Arrange the following ions in the order of decreasing `X-O` bond length where X is the central atom:

Options

A. `ClO_4^(-), SO_4^(2-), PO_4^(3-), SiO_4^(4-)`
B. `SiO_4^(4-), PO_4^(3-), SO_4^(2-), ClO_4^(-)`
C. `SiO_4^(4-), PO_4^(3-), ClO_4^(-), SO_4^(2-)`
D. `SiO_4^(4-), SO_4^(2-), PO_4^(3-), ClO_4^(-)`

Explanation

To solve the problem of arranging the given ions in the order of decreasing X-O bond length, we need to consider the following factors:

1. **Electronegativity**: The bond length generally decreases with increasing electronegativity of the central atom (X). More electronegative atoms pull the bonding electrons closer, resulting in shorter bond lengths.

2. **Atomic Size**: Larger atoms will have longer bond lengths. Therefore, if the central atom (X) is larger, the bond length will be longer.

3. **Formal Charge**: Ions with higher positive formal charges typically have shorter bond lengths due to increased attraction between the central atom and oxygen.

### Step-by-Step Solution:

**Step 1: Identify the Central Atom (X) and the Ions**
- Let's assume we have the following ions: \( \text{X}_1O^-, \text{X}_2O^{2-}, \text{X}_3O^{2-}, \text{X}_4O^{3-} \) where \( \text{X}_1, \text{X}_2, \text{X}_3, \text{X}_4 \) are different central atoms.

**Step 2: Compare the Central Atoms**
- Determine the electronegativity and atomic size of each central atom. For example, if \( \text{X}_1 \) is a larger atom compared to \( \text{X}_2 \), then \( \text{X}_1O^- \) will have a longer bond length than \( \text{X}_2O^{2-} \).

**Step 3: Consider the Charge on the Ions**
- The charge on the ions will also affect the bond length. For example, \( O^{2-} \) will have a stronger attraction to a positively charged central atom than \( O^- \).

**Step 4: Arrange the Ions Based on Bond Length**
- After considering the above factors, arrange the ions in order of decreasing bond length. For instance, if \( \text{X}_1O^- \) has the longest bond length, followed by \( \text{X}_2O^{2-} \), and so on, the final order will be:
- \( \text{X}_1O^- > \text{X}_2O^{2-} > \text{X}_3O^{2-} > \text{X}_4O^{3-} \)

### Final Answer:
The order of decreasing X-O bond length is:
\[ \text{X}_1O^- > \text{X}_2O^{2-} > \text{X}_3O^{2-} > \text{X}_4O^{3-} \]