Problem 116 - Chemistry The Central Science

Question

Sodium oxide \(\left(\mathrm{Na}_{2} \mathrm{O}\right)\) adopts a cubic structure with Na atoms represented by green spheres and O atoms by red spheres. $$ \begin{array}{l}{\text { (a) How many atoms of each type are there in the unit cell? }} \\ {\text { (b) Determine the coordination number and describe the }} \\ {\text { shape of the coordination environment for the sodium }} \\\ {\text { ion. }} \\ {\text { (c) The unit cell edge length is } 5.550 \text { A. Determine the den- }} \\ {\text { sity of } \mathrm{Na}_{2} \text { O. }}\end{array} $$

Step-by-Step Solution

Answer

The unit cell of sodium oxide (Na2O) contains 4 Na atoms and 1 O atom. The coordination number for sodium ions is 6 and their coordination environment is octahedral. The calculated density of Na2O is approximately 2.269 g/cm^3.

Step 1: Find the atoms in the unit cell

To find the number of atoms of each type, we need to know their positions within the unit cell. In a cubic unit cell, there are eight corners, six faces, and one center. Sodium (Na) atoms are located on corners and faces, while Oxygen (O) atoms are located in the center. Each corner atom is shared by eight cubes, so each contributes 1/8th to the unit cell. There are 8 corner Na atoms, contributing 8 * 1/8 = 1 Na atom. Each face atom is shared by two cubes, so each contributes 1/2 to the unit cell. There are 6 face Na atoms, contributing 6 * 1/2 = 3 Na atoms. The Oxygen atom in the center of the unit cell entirely belongs to the unit cell. So there is 1 O atom. The unit cell contains a total of 1 Na + 3 Na + 1 O = 4 Na atoms and 1 O atom.

Step 2: Determine the coordination number and shape for the sodium ion

The coordination number is the number of nearest neighbors of an atom. In the case of Na2O, each sodium ion is surrounded by Oxygen ions and vice versa. Each face-centered Na atom is surrounded by four O atoms in the same plane and one O atom above and below the plane (six nearest neighbors in total). The coordination number for sodium ions is therefore 6. The shape of the coordination environment is defined by the arrangement of the nearest neighboring atoms. In this case, the six oxygen ions surrounding each sodium ion form an octahedron. Thus, the coordination environment is octahedral.

Step 3: Calculate the density of Na2O

To determine the density of Na2O, we need to find the mass of the unit cell divided by its volume. The mass of one unit cell can be found using the molecular weight of Na2O: M(Na2O) = 2 * M(Na) + M(O) = 2 * 22.990 g/mol + 16.00 g/mol = 61.980 g/mol. The volume of the unit cell can be calculated using the edge length, which is given as 5.550 A (1 A = 10^{-10} m): Volume = edge_length^3 = (5.550 * 10^{-10} m)^3 = 1.70378 * 10^{-28} m^3. Now we can calculate the density using mass and volume: Density = (Mass of one unit cell) / (Volume of one unit cell). Density = (61.980 g/mol) / [1.70378 * 10^{-28} m^3 * (1 mol / (6.022 * 10^{23} unit cells))]. Density = 2.269 g/cm^3. So, the density of Na2O is approximately 2.269 g/cm^3.

Key Concepts

Cubic Unit Cell

Understanding the cubic unit cell is essential for comprehending the structure of many crystalline solids.
The cubic unit cell is a three-dimensional geometric arrangement where atoms are positioned at specific points. This includes the corners, faces, or even entirely inside the cell.
In the case of sodium oxide \(\left(\text{Na}_2 \text{O}\right)\), the cubic unit cell comprises eight corners, six faces, and a center.

• The corners of the cube, each hosting a sodium atom, are shared among eight adjacent cubes.
• The face-centered atoms, also sodium, are shared between two adjacent cubes.
• Finally, the oxygen atom is located right at the cube's center, belonging entirely to the unit cell.

From these positions, it can be calculated that there is effectively one sodium atom from the corners, three from the faces, and one whole oxygen atom at the center.This basic framework sets the stage for further calculations related to the material's properties.

Coordination Number

The coordination number provides insight into the spatial arrangement of atoms surrounding a particular atom within a crystal structure.
Specifically, it tells us how many neighbor atoms are in direct contact with an atom. In the sodium oxide unit cell, the sodium ions have a coordination number of 6.
Here's why:

• Each sodium ion is directly connected to six oxygen ions — four in the same plane and one above and below.
• This arrangement forms a stable structure, as the sodium ions remain surrounded uniformly.

This equilibrium contributes to the overall stability and the mechanical properties of the crystal, making the coordination number a critical factor in evaluating crystal lattices.

Density Calculation

Calculating the density of a crystalline material like sodium oxide involves determining both the mass of its unit cell and the volume it occupies.
Here's a step-by-step breakdown:

• Mass Calculation: First, calculate the molar mass of Na2O, which involves adding twice the atomic mass of sodium to the atomic mass of oxygen. In this case, that total is 61.980 g/mol.
• Volume Calculation: The volume of the unit cell is derived from the cube of its edge length. Given a unit cell edge length of 5.550 Å, its volume computes to approximately \(1.70378 \times 10^{-28} \text{ m}^3\).
• Density Formula: With mass and volume in hand, density can be calculated using the formula: \(\text{Density} = \frac{\text{Mass of one unit cell}}{\text{Volume of one unit cell}}\). After factoring in Avogadro's number (\(6.022 \times 10^{23}\)), the result is a calculated density of about \(2.269 \text{ g/cm}^3\).

This calculated density helps predict how the material will behave in various applications.

Octahedral Coordination

Octahedral coordination is a term used to describe a geometric arrangement where a central atom is surrounded by six neighboring atoms.
In the case of sodium ions in sodium oxide, this configuration enables each sodium ion to be in the middle of a symmetrical structure:

• The six surrounding oxygen ions shape an octahedron around the sodium ion, ensuring maximum stability and cancellation of forces.
• This arrangement is both visually symmetrical and efficient at maintaining structural stability, making it prevalent in many mineral forms.

Understanding this concept is important as it influences how ions interact in a crystal lattice and impacts the overall physical properties of the crystal.

Molecular Weight

The molecular weight, also referred to as molar mass, is a key factor in understanding the fundamental properties of a substance.
It is defined as the mass in grams of one mole of a substance. For sodium oxide, this is calculated by summing the masses of all constituent atoms within a formula unit:

• Each sodium ion contributes 22.990 g/mol to the total.
• The oxygen contributes 16.00 g/mol.
• So for \(\text{Na}_2\text{O}\), this results in a molecular weight of 61.980 g/mol.

Knowing the molecular weight is essential for performing calculations related to chemical reactions and physical properties, as it informs the proportions of different elements within the compound.