Carbon exhibits coordination number of six.

Explanation

To determine whether the statement "Carbon exhibits a coordination number of six" is true or false, we can analyze the electronic configuration and bonding capabilities of carbon.

### Step-by-Step Solution:

1. **Identify the Electronic Configuration of Carbon**:
- Carbon has an atomic number of 6, which gives it the electronic configuration of \(1s^2 2s^2 2p^2\).
- This means that in the outer shell (n=2), carbon has a total of 4 electrons: 2 in the 2s orbital and 2 in the 2p orbitals.

2. **Determine the Number of Unpaired Electrons**:
- In the ground state, carbon has 2 unpaired electrons in the 2p orbitals.
- The configuration can be represented as:
- 2s: ↑↓
- 2p: ↑ ↑ (two unpaired electrons in the 2p orbitals).

3. **Excitation of Electrons**:
- To form bonds, carbon can promote one of its 2s electrons to the 2p orbital, resulting in four unpaired electrons:
- 2s: ↑
- 2p: ↑ ↑ ↑ (now there are four unpaired electrons).

4. **Maximum Coordination Number**:
- With four unpaired electrons, carbon can form a maximum of four covalent bonds.
- Each bond corresponds to a coordination number of one, leading to a maximum coordination number of 4.

5. **Availability of Vacant Orbitals**:
- Carbon does not have any vacant d orbitals in its valence shell. The absence of d orbitals limits its ability to expand its coordination number beyond four.

6. **Conclusion**:
- Since carbon can only form a maximum of four bonds due to its electronic configuration and the lack of vacant orbitals, the statement that carbon exhibits a coordination number of six is **false**.

### Final Answer:
The statement "Carbon exhibits a coordination number of six" is **false**.

---