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words only definitions & notes
  1. enthalpy
    (thermodynamics) a thermodynamic quantity equal to the internal energy of a system plus the product of its volume and pressure
    135
    19.2 Enthalpy .
  2. eigenvalue
    (mathematics) any number such that a given square matrix minus that number times the identity matrix has a zero determinant
    If
    this equation is true then the function is an eigenfunction with eigenvalue λ.
  3. solvation
    a chemical process in which solvent molecules and molecules or ions of the solute combine to form a compound
    165
    24 Thermodynamics of Solvation 169
    24.1 The BornModel .
  4. hyperfine
    extremely fine or thin, as in a spectral line split into two or more components
    — Doing this results in the emission or absorption of a photon with energy
    ˜v = 4E
    hc
    (1.5)
    Failure of the Bohr model
    • No fine structure predicted (electron-electron coupling)
    • No hyperfine structure predicted (electron-nucleus coupling)
    • No Zeeman effect predicted (response of spectrum to magnetic field)
    20
    • Spin is not included in theory
    The Bohr quantization idea points to a wavelike behavior for the electron.
  5. quantize
    apply quantum theory to
    • Therefore atomic orbitals must be quantized.
    r =
    4π􀀒0N2~2
    Zmee2 (1.1)
    where Z is the atomic number, me and e are the mass and charge of the
    electron respectively and 􀀒0 is the permittivity of free space.
  6. Schrodinger equation
    the fundamental equation of wave mechanics
    (3.1)
    This equation is the (time independent) Schrödinger equation.
  7. superposition principle
    (geology) the principle that in a series of stratified sedimentary rocks the lowest stratum is the oldest
    187
    27.2 Orthogonality, Completeness, and the Superposition Principle .
  8. quantized
    of or relating to a quantum or capable of existing in only one of two states
    • Therefore atomic orbitals must be quantized.
    r =
    4π􀀒0N2~2
    Zmee2 (1.1)
    where Z is the atomic number, me and e are the mass and charge of the
    electron respectively and 􀀒0 is the permittivity of free space.
  9. equilibrium constant
    (chemistry) the ratio of concentrations when equilibrium is reached in a reversible reaction (when the rate of the forward reaction equals the rate of the reverse reaction)
    Equilibrium constants in terms of KC .
  10. quantization
    the act of dividing into quanta or expressing in terms of quantum theory
    — Doing this results in the emission or absorption of a photon with energy
    ˜v = 4E
    hc
    (1.5)
    Failure of the Bohr model
    • No fine structure predicted (electron-electron coupling)
    • No hyperfine structure predicted (electron-nucleus coupling)
    • No Zeeman effect predicted (response of spectrum to magnetic field)
    20
    • Spin is not included in theory
    The Bohr quantization idea points to a wavelike behavior for the electron.
  11. statistical mechanics
    the branch of physics that makes theoretical predictions about the behavior of macroscopic systems on the basis of statistical laws governing its component particles
    83
    III Statistical Mechanics and The Laws of Thermodynamics
    88
    12 Rudiments of Statistical Mechanics 89
    12.1 Statistics and Entropy .
  12. thermodynamic
    of or concerned with thermodynamics
    83
    III Statistical Mechanics and The Laws of Thermodynamics
    88
    12 Rudiments of Statistical Mechanics 89
    12.1 Statistics and Entropy .
  13. ideal gas
    a hypothetical gas with molecules of negligible size that exert no intermolecular forces
    274
    42.1.1 α and κT for an ideal gas .
  14. Schrodinger
    Austrian physicist who discovered the wave equation
    11.9 Laidler&Meiser ∗ ∗∗
    The must be continuous and single valued
    Particles have wave-like characteristics
    The Bohr atom was an important step towards the formulation of quantum theory
    • Erwin Schrödinger (1887—1961): Wave mechanics
    • Werner Heisenberg (1902—1976): Matrix mechanics
    • Paul Dirac (1902—1984): Abstract vector space approach
    21
    2.
  15. ground state
    the lowest energy state of an atom or other particle
    (1.2)
    Tests of the Bohr atom
    • Ionization energy of Hydrogen atoms
    — The Ionization energy for Hydrogen atoms (Z = 1) is the minium
    energy required to completely remove an electron form it ground state,
    i.e.,
  16. energy state
    a definite stable energy that a physical system can have
    The internal energy state function .
  17. dipole
    an aerial half a wavelength long consisting of two rods connected to a transmission line at the center
    Flaws of the solar system model
    • Newton: OK √
    • Maxwell: problem √
    17
    — As the electron orbits the nucleus, the atom acts as an oscillating dipole
    • — The classical theory of electromagnetism states that oscillating dipoles
    emit radiation and thereby lose energy.
  18. law of thermodynamics
    (physics) a law governing the relations between states of energy in a closed system
    83
    III Statistical Mechanics and The Laws of Thermodynamics
    88
    12 Rudiments of Statistical Mechanics 89
    12.1 Statistics and Entropy .
  19. exclusion principle
    no two electrons or protons or neutrons in a given system can be in states characterized by the same set of quantum numbers
    55
    7.2 The Pauli Exclusion Principle .
  20. zeroth
    preceding even the first
    118
    17 The Zeroth and First Laws of Thermodynamics 119
    17.1 Temperature and the Zeroth Law of Thermodynamics .
  21. entropy
    a numerical measure of the uncertainty of an outcome
    83
    III Statistical Mechanics and The Laws of Thermodynamics
    88
    12 Rudiments of Statistical Mechanics 89
    12.1 Statistics and Entropy .
  22. uncertainty principle
    (quantum theory) the theory that it is impossible to measure both energy and time (or position and momentum) completely accurately at the same time
    29
    3.4 The Heisenberg Uncertainty Principle .
  23. Boltzmann
    Austrian physicist who contributed to the kinetic theory of gases (1844-1906)
    92
    13 The Boltzmann Distribution 94
    13.1 Partition Functions .
  24. thermodynamics
    physics concerned with heat and other forms of energy
    83
    III Statistical Mechanics and The Laws of Thermodynamics
    88
    12 Rudiments of Statistical Mechanics 89
    12.1 Statistics and Entropy .
  25. spectroscopy
    the use of spectroscopes to analyze spectra
    40
    i
    5.2 Spectroscopy (An Introduction) .
  26. normalize
    cause to conform to a standard
    22
    2.2 Howto normalize a wavefunction .
  27. angular momentum
    the product of the momentum of a rotating body and its distance from the axis of rotation
    191
    28 Angular Momentum 192
    28.1 Classical Theory of AngularMomentum .
  28. quantum mechanics
    the branch of quantum physics that accounts for matter at the atomic level; an extension of statistical mechanics based on quantum theory (especially the Pauli exclusion principle)
    Contents
    I Basic Quantum Mechanics 15
    1 Quantum Theory 16
    1.1 The “Fall” of Classical Physics .
  29. Erwin Schrodinger
    Austrian physicist who discovered the wave equation
    11.9 Laidler&Meiser ∗ ∗∗
    The must be continuous and single valued
    Particles have wave-like characteristics
    The Bohr atom was an important step towards the formulation of quantum theory
    • Erwin Schrödinger (1887—1961): Wave mechanics
    • Werner Heisenberg (1902—1976): Matrix mechanics
    • Paul Dirac (1902—1984): Abstract vector space approach
    21
    2.
  30. diatomic
    of or relating to a molecule made up of two atoms
    59
    8 Diatomic Molecules and the Born Oppenheimer Approximation 60
    8.1 Molecular Energy .
  31. oscillator
    generator that produces sonic oscillations or alternating current
    34
    5 The Harmonic Oscillator 38
    5.1 InterestingAspects of theQuantumHarmonicOscillator .
  32. kinetics
    the science concerned with the forces that cause motion
    245
    VII Kinetics and Gases 249
    38 Physical Kinetics 250
    38.1 kinetic theory of gases .
  33. adiabatic
    occurring without loss or gain of heat
    Isothermal and Adiabatic expansions .
  34. wave mechanics
    the modern form of quantum theory
    11.9 Laidler&Meiser ∗ ∗∗
    The must be continuous and single valued
    Particles have wave-like characteristics
    The Bohr atom was an important step towards the formulation of quantum theory
    • Erwin Schrödinger (1887—1961): Wave mechanics
    • Werner Heisenberg (1902—1976): Matrix mechanics
    • Paul Dirac (1902—1984): Abstract vector space approach
    21
    2.
  35. free energy
    (physics) a thermodynamic quantity equivalent to the capacity of a physical system to do work; the units of energy are joules or ergs
    137
    19.3 Helmholtz Free Energy .
  36. coulomb
    a unit of electrical charge equal to the amount of charge transferred by a current of 1 ampere in 1 second
    • The electron orbits the nucleus with the attractive coulomb force balanced
    by the repulsive centrifugal force.
  37. Heisenberg
    German mathematical physicist noted for stating the uncertainty principle (1901-1976)
    29
    3.4 The Heisenberg Uncertainty Principle .
  38. quantum
    the smallest discrete quantity of some physical property
    Contents
    I Basic Quantum Mechanics 15
    1 Quantum Theory 16
    1.1 The “Fall” of Classical Physics .
  39. heuristic
    a commonsense rule to help solve some problem
    Heuristic definition: .
  40. absorption spectrum
    the spectrum of electromagnetic radiation that has passed through a medium that absorbed radiation of certain wavelengths
    Absorption Spectra .
  41. quantum theory
    (physics) a physical theory that certain properties occur only in discrete amounts (quanta)
    Contents
    I Basic Quantum Mechanics 15
    1 Quantum Theory 16
    1.1 The “Fall” of Classical Physics .
  42. physical chemistry
    the branch of chemistry dealing with the physical properties of chemical substances
    Keep in mind this chapter provides some examples of how to solve problems for
    both physical chemistry I and physical chemistry II.
  43. equation
    a mathematical statement that two expressions are the same
    115
    16.3 Equations of State .
  44. energy level
    a definite stable energy that a physical system can have
    We can get the energy levels from kn =
    q
    2mEn
    ~2 and kn = nπ
    a :
    En =
    n2π2~2
    2ma2
    ~= h
    2π =
    n2h2
    8ma2 .
  45. physical property
    any property used to characterize matter and energy and their interactions
    The Average Value Theorem
    Postulate III implies that if ψ is an eigenfunction of a particular operator representing
    a physical observable, then all measurements of that physical property
    will yield the associated eigenvalue.
  46. atomic number
    quantity of protons in the nucleus of an atom of an element
    • Therefore atomic orbitals must be quantized.
    r =
    4π􀀒0N2~2
    Zmee2 (1.1)
    where Z is the atomic number, me and e are the mass and charge of the
    electron respectively and 􀀒0 is the permittivity of free space.
  47. electron
    an elementary particle with negative charge
    49
    6.3 Spin of the electron .
  48. coefficient
    a constant number that serves as a measure of some property
    144
    20.3 Activity and the Activity coefficient .
  49. Joule
    English physicist who established the mechanical theory of heat and discovered the first law of thermodynamics (1818-1889)
    Joule expansion .
  50. hydrogen atom
    an atom of hydrogen
    (1.2)
    Tests of the Bohr atom
    • Ionization energy of Hydrogen atoms
    — The Ionization energy for Hydrogen atoms (Z = 1) is the minium
    energy required to completely remove an electron form it ground state,
    i.e.,
  51. kinetic theory of gases
    (physics) a theory that gases consist of small particles in random motion
    245
    VII Kinetics and Gases 249
    38 Physical Kinetics 250
    38.1 kinetic theory of gases .
  52. Niels Bohr
    Danish physicist who studied atomic structure and radiations
    Bohr’s model: Niels Bohr (1885—1962)
    18
    • Atoms don’t collapse =⇒ what are the consequences
    Experimental clues
    • Atomic gases have discrete spectral lines.
  53. Dirac
    English theoretical physicist who applied relativity theory to quantum mechanics and predicted the existence of antimatter and the positron (1902-1984)
    11.9 Laidler&Meiser ∗ ∗∗
    The must be continuous and single valued
    Particles have wave-like characteristics
    The Bohr atom was an important step towards the formulation of quantum theory
    • Erwin Schrödinger (1887—1961): Wave mechanics
    • Werner Heisenberg (1902—1976): Matrix mechanics
    • Paul Dirac (1902—1984): Abstract vector space approach
    21
    2.
  54. benzene
    a colorless liquid hydrocarbon
    Problems Dealing With Statistical Mechanics and Thermodynamics
    Problem: A vial containing 1020 benzene molecules is at 300K.
  55. ionization
    the process of ionizing
    (1.2)
    Tests of the Bohr atom
    • Ionization energy of Hydrogen atoms
    — The Ionization energy for Hydrogen atoms (Z = 1) is the minium
    energy required to completely remove an electron form it ground state,
    i.e.,
  56. fine structure
    the presence of groups of closely spaced spectrum lines observed in the atomic spectrum of certain elements
    — Doing this results in the emission or absorption of a photon with energy
    ˜v = 4E
    hc
    (1.5)
    Failure of the Bohr model
    • No fine structure predicted (electron-electron coupling)
    • No hyperfine structure predicted (electron-nucleus coupling)
    • No Zeeman effect predicted (response of spectrum to magnetic field)
    20
    • Spin is not included in theory
    The Bohr quantization idea points to a wavelike behavior for the electron.
  57. vector
    a quantity that has magnitude and direction
    To
    solve this problem it is useful to define the set of three dimensional column vectors
    V (j) such that the three elements are the total number of allowed configurations of
    a j-atom chain having the jth atom in state A, B or C. For example,
    V (1) =
    ⎡
    ⎢⎣
    1
    1
    1
    ⎤
    ⎥⎦
    , V (2) =
    ⎡
    ⎢⎣2
    3
    2
    ⎤
    ⎥⎦
    , V (3) =
    ⎡
    ⎢⎣
    5
    7
    5
    ⎤
    ⎥⎦
    , · · · .
  58. conductivity
    the property of transmitting heat, electricity, or sound
    303
    45.3 Thermal conductivity .
  59. electromagnetism
    a physical force produced from the interaction of charged particles
    Flaws of the solar system model
    • Newton: OK √
    • Maxwell: problem √
    17
    — As the electron orbits the nucleus, the atom acts as an oscillating dipole
    • — The classical theory of electromagnetism states that oscillating dipoles
    emit radiation and thereby lose energy.
  60. atom
    the smallest component of an element
    First Attempts at the Structure of the Atom .
  61. derivative
    a compound obtained from another compound
    99
    14 Statistical Thermodynamics 103
    15 Work 107
    15.1 Properties of Partial Derivatives .
  62. chemical reaction
    a process in which substances are changed into others
    153
    22 Chemical Reactions 156
    22.1 Heats of Reactions .
  63. particle
    (nontechnical usage) a tiny piece of anything
    30
    4 Particle inaBox 31
    4.1 The 1DParticle in a Box Problem.
  64. harmonic
    involving or characterized by harmony
    34
    5 The Harmonic Oscillator 38
    5.1 InterestingAspects of theQuantumHarmonicOscillator .
  65. kinetic theory
    (physics) a theory that gases consist of small particles in random motion
    245
    VII Kinetics and Gases 249
    38 Physical Kinetics 250
    38.1 kinetic theory of gases .
  66. commutator
    switch for reversing the direction of an electric current
    The general statement of the Heisenberg uncertainty principle is
    δαδβ ≥
    1
    2
    ¯¯¯
    Dh
    ˆα,ˆβ
    iE¯¯¯ , (3.13)
    where the notation
    h
    ˆα,ˆβ
    i
    means the commutator of ˆα and ˆβ.
  67. kinetic energy
    the mechanical energy that a body has by virtue of motion
    27
    27
    • The total energy for a classical system is
    Ecl = T + V, (3.2)
    where T is the kinetic energy and V is the potential energy.
  68. matrix
    an enclosure within which something originates or develops
    (23)
    The V (j+1) can be found from the V (j) vector using the matrix equation,
    V (j+1) = MV (j), (24)
    where for this example
    M =
    ⎡
    ⎢⎣
    1 1 0
    1 1 1
    0 1 1
    ⎤
    ⎥⎦
    .
  69. molecule
    the simplest structural unit of an element or compound
    42
    II Quantum Mechanics of Atoms and Molecules 45
    6 Hydrogenic Systems 46
    6.1 Hydrogenic systems .
  70. photon
    a tiny bundle of matter that transmits light
    — Doing this results in the emission or absorption of a photon with energy
    ˜v = 4E
    hc
    (1.5)
    Failure of the Bohr model
    • No fine structure predicted (electron-electron coupling)
    • No hyperfine structure predicted (electron-nucleus coupling)
    • No Zeeman effect predicted (response of spectrum to magnetic field)
    20
    • Spin is not included in theory
    The Bohr quantization idea points to a wavelike behavior for the electron.
  71. atomic theory
    a theory of the structure of the atom
    16
    1.2 Bohr’s Atomic Theory .
  72. parameter
    a constant in the equation of a curve that can be varied
    Temperature corrections to the Arrhenious parameters .
  73. minium
    a reddish oxide of lead (Pb3O4) used as a pigment in paints and in glass and ceramics
    (1.2)
    Tests of the Bohr atom
    • Ionization energy of Hydrogen atoms
    — The Ionization energy for Hydrogen atoms (Z = 1) is the minium
    energy required to completely remove an electron form it ground state,
    i.e.,
  74. potential energy
    energy stored by a body or system by virtue of its position
    27
    27
    • The total energy for a classical system is
    Ecl = T + V, (3.2)
    where T is the kinetic energy and V is the potential energy.
  75. rotor
    rotating mechanism consisting of an assembly of rotating airfoils
    The rigid rotor .
  76. photoelectric
    of or pertaining to photoelectricity
    The photoelectric effect
    2.
  77. reversible
    capable of being returned to the original condition
    111
    16 Maximum Work and Reversible changes 113
    16.1 MaximalWork: Reversible versus Irreversible changes .
  78. osmotic pressure
    (physical chemistry) the pressure exerted by a solution necessary to prevent osmosis into that solution when it is separated from the pure solvent by a semipermeable membrane
    Osmotic Pressure .
  79. mechanics
    the branch of physics concerned with the motion of bodies
    Contents
    I Basic Quantum Mechanics 15
    1 Quantum Theory 16
    1.1 The “Fall” of Classical Physics .
  80. orbital
    relating to the path of one body around another
    66
    9 Molecular Orbital Theory and Symmetry 67
    9.1 MolecularOrbital Theory .
  81. oscillate
    move or swing from side to side regularly
    Flaws of the solar system model
    • Newton: OK √
    • Maxwell: problem √
    17
    — As the electron orbits the nucleus, the atom acts as an oscillating dipole
    • — The classical theory of electromagnetism states that oscillating dipoles
    emit radiation and thereby lose energy.
  82. fluorescence
    light emitted during absorption of radiation of some other (invisible) wavelength
    Fluorescence Spectra .
  83. coupling
    the act of pairing a male and female for reproductive purposes
    SpinOrbit Coupling .
  84. momentum
    the product of a body's mass and its velocity
    191
    28 Angular Momentum 192
    28.1 Classical Theory of AngularMomentum .
  85. molecular
    relating to the simplest units of an element or compound
    59
    8 Diatomic Molecules and the Born Oppenheimer Approximation 60
    8.1 Molecular Energy .
  86. kinetic
    relating to the motion of material bodies and their forces
    245
    VII Kinetics and Gases 249
    38 Physical Kinetics 250
    38.1 kinetic theory of gases .
  87. black body
    a hypothetical object capable of absorbing all the electromagnetic radiation falling on it
    Black body radiation and the ultraviolet catastrophe
    16
    16
    5.
  88. linear
    involving a single dimension
    68
    10 Molecular Orbital Diagrams 72
    10.1 LCAO–Linear Combinations of AtomicOrbitals .
  89. symmetry
    balance among the parts of something
    66
    9 Molecular Orbital Theory and Symmetry 67
    9.1 MolecularOrbital Theory .
  90. equilibrium
    a stable situation in which forces cancel one another
    148
    21 Equilibrium 151
    21.0.3
  91. superposition
    the placement of one thing on top of another
    187
    27.2 Orthogonality, Completeness, and the Superposition Principle .
  92. viscosity
    resistance of a liquid to flowing
    301
    45.2 Viscosity .
  93. spectroscopic
    of or relating to or involving spectroscopy
    — Eionize experimentally observed from spectroscopy is 13.605 eV (very
    good agreement)
    • Spectroscopic lines from Hydrogen represent the difference in energy between
    the quantum states
    — Bohr theory: Difference energies
    Ej − Ek =
    e2
    2a0
    μ
    1
    N2
    j −
    1
    N2
    k

    = R
    μ
    1
    N2
    j −
    1
    N2
    k

    (1.4)
    Initial state Nk Final States Nj Series Name
    1 2,3,4,· · · Lyman
    2 3,4,5,· · · Balmer
    3 4,5,6,· · · Pachen
    4 5,6,7,· · · Brackett
    5 6,7,8,· · · Pfund
    • — Since the orbita...
  94. thermal
    relating to or associated with heat
    303
    45.3 Thermal conductivity .
  95. configuration
    an arrangement of parts or elements
    To
    solve this problem it is useful to define the set of three dimensional column vectors
    V (j) such that the three elements are the total number of allowed configurations of
    a j-atom chain having the jth atom in state A, B or C. For example,
    V (1) =
    ⎡
    ⎢⎣
    1
    1
    1
    ⎤
    ⎥⎦
    , V (2) =
    ⎡
    ⎢⎣2
    3
    2
    ⎤
    ⎥⎦
    , V (3) =
    ⎡
    ⎢⎣
    5
    7
    5
    ⎤
    ⎥⎦
    , · · · .
  96. proton
    a stable particle with positive charge
    (16)
    So, about 68% of the time the electron would be found at some distance greater
    then one Bohr radius from the proton.
  97. Planck
    German physicist whose explanation of blackbody radiation in the context of quantized energy emissions initiated quantum theory (1858-1947)
    N is a positive
    real integer called the quantum number. ~ = h/2π is Planck’s constant
    divided by 2π.
  98. radius
    a straight line from the center to the perimeter of a circle
    Problem: What is the probability of finding an electron in the 1s state of hydrogen
    further than one Bohr radius away from the nucleus?
  99. cyclic
    marked by repeated series of events
    Solution: Here we either remember an identity or turn to our handout of partial
    derivative identities to employ the cyclic rule to
    ¡∂P
    ∂T
    ¢
    V :
    μ
    ∂P
    ∂T

    V
    = −
    μ
    ∂P
    ∂V

    T
    μ
    ∂V
    ∂T

    P
    .
  100. theorem
    an idea accepted as a demonstrable truth
    27
    3.3 The Average Value Theorem .
  101. rudiment
    the elementary stage of any subject
    83
    III Statistical Mechanics and The Laws of Thermodynamics
    88
    12 Rudiments of Statistical Mechanics 89
    12.1 Statistics and Entropy .
  102. Pauli
    United States physicist who proposed the exclusion principle
    55
    7.2 The Pauli Exclusion Principle .
  103. Franck
    French composer and teacher who influenced a generation of composers (1822-1890)
    242
    36.2 Franck—Condon activity .
  104. wether
    male sheep especially a castrated one
    • Determine wether you need to approach the problem mathematically
    or conceptually or both.
  105. chain reaction
    a series of chemical reactions in which the product of one is a reactant in the next
    265
    40.4 Chain Reactions .
  106. experimentally
    in an experimental fashion
    — Eionize experimentally observed from spectroscopy is 13.605 eV (very
    good agreement)
    • Spectroscopic lines from Hydrogen represent the difference in energy between
    the quantum states
    — Bohr theory: Difference energies
    Ej − Ek =
    e2
    2a0
    μ
    1
    N2
    j −
    1
    N2
    k

    = R
    μ
    1
    N2
    j −
    1
    N2
    k

    (1.4)
    Initial state Nk Final States Nj Series Name
    1 2,3,4,· · · Lyman
    2 3,4,5,· · · Balmer
    3 4,5,6,· · · Pachen
    4 5,6,7,· · · Brackett
    5 6,7,8,· · · Pfund
    • — Since the orbita...
  107. emission
    the act of causing to flow forth
    Emission Spectra .
  108. nucleus
    a part of the cell responsible for growth and reproduction
    Problem: What is the probability of finding an electron in the 1s state of hydrogen
    further than one Bohr radius away from the nucleus?
  109. spectrum
    a broad range of related objects, values, or qualities
    Absorption Spectra .
  110. freezing point
    the temperature below which a liquid turns into a solid
    Freezing Point Depression .
  111. Euler
    Swiss mathematician (1707-1783)
    Finally for the last function it is best to used Euler’s identity and write
    e−2ix = cos2x + i sin 2x (1)
    The real part of this function, cos 2x, has a period of π as does the imaginary
    part, sin 2x.
  112. differential
    a quality that distinguishes between similar things
    Solve the Schrödinger equation, Hˆ ψ = Eψ, which is now a second order
    differential equation of the form
    ∙
    −~2
    2m ∇2 + V (x, y, z)
    ¸
    ψ = Eψ
    ⇒ −~2
    2m ∇2ψ + (V (x, y, z) − E) ψ = 0 (3.8)
    28
    • Note: It is solely the form of V (x, y, z) which determines whether this
    is easy or hard to do.
  113. spectral
    resembling or characteristic of a phantom
    (29)
    Problem: Using the classical theory of light scattering, calculate the positions of
    the Rayleigh, Stokes and anti-Stokes spectral lines for benzene.
  114. variable
    something that is likely to change
    When P is themore convenient variable .
  115. laser
    an optical device that produces an intense beam of light
    Assume benzene
    has only two active modes (992cm−1 and 3063cm−1) and assume the Laser light
    used to do the scattering is at 20000cm−1 (this is 500nm–green light).
  116. centrifugal force
    the outward force on a body moving in a curved path around another body
    • The electron orbits the nucleus with the attractive coulomb force balanced
    by the repulsive centrifugal force.
  117. atomic
    relating to the smallest component of an element
    16
    1.2 Bohr’s Atomic Theory .
  118. integrating
    the action of incorporating a racial or religious group into a community
    (14)
    Remember the extra r2 sin θ is needed when integrating in spherical polar coordinates.
  119. ultraviolet
    having wavelengths shorter than light but longer than X-rays
    Black body radiation and the ultraviolet catastrophe
    16
    16
    5.
  120. analog
    something having a similarity to something else
    • The kinetic energy is always of the form
    T =
    1
    2m
    ¡
    p2
    x + p2y
    + p2z
    ¢
    (3.3)
    • The potential energy is almost always a function of coordinates only
    V = V (x, y, z) (3.4)
    • Note: Some quantumsystems don’t have classical analogs so the Hamiltonian
    operator must be hypothesized.
  121. theory
    a belief that can guide behavior
    Contents
    I Basic Quantum Mechanics 15
    1 Quantum Theory 16
    1.1 The “Fall” of Classical Physics .
  122. magnetic field
    the lines of force surrounding a permanent magnet or a moving charged particle
    — Doing this results in the emission or absorption of a photon with energy
    ˜v = 4E
    hc
    (1.5)
    Failure of the Bohr model
    • No fine structure predicted (electron-electron coupling)
    • No hyperfine structure predicted (electron-nucleus coupling)
    • No Zeeman effect predicted (response of spectrum to magnetic field)
    20
    • Spin is not included in theory
    The Bohr quantization idea points to a wavelike behavior for the electron.
  123. dynamics
    mechanics concerned with forces that cause motions of bodies
    175
    25 Key Equations for Exam 4 177
    V Quantum Mechanics and Dynamics 180
    26 Particle in a 3D Box 181
    26.1 Particle in a Box .
  124. angular
    having straight lines and sharp points or corners
    191
    28 Angular Momentum 192
    28.1 Classical Theory of AngularMomentum .
  125. hydrogen
    a colorless, odorless gas; the lightest chemical element
    Problem: What is the probability of finding an electron in the 1s state of hydrogen
    further than one Bohr radius away from the nucleus?
  126. chemistry
    the science of matter
    331
    Chemistry 351: Physical
    Chemistry I
    1
    1
    Solved Problems
    I make-up most of the problems on the problems sets, so it might be helpful to
    you to see some of these problems worked out.
  127. residual
    relating to or indicating a remainder
    This residual
    energy is called the zero point energy and is a consequence of the uncertainty
    principle.
  128. generalized
    not biologically differentiated or adapted to a specific function or environment
    Generalized Forces andDisplacements .
  129. Oppenheimer
    United States physicist who directed the project at Los Alamos that developed the first atomic bomb (1904-1967)
    59
    8 Diatomic Molecules and the Born Oppenheimer Approximation 60
    8.1 Molecular Energy .
  130. chemical
    produced by reactions involving atomic or molecular changes
    140
    20 Chemical Potential 142
    20.1 Spontaneity of processes .
  131. condense
    cause a gas or vapor to change into a liquid
    298
    44.4 Equilibria of condensed phases .
  132. temperature
    the degree of hotness or coldness of a body or environment
    118
    17 The Zeroth and First Laws of Thermodynamics 119
    17.1 Temperature and the Zeroth Law of Thermodynamics .
  133. solar system
    the sun with the celestial bodies that revolve around it
    First Attempts at the Structure of the Atom
    The “solar system” model.
  134. amplitude
    greatness of magnitude
    The wavefunction ψ represents a probability amplitude and is not directly observable.
  135. reaction
    an idea evoked by some experience
    153
    22 Chemical Reactions 156
    22.1 Heats of Reactions .
  136. centrifugal
    tending to move away from the middle
    • The electron orbits the nucleus with the attractive coulomb force balanced
    by the repulsive centrifugal force.
  137. vial
    a small bottle that contains liquid medicine
    Problems Dealing With Statistical Mechanics and Thermodynamics
    Problem: A vial containing 1020 benzene molecules is at 300K.
  138. physics
    the science of matter and energy and their interactions
    Contents
    I Basic Quantum Mechanics 15
    1 Quantum Theory 16
    1.1 The “Fall” of Classical Physics .
  139. input
    signal going into an electronic system
    There is only one Rayleigh line and it is at the same frequency at the
    input laser beam which, in this case, is 20000cm−1.
  140. helium
    a light colorless element that is one of the six inert gases
    52
    7 Multi-electron atoms 55
    7.1 Two ElectronAtoms: Helium .
  141. physical
    involving the body as distinguished from the mind or spirit
    245
    VII Kinetics and Gases 249
    38 Physical Kinetics 250
    38.1 kinetic theory of gases .
  142. perturbation
    the act of causing disorder
    205
    30 Approximation Techniques 207
    30.1 Perturbation Theory .
  143. radiation
    the act of spreading outward from a central source
    Black body radiation and the ultraviolet catastrophe
    16
    16
    5.
  144. ion
    a particle that is electrically charged positive or negative
    Ion Transfer Between Phases .
  145. rotation
    the act of turning as if on an axis
    233
    35 Molecular Rotations 235
    35.1 Relaxing the rigid rotor .
  146. procedure
    a particular course of action intended to achieve a result
    Solution: Following our general procedure from the notes if we have some unnormalized
    wavefunction, ψunnorm we know that this function must simply be some
    constant N multiplied by the normalized version of this function:
    ψunnorm = Nψnorm (8)
    We have shown generally that N is given by
    N =
    sZ
    space |ψunnorm(x)|2 dx.
  147. orbit
    the path of a celestial body in its revolution about another
    • The electron orbits the nucleus with the attractive coulomb force balanced
    by the repulsive centrifugal force.
  148. mechanical
    using tools or devices
    24
    3 The Setup of a Quantum Mechanical Problem 27
    3.1 The Hamiltonian .
  149. diffusion
    the act of dispersing something
    300
    45 Transport Properties of Fluids 301
    45.1 Diffusion .
  150. experimental
    of the nature of or undergoing a trial
    The “Fall” of Classical Physics
    A good theory:
    • explain known experimental results
    • self consistent
    • predictive
    • minimal number of postulates
    Around the turn of the century, experiments were being performed in which the results
    defied explanation by means of the current understanding of physics.
  151. periodic
    happening or recurring at regular intervals
    The wave must satisfy periodic boundary conditions much like a vibrating ring
    ∗ ∗ ∗ See Fig.
  152. axis
    a straight line through a body or figure
    (4.1)
    Because of the infinities at x = 0 and x = a, we need to partition the x-axis into
    the three regions shown in the figure.
  153. electronic
    relating to or operating by a controlled current
    237
    36 Electronic Spectroscopy of Molecules 240
    36.1 The Structure of the Electronic State .
  154. liquid
    fluid matter having no fixed shape but a fixed volume
    275
    42.1.2 α and κT for liquids and solids .
  155. precision
    the quality of being exact
    The Heisenberg Uncertainty Principle
    In quantum mechanics certain pairs of variables can not, even in principle, be
    simultaneously known to arbitrary precision.
  156. fluid
    continuous amorphous matter that tends to flow
    288
    VIII More Thermodyanmics 292
    44 Critical Phenomena 293
    44.1 Critical Behavior of fluids .
  157. solar
    relating to the sun or utilizing the energies of the sun
    First Attempts at the Structure of the Atom
    The “solar system” model.
  158. vapor
    a visible suspension in the air of particles of a substance
    Vapor Equilibrium and the Clausius-Clapeyron Equation .
  159. Newton
    English mathematician and physicist
    Flaws of the solar system model
    • Newton: OK √
    • Maxwell: problem √
    17
    — As the electron orbits the nucleus, the atom acts as an oscillating dipole
    • — The classical theory of electromagnetism states that oscillating dipoles
    emit radiation and thereby lose energy.
  160. polar
    of or existing within the Arctic or Antarctic Circles
    (14)
    Remember the extra r2 sin θ is needed when integrating in spherical polar coordinates.
  161. magnetic
    of or relating to or caused by attraction for iron
    — Doing this results in the emission or absorption of a photon with energy
    ˜v = 4E
    hc
    (1.5)
    Failure of the Bohr model
    • No fine structure predicted (electron-electron coupling)
    • No hyperfine structure predicted (electron-nucleus coupling)
    • No Zeeman effect predicted (response of spectrum to magnetic field)
    20
    • Spin is not included in theory
    The Bohr quantization idea points to a wavelike behavior for the electron.
  162. formula
    a group of symbols that make a mathematical statement
    Qcrystal = qN
    HO =
    Ã
    1
    2 sinh β~ω
    2
    !
    (30)
    From our formulas for statistical thermodynamics
    A = −kT ln Qcrystal = +NKT ln
    μ
    2 sinh
    β~ω
    2

    , (31)
    where we used properties of logs to pull the N out front and move the sinh term
    from to the numerator,
    S = −kβ
    ∂Qcrystal
    ∂β
    + k ln Qcrystal (32)
    =
    N kβ~ω
    2
    coth
    β~ω
    2 − k ln
    μ
    2 sinh
    β~ω
    2

    and
    U = −
    ∂Qcrystal
    ∂β
    =
    N~ω
    2
    coth
    β~ω
    2
    .
  163. element
    a substance that cannot be separated into simpler substances
    To
    solve this problem it is useful to define the set of three dimensional column vectors
    V (j) such that the three elements are the total number of allowed configurations of
    a j-atom chain having the jth atom in state A, B or C. For example,
    V (1) =
    ⎡
    ⎢⎣
    1
    1
    1
    ⎤
    ⎥⎦
    , V (2) =
    ⎡
    ⎢⎣2
    3
    2
    ⎤
    ⎥⎦
    , V (3) =
    ⎡
    ⎢⎣
    5
    7
    5
    ⎤
    ⎥⎦
    , · · · .
Created on Thu Dec 15 17:43:41 EST 2011

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