Chapter 2. Atomic Structure

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Chapter 2. Atomic Structure (????)

• Historical Development of Atomic Theory
• In 1808. John Dalton Atomic theory
• All matter is composed of small particles called
  atoms.
• An atom (??) indivisible, retain its identity
  during chemical reactions
• An element (??) A type of matter of only one
  kind of atom, e.g. hydrogen (H), oxygen (O)
• There are sub-atomic particles in the atoms.
• electrons B. neutrons C. protons
• In 1869 Dmitri Mendeleev (Russian)
• J. Lothar Meyer (German)
• ? Periodic Table of Elements (?????)

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Chapter 2 Atomic Structure

• Basic Quantum Mechanics (????)
• Subatomic Particles
• The Electron
• J. J. Thomson credited with the discovery in 1897
• Previously, atoms were believed to be the
  smallest particles
• William Morgan (1785)passed current through a
  vacuum
• Produced a glow
• Luigi Galvani (1800)first produced constant
  current from a battery
• Cu (wet salt solution) Sn Ecell 0.48 V
• Michael Faraday (1836)determines glow comes from
  the cathode
• Eugene Goldstein (1876)coins the term cathode
  ray
• Thomson finds electron is negatively charged with
  1/1836 mass of H

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• Thomsons Experiment
• Repulsion from negative pole of an electric field
  meant that the cathode rays must be negatively
  charged
• The amount of deflection was a function of the
  mass of the ray
• Since many different metals all produced the same
  cathode rays, all atoms must be made up of the
  same /- particles
• G. F. FitzGerald renames cathode rays as
  Electrons

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• The Nucleus
• Ernest Rutherford credited with discovering the
  Nucleus in 1911
• He nuclei (a particle) were deflected as they
  passed through a Gold foil
• Conclusion Heavy, tiny nucleus and much empty
  space in an atom

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• The Proton
• Rutherford and Moseley quickly discovered the
  charge of the nucleus by X-ray emission in 1913
• Z the nuclear charge (or the atomic number)
• Atomic Spectra
• Balmer described the emission spectrum of H in
  1885
• .
• .
• n principle quantum number discrete allowed
  energies for H electrons

nh integer gt 2 quantum numbers RH
Rydberg constant 1.097 x 107 m-1

• h Plancks constant 6.626 x 10-34 J s
• frequency of light s-1
• c speed of light 2.998 x 108 m/s
• wavelength nm
• wavenumber cm-1

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• Bohrs Quantum Theory of the Atom (1913)
• Negative electrons move in stable, circular
  orbits around positive nuclei
• Electrons absorb or emit light by moving out or
  moving in to other orbits
• Bohr replaced Balmers equations with better ones
• Energy levels are far apart at small n, close
  together at large n
• n infinity if the nucleus and electron are
  completely separate
• Only worked for H-atom not a complete
  description of atomic structure

m reduced mass E electron charge Z nuclear
charge 4peo permittivity of vacuum
1.097 x 107 m-1 2.179 x 10-18 J
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Hydrogen Atom Energy Levels
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• Wave nature of particles de Broglie (1924)
• De Broglie Equation relates particles to a
  property of waves
• Massive (visible) particles have short
  wavelengths that cant be observed we call these
  classical particles that obey classical mechanics
• Small particles (electrons) have observable wave
  properties
• Electrons around nuclei are like standing waves
• Heisenbergs Uncertainty Principle (1927)
• Heisenbergs equation relates position and
  velocity
• The better you know velocity (p), the worse you
  can know position (x)
• Dp , then Dx and vice versa
• Cant exactly describe e- orbits, only orbitals,
  which are regions of space where of high
  probability of finding e- in it

h Plancks constant m mass of the particle v
velocity of the particle
p momentum mv
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• The Schrödinger Equation
• Describes wave properties of electrons position,
  mass, total E, potential E
• Y wave function describes an electron wave in
  space describes atomic orbital
• The Schrödinger Equation
• When H is carried out on Y , the result is E
  times Y (E is a constant)
• Different orbitals have different Y and
  consequently, different Es
• The Hamiltonian Operator H

H the Hamiltonian Operator calculus operation
on Y E energy of the electron
Kinetic Energy part
Potential Energy part
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• Definitions
• h Plancks constant
• m mass of the particle
• E total energy of the system
• e charge of electron
• (x2 y2 z2)1/2 r distance to nucleus
• Z charge of the nucleus
• 4peo permittivity of vacuum
• Applied to Y
• V potential E electrostatic attraction
  between electron/nucleus
• Define attractive force negative energy
• Closer to nucleus large V farther from
  nucleus small V

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• There are any number of solutions to the
  Schrödinger equation, each describing an electron
  in an atomic orbital 1s, 2s, 2px, 2py.
• The meaning of Y
• Y has no physical meaning itself, it is just a
  mathematical expression
• Y2 probability of finding the electron at a
  given point in space
• If you consider all space, Y2 100
• Consider where the electron is 90 of the time,
  Y2 atomic orbital (1s)
• Physical Reality imposes some conditions on what
  Y can be
• Y must have only one value because a given
  electron only has one energy
• b) Y and dY must be continuous because the
  electron cant jump
• c) Y ----gt 0 as r ----gt infinity because the
  probability must decrease farther away
• d) The total probability of finding the electron
  somewhere must 1
• Normalization setting values to 1
• 8) All orbitals are orthoganal (perpendicular)

Y is used to make imaginary Y real just use Y2
if not imaginary dt dx dy dz
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• Applications of the Schrödinger Equation
• A Particle in a Box
• A one-dimensional box is a simple case for using
  the Schrödinger Equation
• V(x) 0 inside the box, between x 0 and x a
• V(x) infinity outside the box
• The Particle can never leave the box
• Apply the Schrödinger Equation
• Solve the Schrödinger Equation
• sine/cosine describe waves, so we will make a
  combination of these functions
• Substitute into the Schrödinger Equation and
  solve for r and s

A,B,r,s constants
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• Apply conditions of the particle in a box
• Y must go to 0 at x 0 and x a
• cos sx 1 for x 0
• Y 0 only if B 0
• Y A sin rx
• At x a, Y 0, so sin ra 0
• Only possible if ra np (n integer)
• ra np
• Keep only the values (/- give same results)
• Substitute and solve for E

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• These are the energy levels predicted by the
  Particle in a Box Solutions
• Quantized n 1,2,3,4.
• Substitute r np/a into the wave equation
• Normalization leads to a more complete solution
• Total solution
• Plotting the wave functions
• Classical Mechanics particle has equal
  probability anywhere in the box
• Quantum Mechanics high and low probabilities at
  different locations

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Wave Functions and their Squares for the Particle
in a Box with n 1, 2, and 3