Animated Solution for Chemistry - Periodicity in Properties: The set representing the correct order of ionic radius is

Imagine you are an electron. The nucleus is like a giant magnet pulling you in, while the other electrons are like a crowd of people pushing you away. This delicate balance between the inward pull (nuclear charge) and the outward push (electron-electron repulsion and shielding) determines the size of an atom or ion.

In this problem, we are tasked with arranging four specific ions—${\text{Li}}^{+}$, ${\text{Be}}^{2+}$, ${\text{Na}}^{+}$, and ${\text{Mg}}^{2+}$—in order of their ionic radii.

To solve this, we need to look at their positions in the periodic table. They form a neat little $2\times 2$ square spanning Group 1 and Group 2, and Period 2 and Period 3.

Adding Shells

Let's first look at what happens when we move down a group.

As we go from Lithium ($\text{Li}$) to Sodium ($\text{Na}$) in Group 1, we are adding an entirely new principal quantum shell. It is like adding a new layer to an onion.

Even though the nuclear charge increases, the addition of a new shell physically places the outermost electrons much further away from the nucleus. Furthermore, the inner shells provide a strong shielding effect, canceling out much of the increased nuclear pull.

Therefore, it is a universal rule that ionic radius increases as we move down a group.

This gives us our first two solid inequalities:

Effective Nuclear Charge

Now, let's shift our focus to moving across a period, from left to right.

When we move from Lithium to Beryllium in Period 2, or from Sodium to Magnesium in Period 3, we are adding electrons to the same outer shell.

However, we are also adding protons to the nucleus. Because electrons in the same shell do not shield each other very effectively, the effective nuclear charge ($Z_{\text{eff}}$) increases significantly.

This stronger nuclear magnet pulls the entire electron cloud closer, shrinking the ion.

This gives us our next set of inequalities:

Lithium vs Magnesium

Here is where the problem gets truly interesting. We know that ${\text{Na}}^{+}$ is the largest and ${\text{Be}}^{2+}$ is the smallest. But what about the middle two: ${\text{Li}}^{+}$ and ${\text{Mg}}^{2+}$?

These two elements are placed diagonally from each other in the periodic table. Moving down increases size, but moving right decreases size. These two effects are fighting against each other!

Let's look at the raw physics. ${\text{Mg}}^{2+}$ has the electron configuration of Neon ($1s^{2}2s^{2}2p^6$), meaning it has two occupied shells. ${\text{Li}}^{+}$ has the configuration of Helium ($1s^2$), meaning it only has one occupied shell.

Intuition might tell you that the ion with more shells (${\text{Mg}}^{2+}$) should be larger. But there is a catch!

Magnesium has a massive nuclear charge of $+12$. Lithium only has a nuclear charge of $+3$. The $+12$ charge of the Magnesium nucleus is so overwhelmingly strong that it compresses its two shells into a volume even smaller than Lithium's single shell!

The actual measured radii confirm this: ${\text{Li}}^{+}$ is $0.68\text{A˚}$, while ${\text{Mg}}^{2+}$ is $0.65\text{A˚}$.

Therefore:

Assembling the Order

We now have all the pieces of the puzzle. Let's chain our inequalities together.

We know ${\text{Na}}^{+}$ is larger than ${\text{Li}}^{+}$. We know ${\text{Li}}^{+}$ is larger than ${\text{Mg}}^{2+}$. And we know ${\text{Mg}}^{2+}$ is larger than ${\text{Be}}^{2+}$.

Stringing them together, we get the final, beautiful sequence:

This perfectly matches option (b).

Understanding this interplay between shells and effective nuclear charge is the key to mastering periodic properties. It is not just about memorizing trends; it is about visualizing the tug-of-war happening inside the atom!