A solid is formed and it has three types of atoms X, Y and Z, X forms a fcc lattice with Y atoms occupying all tetrahedral voids and Z atoms occupying half of octahedral voids. The formula of solid is :-

Options

A. `X_(4)YZ_(2)`
B. `X_(4)Y_(2)Z`
C. `XY_(2)Z_(4)`
D. `X_(2)Y_(4)Z`

Explanation

To determine the formula of the solid formed by the atoms X, Y, and Z, we follow these steps:

### Step 1: Determine the number of X atoms in the FCC lattice.
In a face-centered cubic (FCC) lattice:
- There is 1 atom at the center of the cube.
- There are 6 face-centered atoms, each contributing 1/2 to the unit cell.

Calculating the total number of X atoms:
\[ \text{Total X atoms} = 1 + 6 \times \frac{1}{2} = 1 + 3 = 4 \]
So, \( X = 4 \).

### Step 2: Determine the number of Y atoms occupying tetrahedral voids.
In an FCC lattice, there are 8 tetrahedral voids. Since Y occupies all of them:
\[ Y = 8 \]

### Step 3: Determine the number of Z atoms occupying octahedral voids.
In an FCC lattice, the number of octahedral voids is half the number of tetrahedral voids:
\[ \text{Total octahedral voids} = \frac{8}{2} = 4 \]
Since Z occupies half of the octahedral voids:
\[ Z = \frac{4}{2} = 2 \]

### Step 4: Combine the results to write the empirical formula.
Now we have:
- \( X = 4 \)
- \( Y = 8 \)
- \( Z = 2 \)

Thus, the formula can be expressed as:
\[ \text{Formula} = X_4Y_8Z_2 \]

### Step 5: Simplify the formula.
To simplify the formula, we can divide each subscript by 2:
\[ \text{Simplified formula} = X_2Y_4Z \]

### Final Answer:
The formula of the solid is \( X_2Y_4Z \).

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