Consider the isoelectronic species, `Na^(+),Mg^(2+),F^(-)` and `O^(2-)`. The correct order of increasing length of their radii is:

Options

A. `F^(-) lt O^(2-) lt Mg^(2+) lt Na^(+)`
B. `Mg^(2+) lt Na^(+) lt F^(-) lt O^(2-)`
C. `O^(2-) lt F^(-) lt Na^(+) lt Mg^(2+)`
D. `O^(2-) lt F^(-) lt Mg^(2+) lt Na^(+)`

Explanation

To determine the correct order of increasing length of the radii for the isoelectronic species \( \text{Na}^+, \text{Mg}^{2+}, \text{F}^-, \text{O}^{2-} \), we can follow these steps:

### Step 1: Identify the Number of Electrons
All the species are isoelectronic, meaning they all have the same number of electrons. Let's calculate the number of electrons for each ion:

- **Na\(^+\)**: Sodium has an atomic number of 11. Losing 1 electron gives it \( 11 - 1 = 10 \) electrons.
- **Mg\(^{2+}\)**: Magnesium has an atomic number of 12. Losing 2 electrons gives it \( 12 - 2 = 10 \) electrons.
- **F\(^-\)**: Fluorine has an atomic number of 9. Gaining 1 electron gives it \( 9 + 1 = 10 \) electrons.
- **O\(^{2-}\)**: Oxygen has an atomic number of 8. Gaining 2 electrons gives it \( 8 + 2 = 10 \) electrons.

### Step 2: Compare the Number of Protons
Next, we need to determine the number of protons in each ion, as the effective nuclear charge will influence the ionic radius:

- **Na\(^+\)**: 11 protons
- **Mg\(^{2+}\)**: 12 protons
- **F\(^-\)**: 9 protons
- **O\(^{2-}\)**: 8 protons

### Step 3: Analyze the Effective Nuclear Charge
The effective nuclear charge (Z_eff) is the net positive charge experienced by electrons in a multi-electron atom. The more protons present, the stronger the attraction on the electrons, leading to a smaller radius.

- **Mg\(^{2+}\)**: 12 protons (strongest attraction, smallest radius)
- **Na\(^+\)**: 11 protons (next strongest attraction)
- **F\(^-\)**: 9 protons (weaker attraction)
- **O\(^{2-}\)**: 8 protons (weakest attraction, largest radius)

### Step 4: Order the Ions by Radius
Based on the analysis of the effective nuclear charge, we can order the ions from smallest to largest radius:

1. **Mg\(^{2+}\)** (smallest)
2. **Na\(^+\)**
3. **F\(^-\)**
4. **O\(^{2-}\)** (largest)

### Final Answer
The correct order of increasing length of their radii is:
\[ \text{Mg}^{2+} < \text{Na}^+ < \text{F}^- < \text{O}^{2-} \]