From our earlier studies, we know that liquids and gases

are called fluids because of their ability to flow. The

fluidity in both of these states is due to the fact that the

molecules are free to move about. On the contrary, the

constituent particles in solids have fixed positions and

can only oscillate about their mean positions. This

explains the rigidity in solids. These properties depend

upon the nature of constituent particles and the binding

forces operating between them. The correlation between

structure and properties helps in the discovery of new

solid materials with desired properties. For example,

carbon nanotubes are new materials that have potential

to provide material that are tougher than steel, lighter

than aluminium and have more conductive property than

copper. Such materials may play an expanding role in

future development of science and society. Some other

materials which are expected to play an important role

in future are high temperature superconductors,

magnetic materials, biodegradable polymers for

packaging, biocompliant solids for surgical implants, etc.

Thus, the study of this state becomes more important in

the present scenario.

In this Unit, we shall discuss different possible

arrangements of particles resulting in several types of

structures and explore why different arrangements of

structural units lend different properties to solids. We

will also learn how these properties get modified due to

the structural imperfections or by the presence of

impurities in minute amounts.

After studying this Unit, you will be

able to

• describe general characteristics of

solid state;

• distinguish between amorphous

and crystalline solids;

• classify crystalline solids on the

basis of the nature of binding

forces;

• define crystal lattice and unit cell;

• explain close packing of particles;

• describe different types of voids

and close packed structures;

• calculate the packing efficiency of

different types of cubic unit cells;

• correlate the density of a

substance with its unit cell

properties;

• describe the imperfections in

solids and their effect on

properties;

• correlate the electrical and

magnetic properties of solids and

their structure.

Objectives

The vast majority of solid substances like high temperature

superconductors, biocompatible plastics, silicon chips, etc. are destined

to play an ever expanding role in future development of science.

The Solid State

Unit

The Solid State

2020-21

2Chemistry

In Class XI you have learnt that matter can exist in three states namely,

solid, liquid and gas. Under a given set of conditions of temperature and

pressure, which of these would be the most stable state of a given

substance depends upon the net effect of two opposing factors. These

are intermolecular forces which tend to keep the molecules (or atoms

or ions) closer, and the thermal energy, which tends to keep them apart

by making them move faster. At sufficiently low temperature, the thermal

energy is low and intermolecular forces bring them so close that they

cling to one another and occupy fixed positions. These can still oscillate

about their mean positions and the substance exists in solid state. The

following are the characteristic properties of the solid state:

(i) They have definite mass, volume and shape.

(ii) Intermolecular distances are short.

(iii) Intermolecular forces are strong.

(iv) Their constituent particles (atoms, molecules or ions) have fixed

positions and can only oscillate about their mean positions.

(v) They are incompressible and rigid.

Solids can be classified as crystalline or amorphous on the basis of the

nature of order present in the arrangement of their constituent particles.

A crystalline solid usually consists of a large number of small crystals,

each of them having a definite characteristic geometrical shape. The

arrangement of constituent particles (atoms, molecules or ions) in a crystal

is ordered and repetitive in three dimensions. If we observe the pattern in

one region of the crystal, we can predict accurately the position of particles

in any other region of the crystal however far they may be from the place

of observation. Thus, crystal has a long range order which means that

there is a regular pattern of arrangement of particles which repeats itself

periodically over the entire crystal. Sodium chloride and quartz are typical

examples of crystalline solids. Glass, rubber and many plastics do not

form crystals when their liquids solidify on cooling. These are called

amorphous solids. The term amorphous comes from the Greek word

amorphos, meaning no form.The arrangement of constituent particles

(atoms, molecules or ions) in such a solid has only short range order. In

such an arrangement, a regular and

periodically repeating pattern is observed

over short distances only. Regular patterns

are scattered and in between the

arrangement is disordered. The structures

of quartz (crystalline) and quartz glass

(amorphous) are shown in Fig. 1.1 (a) and

(b) respectively. While the two structures

are almost identical, yet in the case of

amorphous quartz glass there is no long

range order. The structure of amorphous

solids is similar to that of liquids. Due to

the differences in the arrangement of the

constituent particles, the two types of solids

differ in their properties.

1.11.1

1.1

GeneralGeneral

General

CharacteristicsCharacteristics

Characteristics

of Solid Stateof Solid State

of Solid State

1.21.2

1.2

AmorphousAmorphous

Amorphous

and Crystallineand Crystalline

and Crystalline

SolidsSolids

Solids

Fig. 1.1: Two dimensional structure of

(a) quartz and (b) quartz glass

2020-21

The Solid State

Crystalline solids have a sharp melting point. At a characteristic

temperature they melt abruptly and become liquid. On the other hand,

amorphous solids soften, melt and start flowing over a range of

temperature and can be moulded and blown into various shapes.

Amorphous solids have the same structural features as liquids and are

conveniently regarded as extremely viscous liquids. They may become

crystalline at some temperature. Some glass objects from ancient

civilisations are found to become milky in appearance because of some

crystallisation. Like liquids, amorphous solids have a tendency to flow,

though very slowly. Therefore, sometimes these are called pseudo

solids or super cooled liquids.

Amorphous solids are isotropic in nature. Their properties such as

mechanical strength, refractive index and electrical conductivity, etc.,

are same in all directions. It is because there is no long range order in

them and arrangement of particles is not definite along all the directions.

Hence, the overall arrangement becomes equivalent in all directions.

Therefore, value of any physical property would be same along

any direction.

Crystalline solids are anisotropic in nature, that

is, some of their physical properties like electrical

resistance or refractive index show different values

when measured along different directions in the

same crystals. This arises from different arrangement

of particles in different directions. This is illustrated

in Fig. 1.2. This figure shows a simple two-

dimensional pattern of arrangement of two kinds of

atoms. Mechanical property such as resistance to

shearing stress might be quite different in two

directions indicated in the figure. Deformation in CD

direction displaces row which has two different types

of atoms while in AB direction rows made of one type

of atoms are displaced. The differences between the

crystalline solids and amorphous solids are

summarised in Table 1.1.

Fig. 1.2: Anisotropy in crystals is due

to different arrangement of

particles along different

directions.

Table 1.1: Distinction between Crystalline and Amorphous Solids

Shape

Melting point

Cleavage

property

Heat of fusion

Definite characteristic geometrical shape

Melt at a sharp and characteristic

temperature

When cut with a sharp edged tool, they

split into two pieces and the newly

generated surfaces are plain and

smooth

They have a definite and characteristic

enthalpy of fusion

Irregular shape

Gradually soften over a range of

temperature

When cut with a sharp edged tool, they

cut into two pieces with irregular

surfaces

They do not have definite enthalpy of

fusion

Property Crystalline solids Amorphous solids

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4Chemistry

Besides crystalline and amorphous solids, there are some solids

which apparently appear amorphous but have microcrystalline

structures. These are called polycrystalline solids. Metals often occur

in polycrystalline condition. Individual crystals are randomly oriented

so a metallic sample may appear to be isotropic even though a single

crystal is anisotropic.

Amorphous solids are useful materials. Glass, rubber and plastics

find many applications in our daily lives. Amorphous silicon is one of the

best photovoltaic material available for conversion of sunlight into electricity.

In Section 1.2, we have learnt about amorphous substances and that

they have only short range order. However, most of the solid substances

are crystalline in nature. For example, all the metallic elements like iron,

copper and silver; non-metallic elements like sulphur, phosphorus and

iodine and compounds like sodium chloride, zinc sulphide and

naphthalene form crystalline solids.

Crystalline solids can be classified in various ways. The method

depends on the purpose in hand. Here, we will classify crystalline solids

on the basis of nature of intermolecular forces or bonds that hold the

constituent particles together. These are — (i) Van der waals forces;

(ii) Ionic bonds; (iii) Covalent bonds; and (iv) Metallic bonds. On this basis,

crystalline solids are classified into four categories viz., molecular, ionic,

metallic and covalent solids. Let us now learn about these categories.

Molecules are the constituent particles of molecular solids. These are

further sub divided into the following categories:

(i) Non polar Molecular Solids

: They comprise either atoms, for example,

argon and helium or the molecules formed by non polar covalent

1.3

ClassificationClassification

Classification

of Crystallineof Crystalline

of Crystalline

Solids

1.3.1 Molecular

Solids

Intext QuestionsIntext Questions

Intext Questions

1.1 Why are solids rigid?

1.2 Why do solids have a definite volume?

1.3 Classify the following as amorphous or crystalline solids: Polyurethane,

naphthalene, benzoic acid, teflon, potassium nitrate, cellophane, polyvinyl

chloride, fibre glass, copper.

1.4 Refractive index of a solid is observed to have the same value along all directions.

Comment on the nature of this solid. Would it show cleavage property?

Anisotropy

Nature

Order in

arrangement

of constituent

particles

Anisotropic in nature

True solids

Long range order

Isotropic in nature

Pseudo solids or super cooled liquids

Only short range order.

2020-21

The Solid State

1.3.3 Metallic

Solids

1.3.4 Covalent or

Network

Solids

1.3.2 Ionic Solids

bonds, for example, H

, Cl

and I

. In these solids, the atoms or

molecules are held by weak dispersion forces or London forces

about which you have learnt in Class XI. These solids are soft and

non-conductors of electricity. They have low melting points and are

usually in liquid or gaseous state at room temperature and pressure.

(ii) Polar Molecular Solids: The molecules of substances like HCl, SO

etc. are formed by polar covalent bonds. The molecules in such

solids are held together by relatively stronger dipole-dipole

interactions. These solids are soft and non-conductors of electricity.

Their melting points are higher than those of non polar molecular

solids yet most of these are gases or liquids under room

temperature and pressure. Solid SO

and solid NH

are some

examples of such solids.

(iii) Hydrogen Bonded Molecular Solids: The molecules of such solids

contain polar covalent bonds between H and F, O or N atoms.

Strong hydrogen bonding binds molecules of such solids like H

(ice). They are non-conductors of electricity. Generally they are

volatile liquids or soft solids under room temperature and pressure.

Ions are the constituent particles of ionic solids. Such solids are formed

by the three dimensional arrangements of cations and anions bound

by strong coulombic (electrostatic) forces. These solids are hard and

brittle in nature. They have high melting and boiling points. Since the

ions are not free to move about, they are electrical insulators in the

solid state. However, in the molten state or when dissolved in water,

the ions become free to move about and they conduct electricity.

Metals are orderly collection of positive ions surrounded by and held

together by a sea of free electrons. These electrons are mobile and are

evenly spread out throughout the crystal. Each metal atom contributes

one or more electrons towards this sea of mobile electrons. These free

and mobile electrons are responsible for high electrical and thermal

conductivity of metals. When an electric field is applied, these electrons

flow through the network of positive ions. Similarly, when heat is

supplied to one portion of a metal, the thermal energy is uniformly

spread throughout by free electrons. Another important characteristic

of metals is their lustre and colour in certain cases. This is also due

to the presence of free electrons in them. Metals are highly malleable

and ductile.

A wide variety of crystalline solids of non-metals result from the

formation of covalent bonds between adjacent atoms throughout the

crystal. They are also called giant molecules. Covalent bonds are

strong and directional in nature, therefore atoms are held very strongly

at their positions. Such solids are very hard and brittle. They have

extremely high melting points and may even decompose before melting.

They are insulators and do not conduct electricity. Diamond (Fig. 1.3)

and silicon carbide are typical examples of such solids. Although

Graphite (Fig. 1.4) also belongs to this class of crystals, but it is soft

and is a conductor of electricity. Its exceptional properties are due to

2020-21

6Chemistry

its typical structure. Carbon

atoms are arranged in

different layers and each

atom is covalently bonded to

three of its neighbouring

atoms in the same layer. The

fourth valence electron of

each atom is present

between different layers and

is free to move about. These

free electrons make graphite

a good conductor of

electricity. Different layers

can slide one over the other.

This makes graphite a soft

solid and a good solid

lubricant.

The different properties

of the four types of solids

are listed in Table 1.2.

Fig. 1.4: Structure of graphite

Fig. 1.3: Network structure

of diamond

Table 1.2: Different Types of Solids

Type of Solid Constituent Bonding/ Examples Physical Electrical Melting

Particles Attractive Nature Conduc- Point

Forces tivity

(1) Molecular solids

(i) Non polar Molecules Dispersion or Ar, CCl

, Soft Insulator Very low

London forces H

, I

, CO

(ii) Polar Dipole-dipole HCl, SO

Soft Insulator Low

interactions

(iii) Hydrogen Hydrogen H

O (ice) Hard Insulator Low

bonded bonding

(2) Ionic solids Ions Coulombic or NaCl, MgO, Hard but Insulators High

electrostatic ZnS, CaF

brittle in solid

state but

conductors

in molten

state and

in aqueous

solutions

(3) Metallic solids Positive Metallic Fe, Cu, Ag, Hard but Conductors Fairly

ions in a bonding Mg malleable in solid high

sea of and state as

delocalised ductile well as in

electrons molten

state

2020-21

The Solid State

You must have noticed that when tiles are placed to cover a floor, a

repeated pattern is generated. If after setting tiles on floor we mark a

point at same location in all the tiles (e.g. Centre of the tile) and see the

marked positions only ignoring the tiles, we obtain a set of points. This

set of points is the scaffolding on which pattern has been developed by

placing tiles. This scaffolding is a space lattice on which two-dimensional

pattern has been developed by placing structural units on its set of

points (i.e. tile in this

case). The structural

unit is called basis or

motif. When motifs are

placed on points in

space lattice, a pattern

is generated. In crystal

structure, motif is

a molecule, atom or

ion. A space lattice,

also called a crystal

lattice, is the pattern of

points representing the

locations of these motifs.

In other words, space

lattice is an abstract

scaffolding for crystal

structure. When we

place motifs in an

identical manner on

points of space lattice,

Intext QuestionsIntext Questions

Intext Questions

1.5 Classify the following solids in different categories based on the nature of

intermolecular forces operating in them:

Potassium sulphate, tin, benzene, urea, ammonia, water, zinc sulphide,

graphite, rubidium, argon, silicon carbide.

1.6 Solid A is a very hard electrical insulator in solid as well as in molten state

and melts at extremely high temperature. What type of solid is it?

1.7 Ionic solids conduct electricity in molten state but not in solid state. Explain.

1.8 What type of solids are electrical conductors, malleable and ductile?

(4) Covalent or Atoms Covalent SiO

Hard Insulators Very

network solids bonding (quartz), high

SiC, C

(diamond),

AlN,

(graphite)

Soft Conductor

(exception)

1.41.4

1.4

CrystalCrystal

Crystal

Lattices andLattices and

Lattices and

Unit CellsUnit Cells

Unit Cells

Lattice

Point

(a) Motif to make

crystal structure

(b) Space lattice or a crystal lattice

(two-dimensional)

Lattice

Point

Lattice

Point

crystal

Fig. 1.5: (a) Motif (b) Space lattice (two-dimensional (c) Hypothetical

two-dimensional crystal structure

2020-21

8Chemistry

we get crystal structure. Fig. 1.5 shows a motif, a two-dimensional

lattice and a hypothetical two-dimensional crystal structure obtained

by placing motifs in the two-dimensional lattice.

Spacial arrangement of lattice points gives rise to different types of

lattices. Fig 1.6 shows arrangement of points in two different lattices.

Fig. 1.6: Arrangement of points in two different lattices

Lattice A Lattice B

In the case of crystalline solids, space lattice is a three-dimensional

array of points. The crystal structure is obtained by associating

structurral motifs with lattice points. Each repeated basis or motif has

same structure and same spacial orientation as other one in a crystal.

The environment of each motif is same throughout the crystal except

for on surface.

Following are the characteristics of a crystal lattice:

(a) Each point in a lattice is called lattice point or lattice site.

(b) Each point in a crystal lattice represents one constituent particle which

may be an atom, a molecule (group of atoms) or an ion.

the lattice.

We need only a small

part of the space lattice of a

crystal to spacify crystal

completely. This small part

is called unit cell. One can

choose unit cell in many

ways. Normally that cell is

chosen which has

perpendicular sides of

shortest length and one can construct entire crystal by translational

displacement of the unit cell in three dimensions. Fig. 1.7 shows

movement of unit cell of a two-dimensional lattice to construct the

entire crystal structure. Also, unit cells have shapes such that these fill

the whole lattice without leaving space between cells.

In two dimensions a parallelogram with side of length ‘a’ and ‘b’

and an angle r between these sides is chosen as unit cell. Possible unit

cells in two dimensions are shown in Fig. 1.8.

Fig. 1.7: Generating

hypothetical two-

dimensional crystal

structure by shifting

square in the

direction of arrows.

Unit

cell

2020-21

The Solid State

A portion of three-dimensional crystal lattice and its

unit cell is shown in Fig. 1.9.

In the three-dimensional crystal structure, unit cell is

characterised by:

(i) its dimensions along the three edges a, b and c. These

edges may or may not be mutually perpendicular.

(ii) angles between the edges, α (between b and c), β (between

a and c) and γ (between a and b). Thus, a unit cell is

characterised by six parameters a, b, c, α

, β and γ.

These parameters of a typical unit cell are shown in

Fig. 1.10.

Unit cells can be broadly divided into two categories, primitive and

centred unit cells.

(a) Primitive Unit Cells

When constituent particles are present only on the corner positions of

a unit cell, it is called as primitive unit cell.

(b) Centred Unit Cells

When a unit cell contains one or more constituent particles present at

positions other than corners in addition to those at corners, it is called

a centred unit cell. Centred unit cells are of three types:

(i) Body-Centred Unit Cells: Such a unit cell contains one constituent

particle (atom, molecule or ion) at its body-centre besides the ones

that are at its corners.

(ii) Face-Centred Unit Cells: Such a unit cell contains one constituent

particle present at the centre of each face, besides the ones that

are at its corners.

(iii) End-Centred Unit Cells: In such a unit cell, one constituent particle

is present at the centre of any two opposite faces besides the ones

present at its corners.

1.4.1 Primitive

and Centred

Unit Cells

Fig. 1.10: Illustration of parameters

of a unit cell

Fig. 1.8: Possible unit cells in two dimensions

90º

60º

Fig. 1.9: A portion of a three-

dimensional cubic space

of a crystal lattice and its

unit cell.

Unit

cell

Lattice

point

Lattice

point

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10Chemistry

Crystal system Possible Axial distances Axial angles Examples

variations or edge lengths

Cubic Primitive, a = b = c α = β = γ = 90° NaCl, Zinc blende,

Body-centred, Cu

Face-centred

Tetragonal Primitive, a = b ≠ c α = β = γ = 90° White tin, SnO

Body-centred TiO

, CaSO

Orthorhombic Primitive, a ≠ b ≠ c α = β = γ = 90° Rhombic sulphur,

Body-centred, KNO

, BaSO

Face-centred,

End-centred

Hexagonal Primitive a = b ≠ c α = β = 90° Graphite, ZnO, CdS,

γ = 120°

Rhombohedral or Primitive a = b = c α = β = γ ≠ 90° Calcite (CaCO

), HgS

Trigonal (cinnabar)

Table 1.3: Seven Primitive Unit Cells and their Possible

Variations as Centred Unit Cells

Fig. 1.11: Seven crystal systems

Inspection of a wide variety of crystals leads to the conclusion that

all can be regarded as conforming to one of the seven regular

figures. These basic regular figures are called seven crystal systems.

To which system a given crystal belongs to is determined by

measuring the angles between its faces and deciding how many

axis are needed to define the principal features of its shape.

Fig. 1.11 shows seven crystal systems.

A French mathematician, Bravais, showed that there are only 14

possible three-dimensional lattices. These are called Bravais lattices.

Unit cells of these lattices are shown in the following box. The

characteristics of their primitive unit cells along with the centred unit

cells that they can form have been listed in Table 1.3.

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The Solid State

Primitive

(or simple)

Body-centred Face-centred

Unit cells of cubic lattices: all sides of same length, angles between faces all 90°

Primitive

Body-centred

Unit cells two tetragonal lattices: one side different in length to the other,

two angles between faces all 90°

Unit cells of four orthorhombic lattices: unequal sides, angles between faces all 90°

Primitive

Body-centred

Face-centred

End-centred

Unit cells of two monoclinic lattices: unequal sides, two faces have angles different to 90°

Primitive

End-centred

More than

90°

Less than

90°

Unit Cells of 14 Types of Bravais LatticesUnit Cells of 14 Types of Bravais Lattices

Unit Cells of 14 Types of Bravais Lattices

Monoclinic Primitive, a ≠ b ≠ c α = γ = 90° Monoclinic sulphur,

End-centred β ≠ 90° Na

.10H

Triclinic Primitive a ≠ b ≠ c α ≠ β ≠ γ ≠ 90° K

, CuSO

. 5H

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12Chemistry

We know that any crystal lattice is made up of a very large number of

unit cells and every lattice point is occupied by one constituent particle

(atom, molecule or ion). Let us now work out what portion of each

particle belongs to a particular unit cell.

We shall consider three types of cubic unit cells and for simplicity

assume that the constituent particle is an atom.

Primitive cubic unit cell has atoms only at its corner. Each atom at

a corner is shared between eight adjacent unit cells as shown in

Fig. 1.12, four unit cells in the same layer and four unit cells of the

upper (or lower) layer. Therefore, only

of an atom (or molecule

or ion) actually belongs to a particular unit cell. In

Fig. 1.13, a primitive cubic unit cell has been depicted

in three different ways. Each small sphere in Fig. 1.13(a)

represents only the centre of the particle occupying

that position and not its actual size. Such structures

are called open structures. The arrangement of

particles is easier to follow in open structures.

Fig. 1.13 (b) depicts space-filling representation of the

unit cell with actual particle size and Fig. 1.13 (c)

shows the actual portions of different atoms present in

a cubic unit cell.

In all, since each cubic unit cell has

8 atoms on its corners, the total number of

atoms in one unit cell is

8 1

× =

atom.

Unit cell of hexagonal

lattice

– one side

different in length to

the other two, the

marked angles on two

faces are 60°

Unit cell of rhombohedral

lattice – all sides of equal

length, angles on two

faces are less than 90°

Unit cell of

triclinic lattice–

unequal sides a, b, c,

A, B, C are unequal

angles with none equal to 90°

less than 90°

1.51.5

1.5

Number ofNumber of

Number of

Atoms in aAtoms in a

Atoms in a

Unit CellUnit Cell

Unit Cell

1.5.1 Primitive

Cubic Unit

Cell

Fig. 1.13: A primitive cubic unit cell (a) open

structure (b) space-filling structure

belonging to one unit cell.

Fig. 1.12: In a simple cubic unit cell,

each corner atom is shared

between 8 unit cells.

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The Solid State

A body-centred cubic (bcc) unit cell has an atom at each of its corners

and also one atom at its body centre. Fig. 1.14 depicts (a) open

structure (b) space filling model and (c) the unit cell with portions of

atoms actually belonging to it. It can be seen that the atom at the

Fig. 1.15: An atom at face

centre of unit cell

is shared between

2 unit cells

(a)

(b)

Fig 1.16: A face-centred cubic unit cell (a) open structure (b) space

filling structure (c) actual portions of atoms belonging to

one unit cell.

1.5.2 Body-

Centred

Cubic Unit

Cell

1.5.3 Face-

Centred

Cubic Unit

Cell

Fig. 1.14: A body-centred cubic unit cell (a) open structure (b) space-

filling structure (c) actual portions of atoms belonging to

one unit cell.

(a)

(b)

(c)

body centre wholly belongs to the unit cell in which it is present. Thus

in a body-centered cubic (bcc) unit cell:

(i) 8 corners ×

per corner atom

= ×

= 1 atom

(ii) 1 body centre atom = 1 × 1 = 1 atom

∴ Total number of atoms per unit cell = 2 atoms

A face-centred cubic (fcc) unit cell contains atoms at all the corners and

at the centre of all the faces of the cube. It can be seen in Fig. 1.15 that

each atom located at the face-centre is shared between two adjacent

unit cells and only

of each atom belongs to a unit cell. Fig. 1.16 depicts

(a) open structure (b) space-filling model and (c) the unit cell with

portions of atoms actually belonging to it. Thus, in a face-centred cubic

(fcc) unit cell:

(i) 8 corners atoms ×

atom per unit cell

= ×

= 1 atom

(ii) 6 face-centred atoms ×

atom per unit cell = 6 ×

= 3 atoms

∴ Total number of atoms per unit cell = 4 atoms

(c)

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14Chemistry

In solids, the constituent particles are close-packed, leaving the

minimum vacant space. Let us consider the constituent particles as

identical hard spheres and build up the three-dimensional structure

in three steps.

(a) Close Packing in One Dimension

There is only one way of arranging spheres in a one-dimensional close

packed structure, that is to arrange them in a row and touching each

other (Fig. 1.17).

In this arrangement, each sphere is in contact

with two of its neighbours. The number of nearest

neighbours of a particle is called its coordination

number. Thus, in one dimensional close packed

arrangement, the coordination number is 2.

(b) Close Packing in Two Dimensions

Two dimensional close packed structure can be generated by stacking

(placing) the rows of close packed spheres. This can be done in two

different ways.

(i) The second row may be placed in contact with the first one such

that the spheres of the second row are exactly above those of the

first row. The spheres of the two rows are aligned horizontally as

well as vertically. If we call the first row as ‘A’ type row, the second

row being exactly the same as the first one, is also of ‘A’ type.

Similarly, we may place more rows to obtain AAA type of

arrangement as shown in Fig. 1.18 (a).

1.6

Close PackedClose Packed

Close Packed

Structures

Intext QuestionsIntext Questions

Intext Questions

1.9 Give the significance of a ‘lattice point’.

1.10 Name the parameters that characterise a unit cell.

1.11 Distinguish between

(i) Hexagonal and monoclinic unit cells

(ii) Face-centred and end-centred unit cells.

1.12 Explain how much portion of an atom located at (i) corner and (ii) body-

centre of a cubic unit cell is part of its neighbouring unit cell.

Fig. 1.17: Close packing of spheres in

one dimension

Fig. 1.18: (a) Square close packing (b) hexagonal close

packing of spheres in two dimensions

2020-21

The Solid State

In this arrangement, each sphere is in contact with four of its

neighbours. Thus, the two dimensional coordination number is 4. Also,

if the centres of these 4 immediate neighbouring spheres are joined, a

square is formed. Hence this packing is called square close packing

in two dimensions.

(ii) The second row may be placed above the first one in a staggered

manner such that its spheres fit in the depressions of the first row.

If the arrangement of spheres in the first row is called ‘A’ type, the

one in the second row is different and may be called ‘B’ type. When

the third row is placed adjacent to the second in staggered manner,

its spheres are aligned with those of the first layer. Hence this layer

is also of ‘A’ type. The spheres of similarly placed fourth row will

be aligned with those of the second row (‘B’ type). Hence this

arrangement is of ABAB type. In this arrangement there is less free

space and this packing is more efficient than the square close

packing. Each sphere is in contact with six of its neighbours and

the two dimensional coordination number is 6. The centres of these

six spheres are at the corners of a regular hexagon (Fig. 1.18 b)

hence this packing is called two dimensional hexagonal close-

packing. It can be seen in Figure 1.18 (b) that in this layer there

are some voids (empty spaces). These are triangular in shape. The

triangular voids are of two different types. In one row, the apex of

the triangles are pointing upwards and in the next layer downwards.

All real structures are three dimensional structures. They can be

obtained by stacking two dimensional layers one above the other. In

the last Section, we discussed close packing in two dimensions which

can be of two types; square close-packed and hexagonal close-packed.

Let us see what types of three dimensional close packing can be obtained

from these.

(i) Three-dimensional close packing forms two-dimensional square

close-packed layers: While placing the second square close-packed

layer above the first we follow the same rule that was

followed when one row was placed adjacent to the other.

The second layer is placed over the first layer such that

the spheres of the upper layer are exactly above those of

the first layer. In this arrangement spheres of both the

layers are perfectly aligned horizontally as well as

vertically as shown in Fig. 1.19. Similarly, we may place

more layers one above the other. If the arrangement of

spheres in the first layer is called ‘A’ type, all the layers

have the same arrangement. Thus this lattice has AAA....

type pattern. The lattice thus generated is the simple

cubic lattice, and its unit cell is the primitive cubic unit

cell (See Fig. 1.19).

(ii) Three dimensional close packing from two

dimensional hexagonal close packed layers: Three

dimensional close packed structure can be generated

by placing layers one over the other.

Fig. 1.19: Simple cubic lattice formed

by A A A .... arrangement

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16Chemistry

(a) Placing second layer over the first layer

Let us take a two dimensional hexagonal close packed layer ‘A’ and

place a similar layer above it such that the spheres of the second layer

are placed in the depressions of the first layer. Since the spheres of the

two layers are aligned differently, let us call the second layer as B. It

can be observed from Fig. 1.20 that all the triangular voids of the first

layer are not covered by the spheres of the second layer. This gives rise

to different arrangements. Wherever a sphere of the second layer is

above the void of the first layer (or vice versa) a tetrahedral void is

Fig. 1.20: A stack of two layers of close packed spheres and voids

generated in them. T = Tetrahedral void; O = Octahedral void

formed. These voids are called tetrahedral voids because a tetrahedron

is formed when the centres of these four spheres are joined. They have

been marked as ‘T’ in Fig. 1.20. One such void has been shown

separately in Fig. 1.21.

Fig 1.21

Tetrahedral and

octahedral voids

(a) top view

(b) exploded side

view and

of the void.

2020-21

The Solid State

At other places, the triangular voids in the second layer are above

the triangular voids in the first layer, and the triangular shapes of these

do not overlap. One of them has the apex of the triangle pointing

upwards and the other downwards. These voids have been marked as

‘O’ in Fig. 1.20. Such voids are surrounded by six spheres and are

called octahedral voids. One such void has been shown separately in

Fig. 1.21. The number of these two types of voids depend upon the

number of close packed spheres.

Let the number of close packed spheres be N, then:

The number of octahedral voids generated = N

The number of tetrahedral voids generated = 2N

(b) Placing third layer over the second layer

When third layer is placed over the second, there are two possibilities.

(i) Covering Tetrahedral Voids: Tetrahedral voids of the second layer

may be covered by the spheres of the third layer. In this case, the

spheres of the third layer are exactly aligned with those of the first

layer. Thus, the pattern of spheres is repeated in alternate layers.

This pattern is often written as ABAB ....... pattern. This structure

is called hexagonal close packed (hcp) structure (Fig. 1.22). This

sort of arrangement of atoms is found in many metals like

magnesium and zinc.

Fig. 1.22

(a) Hexagonal cubic

close-packing

exploded view

showing stacking of

layers of spheres

(b) four layers

stacked in each case

and (c) geometry of

packing.

(ii) Covering Octahedral Voids: The

third layer may be placed above

the second layer in a manner such

that its spheres cover the

octahedral voids. When placed in

this manner, the spheres of the

third layer are not aligned with

those of either the first or the second

layer. This arrangement is called ‘C’

type. Only when fourth layer is

placed, its spheres are aligned with

Fig. 1.23

(a) ABCABC...

arrangement of

layers when

octahedral void is

covered (b) fragment

of structure formed

by this arrangement

resulting in cubic

closed packed (ccp)

or face centred cubic

(fcc) structure.

(a)

(b)

2020-21

18Chemistry

those of the first layer as shown in Figs. 1.22 and 1.23. This pattern

of layers is often written as ABCABC ........... This structure is called

cubic close packed (ccp) or face-centred cubic (fcc) structure. Metals

such as copper and silver crystallise in this structure.

Both these types of close packing are highly efficient and 74%

space in the crystal is filled. In either of them, each sphere is in contact

with twelve spheres. Thus, the coordination number is 12 in either of

these two structures.

Earlier in the section, we have learnt that when particles are close-

packed resulting in either ccp or hcp structure, two types of voids are

generated. While the number of octahedral voids present in a lattice is

equal to the number of close packed particles, the number of tetrahedral

voids generated is twice this number. In ionic solids, the bigger ions

(usually anions) form the close packed structure and the smaller ions

(usually cations) occupy the voids. If the latter ion is small enough

then tetrahedral voids are occupied, if bigger, then octahedral voids. All

octahedral or tetrahedral voids are not occupied. In a given compound,

the fraction of octahedral or tetrahedral voids that are occupied, depends

upon the chemical formula of the compound, as can be seen from the

following examples.

We know that close packed structures have both tetrahedral and octahedral

voids. Let us take ccp (or fcc) structure and locate these voids in it.

(a) Locating Tetrahedral Voids

Let us consider a unit cell of ccp or fcc lattice [Fig. 1(a)]. The unit cell is divided

into eight small cubes.

Example 1.1Example 1.1

Example 1.1

1.6.1 Formula of a

Compound

and Number

of Voids

Filled

A compound is formed by two elements X and Y. Atoms of the element

Y (as anions) make ccp and those of the element X (as cations) occupy

all the octahedral voids. What is the formula of the compound?

The ccp lattice is formed by the element Y. The number of octahedral

voids generated would be equal to the number of atoms of Y present in

it. Since all the octahedral voids are occupied by the atoms of X, their

number would also be equal to that of the element Y. Thus, the atoms

of elements X and Y are present in equal numbers or 1:1 ratio. Therefore,

the formula of the compound is XY.

Atoms of element B form hcp lattice and those of the element A occupy

2/3rd of tetrahedral voids. What is the formula of the compound formed

by the elements A and B?

The number of tetrahedral voids formed is equal to twice the number of

atoms of element B and only 2/3rd of these are occupied by the atoms

of element A. Hence the ratio of the number of atoms of A and B is 2

× (2/3):1 or 4:3 and the formula of the compound is A

Example 1.2Example 1.2

Example 1.2

SolutionSolution

Solution

SolutionSolution

Solution

Locating Tetrahedral and Octahedral VoidsLocating Tetrahedral and Octahedral Voids

Locating Tetrahedral and Octahedral Voids

2020-21

The Solid State

Each small cube has atoms at alternate corners [Fig. 1(a)]. In all, each small

cube has 4 atoms. When joined to each other, they make a regular tetrahedron.

Thus, there is one tetrahedral void in each small cube and eight tetrahedral

voids in total. Each of the eight small cubes have one void in one unit cell of ccp

structure. We know that ccp structure has 4 atoms per unit cell. Thus, the

number of tetrahedral voids is twice the number of atoms.

Fig. 1: (a) Eight tetrahedral voids per unit cell of ccp structure

(b) one tetrahedral void showing the geometry.

(b) Locating Octahedral Voids

Let us again consider a unit cell of ccp or fcc lattice [Fig. 2(a)]. The body centre

of the cube, C is not occupied but it is surrounded by six atoms on face centres.

If these face centres are joined, an octahedron is generated. Thus, this unit cell

has one octahedral void at the body centre of the cube.

Besides the body centre, there is one octahedral void at the centre of each

of the 12 edges [Fig. 2(b)]. It is surrounded by six atoms, four belonging to the

same unit cell (2 on the corners and 2 on face centre) and two belonging to two

adjacent unit cells. Since each edge of the cube is shared between four adjacent

unit cells, so is the octahedral void located on it. Only

of each void belongs

to a particular unit cell.

(a)

(b)

Fig. 2: Location of octahedral voids per unit cell of ccp or fcc lattice (a) at the body centre

of the cube and (b) at the centre of each edge (only one such void is shown).

2020-21

20Chemistry

Thus in cubic close packed structure:

Octahedral void at the body-centre of the cube = 1

12 octahedral voids located at each edge and shared between four unit cells

12 3

× =

∴ Total number of octahedral voids = 4

We know that in ccp structure, each unit cell has 4 atoms. Thus, the number

of octahedral voids is equal to this number.

1.7

Packing

Efficiency

In whatever way the constituent particles (atoms, molecules or ions)

are packed, there is always some free space in the form of voids.

Packing efficiency is the percentage of total space filled by the

particles. Let us calculate the packing efficiency in different types of

structures.

Both types of close packing (hcp

and ccp) are equally efficient. Let us

calculate the efficiency of packing in ccp structure. In Fig. 1.24 let the

unit cell edge length be ‘a’ and face diagonal AC = b.

ABC

∆

= b

= BC

+ AB

= a

= 2a

b =

If r is the radius of the sphere, we find

b = 4r =

or a =

2 2r

(we can also write,

r )

2 2

We know, that each unit cell in ccp structure,

has effectively 4 spheres. Total volume of four

spheres is equal to

(

)

4 4 /3 r

× π

and volume of the

cube is a

( )

2 2r

Therefore,

Volume occupied by four spheres in the unit cell 100

Packing efficiency = %

Total volumeof the unit cell

(

)

( )

4 4 /3 r 100

2 2r

× π ×

(

)

16 /3 r 100

% 74%

16 2r

π ×

= =

1.7.1 Packing

Efficiency in

hcp and ccp

Structures

Fig. 1.24: Cubic close packing other

sides are not provided with

spheres for sake of clarity.

2020-21

The Solid State

From Fig. 1.25, it is clear that the

atom at the centre will be in touch

with the other two atoms diagonally

arranged.

In ∆ EFD,

= a

+ a

= 2a

b =

Now in ∆

AFD

= a

+ b

= a

+ 2a

= 3a

c =

The length of the body diagonal

c is equal to 4r, wher

e r is the radius

of the sphere (atom), as all the three

spheres along the diagonal touch

each other.

Therefore,

= 4r

a =

Also we can write, r =

In this type of structure, total number of atoms is 2 and their volume

(

)

2 r .

× π

Volume of the cube, a

will be equal to

 

 

 

a r

 

 

 

Therefore,

Volume occupied by two spheres in the unit cell 1

Packing efficiency = %

Total volumeof the unit cell

(

)

( )

2 4 /3 r 100

4/ 3 r

× π ×

 

 

(

)

( )

8/3 r 100

% 68%

64/ 3 3 r

π ×

= =

In a simple cubic lattice the atoms are located only on the corners of the

cube. The particles touch each other along the edge (Fig. 1.26).

Thus, the edge length or side of the cube ‘a’, and the radius of each particle,

r are related as

a = 2r

The volume of the cubic unit cell = a

= (2r)

= 8r

Since a simple cubic unit cell contains only 1 atom

The volume of the occupied space =

1.7.2 Efficiency of

Packing in

Body-

Centred

Cubic

Structures

1.7.3 Packing

Efficiency in

Simple Cubic

Lattice

Fig. 1.25: Body-centred cubic unit

cell (sphere along the

body diagonal are shown

with solid boundaries).

2020-21

22Chemistry

∴ Packing efficiency

Volume of one atom

100%

Volume of cubic unit cell

100 100

× = ×

= 52.36% = 52.4 %

Thus, we may conclude that ccp

and hcp structures have maximum

packing efficiency.

From the unit cell dimensions, it is possible to calculate the volume of

the unit cell. Knowing the density of the metal, we can calculate the

mass of the atoms in the unit cell. The determination of the mass of a

single atom gives an accurate method of determination of Avogadro

constant. Suppose, edge length of a unit cell of a cubic crystal

determined by X-ray diffraction is a, d the density of the solid substance

and M the molar mass. In case of cubic crystal:

Volume of a unit cell = a

Mass of the unit cell

= number of atoms in unit cell × mass of each atom = z × m

(Here z is the number of atoms present in one unit cell and m is the

mass of a single atom)

Mass of an atom present in the unit cell:

(M is molar mass)

Therefore, density of the unit cell

3 3 3

A A

mass of unit cell

volume of unit cell

z.m z.M zM

= = or d =

a a .N a N

Remember, the density of the unit cell is the same as the density of

the substance. The density of the solid can always be determined by

other methods. Out of the five parameters (d, z, M, a and N

), if any

four are known, we can determine the fifth.

Fig. 1.26

Simple cubic unit cell.

The spheres are in

contact with each

other along the edge of

the cube.

1.8

Calculations

Involving

Unit Cell

Dimensions

An element has a body-centred cubic (bcc) structure with a cell edge of

288 pm. The density of the element is 7.2 g/cm

. How many atoms are

present in 208 g of the element?

Volume of the unit cell = (288 pm)

= (288×10

-12

= (288×10

-10

cm)

= 2.39×10

-23

Example 1.3Example 1.3

Example 1.3

SolutionSolution

Solution

2020-21

The Solid State

Example 1.4Example 1.4

Example 1.4

Volume of 208 g of the element

208

28.88

7.2 g

−

= = =

mass g

density

Number of unit cells in this volume

23 3

28.88

2.39 10 /

−

cm unit cell

= 12.08×10

unit cells

Since each bcc cubic unit cell contains 2 atoms, therefore, the total number

of atoms in 208 g = 2 (atoms/unit cell) × 12.08 × 10

unit cells

= 24.16×10

atoms

X-ray diffraction studies show that copper crystallises in an fcc unit

cell with cell edge of 3.608×10

-8

cm. In a separate experiment, copper is

determined to have a density of 8.92 g/cm

, calculate the atomic mass

of copper.

In case of fcc lattice, number of atoms per unit cell, z = 4 atoms

Therefore, M =

dN a

− −

× × × ×

–3 23 1 8 3

8.92 g cm 6.022 10 atoms mol (3.608 10 cm)

4 atoms

= 63.1 g/mol

Atomic mass of copper = 63.1u

Silver forms ccp lattice and X-ray studies of its crystals show that the

edge length of its unit cell is 408.6 pm. Calculate the density of silver

(Atomic mass = 107.9 u).

Since the lattice is ccp, the number of silver atoms per unit cell = z = 4

Molar mass of silver = 107.9 g mol

–1

= 107.9×10

-3

kg mol

–1

Edge length of unit cell = a = 408.6 pm = 408.6×10

–12

Density, d =

z.M

a .N

(

)

( ) ( )

3 1

12 23 1

4 107.9 10 kg mol

408.6 10 m 6.022 10 mol

− −

× ×

= 10.5×10

kg m

–3

= 10.5 g cm

-3

Example 1.5Example 1.5

Example 1.5

SolutionSolution

Solution

Intext QuestionsIntext Questions

Intext Questions

1.13 What is the two dimensional coordination number of a molecule in

square close-packed layer?

1.14 A compound forms hexagonal close-packed structure. What is the total

number of voids in 0.5 mol of it? How many of these are tetrahedral voids?

SolutionSolution

Solution

2020-21

24Chemistry

Although crystalline solids have short range as well as long range

order in the arrangement of their constituent particles, yet crystals are

not perfect. Usually a solid consists of an aggregate of large number

of small crystals. These small crystals have defects in them. This

happens when crystallisation process occurs at fast or moderate rate.

Single crystals are formed when the process of crystallisation occurs at

extremely slow rate. Even these crystals are not free of defects. The

defects are basically irregularities in the arrangement of constituent

particles. Broadly speaking, the defects are of two types, namely, point

defects and line defects. Point defects are the irregularities or

deviations from ideal arrangement around a point or an atom in a

crystalline substance, whereas the line defects are the irregularities

or deviations from ideal arrangement in entire rows of lattice points.

These irregularities are called crystal defects. We shall confine our

discussion to point defects only.

Point defects can be classified into three types : (i) stoichiometric defects

(ii) impurity defects and (iii) non-stoichiometric defects.

(a) Stoichiometric Defects

These are the point defects that do not disturb the stoichiometry of

the solid. They are also called intrinsic or thermodynamic defects.

Basically these are of two types, vacancy defects and interstitial defects.

(i) Vacancy Defect: When some of the lattice sites are vacant, the

crystal is said to have vacancy defect (Fig. 1.27). This results in

decrease in density of the substance. This defect can also develop

when a substance is heated.

(ii) Interstitial Defect: When some constituent particles

(atoms or molecules) occupy an interstitial site,

the crystal is said to have interstitial

defect

(Fig. 1.28). This defect increases the density of the

substance.

Vacancy and interstitial defects as explained

above can be shown by non-ionic solids. Ionic

solids must always maintain electrical neutrality.

Rather than simple vacancy or interstitial

defects, they show these defects as Frenkel and

Schottky defects.

1.9

ImperfectionsImperfections

Imperfections

in Solids

1.15 A compound is formed by two elements M and N. The element N

forms ccp and atoms of M occupy 1/3rd of tetrahedral voids. What

is the formula of the compound?

1.16 Which of the following lattices has the highest packing efficiency (i) simple

cubic (ii) body-centred cubic and (iii) hexagonal close-packed lattice?

1.17 An element with molar mass 2.7×10

-2

kg mol

-1

forms a cubic unit cell

with edge length 405 pm. If its density is 2.7×10

kg m

-3

, what is the

nature of the cubic unit cell?

1.9.1 Types of

Point Defects

Fig. 1.27: Vacancy defects

2020-21

The Solid State

(iii) Frenkel Defect: This

defect is shown by ionic

solids. The smaller ion

(usually cation) is

dislocated from its normal

site to an interstitial site

(Fig. 1.29). It creates a

vacancy defect at its

original site and an

interstitial defect at its

new location.

Frenkel defect is also

called dislocation defect.

It does not change the density of the solid. Frenkel defect is

shown by ionic substance in which there is a large difference in

the size of ions, for example, ZnS, AgCl, AgBr and AgI due to

small size of Zn

and Ag

ions.

(iv) Schottky Defect: It is basically a vacancy defect in ionic solids. In

order to maintain electrical neutrality, the number of missing

cations and anions are equal (Fig. 1.30).

Like simple vacancy

defect, Schottky defect also

decreases the density of the

substance. Number of such

defects in ionic solids is quite

significant. For example, in

NaCl there are approximately

Schottky pairs per cm

at room temperature. In

1 cm

there are about

ions. Thus, there is one

Schottky defect per 10

ions.

Schottky defect is shown by

ionic substances in which the

cation and anion are of almost similar sizes.

For example, NaCl, KCl, CsCl and AgBr. It may

be noted that AgBr shows both, Frenkel as

well as Schottky defects.

(b) Impurity Defects

If molten NaCl containing a little amount

of SrCl

is crystallised, some of the sites of

ions are occupied by Sr

(Fig.1.31).

Each Sr

replaces two Na

ions. It occupies

the site of one ion and the other site remains

vacant. The cationic vacancies thus

produced are equal in number to that of

ions. Another similar example is the

solid solution of CdCl

and AgCl.

Fig. 1.28: Interstitial defects

Fig. 1.31: Introduction of cation vacancy in

NaCl by substitution of Na

by Sr

Fig. 1.29: Frenkel defects

Fig. 1.30: Schottky defects

2020-21

26Chemistry

The defects discussed so far do not disturb the stoichiometry of

the crystalline substance. However, a large number of non-

stoichiometric inorganic solids are known which contain the

constituent elements in non-stoichiometric ratio due to defects in

their crystal structures. These defects are of two types: (i) metal

excess defect and (ii) metal deficiency defect.

(i) Metal Excess Defect

Ã Metal excess defect due to anionic vacancies: Alkali halides

like NaCl and KCl show this type of defect. When crystals of

NaCl are heated in an atmosphere of sodium vapour, the

sodium atoms are deposited on the surface of the crystal.

The Cl

–

ions diffuse to the surface of the crystal and

combine with Na atoms to give NaCl. This happens by

loss of electron by sodium atoms to form Na

ions. The

released electrons diffuse into the crystal and occupy

anionic sites (Fig. 1.32). As a result the crystal now has

an excess of sodium. The anionic sites occupied by

unpaired electrons are called F-centres (from the German

word Farbenzenter for colour centre). They impart yellow

colour to the crystals of NaCl. The colour results by

excitation of these electrons when they absorb energy from

the visible light falling on the crystals. Similarly, excess of

lithium makes LiCl crystals pink and excess of potassium

makes KCl crystals violet (or lilac).

Ã Metal excess defect due to the presence of extra cations at

interstitial sites: Zinc oxide is white in colour at room

temperature. On heating it loses oxygen and turns yellow.

heating

ZnO Zn O 2e

+ −

→ + +

Now there is excess of zinc in the crystal and its formula becomes

1+x

O. The excess Zn

ions move to interstitial sites and the electrons

to neighbouring interstitial sites.

(ii) Metal Deficiency Defect

There are many solids which are difficult to prepare in the

stoichiometric composition and contain less amount of the metal as

compared to the stoichiometric proportion. A typical example of

this type is FeO which is mostly found with a composition of Fe

0.95

It may actually range from Fe

0.93

O to Fe

0.96

O. In crystals of FeO

some Fe

cations are missing and the loss of positive charge is

made up by the presence of required number of Fe

ions.

Solids exhibit an amazing range of electrical conductivities, extending

over 27 orders of magnitude ranging from 10

–20

to 10

ohm

–1

Solids can be classified into three types on the basis of their

conductivities.

(i) Conductors: The solids with conductivities ranging between 10

to 10

ohm

–1

are called conductors. Metals have conductivities

in the order of 10

ohm

–1

are good conductors.

1.101.10

1.10

ElectricalElectrical

Electrical

PropertiesProperties

Properties

Fig. 1.32: An F-centre in a crystal

2020-21

The Solid State

(ii) Insulators : These are the solids with very low conductivities

ranging between 10

–20

to 10

–10

ohm

–1

(iii) Semiconductors : These are the solids with conductivities in the

intermediate range from 10

–6

to 10

ohm

–1

A conductor may conduct electricity through movement of electrons or

ions. Metallic conductors belong to the former category and electrolytes

to the latter.

Metals conduct electricity in solid as well as molten state. The

conductivity of metals depend upon the number of valence electrons

available per atom. The atomic orbitals of metal atoms form molecular

orbitals which are so close in energy to each other as to form a

band. If this band is partially filled or it overlaps with a higher

energy unoccupied conduction band, then electrons can flow easily

under an applied electric field and the metal shows conductivity

(Fig. 1.33 a).

If the gap between filled valence band and the next higher

unoccupied band (conduction band) is large, electrons cannot jump to

it and such a substance has very small conductivity and it behaves as

an insulator (Fig. 1.33 b).

In case of semiconductors, the gap between the valence band and

conduction band is small (Fig. 1.33 c). Therefore, some electrons may

jump to conduction band and show some conductivity. Electrical

conductivity of semiconductors increases with rise in temperature, since

more electrons can jump to the conduction band. Substances like

silicon and germanium show this type of behaviour and are called

intrinsic semiconductors.

The conductivity of these intrinsic semiconductors is too low to

be of practical use. Their conductivity is increased by adding an

appropriate amount of suitable impurity. This process is called

Fig. 1.33

Distinction among

(a) metals

(b) insulators and

In each case, an

unshaded area

represents a

conduction band.

1.10.1 Conduction

Electricity

in Metals

1.10.2 Conduction

Electricity

in Semi-

conductors

2020-21

28Chemistry

doping. Doping can be done with an impurity which is electron rich

or electron deficient as compared to the intrinsic semiconductor

silicon or germanium. Such impurities introduce electronic defects

in them.

(a) Electron – rich impurities

Silicon and germanium belong to group 14 of the periodic table and

have four valence electrons each. In their crystals each atom forms

four covalent bonds with its neighbours (Fig. 1.34 a). When doped

with a group 15 element like P or As, which contains five valence

electrons, they occupy some of the lattice sites in silicon or

germanium crystal (Fig. 1.34 b). Four out of five electrons are used

in the formation of four covalent bonds with the four neighbouring

silicon atoms. The fifth electron is extra and becomes delocalised.

These delocalised electrons increase the conductivity of doped silicon

(or germanium). Here the increase in conductivity is due to the

negatively charged electron, hence silicon doped with electron-rich

impurity is called n-type semiconductor.

(b) Electron – deficit impurities

Silicon or germanium can also be doped with a group 13 element

like B, Al or Ga which contains only three valence electrons. The

place where the fourth valence electron is missing is called electron

hole or electron vacancy (Fig. 1.34 c). An electron from a

neighbouring atom can come and fill the electron hole, but in doing

so it would leave an electron hole at its original position. If it

happens, it would appear as if the electron hole has moved in the

direction opposite to that of the electron that filled it. Under the

influence of electric field, electrons would move towards the positively

charged plate through electronic holes, but it would appear as if

electron holes are positively charged and are moving towards

negatively charged plate. This type of semi conductors are called

p-type semiconductors.

Fig. 1.34: Creation of n-type and p-type semiconductors

by doping groups 13 and 15 elements.

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The Solid State

Applications of n-type and p-type semiconductors

Various combinations of n-type and p-type semiconductors are used

for making electronic components. Diode is a combination of n-type

and p-type semiconductors and is used as a rectifier. Transistors are

made by sandwiching a layer of one type of semiconductor between

two layers of the other type of semiconductor. npn and pnp type of

transistors are used to detect or amplify radio or audio signals. The

solar cell is an efficient photo-diode used for conversion of light energy

into electrical energy.

Germanium and silicon are group 14 elements and therefore, have

a characteristic valence of four and form four bonds as in diamond. A

large variety of solid state materials have been prepared by combination

of groups 13 and 15 or 12 and 16 to simulate average valence of four

as in Ge or Si. Typical compounds of groups 13 – 15 are InSb, AlP and

GaAs. Gallium arsenide (GaAs) semiconductors have very fast response

and have revolutionised the design of semiconductor devices. ZnS,

CdS, CdSe and HgTe are examples of groups 12 – 16 compounds. In

these compounds, the bonds are not perfectly covalent and the ionic

character depends on the electronegativities of the two elements.

It is interesting to learn that transition metal oxides show marked

differences in electrical properties. TiO, CrO

and ReO

behave like

metals. Rhenium oxide, ReO

is like metallic copper in its conductivity

and appearance. Certain other oxides like VO, VO

, VO

and TiO

show

metallic or insulating properties depending on temperature.

Every substance has some magnetic properties associated with it.

The origin of these properties lies in the electrons. Each electron in an

atom behaves like a tiny magnet. Its magnetic moment originates

from two types of motions (i) its orbital

motion around the nucleus and (ii) its spin

around its own axis (Fig. 1.35). Electron

being a charged particle and undergoing

these motions can be considered as a small

loop of current which possesses a magnetic

moment. Thus, each electron has a

permanent spin and an orbital magnetic

moment associated with it. Magnitude of

this magnetic moment is very small and

is measured in the unit called Bohr

magneton,

. It is equal to 9.27 × 10

–24

A m

On the basis of their magnetic properties, substances can be

classified into five categories: (i) paramagnetic (ii) diamagnetic

(iii) ferromagnetic (iv) antiferromagnetic and (v) ferrimagnetic.

(i) Paramagnetism: Paramagnetic substances are weakly attracted

by a magnetic field. They are magnetised in a magnetic field in

the same direction. They lose their magnetism in the absence of

magnetic field. Paramagnetism is due to presence of one or

more unpaired electrons which are attracted by the magnetic

field. O

, Cu

, Fe

, Cr

are some examples of such substances.

1.111.11

1.11

MagneticMagnetic

Magnetic

PropertiesProperties

Properties

Fig.1.35: Demonstration of the magnetic moment

associated with (a) an orbiting electron

and (b) a spinning electron.

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30Chemistry

Fig 1.36: Schematic alignment of magnetic moments in (a) ferromagnetic

(b) antiferromagnetic and (c) ferrimagnetic.

(ii) Diamagnetism: Diamagnetic substances are weakly repelled by

a magnetic field. H

O, NaCl and C

are some examples of such

substances. They are weakly magnetised in a magnetic field in

opposite direction. Diamagnetism is shown by those substances

in which all the electrons are paired and there are no unpaired

electrons. Pairing of electrons cancels their magnetic moments

and they lose their magnetic character.

(iii) Ferromagnetism: A few substances like iron, cobalt, nickel,

gadolinium and CrO

are attracted very strongly by a magnetic

field. Such substances are called ferromagnetic substances.

Besides strong attractions, these substances can be permanently

magnetised. In solid state, the metal ions of ferromagnetic

substances are grouped together into small regions called

domains. Thus, each domain acts as a tiny magnet. In an

unmagnetised piece of a ferromagnetic substance the domains

are randomly oriented and their magnetic moments get cancelled.

When the substance is placed in a magnetic field all the domains

get oriented in the direction of the magnetic field (Fig. 1.36 a)

and a strong magnetic effect is produced. This ordering of

domains persist even when the magnetic field is removed and

the ferromagnetic substance becomes a permanent magnet.

(iv) Antiferromagnetism: Substances like MnO showing anti-

ferromagnetism have domain structure similar to ferromagnetic

substance, but their domains are oppositely oriented and cancel

out each other's magnetic moment (Fig. 1.36 b).

(v) Ferrimagnetism: Ferrimagnetism is observed when the magnetic

moments of the domains in the substance are aligned in parallel

and anti-parallel directions in unequal numbers (Fig. 1.36c). They

are weakly attracted by magnetic field as compared to ferromagnetic

substances. Fe

(magnetite) and ferrites like MgFe

and

ZnFe

are examples of such substances. These substances also

lose ferrimagnetism on heating and become paramagnetic.

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The Solid State

Intext QuestionsIntext Questions

Intext Questions

1.18 What type of defect can arise when a solid is heated? Which physical

property is affected by it and in what way?

1.19 What type of stoichiometric defect is shown by:

(i) ZnS (ii) AgBr

1.20 Explain how vacancies are introduced in an ionic solid when a cation

of higher valence is added as an impurity in it.

1.21 Ionic solids, which have anionic vacancies due to metal excess defect,

develop colour. Explain with the help of a suitable example.

1.22 A group 14 element is to be converted into n-type semiconductor by doping

it with a suitable impurity. To which group should this impurity belong?

1.23 What type of substances would make better permanent magnets,

ferromagnetic or ferrimagnetic. Justify your answer.

Solids have definite mass, volume and shape. This is due to the fixed position of

their constituent particles, short distances and strong interactions between them.

In amorphous solids, the arrangement of constituent particles has only short

range order and consequently they behave like super cooled liquids, do not have

sharp melting points and are isotropic in nature. In crystalline solids there is long

range order in the arrangement of their constituent particles. They have sharp

melting points, are anisotropic in nature and their particles have characteristic

shapes. Properties of crystalline solids depend upon the nature of interactions

between their constituent particles. On this basis, they can be divided into four

categories, namely: molecular, ionic, metallic and covalent solids. They differ

widely in their properties.

The constituent particles in crystalline solids are arranged in a regular pattern

which extends throughout the crystal. This arrangement is often depicted in the

form of a three dimensional array of points which is called crystal lattice. Each

lattice point gives the location of one particle in space. In all, fourteen different

types of lattices are possible which are called Bravais lattices. Each lattice can be

generated by repeating its small characteristic portion called unit cell. A unit cell

is characterised by its edge lengths and three angles between these edges. Unit

cells can be either primitive which have particles only at their corner positions or

centred. The centred unit cells have additional particles at their body centre (body-

centred), at the centre of each face (face-centred) or at the centre of two opposite

faces (end-centred). There are seven types of primitive unit cells. Taking centred

unit cells also into account, there are fourteen types of unit cells in all, which

result in fourteen Bravais lattices.

Close-packing of particles result in two highly efficient lattices, hexagonal

close-packed (hcp) and cubic close-packed (ccp). The latter is also called face-

centred cubic (fcc) lattice. In both of these packings 74% space is filled. The

remaining space is present in the form of two types of voids-octahedral voids and

tetrahedral voids. Other types of packing are not close-packings and have less

SummarySummary

Summary

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32Chemistry

Exercises

efficient packing of particles. While in body-centred cubic lattice (bcc) 68% space

is filled, in simple cubic lattice only 52.4 % space is filled.

Solids are not perfect in structure. There are different types of imperfections

or defects in them. Point defects and line defects are common types of defects.

Point defects are of three types - stoichiometric defects, impurity defects and

non-stoichiometric defects. Vacancy defects and interstitial defects are the

two basic types of stoichiometric point defects. In ionic solids, these defects are

present as Frenkel and Schottky defects. Impurity defects are caused by the

presence of an impurity in the crystal. In ionic solids, when the ionic impurity has

a different valence than the main compound, some vacancies are created. Non-

stoichiometric defects are of metal excess type and metal deficient type.

Sometimes calculated amounts of impurities are introduced by doping in

semiconductors that change their electrical properties. Such materials are widely

used in electronics industry. Solids show many types of magnetic properties like

paramagnetism, diamagnetism, ferromagnetism, antiferromagnetism and

ferrimagnetism. These properties are used in audio, video and other recording

devices. All these properties can be correlated with their electronic configurations

or structures.

1.1 Define the term 'amorphous'. Give a few examples of amorphous solids.

1.2 What makes a glass different from a solid such as quartz? Under what

conditions could quartz be converted into glass?

1.3 Classify each of the following solids as ionic, metallic, molecular, network

(covalent) or amorphous.

(i) Tetra phosphorus decoxide (P

) (vii) Graphite

(ii) Ammonium phosphate (NH

)

(viii) Brass

(iii) SiC (ix) Rb

(iv) I

(x) LiBr

(v) P

(xi) Si

(vi) Plastic

1.4 (i) What is meant by the term 'coordination number'?

(ii) What is the coordination number of atoms:

(a) in a cubic close-packed structure?

(b) in a body-centred cubic structure?

1.5 How can you determine the atomic mass of an unknown metal if you know

its density and the dimension of its unit cell? Explain.

1.6 'Stability of a crystal is reflected in the magnitude of its melting points'.

Comment. Collect melting points of solid water, ethyl alcohol, diethyl ether

and methane from a data book. What can you say about the intermolecular

forces between these molecules?

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The Solid State

1.7 How will you distinguish between the following pairs of terms:

(i) Hexagonal close-packing and cubic close-packing?

(ii) Crystal lattice and unit cell?

(iii) Tetrahedral void and octahedral void?

1.8 How many lattice points are there in one unit cell of each of the following

lattice?

(i) Face-centred cubic

(ii) Face-centred tetragonal

(iii) Body-centred

1.9 Explain

(i) The basis of similarities and differences between metallic and ionic

crystals.

(ii) Ionic solids are hard and brittle.

1.10 Calculate the efficiency of packing in case of a metal crystal for

(i) simple cubic

(ii) body-centred cubic

(iii) face-centred cubic (with the assumptions that atoms are touching

each other).

1.11 Silver crystallises in fcc lattice. If edge length of the cell is 4.07 × 10

–8

cm and density is 10.5 g cm

–3

, calculate the atomic mass of silver.

1.12 A cubic solid is made of two elements P and Q. Atoms of Q are at the

corners of the cube and P at the body-centre. What is the formula of the

compound? What are the coordination numbers of P and Q?

1.13 Niobium crystallises in body-centred cubic structure. If density is 8.55

g cm

–3

, calculate atomic radius of niobium using its atomic mass 93 u.

1.14 If the radius of the octahedral void is r and radius of the atoms in close-

packing is R, derive relation between r and R.

1.15 Copper crystallises into a fcc lattice with edge length 3.61 × 10

–8

cm.

Show that the calculated density is in agreement with its measured

value of 8.92 g cm

–3

1.16 Analysis shows that nickel oxide has the formula Ni

0.98

1.00

. What fractions

of nickel exist as Ni

and Ni

ions?

1.17 What is a semiconductor? Describe the two main types of semiconductors

and contrast their conduction mechanism.

1.18 Non-stoichiometric cuprous oxide, Cu

O can be prepared in laboratory.

In this oxide, copper to oxygen ratio is slightly less than 2:1. Can you

account for the fact that this substance is a p-type semiconductor?

1.19 Ferric oxide crystallises in a hexagonal close-packed array of oxide ions

with two out of every three octahedral holes occupied by ferric ions.

Derive the formula of the ferric oxide.

1.20 Classify each of the following as being either a p-type or a n-type

semiconductor:

(i) Ge doped with In (ii) Si doped with B.

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34Chemistry

Answers to Some Intext Questions

1.13 4

1.14 Total number of voids = 9.033 × 10

Number of tetrahedral voids = 6.022 × 10

1.15 M

1.17 ccp

1.21 Gold (atomic radius = 0.144 nm) crystallises in a face-centred unit cell.

What is the length of a side of the cell?

1.22 In terms of band theory, what is the difference

(i) between a conductor and an insulator

(ii) between a conductor and a semiconductor?

1.23 Explain the following terms with suitable examples:

(i) Schottky defect (ii) Frenkel defect (iii) Interstitials and (iv) F-centres.

1.24 Aluminium crystallises in a cubic close-packed structure. Its metallic

radius is 125 pm.

(i) What is the length of the side of the unit cell?

(ii) How many unit cells are there in 1.00 cm

of aluminium?

1.25 If NaCl is doped with 10

–3

mol % of SrCl

, what is the concentration of

cation vacancies?

1.26 Explain the following with suitable examples:

(i) Ferromagnetism

(ii) Paramagnetism

(iii) Ferrimagnetism

(iv) Antiferromagnetism

(v) 12-16 and 13-15 group compounds.

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