BONDING in SOLIDS Engineering Physics

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1.1 Types of Bonding:-

Different charge distributions in the atoms give rise to different types of bonding.

They classified into five classes

1. Ionic bonding

2. Covalent bonding

3. Metallic bonding

4. Hydrogen bonding

5. Vander-Waals bonding

Based on the bond strength atomic bonding can be grouped into “primary” and “secondary” bonding.

Primary bonding:-

 Primary bonding have energies in the range of 1-10 eV.

 Ionic, covalent and metallic bonds are examples of primary bonding

among these ionic and covalent bondings are generally stronger than

the metallic bonding.

Secondary bonding:-

 Secondary bonding have energies in the range of 0.01- 0.5 eV/ bond

 Hydrogen bonding and Vander Waals bonding are examples of

secondary bonding. Generally, Vander Waals bonding is the weakest of

all.

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1.2 Ionic bonding (or) Hetro-Polar bond:-

1. Ionic bond is also called as “Hetro-Polar bond”.

2. Crystals in which ionic bond is present are known as “ionic crystals”.

Ionic crystals are mostly insulating in character.

Ex:-Nacl, NaBr, KBr, Mgo etc...

3. An Ionic bonding can only be formed between two different atoms,

one electropositive and other electronegative.

4. Electropositive elements readily give up electrons and are usually

group Ι (or) ΙΙ Elements e.g. Na, k, Ba.

5. Whereas electronegative elements readily take up electrons and are

usually Group VΙ (or) VΙΙ elements e.g. Cl, Br and O.

An example of the ionic bonding is NaCl. When neutral atom of a Na and Cl are brought close together it is easy for the valence electron of the sodium atom to be transferred to the chlorine atom so that both of them acquire a stable inert gas electronic configuration.

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There exits an electrostatic attraction between positively charged sodium cation and negatively charged chlorine anion. A negative charge attracts all positive charges in the neighbourhood, and vice versa.

Consequently, in the crystalline solid Na+ ions will be surrounded by Cl- and

Cl- ions by the Na+ ions. The attraction between the neighbouring unlike charges exceeds the repulsion due to like charges.

Properties of ionic solids:-

1. Ionic bond is fairly strong

2. They have strong binding energy

3. They have high melting and boiling points

4. Soluble in water

5. Transparent to visible light

6. Conductivity is less

7. The bonding in ionic crystal is non-directional.

1.3 Covalent Bonding (or) Homo-Polar bonds:-

1. Covalent bond is also called “Homo-Polar bonds”.

2. In this bonding the valence electrons are not transferred from one

atom to the other atom as ionic crystals. Because the energy

required to removing all the valence electrons is too large.

3. The covalent bonds are formed by sharing of valence electrons in

the incomplete outer shell of the atoms.

4. Such sharing results in a lowering of the potential energy of the

system. The element of group ΙΙΙ, ΙV and V of the periodic table

shows covalent bonding.

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5. In silicon, germanium and diamond materials the bonds are

covalent in nature. Organic compounds such as CH4 is also having covalent bonding.

6. Polar covalent bonds: A polar bond is a covalent bond in which

there is a separation of charge between one end and the other -

in other words in which one end is slightly positive and the other

slightly negative. Example the hydrogen-chlorine bond in HCl or the

hydrogen-oxygen bonds in water are typical.

7. Coordinate covalent bonds: A covalent bond in which both shared

electrons are donated by the same atom;

8. The simplest case of covalent bonding is hydrogen molecule (H2), in which two hydrogen atom atoms contribute their 1s1 electron to

form covalent bond. In the molecular orbital of H2 the two electrons are equally shared between the nuclei and cannot be

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specifically identified with either nucleus. A covalent bond is also

known as “electron-pair bond”

9. Thus in this example, two atoms are involved in the bond formation

process and they share a single pair of electrons. Hence the

resulting bond is known as a single covalent bond.

10. When the two coordinating atoms share two (or) three pair of

electrons it results in a double (or) a triple bond. In oxygen double

covalent is formed while in nitrogen triple covalent bond is formed.

A covalent bond may also be formed when two (or) more atom of

different non-metals share one (or) more pairs of valence electron.

Ex:- H2o ,Co2 , Hcl and NH3.

Properties of covalent bonding:-

1. Covalent bond crystals are usually hard and brittle

2. Bonding energy is high so that their melting and boiling points are high but low compared to ionic crystals

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3. Covalent bonds are highly directional in character

4. These materials are soluble in non-polar solvents like benzene

5. Pure covalent solids are good insulators of electricity at low temperature. Conductivity increases with increase in temperature.

6. Covalent substances are insoluble in water.

1.4 Metallic Bonding:-

1. The metallic bonding is an electrostatic attraction between positive

metal ions which are fixed in a metal to the electron cloud which are

moving freely in between the metal ions.

2. The bonding in a metal must be considered in terms of all the atoms

of the solid taken together. No valence electron is associated with a

specific atom.

3. The valence electrons from all the atoms belonging to the crystal are

free to move throughout the crystal.

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4. Materials bound in this manner are good conductors of electricity and

heat. The metallic bonding can be considered as a limiting case of

the ionic bonding in which the negative ions are just electrons. For

example sodium chloride contains equal number of Na+ and Cl- , while

metallic sodium contains equal number of Na+ and e-.

5. The crucial difference is that the mass of an electron is very small as

compared to the mass of Cl- ions. As a result of this, its zero point

motion is large so that it is not localized on a lattice. They are held

together by the resulting electrostatic interaction between the

positively charged metal ions and the cloud of negative electrons.

6. The unsaturated nature of metallic bonding accounts for the alloying

properties of metals. If the atoms have more loosely held valence

electrons, the more metallic is the bonding.

7. For example Sodium, Copper, Silver has high electrical and thermal

conductivities because their valence electrons are very mobile. They

are opaque because the free electrons absorb energy from light and

they have high reflectivity, because these free electrons re-emit this

energy as they fall back to lower energy.

Properties of metallic bond materials:-

1. They have high electrical and thermal conductivities

2. The metals are opaque to all electromagnetic radiations

3. The metals have high optical reflection and absorption co-efficient

4. The metallic bond is comparatively weaker than the ionic and covalent bond

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5. Metallic solids have crystalline structure

6. Metallic crystals posses a high degree of crystal symmetry due to symmetrical arrangement of positive ions

1.5 Hydrogen Bonding:-

1. A hydrogen bond is the attractive interaction of a hydrogen atom with

an electronegative atom, like nitrogen, oxygen or fluorine. The

hydrogen must be covalently bonded to another electronegative atom

to create the bond.

2. There exits two types of hydrogen bonds

1). Intermolecular hydrogen bond can occur between molecules

Ex: HF

2). Intramolecular hydrogen bond within different parts of a same

molecule.

Ex: Nitro phenol

Hydrogen bonds are common in covalently bonded molecules which contain hydrogen, such water Ho2 and hydrogen fluoride molecule (HF). Since the

Vignan BONDING IN SOLIDS Engineering Physics bonds are primarily covalent; the electrons are shared between the hydrogen and fluorine atom. However, the electrons tend to spend more time around the fluorine atom. This leads to a small positive charge around the hydrogen atoms, and a negative charge around the fluorine atom. When other molecules with this type of charge are came nearby, the negative charged end of one molecule will be Weakley attracted to the positively charged end of the other molecule. The hydrogen bond (5 to 30 kJ/mole) is stronger than a Vander Waals interaction, but weaker than covalent or ionic bonds.

This type of bond occurs in both inorganic molecules such as water and organic molecules such as DNA.

Properties of hydrogen bonding:-

 The hydrogen bonds are directional

 The bonding is relatively strong as compared to other dipole-diople

interactions

 Hydrogen bonded solids have low melting points

 Since no valence electrons are available in such solids they are good

insulators of electricity

 They are soluble in both polar and non-polar solvents

 They are transparent to the light

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1.6 Vander Waals bonding:-

1. The weak attractive forces that lead to bonding of atoms and

molecules can be formed as the result of dipole interaction. This type

of bonding is called the Vander Walls bonding.

2. A dipole occurs if a certain separation of center of gravity of positive

charge and negative charge exists in a body.

3. There are three types of Vander waals forces

1. Dipole- dipole forces

2. Dipole- induced dipole forces

3. Dispersion forces

Dipole-dipole forces:

It is an interaction between two molecules having permanent dipole moment.

For example polar molecules such as HCl have permanent dipole moment and attract other polar molecule.

Dipole-induces dipole force:

It results when a polar molecule having permanent electric dipole moment induces a dipole moment in non-polar molecules. In this case electric field of polar molecule creates the dipole moment in non-polar molecule. This then results as attractive force between the molecules.

Dispersion force:

It is an attractive force that occurs between two non-polar molecules. When two non-polar molecules near each other tend to have dipole moment due to charge fluctuations. So there exists an attractive Vander walls force.

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Properties of Vander Waals bonding:-

 Vander Waals bonds are non-directional

 Vander Waals bonding is weaker than the hydrogen bonding

 Vander Waals bonded solids have low melting point

 Since no valence electrons are available in such solids they are good

insulators of electricity

 They are soluble in both polar and non-polar solvents

1.7 Forces between Atoms:-

Solids are aggregate of atoms. The arrangement of atoms in any solid material is determined by the character, strength and directionality of the binding forces.

 Electrical forces are responsible for bonding in the atoms, giving

different solids structures

 Magnetic and gravitational forces are negligible in the formation of

solids

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The forces between atoms can be of two types

1. Attractive forces- which keep atoms together

2. Repulsive forces- which prevent the atoms from merging

When the atoms are at infinite separation with each other the potential energy will be zero. From this it is understood that potential energy between the atoms is inversely proportional to some power of the separation.

As the atoms approach the attractive forces initially increase and hence potential energy increase in a negative sense, because the atoms do the work of attraction so potential energy is inversely proportional to some power of the distance between the atoms.

1 Ua  m r

The negative sign indicates forces are attractive from

a From this Ua  m ------(1) r

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Where ‘a’ and ‘m’ are the constants.

When the atoms are at the separation of few atomic diameters repulsive forces exists. The potential energy due to the repulsion can be taken as positive because external work must be done to bring the atoms together.

The repulsive potential energy is inversely proportional to sum power of the distance between the atoms.

1 Ub  n r

b Ub  n ------(2) r

Where ‘r’ is the distance between the atoms.

The net potential energy is sum of the attractive and repulsive potential energies.

U Ua Ur

a b U    ------(3) r m r n

Now the force of an interaction between the two atoms is given by

dU F  dr

From the above eq (3)

d  a b  F    dr  r m r n 

(a)(m) (b)(n) F   r m1 r n1

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am bn F   r m1 r n1

Consider am = A, bm = B, m+1 =M and n+1 = N

A B F   r M r N

Where A, B, M and N are the constants.

Generally N > M. The variation of attractive and repulsive forces with distance of separation is shown above fig. When the separation between atoms is equal to ro the resultant force is equal to zero. This separation is called as “equilibrium separation”.

At equilibrium position F(r  ro )  0 from this

A B F  M  N  0 r0 r0

A B M  N r0 r0

N r0 B M  r0 A

N M B r  0 A

1  B  N M r0     A 

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1.8 Cohesive energy:-

The Interatomic forces are responsible for the crystal formation. A crystal can only be stable if its total energy is lower than the total energy of the atoms when free.

The difference between free atom energy and crystal energy is the cohesive energy or binding energy.

(Free atom energy) – (crystal energy) = (cohesive energy or binding energy)

i.e., the Sodium Chloride crystal is more stable than the collection of equal number of Sodium and Chlorine ions. The potential energy is due to the interaction between the two atoms and it does depend upon interatomic spacing.

The potential energy is given by

U( r ) f ( r ) dr  (1)

AB But fr() rrMN

Substituting f(r) in equation (1) we get

AB U() rMN dr rr

MN Ar Br dr  

Ar11MN Br    c 11MN

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Ar11MN Br     c MN11

ab U() r    c rrMN11

Let m = M-1 and n = N-1

There fore

ab Ur()   rrmn

When the distance of separation is ro , the energy is minimum

ab Umin  mn  ------(2) rroo

At equilibrium distance (ro) the potential energy should be minimum. When the potential energy between atoms is minimum the crystal will be stable.

Thus

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dU  0 dr rr 0

Hence in the formation of chemical compound the repulsive force should be shorter than the attractive forces

dU am bn  mn11   0 dr roo r

am bn mn11 rroo

n1 ro bn m1  ro am

n ro bn m  ro am

nmbn    rroo ------(3) am   

Substituting eq (3) in eq (2) we get

a b a   m  U    min mm    roo r b   n 

a a m Umin  mm  roo r n

an U  1  min m  rmo 

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Thus the minimum value of energy U(r) is negative. At r = ro, the attractive and repulsive forces balance each other. But the attractive and repulsive energies are not equal because m  n.

1.9 Cohesive energy of ionic crystal:-

The cohesive energy of ionic crystals is mainly due to electrostatic interaction and can be calculated on the basis of point charge mode.

We assume that there is a complete transfer of the valence electrons, and then the resulting ions will have spherically symmetric charge distribution which in turn can be treated as a point charge.

From the Born approximation, the cohesive energy of a crystal containing oppositely charged ions is written as the sum of two terms one is due to attractive and other is due to repulsion

Ae2 B U    n ------(1) 4 orr

Here ‘A’ is known as the “Madelung constant” and depends only on geometrical arrangement of ions in the crystal and where ‘B’ is “repulsion constant”.

For example A = 1.748 for NaCl and 1.763 for CsCl

The total energy per kilo mole of the crystal is

B Ae2 U() r N A n ------>(2) rr4 o

26 Where NA = 6.023 10 is the Avogadro’s number.

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The repulsion constant ‘B’ can be evaluated from the fact that the potential energy U is minimum at the equilibrium separation ro between the oppositely charged ions.

dU  0 dr rr o

d B Ae2 N A n 0 dr ro4 o r o

2 Ae d1 d n (roo )  B ( r )  0 4 o dr dr

Ae2 Bn 21n 0 4 orr o o

Ae2 Bn 21 n 4 orr o o

21n Ae ro B  ------(3) 4 on

Substituting eq (3) in eq (2) we get

2 21n NA Ae NAo Ae r U() r  Umin    rr o n 4or o 4  o n r o

2 n1 NA Ae 1 ro Umin    n 4 or o nr o

2 NA Ae 11    4 or o nr o

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2 NA Ae 1 Umin  1  4 oorn

Umin is known as “cohesive energy” of the ionic crystal.

1.10 Evaluation of Madelung constant:-

The madelung constant ‘A’ is a function of crystal structure and can therefore be calculated from the geometrical arrangement of ions of the crystal.

Consider NaCl structure, a central Na+ ions as the reference is having a single positive charge on it.

This ion is surrounded by 6 Cl- ions as first nearest neighbours, 12Na+ ions

as second nearest neighbors at a distance 3ro and so on.

There will be attractive potential energy between central Na+ and Cl- ions and repulsive potential energies between Na+ and Na+ ions.

The attractive potential energy between central Na+ ion and 6Cl – ions which are at a distance of r0 is 6e2 U1  ------(1) 4 oor

The repulsive potential energy between central Na+ ion and 12 Na+ ions at a

distance of 2ro is

12e2 U2  ------(2) 42 oor

The attractive potential energy between central Na+ ion and 8Cl – ions which are at a distance of is 3ro

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8e2 U3  ------(3) 43 oor

The total potential energy U is the sum of all potential energies due to neighboring ions in the solids is

6e2 12 e 2 8 e 2 U     4 oor 4o 2rr o 4  o 3 o

e2 12 8 6 U  6         4 oor 2 3 4

The attractive potential energy equation between two atoms is

Ae2 U  4 oor

Comparing this equation and above equation, we get ‘A’ value as 6 12 8 6 24 24 A        1 2 3 4 5 6

This converges to a value of 1.74756

1e2  6 12 8 6 U           41 oor 2 3 4

For NaCl the value of ‘A’ is 1.74756.