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Early Quantum Theory

Early Quantum Theory. Chapter 27. Discovery and Properties of the Electron. J.J. Thomson (1897) Cathode rays Nearly evacuated tube Anode (+) Cathode (-) Found e/m e/m = E/(B 2 r) Only one quantity was needed to find the other. Discovery and Properties of the Electron.

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Early Quantum Theory

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  1. Early Quantum Theory Chapter 27

  2. Discovery and Properties of the Electron • J.J. Thomson (1897) • Cathode rays • Nearly evacuated tube • Anode (+) • Cathode (-) • Found e/m • e/m = E/(B2r) • Only one quantity was needed to find the other

  3. Discovery and Properties of the Electron • Miliken’s Oil Drop Experiment • Charge occurred in discrete amounts • First evidence that charge is quantized • Balanced fine drops of oil that had been charged between charged plates • Calculated charge of drops when they reached terminal velocity • q = mg/E • e- = -1.6E-19 C • Used this and e/m value to get the mass of the electron (me = 9.11E-31 kg)

  4. Planck’s Quantum Hypothesis • Blackbody • Absorbs all radiation that falls on them • When heated, a blackbody emits radiation • Blackbody radiation is not necessarily black in color • Radiates continuous spectrum • Range of frequencies • Intensity varies with temperature • Intensity reaches that which eyes can see around 1000 K (red) 6000 K Intensity 3000 K Wavelength

  5. Planck’s Quantum Hypothesis • Wien’s Law • λpT = 2.90E-3 m*K • λp = peak wavelength (m) • T = temperature (K) 6000 K Intensity 3000 K Wavelength

  6. Example • Estimate the temperature of the surface of our Sun, given that the Sun emits light whose peak intensity occurs in the visible spectrum at around 500 nm.

  7. Example • Suppose a star has a surface temperature of 32500 K. What color would this star appear?

  8. Planck’s Quantum Hypothesis Cont. • Maxwell’s equations • Energy distributed among oscillating electric charges, or electrons, generated the electromagnetic radiation • Could not predict spectrum emitted by blackbody • Max Planck (1858-1947) • The distributed energy is not continuous • The energy of any molecular vibration is a multiple of a given number • E = nhf, where n = 1, 2, 3, … • E = energy (joules) • h = Planck’s constant = 6.626E-34 J*s • f = frequency of the oscillations (Hz) • n = quantum number

  9. Photon Theory of Light: Photoelectric Effect • Photoelectric Effect • Einstein (1905) • Light comes from radiating source • Light can be thought of as photons with each photon having an energy of E = hf • Photon = packet of light • Light that strikes a metal surface acts as a particle that knocks an electron from the surface of the metal

  10. Photon Theory of Light: Photoelectric Effect • Conservation of energy • Energy of the photon is converted into the work to remove the electron and the KE of the electron • hf = KE + W • hf = KEmax +Wo • Wo = Work function • Minimum energy required to remove an electron • Only valid for least bound electrons • If hf<Wo, then no electrons are ejected • If hf>Wo, then electrons are ejected

  11. Photon Theory of Light: Photoelectric Effect • Wave Theory • An increase in the intensity of light causes an increase in the number of electrons emitted and an increase in their maximum KE • The frequency of light should not affect the KE of the ejected electrons • Photon Theory • An increase in the intensity of light means more electrons are emitted, but the KE of each electron is not changed • An increase in frequency of light produces a higher maximum KE • If the frequency is less than the cut-off frequency, fo, then no electrons will be ejected.

  12. Example • Calculate the energy of a photon of blue light with a wavelength of 450 nm.

  13. Example • What is the kinetic energy and speed of an electron ejected from a sodium surface whose work function is 2.28 eV when illuminated by light of wavelength 410 nm?

  14. Energy, Mass, and Momentum of a Photon • Photons have zero mass • p ≠ mv • p = E/c = hf/c (c = fλ) • p = h/λ • h = Planck’s constant • λ = wavelength (m)

  15. Example • Estimate how many visible light photons a 100 W light bulb emits per second. Assume the bulb has a typical efficiency of 3% (that is, 97% of the energy goes to heat).

  16. Compton Effect • A.H. Compton (1892-1962) • Supported photon theory • Scattered light had a longer wavelength than incident light • Wave theory did not predict a shift in wavelength • λ’ = λ + [h/(moc)](1 – cosφ) • λ’ = shifted wavelength (m) • λ = incident wavelength (m) • h = Planck’s constant • mo = rest mass (kg) • φ = scattered angle

  17. Example • X-rays of wavelength 0.14 nm are scattered from a very thin slice of carbon. What will be the wavelengths of x-rays scattered at a) 0o b)90o c) 180o

  18. Photon Interactions; Pair Production • 4 types of interactions between photon and matter • Photoelectric effect • Photon disappears • Photon knocks an electron to a higher energy level in an atom • Photon disappears • Compton effect • Photon has a new frequency • Pair production: A photon creates matter • Photon disappears • Creates antimatter: conservation of charge

  19. Example • What is the minimum energy of a photon that can produce an electron-positron pair? • What is this photon’s wavelength?

  20. Wave-Particle Duality • Problem? • 2 theories of light • Wave • Particle • Which is correct? • Both according to Niels Bohr • Principle of Complementarity • For experiments, choose only one, but be aware that the other exists for a full picture of light

  21. Wave Nature of Matter • Louis de Broglie (1892-1987) • If light can act as a particle, particles can also act like waves • de Broglie Waves (1923) • λ = h/p • λ = wavelength (m) • h = Planck’s constant • p = momentum of particle (kg*m/s) • Valid for classical and relativistic physics • Not usually noticeable • Size of object ≈ wavelength • Very small masses • Diffraction patterns found with electrons • Confirmed wave-particle duality with all matter

  22. Example • Calculate the de Broglie wavelength of a 0.20 kg ball moving with a speed of 15 m/s.

  23. Example • Determine the wavelength of an electron that has been accelerated through a potential difference of 100 V.

  24. Early Models of the Atom • J.J. Thomson • Plum Pudding Model • First model with subatomic particles

  25. Early Models of the Atom • Rutherford • Gold foil experiment • Nucleus contains positive charge • Planetary, or nuclear, model

  26. Atomic Spectra • Heated gases at low pressure emit spectrum of radiation • Due to oscillations of atoms or molecules • Spectra is discrete, not continuous • If a continuous spectrum of light passes through the gas, certain wavelengths are absorbed • Emission/absorption due to single atoms

  27. Atomic Spectrum • Theory of atom • Must account for discrete spectrum • Begin with hydrogen • J.J. Balmer (1885) • Four visible lines in hydrogen • 1/λ = R(1/22 – 1/n2), n = 3, 4, 5, … • Λ = wavelength (m) • R = rydberg constant = 1.097E7 m-1 • Lyman series • 1/λ = R(1/12 – 1/n2), n = 2, 3, 4, … • UV range • Paschen series • 1/λ = R(1/32 – 1/n2), n = 3, 4, 5, … • Infrared range

  28. Bohr Model • Problems with Rutherford model • Could not explain discrete lines • Curved orbits are accelerating • Give off EM radiation • Lose energy • Spiral into nucleus • Continuous spectra • Unstable atoms

  29. Bohr Model • Quantization of atom • Electrons only allowed in discrete orbits • Electrons would not emit energy in these orbits • Violates classical mechanics • Stationary states • Photons emitted if electron moves to lower energy level • hf = Eu – El • Eu = upper energy • El = lower energy

  30. Bohr Model • Angular momentum is quantized • L = nh/(2π), where n = 1, 2, 3, … • L = angular momentum • Agreed with experimental results of Balmer • Bohr radius • rn = (n2/Z)r1 • rn = radius of nth orbit • n = principle quantum number • Z = number of protons • r1 = Bohr radius = 0.529E-10 m

  31. Bohr Model • Hydrogen • rn = n2r1 • Energy of states • E = KE + PE • En = (Z2/n2)E1 • En = Energy of nth state • E1 = energy of ground (lowest) state • Hydrogen • E1 = -2.17E-18 J = -13.6 eV • En = -13.6 eV/n2 • Why negative? • Derivation of Balmer equation • Bohr model is unsuccessful with heavier atoms • Only seems to work with hydrogen 0 eV n=∞ n=4 -0.85 eV n=3 -1.5 eV Paschen Series n=2 -3.4 eV Balmer Series n=1 -13.6 eV Lyman Series

  32. Example • Determine the wavelength of the first Lyman line, which is the transition from n = 2 to n = 1. In what region of the electromagnetic spectrum does this lie? 0 eV n=∞ n=4 -0.85 eV n=3 -1.5 eV Paschen Series n=2 -3.4 eV Balmer Series n=1 -13.6 eV Lyman Series

  33. Example • Determine the wavelength of light emitted when a hydrogen atom makes a transition from the n = 6 to the n = 2 energy level according to the Bohr model.

  34. Example • Determine the maximum wavelength that hydrogen in its ground state can absorb. What would be the next smaller wavelength that would work? 0 eV n=∞ n=4 -0.85 eV n=3 -1.5 eV Paschen Series n=2 -3.4 eV Balmer Series n=1 -13.6 eV Lyman Series

  35. Example • Use the Bohr model to determine the ionization energy of the He+ ion, which has a single electron. • Calculate the minimum wavelength a photon must have to cause ionization.

  36. de Broglie’s Hypothesis • Electrons have particle-wave duality • Bohr’s theory explained experimental results with the electron as a particle • Could not explain why orbits were quantized or ground state • de Broglie • λ = h/mv • Each electron orbit is a standing wave • Waves persist only if circumference of the orbit contains a whole number of wavelengths • Matches Bohr model • Orbit theory eventually replaced with quantum mechanical theory of electron cloud

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