Assertion: `C_(3)O_(2)` has linear structure.
Reason: Each carbon atom in `C_(3)O_(2)` in `sp` -hydridized.

Options

A. If assertion is true but the reason is false
B. If assertion is false but reason is true
C. If both assertion and reason are true and the reason is the correct explanation of assertion
D. If both assertion and reason are true and the reason is not the correct explanation of assertion

Answer

To analyze the assertion and reason provided in the question regarding the compound \( C_3O_2 \), we will break down the information step by step.

### Step 1: Understanding the Assertion
The assertion states that \( C_3O_2 \) has a linear structure.

- **Explanation**: A linear structure implies that the bond angles between the atoms are 180 degrees. This is typically seen in molecules where the central atom is surrounded by two other atoms with no lone pairs, leading to a straight-line arrangement.

### Step 2: Analyzing the Reason
The reason provided states that each carbon atom in \( C_3O_2 \) is in an \( sp \) hybridized state.

- **Explanation**: Hybridization is a concept used to describe the mixing of atomic orbitals to form new hybrid orbitals. The \( sp \) hybridization occurs when one s orbital and one p orbital mix to form two equivalent \( sp \) hybrid orbitals. This type of hybridization is associated with a linear geometry and a steric number of 2 (which is the sum of sigma bonds and lone pairs).

### Step 3: Determining the Hybridization of Carbon Atoms
To determine the hybridization of the carbon atoms in \( C_3O_2 \):

1. **Count the Valence Electrons**: Each carbon atom has 4 valence electrons. For three carbon atoms, that gives us \( 3 \times 4 = 12 \) valence electrons.
2. **Identify Bonds**: In \( C_3O_2 \), we need to consider how the carbon atoms are bonded to each other and to the oxygen atoms. The structure typically involves carbon-carbon and carbon-oxygen bonds.
3. **Calculate Steric Number**: For a carbon atom involved in two sigma bonds and no lone pairs, the steric number is 2, indicating \( sp \) hybridization.

### Step 4: Confirming the Structure
Given that each carbon atom is \( sp \) hybridized and the steric number is 2, we can conclude that:

- The geometry around each carbon atom is linear, with bond angles of 180 degrees.

### Conclusion
Both the assertion that \( C_3O_2 \) has a linear structure and the reason that each carbon atom is \( sp \) hybridized are correct. The reason provided is also a correct explanation of the assertion.

### Final Answer
Both the assertion and reason are true, and the reason correctly explains the assertion.