KAHIBARO
Table of Contents
Different nuclei with the same mass number
In nuclear physics, isobars are nuclei that have the same mass number, but different atomic numbers. The mass number is written as $A$ and counts the total number of nucleons, meaning protons plus neutrons. The atomic number is written as $Z$ and counts only the protons.
If two nuclei are isobars, they share the same value of $A$, but their values of $Z$ are different. Since
$$
A = Z + N
$$
where $N$ is the number of neutrons, changing $Z$ while keeping $A$ fixed means that $N$ must change in the opposite way.
Isobars are nuclei with the same mass number $A$, but different atomic number $Z$.
Mathematically,
$$
A_1 = A_2, \qquad Z_1 \ne Z_2
$$
How isobars differ from other related terms
It is easy to confuse isobars with isotopes and isotones. The key is to focus on which number stays the same.
| Term | Same quantity | Different quantity |
|---|---|---|
| Isotopes | Atomic number $Z$ | Neutron number $N$ |
| Isotones | Neutron number $N$ | Atomic number $Z$ |
| Isobars | Mass number $A$ | Atomic number $Z$ and neutron number $N$ |
For isobars, the total number of nucleons is unchanged, but the balance between protons and neutrons is different.
Examples of isobars
A simple example is carbon 14 and nitrogen 14:
$$
{}^{14}_{6}\mathrm{C}, \qquad {}^{14}_{7}\mathrm{N}
$$
Both have mass number $A = 14$, so they are isobars. But they have different atomic numbers, $6$ and $7$.
Another example is
$$
{}^{40}_{18}\mathrm{Ar}, \qquad {}^{40}_{19}\mathrm{K}, \qquad {}^{40}_{20}\mathrm{Ca}
$$
All three are isobars because each has $A = 40$.
| Nucleus | $Z$ | $N$ | $A$ |
|---|---|---|---|
| ${}^{14}_{6}\mathrm{C}$ | 6 | 8 | 14 |
| ${}^{14}_{7}\mathrm{N}$ | 7 | 7 | 14 |
| ${}^{40}_{18}\mathrm{Ar}$ | 18 | 22 | 40 |
| ${}^{40}_{19}\mathrm{K}$ | 19 | 21 | 40 |
| ${}^{40}_{20}\mathrm{Ca}$ | 20 | 20 | 40 |
Visual idea of isobars
If the mass number stays fixed, moving from one isobar to another means shifting one nucleon from neutron type to proton type, or the reverse.
Each point on the dashed line has the same total
$$
A = Z + N = 5
$$
so all of them represent isobars of one another.
Why isobars matter
Isobars are important because nuclei with the same mass number can behave differently even though they contain the same total number of nucleons. Their chemical identity changes because chemical properties depend on $Z$, the number of protons. Their nuclear stability can also differ because the proton to neutron balance is different.
A set of isobars is often connected by beta decay, where one neutron can change into a proton, or one proton into a neutron. In such processes, the mass number $A$ stays the same, so the nucleus changes into one of its isobars.
For example,
$$
{}^{14}_{6}\mathrm{C} \to {}^{14}_{7}\mathrm{N} + e^- + \bar{\nu}_e
$$
Here, the nucleus changes from one isobar to another while keeping $A = 14$.
In beta decay, the mass number $A$ remains unchanged, so beta decay commonly transforms one isobar into another.
Recognizing isobars quickly
To decide whether two nuclei are isobars, check the superscript number, which is the mass number. If that number is the same for both nuclei, and the atomic numbers differ, they are isobars.
For nuclei written as
$$
{}^{A}_{Z_1}X, \qquad {}^{A}_{Z_2}Y
$$
they are isobars if
$$
Z_1 \ne Z_2
$$
Even if the element symbols are different, the matching value of $A$ tells you they belong to the same isobar family.
Summary statement
Isobars are nuclei with equal total nucleon number but different proton number. They therefore belong to different elements, have different neutron counts, and often appear in nuclear transformations where the mass number is preserved.
Key identification rule:
$$
\text{Isobars} \iff \text{same } A \text{ and different } Z
$$