Pc Chapter 43

Chapter 43 Molecules and Solids

Molecular Bonds – Introduction 





The bonding mechanisms in a molecule are fundamentally due to electric forces The forces are related to a potential energy function A stable molecule would be expected at a configuration for which the potential energy function has its minimum value

Molecular Bonds – Feature 1 

The force between atoms is repulsive at very small separation distances 





This repulsion is partially electrostatic and partially due to the exclusion principle Due to the exclusion principle, some electrons in overlapping shells are forced into higher energy states The energy of the system increases as if a repulsive force existed between the atoms

Molecular Bonds – Feature 2 

The force between the atoms is attractive at larger distances 



The attractive force (for many molecules) is due to the dipole-dipole interaction between charge distributions within the atoms of the molecules The electric fields of two dipoles will interact, resulting in a force between the dipoles

Potential Energy Function 

The potential energy for a system of two atoms can be expressed in the form A B U (r )   n  m r r    

r is the internuclear separation distance m and n are small integers A is associated with the attractive force B is associated with the repulsive force

Potential Energy Function, Graph 

At large separations, the slope of the curve is positive 



Corresponds to a net attractive force

At the equilibrium separation distance, the attractive and repulsive forces just balance 



At this point the potential energy is a minimum The slope is zero

Molecular Bonds – Types 

Simplified models of molecular bonding include    

Ionic Covalent van der Waals Hydrogen

Ionic Bonding 



Ionic bonding occurs when two atoms combine in such a way that one or more outer electrons are transferred from one atom to the other Ionic bonds are fundamentally caused by the Coulomb attraction between oppositely charged ions

Ionic Bonding, cont. 

When an electron makes a transition from the E = 0 to a negative energy state, energy is released 



The amount of this energy is called the electron affinity of the atom

The dissociation energy is the amount of energy needed to break the molecular bonds and produce neutral atoms

Ionic Bonding, NaCl Example





The graph shows the total energy of the molecule vs the internuclear distance The minimum energy is at the equilibrium separation distance

Ionic Bonding,final 



The energy of the molecule is lower than the energy of the system of two neutral atoms It is said that it is energetically favorable for the molecule to form 

The system of two atoms can reduce its energy by transferring energy out of the system and forming a molecule

Covalent Bonding 





A covalent bond between two atoms is one in which electrons supplied by either one or both atoms are shared by the two atoms Covalent bonds can be described in terms of atomic wave functions The example will be two hydrogen atoms forming H2

Wave Function – Two Atoms Far Apart 

Each atom has a wave function ψ1s (r ) 



1 πao3

e  r ao

There is little overlap between the wave functions of the two atoms

Wave Function – Molecule 





The two atoms are brought close together The wave functions overlap and form the compound wave shown The probability amplitude is larger between the atoms than on either side

Active Figure 43.3

(SLIDESHOW MODE ONLY)

Covalent Bonding, Final 

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

The probability is higher that the electrons associated with the atoms will be located between them This can be modeled as if there were a fixed negative charge between the atoms, exerting attractive Coulomb forces on both nuclei The result is an overall attractive force between the atoms, resulting in the covalent bond

Van der Waals Bonding 

Two neutral molecules are attracted to each other by weak electrostatic forces called van der Waals forces 



Atoms that do not form ionic or covalent bonds are also attracted to each other by van der Waals forces

The van der Waals force is due to the fact that the molecule has a charge distribution with positive and negative centers at different positions in the molecule

Van der Waals Bonding, cont. 



As a result of this charge distribution, the molecule may act as an electric dipole Because of the dipole electric fields, two molecules can interact such that there is an attractive force between them 

Remember, this occurs even though the molecules are electrically neutral

Types of Van der Waals Forces 

Dipole-dipole force 



An interaction between two molecules each having a permanent electric dipole moment

Dipole-induced dipole force 

A polar molecule having a permanent dipole moment induces a dipole moment in a nonpolar molecule

Types of Van der Waals Forces, cont. 

Dispersion force 





An attractive force occurs between two nonpolar molecules The interaction results from the fact that, although the average dipole moment of a nonpolar molecule is zero, the average of the square of the dipole moment is nonzero because of charge fluctuations The two nonpolar molecules tend to have dipole moments that are correlated in time so as to produce van der Waals forces

Hydrogen Bonding 



In addition to covalent bonds, a hydrogen atom in a molecule can also form a hydrogen bond Using water (H2O) as an example 



There are two covalent bonds in the molecule The electrons from the hydrogen atoms are more likely to be found near the oxygen atom than the hydrogen atoms

Hydrogen Bonding – H2O Example, cont. 

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

This leaves essentially bare protons at the positions of the hydrogen atoms The negative end of another molecule can come very close to the proton This bond is strong enough to form a solid crystalline structure

Hydrogen Bonding, Final 

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The hydrogen bond is relatively weak compared with other electrical bonds Hydrogen bonding is a critical mechanism for the linking of biological molecules and polymers DNA is an example

Energy States of Molecules 

The energy of a molecule (assume one in a gaseous phase) can be divided into four categories 

Electronic energy 



Due to the interactions between the molecule’s electrons and nuclei

Translational energy 

Due to the motion of the molecule’s center of mass through space

Energy States of Molecules, 2 

Categories, cont. 

Rotational energy 



Vibrational energy 



Due to the rotation of the molecule about its center of mass Due to the vibration of the molecule’s constituent atoms

The total energy of the molecule is the sum of the energies in these categories: 

E = Eel + Etrans + Erot + Evib

Spectra of Molecules 



The translational energy is unrelated to internal structure and therefore unimportant to the interpretation of the molecule’s spectrum By analyzing its rotational and vibrational energy states, significant information about molecular spectra can be found

Rotational Motion of Molecules 



A diatomic model will be used, but the same ideas can be extended to polyatomic molecules A diatomic molecule aligned along an x axis has only two rotational degrees of freedom 

Corresponding to rotations about the y and x axes

Rotational Motion of Molecules, Energy 

The rotational energy is given by 1 Eω I rot  2



2

I is the moment of inertia of the molecule  m1 m2  2 I   rμr  m1  m2  

2

µ is called the reduced mass of the molecule

Rotational Motion of Molecules, Angular Momentum 



Classically, the value of the molecule’s angular momentum can have any value L = Iω Quantum mechanics restricts the values of the angular momentum to

L  J  J  1 h J  0 , 1, 2,K 

J is an integer called the rotational quantum number

Rotational Kinetic Energy of Molecules, Allowed Levels 

The allowed values are h2 Erot  J  J  1 2I





J  0 , 1, 2,K

The rotational kinetic energy is quantized and depends on its moment of inertia As J increases, the states become farther apart

Allowed Levels, cont. 



For most molecules, transitions result in radiation that is in the microwave region Allowed transitions are given by the condition h2 h2 Erot  J  J 2 I 4π I J  1, 2, 3 ,K 

J is the number of the higher state

Active Figure 43.5

(SLIDESHOW MODE ONLY)

Sample Transitions – CO Example

Vibrational Motion of Molecules 



A molecule can be considered to be a flexible structure where the atoms are bonded by “effective springs” Therefore, the molecule can be modeled as a simple harmonic oscillator

Vibrational Motion of Molecules, Potential Energy 

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A plot of the potential energy function ro is the equilibrium atomic separation For separations close to ro, the shape closely resembles a parabola

Vibrational Energy 

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

Classical mechanics describes the frequency of vibration of a simple harmonic oscillator Quantum mechanics predicts that a molecule will vibrate in quantized states The vibrational and quantized vibrational energy can be altered if the molecule acquires energy of the proper value to cause a transition between quantized states

Vibrational Energy, cont. 

The allowed vibrational energies are Evib 





1   v  h ƒ v  0 , 1, 2,K 2 

v is an integer called the vibrational quantum number

When v = 0, the molecule’s ground state energy is ½hƒ 

The accompanying vibration is always present, even if the molecule is not excited

Vibrational Energy, Final 

The allowed vibrational energies can be expressed as 1 h  Evib   v   2  2π  v  0 , 1, 2,K





k μ

Allowed transitions are Δv = ±1 The energy between states is ΔEvib = hƒ

Some Values for Diatomic Molecules

Molecular Spectra 





In general, a molecule vibrates and rotates simultaneously To a first approximation, these motions are independent of each other The total energy is the sum of the energies for these two motions: 1 h2 E   v  h ƒ  J  J  1 2 2I  

Molecular Energy-Level Diagram 

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For each allowed state of v, there is a complete set of levels corresponding to the allowed values of J The energy separation between successive rotational levels is much smaller than between successive vibrational levels Most molecules at ordinary temperatures vibrate at v = 0 level

Molecular Absorption Spectrum



The spectrum consists of two groups of lines 



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One group to the right of center satisfying the selection rules ΔJ = +1 and Δv = +1 The other group to the left of center satisfying the selection rules ΔJ = -1 and Δv = +1

Adjacent lines are separated by /2πI

Active Figure 43.8

(SLIDESHOW MODE ONLY)

Absorption Spectrum of HCl

 

It fits the predicted pattern very well A peculiarity shows, each line is split into a doublet  Two chlorine isotopes were present in the same sample  Because of their different masses, different I’s are present in the sample

Intensity of Spectral Lines 

The intensity is determined by the product of two functions of J 

The first function is the number of available states for a given value of J 



There are 2J + 1 states available

The second function is the Boltzmann factor 

n  noe

 h2J (J 1)/(2 I kBT )

Intensity of Spectral Lines, cont 

Taking into account both factors by multiplying them, I   2J  1 e  h J (J 1)/(2 I k T ) 2

 



B

The 2J + 1 term increases with J The exponential term decreases

This is in good agreement with the observed envelope of the spectral lines

Bonding in Solids 

Bonds in solids can be of the following types   

Ionic Covalent Metallic

Ionic Bonds in Solids 





The dominant interaction between ions is through the Coulomb force Many crystals are formed by ionic bonding Multiple interactions occur among nearest-neighbor atoms

Ionic Bonds in Solids, 2 

The net effect of all the interactions is a negative electric potential energy Uαk attractive   



e2 e r

is a dimensionless number known as the Madelung constant The value of depends only on the crystalline structure of the solid

Ionic Bonds, NaCl Example

 

 

The crystalline structure is shown (a) Each positive sodium ion is surrounded by six negative chlorine ions (b) Each chlorine ion is surrounded by six sodium ions (c) = 1.747 6 for the NaCl structure

Total Energy in a Crystalline Solid 



As the constituent ions of a crystal are brought close together, a repulsive force exists The potential energy term B/rm accounts for this repulsive force 

This repulsive force is a result of electrostatic forces and the exclusion principle

Total Energy in a Crystalline Solid, cont 

The total potential energy of the crystal is

Utotal 

e2 B = − αk e + m r r

The minimum value, Uo, is called the ionic cohesive energy of the solid 

It represents the energy needed to separate the solid into a collection of isolated positive and negative ions

Properties of Ionic Crystals 



They form relatively stable, hard crystals They are poor electrical conductors  



They contain no free electrons Each electron is bound tightly to one of the ions

They have high melting points

More Properties of Ionic Crystals 

They are transparent to visible radiation, but absorb strongly in the infrared region 



The shells formed by the electrons are so tightly bound that visible light does not possess sufficient energy to promote electrons to the next allowed shell Infrared is absorbed strongly because the vibrations of the ions have natural resonant frequencies in the low-energy infrared region

Final Properties of Ionic Crystals 

Many are quite soluble in polar liquids  



Water is an example of a polar liquid The polar solvent molecules exert an attractive electric force on the charged ions This breaks the ionic bonds and dissolves the solid

Properties of Solids with Covalent Bonds 

Properties include 

Usually very hard 

  

Due to the large atomic cohesive energies

High bond energies High melting points Good electrical conductors

Cohesive Energies for Some Covalent Solids

Covalent Bond Example – Diamond

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

Each carbon atom in a diamond crystal is covalently bonded to four other carbon atoms This forms a tetrahedral structure

Another Carbon Example -Buckyballs 



Carbon can form many different structures The large hollow structure is called buckminsterfullerene 

Also known as a “buckyball”

Metallic Solids 

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

Metallic bonds are generally weaker than ionic or covalent bonds The outer electrons in the atoms of a metal are relatively free to move through the material The number of such mobile electrons in a metal is large

Metallic Solids, cont. 

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The metallic structure can be viewed as a “sea” or “gas” of nearly free electrons surrounding a lattice of positive ions The bonding mechanism is the attractive force between the entire collection of positive ions and the electron gas

Properties of Metallic Solids 

Light interacts strongly with the free electrons in metals 

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

Visible light is absorbed and re-emitted quite close to the surface This accounts for the shiny nature of metal surfaces

High electrical conductivity

More Properties of Metallic Solids 

The metallic bond is nondirectional 

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

This allows many different types of metal atoms to be dissolved in a host metal in varying amounts The resulting solid solutions, or alloys, may be designed to have particular properties

Metals tend to bend when stretched 

Due to the bonding being between all of the electrons and all of the positive ions

Free-Electron Theory of Metals 

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

The quantum-based free-electron theory of electrical conduction in metals takes into account the wave nature of the electrons The model is that the outer-shell electrons are free to move through the metal, but are trapped within a three-dimensional box formed by the metal surfaces Each electron can be represented as a particle in a box

Fermi-Dirac Distribution Function 



Applying statistical physics to a collection of particles can relate microscopic properties to macroscopic properties For electrons, quantum statistics requires that each state of the system can be occupied by only two electrons

Fermi-Dirac Distribution Function, cont. 

The probability that a particular state having energy E is occupied by one of the electrons in a solid is given by ƒ( E ) 





1 e

(E EF ) kBT

1

ƒ(E) is called the Fermi-Dirac distribution function EF is called the Fermi energy

Fermi-Dirac Distribution Function at T = 0 



At T = 0, all states having energies less than the Fermi energy are occupied All states having energies greater than the Fermi energy are vacant

Fermi-Dirac Distribution Function at T > 0 





As T increases, the distribution rounds off slightly States near and below EF lose population States near and above EF gain population

Active Figure 43.15

(SLIDESHOW MODE ONLY)

Electrons as a Particle in a Three-Dimensional Box 



The energy levels for the electrons are very close together The density-of-states function gives the number of allowed states per unit volume that have energies between E and dE: 8 2πme3 2 E 1 2dE g (E )dE  h3 e(E EF ) kBT  1

Fermi Energy at T = 0 K 

The Fermi energy at T = 0 K is 2

23

h  3ne  EF (0)    2mπ 8  e  



The order of magnitude of the Fermi energy for metals is about 5 eV The average energy of a free electron in a metal at 0 K is Eav = (3/5) EF

Fermi Energies for Some Metals

Wave Functions of Solids 



To make the model of a metal more complete, the contributions of the parent atoms that form the crystal must be incorporated Two wave functions are valid for an atom with atomic number A and a single s electron outside a closed shell: ψs (r )   Aƒ(r )e  Zr nao

ψs (r )   Aƒ(r )e  Zr nao

Combined Wave Functions 

The wave functions can combine in the various ways shown 



s + s

+ +

s

-

s

is equivalent to

+

These two possible combinations of wave functions represent two possible states of the two-atom system

Splitting of Energy Levels 





The states are split into two energy levels due to the two ways of combining the wave functions The energy difference is relatively small, so the two states are close together on an energy scale For large values of r, the electron clouds do not overlap and there is no splitting of the energy level

Splitting of Energy Levels, cont. 



As the number of atoms increases, the number of combinations in which the wave functions combine increases Each combination corresponds to a different energy level

Splitting of Energy Levels, final 



When this splitting is extended to the large number of atoms present in a solid, there is a large number of levels of varying energy These levels are so closely spaced they can be thought of as a band of energy levels

Energy Bands in a Crystal 

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In general, a crystalline solid will have a large number of allowed energy bands The white areas represent energy gaps, corresponding to forbidden energies Some bands exhibit an overlap Blue represents filled bands and gold represents empty bands in this example of sodium

Electrical Conduction – Classes of Materials 

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Good electrical conductors contain a high density of free charge carriers The density of charge carriers in an insulator is nearly zero Semiconductors are materials with a charge density between those of insulators and conductors These classes can be discussed in terms of a model based on energy bands

Metals 

To be a good conductor, the charge carriers in a material must be free to move in response to an electric field 





We will consider electrons as the charge carriers

The motion of electrons in response to an electric field represents an increase in the energy of the system When an electric field is applied to a conductor, the electrons move up to an available higher energy state

Metals – Energy Bands 

At T = 0, the Fermi energy lies in the middle of the band 



All levels below EF are filled and those above are empty

If a potential difference is applied to the metal, electrons having energies near EF require only a small amount of additional energy from the applied field to reach nearby higher energy levels

Metals As Good Conductors 



The electrons in a metal experiencing only a small applied electric field are free to move because there are many empty levels available close to the occupied energy level This shows that metals are good conductors

Insulators 





There are no available states that lie close in energy into which electrons can move upward in response to an electric field Although an insulator has many vacant states in the conduction band, these states are separated from the filled band by a large energy gap Only a few electrons can occupy the higher states, so the overall electrical conductivity is very small

Insulator – Energy Bands 





The valence band is filled and the conduction band is empty at T = 0 The Fermi energy lies somewhere in the energy gap At room temperature, very few electrons would be thermally excited into the conduction band

Semiconductors 



The band structure of a semiconductor is like that of an insulator with a smaller energy gap Typical energy gap values are shown in the table

Semiconductors – Energy Bands 



Appreciable numbers of electrons are thermally excited into the conduction band A small applied potential difference can easily raise the energy of the electrons into the conduction band

Semiconductors – Movement of Charges 



Charge carriers in a semiconductor can be positive, negative, or both When an electron moves into the conduction band, it leaves behind a vacant site, called a hole

Semiconductors – Movement of Charges, cont. 

The holes act as charge carriers 



Electrons can transfer into a hole, leaving another hole at its original site

The net effect can be viewed as the holes migrating through the material in the direction opposite the direction of the electrons 

The hole behaves as if it were a particle with charge +e

Intrinsic Semiconductors 



A pure semiconductor material containing only one element is called an intrinsic semiconductor It will have equal numbers of conduction electrons and holes 

Such combinations of charges are called electron-hole pairs

Doped Semiconductors 

 

Impurities can be added to a semiconductor This process is called doping Doping 

 

Modifies the band structure of the semiconductor Modifies its resistivity Can be used to control the conductivity of the semiconductor

n-Type Semiconductors 





An impurity can add an electron to the structure This impurity would be referred to as a donor atom Semiconductors doped with donor atoms are called ntype semiconductors

n-Type Semiconductors, Energy Levels 



The energy level of the extra electron is just below the conduction band The electron of the donor atom can move into the conduction band as a result of a small amount of energy

p-Type Semiconductors 

An impurity can add a hole to the structure 





This is an electron deficiency

This impurity would be referred to as a acceptor atom Semiconductors doped with acceptor atoms are called p-type semiconductors

p-Type Semiconductors, Energy Levels 







The energy level of the hole is just above the valence band An electron from the valence band can fill the hole with an addition of a small amount of energy A hole is left behind in the valance band This hole can carry current in the presence of an electric field

Extrinsic Semiconductors 



When conduction in a semiconductor is the result of acceptor or donor impurities, the material is called an extrinsic semiconductor Doping densities range from 1013 to 1019 cm-3

Semiconductor Devices 



Many electronic devices are based on semiconductors These devices include    

Junction diode Light-emitting and light-absorbing diodes Transistor Integrated Circuit

The Junction Diode 

 



A p-type semiconductor is joined to an n-type This forms a p-n junction A junction diode is a device based on a single p-n junction The role of the diode is to pass current in one direction, but not the other

The Junction Diode, 2 

The junction has three distinct regions   



a p region an n region a depletion region

The depletion region is caused by the diffusion of electrons to fill holes 

This can be modeled as if the holes being filled were diffusing to the n region

The Junction Diode, 3 



Because the two sides of the depletion region each carry a net charge, an internal electric field exists in the depletion region This internal field creates an internal potential difference that prevents further diffusion and ensures zero current in the junction when no potential difference is applied

Junction Diode, Biasing 

A diode is forward biased when the p side is connected to the positive terminal of a battery 



This decreases the internal potential difference which results in a current that increases exponentially

A diode is reverse biased when the n side is connected to the positive terminal of a battery 

This increases the internal potential difference and results in a very small current that quickly reaches a saturation value

Junction Diode: I-∆V Characteristics

LEDs and Light Absorption







Light emission and absorption in semiconductors is similar to that in gaseous atoms, with the energy bands of the semiconductor taken into account An electron in the conduction band can recombine with a hole in the valance band and emit a photon An electron in the valance band can absorb a photon and be promoted to the conduction band, leaving behind a hole

Transistors 

A transistor is formed from two p-n junctions 



A narrow n region sandwiched between two p regions or a narrow p region between two n regions

The transistor can be used as  

An amplifier A switch

Integrated Circuits 



An integrated circuit is a collection of interconnected transistors, diodes, resistors and capacitors fabricated on a single piece of silicon known as a chip Integrated circuits 



Solved the interconnectedness problem posed by transistors Possess the advantages of miniaturization and fast response

Superconductivity 



A superconductor expels magnetic fields from its interior by forming surface currents Surface currents induced on the superconductor’s surface produce a magnetic field that exactly cancels the externally applied field

Superconductivity and Cooper Pairs 





Two electrons are bound into a Cooper pair when they interact via distortions in the array of lattice atoms so that there is a net attractive force between them Cooper pairs act like bosons and do not obey the exclusion principle The entire collection of Cooper pairs in a metal can be described by a single wave function

Superconductivity, cont. 



Under the action of an applied electric field, the Cooper pairs experience an electric force and move through the metal There is no resistance to the movement of the Cooper pairs  

They are in the lowest possible energy state There are no energy states above that of the Cooper pairs because of the energy gap

Superconductivity Critical Temperatures 





A new family of compounds was found that was superconducting at “high” temperatures The critical temperature is the temperature at which the electrical resistance of the material decreases to virtually zero Critical temperatures for some materials are shown