How many type of bond lengths are there in `SO_(3) ^(2-)` ?

Options

A. one
B. two
C. three
D. four

Explanation

To determine how many types of bond lengths are present in the sulfate ion, \( SO_3^{2-} \), we can follow these steps:

### Step 1: Draw the Lewis Structure
The first step is to draw the Lewis structure for \( SO_3^{2-} \). The sulfur atom (S) is the central atom, surrounded by three oxygen atoms (O). The overall charge of -2 indicates that there are two additional electrons.

- Sulfur has 6 valence electrons.
- Each oxygen has 6 valence electrons, and there are three oxygen atoms, contributing a total of 18 electrons.
- The two extra electrons from the -2 charge add 2 more electrons.

Total valence electrons = \( 6 + 18 + 2 = 26 \) electrons.

### Step 2: Assign Bonds and Lone Pairs
In the Lewis structure, sulfur forms a double bond with one oxygen atom and single bonds with the other two oxygen atoms. The remaining electrons are placed as lone pairs on the oxygen atoms.

### Step 3: Identify Resonance Structures
The next step is to identify the resonance structures. In \( SO_3^{2-} \), the double bond can shift among the three oxygen atoms. This means we can draw multiple resonance structures where the position of the double bond changes.

1. **Resonance Structure 1**: S=O (double bond) with O and O (single bonds).
2. **Resonance Structure 2**: S=O (double bond) with O and O (single bonds) shifted.
3. **Resonance Structure 3**: S=O (double bond) with O and O (single bonds) shifted again.

### Step 4: Calculate Bond Order
The bond order for each sulfur-oxygen bond can be calculated as follows:

\[ \text{Bond Order} = \frac{\text{Total number of bonds in all resonance structures}}{\text{Number of resonance structures}} \]

In this case, each resonance structure contributes to the bond order. There are three resonance structures, and each bond contributes to the overall bond order.

- Total bonds = 4 (1 double bond + 2 single bonds in each structure)
- Number of resonance structures = 3

Thus, the bond order for each \( SO \) bond is:

\[ \text{Bond Order} = \frac{4}{3} \approx 1.33 \]

### Step 5: Determine Bond Lengths
Since all sulfur-oxygen bonds have the same bond order (1.33), they will also have the same bond length. The bond length is inversely related to bond order; thus, if the bond order is the same, the bond lengths will also be the same.

### Conclusion
In \( SO_3^{2-} \), there is only **one type of bond length** present because all sulfur-oxygen bonds are resonance stabilized and have the same bond order.

---