In a crystalline solid, atoms of X form fcc packing and atoms of Y occupy all octahedral voids. If all the atoms along one body diagonal are removed, what is the simplest formula of the crystalline lattice solid?

Explanation

To determine the simplest formula of the crystalline lattice solid when atoms along one body diagonal are removed, we can follow these steps:

### Step 1: Understand the FCC Structure
In a face-centered cubic (FCC) structure, the atoms of X occupy:
- 8 corner positions, each contributing \( \frac{1}{8} \) of an atom to the unit cell.
- 6 face-centered positions, each contributing \( \frac{1}{2} \) of an atom to the unit cell.

### Step 2: Calculate the Total Number of X Atoms
The total contribution of X atoms in the FCC unit cell can be calculated as follows:
- Contribution from corners: \( 8 \times \frac{1}{8} = 1 \)
- Contribution from face centers: \( 6 \times \frac{1}{2} = 3 \)

Thus, the total number of X atoms in the unit cell is:
\[ 1 + 3 = 4 \]

### Step 3: Understand the Octahedral Voids
In the FCC structure, the octahedral voids are located at:
- 12 edge centers (each contributing \( \frac{1}{4} \) of an atom to the unit cell).
- 1 body center (contributing 1 atom).

### Step 4: Calculate the Total Number of Y Atoms
The total contribution of Y atoms occupying the octahedral voids can be calculated as follows:
- Contribution from edge centers: \( 12 \times \frac{1}{4} = 3 \)
- Contribution from body center: \( 1 \)

Thus, the total number of Y atoms in the unit cell is:
\[ 3 + 1 = 4 \]

### Step 5: Remove Atoms Along the Body Diagonal
When all atoms along one body diagonal are removed, we need to identify which atoms are affected:
- The body diagonal connects two corner atoms and the body center atom.
- Therefore, we remove:
- 2 X atoms from the corners (each corner contributes \( \frac{1}{8} \)).
- 1 Y atom from the body center.

### Step 6: Calculate Remaining Atoms After Removal
After removing the atoms:
- Remaining X atoms:
\[ 4 - 2 = 2 \]
- Remaining Y atoms:
\[ 4 - 1 = 3 \]

### Step 7: Write the Molecular Formula
The simplest formula can be written as:
\[ \text{X}_2\text{Y}_3 \]

### Step 8: Simplify the Formula
To express the formula in its simplest form, we can divide both subscripts by their greatest common divisor (which is 1 in this case):
\[ \text{X}_2\text{Y}_3 \]

### Final Answer
The simplest formula of the crystalline lattice solid is:
\[ \text{X}_2\text{Y}_3 \]

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