Problem 22 - Chemistry in 30 days

Question

A solid has a structure in which \(\mathrm{W}\) atoms are located at the corners of the cubic lattice, \(\mathrm{O}\) atoms at the center of the edges and \(\mathrm{Na}\) atom at the center of the cube. The formula of the compound is (a) \(\mathrm{NaWO}_{2}\) (b) \(\mathrm{Na}_{2} \mathrm{WO}_{3}\) \(\mathrm{NaWO}_{3}\) (d) \(\mathrm{NaWO}_{4}\)

Step-by-Step Solution

Answer

The formula of the compound is \( \mathrm{NaWO}_3 \).

Step 1: Identifying Atoms and Their Positions

We have three types of atoms: \( \mathrm{W} \) at the corners, \( \mathrm{O} \) at the edge centers, and \( \mathrm{Na} \) at the cube center. We need to determine how they contribute to the unit cell.

Step 2: Counting W Atoms

\( \mathrm{W} \) atoms are located at the corners of the cubic lattice. Each corner atom is shared by 8 adjacent cubes. Thus, contribution from each \( \mathrm{W} \) atom is \( \frac{1}{8} \). Since there are 8 corners, the total contribution is \( 8 \times \frac{1}{8} = 1 \) \( \mathrm{W} \) atom per unit cell.

Step 3: Counting O Atoms

\( \mathrm{O} \) atoms are at the center of each edge. Each edge atom is shared by 4 adjacent cubes. There are 12 edges, and the contribution from each \( \mathrm{O} \) atom is \( \frac{1}{4} \). Therefore, the total contribution is \( 12 \times \frac{1}{4} = 3 \) \( \mathrm{O} \) atoms per unit cell.

Step 4: Counting Na Atoms

The \( \mathrm{Na} \) atom is located at the center of the cube and is not shared. Therefore, it fully belongs to the unit cell, contributing 1 \( \mathrm{Na} \) atom per unit cell.

Step 5: Writing the Chemical Formula

Based on the counts, the chemical formula of the compound is derived. We have 1 \( \mathrm{Na} \), 1 \( \mathrm{W} \), and 3 \( \mathrm{O} \) atoms per unit cell. Therefore, the formula is \( \mathrm{NaWO}_3 \).

Key Concepts

Cubic Lattice

A cubic lattice is a three-dimensional arrangement of points (atoms, molecules, or ions) repeated periodically in space. Think of it as a grid made up of cubes stacked together. This kind of structure is common in crystals like salt or metals.

An important aspect of the cubic lattice is its symmetry and uniformity, which makes it easy to predict how atoms are positioned.

• Atoms are placed at specific, repeating positions.
• Because of its regularity, scientists can quickly understand how each atom affects the overall structure.

In a cubic lattice, the corners often host atoms since they connect to many adjacent unit cells. This arrangement is essential for understanding how atoms contribute to the overall properties of the material.

Unit Cell

The unit cell is the simplest repeating structure within a crystal. It's like the building block of the crystal, which repeats over and over to create the entire crystal lattice.

Each unit cell in a cubic lattice is shaped like a cube and contains the positions where different atoms sit. The example from the exercise demonstrates a cubic unit cell, which is influential in describing various important properties of materials.

• The unit cell helps in determining the crystal’s overall symmetry.
• It defines how many atoms of each type are included in the crystal's formula.

Understanding the unit cell is vital because it provides essential data about the proportions of different atoms in a compound, directly helping to establish the material's chemical formula.

Chemical Formula

A chemical formula is a simple way of indicating which atoms, and how many of each, are in a compound. It offers a glimpse into the "recipe" of a compound's structure.

From the exercise, the crystal is composed of tungsten (W), oxygen (O), and sodium (Na). By analyzing the contributions of each atom from the unit cell structure, you can derive the chemical formula of the compound.

• Only whole numbers are used to represent the count of atoms in the most reduced form.
• The derived formula summarizes the entire makeup of the unit cell, providing insights into stoichiometry and compound formation.

In this case, although the arrangement seems complex, the final chemical formula is relatively straightforward, indicating the specific ratios of atoms involved.

Atom Contribution

Understanding atom contribution refers to determining how each atom adds to a unit cell's structure. Different positions within the unit cell mean that atoms may 'belong' to the unit cell in varying proportions.

For instance, in a cubic lattice:

• Corner atoms contribute partially because they are shared with other neighboring cells.
• Edge atoms also share their presence between cells; hence, their contributions are fractional.
• Atoms fully inside the cell, like those at the center, contribute completely to the unit cell.

This calculation is essential in resolving the formula of compounds from their solid structures. By analyzing each atom's contribution precisely, scientists gather clues about the actual number of atoms per unit cell, which directly impacts the compound's chemical formula and its properties.