If in diamond, there is a unit cell of carbon atoms as fcc and if carbon atom is `sp^(3)` hybridized, what fractions of void are occupied by carbon atom.

Options

A. 25 % tetrahedral
B. 50 % tetrahedral
C. 25 % octahedral
D. 50 % octahedral

Answer

To solve the problem, we need to analyze the structure of diamond, which is based on a face-centered cubic (FCC) lattice of carbon atoms. The carbon atoms in diamond are sp³ hybridized, leading to a tetrahedral arrangement. Let's break down the solution step by step:

### Step 1: Understand the FCC Unit Cell
In a face-centered cubic (FCC) unit cell, there are atoms located at each of the corners and the centers of each face. The number of atoms per FCC unit cell can be calculated as follows:
- Each corner atom is shared by 8 unit cells, contributing \( \frac{1}{8} \) of an atom per unit cell.
- Each face-centered atom is shared by 2 unit cells, contributing \( \frac{1}{2} \) of an atom per unit cell.

Thus, the total number of carbon atoms in an FCC unit cell is:
\[ \text{Number of atoms} = 8 \times \frac{1}{8} + 6 \times \frac{1}{2} = 1 + 3 = 4 \text{ atoms} \]

### Step 2: Determine the Number of Tetrahedral Voids
In an FCC lattice, the number of tetrahedral voids can be calculated using the formula:
\[ \text{Number of tetrahedral voids} = 2n \]
where \( n \) is the number of atoms per unit cell. Since we have 4 carbon atoms in the unit cell:
\[ \text{Number of tetrahedral voids} = 2 \times 4 = 8 \text{ tetrahedral voids} \]

### Step 3: Calculate the Occupied Tetrahedral Voids
In diamond, each carbon atom occupies one tetrahedral void. Since there are 4 carbon atoms in the unit cell, they will occupy 4 of the available tetrahedral voids:
\[ \text{Occupied tetrahedral voids} = 4 \]

### Step 4: Calculate the Fraction of Voids Occupied
To find the fraction of the tetrahedral voids that are occupied by carbon atoms, we use the formula:
\[ \text{Fraction of voids occupied} = \frac{\text{Number of occupied voids}}{\text{Total number of voids}} = \frac{4}{8} = \frac{1}{2} \]

### Conclusion
The fraction of voids occupied by carbon atoms in the diamond structure is \( \frac{1}{2} \) or 50%.

### Final Answer
Thus, the fraction of voids occupied by carbon atoms in diamond is \( \frac{1}{2} \).

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