Animated Solution for Chemistry - Periodicity in Properties: The correct sequence which shows decreasing order of the ionic radii of the elements is

When we look at the ions given in the problem—${\text{O}}^{2-}$, ${\text{F}}^{-}$, ${\text{Na}}^{+}$, ${\text{Mg}}^{2+}$, and ${\text{Al}}^{3+}$—they might seem completely different at first glance. However, they share a hidden, unifying secret. If we calculate the total number of electrons for each species, we find a fascinating pattern.

Oxygen normally has $8$ electrons, but gaining two gives it $10$. Fluorine has $9$, and gaining one gives it $10$. Sodium has $11$, but losing one leaves it with $10$. Magnesium ($12$) loses two, and Aluminum ($13$) loses three, both ending up with exactly $10$ electrons. Because they all possess the exact same number of electrons, they belong to a special category known as isoelectronic species.

Now, how do we determine which of these ions is the largest and which is the smallest? Imagine a microscopic tug-of-war. On one side, you have the electron cloud, which is identical for all these ions (exactly $10$ electrons). On the other side, you have the nucleus, pulling those electrons inward with its positively charged protons.

Since the electron cloud is constant, the size of the ion depends entirely on the strength of the nucleus. The more protons a nucleus has, the stronger its inward pull. This relationship can be mathematically expressed as:

where $r$ is the ionic radius and $Z$ is the atomic number (number of protons).

Let's line up our competitors based on their atomic numbers ($Z$): - $\text{O}=8$ - $\text{F}=9$ - $\text{Na}=11$ - $\text{Mg}=12$ - $\text{Al}=13$

Oxygen has the weakest team of protons ($8$), meaning it cannot pull its $10$ electrons very tightly. As a result, its electron cloud expands the most, making ${\text{O}}^{2-}$ the largest ion. On the opposite end of the spectrum, Aluminum has a powerhouse team of $13$ protons. It yanks those $10$ electrons inward with tremendous force, compressing the electron cloud and making ${\text{Al}}^{3+}$ the smallest ion.

Following this logic, the decreasing order of ionic radii perfectly mirrors the increasing order of their atomic numbers:

This elegant sequence matches option (d). Whenever you encounter isoelectronic species, always remember the golden rule: more protons mean a smaller size.