Classification and periodicity of elemaent

Unit 2.1 SOLUTION

LeC 4                          SOLUTIONS

Mostly of the reaction in laboratories are carried out in the form of solution so we have to study  solution. we will consider mostly liquid solutions and their formation. This will be followed by studying the properties of the solutions, like vapour pressure and colligative properties. We will begin with types of solutions and then various alternatives in which concentrations of a solute can be expressed in liquid solution.

Solutions are homogeneous mixtures of two or more than two components. By homogenous mixture we mean that its composition and properties are uniform throughout the mixture. Generally, the component that is present in the largest quantity is known as solvent. Solvent determines the physical state in which solution exists. One or more components present in the solution other than solvent are called solutes. In this section we shall consider only binary solutions i.e. solution of two componenets. 

Therefore, it is important to understand as how the amount of substance is expressed when it is present in the form of a solution. The concentration of a solution or the amount of substance present in its given volume can be expressed in any of the following ways.

1. Mass per cent or weight per cent (w/w %) or Volume Per cent (v/V%)

2. Mole fraction

3. Molarity

4. Molality

5. Part per Million

6. Normality

Let us now study four of them in detail, we will Normality study in  12^th standard.

1. Mass per cent or weight per cent (w/w %) or Volume Per cent (v/V%)

The mass percentage of a component of a solution is defined as:

For example, if a solution is described by 10% glucose in water by mass, it means that 10 g of glucose is dissolved in 90 g of water resulting in a 100 g solution

similarly we can define the Volume percent as

For example, 10% ethanol solution in water means that 10 mL of ethanol is dissolved in 90 mL water such that the total volume of the solution is 100 mL.

2. Mole fraction

Commonly used symbol for mole fraction is x and subscript used on the right hand side of x denotes the component. It is defined as:

For example, in a binary mixture, if the number of moles of A and B are nA and nB respectively, the mole fraction of A  & B will be

if there are i component in the solution then there is i mole fraction and sum of all mole fraction is unity.

3. Molarity

Molarity (M) is defined as number of moles of solute dissolved in one litre (or one cubic decimetre) of solution,

For example, 0.25 mol/L (or 0.25 M) solution of NaOH means that 0.25 mol of NaOH has been dissolved in one litre (or one cubic decimetre).

4. Molality

Molality (m) is defined as the number of moles of the solute per kilogram (kg) of the solvent and is expressed as:

For example, 1.00 mol/kg (or 1.00 m) solution of KCl means that 1 mole (74.5 g) of KCl is dissolved in 1 kg of water.

Each method of expressing concentration of the solutions has its own merits and demerits. Mass %, mole fraction and molality are independent of temperature, whereas Molarity is a function of temperature. This is because volume depends on temperature and the mass does not.

Problem: Calculate (a) molality (b) molarity and (c) mole fraction of KI if the density of 20% (mass/mass) aqueous KI is 1.202 g/mL.

5. Part per Million

Parts per million: When a solute is present in trace quantities, it is convenient to express concentration in parts per million (ppm) and is defined as:

As in the case of percentage, concentration in parts per million can also be expressed as mass to mass, volume to volume and mass to volume. A litre of sea water (which weighs 1030 g) contains about 6 × 10–3 g of dissolved oxygen (O2). Such a small concentration is also expressed as 5.8 g per 106 g (5.8 ppm) of sea water. The concentration of pollutants in water or atmosphere is often expressed in terms of μg/mL or ppm.

Solubility 

Solubility of a substance is its maximum amount that can be dissolved in a specified amount of solvent. It depends upon the nature of solute and solvent as well as temperature and pressure. this is the reason that we can mix a certain amount of sugar or salt in water, upto its solubility we cant mix more salt or sugar at that temperature and pressure. if we want to mix more amount of sugar then we have to change its temperature and pressure.

Let us discuss solution of solid in liquid, gas in liquid and liquid in liquid.

Solubility of a Solid in a Liquid

Every solid does not dissolve in all liquid,  a solute dissolves in a solvent if the intermolecular interactions are similar in the both solute and solvent.we may say like dissolves like .It is observed that polar solutes dissolve in polar solvents and non polar solutes in nonpolar solvents.

for example sodium chloride and sugar dissolve readily in water, naphthalene and anthracene do not. On the other hand, naphthalene and anthracene dissolve readily in benzene but sodium chloride and sugar do not. 

When a solid solute is added to the solvent, some solute dissolves and its concentration increases in solution. This process is known as dissolution. Some solute particles in solution collide with the solid solute particles and get separated out of solution. This process is known as crystallisation. 

A stage is reached when the two processes occur at the same rate. Under such conditions, number of solute particles going into solution will be equal to the solute particles separating out and a state of dynamic equilibrium is reached.At this stage the concentration of solute in solution will remain constant under the given conditions, i.e., temperature and pressure.

Solute + Solvent -------------> Solution + Heat

Similar process is followed when gases are dissolved in liquid solvents. Such a solution in which no more solute can be dissolved at the same temperature and pressure is called a saturated solution. An unsaturated solution is one in which more solute can be dissolved at the same temperature.

The solution which is in dynamic equilibrium with undissolved solute is the saturated solution and contains the maximum amount of solute dissolved in a given amount of solvent. Thus, the concentration of solute in such a solution is its solubility.

we have observed that solubility of one substance into another depends on the nature of the substances. In addition to these variables, two other parameters, i.e., temperature and pressure also control this phenomenon.

Effect of Temperature & Pressure 

The solubility of a solid in a liquid is significantly affected by temperature changes and must follow Le Chateliers Principle. In general, if in a nearly saturated solution, the dissolution process is endothermic (Δsol H > 0), the solubility should increase with rise in temperature and if it is exothermic (Δsol H > 0) the solubility should decrease. These trends are also observed experimentally.

Pressure does not have any significant effect on solubility of solids in liquids. It is so because solids and liquids are highly incompressible and practically remain unaffected by changes in pressure

before discuss gas in liquid and liquid in liquid lets discuss vapour pressure.

vapour pressure

The vapor pressure of a liquid is the equilibrium pressure of a vapor above its liquid (or solid), that is, the pressure of the vapor resulting from evaporation of a liquid (or solid) above a sample of the liquid (or solid) in a closed container. The vapor pressure of a liquid varies with its temperature, As the temperature of a liquid or solid increases its vapor pressure also increases

Factors That Affect Vapor Pressure:

1) Surface Area: the surface area of the solid or liquid in contact with the gas has no effect on the vapor pressure.

2)Types of Molecules: the types of molecules that make up a solid or liquid determine its vapor pressure. If the intermolecular forces between molecules are relatively strong, the vapor pressure will be relatively low and if relatively weak, intermolecular forces between molecules are weak the vapor pressure will be relatively high.

3) Temperature: at a higher temperature, more molecules have enough energy to escape from the liquid or solid

Solubility of a Gas in a Liquid

Many gases dissolve in water. Oxygen dissolves only to a small extent in water. It is this dissolved oxygen which sustains all aquatic life. Solubility of gases in liquids is greatly affected by pressure and temperature. The solubility of gases increase with increase of pressure. For solution of gases in a solvent, consider a system as shown in Fig.. The lower part is solution and the upper part is gaseous system at pressure p and temperature T. Assume this system to be in a state of dynamic equilibrium, i.e., under these conditions rate of

gaseous particles entering and leaving the solution phase is the same. Now increase the pressure over the solution phase by compressing the gas to a smaller volume. 

This will increase the number of gaseous particles per unit volume over the solution and also the rate at which the gaseous particles are striking the surface of solution to enter it. The solubility of the gas will increase until a new equilibrium is reached resulting in an increase in the pressure of a gas above the solution and thus its solubility increases.

Henry was the first to give a quantitative relation between pressure and solubility of a gas in a solvent which is known as Henry’s law. The law states that at a constant temperature “the

partial pressure of the gas in vapour phase (p) is proportional to the mole fraction of the gas (x) in the solution” and is expressed as:

p = KH x

Here KH is the Henry’s law constant. If we draw a graph between partial pressure of the gas  HCl (in torr) versus mole fraction of the gas in solution, then we should get a plot of the type as shown in Fig. 

Different gases have different KH values at the same temperature. This suggests that KH is a function of the nature of the gas.

It is obvious from equation that higher the value of KH at a given pressure, the lower is the solubility of the gas in the liquid.

It can be seen that KH values for both N2 and O2 increase with increase of temperature indicating that the solubility of gases increases with decrease of temperature. It is due to this reason that aquatic species are more comfortable in cold waters rather than in warm waters.

we can see application of Henry's laws in soft drinks, scuba drivers, people living at high altitude.

Effect of Temperature:

When dissolved, the gas molecules are present in liquid phase and the process of dissolution can be considered similar to condensation and heat is evolved in this process. We have learnt in the last Section that dissolution process involves dynamic equilibrium and thus must follow Le Chatelier’s Principle. As dissolution is an exothermic process, the solubility should decrease with increase of temperature.

Liquid - Liquid Solution

Let us consider a binary solution of two volatile liquids and denote the two components as 1 and 2. When taken in a closed vessel, both the components would evaporate and eventually an equilibrium would be established between vapour phase and the liquid phase. Let the total vapour pressure at this stage be ptotal and p1 and p2 be the partial vapour pressures of the two components 1 and 2 respectively. These partial pressures are related to the mole fractions x1 and x2 of the two components 1 and 2 respectively.

The relationship is known as the Raoult’s law which states that for a solution of volatile liquids, the partial vapour pressure of each component in the solution is directly proportional to its mole fraction.

Thus, for component 1

Following conclusions can be drawn from equation:

(i) Total vapour pressure over the solution can be related to the mole fraction of any one component.

(ii) Total vapour pressure over the solution varies linearly with the mole fraction of component 2.

(iii) Depending on the vapour pressures of the pure components 1 and 2, total vapour pressure over the solution decreases or increases with the increase of the mole fraction of component 1.

A plot of p1 or p2 versus the mole fractions x1 and x2 for a solution gives a linear plot as shown in Fig. These lines (I and II) pass through the points and respectively when x1 and x2 equal

unity. Similarly the plot (line III) of ptotal versus x2 is also linear. 

The composition of vapour phase in equilibrium with the solution is determined by the partial pressures of the components. If y1 and y2 are the mole fractions of the components 1 and 2 respectively in the vapour phase then, using Dalton’s law of partial pressures

UNIT 1.3 THE SOLID STATE: DEFECT, ELECTRONIC AND MAGNETIC PROPERTIES

LEC 3   

1.Imperfection in lattice

An ionic crystal is perfect when it contains same unit cell containing same lattice point thoughout the crystal .but it is possible at absolute zero temperature [i.e 0 k]. above this deviation in ideal crystal lattice is seen called as defect or imperfection.

 Point defect :-  the irregularities or deviation from ideal arrangement around a point or atom.

 Line defect :-  the deviations from ideal arrangement around entire rows of lattice.

Defect is of 3 types:- 

A} Stoichiometric defect   _  in this type of defect equal no. of anion and cation are missing from there lattice sites. so, in defective crystal also the molecular formulae remain same and electrical neutrality remains maintained. Example of this type of defect are vacancy defect , interstitial defect ,schottky defect ,frenkel defect.........

1.1 Defect in non ionic crystal

vacancy defect - in this type of defect atoms[lattice point] are missing from their lattice site. this is very common and simplest type of stoichometric defect found in non ionic crystal.

interstitial defect- in this type of defect atoms or molecule comes into interstital space of the lattice but reamins in the crystal lattice.

1.2 Defect in ionic crystal

schottky defect-  in this type of defect equal no. of anian and cation are missing from their lattice site creating holes at the site of vacancy. this is very common and simple type of stoichiometric defect found in ionic crystal. These defect are found in compounds having high coordination number and anion and cation are of equal size. 
example are- NaCl,CsCl.

1.  Electrical neutrality is maintained becoause of the fact that equal no. of anion and cation are missing.

2.  Density decrease because holes are created.

3.  Stability of the crystal is decreased because holes are created and deviated from ideal structure.

4.  Electrical conductivity is increased because of the presence of holes . when electric field is applied than ion from nearby lattice site fills the hole and hole is created at that site .In this way holes moves from one end to other and electrical conductivities increases.

frenkel defect- in this type of defect ion is missing from its normal lattice site and occupy a interstitial  position.this defect is seen in compounds with low coordination number and anion is larger in size than cation. 
example are-all silver halide.
1.Electrical neutrality is maintained because anion and cation are  not missing from lattice.

2.Density remain the same .

3.Stability decreases.[same reason]
4.Electrical conductivity increases[same reason].

B}Non stoichiometric defect - in this type of defect the no. of anion and cation missing are not equal so chemical formulae is changed.

 This defect is of two types:-

1. Metalal excess defects -This occur due to following reason:-

1.1 Due to anion vacancy-  This type of defect is mostly seen in compounds having schttky defect[silver halides]. It involves removal of anion from lattice site leaving electron at that site so that electrical neutrality is maintained called as F- Centre or ferbenzenter centre.  In this way electrical neutrality is maintained.

When alkali metal halide are heated in vapour of alkali metal the metal atoms get deposited on the surface of alkali metal halide crystal and halide ion  get diffused into the surface and combine with metal atoms. the electron thus produced due to ionization of metal atom comes at the site from where halide ion is removed called as F-Centre giving coulered crystal. 

LiCl [pink], KCl[violet], NaCl[yellow].

1.2 Due to extra cation in interstitial position- In this cation occupy extra interstial site and to maintain electrical neutrality electron is present in other interstitial site.
Zinc oxide is white in colour at room temperature. On heating it loses oxygen and turns yellow.
Now there is excess of zinc in the crystal and its formula becomes Zn1+xO. The excess Zn2+ ions move to interstitial sites and the electrons to neighbouring interstitial sites. In this way electrical neutrality is maintained.

2 Metal deficiency defect- this defect occur due to following two reasons:-

2.1 Due to cation vacancy- In this cation is missing from their lattice point and another cation occupy extra postive charge to balance the cation loss .  In this way electrical neutrality is maintained.this is found in those cation in which variable oxidation state is found like ferrous oxide,ferrous sulphite,nickel oxide etc....

2.1 due to extra anion occupying interstitial site-  In this type of defect extra anion is present in interstitial space and charge is balanced by oxidation ofequal no. of  adjacent cation to higher oxidation state. this defect is very common because the size of anion is large enough to fit in interstitial site. 

C}Impurity defect

If we add impurity in crystal lattice than its electrical property changes called impurity defect.these defect is easily seen in compound of 14 group mostly silicon and germanium element in which every atom is bonded with with four other atom to complete its octate. If in this compound small amount of group 13 or15 is added [process is known as doping] than its electrical conductiviy increase and compound is known as P type and N type semiconductor respectively.

P Type semiconductor- If doping is done with 13 group element[electron deficit impurity] than this type of semiconductor is formed . In this the 14 group element form 3 covalant bond with 13 group element [as its valancy is3] and one hole is created at the place of 4th covalant bond . when electric field is applied hole is filled by adjacent electron  so hole and electron moves in opposite direction .since current is carried by positive hole it is called as P- Type semiconductor.

N Type semiconductor- If doping is done by 15 group element [electron rich impurity] than this type of semiconductor is formed. In this the 14 group element form 4 covalant bond with 15 group element  and one electron is left free which is responsible for its extra conductivity.since current is carried by electron negative  it is called as N- Type semiconductor.

2. Electrical property

Solids exhibit an amazing range of electrical conductivities, extending over 27 orders of magnitude ranging from 10–20 to 107 ohm–1 m–1.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–1m–1 are called conductors. Metals have conductivities in the order of 107 ohm–1m–1 are good conductors.

(ii) Insulators : These are the solids with very low conductivities ranging between 10–20 to 10–10 ohm–1m–1.
(iii) Semiconductors : These are the solids with conductivities in the intermediate range from 10–6 to 104 ohm–1m–1.



3.Magnetic properties

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 .Electron being a charged particle and undergoing these motion can be considered as a small loop of current which possesses a magnetic. 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.

(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.

(ii) Diamagnetism: Diamagnetic substances are weakly repelled by a magnetic field. They are weakly magnetised in a magnetic field inopposite 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 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 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 antiferromagnetism have domain structure similar to ferromagnetic substance, but their domains are oppositely oriented and cancel out each other's magnetic moment.

(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 . They are weakly attracted by magnetic field as compared to ferromagnetic substances.These substances also lose ferrimagnetism on heating and become paramagnetic.

UNIT 1.2 THE SOLID STATE: CRYSTAL LATTICE & UNIT CELL

LEC 2            CRYSTAL LATTICE & UNIT CELL

all crystal have a regular repetition of some constituent particles (atoms, molecules or ions). to represent the relative arrangement of these particles in a crystal, each particles (atom, molecule or ion) is considered as a point, position taken by these particles in three dimension crystal are called as lattice points or lattice sites.

the representation of crystal in which the location of the ions or atoms are shown by lattice points is called crystal lattice or space lattice. space lattice or crystal lattice can be define as an array of lattice points showing the arrangement of constituent particles in different positions in three dimensional space. There are only 14 possible three dimensional lattices. These are called Bravais Lattices. The 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.

(c) Lattice points are joined by straight lines to bring out the geometry of the lattice.

in space lattice, a certain group of lattice points set the pattern of whole lattice. the smallest group of lattice points is called unit cell. so the Unit Cell can be defined as the smallest portion of the crystal lattice which define completely the repeating pattern in the crystal in all directions.

A 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.

Unit cells can be broadly divided into two categories:

1) Primitive / simple Unit Cells:  When constituent particles are present only on the corner positions of a unit cell, it is called as primitive unit cell.

2) 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.

On the basis of geometrical consideration edge length and axial angle (a, b, c, ∝, β, γ), main seven crystal system system are present also known as seven primitive Unit Cell and on the basis of arrangement of particles in primitive or centred unit cell.

Number of Atom in a Unit Cell & Packaging Efficiency

we know that any crystal lattice is made up of a very large number of unit cell and every lattice point is occupied by one constituent particles (atom, molecule or ion).
In whatever way the constituent particles are packed there is always some free space in the form of voids. Packaging Efficiency is the percentage of total space filled by the Particles in a unit cell.
 now we are going to study how many atoms are present in different type of unit cell and what is the Packaging Efficiency of the different Structure.

1) Primitive Cubic Unit Cell:

In primitive Unit cell has atoms only at its corner. each atom at a corner is shared by eight adjacent unit cell and a unit cell has 8 corners so

Now for packing efficiency lets consider atom is a spherical type of radius "r" and unit cell is a cube of edge length "a" then

2) Body Centred Unit Cell

A body-centred cubic (BCC) unit cell has an atom at each of its corners and also one atom at its body centre, It can be seen that the atom at the body centre wholly belongs to the unit cell in which it is present.

The atom at the centre will be in touch with the other two atoms diagonally arranged as in fig shows

3) Face Centred Cubic Unit Cell (FCC)

A face centred cubic (fcc) unit cell contains atoms at all the corners and at the centre of all the faces of the cube and each atom located at the face centre is shared between two adjacent unit cells.
No of atoms present in one unit cell is given as

now packing Efficiency is given as

it means that FCC structure have maximum Efficiency of packaging.

Packing of Constituent in Metallic Crystal

In the formation of Crystals, the constituent particles may be atoms, ions or molecules. these constituent particles may be of different size and so it may give different mode of particles in the crystal.
The concept of packing of constituent in metallic crystal is different from Ionic crystal but in this class we only study about metallic crystal according to syllabus.
The number of nearest neighbours of a particle is called its coordination number. 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

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.

The two dimensional coordination number is 4.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 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 hence this packing is called two dimensional hexagonal close packing.

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.

c) Close Packing in Three Dimensions

All real structures are three dimensional structures. They can be obtained by stacking two dimensional layers one above the other. Let us see what types of three dimensional close packing can be obtained from these.

(i) Three dimensional close packing from 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. 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.

(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.

(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 that not all the triangular voids of the first layer are 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 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’.

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’ Such voids are surrounded by six spheres and are called octahedral voids. 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. This sort of arrangement of atoms is found in many metals like magnesium and zinc.

(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 those of the first layer as shown in Figs. 1.18 and 1.19. 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.

Formula of a Compound and Number of Voids Filled

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. Not all octahedral or tetrahedral voids are occupied. In a given compound, the fraction of octahedral or tetrahedral voids that are occupied, depends upon the chemical formula of the compound. for example

Problem: 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?

Solution: 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.
No of atom of B at HCP lattice = N
No of Tetrahedral Voids = 2N
only 2/3 Tetrahedral voids are accupied by A then No of atom A = 2N x 2/3 = 4N/3
Ratio of A and B is  4:3 hence the formula of the compound is A4B3.