Band Theory of Solids, Classification of Crystals on the Basis of Bonding, Conductometric Titrations and Potentiometric Titrations

 Band theory of solids 

Band theory of solids describes the quantum state that an electron takes inside a metal solid. Every molecule comprises various discrete energy levels. The way electrons behave inside a molecule is well explained through band theory. Band Theory was developed from the knowledge gained during the quantum revolution in science. In 1928, Felix Bloch applied quantum theory to solids.

In atoms, electrons are filled in respective energy orbits following Pauli’s exclusion principle. Two atomic orbitals combine to form a molecular orbit with two distinct energy levels. In solids, 1023 stacked up lines confined in a tiny space would look like a band. Thereby forming an energy continuum called energy bands. Band theory helps to visualize the difference between conductor, semiconductor, and an insulator by plotting available energies for an electron in a material.

Energy Bands in Solids:

In band theory of solids, there are many energy bands but the following are the three most important energy bands in solids:

• Valence Band

• Conduction Band

• Forbidden Band

Valence band

The energy band that consists of valence electrons energy levels, is known as the valence band. The valence band is present below the conduction band and the electrons of this band are loosely bound to the nucleus of the atom.

Conduction band

The energy band that consists of free electrons energy levels, is known as the conduction band. For electrons to be free, external energy must be applied such that the valence electrons get pushed to the conduction band and become free.

Forbidden band

The energy gap between the valence band and the conduction band is known as the forbidden band which is also known as the forbidden gap. The electrical conductivity of a solid is determined from the forbidden gap and also the classification of the materials as conductors, semiconductors, and insulators.

Solids are classified into conductors, insulators, and semiconductors based on the band theory.

Conductors

In a conductor there are no band gaps between the valence and conduction bands. In some metals the conduction and valence bands partially overlap. This means that electrons can move freely between the valence band and the conduction band.

The conduction band is only partially filled. This means there are spaces for electrons to move into. When electrons for the valence band move into the conduction band they are free to move. This allows conduction.

Insulators

An insulator has a large gap between the valence band and the conduction band. The valence band is full as no electrons can move up to the conduction band. As a result, the conduction band is empty.

Only the electrons in a conduction band can move easily, so because there aren't any electrons in an insulator's conduction band, the material can't conduct.

Semiconductors

In a semiconductor, the gap between the valence band and conduction band is smaller. At room temperature there is sufficient energy available to move some electrons from the valence band into the conduction band. This allows some conduction to take place.

An increase in temperature increases the conductivity of a semiconductor because more electrons will have enough energy to move into the conduction band.The difference between insulators and semiconductors is due to a small amount of impurity added to a semiconductor which affects the energy bands. This process is called doping.

Classification of Crystals on the Basis of Bonds

Crystals can also be classified on the basis of the bonds that hold the ions, molecules or atoms

together in the crystal lattice. Thus we have :

(a) Ionic crystals

(b) Molecular crystals

(c) Network covalent crystals

(d) Metallic crystals

Ionic Crystals

In an ionic crystal the lattice is made of positive and negative ions. These are held together by ionic bonds – the strong electrostatic attractions between oppositely charged ions. Consequently, the cations and anions attract one another and pack together in an arrangement so that the attractive forces maximise. The sodium chloride lattice shown in figure is an example. Each ion is surrounded by neighbours of opposite charge and there are no separate molecules. Since the ions are fixed in their lattice sites, typical ionic solids are hard and rigid with high melting points. In spite of their hardness, ionic solids are brittle. They shatter easily by hammering . By hammering, a layer of ions slips away from their oppositely charged neighbours and brings them closer to ions of like charge. The increase of electrostatic repulsions along the displaced plane causes the crystal to break. Figure of Sodium chloride crystal lattice in the below.

Ionic solids are non-conducting because the ions are in fixed positions. However, in the fused state the ions are allowed freedom of movement so that it becomes possible for them to conduct electricity.

Lattice Energy of an Ionic Crystal (Born-Haber Cycle)

The positive and negative ion in an ionic crystal are held together by electrostatic forces. The bond energy is expressed in terms of the lattice energy which may be defined as : the change in enthalpy (heat change) that occurs when 1 mole of a solid crystalline substance is formed from its gaseous ions. 

Figure: A Born-Haber cycle for the formation of NaCl crystal from its elements.

The lattice energy of NaCl, for example, is the change in enthalpy, ΔHº, when Na+ and Cl– ions in the gas phase come together to form 1 mole of NaCl crystal.

Molecular Crystals

In molecular crystals, molecules are the structural units. These are held together by van der Waals’ forces. As in case of ionic crystals, the molecules are packed together in a tightly packed pattern because the forces of attraction are non-directional. When this type of crystal melts, it is only the weak van der Waals forces that must be overcome. Therefore molecular solids have low melting points. Most organic substances are molecular solids. Crystal lattice of dry CO2. Dry ice, or frozen carbon dioxide, is the best example of a molecular solid. The van der Waals’ forces holding the CO2 molecules together are weak enough so that dry ice passes from solid state to gaseous state at –78ºC. 

Figure: Arrangement of CO2 molecules in the crystal lattice.

Network Covalent Crystals

In this type of crystals atoms occupy the lattice sites. These atoms are bonded to one another by covalent bonds. The atoms interlocked by a network of covalent bonds produce a crystal which is considered to be a single giant molecule. Such a solid is called a network covalent solid or simply covalent solid. Since the atoms are bound by strong covalent bonds, these crystals are very hard and have very high melting points.

Figure: Crystal structure of diamond.

Metallic Crystals

The crystals of metals consist of atoms present at the lattice sites. The atoms are arranged in different patterns, often in layers placed one above the other. The atoms in a metal crystal are viewed to be held together by a metallic bond. The valence electrons of the metal atoms are considered to be delocalised leaving positive metal ions. The freed electrons move throughout the vacant spaces between the ions. The electrostatic attractions between the metal ions and the electron cloud constitute the metallic bond. Thus a metal crystal may be described as having positive ions at the lattice positions surrounded by mobile electrons throughout the crystal. The electron sea model explains well the properties of metals. The mobile electrons in the crystal structure make metals excellent conductors of heat and electricity. On application of force, say with a hammer, metals can be deformed. The metal ions in the crystal change positions without making material difference in the environments. The attractive force between ions and the electron cloud remains the same. The crystal, therefore, does not break.

Figure: A representation of a metallic crystal structure. 

Structure of Metal Crystals

The individual atoms in a metallic crystal lattice can be thought of as hard spheres. The spherical atoms are packed together in the lattice very efficiently in geometrical arrangements so as to leave minimum interspaces. A layer of uniform spheres can be arranged either as in Figure (a) or (b). Clearly the second of the patterns uses space more efficiently. Here the spheres fit into the hollows between the adjacent spheres. Thus the vacant spaces (voids) between the spheres are smaller than in the first pattern. The metallic crystals are of the second type i.e., close packing. As clear from figure (b), each sphere in a closely packed layer is in contact with four others. Thus each ball touches six other at the corners of a hexagon. Three dimensional metallic crystals

consist of closely packed layers stacked one over the other. The spheres forming the second layer fill the holes or voids in the first layer and the spheres of the third layer fill the voids in the second layer. Depending upon the geometrical arrangements of spheres in the three layers, the close-packed metallic crystals are of two types :

(a) Hexagonal close-packed (hcp)

(b) Cubic close-packed (ccp)

Figure: Two packing patterns of spheres.

References 

1. Essentials of Physical Chemistry by Arun Bahl, B.S Bahl, G.D. Tuli

2. https://en.m.wikipedia.org/

3. https://www.britannica.com/

4. https://www.bbc.com/

5. https://www.khanacademy.org/ 

Conductometric Titrations

Titrations in which conductance measurements are made use of in determining the end-point of acid-alkali reactions, some displacement reactions or precipitation reactions are called Conductometric titrations. In these titrations, advantage is taken of the fact that the conductance of a solution at a constant temperature depends upon the number of ions present in it and their mobility. For this purpose, the titrant is added from a burette into a measured volume of the solution to be titrated which is taken in a conductance cell and the conductance readings corresponding to the various additions are plotted against the volume of the titrant. In this way two linear curves are obtained, the point of intersection of which is the end-point. Several phenomena like hydrolysis of reactants or products or partial solubility of a precipitated product give rise, however to a curvature in the curves.

The shapes of curves obtained in certain types of titration are discussed below :

(1) Titration of a Strong acid against a Strong base

Consider the reaction in which hydrochloric acid solution is titrated against a solution of sodium hydroxide. Take 20 ml of the acid solution in the conductance cell placed in a thermostat and determine its conductance. Now add 1 ml sodium hydroxide solution from the burette at a time. After each addition, determine the conductance of the solution after through mixing and plot the conductance of the solution against the volume of the alkali added. It will be observed that the points lie on two lines which are almost straight. The point of intersection of the interpolated lines will be the end point and the volume of alkali corresponding to this point is the volume of alkali required to neutralise 20 ml of the acid . The reason for this is that before the addition of alkali, the conductance of the solution is due to presence of H+ and Cl- ions. Since hydrogen ions possess the greatest mobility of any ion, the greater part of the conductance is due to it. As alkali solution is added, the hydrogen ions are removed by combination with the hydroxyl ions forming feebly ionised water molecules and their place is taken up by comparatively slow moving Na+ ions.

H+ + Cl- + Na+ + OH- ⎯⎯→ Na+ + Cl- + H2O (feebly ionised)

As a result of this, the conductance of the solution decreases and continues to fall with every subsequent addition of alkali till the end-point is reached. After the equivalence point, the further addition of sodium hydroxide solution results in an increase of conductance since the hydroxyl ions are no longer removed in the chemical reaction in the form of feebly ionised water. The point of minimum conductance, therefore, coincides with the end-point of the titration. In order to get accurate results, the volume change during titration should be as little as possible. The titrant should, therefore, be about 10 times as strong as the acid solution in the conductance cell in order to keep the volume change small. If this is not so, a correction to the readings has to be applied viz, 

actual conductance = v+VV× observed conductance

where v is the volume of the titrant and V is the original volume of the solution to be titrated.

Figure: Conductometric titration curve for strong acid and strong base.

(2) Titration of a Weak acid against a Strong alkali

When a weak acid like acetic acid is titrated against a strong alkali like sodium hydroxide, we get a curve of the type shown in the Figure. The initial conductance of the solution is low because of the poor dissociation of the weak acid. On adding alkali, highly ionised sodium acetate is formed. The acetate ions at first tend to suppress the ionisation of acetic acid still further due to Common Ion Effect but after a while the conductance begins to increase because the conducting power of highly ionised salt exceeds that of the weak acid.

CH3COOH + Na+ + OH- ⎯⎯→ CH3COO- + Na+ + H2O (feebly ionised)

Immediately after the end point, further addition of sodium hydroxide introduces the fast moving hydroxyl ions. Thus, the conductance value shows a sharp increase. The point of intersection of the two curves, gives the end-point.

H+ + Cl- + NH4OH ⎯⎯→ NH4+ + Cl- + H2O (feebly ionised)

After the end-point has been reached, the addition of ammonium hydroxide will not cause any appreciable change in conductance value as it is a weak electrolyte and its conductance is very small compared with that of the acid or its salt. The shape of this part of the curve will, therefore, be as shown in the figure.

Figure: Curve for titration of a weak acid against a strong base.

(3) Titration of a Strong acid against a Weak base

The curve obtained for the titration of a strong acid against a weak base is shown in Figure. In this case, the conductance of the solution will first decrease due to the fixing up of the fast moving H+ ions and their replacement by slow moving NH4+ions.

Figure: Curve for titration of a strong acid against a weak base.

(4) Titration of a Weak acid against a Weak base

The conductometric method is particularly suitable as such titrations do not give a sharp end-point with indicators. Consider the titration of acetic acid with ammonium hydroxide. The complete titration curve is shown in Figure. The initial conductance of the solution in this case is also low due to the poor dissociation of the weak acid. But it starts increasing as the salt CH3COONH4 is formed. After the equivalence point, the conductivity remains almost constant because the free base NH4OH is a weak electrolyte. The end-point is quite sharp.

Figure:Curve for titration of acetic acid against ammonium hydroxide.

(5) Precipitation reactions

The end-point in precipitation reactions can be accurately determined by conductometric titration. The accuracy is further increased by working with fairly dilute solutions and addition of alcohol which reduces the solubility of the precipitate and prevents adsorption. In the titration of potassium chloride against silver nitrate, for example, the change in conductance on the addition of silver nitrate is not much since the mobility of the potassium ion and the silver ion is of the same order. Thus the curve is nearly horizontal.

Ag+ + NO3- + K+ + Cl- ⎯⎯→ K+ +  NO3-   + AgCl (ppt)

After the end-point, there is a sharp increase in conductance due to an increase in the number of free ions in solution.

Figure: Titration of potassium chloride against silver nitrate.

Potentiometric Titrations

In a potentiometric titration, a suitable electrode immersed in the solution to be titrated acts as the ‘indicator’. The indicator electrode is paired with a reference electrode and the two electrodes are connected to an electronic voltmeter. The emf of the indicator electrode changes gradually with the change of concentration of ions caused by the addition of titrant from the burette. The equivalence point is indicated by a sharp change in electrode potential.Since the reference electrode potential has a constant value, any change in the indicator electrode potential is reflected by a similar change in the cell potential. Therefore, the equivalence point can be found by plotting a graph between the cell emf and the volume of titrant added from the burette. A sharp rise of the curve shows the equivalence point and the corresponding volume on the graph is the volume of the solution used for the titration.

The potentiometric titrations may be of three types :

(a) Acid-base titrations

(b) Oxidation-reduction titrations

(c) Precipitation titrations

Acid-base Titrations

A hydrogen electrode or a glass electrode is immersed in solution of the acid whose strength is to be determined. The glass electrode is coupled with a standard calomel electrode. The cell thus formed is connected to the potentiometer or electronic voltmeter. When alkali is added, pH of the solution changes. The emf of the cell also changes with pH of the solution in accordance with the relation. The standard alkali solution is then added from the burette in small volumes. After each addition, the emf of the cell is recorded. The emf is then plotted against the volume of alkali added. The shape of the curve for the titration of a strong acid against strong alkali (HCl versus NaOH) is shown in Figure (a). The steepest portion of the curve indicates the equivalence point. However, when the solutions are very dilute, or weak acids or bases, are involved, the steepness of the curve is less marked and it is difficult to judge the end-point. In such a case, we plot the slope of the curve, ΔE/ΔV against the volume of alkali used. The maximum of the curve indicates the end-point.

Figure: Potentiometric titration curve of an acid and a base.

Oxidation-reduction Titrations

The titration of ferrous ions (Fe2+) with ceric ions (Ce4+) is an example of oxidation-reduction (or redox) titration. Fe2+ ion is oxidised to Fe3+ ion, while Ce4+ is reduced to Ce3+ ion.

Fe2+ + Ce4+ ⎯⎯→ Fe3+ + Ce3+

The indicator electrode is a shiny platinum strip dipping in the solution of Fe2+ ions, and it is connected to a standard calomel electrode. The Ce4+ solution is added from the burette and the cell potential, E, recorded after each addition.The potential of the platinum electrode depends on the ratio [Fe3+]/[Fe2+]. The potential of the cell, E, also changes with the change of the ratio [Fe3+]/[Fe2+]. Therefore, the cell potential changes with the addition of Ce4+ ions from the burette. Figure shows how the potential of the cell changes during the titration. At the equivalence point there is a sharp rise of potential which indicates the end- point. Potentiometric titrations of this type are particularly useful for coloured solutions in which an indicator cannot be employed.

Figure: Potentiometric titration curve of Fe2+ ions and Ce4+ ions.

Precipitation Titration

A typical precipitation titration is that of sodium chloride solution against silver nitrate solution.  A silver electrode dipping in the unknown sodium chloride solution is coupled with a calomel electrode through a salt bridge. However, if the calomel electrode were in direct contact with a solution containing excess silver ions, chloride would seep through the sintered base and react to form an insoluble layer of silver chloride. Any change in the cell potential is due to changes in concentration of Ag+ ions around the silver electrode. Initially the concentration of Ag+ ions will be zero. But as silver nitrate is added from the burette, silver chloride is precipitated. Now the solution will contain a small concentration of Ag+ ions formed by the slight dissociation of silver chloride. This concentration will increase slightly as Cl– ions are removed in order to maintain the solubility product Ksp = [Ag+][Cl–]. After the equivalence point, the concentration of Ag+ ions and, therefore the silver electrode potential will rise very sharply owing to the presence of excess of Ag+ ions. The volume of AgNO3 solution used to reach the equivalence point as shown in Figure.

Figure: Potentiometric titration curve.

Difference between Potentiometric Titrations and Conductometric Titrations

Potentiometric Titrations	Conductometric titrations
Potentiometric titrations are analytical techniques that help us to measure the potential across the analyte.	Conductometric titrations are analytical techniques that help us to measure the conductivity of an analyte
Can observe a sudden change in potential	Can observe a sudden change in conductivity
No need for an indicator, far more accurate, can be automated, etc.	No need for an indicator, can be suitable for coloured analytes and suspensions, results are accurate, etc.
Highly pH sensitive	Increased levels of salt can cause errors in the final result

References 

1. Essentials of Physical Chemistry by Arun Bahl, B.S Bahl, G.D. Tuli

2. https://en.m.wikipedia.org/

3. https://www.britannica.com/

4. https://www.bbc.com/

5. https://www.khanacademy.org/