What is the formula of the compound formed by elements X and Y if atoms of element Y are in hcp arrangement and those of the element X occupy `2//3`rd of tetrahedral voids?

Explanation

To find the formula of the compound formed by elements X and Y, we will follow these steps:

### Step 1: Determine the number of atoms of Y in the hcp arrangement.
In a hexagonal close-packed (hcp) arrangement, the effective number of atoms can be calculated as follows:
- There are 12 corner atoms, each shared by 6 unit cells, contributing \( \frac{1}{6} \) of an atom per unit cell.
- There are 2 face-centered atoms, each contributing \( \frac{1}{2} \) of an atom.
- There is 1 atom in the middle layer contributing fully.

The total contribution from these atoms is:
\[ \text{Effective number of Y atoms} = \left(\frac{1}{6} \times 12\right) + \left(\frac{1}{2} \times 2\right) + (1 \times 3) = 2 + 1 + 3 = 6 \]

### Step 2: Calculate the number of tetrahedral voids.
In a close-packed structure, the number of tetrahedral voids is given by:
\[ \text{Number of tetrahedral voids} = 2 \times \text{Number of atoms in the lattice} \]
Here, the number of atoms in the lattice is 6 (from step 1), so:
\[ \text{Number of tetrahedral voids} = 2 \times 6 = 12 \]

### Step 3: Determine the number of atoms of X occupying the tetrahedral voids.
According to the problem, atoms of element X occupy \( \frac{2}{3} \) of the tetrahedral voids. Therefore, the number of atoms of X is:
\[ \text{Number of X atoms} = \frac{2}{3} \times 12 = 8 \]

### Step 4: Find the ratio of atoms of X to Y.
Now, we have:
- Number of atoms of X = 8
- Number of atoms of Y = 6

The ratio of X to Y is:
\[ \text{Ratio of X to Y} = \frac{8}{6} = \frac{4}{3} \]

### Step 5: Write the formula of the compound.
From the ratio of X to Y, we can write the formula of the compound as:
\[ \text{Formula} = X_4Y_3 \]

### Final Answer:
The formula of the compound formed by elements X and Y is \( X_4Y_3 \).

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