To determine how many types of bond lengths are present in the sulfate ion \( SO_3^{2-} \), we can follow these steps:
### Step 1: Determine the Valence Electrons
- Sulfur (S) has 6 valence electrons.
- Each Oxygen (O) has 6 valence electrons, and there are three Oxygens contributing a total of 18 valence electrons.
- Additionally, the \( SO_3^{2-} \) ion has 2 extra electrons due to the 2- charge.
- Total valence electrons = \( 6 + 18 + 2 = 26 \) electrons.
### Step 2: Draw the Lewis Structure
- In the Lewis structure, sulfur is the central atom and forms bonds with the three oxygen atoms.
- To satisfy the octet rule, sulfur forms double bonds with two oxygen atoms and a single bond with the third oxygen atom, which carries a negative charge.
- The structure can be represented with resonance, where the double bond can be between sulfur and any of the three oxygen atoms.
### Step 3: Identify Resonance Structures
- The \( SO_3^{2-} \) ion has three resonance structures:
1. One structure with a double bond between S and O1, and single bonds with O2 and O3.
2. Another structure with a double bond between S and O2, and single bonds with O1 and O3.
3. The last structure has a double bond between S and O3, and single bonds with O1 and O2.
### Step 4: Calculate Bond Order
- The bond order is calculated as follows:
- For each S-O bond, count the total number of bonds in all resonance structures and divide by the number of resonance structures.
- For each bond (S-O1, S-O2, S-O3), the total number of bonds in resonance structures is 4 (2 double bonds and 1 single bond for each resonance structure).
- Thus, the bond order for each S-O bond = \( \frac{4}{3} \approx 1.33 \).
### Step 5: Determine Bond Lengths
- Since the bond order for all S-O bonds is the same (1.33), the bond lengths will also be the same.
- The bond length is inversely proportional to the bond order, meaning that if the bond order is the same, the bond lengths will also be the same.
### Conclusion
- Therefore, there is **only one type of bond length** in \( SO_3^{2-} \).
