(1) Number of atoms in per unit cell
The total number of atoms contained in the unit cell for a simple cubic called the unit cell content.
The simplest relation can determine for it is, \[\frac{{{n}_{c}}}{8}+\frac{{{n}_{f}}}{2}+\frac{{{n}_{i}}}{1}\]
Where \[{{n}_{c}}=\] Number of atoms at the corners of the cube=8
\[{{n}_{f}}=\] Number of atoms at six faces of the cube = 6
\[{{n}_{\,i}}=\] Number of atoms inside the cube = 1
Cubic unit cell
nc
nf
ni
Total atom in per unit cell
Simple cubic (sc)
8
0
0
1
body centered cubic (bcc)
8
0
1
2
Face centered cubic (fcc)
8
6
0
4
(2) Co-ordination number (C.N.) : It is defined as the number of nearest neighbours or touching particles with other particle present in a crystal is called its co-ordination number. It depends upon structure of the crystal.
For simple cubic system C.N. = 6.
For body centred cubic system C.N. = 8
For face centred cubic system C.N. = 12.
(3) Density of the unit cell \[(\rho )\] : It is defined as the ratio of mass per unit cell to the total volume of unit cell.
\[\rho =\]\[\frac{Z\times M}{{{a}^{3}}\times {{N}_{0}}}\]
Where Z = Number of particles per unit cell
M = Atomic mass or molecular mass
\[{{N}_{0}}=\] Avogadro number \[(6.023\times {{10}^{23}}mo{{l}^{-1}})\]
\[a=\] Edge length of the unit cell= \[a\ pm=a\times {{10}^{-10}}cm\]
\[{{a}^{3}}=\] volume of the unit cell
i.e. \[\rho =\frac{Z\times M}{{{a}^{3}}\times {{N}_{0}}\times {{10}^{-30}}}g/c{{m}^{3}}\]
The density of the substance is same as the density of the unit cell.
(4) Packing fraction (P.F.) : It is defined as ratio of the volume of the unit cell that is occupied by spheres of the unit cell to the total volume of the unit cell.
Let radius of the atom in the packing = r
Edge length of the cube = a
Volume of the cube V = \[{{a}^{3}}\]
Volume of the atom (spherical) \[\nu =\frac{4}{3}\pi {{r}^{3}}\]
Packing density \[=\frac{\nu Z}{V}=\frac{\frac{4}{3}\pi {{r}^{3}}Z}{{{a}^{3}}}\]
The crystals of the substance are obtained by cooling the liquid (or the melt) of the solution of that substance. The size of the crystal depends upon the rate of cooling. If cooling is carried out slowly, crystals of large size are obtained because the particles (ions, atoms or molecules) get sufficient time to arrange themselves in proper positions.
Atoms of molecules \[\xrightarrow{\text{Dissolved }}\] cluster \[\xrightarrow{\text{dissolved}}\] dissolved embryo \[\to \underset{\text{(unstable)}}{\mathop{\text{nucleus }}}\,\] \[\to \] crystal
(If loosing units dissolves as embryo and if gaining unit grow as a crystals).
Bravais (1848) showed from geometrical considerations that there can be only 14 different ways in which similar points can be arranged. Thus, there can be only 14 different space lattices. These 14 types of lattices are known as Bravais Lattices. But on the other hand Bravais showed that there are only seven types of crystal systems.
Bravais lattices corresponding to different crystal systems
Crystal is a homogeneous portion of a crystalline substance, composed of a regular pattern of structural units (ions, atoms or molecules) by plane surfaces making definite angles with each other giving a regular geometric form.
A regular array of points (showing atoms/ions) in three dimensions is commonly called as a space lattice, or lattice.
Each point in a space lattice represents an atom or a group of atoms.
Each point in a space lattice has identical surroundings throughout.
A three dimensional group of lattice points which when repeated in space generates the crystal called unit cell.
The unit cell is described by the lengths of its edges, a, b, c (which are related to the spacing between layers) and the angles between the edges, \[\alpha ,\,\beta ,\gamma .\]
Types of units cells
A units cell is obtained by joining the lattice points. The choice of lattice points to draw a unit cell is made on the basis of the external geometry of the crystal, and symmetry of the lattice. There are four different types of unit cells. These are,
(1) Primitive or simple cubic (sc) : Atoms are arranged only at the corners of the unit cell.
(2) Body centred cubic (bcc) : Atoms are arranged at the corners and at the centre of the unit cell.
(3) Face centred cubic (fcc) : Atoms are arranged at the corners and at the centre of each faces of the unit cell.
(4) Side centered : Atoms are arranged at the centre of only one set of faces in addition to the atoms at the corner of the unit cell.
“The branch of science that deals with the study of structure, geometry and properties of crystals is called crystallography”.
(1) Symmetry in Crystal : A crystal possess following three types of symmetry,
(i) Plane of symmetry : It is an imaginary plane which passes through the centre of a crystal can divides it into two equal portions which are exactly the mirror images of each other.
(ii) Axis of symmetry : An axis of symmetry or axis of rotation is an imaginary line, passing through the crystal such that when the crystal is rotated about this line, it presents the same appearance more than once in one complete revolution i.e., in a rotation through 360°. Suppose, the same appearance of crystal is repeated, on rotating it through an angle of 360°/n, around an imaginary axis, is called an n-fold axis where, n is known as the order of axis. By order is meant the value of n in \[2\pi /n\] so that rotation through \[2\pi /n,\] gives an equivalent configuration.
(iii) Centre of symmetry : It is an imaginary point in the crystal that any line drawn through it intersects the surface of the crystal at equal distance on either side.
Only simple cubic system have one centre of symmetry. Other system do not have centre of symmetry.
The total number of planes, axes and centre of symmetries possessed by a crystal is termed as elements of symmetry.
A cubic crystal possesses total 23 elements of symmetry.
\[\frac{\begin{align} & Plane\,\,of\,\,symmetry(3+6)=9 \\ & Plane\,\,of\,\,symmetry(3+4+6)=13 \\ & Axis\,\,of\,\,symmetry(1)=1 \\ \end{align}}{Total\,\,elements\,\,of\,symmetry\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\,23}\]
(2) Laws of crystallography : Crystallography is based on three fundamental laws.
(i) Law of constancy of interfacial angles : This law states that angle between adjacent corresponding faces is inter facial angles of the crystal of a particular substance is always constant inspite of different shapes and sizes and mode of growth of crystal. The size and shape of crystal depend upon the conditions of crystallisation. This law is also known as Steno's Law.
(ii) Law of rational indices : This law states that the ratio of intercepts of different faces of a crystal with the three axes are constant and can be expressed by rational numbers that the intercepts of any face of a crystal along the crystallographic axes are either equal to unit intercepts (i.e., intercepts made by unit cell) a, b, c or some simple whole number multiples of them e.g., na, n' b, n''c, where n, n' and n'' are simple more...
(1) Types of solids
Solids can be broadly classified into following two types,
(i) Crystalline solids/True solids,
(ii) Amorphous solids/Pseudo solids
Crystalline solids
Amorphous solids
They have long range order.
They have short range order.
They have definite melting point
Not have definite melting point
They have a definite heat of fusion
Not have definite heat of fusion
They are rigid and incompressible
Not be compressed to any appreciable extent
They are given cleavage i.e. they break into two pieces with plane surfaces
They are given irregular cleavage i.e. they break into two pieces with irregular surface
They are anisotropic because of these substances show different property in different direction
They are isotropic because of these substances show same property in all directions
There is a sudden change in volume when it melts.
There is no sudden change in volume on melting.
These possess symmetry
Not possess any symmetry.
These possess interfacial angles.
Not possess interfacial angles.
(2) Crystalline and amorphous silica \[(Si{{O}_{2}})\]
Silica occurs in crystalline as well as amorphous states. Quartz is a typical example of crystalline silica. Quartz and the amorphous silica differ considerably in their properties.
Quartz
Amorphous silica
It is crystalline in nature
It is light (fluffy) white powder
All four corners of \[SiO_{4}^{4-}\] tetrahedron are shared by others to give a network solid
The \[SiO_{4}^{4-}\] tetrahedra are randomly joined, giving rise to polymeric chains, sheets or three-dimensional units
It has high and sharp melting point (1710°C)
It does not have sharp melting point
(3) Diamond and graphite
Diamond and graphite are tow allotropes of carbon. Diamond and graphite both are covalent crystals. But, they differ considerably in their properties.
Molecular masses can be calculated by measuring any of the colligative properties. The relation between colligative properties and molecular mass of the solute is based on following assumptions.
(1) The solution is dilute, so that Raoult’s law is obeyed.
(2) The solute neither undergoes dissociation nor association in solution.
In case of solutions where above assumptions are not valid we find discrepencies between observed and calculated values of colligative properties. These anomalies are primarily due to
(i) Association of solute molecules.
(ii) Dissociation of solute molecules.
(i) Association of solute molecules : Certain solutes in solution are found to associate. This eventually leads to a decrease in the number of molecular particles in the solutions. Thus, it results in a decrease in the values of colligative properties.
Colligative property\[\propto \frac{1}{\text{molecular mass of solute}}\]
therefore, higher values are obtained for molecular masses than normal values for unassociated molecules.
(ii) Dissociation of solute molecules : A number of electrolytes dissociate in solution to give two or more particles (ions). Therefore, the number of solute particles, in solutions of such substances, is more than the expected value. Accordingly, such solutions exhibit higher values of colligative properties. Since colligative properties are inversely proportional to molecular masses, therefore, molecular masses of such substances as calculated from colligative properties will be less than their normal values.
Van’t Hoff’s factor (i) : In 1886, Van’t Hoff introduced a factor ‘i’ called Van’t Hoff’s factor, to express the extent of association or dissociation of solutes in solution. It is ratio of the normal and observed molecular masses of the solute, i.e.,
\[i=\frac{\text{Normal molecular mass}}{\text{Observed molecular mass}}\]
In case of association, observed molecular mass being more than the normal, the factor i has a value less than 1. But in case of dissociation, the Van’t Hoff’s factor is more than 1 because the observed molecular mass has a lesser value than the normal molecular mass. In case there is no dissociation the value of ‘i’ becomes equal to one.
Since colligative properties are inversely proportional to molecular masses, the Van’t Hoff’s factor may also be written as,
\[i=\frac{Observed\text{ }value\text{ }of\text{ }colligative\text{ }property}{\begin{align} & Calculated\text{ }value\text{ }of\text{ }colligative\text{ }property \\ & assuming\text{ }no\text{ }association\text{ }or\text{ }dissociation \\ \end{align}}\]
\[i=\frac{\text{No}\text{. of particles after association or dissociation}}{\text{No}\text{. of particles before association or dissociation}}\]
Introduction of the Van’t Hoff factor modifies the equations for the colligative properties as follows,
Relative lowering of vapour pressure\[=\frac{P_{A}^{o}-{{P}_{A}}}{P_{A}^{o}}=i{{X}_{B}}\]
Elevation of boiling point, \[\Delta {{T}_{b}}=i{{k}_{b}}m\]
Depression in freezing point, \[\Delta {{T}_{f}}=i{{k}_{f}}m\]
Osmotic pressure, \[\pi =\frac{inRT}{V}\]; \[\pi =iCRT\]
From the value of ‘i’, it is possible to calculate degree of dissociation or degree of association of substance.
Degree of dissociation (a) : It is defined as the fraction of total molecules which dissociate into simpler molecules or ions.
\[\alpha =\frac{i-1}{m-1}\]; m= number of particles in solution
Degree of association (a) : It is defined as the fraction of the total number of molecules which associate or combine together resulting in the formation of a bigger molecules.
\[\alpha =\frac{i-1}{1/m-1}\]; m more...
The colligative properties of solutions, viz. lowering of vapour pressure, osmotic pressure, elevation in b.p. and depression in freezing point, depend on the total number of solute particles present in solution. Since the electrolytes ionise and give more than one particle per formula unit in solution, the colligative effect of an electrolyte solution is always greater than that of a non-electrolyte of the same molar concentration. All colligative properties are used for calculating molecular masses of non-volatile solutes. However osmotic pressure is the best colligative property for determining molecular mass of a non-volatile substance.
Colligative properties are depending on following factory
(1) Colligative properties \[\propto \] Number of particles
\[\propto \] Number of molecules
(in case of non-electrolytes)
\[\propto \] Number of ions
(In case of electrolytes)
\[\propto \] Number of moles of solute
\[\propto \] Mole fraction of solute
(2) For different solutes of same molar concentration, the magnitude of the colligative properties is more for that solution which gives more number of particles on ionisation.
(3) For different solutions of same molar concentration of different non-electrolyte solutes, the magnitude of the colligative properties will be same for all.
(4) For different molar concentrations of the same solute, the magnitude of colligative properties is more for the more concentrated solution.
(5) For solutions of different solutes but of same percent strength, the magnitude of colligative property is more for the solute with least molecular weight.
(6) For solutions of different solutes of the same percent strength, the magnitude of colligative property is more for that solute which gives more number of particles, which can be known by the knowledge of molecular weight and its ionisation behaviour.
Freezing point is the temperature at which the liquid and the solid states of a substance are in equilibrium with each other or it may be defined as the temperature at which the liquid and the solid states of a substance have the same vapour pressure. It is observed that the freezing point of a solution is always less than the freezing point of the pure solvent. Thus the freezing point of sea water is low than that of pure water. The depression in freezing point \[(\Delta T\] or \[\Delta {{T}_{f}})\] of a solvent is the difference in the freezing point of the pure solvent \[({{T}_{s}})\] and the solution \[({{T}_{sol.}})\].
\[{{T}_{s}}-{{T}_{sol}}=\Delta {{T}_{f}}\] or \[\Delta T\]
\[NaCl\] or \[CaC{{l}_{2}}\] (anhydrous) are used to clear snow on roads. They depress the freezing point of water and thus reduce the temperature of the formation of ice.
Depression in freezing point is determined by Beckmann’s method and Rast’s camphor method. Study of depression in freezing point of a liquid in which a non-volatile solute is dissolved in it is called as cryoscopy.
Important relations concerning depression in freezing point.
(1) Depression in freezing point is directly proportional to the lowering of vapour pressure. \[\Delta {{T}_{f}}\propto {{p}^{0}}-p\]
(2) \[\Delta {{T}_{f}}={{K}_{f}}\times m\]
where \[{{K}_{f}}=\] molal depression constant or cryoscopic constant; \[m=\] Molality of the solution (i.e., no. of moles of solute per \[1000g\] of the solvent); \[\Delta {{T}_{f}}=\]Depression in freezing point
(3) \[\Delta {{T}_{f}}=\frac{1000\times {{K}_{f}}\times w}{m\times W}\] or \[m=\frac{1000\times {{K}_{f}}\times w}{\Delta {{T}_{f}}\times W}\]
where \[{{K}_{f}}\] is molal depression constant and defined as the depression in freezing point produced when 1 mole of the solute is dissolved in \[1kg\] of the solvent. \[w\] and \[W\] are the weights of solute and solvent and \[m\] is the molecular weight of the solute.
(4) \[{{K}_{f}}=\frac{R{{({{T}_{0}})}^{2}}}{{{l}_{f}}1000}=\frac{0.002{{({{T}_{0}})}^{2}}}{{{l}_{f}}}\]
where \[{{T}_{0}}=\]Normal freezing point of the solvent; \[{{l}_{f}}=\]Latent heat of fusion/g of solvent; \[{{K}_{f}}\] for water is \[1.86\ \deg -kg\ mo{{l}^{-1}}\]
Boiling point of a liquid may be defined as the temperature at which its vapour pressure becomes equal to atmospheric pressure, i.e., 760 mm. Since the addition of a non-volatile solute lowers the vapour pressure of the solvent, solution always has lower vapour pressure than the solvent and hence it must be heated to a higher temperature to make its vapour pressure equal to atmospheric pressure with the result the solution boils at a higher temperature than the pure solvent. Thus sea water boils at a higher temperature than distilled water. If Tb is the boiling point of the solvent and T is the boiling point of the solution, the difference in the boiling point (DT or D Tb) is called the elevation of boiling point.
\[T-{{T}_{b}}=\Delta {{T}_{b}}\] or \[\Delta T\]
Elevation in boiling point is determined by Landsberger’s method and Cottrell’s method. Study of elevation in boiling point of a liquid in which a non-volatile solute is dissolved is called as ebullioscopy.
Important relations concerning elevation in boiling point
(1) The elevation of boiling point is directly proportional to the lowering of vapour pressure, i.e., \[\Delta {{T}_{b}}\propto {{p}^{0}}-p\]
(2) \[\Delta {{T}_{b}}={{K}_{b}}\times m\]
where \[{{K}_{b}}=\] molal elevation constant or ebullioscopic constant of the solvent; \[m=\] Molality of the solution, i.e., number of moles of solute per \[1000g\] of the solvent; \[\Delta {{T}_{b}}=\] Elevation in boiling point
(3) \[\Delta {{T}_{b}}=\frac{1000\times {{K}_{b}}\times w}{m\times W}\] or \[m=\frac{1000\times {{K}_{b}}\times w}{\Delta {{T}_{b}}\times W}\]
where, \[{{K}_{b}}\] is molal elevation constant and defined as the elevation in b.pt. produced when 1 mole of the solute is dissolved in 1 kg of the solvent.
\[w\] and \[W\] are the weights of solute and solvent and \[m\] is the molecular weight of the solute.
(4) \[{{K}_{b}}=\frac{0.002{{({{T}_{0}})}^{2}}}{{{l}_{V}}}\]
where \[{{T}_{0}}=\] Normal boiling point of the pure solvent; \[{{l}_{V}}=\]Latent heat of evaporation in \[cal/g\] of pure solvent; \[{{K}_{b}}\] for water is \[0.52\ \deg -kg\ mo{{l}^{-1}}\].
(1) Osmosis : The flow of solvent from pure solvent or from solution of lower concentration into solution of higher concentration through a semi-permeable membrane is called Osmosis. Osmosis may be divided in following types,
(i) Exo-Osmosis : The outward osmotic flow of water from a cell containing an aqueous solution through a semi-permeable membrane is called as Exo-osmosis. For example, egg (after removing hard shell) placed in conc. NaCl solutions, will shrink due to exo-osmosis.
(ii) Endo-osmosis : The inward flow of water into the cell containing an aqueous solution through a semi-permeable membrane is called as endo-osmosis. e.g., an egg placed in water swells up due to endo-osmosis.
(iii) Reverse osmosis : If a pressure higher than osmotic pressure is applied on the solution, the solvent will flow from the solution into the pure solvent through the semi-permeable membrane. Since here the flow of solvent is in the reverse direction to that observed in the usual osmosis, the process is called reverse osmosis.
Differences between osmosis and diffusion
Osmosis
Diffusion
In osmosis movement of molecules takes place through a semi-permeable membrane.
In diffusion there is no role of semi-permeable membrane.
It involves movement of only solvent molecules from one side to the other.
It involves passage of solvent as well as solute molecules from one region to the other.
Osmosis is limited to solutions only.
Diffusion can take place in liquids, gases and solutions.
Osmosis can be stopped or reversed by applying additional pressure on the solution side.
Diffusion can neither be stopped nor reversed
(2) Osmotic pressure (p)
The osmotic pressure of a solution at a particular temperature may be defined as the excess hydrostatic pressure that builds up when the solution is separated from the solvent by a semi-permeable membrane. It is denoted by p.
or
Osmotic pressure may be defined as the excess pressure which must be applied to a solution in order to prevent flow of solvent into the solution through the semi-permeable membrane.
or
Osmotic pressure is the excess pressure which must be applied to a given solution in order to increase its vapour pressure until it becomes equal to that of the solution.
(i) Measurements of osmotic pressure : Following methods are used for the measurement of osmotic pressure,
(a) Pfeffer’s method, (b) Morse and Frazer’s method, (c) Berkeley and Hartley’s method, (d) Townsend’s negative pressure method, (e) De Vries plasmolytic method.
(ii) Determination of molecular mass of non-volatile solute from osmotic pressure (p) : The osmotic pressure is a colligative property. For a given more...
Types of units cells
A units cell is obtained by joining the lattice points. The choice of lattice points to draw a unit cell is made on the basis of the external geometry of the crystal, and symmetry of the lattice. There are four different types of unit cells. These are,
(1) Primitive or simple cubic (sc) : Atoms are arranged only at the corners of the unit cell.
(2) Body centred cubic (bcc) : Atoms are arranged at the corners and at the centre of the unit cell.
(3) Face centred cubic (fcc) : Atoms are arranged at the corners and at the centre of each faces of the unit cell.
(4) Side centered : Atoms are arranged at the centre of only one set of faces in addition to the atoms at the corner of the unit cell. 
(ii) Axis of symmetry : An axis of symmetry or axis of rotation is an imaginary line, passing through the crystal such that when the crystal is rotated about this line, it presents the same appearance more than once in one complete revolution i.e., in a rotation through 360°. Suppose, the same appearance of crystal is repeated, on rotating it through an angle of 360°/n, around an imaginary axis, is called an n-fold axis where, n is known as the order of axis. By order is meant the value of n in \[2\pi /n\] so that rotation through \[2\pi /n,\] gives an equivalent configuration. 
(iii) Centre of symmetry : It is an imaginary point in the crystal that any line drawn through it intersects the surface of the crystal at equal distance on either side.
Only simple cubic system have one centre of symmetry. Other system do not have centre of symmetry.
The total number of planes, axes and centre of symmetries possessed by a crystal is termed as elements of symmetry.
A cubic crystal possesses total 23 elements of symmetry.
\[\frac{\begin{align} & Plane\,\,of\,\,symmetry(3+6)=9 \\ & Plane\,\,of\,\,symmetry(3+4+6)=13 \\ & Axis\,\,of\,\,symmetry(1)=1 \\ \end{align}}{Total\,\,elements\,\,of\,symmetry\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\,23}\]
(2) Laws of crystallography : Crystallography is based on three fundamental laws.
(i) Law of constancy of interfacial angles : This law states that angle between adjacent corresponding faces is inter facial angles of the crystal of a particular substance is always constant inspite of different shapes and sizes and mode of growth of crystal. The size and shape of crystal depend upon the conditions of crystallisation. This law is also known as Steno's Law.
(ii) Law of rational indices : This law states that the ratio of intercepts of different faces of a crystal with the three axes are constant and can be expressed by rational numbers that the intercepts of any face of a crystal along the crystallographic axes are either equal to unit intercepts (i.e., intercepts made by unit cell) a, b, c or some simple whole number multiples of them e.g., na, n' b, n''c, where n, n' and n'' are simple more... 
