Three atoms P, Q and R crystallize in a cubic solid lattice where P atoms are at the altemate 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 fomula 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 determine the formula of the compound formed by the atoms P, Q, and R in a cubic solid lattice, we need to analyze the positions and contributions of each type of atom in the lattice.

### Step 1: Determine the contribution of P atoms
- P atoms are located at alternate faces of the cube.
- In a cube, there are 6 faces. Since P atoms are at alternate faces, we have 3 faces with P atoms.
- The contribution of each P atom at a face is \( \frac{1}{2} \) (since each face is shared with another cube).
- Therefore, the total contribution of P atoms is:
\[ \text{Contribution of P} = 3 \times \frac{1}{2} = \frac{3}{2} \]

### Step 2: Determine the contribution of Q atoms
- Q atoms are located at \( \frac{2}{3} \) of the total corners present.
- A cube has 8 corners, so the number of Q atoms is:
\[ \text{Number of Q atoms} = \frac{2}{3} \times 8 = \frac{16}{3} \]
- Each corner atom contributes \( \frac{1}{8} \) to the unit cell, but since we are counting the number of atoms, we will keep it as \( \frac{16}{3} \).

### Step 3: Determine the contribution of R atoms
- R atoms are located at the center of the edges of the cube.
- A cube has 12 edges, and each edge contributes \( \frac{1}{4} \) (since each edge is shared between two adjacent cubes).
- Therefore, the total contribution of R atoms is:
\[ \text{Contribution of R} = 12 \times \frac{1}{4} = 3 \]

### Step 4: Combine the contributions
Now we have:
- Contribution of P = \( \frac{3}{2} \)
- Contribution of Q = \( \frac{16}{3} \)
- Contribution of R = \( 3 \)

### Step 5: Write the empirical formula
To write the empirical formula, we need to express the contributions in whole numbers. We can multiply each contribution by a common factor to eliminate the fractions. The least common multiple of the denominators (2, 3, and 1) is 6.

- For P:
\[ \frac{3}{2} \times 6 = 9 \]
- For Q:
\[ \frac{16}{3} \times 6 = 32 \]
- For R:
\[ 3 \times 6 = 18 \]

Thus, the empirical formula can be represented as:
\[ P_9Q_{32}R_{18} \]

### Step 6: Simplify the formula
To simplify the formula, we can divide all coefficients by the greatest common divisor (which is 9 in this case):
\[ P_1Q_{\frac{32}{9}}R_2 \]

However, since we need to express it in whole numbers, we can keep it as \( P_9Q_{32}R_{18} \) for clarity.

### Final Answer
The formula of the compound is \( P_3Q_{16}R_9 \).

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