Question

A compound formed by elements \(X\) and \(Y\) crystallizes in cubic structure where \(X\) atoms are at the corner of cube and \(Y\) atoms are at face centre. What is the formula of compound?

Short answer

The empirical formula of the compound is \( XY_3 \).

Step-by-step solution

Determine the number of atoms of element X

In a cubic structure, atoms of element X are at the corners of the cube. Each corner atom is shared among 8 adjacent cubes. Therefore, each corner contributes \frac{1}{8} of an atom to the unit cell. Since there are 8 corners in a cube: \[ \text{Number of X atoms} = 8 \times \frac{1}{8} = 1 \]

Determine the number of atoms of element Y

Atoms of element Y are located at the face centers of the cube. Each face center atom is shared between 2 adjacent cubes. Therefore, each face center atom contributes \frac{1}{2} of an atom to the unit cell. Since there are 6 faces in a cube: \[ \text{Number of Y atoms} = 6 \times \frac{1}{2} = 3 \]

Derive the empirical formula

From the previous steps, it's found that there is 1 atom of X and 3 atoms of Y in the unit cell. Thus, the empirical formula of the compound is \[ XY_3 \]

Key concepts

Unit Cell Calculation

When studying crystalline structures, the unit cell is the smallest repeating unit that shows the entire symmetry of the arrangement of atoms in the crystal. It's like a building block. To calculate the number of atoms in a unit cell, we need to consider the position and sharing of these atoms. For element X, the atoms are at the corners of the cube. Each corner is shared with 8 adjacent cubes, so each corner atom contributes \frac{1}{8} of an atom to the unit cell. Given there are 8 corners: \( 8 \times \frac{1}{8} = 1 \). This means the unit cell has one whole atom of X. For element Y, atoms are at the face centers. Each face center atom is shared between 2 cubes, contributing \(\frac{1}{2}\) of an atom to the unit cell. With 6 faces, we have: \( 6 \times \frac{1}{2} = 3 \). Hence, the unit cell has three whole atoms of Y.

Empirical Formula

The empirical formula represents the simplest whole-number ratio of elements in a compound. We've determined that the unit cell contains one atom of X and three atoms of Y. The empirical formula shows how these atoms combine in the smallest ratio. Thus, for this compound, since there is 1 atom of X and 3 atoms of Y, the empirical formula is \(XY_3\). This formula tells us that for every 1 atom of X, there are 3 atoms of Y in the compound.

Crystallography

Crystallography is the study of crystal structures and the arrangement of atoms within solids. In our exercise, the compound of elements X and Y forms a cubic crystal structure. The unit cell is repeated in three-dimensional space to create the entire crystal. By studying the positions of X and Y within the cube, we can determine the overall structure and the empirical formula. Understanding crystallography helps in various fields, from materials science to chemistry, as it explains the properties and behaviors of materials based on their atomic structure.