Chapter 43
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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

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Chapter 43

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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

Features of Molecular Bonds

  • 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

  • The force between the atoms is attractive at larger distances

Potential Energy Function

  • The potential energy for a system of two atoms can be expressed in the form

    • 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

  • There is little overlap between the wave functions of the two atoms when they are far away from each other

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

  • Use the active figure to move the individual wave functions

  • Observe the composite wave function



Covalent Bonding, Final

  • 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 Waalsforces

    • 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.

  • 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

  • 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

      • Due to the rotation of the molecule about its center of mass

    • Vibrational energy

      • 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

  • I is the moment of inertia of the molecule

    • 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

    • J is an integer called the rotational quantum number

Rotational Kinetic Energy of Molecules, Allowed Levels

  • The allowed values are

  • 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

    • J is the number of the higher state

Active Figure 43.5

  • Use the active figure to adjust the distance between the atoms

  • Choose the initial rotational energy state of the molecule

  • Observe the transition of the molecule to lower energy states



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

  • 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

  • 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

    • 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

  • Selection rule for allowed transitions is Δv = ±1

  • The energy of an absorbed photon is Ephoton = ΔEvib = hƒ

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:

Molecular Energy-Level Diagram

  • 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

    • 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 h/2πI

Active Figure 43.8

  • Use the active figure to adjust the spring constant and the moment of inertia of the molecule

  • Observe the effect on the energy levels and the spectral lines



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

Intensity of Spectral Lines, cont

  • Taking into account both factors by multiplying them,

    • 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

    • α is a dimensionless number known as the Madelung constant

    • The value of α 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)

  • α = 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

  • 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

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

  • 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

  • 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.

  • 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

    • 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

    • 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 stressed

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

Free-Electron Theory of Metals

  • 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) 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

  • Use the active figure to adjust the temperature

  • Observe the effect on the Fermi-Dirac distribution function



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 E + dE:

Fermi Energy at T = 0 K

  • The Fermi energy at T = 0 K is

  • 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 Eavg = (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 Z and a single s electron outside a closed shell:

Combined Wave Functions

  • The wave functions can combine in the various ways shown

    • ψs- + ψs- is equivalent to ψs+ + ψs+

  • 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

  • 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

  • Good electrical conductors contain a high density of free charge carriers

  • The density of free 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


  • 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 empty states above the Fermi energy

Metals As Good Conductors

  • The electrons in a metal experiencing only a weak 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 excellent electrical conductors


  • 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


  • 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 n-typesemiconductors

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-typesemiconductors

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-DV 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


  • A junction 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


  • 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

  • The critical temperature is the temperature at which the electrical resistance of the material decreases to virtually zero

  • A new family of compounds was found that was superconducting at “high” temperatures

    • First discovered in 1986

    • Found materials that are superconductive up to temperatures of 150 K

    • Currently no widely accepted theory for high-temperature superconductivity

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