Question

A compound is formed by two elements A and B. The element B forms cubic close packed structure and atoms of A occupy \(\frac 13\) of tetrahedral voids. If the formula of the compound is A,By. then the value of x +y is in option

• A. 4
• B. 3
• C. 2
• D. 5

Answer 1

The Correct Option is D

To find the value of \( x + y \) in the given compound \( A_xB_y \), we start by understanding the structure of the compound. Element \( B \) forms a cubic close packed (ccp) structure. In a ccp lattice, there are 4 atoms per unit cell. This means there are 4 \( B \) atoms in the lattice.

In a ccp structure, there are tetrahedral voids equal to twice the number of atoms of the element forming the ccp structure. Thus, there are \( 2 \times 4 = 8 \) tetrahedral voids. The problem states that atoms of element \( A \) occupy \(\frac{1}{3}\) of these tetrahedral voids.

Let's calculate the number of \( A \) atoms: \( 8 \times \frac{1}{3} = \frac{8}{3} \) atoms of \( A \).

The empirical formula ratio between \( A \) and \( B \) is \( A_{\frac{8}{3}}B_4 \). To express this in simplest whole numbers, we multiply through by 3 to clear the fraction: \( A_8B_{12} \).

Simplifying this ratio gives us \( A_2B_3 \). 

Therefore, \( x = 2 \) and \( y = 3 \).

Finally, calculating \( x + y \):

\( x + y = 2 + 3 = 5 \).

Thus, the value of \( x + y \) is 5.

Answer 2

Cubic Close-Packed (CCP) Structure:
In a cubic close-packed (ccp) structure, also known as a face-centered cubic (fcc) structure, there are 4 atoms per unit cell.

Tetrahedral Voids:
In a ccp structure, there are 2 tetrahedral voids for each atom in the structure.
Since there are 4 atoms of B per unit cell in the ccp structure, there are \(4 \times 2 = 8\) tetrahedral voids.

Atoms of A:
Atoms of element A occupy \(\frac{1}{3}\) of the tetrahedral voids.
Therefore, the number of A atoms per unit cell is:
\(8 \times \frac{1}{3} = \frac{8}{3}\)

Atoms of B:
The number of B atoms per unit cell is 4, as it forms the ccp structure.

Formula of the Compound:
The ratio of A to B atoms is \(\frac{8}{3}\) to 4.
To simplify the ratio, we can multiply both parts by 3 to eliminate the fraction:
\(A : B = \frac{8}{3} : 4 = 8 : 12 = 2 : 3\)
Therefore, the formula of the compound is \(A_2B_3\).

The sum of x and y in the formula\(A_xB_y\):
In the formula \(A_2B_3\), x = 2 and y = 3.
Thus, \(x + y = 2 + 3=5\).

So, the correct option is (D): 5.

Answer 3

Number of tetrahedral voids = 2 x Number of atoms of B
Number of atoms of A = \(\frac 13\) x Number of tetrahedral voids
Let the number of atoms of B be "x" and the number of atoms of A be "y."
From the given information, we have the equations:
x = Number of atoms of B
y = \(\frac 13\) * 2x = \(\frac {2x}{3}\)
Since the formula of the compound is AB, the total number of atoms in the compound is x + y.
Total number of atoms = x + y = x + \(\frac {2x}{3}\) = \(\frac {2x}{3}\)
So, the value of x + y is \(\frac {5x}{3}\).
Since we are looking for the value of x + y in terms of a constant value, we can simplify the fraction by choosing a suitable value for x. Let's choose x = 3 (which is a common choice in these types of problems):
x = 3y = \(\frac {2x}{3}\) = 2
So, x + y = 3 + 2 = 5.

Therefore, the correct option is (D): 5