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What is a lattice? What is a unit cell? Describe a simple cubic unit cell. How many net atoms are contained in a simple cubic unit cell? How is the radius of the atom related to the cube edge length for a simple cubic unit cell? Answer the same questions for the body-centered cubic unit cell and for the face-centered unit cell.

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Chemistry: An Atoms First Approach

2nd Edition
Steven S. Zumdahl + 1 other
Publisher: Cengage Learning
ISBN: 9781305079243

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Section
BuyFindarrow_forward

Chemistry: An Atoms First Approach

2nd Edition
Steven S. Zumdahl + 1 other
Publisher: Cengage Learning
ISBN: 9781305079243
Chapter 9, Problem 5RQ
Textbook Problem
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What is a lattice? What is a unit cell? Describe a simple cubic unit cell. How many net atoms are contained in a simple cubic unit cell? How is the radius of the atom related to the cube edge length for a simple cubic unit cell? Answer the same questions for the body-centered cubic unit cell and for the face-centered unit cell.

Interpretation Introduction

Interpretation:

The following concepts has to be described –

  • Lattice
  • Simple cubic Unit cell
  • Number of net atoms in a simple cubic unit cell
  • Relationship between atomic radius and cube edge length in a simple cubic unit cell
  • Body-centered cubic unit cell
  • Number of net atoms in a Body-centered cubic unit cell
  • Relationship between atomic radius and cube edge length in a Body-centered cubic unit cell

Concept Introduction:

In packing of crystalline solids, the components – ions, atoms or molecules are designated as spheres. The components arrange themselves in a well-defined, compact, regular and three – dimensional pattern.

The simplest and basic unit of a crystalline solid is known as unit cell.  It is cubic in shape. It is the building block of crystalline solids. The unit cells repeat themselves to build a lattice. Crystalline solids consist of many of such lattices.

There are basically three types of unit cells – Simple cubic unit cell, Body-centered cubic unit cell and Face-centered cubic unit cell, they form Simple cubic lattice, Body-centered cubic lattice and Face-centered cubic lattice respectively. These three types of unit cells differ in the arrangement of the components.

Explanation of Solution

Unit cell is the simplest form of a lattice. The unit cell has certain arrangement of atoms in it. These unit cells repetitively connect with other unit cells. As a result a network like macrostructure is formed consisting of arrays of the components in a three dimensional pattern called lattice. A typical lattice looks like –

                            

                                               Figure 1

A description of Lattice is given.

Describe what a simple cubic unit cell is.

A simple cubic cell is a type of unit cell with atoms arranged in all the corners of the unit cell.

A simple cubic unit cell is the simplest form of a cubic unit cell. A cube has eight vertices, twelve edges and six faces. Similarly a cubic unit cell has eight vertices, twelve edges and six faces. If in a cubic unit cell, the components occupy only the eight vertices, then the unit cell is known as simple cubic unit cell.

                                        

                                                  Simple Cubic unit cell

                                                             Figure 2

A description of Simple cubic unit cell is given.

Analyze the number of atoms a simple cubic unit cell contains.

A simple cubic unit cell contains one atom per unit cell.

As we know, a simple cubic unit cell has its all the eight vertices arranged with atoms in it. Thus eight atoms are arranged in a simple cubic unit cell in eight vertices. Each atom in a vertex is shared by eight neighboring unit cells. Each layer of a lattice has four unit cells and the upper or lower to that layer has four unit cells. So, each simple cubic unit cell has 18th of an atom at each vertex. So 8 vertices have 8×18=1atom per unit cell.

The total number of atoms present in a simple cubic unit cell is analyzed and identified.

Analyze the relationship between atomic radius and cube edge length in a simple cubic unit cell.

The relationship between atomic radius ‘r’ and cube edge length ‘l’ in a simple cubic unit cell is given as - l=2r .

In a simple cubic unit cell, the atoms touch along the edge of the cube of length ‘l’. Consider a triangular part in a simple cubic unit cell as follows –

                               

                                                          Figure 3

Thus the edge length ‘l’ of the simple cubic unit cell is related to the atomic radius ‘r’ as, l=2r .

The relationship between atomic radius and cube edge length in a simple cubic unit cell is described.

Describe what a body – centered cubic unit cell is.

A body – centered cubic cell is a type of unit cell with atoms arranged in all the vertices of the unit cell and one atom at the center of the cube.

In a unit cell, if the components occupy all the eight vertices and one atom at the center of the cube. Then the unit cell is known as body – centered cubic unit cell.

                                          

                                              Body – centered cubic unit cell

                                                             Figure 4

A description of Body – centered cubic unit cell is given.

To analyze the number of atoms present in a Body – centered cubic unit cell.

A body – centered cubic unit cell contains 2 atoms.

A body – centered cubic unit cell has its all the eight vertices arranged with atoms in it. Eight atoms are arranged in a simple cubic unit cell in eight vertices. Each atom in a vertex is shared by eight neighboring unit cells such that each cubic unit cell has 18th of an atom at each vertex. Further, each body – centered cubic unit cell has one atom at the center.  So 8 vertices have 8×18=1atom+1 atom at the body of the cube gives 2 atoms per body – centered cubic unit cell

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Chapter 9 Solutions

Chemistry: An Atoms First Approach
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