Chapter 7: The Quantum-Mechanical Model of the Atom - Full Chapter Notes PDF

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Chapter 7: The Quantum-Mechanical Model of the Atom - Full Chapter Notes - 2021/2022 academic year - CHEM 1000H Introduction to Chemistry - September 21 2021 (includes seminar)...

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Chapter 7: The Quantum-Mechanical Model of the Atom 7.1: Quantum Mechanics and Light ● Quantum Mechanics ○ Central to our conception of molecular structure and “atomic scale” Matter ○ Key concepts: ■ Particles exhibit wave properties: electrons diffract like waves ■ Waves exhibit particle properties: light absorbed and emitted by matter in discrete units ■ Matter exhibits non-continuous energy when confined to small regions of space: electrons are found to have discrete energies = energy quantization ■ Indeterminacy: there is a limit to how precisely you can simultaneously measure some things like position and energy ● The Nature of Light: A Particle Wave ○ Light exhibits “wave-particle” duality: ○ Hence, light acts as an electromagnetic (EM) wave: ■

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Light as a Wave: Diffraction ○ Waves bend around an object of a size similar to its wavelength: ■ - Plane waves of light diffract after passing through a slit the widths of the light’s wavelength - Numerical approximation of diffraction pattern from a slit of width equal to wavelength of an incident plane wave in 3D spectrum visualization

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Light as a Wave: Interference ○ Light waves “add up” when they overlap, to make a new “combined” wave: ■ - Two diffracted “wave fronts” interfere to make regions of high and low amplitude or intensity

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Lights as a Particle: Photoelectric Effect ○ Phenomena like “The Photoelectric Effect” turned physics thinking upside down at the dawn of the 20th century: ■

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The Particle Nature of Light ○ The photoelectric effect could not be understood “classically” since: ■ A more intense low-frequency light should eventually have enough energy to free electrons, not seen ■ Light that does generate electrons should give faster electrons when intensity is increased, but instead just gave more electrons ○ Only by thinking of light as being comprised of particles of a fixed energy could the experiment be understood Particle Nature of Light Explanation ○

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If the intensity of light in a photoelectric effect experiment is doubled, twice as many electrons will be ejected per second The Energy of a Photon ○ The particle properties of light give rise to the Planck-Einstein Relation ■

The Planck-Einstein Relation in Disguise ○ Since frequency and wavelength are related through c = λv, we also have that: ■

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Later, it will prove useful to define “wavenumber”, the reciprocal of wavelength, so that: ■

7.2: Spectroscopy ● Excited States of Atoms ○ Electrons in atoms exist in specific configurations of different energies ○

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In both theories these are called “stationary states”, the “allowed” energy states of an atom or molecule Atomic Absorption ○ During a photon absorption, the photon is annihilated and an electron “jumps” to a higher-energy orbital and a new stationary state is formed ■

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When a photon is absorbed by an atom, its energy is converted to electric potential energy Atomic Emission Spectra ○ At some point in time later, this energy may be released in the form of a new photon: ■

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The energy of the photon this generated must equal to the difference between the energy of the two stationary states: ■

The Bohr Frequency Condition ○

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Since from the Planck-Einstein relation: ■

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Bohr proposed that: ■

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Spectroscopy ○ The wavelength of absorbed and emitted photons by matter gives the energy difference between QM energy states of atoms and molecules ○ The is the basis of “Spectroscopy”, the study of light absorption and emission to determine the energy spectrum of a system: ■

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The Bohr Model of the Atom ○ Bohr developed a model for the structure of the atom that gives the right answer for one-electron atoms such as H, He+, Li2+, etc. ○ Although conceptually naive, it gave the correct expression for the energy of stationary states of H-atoms, and quantitatively accounted for the wavelength of emission line in the H-atom spectrum ■

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The Bohr Model and H-atom Emission ○ H-atom electronic emission spectra: ■

7.3: The Schrodinger Equation ● Wave Properties of Matter: A Wave Particle ○ Early in the 20th century, it was recognized that matter had both wave and particle properties ○ For example, it was observed by Davisson and Germer in 1927 that electrons “diffracted”, a wave property. ○ Modern version result with a “one-at-a-time” apparatus: ■

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Since electrons are detected one by one as particles, we have to conclude that each electron must have passed through at random on either side of the biprism, this creating a uniform distribution, without any interference when accumulated The Dawn of Quantum Mechanics ○ de Broglie postulated that free particles have an intrinsic wavelength, expressed as: ■

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The de Broglie Relationship ○

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The Schrodinger Equation ○ Legend has the someone at a conference in the 1920’s posited that, if matter has wave properties, there should be a wave equation to describe it ○ Legend also has the Erwin Schrodinger postulated just such an equation: ■ Ĥψ = Eψ ○ Solutions to the schrodinger equation are: ■

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The Wavefunction ○ Solutions to the Schrodinger Equation are mathematical function called “wave functions” ■

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A stationary state wavefunction has a fixed shape, but an amplitude and phase that vary in space and time ○ We usually see stationary state wave functions as a “snapshot” in time, without the time variation of amplitude or phase Sine Functions ○ The “sine” function has the required differentiation property: ■

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So, a general solution is: ■ Ψ = Asinkx ○ A and k are determined by “boundary conditions” Boundary Conditions ○ The wave function must be 0 at the “ends of the box”, i.e. Ψ(x=0) and Ψ(x=a) = 0 ■ Ψ(x=0) = Asink0 = A ⋅ 0 = 0 ■ Ψ(x=a) = Asinka ○ This second condition can only be true if A = 0 (no wave function at all) or when sin(ka) = 0 (i.e. when ka = nπ n = 1, 2, 3, …) The Solution ○

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The Energy Level Diagram ○

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Electron Energy Equations ○ Solving the Schrodinger equation always yields one or more statements of the particle’s energy ○ Generally a function of: ■ Integer “quantum numbers” ■ Physical constants ○ For example, the “allowed energies” of a QM oscillator are: ■

7.4: Wavefunctions ● The Wavefunctions ○

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Probability Distributions ○ Wavefunctions are not themselves directly physically interpretable ○ Max Born suggested that the correct interpretation of the wavefunction’s significance was that: ■ Ψ2 = Probability of finding a particle in a particular region of space per unit of volume The Particle in a Box Probability Distributions ○

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General Conclusions ○ This solution reflects these general facts about QM systems ○ Energy is quantized, and may only be found to be equal to the “eigenvalues” of the Schrodinger Equation for the system ○ The probability of finding the particle at a location in space is given by the square of the wavefunction at that point The H-atom: A Particle in a Spherical Coulomb Field ○ The hydrogen atom is the simplest atomic quantum mechanics problems ○ It is the “particle in a spherical centrosymmetric (Coulomb) field” problem Atomic Wavefunctions ○ The wave function solutions to the Schrodinger Equation for atoms are called Atomic Orbitals ○ Orbitals are mathematics functions that describe the properties of a single electron in an atom or molecule The Hydrogen Atom Orbitals ○ The H-atom has several quantum numbers associated with its solution: ■

The Principal Quantum Number “n” ○ “N” determines the total energy of the electron, and the size of the orbital The Angular Momentum Quantum Number “l” ○ “L” determines the shape of the orbital, and is usually symbolized by a letter: ■

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“l” also determines how much of the electron’s energy is due to angular motion

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The Magnetic Quantum Number “ml” ○ “Ml” determines the orientation of the orbital in space, and is also indicated by a function or x, y, and/or z ■

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What Atomic Orbitals “Look Like” ○ Combining the radial (n), angular (l), and magnetic (ml) parts of the wave function, we get the complete orbitals of the H-atom ■

7.5: Phase and Spin ● Phase ○ The wave function has positive and negative amplitude: ■

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The sign of the amplitude is called the phase The time-dependence of wave functions shows that these phases oscillate in time

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○ Phase is a “relative” difference, like a snapshot of an oscillating classical wave “Relative” Phases of a Classical Wave ○

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Orbital Phase ○ The phase of atomic orbitals is usually indicated with either colour or a +/- sign: ■

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The Stern-Gerlach Experiment (1921) ○

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Ground-state silver atoms ([Kr]4d105s1) passed through an inhomogeneous magnetic field deflect in one of two directions

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Deflection of a particle in such a field demonstrates that the particle has a magnetic moment, which causes it to follow one of two directions in the field ○ Magnetic moment must be due to some form of angular momentum ○ Problem: Silver atom electrons have no net angular momentum arising from rotation in orbital, Ag: [Ar]4d 105s1 ○ Fact: All paired electrons in closed shells have no net orbital angular momentum, since the OAM of each electron is exactly cancelled by that of the paired electron ■ Filled orbitals → no net (orbital) angular momentum ○ Silver atoms have all closed shells (the [Ar]4d10 part) plus an electron in the 5 ”s” (l = 0, i.e. no angular momentum) orbital ○ Angular momentum of silver atom arises from some “intrinsic” source Classical Analog ○ The only classical analog that provides a consistent picture of the properties of this phenomenon is the concept of a charged sphere rotating about an axis ○ This rotational motion generates a magnetic moment aligned along the axis around which the object rotates, hence “spin” Classical Magnetic Moment Due to Spinning Charged Sphere ○

What We Know About QM Spin: ○ We know nothing about what “causes” spin, or what it is ○ However, we do know some things about spin AM ○ For electron spin, we have two new quantum number, which have similar properties to the quantum numbers ‘l’ and ‘m l’ (Orbital AM) ■ The spin angular momentum quantum number: ● s = ½ (the only value) ■ The spin magnetic quantum number: ● ms = s, s-1, s-2, … -s = ½, -½ In Words, Spin Properties: ○ All electrons have the same amount of total spin angular momentum ○ The electrons can take only one of two orientations with respect to an applied magnetic field: ■

7.6: Many-Electron Atoms ● Atoms With Many Electrons ○ The concepts and solutions to the one-electron H-atom problem are used to solve the “many-electron atom” problem ○ Electrons are assigned “one-electron wave-function” or orbitals, and an electronic state description is built by a product of these orbitals ■

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The “Orbital” Approximation: A wave function for a multielectron atom or molecule is constructed from a combination of one-electron wave functions (orbitals) Shielding in Multi-Electron Atoms ○ The nuclear charge “felt” by an electron is diminished by the presence of other electrons ○ The extent of “shielding” felt by an electron in different types of orbitals follows a pattern: ■ s...