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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

Chapter 43

Molecules and Solids

molecular bonds introduction
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
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
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
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
Molecular Bonds – Types
  • Simplified models of molecular bonding include
    • Ionic
    • Covalent
    • van der Waals
    • Hydrogen
ionic bonding
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
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
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
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
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
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
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
Active Figure 43.3
  • Use the active figure to move the individual wave functions
  • Observe the composite wave function

PLAY

ACTIVE FIGURE

covalent bonding final
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
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
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
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
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
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 h 2 o example cont
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
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
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
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
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
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
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
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
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
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
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

PLAY

ACTIVE FIGURE

vibrational motion of molecules
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
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
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
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
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
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
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
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
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

PLAY

ACTIVE FIGURE

absorption spectrum of hcl
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
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
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
Bonding in Solids
  • Bonds in solids can be of the following types
    • Ionic
    • Covalent
    • Metallic
ionic bonds in solids
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
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
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
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
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
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
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 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
covalent bond example diamond
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
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 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
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
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
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
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
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
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
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 01
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
Active Figure 43.15
  • Use the active figure to adjust the temperature
  • Observe the effect on the Fermi-Dirac distribution function

PLAY

ACTIVE FIGURE

electrons as a particle in a three dimensional box
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
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
wave functions of solids
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
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
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
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
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
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
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
metals
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
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
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
insulators
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 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
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
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
leds and light absorption
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
Transistors
  • 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
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
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
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
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
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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