`Li^(+),Mg^(2+),K^(+),Al^(3+)` (Arrange in increasing order of radii)

Answer

To arrange the ions \( \text{Li}^+, \text{Mg}^{2+}, \text{K}^+, \text{Al}^{3+} \) in increasing order of their radii, we need to consider the following factors:

1. **Charge of the Ions**: The greater the positive charge on the ion, the smaller the radius. This is because a higher positive charge pulls the electrons closer to the nucleus, resulting in a smaller radius.

2. **Position in the Periodic Table**: Ions in the same group will have increasing radii as you move down the group due to the addition of electron shells.

Now, let's analyze each ion:

- **\( \text{Li}^+ \)**: Lithium has a +1 charge and is in the second period of the periodic table.

- **\( \text{Mg}^{2+} \)**: Magnesium has a +2 charge and is in the third period. The +2 charge means it has a smaller radius than \( \text{Li}^+ \).

- **\( \text{K}^+ \)**: Potassium has a +1 charge and is in the fourth period. Being lower in the group, it has a larger radius than \( \text{Li}^+ \).

- **\( \text{Al}^{3+} \)**: Aluminum has a +3 charge and is also in the third period. The +3 charge means it has a smaller radius than \( \text{Mg}^{2+} \).

Now, let's summarize the relative sizes based on their charges and positions:

- \( \text{Al}^{3+} \) < \( \text{Mg}^{2+} \) < \( \text{Li}^+ \) < \( \text{K}^+ \)

Thus, the increasing order of radii is:

**Increasing order of radii:** \( \text{Al}^{3+} < \text{Mg}^{2+} < \text{Li}^+ < \text{K}^+ \)