The Third Dimension

By Lesley Smart, Michael Gagan

The Royal Society of Chemistry

Copyright © 2002 The Open University
All rights reserved.
ISBN: 978-0-85404-660-7

Contents

PART 1 CRYSTALS Lesley Smart,
1 INTRODUCTION, 11,
2 STRUCTURES OF METALS, 15,
3 THE INTERNAL STRUCTURE OF A CRYSTAL, 29,
4 IONIC SOLIDS, 46,
5 IONIC RADII, 69,
6 EXTENDED COVALENT STRUCTURES, 82,
7 MOLECULAR CRYSTALS, 89,
8 DEFECTS IN CRYSTALS, 100,
9 SUMMARY OF THE THIRD DIMENSION: CRYSTALS, 105,
LEARNING OUTCOMES, 106,
QUESTIONS: ANSWERS AND COMMENTS, 109,
ANSWERS TO COMPUTER ACTIVITIES, 117,
FURTHER READING, 118,
ACKNOWLEDGEMENTS, 118,
PART 2 MOLECULAR SHAPE Michael Gagan,
1 THE TETRAHEDRAL CARBON ATOM, 121,
2 MOLECULAR CONFORMATION, 126,
3 THE REPRESENTATION OF MOLECULES, 128,
4 CONSTITUTIONAL (STRUCTURAL) ISOMERISM, 136,
5 STEREOISOMERS OF MOLECULES CONTAINING DOUBLE BONDS, 139,
6 CHIRALITY, 148,
7 MOLECULES WITH MORE THAN ONE CHIRAL ATOM, 160,
8 STEREOCHEMISRTY OF SATURATED RING COMPOUNDS, 165,
9 CONCLUSION, 167,
APPENDIX A BRIEF SURVEY OF ORGANIC FUNCTIONAL GROUPS, 169,
LEARNING OUTCOMES, 175,
QUESTIONS: ANSWERS AND COMMENTS, 177,
ANSWERS TO EXERCISES, 186,
ANSWERS TO MODEL EXERCISES, 188,
FURTHER READING, 194,
ACKNOWLEDGEMENTS, 194,
CASE STUDY: LIQUID CRYSTALS — The fourth state of matter Corrie Imrie,
1 THE DISCOVERY, 197,
2 THE FOURTH STATE OF MATTER, 202,
3 TYPES OF LIQUID CRYSTALS, 203,
4 LIQUID CRYSTAL PHASES, 206,
5 IDENTIFYING LIQUID CRYSTAL PHASES, 208,
6 MOLECULAR STRUCTURE AND LIQUID CRYSTALLINITY, 210,
7 LIQUID CRYSTAL DISPLAY DEVICES, 212,
8 CHIRALITY AND LIQUID CRYSTALS, 215,
9 DISCOTIC LIQUID CRYSTALS, 217,
10 LIQUID CRYSTAL POLYMERS, 219,
11 SIDE-GROUP LIQUID CRYSTAL POLYMERS, 223,
12 LYOTROPIC LIQUID CRYSTALS, 226,
13 LIQUID CRYSTALS AND LIFE, 228,
14 WHAT NEXT?, 230,
FURTHER READING, 231,
ACKNOWLEDGEMENTS, 231,
INDEX, 232,
CD-ROM INFORMATION, 240,

CHAPTER 1

Part 1

Crystals

Lesley Smart

1 INTRODUCTION

You will be aware how atoms sometimes bond together to make individual molecules, as in carbon dioxide, chlorine and iodine for instance, but how in other cases, like diamond, silica, and sodium chloride, it is not possible to distinguish individual molecules (Figures 1.1a-d). You should also remember that bonding, and thus the arrangement of electrons, influences the shape of a molecule such as ammonia (Figure 1.1e). Many chemicals are solids at normal temperatures, and in the first part of this Book we are going to investigate the variety of structures adopted by elements and compounds in the crystalline solid state. The second part of the Book will concentrate only on the structure and shape of individual molecules.

To begin with, we are going to look at the structure of metals and of ionic solids. Such materials are very different from those containing individual molecules, in that they comprise extended arrays of atoms or ions in which discrete molecules are not identifiable. For instance, as well as diamond (Figure 1.1c) we can consider another familiar example, sodium chloride (common salt, Figure 1.2a), which has the empirical formula NaCl. Although the formula gives us the highly important information that there is one sodium atom present for every chlorine atom, it disguises the fact that crystalline sodium chloride does not comprise discrete NaCl molecules.

* What is the structure of sodium chloride?

* It is an aggregate of Na+ ions and Cl- ions arranged together in a regular, repeating three-dimensional pattern, with six ions of one type octahedrally surrounding an ion of the other type (Figure 1.2b).

The structure adopted by NaCl is also adopted by many other compounds, but it is not the only type of crystal structure. Ionic solids adopt a variety of patterns in the arrangements of their constituent ions. We shall show you a few of the ones that are more commonly found, and discuss the reasons for the variations.

We shall then move from ionic solids to look at a variety of structures adopted by solids where covalent rather than ionic bonding is predominant. For instance, silica (quartz), with empirical formula SiO2 (Figure 1.3), is covalently bonded: in its crystals we find an extended structure of covalently linked silicon and oxygen atoms, but no distinguishable individual molecules of SiO2.

In the final kind of solid state structure that we consider — molecular structures — small individual, covalently bound molecules such as iodine form crystals in which the molecules are held together by weak bonding (Figure 1.4).

Not everyone finds it an easy matter to visualize things in three dimensions, and so we have devised several ways of demonstrating the structures; you will have to find out which method helps you most. First and foremost is the use of WebLab Viewer Lite™ to look at the structures. Most of the diagrams of crystal structures in this Book can be viewed using WebLab ViewerLite. You will find the files on one of the CD-ROMs associated with this Book; you can access them via the CD-ROM Guide (p. 240) under ‘Figures’. WebLab ViewerLite allows you to view a structure in any orientation: the instructions for using it can be found on one of the CD-ROMs, and if you haven’t used it before, you should practise now with the four structures in this introduction. In addition to WebLab ViewerLite on one of the CD-ROMs, there is a program Virtual Crystals to work through, model-building exercises to view, and a program entitled Crystals, which contains some computer animations. If at all possible, try to study this Book near your computer, so that you can more easily use some or all of these aids.

Many ionic crystal structures can be regarded as based on a few very simple ways of packing spheres together in three dimensions. The simple sphere-packing schemes are best illustrated by the structures of metals, and so it is with these that we begin in the next Section.

2

STRUCTURES OF METALS

2.1 Close-packing in two dimensions

Before moving on to three-dimensional structures, it’s convenient first to take a look at the efficiency of close-packing in two dimensions, by looking at two different ways in which circles may be arranged. Figure 2.la shows a close-packed arrangement, whereas Figure 2.1 b contains a square arrangement. The first apparent difference is that each circle touches six others in Figure 2.la, but only four others in Figure 2.1b. How do the two arrangements compare for efficient use of space? The bounded areas highlighted on the two diagrams both enclose a total of four complete circles. If we overlay these two areas (Figure 2.lc), we can see clearly that the four circles (comprising two whole circles, two half circles, and four quarter circles) in Figure 2.la occupy less total area than the four circles in Figure 2.lb (comprising one whole circle, four half circles, and four quarter circles) — about 15% less in fact. Now let’s move on to the three-dimensional case.

COMPUTER ACTIVITY 2.1 Using computer animation and models to study close-packing

The close-packing of spheres is demonstrated by building models in Model Building on one of the CD-ROMs associated with this Book.

There is a computer animation of close-packing in the first part of Crystals on the CD-ROM.

Close-packing is also demonstrated on the CD-ROM in the program Virtual Crystals.

You should try these activities as soon as possible. Each of them should take you 15-20 minutes to complete.

2.2 Close-packing in three dimensions

A large number of metals adopt what is known as a close-packed structure (Figure 2.2). We model these structures by treating the atoms as though they are hard spheres. This is a convenient model to use, and modern science shows that it is not unreasonable: Figure 2.3 shows the most detailed pictures of atoms that are currently available, using electron microscopy. In Figure 2.3a we see the image of a single gold atom, and in Figure 2.3b the arrangement of gold atoms on the surface of a gold crystal. Notice the way the gold atoms are arranged on the surface; such an arrangement of atoms makes the most efficient use of the space available.

We are interested in the structure of solids, which are three-dimensional, and thus in the three-dimensional packing of spheres. We shall concentrate first on a layer of spheres. As with circles, spheres pack together in a layer most efficiently if they are close-packed (Figure 2.4a and b). Looked at from above, the spheres in the close-packed layer have ‘curved triangular’ spaces between them; for the sake of simplicity we shall refer to them as triangles. These are marked with crosses and dots in Figure 2.4c.

* In an infinite layer of spheres, what is the ratio of the number of spaces to the number of spheres?

* There are twice as many spaces as there are spheres.

This is easy to check by counting the number of spaces around a sphere in the centre of Figure 2.4c: there are six. Now count how many spheres there are around one space: the answer is three.

* Do the spaces in Figure 2.4~ differ in any way?

* The spaces marked with a cross are characterized by the tips of the triangles pointing up the page. The other triangles, with tips pointing down the page, are marked with dots, and are equal in number to those marked with crosses.

At this stage in our argument these two types are equivalent: Figure 2.4c would look exactly the same if viewed from the other side.

To start to build up the three-dimensional array, a similar close-packed layer is placed on the first. The layers could be stacked so that the spheres in the second layer are placed directly on top of those in the first, but this is not the most efficient use of space. Close-packing is achieved when the spheres of the upper layer are placed over the hollows of the lower layer. In Figure 2.5 the green spheres represent the upper layer (layer B), and the original layer is represented by the blue spheres (layer A). Look carefully at Figure 2.5. The spheres of layer B rest in the hollows or triangular spaces marked with a cross in layer A — that is, in the spaces with the tips pointing toward the top of the diagram. The hollows marked with a spot, with their tips pointing downward, remain uncovered. (We could equally well have used the spheres in layer B to cover the spaces marked with dots, and left uncovered the spaces marked with crosses; it would not have affected the efficiency of the packing.)

So far, we have talked of the spaces between the spheres as being curved triangles, but, of course, they are no such thing. Look carefully at Figure 2.6a: the spaces now formed by the layers are of two types. The spaces marked by crosses in Figure 2.6a are bounded by four spheres, with their centres arranged at the corners of a tetrahedron. These spaces are known as tetrahedral holes; one is shown in a side view as Figure 2.6b. The holes marked by spots in Figure 2.6c are bounded by six spheres, with their centres located at the corners of an octahedron: these are called octahedral holes; one is shown in a side view as Figure 2.6d.

* What do you notice immediately about the relative sizes of the octahedral and tetrahedral holes?

* The octahedral holes are much bigger than the tetrahedral holes. (This is obvious when you think that it takes six spheres to enclose an octahedral hole, and only four to enclose a tetrahedral hole.) It can be calculated that the radius of a sphere that will just fit in a tetrahedral hole in a close-packed array of spheres of radius r, is 0.225r (Figure 2.7a). For an octahedral hole the radius is 0.414r (Figure 2.7b).

* Now look at Figures 2.4 and 2.5 again, and try to work out the proportion of octahedral holes to spheres in the structure.

* From Figure 2.4 you can see that there would be twice as many triangles as spheres in an infinite array. When the second layer of spheres is added in Figure 2.5, half of these triangles become octahedral holes (the ones with the dots), so there is one octahedral hole for each sphere, and in an infinite three-dimensional array of n spheres, there will be n octahedral holes.

From Figure 2.6a, you can see that in an infinite array there would be twice as many tetrahedral holes as octahedral holes. So, in an infinite three-dimensional array of n spheres, there will be 2n tetrahedral holes.

So far we have only two layers. When we add a third layer, there is a choice: there are two types of hollow on layer B of Figure 2.5 in which the spheres of the next layer can rest. In the first alternative, the spheres in layer C can be arranged so that they are directly over the spheres in layer A; they are placed over the tetrahedral holes (Figure 2.8). This type of packing is known as hexagonal close-packing, hcp, and is adopted by many of the metallic elements in the Periodic Table, such as Be, Mg, Sc, Ti, Co and Zn, to name but a few (Figure 2.9). The layers in a hexagonal close-packed structure are designated AB AB. … Notice that in hexagonal close-packing the positions marked with a dot are always occupied by spaces and never by spheres. Thus, in models of this type of packing, a thin wire could be drawn vertically through the layers using these narrow channels.

The alternative arrangement for a third layer on Figure 2.5, is for the layer C spheres not to lie directly over the spheres of layer A, but over the spaces marked with spots — that is, over the octahedral holes (Figure 2.10). This is known as cubic close-packing, ccp, and is designated ABC ABC. … An easy way to remember the difference between hexagonal and cubic close-packing, is that both the word ‘cubic’ and the sequence ABC, end with the letter ‘c’.

This cubic close-packed structure is also adopted by many of the metallic elements, of which gold is one example. The close-packed planes of atoms are to be found parallel to the body-diagonal in the cubes (Figures 2.11a and b), and one of these planes on the surface of a gold sample has been captured dramatically by an STM picture (Figure 2.3b).

There are innumerable stacking sequences possible that may be described as close-packed, where the repeating sequence of layers is not as simple as in the two cases described above. The hexagonal close-packed sequence (ABABAB …) and the cubic close-packed sequence (ABCABC …) are the two simplest sequences, and are those most commonly encountered in the crystal structures of the metallic elements.

Now look carefully at Figures 2.8a and 2.10a again. Select a sphere in the middle of layer B, and in each case count the total number of spheres actually touching it. (You can do this in WebLab ViewerLite by clicking on a particular atom to select it.)

* How many spheres did you count?

* You should have counted twelve: in both cases there are six in the second layer, three in the plane above and three in the plane below.

This number of equidistant nearest neighbours is known as the coordination number. You will encounter this term many times in chemistry.

QUESTION 2.1

How many tetrahedral holes and how many octahedral holes are there in a ccp array of n spheres?

QUESTION 2.2

What is the coordination number of any atom in an infinite ccp array?

2.3 Body-centred and primitive cubic structures

Some metals, such as the alkali metals of Group 1 (Figure 2.13), adopt a structure that is somewhat less efficiently packed than hcp or ccp. This is the body-centred cubic (bcc) structure, in which an atom in the centre of a cube is surrounded by eight equidistant nearest neighbours, situated at the corners of the cube. A portion of a body-centred cubic array is shown in Figure 2.14a. You should be able to see that the cube formed by eight spheres contains a sphere occupying the centre of the cube, thus giving the structure its name.

It is common in structures for the positions of the atoms to be represented by small spheres that do not touch (Figure 2.14 b and c); such diagrams are commonly known as ball-and-stick models: this is merely a device to open up the structure and allow it to be seen more clearly, even though it is a less realistic representation of a real crystal than the space-filling model we used in Figure 2.14a. We shall discuss the relative sizes of atoms and ions in Section 5.

In the bcc structure, each atom is surrounded by eight atoms at the corners of the cube, but it has six other atoms only slightly more remote at the body-centres of the adjacent cubes (Figure 2.15). This means that the packing efficiency in bcc is not very different from that in close-packed structures: 68% of the volume is occupied, compared with the 74% of volume occupied in close-packing, regardless of the atom sizes.

QUESTION 2.3

If the length of a side of the cube in Figure 2.14b is a, what is the shortest distance from the atom at the cube centre to (a) an atom at a corner of the cube, and (b) the equivalent position in an adjacent cube? (See the Maths Help overleaf.)

(Continues…)Excerpted from The Third Dimension by Lesley Smart, Michael Gagan. Copyright © 2002 The Open University. Excerpted by permission of The Royal Society of Chemistry.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.