8 Elements
Topics
8.1
.
Close-packing of spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
Solid structures of group 18 elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
Solids with molecular units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
Solids with in®nite covalent lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
Metallic bonding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction
In Section 2.21, we discussed the van der Waals forces that operate between
atoms of a noble gas and noted that the weakness of these forces is re¯ected
in some of the physical properties of the noble gases (Table 2.6). We have
already observed that these elements form solids only at very low temperatures, and that helium can only be solidi®ed under pressures greater than
atmospheric pressure. When a group 18 element solidi®es, the atoms form
an ordered structure in which they are close-packed. Before considering the
speci®cs of the solid state structures of the noble gases, we discuss what is
meant by the close-packing of spheres. This concept may be familiar to
some, but not all, readers and a brief discussion of the topic is given in
Section 8.2. In Section 8.3, we consider simple and body-centred cubic
packing of spheres.
In this chapter we consider not only elemental solids that consist of closepacked atoms, but also the structures of those elements which contain
molecular units, small or large, or in®nite covalent lattices in the solid state.
8.2
Close-packing of spheres
Hexagonal and cubic close-packing
Suppose we have a rectangular box and place in it some spheres of equal size.
If we impose a restriction that there must be a regular arrangement of the
spheres, then the most ecient way in which to cover the ¯oor of the box
is to pack the spheres as shown in Figure 8.1. This is part of a close-packed
arrangement, and spheres not on the edge of the assembly are in contact
with six other spheres within the single layer. Figure 8.1 emphasizes that a
motif of hexagons is visible.
If we now add a second layer of close-packed spheres to the ®rst, allowing
the spheres in the second layer to rest in the hollows between those in the ®rst
Close-packing of spheres
265
Fig. 8.1 The aim is to cover the
¯oor of a rectangular box with
spheres which are of equal size,
and the criterion is that the
arrangement must be an ordered
one. The most ecient
arrangement is a close-packed
one. The repeating hexagonal
motif is shown in red.
Close-packing of spheres
represents the most ecient
use of space.
"
Unit cell: see Section 7.7
layer, there is only enough room for every other hollow to be occupied
(Figure 8.2). When a third layer of spheres is added, there is again only
room to place them in every other hollow, but now there are two distinct
sets of hollows. In the ®rst set, the hollows lie directly over the spheres of
the ®rst layer. These are the hollows marked A in the top diagram in
Figure 8.3. The hollows of the second set lie over the unoccupied hollows of
the ®rst layer. These are the hollows marked C at the top of Figure 8.3.
Depending upon which set of hollows is occupied as the third layer of spheres
is put in place, one of two close-packed structures results.
In Figure 8.3a, the third layer of spheres lies directly over the top of the
®rst layer and this is emphasized in a side view of the same arrangement in
Figure 8.3c. The layers are labelled A and B, and the repetition of layers
gives rise to an ABABAB . . . arrangement as we continue packing the spheres
in this manner.
In Figures 8.3b and 8.3d, the third layer of spheres does not lie directly
over the ®rst layer. The layers are labelled A, B and C, and the repetition
of layers produces an ABCABC . . . arrangement.
In both of these close-packed structures, each sphere is in contact with six
spheres within its layer plus three below and three above, giving twelve
nearest-neighbours, i.e. each sphere is 12-coordinate. Figure 8.4a shows
this for an ABA arrangement and Figure 8.4b depicts it for an ABC arrangement. The two ®gures di€er only in the mutual orientations of the two sets of
three atoms at the top and bottom of the diagrams.
The ABABAB . . . arrangement is called hexagonal close-packing (hcp) of
spheres, while the ABCABC . . . arrangement is known as cubic close-packing
(ccp). The unit cells that characterize these arrangements of spheres are
shown in Figure 8.5.
Fig. 8.2 When three spheres are arranged in a triangle and touch one another, there is a hollow at the centre of the triangle. A
single layer of close-packed spheres possesses hollows which form a regular pattern. (a) There are six hollows in between the
seven close-packed spheres. When a second layer of spheres is placed on top of the ®rst so that the new spheres lie in the hollows
in the ®rst layer, there is only room for every other hollow to be occupied. This is emphasized in (b) which gives a side view of the
arrangement as the second layer of spheres is added to the ®rst.
266
CHAPTER 8 . Elements
Fig. 8.3 The second layer in a close-packed array of spheres possesses two types of hollow. Those labelled A lie directly over
spheres in the ®rst layer, while a hollow of type C lies over a hollow in the ®rst layer. This leads to two possible arrangements
of the spheres in the third layer. The sequences (a) ABA and (b) ABC viewed from above, and (c) ABA and (d) ABC viewed
from the side. Notice in (d) that the ABC sequence of layers generates a plane consisting of repeating square units.
The unit cell shown in Figure 8.5a illustrates the presence of the cubic unit
that lends its name to cubic close-packing. It consists of eight spheres at the
corners of the cube, with a sphere at the centre of each square face. An alternative name for cubic close-packing is a face-centred cubic (fcc) arrangement.
It is easy to relate Figure 8.5b to the ABABAB . . . arrangement shown in
Figure 8.3 but it is not so easy to recognize the relationship between the unit
cell in Figure 8.5a and ABCABC . . . layer arrangement. The ®rst diagram in
Figure 8.6a shows three adjacent face-centred cubic units. If the structure
is rotated through 458, the close-packed layers discussed above become
apparent.
Close-packing of spheres
267
Fig. 8.4 In a close-packed assembly of spheres, each sphere has twelve nearest-neighbours. This is true for both (a) the ABA
arrangement and (b) the ABC arrangement.
In the last diagram in Figure 8.6a, the spheres are colour-coded according
to the layer in which they belong. This same diagram is redrawn in a di€erent
orientation in Figure 8.6b to show the A, B and C layers more clearly. The
relationship is best con®rmed by building a three-dimensional model or
using a computer modelling program.
Fig. 8.5 The ABCABC. . . and ABABAB. . . close-packed arrangements of spheres are termed (a) cubic and (b) hexagonal closepacking respectively. If the spheres are `pulled apart', (c) the cubic and (d) the hexagonal units become clear. Cubic close-packing
is also called a face-centred cubic arrangement. In diagrams (a) and (c), each face of the cubic unit possesses a sphere at its centre.
268
CHAPTER 8 . Elements
Fig. 8.6 (a) The relationship
between face-centred cubic units
(three are shown) and the
ABCABC. . . sequence of layers
of close-packed spheres can be
seen by tilting the cubic units
through 458. (b) The last
structure from (a) is redrawn to
show the ABCABC. . . sequence
of layers more clearly.
Spheres of equal size may be close-packed in (at least) two ways.
In hexagonal close-packing (hcp), layers of close-packed spheres pack in an
ABABAB . . . pattern.
In cubic close-packing (ccp) or face-centred cubic (fcc) arrangement, layers of
close-packed spheres are in an ABCABC . . . pattern.
In both an hcp and ccp array, each sphere has twelve nearest-neighbours.
Interstitial holes
A close-packed arrangement
of spheres contains
octahedral and tetrahedral
holes.
Placing one sphere on top of three others in a close-packed array gives a
tetrahedral unit (Figure 8.7a) inside which is a cavity called an interstitial
hole ± speci®cally in this case, a tetrahedral hole.
A close-packed array contains tetrahedral holes but in addition there are
octahedral holes (Figure 8.7b). Both types of interstitial hole are present in
the unit cell shown in Figure 8.5a, and this is further explored in Box 8.1.
It is not necessarily obvious why octahedral holes should arise in a closepacked arrangement, but if we look at Figures 8.3 and 8.6, we see that the
Fig. 8.7 (a) A tetrahedral hole (found in both ccp and hcp arrangements) is formed when one sphere rests in the hollow between
three others which are close-packed. (b) An octahedral hole is formed when two spheres stack above and below a square unit of
spheres. Such an arrangement is found in ccp and hcp assemblies of spheres.
Simple cubic and body-centred cubic packing of spheres
269
THEORETICAL AND CHEMICAL BACKGROUND
Box 8.1 An alternative description of ionic lattices
A face-centred cubic (fcc) arrangement of spheres
(cubic close-packing) is shown above. The left-hand
diagram emphasizes the octahedral hole present at the
centre of the fcc unit. The right-hand diagram shows
one tetrahedral hole; there are eight within the fcc unit.
For an array that is made up of spheres of equal sizes,
the interstitial holes remain empty.
An ionic lattice contains spheres (i.e. ions) which are
not all the same size. In many cases (e.g. NaCl) the
anions are larger than the cations, and it is convenient
to consider a close-packed array of anions with the
smaller cations in the interstitial holes.
Consider a cubic close-packed array of sul®de anions.
They repel one another when they are in close
proximity, but if zinc cations are placed in some of the
interstitial holes, we can generate an ionic lattice that
is stabilized by cation±anion attractive forces. A unit
cell of the zinc blende (ZnS) structure is shown opposite.
The S2 ions are in an fcc arrangement. Half of
the tetrahedral holes are occupied by Zn2‡ ions. The
organization of the cations is such that every other
tetrahedral hole is ®lled with a cation.
This description of the structure of zinc blende relies
on the idea that small cations can ®t into the interstitial
holes which lie between large anions which are closepacked. Note that the anions thus de®ne the corners
of the unit cell.
Similarly, it is possible to describe the structures of
NaCl and CaF2 in terms of cations occupying holes
in a close-packed array of anions.
sequence of three repeating layers in the ccp structure leads to planes of
spheres in square motifs. Such a plane is visible on the left-hand side of
Figure 8.3d, and is viewed `face-on' in the second two diagrams in Figure
8.6a. Stacking a sphere above and below each square array leads to the
formation of an octahedral hole (Figure 8.7b). There is no substitute for
model-building or using computer graphics to clarify these arguments!
8.3
Simple cubic and body-centred cubic packing of spheres
In cubic or hexagonal close-packed assemblies, each sphere has twelve
nearest-neighbours, and spheres are packed eciently so that there is the
minimum amount of `wasted' space. It is, however, possible to pack spheres
of equal size in an ordered assembly that is not close-packed and one such
method is that of simple cubic packing (Figure 8.8). The unit cell contains
270
CHAPTER 8 . Elements
Fig. 8.8 (a) The arrangement of
spheres of equal size in a simple
cubic packing. (b) The repeating
unit in this arrangement consists
of one sphere at each corner of
a cube.
eight spheres arranged at the corners of a cube, and when these units are
placed next to one another in an extended array, each sphere has six nearest-neighbours and is in an octahedral environment.
Simple cubic packing of spheres provides an ordered arrangement but there
is a signi®cant amount of unused space. Each interstitial hole (a `cubic' hole) is
too small to accommodate another sphere of equal size. If we force a sphere (of
equal size) into each interstitial hole in the simple cubic array, the original
spheres are pushed apart slightly. The result is the formation of a body-centred
cubic (bcc) arrangement. The di€erence between simple and body-centred
cubic structures can be seen by comparing Figures 8.8a and 8.9a. Although
bcc packing makes better use of the space available than simple cubic packing,
it is still less ecient than that in ccp or hcp assemblies.
The repeating unit in a bcc assembly is shown in Figure 8.9b, and
the number of nearest-neighbours in the bcc arrangement is eight (Figure
8.9c).
Fig. 8.9 (a) The packing of spheres of equal size in a body-centred cubic (bcc) arrangement. (b) The unit cell in a bcc arrangement of spheres is a cube with one sphere at the centre. The central sphere touches each corner sphere, but the corner spheres do
not touch one another. (c) Each sphere in a bcc arrangement has eight nearest-neighbours.
Crystalline and amorphous solids
271
The repetition of cubic units of spheres of equal size gives simple cubic packing,
and each sphere has six nearest-neighbours.
In a body-centred cubic (bcc) arrangement, the unit cell consists of a cube of
eight, non-touching spheres with one sphere in the centre. Each sphere has
eight nearest-neighbours.
The simple and body-centred cubic arrangements are not close-packed.
8.4
A summary of the similarities and differences between closepacked and non-close-packed arrangements
Hexagonal close-packing (Figures 8.5b and 8.5d)
.
.
.
Layers of close-packed atoms are arranged in an ABABAB . . . manner.
Each sphere has twelve nearest-neighbours.
The arrangement contains tetrahedral and octahedral interstitial sites.
Cubic close-packing or face-centred cubic (Figures 8.5a and 8.5c)
.
.
.
Layers of close-packed atoms are arranged in an ABCABC . . . manner.
Each sphere has twelve nearest-neighbours.
The arrangement contains tetrahedral and octahedral interstitial sites.
Simple cubic packing (Figure 8.8)
.
.
The spheres are not close-packed.
Each sphere has six nearest-neighbours.
Body-centred cubic packing (Figure 8.9)
.
.
The spheres are not close-packed.
Each sphere has eight nearest-neighbours.
Relative ef®ciency of packing
The relative eciency with which spheres of equal size are packed follows the
sequence:
body-centred
cubic
hexagonal
> simple cubic packing
>
ˆ
cubic packing
close-packing
close-packing
8.5
Crystalline and amorphous solids
In a crystalline solid, atoms, molecules or ions are packed in an ordered
manner, with a unit cell that is repeated throughout the crystal lattice. For
272
CHAPTER 8 . Elements
"
X-ray diffraction:
see Section 3.2
In a crystalline solid, atoms,
molecules or ions are packed
in an ordered lattice with a
characteristic unit cell.
8.6
an X-ray di€raction experiment, a single crystal is usually required. If a single
crystal shatters, it may cleave along well-de®ned cleavage planes. This leads
to particular crystals possessing characteristic shapes.
In an amorphous solid, the particles are not arranged in an ordered or
repetitive manner. Crushing an amorphous solid leads to the formation of
a powder, whereas crushing crystals leads to microcrystalline materials. However, microcrystals may look like powders to the naked eye!
Solid state structures of the group 18 elements
The elements in group 18 are referred to as noble gases, and it is not usual to
think of them in other states. The group 18 elements solidify only at low temperatures (Table 2.6) and the enthalpy change that accompanies the fusion
(melting) of one mole of each element is very small, indicating that the van
der Waals forces between the atoms in the solid state are very weak. In the
crystalline solid, the atoms of each group 18 element are close-packed.
Cubic close-packing is observed for the atoms of each of solid neon,
argon, krypton and xenon.
COMMERCIAL AND LABORATORY APPLICATIONS
Box 8.2 Liquid helium: an important coolant
Although the group 18 elements are usually encountered in the gas phase, they have a number of important
applications. One such is the use of liquid helium as a
coolant.
Liquid nitrogen (bp 77 K) is frequently used as a
coolant in the laboratory or in industry. However, it
is not possible to reach extremely low temperatures
by using liquid N2 alone; liquefaction of gaseous N2
under pressure provides a liquid at a temperature just
below the boiling point. In order to reach lower temperatures it is necessary to use a liquid with a boiling
point that is much lower than that of N2 , and liquid
helium is widely used for this purpose. As normally
8.7
found, helium boils at 4.2 K, and the use of liquid
helium is the most important method for reaching temperatures which approach absolute zero. Below 2.2 K,
isotopically pure 4 He undergoes a transformation into
the so-called He II. This is a remarkable liquid which
possesses a viscosity close to zero, and a thermal conductivity which far exceeds that of copper.
Until recently it was necessary to resort to cooling
potential superconducting materials in liquid helium
in order to observe the superconductivity, although
higher-temperature superconductors are now known
(see Box 8.4).
Elemental solids containing diatomic molecules
Figure 3.2 showed a selection of covalent homonuclear molecules (H2 , O2 ,
O3 , I2 , P4 , S6 and S8 ). Each is a molecular form of an element. In the gas
phase, these molecules are separate from one another, but in the solid
state, they pack together with van der Waals forces operating between
them. In this and the next two sections we consider the solid state
structures of H2 , elements from groups 17 and 16, an allotrope of phosphorus, and one group of allotropes of carbon ± all are molecular solids
and non-metals.
Elemental solids containing diatomic molecules
273
Fig. 8.10 Molecules of H2 rotate
freely in the solid state. Some
possible orientations for H2
molecules with respect to the
mid-point of the H H bond are
shown. Taking these and all
other possible orientations leads
to a description of H2 as a sphere.
Dihydrogen and di¯uorine
"
Metal lattices:
see Section 8.11
When gaseous H2 is cooled to 20.4 K, it lique®es.§ Further cooling to 14.0 K
results in the formation of solid dihydrogen. Even approaching absolute
zero (0 K), molecules of H2 possess sucient energy to rotate about a
point in the solid state lattice. Figure 8.10 shows that the result is that
each H2 molecule is described by a single sphere, the centre of which coincides with the midpoint of the H H bond. Solid H2 possesses an hcp
arrangement of such spheres, each of which represents one H2 molecule.
It is possible to apply the model of close-packed spheres because the H2
molecules are rotating at the temperature at which the solid state structure
has been determined.
Molecular F2 solidi®es at 53 K. Below 45 K, the molecules of F2 can freely
rotate, and the structure is described as distorted close-packed, with each F2
molecule being represented by a sphere.² The enthalpy of fusion of F2 is
0.5 kJ mol 1 and this low value suggests that only van der Waals forces
must be overcome in order to melt the solid.
The situation described for crystalline H2 and F2 is unusual. In the solid
state, molecules of most elements or compounds are not freely rotating.
Thermal motion such as the vibration of bonds does occur, but the positions
of the atoms in a molecule can often be de®ned to a reasonable degree of
accuracy. This means that the packing of spheres is not an appropriate
model for most solid state structures because the component species are
not spherical. It is applicable to elements of group 18 because they are monatomic, to H2 and F2 because the molecules are freely rotating, and to
metals.
Dichlorine, dibromine and diiodine (group 17)
Some physical and structural properties of Cl2 , Br2 and I2 are given in
Table 8.1. At 298 K (1 bar pressure), I2 is a solid, but Br2 is a liquid and
Cl2 a gas. Solid Cl2 , Br2 and I2 share common structures that di€er from
those of F2 .
In the crystalline state, molecules of Cl2 (or Br2 or I2 ) are arranged in a
zigzag pattern within a layer (Figure 8.11) and these layers of molecules are
stacked together. There are three characteristic distances in the structure
§
²
Here, and throughout the chapter, we consider phase changes at atmospheric pressure, unless
otherwise stated.
Above 45 K, a second phase with a more complicated structure exists.
274
CHAPTER 8 . Elements
Table 8.1 Some physical and structural properties of dichlorine, dibromine and diiodine. For details of the solid state structure
of these elements, refer to the text and Figure 8.11.
Element
Melting
point / K
Chlorine 171.5
Bromine 265.8
Iodine
386.7
fus H
/ kJ mol
1
6.4
10.6
15.5
Covalent
radius
(rcov ) / pm
Van der
Waals
radius
(rv ) / pm
Intramolecular
distance, a in
Figure 8.11
/ pm
Intermolecular
distance within
a layer, b in
Figure 8.11
/ pm
Intermolecular
distance
between
layers / pm
Intramolecular
distance for
molecule in
the gaseous
state / pm
99
114
133
180
195
215
198
227
272
332
331
350
374
399
427
199
228
267
Fig. 8.11 The solid state
structure of Cl2 , Br2 and I2
consists of X2 (X ˆ Cl, Br or I)
molecules arranged in zigzag
chains within layers. Part of one
layer is shown. Values of the
intramolecular X X distance,
a, and the intermolecular
distance, b, are listed in Table 8.1.
which are particularly informative. Consider the structure of solid Cl2 .
Within a layer (part of which is shown in Figure 8.11), the intramolecular
Cl Cl distance is 198 pm (a in Figure 8.11). The measured Cl Cl bond
distance is twice the covalent radius (Table 8.1). Also, within a plane,
we can measure intermolecular Cl:::Cl distances, and the shortest such
distance (b in Figure 8.11) is 332 pm. This is shorter than twice the van
der Waals radius of chlorine and suggests that there is some degree of interaction between the Cl2 molecules in a layer. The shortest intermolecular
Cl:::Cl distance between layers of molecules is 374 pm. The degree of intermolecular interaction becomes more pronounced in going from Cl2 to Br2 ,
and from Br2 to I2 as the distances in Table 8.1 indicate. Note also that the
I I bond length in solid I2 is longer than in a gaseous molecule (Table 8.1)
although there is little change in either the Cl Cl or Br Br bond length in
going from gaseous to solid Cl2 or Br2 . In solid I2 , the bonding interaction
between molecules is at the expense of some bonding character within each
I2 molecule.
8.8
Elemental molecular solids in groups 15 and 16
Sulfur (group 16)
Sulfur forms S S bonds in a variety of cyclic and chain structures and Table
3.2 listed a range of allotropes of sulfur. One allotrope is S6 which has a cyclic
structure with a chair conformation (Figure 8.12). When S6 crystallizes, the
rings pack together eciently to give a solid which is the highest density
form of elemental sulfur (2.2 g cm 3 ). Only van der Waals forces operate
between the rings.
Elemental molecular solids in groups 15 and 16
275
Fig. 8.12 Two views of an S6
molecule ± one view emphasizes
the chair conformation of the
ring.
"
Chair and boat conformers:
see Section 24.2
"
rcov (S) = 103 pm
Fig. 8.13 Two views of an S8
molecule. The shape of the ring is
often called a `crown'. The Se8
molecule also has this geometry.
The conformation of a molecule describes the relative spatial arrangement of the
atoms. Two commonly observed conformations of six-membered rings are the
chair and boat forms.
Crystalline orthorhombic sulfur (the a-form, and the standard state of the
element) consists of S8 rings (Figure 8.13) which are packed together with
van der Waals interactions between the rings. The average S S bond
length within each ring is 206 pm, consistent with the presence of single
bonds. The organization of the S8 rings in the crystalline state is shown in
Figure 8.14. The rings do not simply stack immediately on top of each other.
Monoclinic sulfur (the b-form) also contains S8 rings but these are less
eciently packed in the solid state (density ˆ 1:94 g cm 3 ) than are those
in orthorhombic sulfur (density ˆ 2:07 g cm 3 ). When orthorhombic sulfur
is heated to 368 K, a reorganization of the S8 rings in the lattice occurs
and the solid transforms into the monoclinic form. Single crystals of orthorhombic sulfur can be rapidly heated to 385 K, when they melt instead of
undergoing the orthorhombic to monoclinic transformation. If crystallization takes place at 373 K, the S8 rings adopt the structure of monoclinic
sulfur, but the crystals must be cooled rapidly to 298 K. On standing at
room temperature, monoclinic sulfur crystals change into the orthorhombic
allotrope within a few weeks.
276
CHAPTER 8 . Elements
Fig. 8.14 The arrangement of S8
rings in the solid state of
orthorhombic sulfur (the
standard state of the element).
Most allotropes of sulfur contain cyclic units (Table 3.2) but in some, Sx
chains of various lengths are present. Each chain contains S S single
bonds and forms a helix (Figure 8.15). An important property of a helix is
its handedness. It can turn in either a right-handed or left-handed manner;
each form is distinct from the other and they cannot be superimposed.
There are di€erent forms of polycatenasulfur which contain mixtures of
rings and chains, and these include rubbery and plastic sulfur. Filaments
of these can be drawn from molten sulfur; their compositions alter with
time, and at 298 K, transformation into orthorhombic sulfur eventually
occurs. Two examples of well-characterized allotropes which contains helical
chains are ®brous and laminar sulfur. In ®brous sulfur, the chains lie parallel
to one another and equal numbers of left- and right-handed helices are
present. In laminar sulfur, there is some criss-crossing of the helical chains.
The pre®x catena is used
within the IUPAC
nomenclature to mean a
chain structure.
Selenium and tellurium (group 16)
"
Se1 or Te1 = chain of
in®nite length
Fig. 8.15 A strand of helical
sulfur (S1 ) has a handedness:
(a) a right-handed helix and
(b) a left-handed helix. The two
chains are non-superimposable.
Elemental selenium and tellurium form both rings and helical chains, and
selenium possesses several allotropes. Crystalline monoclinic selenium is
red and contains Se8 rings with the same crown shape as S8 (Figure 8.13).
The standard state of the element is grey (or metallic) selenium, and in
the crystalline state, it contains helical chains of selenium atoms (Se1 ).
Tellurium has one crystalline form and this contains helical Te1 chains. In
both this and grey selenium, the axes of the chains lie parallel to each
A molecular allotrope of carbon: C60
277
other, and a view through each lattice down the axes shows the presence of a
hexagonal network.
Phosphorus (group 15)
Fig. 8.16 The tetrahedral P4
molecular unit present in white
phosphorus. All the P P
distances are equal.
8.9
The standard state of phosphorus is `white phosphorus'. This allotrope is not
the thermodynamically most stable state of the element but has been de®ned
as the standard state (see Section 1.17). The most stable crystalline form of
the element is black phosphorus, and this, and red phosphorus, are described
in Section 8.10.
Crystalline white phosphorus contains tetrahedral P4 molecules (Figure
8.16). The intramolecular P P distance is 221 pm, consistent with the
presence of P P single bonds (rcov ˆ 110 pm).
A molecular allotrope of carbon: C60
"
Diamond and graphite:
see Section 8.10
"
Restricted geometry of
carbon: see Section 5.14
When crystals of a
substance are grown from a
solution, they may contain
solvent of crystallization, the
presence of which is
indicated in the molecular
formula.
Fig. 8.17 One of the fullerenes ±
C60 . (a) The C60 molecule is made
up of fused ®ve- and sixmembered rings of carbon atoms
which form an approximately
spherical molecule. (b) A
representation of C60 showing
only the upper surface (in the
same orientation as in (a))
illustrating the localized single
and double carbon±carbon
bonds.
The allotropes of carbon that have, in the past, been most commonly cited
are diamond and graphite. Since the mid-1980s, new allotropes of carbon
± the fullerenes ± have been recognized.
The fullerenes are discrete molecules, and the most widely studied is
C60 (Figure 8.17a). The spherical shell of 60 atoms is made up of ®ve- and
six-membered rings and the carbon atoms are equivalent. Each ®vemembered ring (a pentagon) is connected to ®ve six-membered rings (hexagons). No ®ve-membered rings are adjacent to each other.
The geometry about a carbon atom is usually either linear, trigonal planar
or tetrahedral, and although apparently complex, the structure of C60
complies with this restriction. Each carbon atom in C60 is covalently
bonded to three others in an approximately trigonal planar arrangement.
Since the surface of the C60 molecule is relatively large, the deviation from
planarity at each carbon centre is small. The C C bonds in C60 fall into
two groups ± the bonds at the junctions of two hexagonal rings (139 pm)
and those at the junctions of a hexagonal and a pentagonal ring (145 pm).
Figure 8.17b shows the usual representation of C60 , with carbon±carbon
double and single bonds.
In the solid state at 298 K, the spherical C60 molecules are arranged in
a close-packed structure. However, most single crystal X-ray di€raction
studies of C60 have involved solvated samples rather than the pure solid
element. For example, C60 is soluble in benzene …C6 H6 †, and single crystals
278
CHAPTER 8 . Elements
THEORETICAL AND CHEMICAL BACKGROUND
Box 8.3 Why the name fullerene?
The geodesic dome housing Vancouver's interactive Science
World. The dome was designed for Expo '86.
# Dorling Kindersley.
The architect Richard Buckminster Fuller has designed
geodesic domes such as the one on the left and the one
built at EXPO '67 in Montreal. This geodesic dome was
constructed of hexagonal motifs but on its own, the
network can only lead to a planar sheet. The placement
of pentagonal panels at intervals in the structure leads
to a curvature of the surface, suf®cient to construct a
dome. The structure of C60 is also represented in a football (soccer ball), which has pentagonal (often black)
and hexagonal (often white) panels. C60 ± buckminsterfullerene ± has also been christened `bucky-ball'.
The name `fullerene' has been given to the class of
near-spherical Cn allotropes which include C60 , C70
and C84 .
A very readable article that conveys the excitement of
the discovery of C60 has been written by Harold W.
Kroto (1992): `C60 : Buckminsterfullerene, The Celestial
Sphere that Fell to Earth,' Angewandte Chemie, International Edition in English, vol. 31, p. 111.
In October 1996, Sir Harry Kroto of Sussex University, UK, and Professors Richard Smalley and Robert
Curl of Rice University, USA, were awarded the Nobel
Prize for Chemistry for their pioneering work on C60 .
grown by evaporating solvent from a solution of C60 in benzene have the
composition C60 :4C6 H6 . Figure 8.18 shows part of the crystal lattice of
C60 :4C6 H6 . The C60 molecules are arranged in an ordered manner with the
benzene molecules occupying the spaces between them.
Fig. 8.18 Part of the solid state
structure of C60 :4C6 H6 . The C60
molecules form an ordered array
with the benzene molecules in
between them. The formula
C60 :4C6 H6 indicates that one
mole of C60 crystallizes with four
moles of benzene; this ratio is
apparent in the diagram.
Structures of solids containing in®nite covalent lattices
279
THEORETICAL AND CHEMICAL BACKGROUND
Box 8.4 Superconductivity: alkali metal fullerides M3 C60
Alkali metals, M, reduce C60 to give fulleride salts of
type [M‡ ]3 [C60 ]3 and at low temperatures, some of
these compounds become superconducting. A superconductor is able to conduct electricity without
resistance and, until 1986, no compounds were known
that were superconductors above 20 K. The temperature at which a material becomes superconducting is
called its critical temperature (Tc ), and in 1987, this
barrier was broken ± high-temperature superconductors
were born. Many high-temperature superconductors
are metal oxides, for example YBa2 Cu4 O8
(Tc ˆ 80 K), YBa2 Cu3 O7 (Tc ˆ 95 K), Ba2 CaCu2 Tl2 O8
(Tc ˆ 110 K) and Ba2 Ca2 Cu3 Tl2 O10 (Tc ˆ 128 K).
The M3 C60 fulleride superconductors are structurally simpler than the metal oxide systems, and can
be described in terms of the alkali metal cations
occupying the interstitial holes in a lattice composed
of close-packed C60 cages. Each [C60 ]3 anion is
approximately spherical and a close-packing of spheres
approach is valid. In K3 C60 and Rb3 C60 , the [C60 ]3
cages are arranged in a face-centred cubic (fcc)
arrangement:
If you look back at Box 8.1, you will see that the
fcc unit cell contains an octahedral hole and eight
tetrahedral holes. There are also twelve octahedral
holes shared between adjacent unit cells. The alkali
metal cations in K3 C60 and Rb3 C60 completely
occupy the octahedral (grey) and tetrahedral (red)
holes:
The values of Tc for K3 C60 and Rb3 C60 are 18 K and
28 K respectively, but for Cs3 C60 (in which the C60
cages adopt a body-centred cubic lattice), Tc ˆ 40 K.
Cs3 C60 is (at present) the highest temperature superconductor of this family of alkali metal fullerides.
[What kind of interstitial holes can the Cs‡ ions
occupy in the bcc lattice?] Na3 C60 is structurally related
to its potassium and rubidium analogues, but it is not
superconducting. This area of fullerene chemistry is
actively being pursued with hopes of further raising
the Tc barrier.
A series of well-illustrated articles describing various
aspects of superconductivity can be found in Chemistry
in Britain (1994), vol. 30, pp. 722±748.
8.10
Structures of solids containing in®nite covalent lattices
Some non-metallic elements in the p-block crystallize with in®nite lattice
structures. (These are also called giant, or extended, lattices.) Diamond
and graphite are well-known examples and are described below along with
allotropes of boron, silicon, phosphorus, arsenic and antimony. When
these elements melt, covalent bonds are broken.
280
CHAPTER 8 . Elements
Boron (group 13)
Fig. 8.19 The B12 -icosahedral
unit that is the fundamental
building block in both a- and
b-rhombohedral boron. These
allotropes possess in®nite
covalent lattices in the solid state.
Fig. 8.20 Part of one layer of the
in®nite lattice of a-rhombohedral
boron. The building-blocks are
B12 -icosahedra. The overall
structure may be considered to
consist of spheres in a cubic
close-packed arrangement.
Delocalized, covalent bonding
interactions between the B12 units support the framework of
the in®nite lattice making it rigid.
The standard state of boron is the b-rhombohedral form. The structure of
this allotrope is complex and we begin the discussion instead with a-rhombohedral boron. The basic building-block of both a- and b-rhombohedral
boron is an icosahedral B12 -unit (Figure 8.19). Each boron atom is covalently
bonded to another ®ve boron atoms within the icosahedron, despite the fact
that a boron atom has only three valence electrons. The bonding within each
B12 -unit is delocalized and it is important to remember that the B B
connections in Figure 8.19 are not two-centre two-electron bonds.
The structure of a-rhombohedral boron consists of B12 -units arranged
in an approximately cubic close-packed manner. The boron atoms of
the icosahedral unit lie on a spherical surface, and so the close-packing of
spheres is an appropriate way in which to describe the solid state structure.
However, unlike the close-packed arrays described earlier, the `spheres' in
a-rhombohedral boron are covalently linked to each other. Part of the
structure (one layer of the in®nite lattice) is shown in Figure 8.20, and
such layers are arranged in an ABCABC . . . fashion (Figures 8.3 to 8.5).
The presence of B B covalent bonding interactions between the B12 -units
distinguishes this as an in®nite covalent lattice rather than a true closepacked assembly.
The structure of b-rhombohedral boron consists of B84 -units, linked
together by B10 -units. Each B84 -unit is conveniently described in terms of
three subunits ± …B12 †…B12 †…B60 †. At the centre of the B84 -unit is a B12 -icosahedron (Figure 8.21a) and radially attached to each boron atom is another
boron atom (Figure 8.21b). The term `radial' is used to signify that the
bonds that connect the second set of twelve boron atoms to the central
B12 -unit point outwards from the centre of the unit. The …B12 †…B12 † subunit so-formed lies inside a B60 -cage which has the same geometry as a C60
molecule (compare Figures 8.21c and 8.17a). The whole B84 -unit is
shown in Figure 8.21d. Notice that each of the boron atoms of the second
Structures of solids containing in®nite covalent lattices
281
Fig. 8.21 The construction of the B84 -unit that is the main building block in the in®nite lattice of b-rhombohedral boron. (a) In
the centre of the unit is a B12 -icosahedron. (b) To each boron atom in the central icosahedron, another boron atom is attached.
(c) A B60 -cage is the outer `skin' of the B84 -unit. (d) The ®nal B84 -unit contains three covalently bonded sub-units ±
(B12 )(B12 )(B60 ).
B12 -unit is connected to a pentagonal ring of the B60 -sub-unit. In the in®nite
lattice of b-rhombohedral boron, the B84 -units shown in Figure 8.21d are
interconnected by B10 -units. The whole network is extremely rigid, as is
that of the a-rhombohedral allotrope, and crystalline boron is very hard,
and has a particularly high melting point (2453 K for b-rhombohedral
boron).
Carbon (group 14)
Fig. 8.22 Part of the in®nite
covalent lattice of diamond, an
allotrope of carbon.
"
rcov (C) = 77 pm
rv (C) = 185 pm
Diamond and graphite are allotropes of carbon and possess in®nite covalent
lattices in the solid state. They di€er remarkably in their physical appearance
and properties. Crystals of diamond are transparent, extremely hard and are
highly prized for jewellery. Crystals of graphite are black and have a slippery
feel.
Figure 8.22 shows part of the crystalline structure of diamond. Each tetrahedral carbon atom forms four single C C bonds and the overall lattice is
very rigid.
The standard state of carbon is graphite. Strictly, this is a-graphite, since
there is another allotrope called b-graphite. The structures of both a- and bgraphite are in®nite covalent lattices, composed of parallel planes of fused
hexagonal rings. In a-graphite, the assembly contains two repeating layers,
whereas there are three repeat units in the b-form. Heating b-graphite
above 1298 K brings about a change to a-graphite.
The structure of a-graphite (`normal' graphite) is shown in Figure 8.23.
The C C bond distances within a layer are equal (142 pm) and the distance
between two adjacent layers is 335 pm, indicating that inter-layer interactions
are weak. The physical nature of graphite re¯ects this; it cleaves readily
between the planes of hexagonal rings. This property allows graphite to be
used as a lubricant. The carbon atoms in normal graphite are less eciently
packed than in diamond; the densities of a-graphite and diamond are 2.3 and
3.5 g cm 3 respectively.
The electrical conductivity of graphite is an important property of this
allotrope, and can be explained in terms of the structure and bonding. The
valence electronic con®guration of carbon is [He]2s2 2p2 , and as each
carbon atom forms a covalent bond to each of three other atoms in the
282
CHAPTER 8 . Elements
THEORETICAL AND CHEMICAL BACKGROUND
Box 8.5 The relationship between the structure of diamond and zinc blende
The structure of zinc blende (ZnS) was discussed in
Section 7.13, and the unit cell was shown in Figure
7.17. If all the zinc(II) and sul®de centres in this structure are replaced by carbon atoms, the unit cell shown
below results:
Figure 8.22 becomes apparent:
If we view this unit cell from a different angle, the same
representation of the diamond lattice that we showed in
We can therefore see that the diamond and zinc blende
structures are related.
same layer, one valence electron remains unused. The odd electrons are
conducted through the planes of hexagonal rings. The electrical resistivity
of a-graphite is 1:3 10 5 m at 293 K in a direction through the planes
of hexagonal rings, but is about 1 m in a direction perpendicular to the
planes. The electrical resistivity of diamond is 1 1011 m, and diamond
is an excellent insulator. All valence electrons in diamond are involved in
localized C C single bond formation.
Fig. 8.23 Part of the in®nite
covalent lattice of normal (a)
graphite. This allotrope is the
standard state of carbon. There
are two repeating layers
consisting of fused hexagonal
rings. The red lines between the
layers indicate which carbon
atoms lie directly over which
other atoms.
Structures of solids containing in®nite covalent lattices
283
COMMERCIAL AND LABORATORY APPLICATIONS
Box 8.6 Carbon nanotubes
We usually think of graphite sheets as being ¯at (see
Figure 8.23) but research over the last few years has
shown that it is possible to make `tube-like' structures
about 10 nm in diameter consisting of rolled graphite
sheets ± these are known as nanotubes. Carbon nanotubes can now be made on a macroscopic scale ± an
electric arc between a graphite anode and graphite
cathode in a helium atmosphere leads to the formation
of the tubes on the cathode surface. Collaborative work
between research groups in Switzerland and Brazil has
resulted in the production of a single layer of carbon
nanotubes aligned vertically on a polymer surface.
"
See also Section 8.13
The monolayer can be made to emit electrons and
these can be detected at an anode. The potential
application of this device is in the construction of ¯atscreen visual displays such as those in portable laptop computers. It is hoped that carbon nanotubes will
be one answer to the enhancement of the quality of
such ¯at screens.
An article that presents the three-dimensional beauty
of nanotubes and other Cn -species is: `Beyond C60 :
graphite structures for the future', Chemical Society
Reviews (1995), vol. 24, p. 341.
The electrical resistivity of a material measures its resistance (to an electrical
current). A good electrical conductor has a very low resistivity, and the reverse
is true for an insulator.
For a wire of uniform cross section, the resistivity () is given in units of ohm
metre (
m) where the resistance is in ohms, and the length of the wire is
measured in metres.
Resistance (in ) ˆ
Rˆ
resistivity (in m) length of wire (in m)
cross section (in m2 )
l
a
Silicon, germanium and tin (group 14)
In the solid state at 298 K silicon crystallizes with a diamond-type lattice
(Figure 8.22). Germanium and the grey allotrope of tin also adopt this
in®nite structure but the character of these elements puts them into the
category of semi-metals rather than non-metals. We discuss these allotropes
further in Section 8.11, but it is instructive here to compare some of their
physical properties (Table 8.2), particularly their electrical resistivities, to
understand why a distinction is made between the group 14 elements.
Phosphorus, arsenic, antimony and bismuth (group 15)
"
White phosphorus: see
Section 8.8
Although it is de®ned as the standard state of the element, white phosphorus
is a metastable state. Other allotropes of phosphorus are either amorphous,
or possess in®nite covalent lattices, but on melting, all allotropes give a liquid
containing P4 molecules.
Amorphous red phosphorus is more dense and less reactive than white
phosphorus. Crystallization of red phosphorus in molten lead produces a
monoclinic allotrope called Hittorf 's (violet) phosphorus. The solid state
284
CHAPTER 8 . Elements
Table 8.2 Some physical and structural properties of diamond, silicon, germanium and grey tin. All share a common in®nite
lattice type (Figure 8.22).
a
Element
Appearance of
crystalline solid
Melting
point
/ Ka
Enthalpy of
fusion / kJ mol
Density
/ g cm 3
Interatomic
distance in the
crystal lattice / pm
Electrical
resistivity / m
(temperature)
Carbon (diamond)
Transparent
3820
105
3.5
154
1 1011 (293 K)
Silicon
Blue-grey,
lustrous
1683
40
2.3
235
1 10
Germanium
Grey-white,
lustrous
1211
35
5.3
244
0.46 (295 K)
Tin (grey allotrope)
Dull grey
±
5.75
280
11 10
±
1
3
(273 K)
8
(273 K)
The grey allotrope of tin is the low-temperature form of the element. Above 286 K the white form is stable; this melts at 505 K.
structure is a complicated in®nite lattice and two views of the repeat unit are
shown in Figure 8.24. Units of this type are connected end-to-end to form
chain-like arrays of 3-coordinate phosphorus atoms. These chains lie
parallel to each other forming layers, but within a layer, the chains are not
bonded together. An in®nite lattice is created by placing the layers one on
top of another, such that an atom of the type labelled P 0 in Figure 8.24 is
covalently bonded to another similar atom (labelled P 00 ) in an adjacent
sheet. The chains in one layer lie at right angles to the chains in the next
bonded layer, giving a criss-cross network overall. The P P bond distances
in the lattice are all similar (222 pm) and are consistent with single covalent
bonds.
Black phosphorus is the most thermodynamically stable form of the
element. The solid state structure of the rhombohedral form of black phosphorus consists of a hexagonal net of P6 -rings (Figure 8.25) and hexagonal
P6 -units are also present in the orthorhombic form of the element.
In the solid state, the rhombohedral allotropes of arsenic, antimony and
bismuth are isostructural with the rhombohedral form of black phosphorus.
Down group 15, there is a tendency for the coordination number of each
atom to change from 3 (atoms within a layer) to 6 (three atoms within a
layer and three in the next layer). These allotropes of arsenic, antimony
and bismuth are known as the `metallic' forms, and the metallic character
of the element increases as group 15 is descended.
Fig. 8.24 Part of the in®nite
lattice of Hittorf 's phosphorus.
(a) Part of the chain-like arrays
of atoms; the repeat unit contains
21 atoms, and atoms P 0 and P 00
are equivalent atoms in adjacent
chains; the chains are linked
through the P 0 P 00 bond. (b) The
same unit viewed from the end,
emphasizing the channels that
run through the structure.
The structures of metallic elements at 298 K
285
Fig. 8.25 Layers of puckered
six-membered rings are present in
the structures of black
phosphorus and the
rhombohedral allotropes of
arsenic, antimony and bismuth,
all group 15 elements.
8.11
The structures of metallic elements at 298 K
Elements in groups 1 and 2 (the s-block) and the d-block are metallic. In the
p-block, a diagonal line approximately separates non-metallic from metallic
elements (Figure 8.26) although the distinction is not clear-cut. The solid
state structures of metals are readily described in terms of the packing of
their atoms. Table 8.3 lists lattice types for metals of the s- and d-blocks,
and also gives the melting points of these elements.
s-Block metals
In the solid state at 298 K, the atoms of each alkali metal (group 1) are
packed in a body-centred cubic arrangement (Figure 8.9). All these metals
are soft and have relatively low melting points (Table 8.3). The enthalpies
of fusion are correspondingly low, decreasing down group 1 from
3.0 kJ mol 1 for lithium to 2.1 kJ mol 1 for caesium.
With the exception of barium which has a bcc lattice, the alkaline earth
metals (group 2) possess hexagonal close-packed structures, and their
melting points are higher than those of the group 1 metals.
Fig. 8.26 A `diagonal' line is
often drawn through the p-block
to indicate approximately the
positions of the non-metals
(shown in green) and the metals
(shown in pink). The distinction
is not clear-cut and elements
lying along the line may show the
characteristics of both, being
classed as semi-metals.
286
CHAPTER 8 . Elements
Table 8.3 Structures (at 298 K) and melting points (K) of the metallic elements: g, hexagonal close-packed; g, cubic
close-packed (face-centred cubic); g, body-centred cubic.
1
2
3
4
5
6
7
8
9
10
11
12
Li
Be
454
1560
Na
Mg
371
923
K
Ca
Sc
Ti
V
Cr
Fe
Co
Ni
Cu
Zn
337
1115
1814
1941
2183
2180
Mn
see text
1519
1811
1768
1728
1358
693
Rb
Sr
Y
Zr
Nb
Mo
Tc
Ru
Rh
Pd
Ag
Cd
312
1050
1799
2128
2750
2896
2430
2607
2237
1828
1235
594
Cs
Ba
La
Hf
Ta
W
Re
Os
Ir
Pt
Au
301
1000
1193
2506
3290
3695
3459
3306
2719
2041
1337
Hg
see text
234
d-Block metals
An element or compound is
polymorphic if it exhibits
more than one crystalline
form depending on
conditions (e.g.
temperature, pressure).
Each structural form is a
polymorph.
At 298 K, the structures of the metals in the d-block are either hcp, ccp or bcc,
with the exceptions of mercury and manganese (see below). Table 8.3 shows
that the lattice type adopted depends, in general, on the position of the metal
in the periodic table. For most of the d-block metals, a close-packed structure
is observed at 298 K, and a bcc structure is present as a high-temperature form
of the element. Iron is unusual in that it adopts a bcc structure at 298 K, transforms to an fcc lattice at 1179 K, and reverts to a bcc structure at 1674 K.
The melting points of the metals in groups 3 to 11 are far higher than those of
the s-block metals, and enthalpies of fusion are correspondingly higher. Two dblock elements are worthy of special note. The ®rst is mercury. All three metals
in group 12 stand out among the d-block elements in possessing relatively low
melting points, but mercury is well known for the fact that it is a liquid at 298 K.
Its enthalpy of fusion is 2.3 kJ mol 1 , a value that is atypical of the d-block
elements but is, rather, similar to those of the alkali metals. In the crystalline
state, mercury atoms are arranged in a distorted simple cubic lattice (Figure
8.8). The second metal of interest is manganese. Atoms in the solid state are
arranged in a complex cubic lattice in such a way that there are four atom
types with coordination numbers of 12, 13 or 16. The reasons for this deviation
from one of the more common structural types are not simple to understand.
Metals and semi-metals in the p-block
"
Elemental boron:
see Section 8.10
The physical and chemical properties of the heavier elements in groups 13, 14
and 15 indicate that these elements are metallic; elements intermediate
between metals and non-metals are termed semi-metals, for example germanium. The structures described below are those observed at 298 K.
Aluminium possesses a cubic close-packed lattice, typical of a metallic
element. The melting point (933 K) is only slightly higher than that of
magnesium, the element preceding it in the periodic table, and is dramatically
lower than that of boron (2453 K). Atoms of thallium form an hcp structure
287
Metallic radius
typical of a metal, and indium possesses a distorted close-packed arrangement of atoms. The solid state structure of gallium is not so easily described;
there is one nearest-neighbour (247 pm) and six other close atoms at
distances between 270 and 279 pm. Gallium has a low melting point
(303 K) which means that it is a liquid metal in some places in the world
but a solid in others! The crystalline state of gallium is in between that of a
metal and a molecular solid containing Ga2 units.
The `diagonal' line in Figure 8.26 passes through group 14 between silicon
and germanium, suggesting that we might consider silicon to be a non-metal,
and germanium a metal. But the distinction is not clear-cut. In the solid state,
both elements have the same in®nite covalent lattice as diamond, but their
electrical resistivities are signi®cantly lower than that of diamond (Table
8.2), indicating metallic behaviour. The heaviest element, lead, possesses a
ccp lattice. The intermediate element is tin. White (b) tin is the stable
allotrope at 298 K, but at temperatures below 286 K, this transforms into
the grey a-form which has a diamond-type lattice (Table 8.2). The structure
of white tin is related to that of the grey allotrope by a distortion of the lattice
such that each tin atom goes from having four to six nearest-neighbours. The
density of white tin (7.31 g cm 3 ) is greater than that of the grey allotrope
(5.75 g cm 3 ). Tin is an unusual metal: the density decreases on going from
b- to a-Sn, whereas it is more usual for there to be an increase in density
on going from a higher to lower temperature polymorph. The transition
from white to grey tin is quite slow, but it can be dramatic. In the 19th century, military uniforms used tin buttons which crumbled in exceptionally
cold winters. Similarly, in 1851, the citizens of Zeitz were alarmed to discover
that the tin organ pipes in their church had crumbled to powder!
The structure of bismuth was described in Section 8.10.
8.12
Metallic radius
The metallic radius is half of the distance between the nearest-neighbour
atoms in a solid state metallic lattice, and Table 8.4 lists metallic radii for
the s- and d-block elements. Atom size increases down each of groups 1
Table 8.4 Metallic radii (pm) of the s- and d-block metals; lanthanum (La) is usually classi®ed with the f-block elements.
1
2
3
4
5
6
7
8
9
10
11
12
Li
Be
157
112
Na
Mg
191
160
K
Ca
Sc
Ti
V
Cr
Mn
Fe
Co
Ni
Cu
Zn
235
197
164
147
135
129
137
126
125
125
128
137
Rb
Sr
Y
Zr
Nb
Mo
Tc
Ru
Rh
Pd
Ag
Cd
250
215
182
160
147
140
135
134
134
137
144
152
Cs
Ba
La
Hf
Ta
W
Re
Os
Ir
Pt
Au
Hg
272
224
187
159
147
141
137
135
136
139
144
±
288
CHAPTER 8 . Elements
"
Van der Waals radius:
see Section 2.21
Covalent radius:
see Section 3.3
Ionic radius:
see Section 7.14
and 2. In each of the triads of the d-block elements, there is generally an
increase in radius in going from the ®rst to second row element, but very
little change in size in going from the second to the third row metal. This
latter observation is due to the presence of a ®lled 4f level, and the socalled lanthanoid contraction ± the ®rst row of lanthanoid elements lies
between lanthanum (La) and hafnium (Hf). The poorly shielded 4f electrons
are relatively close to the nucleus and have little e€ect on the observed radius.
The metallic radius is half of the distance between the nearest-neighbour atoms
in a solid state metal lattice.
8.13
An electrical conductor
o€ers a low resistance
(measured in ohms, ) to
the ¯ow of an electrical
current (measured in
amperes, A). An insulator
o€ers a high resistance.
Worked example 8.1
Metallic bonding
Metals are electrical conductors
One physical property that characterizes a metal is its low electrical resistivity
± that is, a metal conducts electricity very well. With the exception of
mercury, all elements that are metallic at 298 K are solid. Although the packing of atoms is a convenient means of describing the solid state structure of
metals, it gives no feeling for the communication that there must be between
the atoms. Communication is implicit in the property of electrical conductivity, since electrons must be able to ¯ow through an assembly of atoms
in a metal. In order to understand why metals are such good electrical
conductors, we must ®rst consider the bonding between atoms in a metal.
The electrical resistivity of titanium is 4:3 10 7 m. What is the resistance of
a 0.50 m strip of titanium wire with cross section 8:0 10 7 m2 ?
Equation needed:
Resistance (in † ˆ
resistivity (in m† length of wire (in m)
cross section (in m2 )
Ensure all units are consistent ± in this example, they are.
Resistance ˆ
4:3 10 7 †…0:50†
8:0 10 7
ˆ 0:27 A `sea of electrons'
An early approach to metallic bonding (the Drude±Lorentz theory) was to
consider a model in which the valence electrons of each metal atom were
free to move in the crystal lattice. Thus, instead of simply being composed
of neutral atoms, the metal lattice is considered to be an assembly of positive
ions (the nuclei surrounded by their core electrons) and electrons (the valence
electrons). When a potential di€erence§ is applied across a piece of metal, the
valence electrons move from high to low potential and a current ¯ows.
§
Ohm's Law states: V ˆ IR (potential di€erence (in V) ˆ current (in A) resistance (in )).
Metallic bonding
289
In this model, a metallic element is considered to consist of positive ions
(arranged, for example, in a close-packed manner) and a `sea of electrons'.
The theory gives a satisfactory general explanation for the conduction of
electricity but cannot account for the detailed variation of electrical conductivities amongst the metallic elements. Several other theories have been
described, of which band theory is the most general.
Band theory
"
LCAO: see Section 3.12
Fig. 8.27 The interaction of two
2s atomic orbitals in Li2 leads to
the formation of two MOs. If
there are three lithium atoms,
three MOs are formed, and so on.
For Lin , there are n molecular
orbitals, but because the 2s
atomic orbitals were all of the
same energy, the energies of these
MOs are very close together and
are described as a band of
orbitals.
Band theory follows from a consideration of the energies of the molecular
orbitals of an assembly of metal atoms. In constructing the ground state
electronic con®guration of a molecule, we have previously applied the
aufbau principle and have arranged the electrons so as to give the lowest
energy state. In the case of a degeneracy, this may give singly occupied highest lying molecular orbitals (for example in the B2 and O2 molecules).
A molecular orbital diagram that describes the bonding in a metallic solid is
characterized by having a large number of orbitals which are very close in
energy. In this case, they form a continuous set of energy states called a band.
These arise as follows. In an LCAO approach, we consider the interaction
between similar atomic orbitals, for example 2s with 2s, and 2p with 2p.
Figure 8.27 shows the result of the interaction of 2s atomic orbitals for
di€erent numbers of lithium atoms. The energies of these 2s atomic orbitals
are the same. If two Li atoms combine, the overlap of the two 2s atomic
orbitals leads to the formation of two MOs. If there are three lithium atoms,
three MOs are formed, and if there are four metal atoms, four MOs result.
For an assembly containing n lithium atoms, there must be n molecular orbitals,
but because the 2s atomic orbitals are all of the same energy, the energies of the
resultant MOs are very close together and can be described as a band of orbitals.
The occupation of the band depends upon the number of valence electrons
available. Each Li atom provides one valence electron and the band shown
in Figure 8.27 is half-occupied. This leads to a delocalized picture of the bonding in the metal, and the metal±metal bonding is non-directional.
When metal atoms have more than one type of atomic orbital in the
valence shell, correspondingly more bands are formed. If two bands are
290
CHAPTER 8 . Elements
close together they will overlap, giving a single band in which there is mixed
orbital character (for example s and p character). Some bands will be separated from other bands by de®ned energy gaps, as shown in Figure 8.28.
The lowest band is fully occupied with electrons, while the highest band is
empty. The central band is partially occupied with electrons, and because
the energy states that make up the band are so close, the electrons can
move between energy states in the same band. In the bulk metal, electrons
are therefore mobile. If a potential di€erence is applied across the metal,
the electrons move in a direction from high to low potential and a current
¯ows. The energy gaps between bands are relatively large, and it is the
presence of a partially occupied band that characterizes a metal.
A band is a group of MOs that are extremely close in energy. The energy di€erences are so small that the system behaves as if a continuous, non-quantized
variation of energy within the band is possible.
A band gap occurs when there is a signi®cant energy di€erence between two
bands.
Fig. 8.28 The molecular orbitals
that describe the bonding in a
bulk metal are very close in
energy and are represented by
bands. Bands may be fully
occupied with electrons (shown
in blue), unoccupied (shown in
white), or partially occupied. The
®gure shows a schematic
representation of these bands for
a metal.
Fig. 8.29 The energy di€erence
(the band gap) between occupied
(shown in blue) and unoccupied
(shown in white) bands of MOs
decreases in going down group
14. This allows a change from
non-metallic towards metallic
character in going down the
group.
Semiconductors
In going down each of groups 13, 14, 15 and 16, there is a transition from a
non-metal to a metal, passing through some intermediate stage characterized
by the semi-metals. As we have already seen for the group 14 elements, the
metal/non-metal boundary is not well de®ned.
Figure 8.29 gives a representation of the bonding situation for these
allotropes. The MO diagram for bulk diamond can be represented in terms
of a fully occupied and an unoccupied band. There is a large band gap
(520 kJ mol 1 ) and diamond is an insulator. The situation for silicon and
Problems
291
germanium can be similarly represented, but now the band gaps are much smaller (106 and 64 kJ mol 1 respectively). In grey tin, only 8 kJ mol 1 separates the
®lled and empty bands, and here the situation is approaching that of a single
band which is partly occupied. The conduction of electricity in silicon, germanium and grey tin depends upon thermal population of the upper band
(the conduction band) and these allotropes are classed as semiconductors. As
the temperature is increased, some electrons will have sucient energy to
make the transition from the lower to higher energy band. The smaller the
band gap, the greater the number of electrons that will possess sucient
energy to make the transition, and the greater the electrical conductivity.
SUMMARY
In this chapter, we have discussed the structures of elements in the solid state.
Do you know what the following terms mean?
.
close-packing of spheres
.
crystalline solid
.
electrical resistivity
.
cubic close-packing
.
amorphous solid
.
insulator
.
hexagonal close-packing
.
metallic radius
simple cubic lattice
chair and boat conformers of a
six-membered ring
.
.
.
metallic bonding
.
body-centred cubic lattice
.
.
interstitial hole
.
handedness of a helical chain
.
catena
band gap
.
semiconductor
You should be able:
.
to discuss how the close-packing of spheres can give
rise to at least two assemblies
.
.
.
.
.
to discuss the relationship between simple and bodycentred cubic packing of spheres
to distinguish between intra- and intermolecular
bonds in solid state structures (are both types of
bonding always present?)
.
to state how many nearest-neighbours an atom has
in ccp, hcp, simple cubic and bcc arrangements
to describe structural variation among the
allotropes of boron, carbon, phosphorus and sulfur
.
to compare the eciencies of packing in ccp, hcp,
simple cubic and bcc arrangements
to describe similarities and di€erences in the solid
state structures of the elements of group 17
.
to give examples of metals with di€erent types of
lattice structures
.
to describe using simple band theory how a metal
can conduct electricity
to appreciate why the packing of spheres is an
appropriate model for some but not all solid state
lattices
PROBLEMS
8.1 When spheres of an equal size are close-packed,
what are the features that characterize whether the
arrangement is hexagonal or cubic close-packed?
Draw a representation of the repeat unit for each
arrangement.
8.2 What is an interstitial hole? What types of holes are
present in hcp and ccp arrangements?
8.3 How are spheres organized in a body-centred cubic
arrangement? How does the body-centred cubic
arrangement di€er from a simple cubic one?
8.4 What is meant by a nearest-neighbour in an
assembly of spheres? How many nearestneighbours does each sphere possess in a (a) cubic
close-packed, (b) hexagonal close-packed, (c)
simple cubic and (d) body-centred cubic
arrangement?
8.5 Write down the ground state electronic
con®guration of Cl, S and P and use these data to
describe the bonding in each of the molecular units
present in (a) solid dichlorine, (b) orthorhombic
sulfur, (c) ®brous sulfur and (d) white phosphorus.
292
CHAPTER 8 . Elements
8.6 What is meant by electrical resistivity? Use the data
in Table 8.5 to discuss the statement: `All metals are
good electrical conductors.'
Table 8.5 Table of data for problem 8.6. All the
resistivities are measured for pure samples at 273 K
unless otherwise stated.
Element
Electrical resistivity / m
Copper
Silver
Aluminium
Iron
Gallium
Tin
Mercury
Bismuth
Manganese
Silicon
Boron
Phosphorus
1:5 10 8
1:5 10 8
2:4 10 8
8:6 10 8
1:4 10 7
3:9 10 7
9:4 10 7
1:1 10 6
1:4 10 6
1:0 10 3
1:8 104
1:0 109 (293 K)
8.10 Comment on the relative values of the metallic
(197 pm) and ionic radii (100 pm) of calcium.
8.11 Account for the following observations:
(a) The density of a-graphite is less than that of
diamond.
(b) In group 13, the melting point of brhombohedral B is much higher (2453 K) than that
of Al (933 K).
(c) The group 1 metals tend to exhibit lower values
of a H o (298 K) than metals in the d-block.
8.12 To what processes do the values of (a)
fus H ˆ 0:7 kJ mol 1 , (b) vap H ˆ 5:6 kJ mol 1
and (c) a H o ˆ 473 kJ mol 1 for nitrogen refer?
Discuss the relative magnitudes of the values.
8.13 A localized covalent -bond is directional, while
metallic bonding is non-directional. Discuss the
features of covalent and metallic bonding that lead
to this di€erence.
8.7 Brie¯y discuss allotropy with respect to carbon.
What is the standard state of this element? Explain
why a-graphite and diamond show such widely
di€ering electrical resistivities, and suggest why the
electrical resistivity in a graphite rod is directiondependent.
8.8 What is meant by solvent of crystallization? The
crystallization of C60 from benzene and
diiodomethane yields solvated crystals of formula
C60 :xC6 H6 :yCH2 I2 . If the loss of solvent leads to a
32.4% reduction in the molar mass, what is a
possible stoichiometry of the solvated compound?
8.9 What lattice structure is typical of an alkali metal at
298 K? Table 8.6 lists values of the enthalpies of
fusion and vaporization for the alkali metals.
Describe what is happening in each process in terms
of interatomic interactions. Use the data in Table
8.6 to plot graphs which show the trends in melting
point, and enthalpies of fusion and atomization
down group 1. How do you account for the trends
observed, and any relationships that there may be
between them?
Table 8.6 Table of data for problem 8.9.
Alkali
metal
Melting
point / K
Enthalpy
of fusion
/ kJ mol 1
Enthalpy of
atomization
/ kJ mol 1
Lithium
Sodium
Potassium
Rubidium
Caesium
454
371
337
312
301
3.0
2.6
2.3
2.2
2.1
162
108
90
82
78
8.14 Using simple band theory, describe the di€erences
between electrical conduction in a metal such
as lithium and in a semiconductor such as
germanium.
8.15 (a) What is a band gap? (b) Which of the
following would you associate with a metal, a
semiconductor and an insulator: (i) a large band
gap, (ii) a very small band gap, (iii) a partially
occupied band?
Additional problems
8.16 ReO3 is a structural type. The structure can be
described as a simple cubic array of Re atoms with
O atoms located in the middle of each of the cube
edges. Construct a unit cell of ReO3 . What are the
coordination numbers of Re and O? Con®rm that
your unit cell gives the correct stoichiometry for
ReO3 .
8.17 Two types of semiconductors, n- and p-types, are
made by doping a host such as silicon with a small
amount of an element that has more or fewer
valence electrons than the host. How do you think
doping Si with As to give an n-type semiconductor
would change the electronic conductivity of the
material?