z$ is the atomic number, and $A$ is the mass number.
For example, carbon has $Z = 6$. Its isotopes include
$$
{}^{12}_{6}\mathrm{C}, \quad {}^{13}_{6}\mathrm{C}, \quad {}^{14}_{6}\mathrm{C}
$$
All of these are carbon because each has 6 protons. They differ in neutron number:
$$
N = A - Z
$$
So for these three nuclei,
$$
N = 12 - 6 = 6
$$
$$
N = 13 - 6 = 7
$$
$$
N = 14 - 6 = 8
$$
Examples of common isotopes
Many elements have more than one isotope. Some are stable, and some are unstable. The topic of radioactive decay belongs elsewhere, so here we focus only on the meaning of isotopes themselves.
| Element | Symbol | Atomic number $Z$ | Example isotope | Neutrons $N$ |
|---|---|---|---|---|
| Hydrogen | H | 1 | ${}^{1}_{1}\mathrm{H}$ | 0 |
| Hydrogen | H | 1 | ${}^{2}_{1}\mathrm{H}$ | 1 |
| Hydrogen | H | 1 | ${}^{3}_{1}\mathrm{H}$ | 2 |
| Carbon | C | 6 | ${}^{12}_{6}\mathrm{C}$ | 6 |
| Carbon | C | 6 | ${}^{14}_{6}\mathrm{C}$ | 8 |
| Uranium | U | 92 | ${}^{235}_{92}\mathrm{U}$ | 143 |
| Uranium | U | 92 | ${}^{238}_{92}\mathrm{U}$ | 146 |
Hydrogen gives a very clear example. Its three well known isotopes are protium, deuterium, and tritium. They all have one proton, but they contain 0, 1, and 2 neutrons respectively.
Why isotopes matter
Because isotopes differ in neutron number, they have different nuclear masses and often different nuclear stability. Even though they belong to the same element, their nuclei are not identical. This matters strongly in nuclear physics, where neutron number affects binding, reactions, and stability.
In chemistry, isotopes of the same element behave very similarly because chemical behavior depends mainly on electrons, and the number of electrons in a neutral atom is tied to the proton number. In nuclear physics, however, isotopic differences are very important because nuclear behavior depends directly on the composition of the nucleus.
Rule for identifying isotopes: if two nuclei have the same $Z$ and different $A$, they are isotopes of the same element.
Equivalently, they have the same number of protons and different numbers of neutrons.
Isotopes compared with related terms
It is useful to distinguish isotopes from other families of nuclei.
| Relationship | Same $Z$? | Same $N$? | Same $A$? |
|---|---|---|---|
| Isotopes | Yes | No | No |
| Isotones | No | Yes | Usually no |
| Isobars | No | Usually no | Yes |
So isotopes are grouped by equal proton number. This is the key feature.
A simple nuclear picture
The idea of isotopes can be visualized by keeping the proton count fixed and changing only the neutron count.
This drawing shows that the element remains carbon throughout because the proton number does not change.
Reading isotope names
Isotopes are often named by the element followed by the mass number. For example,
$$
{}^{14}_{6}\mathrm{C}
$$
is called carbon 14, and
$$
{}^{235}_{92}\mathrm{U}
$$
is called uranium 235.
This naming style is simple and very common in nuclear physics.
A quick identification method
When you see a nucleus, you can decide whether two nuclei are isotopes by following a short check. First compare their atomic numbers. If the atomic numbers match, they are the same element. Then compare their mass numbers. If those differ, the nuclei are isotopes of that element.
For example, compare
$$
{}^{16}_{8}\mathrm{O} \quad \text{and} \quad {}^{18}_{8}\mathrm{O}
$$
Both have $Z = 8$, so both are oxygen. Their mass numbers are different, so they are isotopes.
But compare
$$
{}^{14}_{6}\mathrm{C} \quad \text{and} \quad {}^{14}_{7}\mathrm{N}
$$
These have the same mass number but different atomic number, so they are not isotopes.
To test for isotopes, compare proton numbers first.
Same $Z$, different $N$ or $A$ means isotopes.
Different $Z$ means different elements, not isotopes.
Summary in formula form
The defining relation for isotopes is best expressed as
$$
Z_1 = Z_2, \qquad N_1 \ne N_2
$$
which also implies
$$
A_1 \ne A_2
$$
for two isotopes of the same element.
This simple idea is one of the basic classification rules in nuclear structure.