Which one of the following statements is an incorrect?

Options

A. The fraction of the total volume occupied by the atoms in a primitive cell is 0.48
B. Molecular solids are generally volatile
C. The number of carbon atoms in an unit cell of diamond is 4
D. The number of Bravais lattices in which a crystal can be categorised is 14

Explanation

To determine which statement is incorrect among the given options, we will analyze each statement step by step.

### Step 1: Analyze the first statement
**Statement:** The fraction of total volume occupied by the atoms in a primitive cell is 0.48.

- A primitive cell (or simple cubic cell) contains 1 atom, calculated as:
- Each corner atom contributes \( \frac{1}{8} \) of its volume to the cell.
- There are 8 corners in a cube, so the total contribution is \( 8 \times \frac{1}{8} = 1 \) atom.

- The volume of the atom (assuming it is spherical) is given by:
\[ V_{atom} = \frac{4}{3} \pi r^3 \]

- The edge length \( a \) of the cube is \( 2r \), hence the volume of the cube is:
\[ V_{cube} = a^3 = (2r)^3 = 8r^3 \]

- The fraction of volume occupied by the atom is:
\[ \text{Fraction} = \frac{V_{atom}}{V_{cube}} = \frac{\frac{4}{3} \pi r^3}{8r^3} = \frac{\pi}{6} \approx 0.52 \]

**Conclusion:** The first statement is incorrect because the fraction is approximately 0.52, not 0.48.

### Step 2: Analyze the second statement
**Statement:** Molecular solids are generally volatile.

- Molecular solids are characterized by being soft, having low melting points, and being electrical insulators. They can also be volatile due to weak intermolecular forces.

**Conclusion:** The second statement is correct.

### Step 3: Analyze the third statement
**Statement:** The number of carbon atoms in a unit cell of diamond is 4.

- Diamond has a face-centered cubic (FCC) structure. The number of atoms in an FCC unit cell is calculated as:
\[ \text{Number of atoms} = 8 \times \frac{1}{8} + 6 \times \frac{1}{2} = 1 + 3 = 4 \]

- However, in diamond, carbon atoms occupy half of the tetrahedral voids in addition to the atoms at the corners and faces. There are 8 tetrahedral voids in an FCC unit cell, and diamond has 4 carbon atoms in these voids.

**Conclusion:** The third statement is incorrect because it does not account for the additional carbon atoms in the tetrahedral voids, leading to a total of 8 carbon atoms in the diamond unit cell.

### Step 4: Analyze the fourth statement
**Statement:** The number of Bravais lattices in which a crystal can be categorized is 14.

- This statement is correct as there are indeed 14 distinct Bravais lattices in three-dimensional space.

### Final Conclusion
The incorrect statements are the first and third.