The number and type of bonds between two carbon atoms in `C_(2)` are:

Options

A. one sigma `(sigma)` and one pi `(pi)` bonds
B. two pi `(pi)` bonds
C. one sigma `(sigma)` and one half pi `(pi)` bonds
D. one sigma `(sigma)` bond

Explanation

To determine the number and type of bonds between two carbon atoms in \( C_2 \), we can analyze the molecular orbital (MO) configuration of the molecule. Here’s a step-by-step breakdown of the solution:

### Step 1: Determine the Total Number of Electrons
Each carbon atom has 6 electrons. Therefore, in \( C_2 \), the total number of electrons is:
\[ 6 \text{ (from one carbon)} \times 2 \text{ (two carbons)} = 12 \text{ electrons} \]

### Step 2: Fill the Molecular Orbitals
According to molecular orbital theory, the molecular orbitals for diatomic molecules like \( C_2 \) are filled in the following order:
1. \( \sigma_{1s} \)
2. \( \sigma_{1s}^* \) (anti-bonding)
3. \( \sigma_{2s} \)
4. \( \sigma_{2s}^* \) (anti-bonding)
5. \( \sigma_{2p_z} \)
6. \( \pi_{2p_x} \) and \( \pi_{2p_y} \) (degenerate orbitals)
7. \( \pi_{2p_x}^* \) and \( \pi_{2p_y}^* \) (anti-bonding degenerate orbitals)

For \( C_2 \), we fill the molecular orbitals with the 12 electrons:
- \( \sigma_{1s}^2 \)
- \( \sigma_{1s}^* \) (0 electrons)
- \( \sigma_{2s}^2 \)
- \( \sigma_{2s}^* \) (0 electrons)
- \( \sigma_{2p_z}^2 \) (0 electrons)
- \( \pi_{2p_x}^2 \) and \( \pi_{2p_y}^2 \) (4 electrons total)

### Step 3: Count the Bonds
- The \( \sigma_{1s} \) and \( \sigma_{2s} \) orbitals are fully filled, but their anti-bonding counterparts \( \sigma_{1s}^* \) and \( \sigma_{2s}^* \) have no electrons, meaning they do not contribute to bonding.
- The \( \pi_{2p_x} \) and \( \pi_{2p_y} \) orbitals are filled with 2 electrons each, contributing to bonding.

### Conclusion
In \( C_2 \), there are:
- No sigma bonds formed (as the bonding and anti-bonding orbitals cancel each other out).
- Two pi bonds formed (from the filled \( \pi_{2p_x} \) and \( \pi_{2p_y} \) orbitals).

Thus, the number and type of bonds between the two carbon atoms in \( C_2 \) are:
\[ \text{2 pi bonds} \]