Titanium crystallizes in a face centred cubic lattice. It reacts with carbon or hydrogen interstitially by allowing atoms of these elements to occupy holes in the host lattice. Hydrogen occupies tetrahedral holes but carbon occupies octahedral holes the formula of titanium carbide and hydride are

Options

A. `TiC_(2), TiH_(4)`
B. `TiC, TiH_(2)`
C. `Ti_(3)C, TiH_(2)`
D. `TiC_(2), TiH`

Answer

To determine the formulas of titanium carbide and titanium hydride, we need to analyze the face-centered cubic (FCC) lattice of titanium and the positions of carbon and hydrogen in the lattice.

### Step-by-Step Solution:

1. **Understanding the FCC Lattice**:
- Titanium crystallizes in a face-centered cubic (FCC) lattice.
- In an FCC unit cell, there are:
- 8 corner atoms, each shared by 8 unit cells (contributing \( \frac{1}{8} \times 8 = 1 \) atom).
- 6 face-centered atoms, each shared by 2 unit cells (contributing \( \frac{1}{2} \times 6 = 3 \) atoms).
- Therefore, the total number of titanium atoms per unit cell is:
\[ 1 + 3 = 4 \text{ titanium atoms} \]

2. **Calculating Tetrahedral and Octahedral Voids**:
- In an FCC lattice, the number of tetrahedral voids is twice the number of atoms per unit cell:
\[ \text{Number of tetrahedral voids} = 2 \times 4 = 8 \]
- The number of octahedral voids is equal to the number of atoms per unit cell:
\[ \text{Number of octahedral voids} = 4 \]

3. **Determining the Formula for Titanium Carbide (TiC)**:
- Carbon occupies the octahedral voids. Since there are 4 octahedral voids, we can place 4 carbon atoms in these voids.
- The ratio of titanium to carbon in the unit cell is:
\[ \text{Ti: C} = 4:4 = 1:1 \]
- Therefore, the formula for titanium carbide is:
\[ \text{TiC} \]

4. **Determining the Formula for Titanium Hydride (TiH₂)**:
- Hydrogen occupies the tetrahedral voids. Since there are 8 tetrahedral voids, we can place 8 hydrogen atoms in these voids.
- The ratio of titanium to hydrogen in the unit cell is:
\[ \text{Ti: H} = 4:8 = 1:2 \]
- Therefore, the formula for titanium hydride is:
\[ \text{TiH}_2 \]

### Final Formulas:
- The formula for titanium carbide (TiC) is **TiC**.
- The formula for titanium hydride (TiH₂) is **TiH₂**.