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 the molecule C₂, we can follow these steps:

### Step 1: Determine the Total Number of Electrons
Each carbon atom has 6 electrons. Therefore, in the C₂ molecule, the total number of electrons is:
\[ \text{Total electrons} = 2 \times 6 = 12 \]

### Step 2: Understand Molecular Orbital Theory
According to molecular orbital theory, molecular orbitals are filled in order of increasing energy. For molecules with up to 14 electrons, the order of filling is as follows:
- σ(1s)
- σ*(1s)
- σ(2s)
- σ*(2s)
- σ(2p_z)
- π(2p_x) = π(2p_y)
- π*(2p_x) = π*(2p_y)

### Step 3: Fill the Molecular Orbitals
We will fill the 12 electrons into the molecular orbitals according to the order mentioned above:
1. Fill σ(1s) with 2 electrons.
2. Fill σ*(1s) with 2 electrons.
3. Fill σ(2s) with 2 electrons.
4. Fill σ*(2s) with 2 electrons.
5. Fill σ(2p_z) with 2 electrons.
6. Fill π(2p_x) and π(2p_y) with the remaining 2 electrons, following Hund's rule (one electron in each before pairing).

The filling will look like this:
- σ(1s): 2
- σ*(1s): 2
- σ(2s): 2
- σ*(2s): 2
- σ(2p_z): 2
- π(2p_x): 1
- π(2p_y): 1

### Step 4: Calculate Bond Order
The bond order can be calculated using the formula:
\[ \text{Bond Order} = \frac{(\text{Number of electrons in bonding orbitals}) - (\text{Number of electrons in anti-bonding orbitals})}{2} \]
In C₂:
- Electrons in bonding orbitals (σ(1s), σ(2s), σ(2p_z), π(2p_x), π(2p_y)) = 2 + 2 + 2 + 1 + 1 = 8
- Electrons in anti-bonding orbitals (σ*(1s), σ*(2s), π*(2p_x), π*(2p_y)) = 2 + 2 = 4

Thus, the bond order is:
\[ \text{Bond Order} = \frac{8 - 4}{2} = 2 \]

### Step 5: Identify the Type of Bonds
In C₂, since we have a bond order of 2, this indicates that there are two bonds between the two carbon atoms. The presence of unpaired electrons in the π orbitals suggests that these bonds are π bonds.

### Final Answer
The number of bonds between the two carbon atoms in C₂ is 2, and the type of bonds are π bonds.

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