The correct order of increasing radii of the ions `Br^(-), F^(-), O^(2-) and S^(2-)` is as follows

Options

A. `Br^(-)ltF^(-)ltO^(2-)ltS^(2-)`
B. `S^(2-)ltO^(2-)ltF^(-)ltBr^(-)`
C. `F^(-)ltO^(2-)ltS^(2-)ltBr^(-)`
D. `F^(-)ltBr^(-)ltO^(2-)ltS^(2-)`

Explanation

To determine the correct order of increasing radii of the ions \( \text{Br}^- \), \( \text{F}^- \), \( \text{O}^{2-} \), and \( \text{S}^{2-} \), we can follow these steps:

### Step 1: Identify the Ions and Their Charges
We have four ions to compare:
- \( \text{Br}^- \) (Bromide ion)
- \( \text{F}^- \) (Fluoride ion)
- \( \text{O}^{2-} \) (Oxide ion)
- \( \text{S}^{2-} \) (Sulfide ion)

### Step 2: Determine Isoelectronic Species
The ions \( \text{F}^- \), \( \text{O}^{2-} \), \( \text{S}^{2-} \), and \( \text{Br}^- \) can be analyzed based on their electron configurations. The ions \( \text{F}^- \), \( \text{O}^{2-} \), and \( \text{S}^{2-} \) are isoelectronic with \( \text{Ne} \) (all have 10 electrons).

### Step 3: Compare Nuclear Charges
- **Nuclear charge of \( \text{F}^- \)**: 9 (protons)
- **Nuclear charge of \( \text{O}^{2-} \)**: 8 (protons)
- **Nuclear charge of \( \text{S}^{2-} \)**: 16 (protons)
- **Nuclear charge of \( \text{Br}^- \)**: 35 (protons)

### Step 4: Analyze the Relationship Between Nuclear Charge and Ionic Radius
For isoelectronic species, the ionic radius decreases with an increase in nuclear charge. Therefore:
- \( \text{F}^- \) (9 protons) will have a larger radius than \( \text{O}^{2-} \) (8 protons).
- \( \text{O}^{2-} \) will have a larger radius than \( \text{S}^{2-} \) (16 protons) because \( \text{S}^{2-} \) has a higher nuclear charge, pulling the electrons closer.

### Step 5: Compare \( \text{Br}^- \) with Others
Since \( \text{Br}^- \) has the highest nuclear charge (35 protons), it will have the largest radius among the four ions.

### Step 6: Final Order of Increasing Ionic Radii
Based on the above analysis, the order of increasing ionic radii is:
\[ \text{F}^- < \text{O}^{2-} < \text{S}^{2-} < \text{Br}^- \]

### Conclusion
The correct order of increasing radii of the ions is:
\[ \text{F}^- < \text{O}^{2-} < \text{S}^{2-} < \text{Br}^- \]