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 center of cube $Y$ is at $\frac{1}{3} rd$ of the total faces. The empirical formula of the compound is

• A. $X _2 Y _{1.5}$
• B. $X_{3} Y_2$
• 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, we need to calculate the number of atoms of elements \(X\) and \(Y\) in the unit cell of the cubic structure in question.

1. Atoms of \(X\):
   • Atoms of \(X\) are located on every alternate corner and one at the center of the cube.
   • In a cubic structure, there are 8 corners, but since atoms are present at every alternate corner, there are 4 corner atoms of \(X\).
   • Each corner atom is shared by 8 adjacent unit cells, so the contribution of corner atoms is: \(\frac{1}{8} \times 4 = \frac{1}{2}\)
   • Plus, there's 1 atom of \(X\) at the center, which is wholly included in the unit cell.
   • Thus, total \(X\) atoms per unit cell = \(\frac{1}{2} + 1 = \frac{3}{2}\)
2. Atoms of \(Y\):
   • Atoms of \(Y\) are at \(\frac{1}{3}\) of the total faces.
   • There are 6 faces on the cube, hence atoms of \(Y\) are present in \(\frac{1}{3} \times 6 = 2\) faces.
   • Each face-centered atom provides \(0.5\) to the unit cell as it is shared between two cells.
   • Therefore, total \(Y\) atoms per unit cell = \(2 \times 0.5 = 1\)
3. Empirical Formula Calculation:
   • The ratio of \(X\) to \(Y\) in the unit cell is \((\frac{3}{2}):1\).
   • To express as a whole number ratio, multiply each by 2: \((3:2)\).
   • Thus, the empirical formula is \(X_3Y_2\).

From the given options, the correct empirical formula is \(X_3Y_2\), matching with option 2.