A solid is formed by two elements P and Q . The element Q forms cubic close packing and atoms of P occupy one third of tetrahedral voids. The formula of the compound is

Options

A. `P_(7)Q_(3)`
B. `P_(3)Q`
C. `P_(2)Q_(3)`
D. `P_(3)Q_(2)`

Answer

To determine the formula of the compound formed by elements P and Q, we will follow these steps:

### Step 1: Understand the structure of Q
Element Q forms a cubic close packing (CCP) structure, which is also known as face-centered cubic (FCC). In this arrangement, we need to calculate the total number of atoms of Q in one unit cell.

**Hint:** Remember that in an FCC structure, the contribution of atoms from face centers and corners is crucial for calculating the total number of atoms.

### Step 2: Calculate the number of Q atoms in the unit cell
In an FCC unit cell:
- There are 6 face-centered atoms, each contributing \( \frac{1}{2} \) atom to the unit cell (since each face atom is shared between 2 unit cells).
- There are 8 corner atoms, each contributing \( \frac{1}{8} \) atom to the unit cell (since each corner atom is shared among 8 unit cells).

Calculating the total:
- Contribution from face-centered atoms: \( 6 \times \frac{1}{2} = 3 \)
- Contribution from corner atoms: \( 8 \times \frac{1}{8} = 1 \)

Total number of Q atoms in the unit cell = \( 3 + 1 = 4 \).

**Hint:** Add contributions from both face-centered and corner atoms to find the total number of atoms.

### Step 3: Determine the number of tetrahedral voids
In a cubic close packing (FCC) structure, the number of tetrahedral voids is given by \( 2n \), where \( n \) is the number of atoms present in the unit cell.

Since we have 4 atoms of Q:
- Number of tetrahedral voids = \( 2 \times 4 = 8 \).

**Hint:** The relationship between atoms and tetrahedral voids is a key concept in solid-state chemistry.

### Step 4: Calculate the number of P atoms
According to the problem, atoms of P occupy one third of the tetrahedral voids. Therefore, the number of P atoms is:
- Number of P atoms = \( \frac{1}{3} \times 8 = \frac{8}{3} \).

**Hint:** Make sure to apply the fraction correctly when determining how many voids are occupied.

### Step 5: Write the ratio of P to Q
Now we have:
- Number of P atoms = \( \frac{8}{3} \)
- Number of Q atoms = 4

To express this in a simple whole number ratio, we can write:
- Ratio of P to Q = \( \frac{8/3}{4} = \frac{8}{3} \times \frac{1}{4} = \frac{8}{12} = \frac{2}{3} \).

**Hint:** Simplifying ratios is essential to finding the empirical formula.

### Step 6: Find the empirical formula
To convert the ratio into whole numbers, we can multiply both parts of the ratio by 3:
- P = \( 8 \) (from \( \frac{8}{3} \times 3 \))
- Q = \( 12 \) (from \( 4 \times 3 \))

Now, we can simplify:
- The ratio of P to Q is \( 8:12 \), which simplifies to \( 2:3 \).

Thus, the empirical formula of the compound is \( P_2Q_3 \).

**Final Answer:** The formula of the compound is \( P_2Q_3 \).