Chapter 1 The Solid State

The stability and existence of solid-state favours lower temperatures as it depends on 2 rival forces.

When the temperature is low, under these circumstances the thermal energy of molecules ( energy which tends to move them faster and makes them apart; sometimes results in a change of state) is sufficiently low and intermolecular forces (forces b/w molecules, ions, atoms that tend to keep them closer) are high enough to make molecules bring in a closer contact that they attach oneself tightly to another and occupy fixed position (in solid-state).

• Isotropicity (equal values of physical properties) or anisotropicity (unequal values of symmetrical properties) is the property that deals with distribution of inherent physical property( eg. - mechanical strength, refractive index) of any substance to the different molecular axes in accordance with the symmetry of the molecules of that substance.

• Crystalline solids are anisotropic in nature i.e. they show different values of physical properties like electrical resistance or refractive index in different directions, unlike the amorphous solids. This is primarily because of their uneven arrangement of particles in different directions (x, y and z axes). It depends upon the symmetry of the molecules which are arranged in a different way in all the planes (3-dimensional x, y and z planes ).

This figure above shows an example of a simple 2-dimensional arrangement of 2 different types of atoms in a crystal. Here the CD direction row there is 2 types of atoms whereas in AB direction row is made of only one type of atoms – this arrangement will result in anisotropicity and different values for physical properties in different rows of directions.

• Quartz glass is an amorphous or non-crystalline solid because it lacks in a long-range order i.e. a regular long pattern of constituent particles in three dimensions which periodically repeats itself throughout the whole crystal and have a definite geometrical shape which is the case for other 3 options (i) Graphite (C), (iii) Chrome alum and (iv) Silicon carbide (SiC).

• Instead of that quartz glass has a short-range order i.e. the repetitive pattern falls through a short distance only and the regular pattern is scattered and they are disarrayed in nature (resemblances that of a liquid) unlike quartz itself which is a crystalline solid.

• Antiferromagnetism is the case where the magnetic moment of the atoms or molecules occur in accordance with spins of their electrons (usually in low temperatures) align themselves with a regular pattern with the neighbouring spins oriented in opposite directions cancelling each other’s magnetic moments which is option (iv) .

• Options (i) and

(ii) are simply the case for ferromagnetism when the substance is placed in a magnetic field the domains (substance particles grouped in different regions) orient themselves in the direction of the magnetic field which results in strong magnetism.

• Option (iii) is clearly the case of ferrimagnetism where the domains (tiny magnets) are aligned in parallel and anti-parallel orientation in unequal numbers which result in very weak magnetism.

Since, quartz glass in an amorphous solid having short range order of constituents. Hence value of refractive index is same in all direction, can be measured and not be equal to zero.

• Amorphous solids usually have definite values of various physical properties like mechanical strength or refractive index, electrical conductivity etc.(isotropic not anisotropic) through all the directions. This is due to lack of long-range repetitive arrangement or order unlike crystalline solids; arrangement of particles is indefinite in different directions (x,y & z planes) which results in an overall equivalent arrangement in all directions, therefore giving birth to isotropicity and the value of physical properties will be same along all directions.

• Other 3 options (i) On heating they may become crystalline at certain temperature , (ii) They may become crystalline on keeping for long time and

(iii) Amorphous solids can be moulded by heating are possible cases for amorphous solids.

• Crystalline solids usually have sharp and well-defined melting points because they have a regular pattern of constituent particles which repeats itself periodically throughout the crystal i.e. long-range order. Therefore they have same distance between them or they have same kind of adjacent neighbours and this regularity in crystal lattices create equal environments for all the particles. Thus the intermolecular forces among them are evenly equal and same amount of thermal energy will be needed to break each interaction b/w particles (atoms, molecules or ions) at the same time.

• Options (i) a regular arrangement of constituent particles observed over a short distance in the crystal lattice and (iii) same arrangement of constituent particles in different directions are entirely untrue for crystalline solids.

• And option (iv) different arrangement of constituent particles in different directions cannot explain the phenomena of sharp melting points of crystalline solids because this is in the side of contrary.

• Iodine molecules are a form of non-polar constituent particles which form crystals and held together by London force (or induced dipole-dipole interaction) which is a weakly dispersion force. It is possible when 2 adjacent atoms of molecules come in a position where a temporary dipole is formed, it is the weakest intermolecular force.

Other 3 options (ii) dipole-dipole interactions (iii) covalent bonds and (iv) coulombic forces are not the case of iodine molecules because they require polarity or ionic entities to occur.

• Diamond is a crystalline solid of non-metal Carbon and it has covalent bonding throughout the crystal. Between two adjacent atoms of the crystal there is a strong and directional interaction which results in a covalent bond which helps the atoms to held strongly at their positions. Therefore resulting in a very high melting point which indicates that diamond is a network solid.

The network structure of diamond

• The other 3 options (i) SO₂ (Solid) (ii) I₂ and (iv) H₂O (Ice) are examples of molecular crystalline solids

• in which SO₂ (Solid) is a type of polar molecular solid (strong dipole-dipole interaction b/w S and O results in stronger polar covalent bonds).

• I₂ can be classified as non-polar molecular solid (formed by weakest dispersion force, London force due to temporary induced dipole-dipole interaction.

• And at last H₂O (Ice) is hydrogen-bonded molecular solid (polar covalent strong interaction b/w H and O reinforces the formation of hydrogen bond.

• I₂ (s) is a non-polar molecular crystalline solid obviously not an electrical conductor i.e. they are non-conductors of electricity because it is formed by weakly dispersion forces; London forces and there is no entity (like ions or ) to conduct electricity.

H₂O (s), which is hydrogen-bonded molecular crystalline solid also do not conduct electricity due to stronger hydrogen-bonding between adjacent water molecules there are no free ions left to conduct electricity.

• On the other hand option (A) Mg (s), which is metal and eventually has electrical conductivity properties because it has free and mobile electron (valence shell) which are cast evenly throughout the crystal.

• And in the case of (B) TiO (s) which behaves like a metal also can conduct electricity because the d-electrons are partially filled and leads to electrical conductivity.

• Although the component ions of the ionic solids are not free for movement in the solid-state as they are held together by strong electrostatic forces ; they become free to move about in the molten state or aqueous state (when they get dissolved in water because) they can overcome coulombic attraction in that state and hence they have a certain amount of electrical conductivity which is high in the molten state.

• Other 3 options (ii) Brittle nature (iii) Very strong forces of interactions and (iv) Anisotropic nature are 3 important characteristics of ionic solids

• Graphite has an exceptional but typical structure which leads to conductance.

• In this structure, the carbon atoms are arranged in multiple layers and each C atom is covalently bonded to another 3 adjacent C atoms residing in the same layer.

• And the 4^th valence electrons which were left out of the C atoms are present among different layers and are therefore free for movement. These free electrons result in graphite in a good conductor.

The other 3 options, (i) lone pair of electrons (iii) cations and (iv) anions are not possible for neutral carbon atoms present in graphite.

• TiO₃ behaves as conductor or insulator mainly because of the energy gap variability between valence band and conduction band. This gap changes in accordance with temperature; as the temperature increases so decreases the gap and it leads to more conductive nature which is opposite in case of lowering of temperatures.

• Options (i) TiO (ii) SiO₂ and (iv) MgO are incorrect .

• TiO which behaves like a metal also can conduct electricity because the d-electrons are partially filled and leads to electrical conductivity.

• (ii) SiO₂ (quartz) is a network -solid which is covalent in nature and always behave as an insulator.

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• And (iv) MgO is an ionic crystalline solid which means it acts as an insulator in the solid-state and conducts electricity in the molten or aqueous state.

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• CrO₂ behaves like a metal also can conduct electricity. Because Cr is a transition metal and has d-electrons (electrons present in d-orbitals) which are partially filled and leads to electrical conductivity just like metals having free electrons.

(i) SiO₂ (quartz) is a network solid which is covalent in nature and always behave as an insulator ( no free electrons to conduct electricity).

And (ii) MgO is an ionic crystalline solid which means it acts as an insulator in the solid-state( as there is no free movement for ions) and conducts electricity in the molten or aqueous state(ions overcome electrostatic attractions and are free to move).

(iii) SO₂(s) is a polar molecular crystal and therefore is formed by stronger dipole-dipole interactions which make them non-conductors of electricity, unlike metals.

The lattice site in a pure crystal cannot be occupied by electrons; because in each lattice point of a pure crystal (no impurity) there is only space for either molecule or atom or ions which are essentially joined together in straight lines and leads to a specific geometrical shape for each crystal. And they are also arranged in stoichiometric ratio which is fixed in proportion.

• But electrons can only occupy when there is a vacancy or defect in the corresponding crystal, i.e. crystal is no more in pure form it has impurities.

Hence, options (i) molecule, (ii) ion and (iv) atom are not correct because they can occupy the lattice site of a pure crystal.

• Graphite cannot be classified as an ionic solid because it does not have the characteristic of an ionic solid-

• it does not have ions as constituent particles rather neutral C atoms.

• Even in water, it does not form any sort of ions for conductivity.

• Whereas, graphite belongs to the classes (i) conducting solid (having multiple-layered structures and 3 fold bonding it is left with free electrons for conduction.

• (ii) network solid (carbon atoms are covalently bonded and form network type crystalline structure) and

• (iii) covalent solid ( constituent C atoms are covalently bonded with each other in multiple layers).

• In Frenkel defect which is also known as dislocation defect, the smaller ions (usually the cations) get dislocated from its normal lattice site to an interstitial site, therefore resulting a vacancy defect in the normal site (vacancy for cation) and simultaneously an interstitial defect in the new location of the cation.

• On the other hand (ii) Schottky defect (iii) Vacancy defect (iv) Metal deficiency defect do not necessarily deal with cations.

• In (ii) Schottky defect the number of missing cations and anions are equal in order to maintain electrical neutrality in the crystal and thus decreases density.

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• (iii)Vacancy defect occurs in case of any kind of vacancy in the crystal.

• And last (iv) Metal deficiency defect is not even a stoichiometric defect like the other options given. It is rather a non- stoichiometric defect that deals with less amount of metal as compared to the ideal stoichiometric proportion (loss of positive charge).

•The Schottky defect is basically a kind of vacancy defect with the tendency to maintain electrical neutrality, therefore same no.s of cations are anions are missing from the crystal lattice.

• Other 3 options are incorrect.

• (i) some of the cations move from their lattice site to interstitial sites – this occurs in the case for Frankel defect only.

• some lattice sites are occupied by electrons- this is the case of metal excess defect.

• (iv) some impurity is present in the lattice – this is observed in crystals with impurity defects.

• P-type semiconductors itself have positive charge carriers (holes) which are relatively free to move from one site to another (although the holes do not move actually , it is the electrons that move to holes and create a new hole in its original position).

• But at last it is neutral because there are fixed acceptors ( atoms ) who accept the electrons and become negative ; therefore a neutrality is worked out between the holes (positively charged) and the accepting atoms (negatively charged).

• Hence, options (i) , (iii) , (iv) are in correct.

• In order to build a n-type semiconductor from silicon, it has to be doped with electron-rich impurities (element having more valence electrons).

• As group 14 element Si has 4 valence electrons, therefore to get an extra electron carrier for the semiconductor it has to be doped with elements of valency 5 i.e. group 15 elements such as P or As.

• Four out of five electrons of group 15 element (P or As) will be forming four covalent bonds with the four silicon atoms which are adjacent.

• The fifth valence electron of the group 15 element now becomes an extra and gets freed and delocalised. These delocalised electrons enhance the conductivity of doped silicon.

Hence, other 3 options (i) 2 (ii) 1(iii) 3 are incorrect, because their doping would not form any n type semiconductor.

• We know that ,if n be the number of atoms in a crystal,

then the number of tetrahedral voids must be 2n.

• And for a face centred unit cell -

(i)there are 8 Corner atoms i.e. (8 X1/8) atoms per unit cell=1atom and

B. face-centred atom number is 6 i.e.( 6 X 1/2) atoms per unit cell= 3 atom (because each atom at the face centre is used by to adjacent unit cells)

Hence no. Of tetrahedral voids for face centred unit cell = 2(4) = 8.

Hence other 3 options (i) , (iii) and (iv) cannot be correct.

• AgBr(s) has CCP(cubic close-packed) type crystals and shows both of the Schottky defect and the Frenkel defect.

• this is mainly due to the intermediate sizes of the radius of Ag⁺ and Br^- ions for which they are able to show defect with cation dislocation in the lattice (Frankel ) and the missing of equal no.s of cations and anions (Schottky) as well.

• AgBr(s) crystals do not show (C) Metal excess defect or (D) Metal deficiency defect hence other options (ii) (C) and (D) (iii) (A) and (C) (iv) (B) and (D) are all incorrect.

• In hcp (hexagonal close packing), the third layer and the first layer resembles each other in arrangement of the spheres and is able to cover all the tetrahedral voids.

• Since the pattern of spheres is repetitive in alternative layers, the stacking for hcp may be described as "A-B-A-B-A-B."

• The atoms in a hexagonal closest packed structure efficiently occupy 74% of space while 26% is empty space which is one of the efficient packings.

• The arrangement of ccp (cubic close packing )also efficiently fills up 74% of space( Similar to hexagonal closest packing).

• In ccp the 2nd layer of the spheres is placed on half of the depressions of the first layer and the third layer is completely different than that of the first two layers and is stacked in the depressions of the second layer, thus it covers all of the octahedral voids.

• The spheres in the third layer are not in line with those in layer 1, and the pattern of spheres does not repeat until a fourth layer is added. The fourth layer is the same as the first layer, so the arrangement of layers is "A-B-C-A-B-C."

• In a body-centred cubic arrangement of lattices, the constituent atoms are located on the 8 corners of the cube with one atom at the centre of the cube.

Let the edge length of the cube be ‘a’ (ED or BC in the fig) and radius of each constituent spherical particle be ‘r’ and the arrangement is such as that atom at the centre is in touch with other two atoms diagonally.

• Hence the body diagonal AF will be equal to = r +2r + r = 4r (1)

In the right angle ∆ EFD,

FD² or b²⁼ FE² + ED² ( by Pythagoras’s theorem ) = a² + a² = 2a²

Or, FD or b = √2a

• In the right-angled triangle, ∆ AFD

AF² or c²= AD² + FD² = a² + b² = a² + 2a² (as b²= 2a² ) =3a²

Or, AF or c =√3a (2)

Hence, from equations (1) and (2) we can get ,

AF = 4r =√3a a

Or, r = √3a/4

And a = 4r/√3

• Then, the volume of 1 cubic unit cell is a³ = (4r/√3) = 64r³/ 3√3

• Since in body centred arrangement , there are 2 atoms per unit cell

Volume occupied by the atoms = 2 X 4/3 π r³ = 8/3 π r³

Since, Packing efficiency = Volume occupied by two spheres in the unit cell X 100 %

Total volume of the unit cell

=8/3 π r³X 100 % = 68.04 % = 68%_appx.

64r³/ 3√3

Therefore , Packing efficiency in a body centred cubic lattice is 68 %

So, empty space or void space will be = (100 -68 )% = 32 %

• Hexagonal close-packed crystal structure forms only when the centres of spheres in the 3^rd layer are vertically above the centres of first layer (A) as the 2^nd layer is placed in only one type of depressions of the 1^st layer( at each junction of spheres ).

• This depression gives rise to voids which by placing new layer of spheres i.e. the 3^rd one (also A) can be covered well.

• This third layer has to resemble the first layer because layer 1 and layer must be in the same alignment and thus the pattern of spheres is repeated in alternate layers. That is why, this pattern is often written as ABAB .......pattern.

Unlike in ccp which is the actual case for (iv).

• The coordination number is 12 – it is true as in hcp each sphere in a layer is surrounded by 6 spheres in the same plane , 3 spheres above it and 3 spheres below it – therefore making a total 12 which is the coordination number for this arrangement.

• (iii) It has 74% packing efficiency – also true as it resembles with ccp in efficiency it can be calculated in accordance with ccp structure ;

If the unit cell edge length= a and face diagonal AC=b

Then in right angled triangle ∆ ABC ,AC² or b² =BC²+AB²= (a²+a²⁾=2a²

Or , b=√2a

if r is the radius of the sphere, then b=r +2r+r=4r=√2a

or, a=4r/√2=2√2r (i.e. r=a/2√2)

it is known that each unit cell in ccp/hcp arrangement has effectively four spheres.

Hence , the total volume occupied by four sphere is = to 4×(4/3)πr³ and

Therefore, volume of the cube = a³ = (2√2r)³.

Therefore, packing efficiency= (volume occupied by 4 spheres in unit cell/ total volume of the unit cell )×100℅

= 4×(4/3)πr³/(2√2r)³ ×100%=74%

• Tetrahedral voids of the second layer are covered by the spheres of the third layer –

this is also a true fact for hcp , because When a sphere of the second layer(B) is placed above the void of the first layer (A) always tetrahedral void is formed this is because a tetrahedron forms when the centres of four spheres are joined and this tetrahedral voids can be totally covered by the placement of a 3^rd layer (A) above them.

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In NaCl, the Cl^- ions are arranged in ccp arrangement where Cl^- ions are present at the corners as well as the center of the face of the cube. The Na⁺ ions are located such that each Na⁺ ion is surrounded with six chloride ions. Na⁺ ions occupy all the octahedral voids. Each chloride ion is also surrounded by six Na⁺ ions and the stoichiometry is 1:1. Thus the coordination number for both species is the same.

In CaF₂ arrangement, the Ca²⁺ ions are arranged in ccp arrangement where Ca²⁺ ions are placed at the corners as well as the center of the face of the cube. Fluoride ions occupy tetrahedral voids and in way that each ion is surrounded by four calcium ions. The coordination number of F^- is 4. There are two tetrahedral sites available for each calcium ion, and thus the F^- ions occupy all eight tetrahedral voids. However, the stoichiometry is 1:2, and the coordination number of calcium ion is 8, therefore the coordination number for these species is not the same.

In Na₂O arrangement, the O^2- ions are arranged in ccp arrangement where O^2- ions are placed at corners as well as the center of the face of the cube. Na⁺ ions occupy all tetrahedral voids surrounding O^2- ion. Each sodium ion is surrounded by 4 oxide ions and each oxide ion is surrounded by 8 sodium ions. The coordination number of Na⁺ is 4 and O^2- is 8, therefore the coordination number of these species is not the same.

In Zinc blende ZnS arrangement, the S^2- atoms are in ccp arrangement, where each sulphideion is surrounded by four zinc ions. There are eight tetrahedral voids and four sulphide ions occupy half of the tetrahedral holes. Each sulphide ion itself is surrounded by four zinc ions, thus making both of their coordination numbers 4.

Thus the correct answer is (i).

Two dimension close packed structures, where rows of identical spherical molecules are stacked on top of each other, can be