Which of the following statements are correct about `CO_(3)^(2-)`?

Options

A. The hybridisation of central atom in `sp^(3)`
B. Its resonance structure has one C-O single bond and two C=O double bonds
C. The average formal charge on each oxygen atom is `0.67` units
D. All `C-O` bond lengths are equal.

Explanation

To determine which statements are correct about the carbonate ion \( CO_3^{2-} \), we will analyze each statement step-by-step.

### Step 1: Determine the Hybridization of the Central Atom (Carbon)
1. **Identify the valence electrons of carbon**: Carbon has 4 valence electrons.
2. **Count the total number of electrons in the carbonate ion**: The carbonate ion has a total charge of -2, and there are 3 oxygen atoms (each contributing 6 valence electrons). Therefore, the total number of valence electrons is:
\[ \text{Total valence electrons} = 4 \, (\text{from C}) + 3 \times 6 \, (\text{from O}) + 2 \, (\text{from charge}) = 4 + 18 + 2 = 24 \]
3. **Calculate the hybridization**: The hybridization can be determined using the formula:
\[ \text{Hybridization} = \frac{\text{Number of valence electrons}}{2} + \text{Number of lone pairs} \]
In the case of \( CO_3^{2-} \), carbon forms three sigma bonds (with three oxygen atoms) and has no lone pairs. Thus:
\[ \text{Hybridization} = \frac{6}{2} = 3 \quad \text{(indicating sp}^2\text{ hybridization)} \]
**Conclusion**: The hybridization of the central atom (carbon) is \( sp^2 \), so the first statement is incorrect.

### Step 2: Analyze the Resonance Structures
1. **Draw the resonance structures**: The carbonate ion has three resonance structures, each showing one double bond and two single bonds between carbon and oxygen. The structures can be represented as:
- Structure 1: \( O=C-O^- \) with a double bond to one oxygen and single bonds to the other two.
- Structure 2: \( O^- - C=O \) with a double bond to another oxygen.
- Structure 3: \( O=C-O^- \) with the double bond to the third oxygen.
2. **Evaluate the statement about bond types**: The statement claims there is one single bond and two double bonds in the resonance structure. This is incorrect as each resonance structure has one double bond and two single bonds.

### Step 3: Calculate the Average Formal Charge on Each Oxygen Atom
1. **Calculate the formal charge**: The formal charge can be calculated using the formula:
\[ \text{Formal Charge} = \text{Valence electrons} - \left( \text{Non-bonding electrons} + \frac{1}{2} \times \text{Bonding electrons} \right) \]
For the carbonate ion, the total charge is -2, distributed among three oxygen atoms:
\[ \text{Average formal charge on each oxygen} = \frac{-2}{3} \approx -0.67 \]
**Conclusion**: This statement is correct.

### Step 4: Determine the Bond Lengths
1. **Analyze bond lengths**: Due to resonance, all \( C-O \) bonds in the carbonate ion are equivalent. The bond order is calculated as:
\[ \text{Bond order} = \frac{\text{Number of bonds}}{\text{Number of resonance structures}} = \frac{4}{3} \, \text{(since there are 4 bonds in total across 3 structures)} \]
Since all bonds are equivalent, the statement that all \( C-O \) bond lengths are equal is correct.

### Final Conclusion
The correct statements about \( CO_3^{2-} \) are:
- The average formal charge on each oxygen atom is approximately -0.67 (correct).
- All \( C-O \) bond lengths are equal (correct).

### Summary of Correct Statements
- Statement 3: Average formal charge on each oxygen atom is 0.67 units (correct).
- Statement 4: All \( C-O \) bond lengths are equal (correct).