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Quantum Mechanics and Atomic Theory

Quantum Mechanics and Atomic Theory. Wave models for electron orbitals. Electromagnetic Radiation. Radiant energy that exhibits wavelength-like behavior and travels through space at the speed of light in a vacuum. Waves. Waves have 3 primary characteristics: 1. Wavelength :

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Quantum Mechanics and Atomic Theory

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  1. Quantum Mechanics and Atomic Theory Wave models for electron orbitals

  2. Electromagnetic Radiation • Radiant energy that exhibits wavelength-like behavior and travels through space at the speed of light in a vacuum.

  3. Waves • Waves have 3 primary characteristics: • 1. Wavelength: • 2. Frequency: • 3. Speed: speed of light is 2.9979  108 m/s.

  4. The Nature of Waves

  5. Wavelength and frequency can be interconverted. •  = frequency (s1) • = wavelength (m) • c = speed of light (m s1)

  6. Planck’s Constant Transfer of energy is quantized, and can only occur in discrete units, called quanta. • E = change in energy, in J • h = Planck’s constant, 6.626  1034 J s • = frequency, in s1 • = wavelength, in m

  7. Energy and Mass • Einstein- When a system loses energy, it loses mass • m = E/c2 • E = energy • m= mass • c = speed of light

  8. Energy and Mass (Hence the dual nature of light.) Einstein’s calculations show that photons do exhibit momentum, and are affected by gravity. However, it is important to recognize that the photon is in no sense a typical particle. A photon has mass only in a relativistic sense- it has no rest mass.

  9. Electromagnetic Radiation

  10. Dual Nature of Light • Energy is quantized. It can be transferred only in discrete units called quanta. • Electromagnetic radiation, which was previously though to exhibit only wave properties, seems to show certain characteristics of particulate matter as well.

  11. Wavelength and Mass- Do all particles exhibit wave properties? de Broglie’s Equation- relates the wavelength of a particle to its momentum. •  = wavelength, in m • h = Planck’s constant, 6.626  1034 J s = kg m2 s1 • m = mass, in kg • = frequency, in s1

  12. Problem 12.2, pg. 509de Broglie’s Equation • Compare the wavelength for an electron (mass = 9.11 X 10-31 kg) traveling at a speed of 1.0 X 107 m/s with that for a baseball (mass = 0.10 kg) traveling at 35 m/s.

  13. The Constructive and Destructive Interference of Waves Diffraction results when electromagnetic energy (light) is scattered from a regular array of points or lines.

  14. Confirmation of Wavelike Properties of Electrons • 1927- Davisson and Germer at Bell Labs observed diffraction pattern with a beam of electrons directed through a nickel crystal similar to that seen from the diffraction of X-rays (electromagnetic radiation). X-ray diffraction has been a useful tool in understanding the structure of solids. The lattice of atoms in a molecular crystal serves as a series of barriers and openings that diffracts X rays as they pass through. The diffracted X rays form an interference pattern that can be used to determine the spacing of atoms in the crystal.

  15. Atomic Spectrum of Hydrogen • Continuous spectrum: Contains all the wavelengths of light. • Line (discrete) spectrum: Contains only someof the wavelengths of light.

  16. The Bohr Model- Quantum Model for the Hydrogen Atom The electron in a hydrogen atom moves around the nucleus only in certain allowed circular orbits. Quantized! • E= energy of the levels in the H-atom • z= atomic number (for H, z = 1) • n = an integer- orbit radius

  17. The Bohr Model E = Efinal stateEinitial state By combining Einstein’s and de Broglie’s equations, the wavelength of light emitted when and electron moves can be predicted (calculated).

  18. Problem 12.3 pg. 514Bohr Model Calculations • Calculate the energy required to excite the hydrogen electron from level n=1 to level n=2. Also, calculate the wavelength of light that must be absorbed by a hydrogen atom in its ground state.

  19. Bohr’s Model Fails • Bohr’s model predicted emitted wavelengths for the movement of the one electron in the hydrogen atom. His calculations fail when applied to other atoms. Therefore, something is missing from Bohr’s interpretation of the atom.

  20. Quantum Mechanics • Erin Schrodinger equation- based on the wave propertiesof the atom •  = wave function • = mathematical operator • E = total energy of the atom • A specific wave function is often called an orbital.

  21. The Standing Waves Caused by the Vibration of a Guitar String Fastened at Both Ends The Hydrogen Electron Visualized as a Standing Wave Around the Nucleus

  22. Heisenberg Uncertainty Principle • x= position • mv = momentum • h = Planck’s constant • The more accurately we know a particle’s position, the less accurately we can know its momentum.

  23. Example Problem 12.5, pg 52The Heisenberg Uncertainty Principle • The hydrogen atom has a radius on the order of 0.05 nm. Assuming that we know the position of an electron to an accuracy of 1% of the hydrogen radius, calculate the uncertainty in the velocity of the electron. Then compare this value with the uncertainty in the velocity of a ball of mass 0.2 kg and radius 0.05 m whose position is known to an accuracy of 1% of its radius.

  24. Probability Distribution Probability Distribution for the 1s Wave Function • square of the wave function • probability of finding an electron at a given position Radial probability distribution is the probability distribution in each spherical shell.

  25. Radial Probability Distribution

  26. Two Representations of the Hydrogen 1s, 2s, and 3s Orbitals

  27. The Boundary Surface Representations of All Three 2p Orbitals

  28. The Boundary Surfaces of All of the 3d Orbitals

  29. Representation of the 4f Orbitals in Terms of Their Boundary Surfaces

  30. Quantum Numbers (QN) • Principal QN(n = 1, 2, 3, . . .) - • Angular Momentum QN(l = 0 to n 1) – • Magnetic QN(ml = l to l) - • 4.Electron Spin QN(ms= +1/2, 1/2) -

  31. Quantum Numbers • Determine the quantum numbers from the outermost electron in each of the following: • An atom of Ne • An atom of S • Mg2+ ion • O2- ion

  32. Pauli Exclusion Principle

  33. Aufbau Principle • As protons are added one by one to the nucleus to build up the elements, electrons are similarly added to these hydrogen-like orbitals.

  34. Hund’s Rule • When an electron is added to a subshell, it will always occupy an empty orbital if one is available.

  35. Diamagnetism and Paramagnetism • Diagmagnetic • Paramagnetic Orbital Notation of Phosphorous- Diamagnetic or Paramagnetic?

  36. Valence Electrons The electrons in the outermost principle quantum level of an atom. Inner electrons are called core electrons.

  37. Broad Periodic Table Classifications • Representative Elements(main group): filling s and p orbitals (Na, Al, Ne, O) • Transition Elements: filling dorbitals (Fe, Co, Ni) • Lanthanide and Actinide Series(inner transition elements): filling 4fand 5forbitals (Eu, Am, Es)

  38. The Orbitals Being Filled for Elements in Various Parts of the Periodic Table

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