Three atoms P, Q and R crystallize in a cubic solid lattice where P atoms are at the alternate faces, R atoms are at the centre of edges and Q atoms are at the 2/3 rd of the total corners present, hence the formula of the compound is

Options

A. `P_2 Q_9 R_2`
B. `P_3 Q_2 R_9`
C. `P_3 Q_4 R_1`
D. `P_2 Q_3 R_4`

Answer

To solve the problem of determining the formula of the compound formed by atoms P, Q, and R in a cubic solid lattice, we will follow these steps:

### Step 1: Calculate the number of P atoms per unit cell
- P atoms are located at alternate faces of the cubic unit cell.
- A cube has 2 faces that can be considered alternate (one face and its opposite).
- Each face atom contributes \( \frac{1}{2} \) to the unit cell because it is shared with another unit cell.

**Calculation:**
\[ \text{Number of P atoms per unit cell} = 2 \times \frac{1}{2} = 1 \]

### Step 2: Calculate the number of R atoms per unit cell
- R atoms are located at the center of the edges of the cube.
- A cube has 12 edges, and each edge atom is shared by 4 unit cells.

**Calculation:**
\[ \text{Number of R atoms per unit cell} = 12 \times \frac{1}{4} = 3 \]

### Step 3: Calculate the number of Q atoms per unit cell
- Q atoms are present at \( \frac{2}{3} \) of the total corners.
- A cube has 8 corners, so \( \frac{2}{3} \) of 8 corners gives us:
\[ \text{Number of Q atoms} = \frac{2}{3} \times 8 = \frac{16}{3} \]
- Since corner atoms are shared by 8 unit cells, the contribution per unit cell is \( \frac{1}{8} \).

**Calculation:**
\[ \text{Contribution of Q atoms per unit cell} = \frac{16}{3} \times \frac{1}{8} = \frac{2}{3} \]

### Step 4: Determine the ratio of P, Q, and R
Now we have:
- P = 1
- Q = \( \frac{2}{3} \)
- R = 3

To find a common ratio, we can multiply each by 3 to eliminate the fraction:
\[ \text{P} : \text{Q} : \text{R} = 3 : 2 : 9 \]

### Step 5: Write the empirical formula
From the ratio we derived:
- P = 3
- Q = 2
- R = 9

Thus, the empirical formula of the compound is:
\[ \text{P}_3\text{Q}_2\text{R}_9 \]

### Final Answer
The formula of the compound is \( \text{P}_3\text{Q}_2\text{R}_9 \).

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