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 two separate `pi`-bonds
D. all are correct

Explanation

To determine which statement is correct regarding the allene molecule, \(C_3H_4\), we will analyze the structure and hybridization of the carbon atoms in the molecule step by step.

### Step 1: Understand the Structure of Allene
Allene, \(C_3H_4\), has a unique structure where it consists of three carbon atoms. The central carbon atom is bonded to two terminal carbon atoms, each of which is also bonded to two hydrogen atoms. The structure can be represented as follows:

```
H H
\ /
C
//\
C C
| |
H H
```

### Step 2: Determine the Hybridization of the Central Carbon Atom
The central carbon atom in allene is bonded to two other carbon atoms through double bonds. In this case, the central carbon atom undergoes **sp hybridization** because it forms two sigma bonds and has two unhybridized p-orbitals available for pi bonding.

### Step 3: Determine the Hybridization of the Terminal Carbon Atoms
The terminal carbon atoms are each bonded to one hydrogen atom and one carbon atom through a double bond. Therefore, these terminal carbon atoms undergo **sp² hybridization** since they form three sigma bonds (one with hydrogen and one with the central carbon) and have one unhybridized p-orbital.

### Step 4: Analyze the Orientation of the Hybrid Orbitals
The two unhybridized p-orbitals on the central carbon atom are perpendicular to each other. This arrangement allows the formation of two separate pi bonds with the terminal carbon atoms. Consequently, the planes containing the CH₂ groups (attached to the terminal carbons) are mutually perpendicular to each other.

### Step 5: Evaluate the Given Statements
Now, we can evaluate the correctness of the provided statements:
1. The central carbon atom is sp hybridized. **(Correct)**
2. The terminal carbon atoms are sp² hybridized. **(Correct)**
3. The planes containing the CH₂ groups are mutually perpendicular, allowing the formation of two separate pi bonds. **(Correct)**
4. All of the above statements are correct. **(Correct)**

### Conclusion
Since all the statements are correct, the answer is that the fourth option, "All are correct," is the correct statement regarding the allene molecule \(C_3H_4\).

---