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UNIT I Syllabus : THE SOLID STATE Marks-4Classification of Solids based on different binding forces : molecular, Ionic,
covalent and metallic solids, Amorphous and crystalline solids (elementary
idea), unit cell in two dimensional and three dimensional lattices, calculation
of density of unit cell, packing in solids, voids number of atoms per unit cell
in a cubic unit cell, point defects, electrical and magnetic properties.
1. Gystalline and Amorphous solids (comparison)
Property Crystalline Solids Amorphous Solids
* Shape
* Melting
point
*Cleavage
property
* Heat of
fusion
*
Anisotropy
* Nature
* Order in
the
arrangement
of
constituent
particles
Examples
Definite characteristic shape.
Melt at a sharp and characteristic
temperature
When cut with a sharpedged tool,
they split into two pieces and the
newly generated surfaces are
plane and smooth.
They have a definite and
characteristic heat of fusion.
They are anisotropic in nature.
True-solids
They have long range order.
All solid elements and solid
compds : eg. Cu, Al, NaCl, Sugar
etc.
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
heat of fusion.
Isotropic in nature.
Pseudo solids or super
cooled liquids
They have only short range
order.
Glass, plastics, Rubber,
charcoal, lamp black etc.
1
2. Classification of solids on the basis of different binding forces.
Type of
solid
Constituent
particles
Nature of
bond
Examples Physical
Nature
Electrical
Conductivity
Melting
point
1.
Molecular
Solids
(i) Non
Polar
(ii) Polar
(iii)
Hydrogen
bonded
Non polar
molecules
Polar
molecules
H-bonded
molecules
Dispersion
or London
forces
Dipole-
dipole
force
Hydrogen
bonding
Ar., CCl4,
H2, Solid
CO2, I2
HCl, HBr,
SO2
H2o (ice)
NH3(S)
HF (S)
Soft
Soft
Hard
Insulator
Insulator
Insulator
Very
low
Low
Low
2. Ionic
Solids
Ions Coulomic
forces
NaCl, Mgo,
ZnS, CaF2
Hard but
brittle
Insulator in
solid state but
conductors in
molten state
and aqueous
state
High
3. Metallic solids
Positive ions in a sea of delocalized electrons
Metallic bonding.
Fe, Cu, Ag, Mg
Hard but malleable and ductile
Conductor in solid as well as in molten state.
Fairly high.
4. Covalent or Network solids
Atoms Covalent bonding
SiO2(quartz) SiC ; C (diamond) graphite
Hard
Soft
Insulator
Conductor (exception)
Very high
3. *Some Important Terminology.
* Space lattice/Crystal lattice : A well defined ordered and regular
arrangement of atoms, molecules or ions in the three dimensional
space is called crystal or space lattice. There are fourteen types of
space lattices possible, they are also called bravais lattices.
2
* Lattice Point : The atoms, molecules or ions in a crystalline
substance are shown by points in its space lattice and are called lattice
points.
* Lattice Site : The position occupied by a constituent particle in a
space lattice is called lattice site. Eg. Corner, Body centre, Face centre
etc.
* Unit Cell : The smallest part of the crystal lattice which when
repeated over and over again produces the complete crystal is called
unit cell.
4. Calculation of number of atoms (z) per unit cell/rank of Unit Cell
of a Cubic crystal system :
Unit Cell No. of
atoms at
corners
No. of atoms
at face centres
No.of atoms at body
centre
Total
Primitive/Simple
Cubic U.C.
8x1/8 = 1 0 0 1
Body centered
Cubic U.C.
8x1/8=1 0 1 2
Face Centred
Cubic U.C.
8x1/8=1 6x1/2=3 0 4
5. Seven Primitive unit cells and their possible variations as centred unit
cells:
Crystal Possible Axial Axial angles Examples
3
System Variations distances or
Edge
Lengths
* Cubic Primitive,
Body
centred, face
centred
a=b=c £=ß=γ=900 NaCl, ZnS, Cu
* Tetragonal
*Orthorhom
-bic
* Hexagonal
*Rhombohe
d
-ral or
Trigonal
*Monoclinic
Tridinic
Primitive,
Body
centred
Primitive,
Body
Centred,
Face
Centred, End
Centred
Primitive
Primitive
Primitive
End Centred
Primitive
a=b=c
a=b = c
a=b = c
a=b=c
a=b=c
a=b=c
£=ß=r=900
£= ß=r=900
£=ß=900
r=1200
£=ß= r=900
£=r=900
ß = 1200
£=ß=r=900
White Tin, SnO2,
TiO2, CaSO4
Rhombic-
Sulphur, KNO3,
BaSO4
Graphite,
ZnO, CdS
Calcite
(CaCO3), HgS
Mono Clinic
Sulphur,
Na2SO4.10 H2O
K2Cr207,
CuSO4.5 H2O,
H3BO3
6. Close Packing in Crystals :
Two dimensional close packing in metallic crystals.
4
Square close Packing Hexagonal Close Packing
* The particles of adjacent rows
show vertical as well as horizontal
alignment.
* Coordination number of each
particle is four.
* Packing fraction or packing
efficiency is about 52.4%
* The shape of void is tetraangular
The particles in every alternate row are
vertically aligned
Coordination number of each particle
is six.
Packing fraction or packing efficiency
is about 60.4%
The shape of void is triangular.
As the packing efficiency of hexagonal close packing is more than square
close packing. Hence H.C.P. is more stable arrangement in two dimensions.
Three dimensional close packing is metallic crystal.
(i) Three dimensional close packing from two dimensional square Close-
packed layers :-
In this packing the square close-packed layers of particles are stacked
one over the other so that all the atoms are vertically as well as
horizontally aligned.
5
The lattice thus generated is the simple cubic lattice, and its unit cell is
the primitive cubic unit cell.
(ii) Three dimensional close packing from two dimensional hexagonal
close-packed layers :
In three dimentional close packed structures the arrangement of hexagonal layers
can be done in two ways :
ABAB….Packing ABC ABC….Packing
ABAB….or H.C.P. packing is generated by
stacking H.C.P.-layers one over the other
such that spheres in each alternate layer are
vertically aligned
Hexagonal close packing, e.g.
Mo, Mg, Be etc.
Packing efficiency is 74%
ABC ABC or C.C.P. is
generated by stacking
H.C.P. layers one over
the other such that
spheres in every fourth
layer are vertically
aligned
Cubic close packing,
e.g.
Cu, Ag, Au, Fe, Ni, Al,
etc.
Packing efficiency is
74%
6
7. VOIDS : The hollow or vacant spaces among the constituent particles in
close packed structures are called voids or interstitials.
Two types of voids in crystals :
Tetrahedral void Octa-hedral void
*The void is surrounded by four
spheres arranged tetrahedral
*Coordination number of letra
hadral void is 4.
* The number of tetra hedral voids
per particle is 2.
Radius ratio in tetrahedral void is
0.225.
The void is surrounded by size spheres
arranged octahedrally
Coordination number of octahedral void
is 6.
The number of octahedral voids per
particle is one.
Radius ratio in octahedral void is 0.414.
8. Relation ship between density and edge length of a unit Cell of a
crystal.
Donsity ‘d’ of unit cell Mass of Unit cell =
Volume of unit cell
No. of atoms x Mass of each atom =
Volume of Unit cell
d = Z x M /a3 x NA
Z – No. of atom/U.C., M – Molar mass of element
a—edge length of U.C. and NA—Avogadro’s No.
7
9. Packing efficiency in case of metal crystals with the assumption that
atoms are touching each other.
Packing efficiency = Volume occupied by all the spheres
in the unit cell x 100 / volume of U.C.
For simple cubic system :-
a = 2 r
No. of spheres/U.C. = one
Packing fraction = 1 x 4/3 kk r3x100 ---------------------- (2r)3
= 52.4%
For body centered Cubic system :-
BC = 2 a
CD = 4 r = 3 a or a = 4r/3
No. of spheres per U.C. = 2
Rocking efficiency = 2 x 4/3 kk r3 x 100 ----------------------
(4r/ 3)3
= 68%
For Face centered cubic system :
AC = 2 a
AC = 4r = 2 a or a = 2 2 r
No. of spheres per U.C. = 4
Packing efficiency = 4x4/3 kk r3 x 100 ------------------- (2 2 r)3
= 74%
10. Important Questions (Sold)
1. Some glass objects from ancient civilizations are found to become
milky in appearance, why ?
8
Ans. At some temperature they under go crystallization, which make them
milky in appearance.
2. Glass panes fixed to windows or doors of old buildings are found to be
thicker at bottom than at the top. Why ?
Ans. Glass is an amorphous solid, which has property of fluidity.
3. A solid has a cubic structure in which ‘x’ atoms are located at the
corners of the cube ‘y’ atoms are at the cube centres and ‘o’ atoms are
at the edge centres. What is the formula of the compound ?
Ans. No. of ‘x’ atoms = 8 x 1/8 = 1
No. of ‘y’ atoms = 1
No. of ‘o’ atoms = 12 x ¼ = 3
Formula of compound = XYO3
4. Potassium crystallizes in a body centred cubic lattice. What is the
approximate number of unit cells in 11.7 g of potassium ? [ At. Mass
of k = 39 U ]
Ans. No. of moles of ‘k’ = 11.7/39 = 0.3
No. of atoms of ‘k’ = 0.3 x 6.022 x 1023
No. of atoms of ‘k’ per B.C.C. Unit Cell = 2
No. of Unit cells = 0.3 x 6.022x1023 = 9.03 x 1022
25. 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 ?
Ans. No. of ‘N’ atoms = 8 x 1/8 + 6 x ½ = 4
No. of ‘M’ atoms = 8 x 1/3 = 8/3
Formula of compound = M8/3 N4 = M2N3
6. Niobium Crystallizes in body centered cubic structure of density 8.55
g cm-3. Calculate atomic radius of niobium.
[Atomic mass of Nb = 92.91 U ]
Ans. d = Z x M / a3 x NA or a3 = 2 x 92.91
6.0222 x 1023x8.55
9
= 3.609 x 10-23 = 36.09 x 10-24
a = 3.305 x 10-8 cm = 330.5 pm
4r = 3 a or r= 1.732 x 330.5 ----------------
4 = 143 pm.
7. An element crystallizes in a structure having a FCC unit cell of an
edge 200 pm. Calculate the density, if 200 g of this element contains
2.4 x 1024 atoms.
Ans. Z x M d = a3 x NA
Here we can take M/NA = 200/2.4 x 1024
4 x 200 d = = 41.67 g/cc (200)3 x (10-10)3 x 2.4 x 1024
8. The density of chromium metal is 7.2 g/cc. If the unit cell is cubic
with an edge length of 289 pm, determine the type of the unit cell
present in its crystals.
[Atomic mass of Cr = 52 U ; NA = 6.02 x 1023 ]
Ans. Z x M or d x a3 x NA d = Z = a3 x NA M
7.2 x (289)3 x (10-10)3 x 6.02 x 1023
or Z = 52
= 2.011 = 2
Unit Cell is B.C.C.
10
Important Questions (Un sold)
1. What is unit cell ? calculate the number of atoms in a f.c.c. unit cell of
an element.
2. Two elements P and Q have B.C.C. and F.C.C. structures. What will
be the number of tetrahendral and octahedral voids per unit cell ?
3. Give the coordination number of tetrahedral and octahedral voids.
4. An element exists as hexagonal close packed structures as well as
cubic close packed structure. In which case the element would have
higher density ?
5. In a compound, oxide ions have CCP arrangement. Cations A are
present in one eighth of the tetrahedral voids and cations B occupy
half the octahedral voids. What is the simplest formula of the
compound?
6. If the radius of bromide ion is 0.182 pm, how large a cation can fit in
each of the letrahedral void ?
7. Iron has body centred cubic structure. The edge length of the unit cell
is found to be 286 pm. What is the radius of an iron atom ?
8. Calculate the density of silver which crystallizes in the F.C.C.
structure. The distance between the nearest silver atoms is 287 pm.
(Molar mass of silver = 107.87 g/mol).
Imperfections in Crystals
11. Any deviation from perfectly ordered arrangement of constituent
particles in a crystal is called imperfection or defect.
A defect in a crystal arise due to Irregularity in the arrangement of
particles or presence of impurities or rise in temperature of the crystal.
Defects in crystals modify the existing properties or even introduce
new properties in a crystal.
Broadly, we can classify the defects into two categories :
(i) Point defects or atomic defects (ii) Line defects.
11
Point defects arise due to irregularities from ideal arrangement
around a point or an atom in a crystal, where as line defects arise due
to deviation from ideal arrangement in the entire rows of lattice point.
Point defects are of three types :
(i) Stoichiometric defect
(ii) Non stoichiometric defect and (iii) Impurity defects.
Stolchiometric defects :
A defect in a crystal due to which the stoichiometry of the crystal
remains unaltered. These defects are of two types.
Property Schott Ky defect Frenkel defect
Definition
Density
Conductivity
Mechanical
Stress
Type of
Crystal in
which the
defect arises
Examples
A point defect which arises due to
complete missing of ion pairs.
It decreases
Increases
Decreases
Ionic crystals with high
coordination number and having
comparable ionic radii.
NaCl, KCl, CsCl and Ag Br
Ag Br shows both type of defects
A point defect which arises
due to displacement of a
smaller ion from its lattice
site to interstitial hole.
It remains the same.
Increases
Decreases
Ionic crystals with low
coordination number and
having large difference in
ionic radii
Zn S, Agcl, Ag Br, AgI
12
Exeption -
Non stoichiometric defects : A defect in a crystal due to which
stoichiometry (ie ratio of cation to anion) of the compound is changed.
These defects arise in two ways :-
(i) By metal excess
(ii) Metal deficiency
* Metal Excess defect can arise in a crytal by anion vacancy or by extra
cation occupying Interstitial hole.
Metal excess defect due to anion
vacancy
Metal excess defect due to interstitial
cations
* A compound may have excess
metal ion if an anion is absent from
its lattice site leaving a hole which is
occupied by the electron to maintain
the electrical neutrality.
* This defect is generally developed
when an alkali metal halide (NaCl,
KCl) is heated in an atmosphere of
alkali metal vapours. Some of the
anions diffuse in to the surface of the
crystal and combine with alkali metal
atoms to give NaCl. The electron
released are trapped in anionic
vacancy
If an extra cation is present in an
interstitial site and electrical neutrality
is maintained by the presence of
electron in the nearby interstitial sites.
This defect arises in ZnO, when it is
heated, it loses oxygen and Zn2+ ions
are trapped is interstitial holes along
with electrons in the neighboring sites
to maintain electrical neutrality.
*These defects are found in the crystals
which is likely to possesses frenkel
defect.
13
* These types of defects are found is
the crystal which is likely to possess
schottky defect.
* F-Centre : An electron trapped in
an anion vacancy is called F-centre.
F-centre in a crystal make it coloured
& para-magnetic. E.g. NaCl becomes
yellow, KCl – Violet and LiCl—
pink.
* Zn O becomes yellow other wise it is
white. The formula of Zinc oxide
becomes Zn(1+x) O.
* Metal deficiency defect :
There are many solids which are difficult to prepare in stoichiometric
composition and contain less amount of metal as compared to the
stoichiometric proportion.
This defect generally found in compounds in which metal exhibits
variable valency e.g. FeO, FeS and NiO. Incase of FeO some of Fe2+ ions are
missing and loss of +ve charge is balanced by the presence of required
number of Fe3+ ions. For the loss three Fe2+ ions in a crystal there will be two
Fe3+ ions in the cation vacancies and one cation vacancy will be unoccupied.
The formula of nonstoichiometric FeO is fe0.95 O1.0
Impurity defects :
These defects in the Ionic solids may be introduced by adding
impurity ions having different valence state than that of the host ions and
creating cation vacancy. For example, addition of SrCl2 to NaCl or CdCl2 to
14
AgCl yields solid solutions in which there will be one cation vacancy for
every impurity ion added.
(Na+) (Cl-) (Na+) (Cl-) (Na+) Cation vacancy(Cl) (Na+) (Cl-) (Cl-)
(Sr2+) (Cl-) (Na+) (Cl-) (Na+)
(Cl-) (Na+) (Cl-) (Na+) (Cl-)
12. Properties of Solids :
(a) Electrical properties of Solids
Solids can be classified into three types on the basis of their
conductivities.
(i) Conductors : The solids with conductivities ranging between 104 to
107 ohm-1 m-1 are called conductors. Metals have conductivities in the
order of 107 ohm-1 m-1 are good conductors.
(ii) Semiconductors : These are the solids with conductivities ranging
between 10-6 to 104 ohm-1 m-1.
(iii) Insulators : These are the solids with very low conductivities ranging
from 10-20 to 10-10 ohm-1 m-1
Size of energy gap in conductors, semiconductoss and Insulators:-
In metal (conductors) there in no energy gap due to overlapping
between valence band and conduction band and it decreases with
increase in temperature. Sn (Tin) Eg = 8 kg/mol .
In case of semi conductors like germanium and silicon, the energy gap
is very small and conductivity increase with increase in temperature
eg. Si and Ge (Eg = 111 kj/mol).
In case of Insulators, the energy gap is very large and therefore the
vacant conduction bond is not available to the electrons of the
completely filled valence band. E.g. Diamond (Eg=511 kg/mol has
large energy gap. It is an Insulator.
15
Diagrammatical representation of energy gap (Eg) :
Electrical conductivity of semiconductors increase with rise in
temperature, sincemore electons can jump to the conduction bond. Si and Ge
show this type of behavior and are called intrinsic semi conductors. The
conductivity of Semi conductors can be increased by adding an appropriate
amount of suitable impurity. This process is called doping.
Doping can be done with an imparity which is electron rich or electron
deficient as compared to the intrinsic semi conductor Si or Ge. Such
impurities introduce electronic defects in them.
(a) Electron-rich impurities : Si/Ge (Group -14) doped with electron-rich
impurity P/As (Group -15) is called n-type semi conductor. Fig. (b).
(b) Electron-deficient impurities : Si/Ge (Group-14) doped with electron-
deficient impurity B/Al (Group-13) is called p-type semi conductor. Fig.
16
(c).
Creation of n-type and p-type semi conductors by doping groups 13 and 15
elements.
(b) Magnetic properties :
Every substance has some magnetic property associated with it. The
origion of these properties lies in the motion of an electron in a atom. Each
electron therefore behave like a tiny magnet. The magnetic moments which
originate from two types of motions.
(i) Orbital motion of electron around the nucleus
(ii) Spin motion of electron around its own axis.
Magnetic moment Magnetic moment
17
On the basis of magnetic properties, substances can be classified into
five categories :-
(a) Diamagnetic solids : These are weakly repelled by external magnetic
field and do not have unpaired electron e.g. TiO2, V2O3, NaCl, C6H6.
(b) Paramagnetic solids : These are weakly attracted by external magnetic
field and have unpaired electrons. They lose magnetism in the absence of
external magnetic field e.g. Cu0, 02, Fe3+, TiO.
(c) Ferromagnetic solids : These are strongly attracted by the external
magnetic field and show permanent magnetism even if external magnetic
field is removed e.g. Fe, Co, Ni, CrO2.
(d) Anti ferromagnetism :- In these type of solids the domain (A group of
metal ions in a small region) orient in anti parallel direction in equal
number so that the net magnetic moment is zero eg. MnO.
(e) Ferromagnetic solids :- In these type of solids the magnetic moment of
domains are oriented in anti parallel direction in unequal number so that
the net magnetic moment is very small e.g. Fe3O4 (Magnetite) and all
ferrites like MgFe2O4 and Zn Fe2O4.
(a)
(b)
(c)
Schematic alignment of magnetic moments in
(a) Ferromagnetic (b) anti ferromagnetic and
(c) Ferromagnetic substances
18
Important questions (Un solved)
1. State difference between schottky and frenkel defects.
2. Cite one example each of following : (i) Solid which exhibit schottky
defect and (ii) Frenkel defect. (iii) Schottky and frenkel defects both.
3. What is F-centre ? What are its effect on the properties of solid.
4. What happens when Zn O is heated strongly ? Point out the Kind of defect
developed in ZnO as a result of heating.
5. Which of these two CdCl2 and NaCl will produce schottky defect ?
6. What happens when Fe3o4 is heated to 850 K and why ?
7. Analysis shows that Nickel oxide has formula Ni0.98 O1.0
What fraction of Nickel exists as Ni2+ and Ni3+ ions ?
8. Give at least two important application of n and p-type semi-conductors.
9. Define the terms “ Valence band, conduction band, forbiddon zone,
ferrimagnetic solids and ferro magnetic solids.
10. If NaCl is doped with 10-2 mol% of SrCl2. What is the concentration of
cation vacancy and actual number of cation vacancies ?
For Bright Students (Add to your Knwolwedge)
1. Effect of temperature and pressure on crystal structure : with the
increase in pressure the corrdination number increases resulting is change
in the crystal structure e.g. when NaCl (6 : 6 coordination) crystal is
subjected to high pressure it changes to CsCl-type structure (8:8-
coordinations)
If temperature is increased, the coordination number decreases e.g.. When
CsCl (8:8-coordination) crystal is heated to 760 K it changes to NaCl
type structure (6:6 – coordination)
2. Relation between radius ratio and coordination number for Ionic
crystals :
Radius ratio : It is the ratio of radius of cation to radius of anion in an
ionic crystal.
Radius ratio r+/r- Structure Coordination No. Examples
19
0-0.155
0.155-0.225
0.225-0.414
0.414-0.732
0.732-1.00
Linear
Triangular
Tetra hedral
O ctahedral
Cubic (b.c.c.)
2
3
4
6
8
HF2
B2O3, BN
SiO44-; ZnS
MgO, NaCl
CsCl, Cs Br.
3. Structure of simple cubic Ionic compounds (AB, AB2, A2B-Types)
Type Structure Example Co.No. No. of formula units per U.C.
Calculation of no. of cations and onions per unit cell.
(i) AB Rock Salt-type (NaCl)
F.C.C. Halides of Li, Na, K and Rb Mgo, AgF, NH4Cl
Na+=6Cl- = 6 4
Na+= 12x1/4+Body centre = edge centre+Body centre = 3+1 = 4Cl = 8x1/8+6x1/2 corners + face centres = 1 + 3 = 4
(ii) CsCl B.C.C. Cs Br, Cs I, TlCl,Te Br, TlI, TlCN
Cs+=8Cl-=8 1
Cs+ = Body centre = 1Cl- = Corners = 8x1/8=1
(iii) Zn S(Zinc blende)
F.C.C. CuCl, Cu Br, CuI AgI, BeS.
Zn2+=4S2-=4 4
Zn2+=4 (50% of Tetra hedral vids)S2-=4 (at corners and face cntre)
AB2-TypeCaF2
(Fluorite type)
F.C.C. BaF2, BaCl2,SrF2, SrCl2
CdF2, PbF2
Ca2+=8F-=4 4
Ca2+=4(one at corner + three At face centres)F = 8 (at 100% Tetra hedral voids)
A2B- F.C.C. K2O, Li2O, Na+=4 Na+=8 (one at each
20
Type Na2O(Anti fluo rite)
Rb2O, Rb2S O2-=8 4 Tetra hedral void)O2-=4 (One at corner three at Face centres.)
Important Question (Un solved)
1. What are the coordination no. of Cation and anions in CaF2 and K2O.
2. Mgo has NaCl-type structure. What is the coordination no. of oxide
ion?
3. The ionic radii of Mg2+ and O2 ions are 66 pm and 140 pm
respectively. What type of interstial void Mg2+ is likely to occupy ?
What is the probable coordination number of Mg2+ ion ?
4. If the radius of Bromide ion is 0.182 nm, how large a cation can fit in
each of the Tetra hedral void ?
5. Lead II sulphide crystal has NaCl structure. What is its density ? The
edge length of the uniti cell of PbS crystal is 500 pm.
(NA=6.0222x1023 mol-1, at masses : Pb = 207, S = 32)
6. Caesium chloride crystallizes as a body centred cubic lattice and has a
density 4.0 g/cc. Calculate the edge length of unit cell. Also distance
between cesium and chloride ions. [Molar Mass of CsCl = 168.5,
NA=6.022 x 1023]
21