Q.A nuclide 1 is said to be the mirror isobar of nuclide 2 if $Z_1 = N_2$ and $Z_2 = N_1$.

Concept: Mirror Isobars and Binding Energy

1. For $^{23}_{11}\text{Na}$, $Z = 11$, $N = 23 - 11 = 12$. A mirror isobar must have $Z' = N = 12$ and $N' = Z = 11$, so its mass number is $A' = Z' + N' = 12 + 11 = 23$. Thus the mirror isobar is $^{23}_{12}\text{Mg}$.
2. Binding energy depends on the balance of proton-proton Coulomb repulsion and nuclear force. $^{23}_{11}\text{Na}$ has 11 protons and 12 neutrons; $^{23}_{12}\text{Mg}$ has 12 protons and 11 neutrons. …

Why Mirror Isobars Matter

The idea of mirror isobars is a beautiful symmetry in nuclear physics. Two nuclei are mirror isobars if the number of protons in one equals the number of neutrons in the other, and vice versa — essentially, they are "reflections" of each other across the proton-neutron line. For a given mass number $A$, if one nucleus has $(Z, N)$, the other has $(N, Z)$.

This symmetry lets us isolate the effect of the Coulomb (electrostatic) force on nuclear binding. Since the strong nuclear force is nearly charge-independent (it treats protons and neutrons almost identically), the difference in binding energy between two mirror isobars comes almost entirely from the extra Coulomb repulsion in the nucleus with more protons.

Step-by-Step Solution

1. Identify the given nuclide and its composition

We have $^{23}_{11}\text{Na}$.

• Mass number $A = 23$
• Atomic number $Z_1 = 11$ (protons)
• Neutron number $N_1 = A - Z_1 = 23 - 11 = 12$

2. Apply the mirror isobar condition

For the mirror isobar (nuclide 2), we require:

• $Z_2 = N_1 = 12$
• $N_2 = Z_1 = 11$

So the mirror isobar has $Z_2 = 12$ and $N_2 = 11$, with the same $A = 23$.

3. Identify the element

$Z = 12$ is magnesium (Mg). Therefore the mirror isobar is $^{23}_{12}\text{Mg}$.

4. Compare binding energies

Now, which one has greater binding energy?

Binding energy is the energy required to break the nucleus into its constituent nucleons. It comes from two main contributions:

• The strong nuclear force (attractive, roughly the same for both, since the total number of nucleons $A$ is identical)
• The Coulomb repulsion (repulsive, depends on $Z^2$) …

Method: Mirror Isobars — Finding the Partner Nuclide and Comparing Binding Energies

This method applies to any question involving mirror isobars: nuclide pairs with the same mass number whose proton and neutron counts are swapped.

Steps

Step 1: Find $Z$ and $N$ of the given nuclide

For a nuclide $^A_Z X$, the proton number is $Z$ (given by the notation) and the neutron number follows from $N = A - Z$.

Step 2: Apply the mirror-isobar definition to find the partner nuclide

By definition, a mirror isobar has its proton and neutron counts exchanged relative to the original:

The mass number is unchanged (both nuclei are isobars), but the new proton number $Z'$ identifies a different element — look it up on the periodic table to name the partner nuclide.

Step 3: Compare binding energies using the Coulomb-repulsion argument, not by calculating from scratch …