In the crystalline solids the smallest repeating part in the lattice is known as unit cell. The unit cells are described as simple (points at all corners), body centred (points at all the corners and it the centre), face centred (points at all the corners and centre of all faces), and end centred (points at all hte corners and centres of two opposite and faces) unit cells. In two common tyupes of packing ccp and hcp, 26% of space is left unoccupied in the form of interstitial sites. For the stable ionic crystalline structures, there is difinite radius ratio limit for a cation to fit perfectly in the lattice of anions, called radius ratio rule. This also defines the coordination number of an ion, which is the number of nearest neighbours of opposite charges. This depeds upon the ratio of radii of two types of ions, `r_(+)//r_(-)`. This ratio for coordination numbers 3,4,6,and 8 is respectively 0.155 - 0.225, 0.225 - 0.414, 0.414 - 0.732 and 0.732 - 1 respectively.
In a cubic lattice of XYZ, X atoms are present at all corners except one corner which is occupied by Y atoms. Z atoms are present at face centres . The formula of the compound is

Options

A. `X_8YZ_(24)`
B. `XYZ_3`
C. `X_7Y_(24)Z`
D. `X_7 YZ_(24)`

Explanation

To find the formula of the compound formed by the atoms in the cubic lattice described in the question, we will follow these steps:

### Step 1: Identify the positions of the atoms in the unit cell
- X atoms are present at all corners except one corner.
- Y atom occupies the one corner that is not occupied by X.
- Z atoms are present at the face centers of the cube.

### Step 2: Count the contribution of X atoms
- In a cubic lattice, there are a total of 8 corners.
- Since X occupies 7 corners, the contribution of X atoms from the corners is calculated as follows:
\[ \text{Contribution of X} = 7 \times \frac{1}{8} = \frac{7}{8} \]

### Step 3: Count the contribution of Y atoms
- The Y atom occupies 1 corner, so its contribution is:
\[ \text{Contribution of Y} = 1 \times \frac{1}{8} = \frac{1}{8} \]

### Step 4: Count the contribution of Z atoms
- There are 6 faces in a cube, and Z atoms are located at the centers of these faces.
- The contribution of Z atoms from the face centers is calculated as follows:
\[ \text{Contribution of Z} = 6 \times \frac{1}{2} = 3 \]

### Step 5: Combine the contributions to find the formula
- Now we can summarize the contributions:
- X: \(\frac{7}{8}\)
- Y: \(\frac{1}{8}\)
- Z: \(3\)

### Step 6: Normalize the contributions to find the simplest ratio
- To find the simplest ratio, we can multiply all contributions by 8 to eliminate the fractions:
\[ \text{X} = 7, \quad \text{Y} = 1, \quad \text{Z} = 3 \times 8 = 24 \]

### Step 7: Write the final formula
- Therefore, the formula of the compound is:
\[ \text{Formula} = \text{X}_7\text{Y}_1\text{Z}_{24} \]

### Final Answer:
The formula of the compound is \( \text{X}_7\text{Y}_1\text{Z}_{24} \).

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