Which of the following statements is correct in the context of the allene molecule, `C_(3)H_(4)` ?

Options

A. The central carbon is sp hybridized
B. The terminal carbon atoms are `sp^(2)` hybridized
C. The planes containing the `CH_(2)` groups are mutually perpendicular to permit the formations of two separate `pi`-bonds
D. All are correct

Explanation

To determine which statements are correct regarding the allene molecule, \( C_3H_4 \), we need to analyze its structure and hybridization.

### Step-by-Step Solution:

1. **Identify the Structure of Allene (\( C_3H_4 \))**:
- The allene molecule consists of three carbon atoms and four hydrogen atoms. The structure can be represented as:
```
H H
\ /
C = C
/ \
H C
/ \
H H
```
- In this structure, the central carbon atom is double bonded to the two terminal carbon atoms.

2. **Determine the Hybridization of Carbon Atoms**:
- **Central Carbon Atom**: The central carbon atom is bonded to two other carbon atoms via double bonds. It forms two sigma bonds and has no lone pairs. Therefore, it is **sp hybridized**.
- **Terminal Carbon Atoms**: Each terminal carbon atom is bonded to the central carbon atom via a double bond and to two hydrogen atoms via single bonds. Each terminal carbon forms three sigma bonds and has no lone pairs, making them **sp² hybridized**.

3. **Analyze the Orientation of the CH₂ Groups**:
- The two terminal CH₂ groups (attached to the terminal carbon atoms) lie in one plane, while the central carbon atom, which is sp hybridized, has two unhybridized p-orbitals that are perpendicular to each other. This results in the two planes containing the CH₂ groups being mutually perpendicular.

4. **Evaluate the Statements**:
- **Statement 1**: The central carbon is sp hybridized. **(True)**
- **Statement 2**: The terminal carbon atoms are sp² hybridized. **(True)**
- **Statement 3**: The planes containing the CH₂ groups are mutually perpendicular to permit the formations of two separate π-bonds. **(True)**

5. **Conclusion**:
- Since all the statements about the allene molecule are correct, the correct option is that **all statements are correct**.

### Final Answer:
**Option 4: All statements are correct.**