Question

A cubic solid is made up of two elements $X$ and $Y$ Atoms of $X$ are present on every alternate corner and one at the enter of cube $Y$ is at $\frac{1}{3} td$ of the total faces The empirical formula of the compound is

• A. $X _2 Y _{1.5}$
• B. $X _{2.5} Y$
• C. $XY _2, 5$
• D. $X _{1.5} Y _2$

Answer 1

The Correct Option is B

To determine the empirical formula of the compound made of elements \( X \) and \( Y \) in the given cubic solid, let's analyze the distribution of atoms as mentioned in the question:

1. Atoms of \( X \):
   • Atoms of \( X \) are present on every alternate corner. In a cube, there are 8 corners. Therefore, atoms will be at 4 corners (since it's every alternate corner). Each corner atom is shared among 8 adjacent unit cells, contributing \( \frac{1}{8} \) atom to the unit cell. Hence, atoms on corners contribute:

\(\text{Total corner contribution of } X = 4 \times \frac{1}{8} = \frac{1}{2} \text{ atom/unit cell}\)

• There's one atom of \( X \) at the center of the cube. This atom is not shared with any other cells, thus contributing:

\(\text{Center contribution of } X = 1 \text{ atom/unit cell}\)

• The total contribution of \( X \) in the cube is:

\(\text{Total } X = \frac{1}{2} + 1 = \frac{3}{2} = 1.5 \text{ atoms/unit cell}\)

1. Atoms of \( Y \):
   • Atoms of \( Y \) are present at \(\frac{1}{3}\) of the total faces. A cube has 6 faces. Hence, the number of faces occupied by \( Y \) is:

\(\text{Number of } Y \text{ atoms on faces} = \frac{1}{3} \times 6 = 2 \text{ faces}\)

• Each face atom is shared between 2 adjacent unit cells, contributing \( \frac{1}{2} \) atom to the unit cell for each face. So, the total contribution of \( Y \) is:

\(\text{Total face contribution of } Y = 2 \times \frac{1}{2} = 1 \text{ atom/unit cell}\)

Therefore, the ratio of \( X \) to \( Y \) is \( \frac{1.5}{1} : 1 \). To simplify, multiply both by 2:

\(X_{2.5} Y\)

The correct empirical formula of the compound, therefore, is \( X_{2.5} Y \). Hence, the correct answer is:

\(X_{2.5} Y\)