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Ch.12 - Solids and Modern Materials
All textbooksBrown 14th EditionCh.12 - Solids and Modern MaterialsProblem 116c
Chapter 12, Problem 116c

Sodium oxide (Na2O) adopts a cubic structure with Na atoms represented by green spheres and O atoms by red spheres.
Cubic structure of sodium oxide with green Na and red O spheres, edge length 5.550 Å.
(c) The unit cell edge length is 5.550 Å. Determine the density of Na2O.

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1
Step 1: Identify the type of unit cell and the number of formula units per unit cell. For Na2O, it adopts a cubic structure, and typically, a cubic unit cell contains 4 formula units (Z = 4).
Step 2: Calculate the molar mass of Na2O. The molar mass of Na is approximately 22.99 g/mol, and the molar mass of O is approximately 16.00 g/mol. Therefore, the molar mass of Na2O is 2(22.99) + 16.00 g/mol.
Step 3: Convert the unit cell edge length from Ångströms to centimeters. The given edge length is 5.550 Å. Since 1 Å = 1 x 10^-8 cm, convert the edge length to cm.
Step 4: Calculate the volume of the unit cell. The volume of a cubic unit cell is given by the cube of the edge length (a^3).
Step 5: Determine the density of Na2O using the formula: density = (mass of unit cell) / (volume of unit cell). The mass of the unit cell can be found by multiplying the number of formula units per unit cell (Z) by the molar mass of Na2O and dividing by Avogadro's number (6.022 x 10^23).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Unit Cell

A unit cell is the smallest repeating unit in a crystal lattice that reflects the overall symmetry and structure of the entire crystal. In the case of sodium oxide (Na2O), the cubic unit cell contains a specific arrangement of sodium (Na) and oxygen (O) atoms, which is crucial for calculating properties like density. Understanding the unit cell helps in visualizing how atoms are packed in a solid.
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Density Calculation

Density is defined as mass per unit volume and is a key property of materials. To calculate the density of Na2O, one must determine the mass of the formula unit and the volume of the unit cell. The formula for density is given by ρ = mass/volume, where the mass can be derived from the molar mass of Na2O and the volume is calculated from the edge length of the cubic unit cell.
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Molar Mass

Molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). For sodium oxide (Na2O), the molar mass is calculated by summing the atomic masses of its constituent elements: sodium (Na) and oxygen (O). This value is essential for determining the mass of the Na2O present in the unit cell, which is necessary for the density calculation.
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Related Practice
Textbook Question

Energy bands are considered continuous due to the large number of closely spaced energy levels. The range of energy levels in a crystal of copper is approximately 1 × 10–19 J. Assuming equal spacing between levels, the spacing between energy levels may be approximated by dividing the range of energies by the number of atoms in the crystal. (b) Determine the average spacing in J between energy levels in the copper metal in part (a).