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Ch 27 1 Chapter 27 Early Quantum Theory and Models of the Atom © 2006, B.J. Lieb Some figures electronically reproduced by permission of Pearson Education, Inc., Upper Saddle River, New Jersey Giancoli, PHYSICS,6/E © 2004.

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Ch 27 2 Properties of the Electron Called “cathode ray” because appeared to come from the cathode J. J. Thomson did the first experiments to discover its properties and received Nobel Prize for work. First discovered in experiments where electricity is discharged through rarefied gases. First experiments measured charge /mass. Another experiment measured e and thus mass could be determined

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Ch 27 3 Electron Charge to Mass Ratio Magnetic Force = Centripetal Force The electric field is adjusted until it balances the magnetic field and thus eE = evB. This gives This is a velocity selector since all electrons that pass through have the same velocity. The final equation is

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Ch 27 4 Blackbody Radiation Intensity vs. wavelength is shown above Radiation emitted by any “hot” object In Ch 14, we learned Intensity T 4 The wavelength of the peak of the spectrum P depends on the Kelvin temperature by

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Ch 27 5 Planck’s Quantum Hypothesis Attempts to explain the shape of the blackbody curve were unsuccessful In 1900 Max Planck proposed that radiation was emitted in discrete steps called quanta instead of continuously He did not see this as revolutionary Introduced a new constant-now called Planck’s constant h =6.626 x J s = 4.14 x eV·s

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Ch 27 6 Planck’s Quantum Hypothesis We can understand quantized energy by considering the energy of a box on stairs vs. box on a ramp. E = m g y If the height of each step is Δ y, can you derive an equation for the quantized potential energy of the box on the steps. E = m g ( n Δ y ) where n is an integer.

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Ch 27 7 Photoelectric Effect Light shines on a metal and electrons (called photoelectrons ) are given off Easy to measure kinetic energy of electrons There was a threshold frequency below which no electrons were emitted

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Ch 27 8 Photoelectric Effect Wave theory predicts that: Number of electrons Intensity Maximum electron kinetic energy intensity Frequency of light should not affect kinetic energy No threshold frequency This can not explain the photoelectric effect

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Ch 27 9 Photon Theory of Light In 1905 Einstein proposed that in some experiments light behaved like particles instead of waves Light consisted as stream of photons, each with energy: Where h is Planck’s constant Each photon had wavelike properties

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Ch Explanation of Photoelectric Effect Work function W 0 is the energy necessary to free the least tightly bound electron A single photon with energy hf gives all of this energy to a single electron A photon with frequency below the threshold lacks sufficient energy to free the electron, so hf 0 = W 0 This electron thus escapes from the metal with kinetic energy

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Ch Photons where h is Planck’s Constant The rest mass of a photon must be zero because it travels at the speed of light The photon has momentum (p) because if we substitute m 0 = 0 in We get the following equation and thus The energy of a photon is given by

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Ch Example 27-1 (20) In a photoelectric effect experiment it is observed that no current flows unless the wavelength is less than 570 nm. What is the work function of the material? What is the stopping voltage if light of wavelength 400 nm is used? So the stopping voltage is 0.93 volts.

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Ch Evidence for Photon Nature of Electromagnetic Radiation Compton Effect: photon scatters off of electron, photon looses energy and electron gains energy. This effect shows that momentum and energy is conserved.

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Ch Further Evidence for Photon Nature of Electromagnetic Radiation Pair Production: a photon passing a nucleus is converted into an electron-positron pair. (A positron is a positive particle with all the other properties of an electron.) Since the mass of the electron is 0.511MeV/c 2 and two electrons must be produced, the kinetic energy shared by the two electrons is

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Ch Electron Microscopes Electrons accelerated through 100,000 V have 0.004nm and can achieve a resolution of 0.2 nm which is a factor of 1000x better than optical microscopes. Use magnetic lenses to focus electron beam. Scanning Tunneling Microscope has a probe that moves up and down to maintain a constant tunneling current.

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Ch Thomson Model of the Atom J.J. Thomson Model: had negatively charged electrons inside a sphere of positive charge. Assumed that the electrons would oscillate due to electric forces. An oscillating charge produces electromagnetic radiation which should match agree with atomic spectra.

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Ch Rutherford Experiment Ernest Rutherford performed an experiment to probe the structure of the atom. He aimed a beam of alpha particles at a thin gold foil and measured how they were scattered. Alpha particle is the nucleus of a He atom (two protons and two neutrons) and thus was positively charged. Alpha particles are emitted by a radioactive nucleus.

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Ch Rutherford Experiment Results Most alpha particles passed through foil without scattering A few were scattered through large angles

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Ch Rutherford Experiment Results Concluded that foil was mainly empty space with some small but massive concentrations of positive charge. An alpha particle that happened to pass near a nucleus was repelled without ever touching the nucleus. Rutherford proposed a positive heavy nucleus with radius of m with electrons in orbit Problem was electrons should radiate energy away.

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Ch Hydrogen Spectra The visible spectra from hydrogen gas has a very distinctive pattern that can be represented by the Balmer formula where R = x10 -7 m -1.

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Ch Hydrogen Spectra The Balmer formula could also be modified to fit the Lyman series that was discovered in the ultraviolet And the Paschen series in the infrared

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Ch Bohr Model of the Atom Niels Bohr was a Danish physicist who studied at the Rutherford lab. He decided to try to add the quantum effects of Planck and Einstein to the Rutherford planetary model of the atom He knew that the answer had to be the Balmer formula but the task was to develop a set of assumptions that would lead to it. Problem was that a charged particle in orbit is like an antenna--it should emit radiation and gradually loose energy until it fell into the nucleus The discrete wavelengths emitted by hydrogen suggested a quantum effect as in the stair example

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Ch Bohr’s Assumptions Bohr was like a student who looked up the answer in the back of the book and needed to find a way to get that answer He said electron could remain in possible orbit called a stationary state without emitting any radiation Each stationary state is characterized by a definite energy E n When electron changes from the upper to the lower stationary state (or orbit) it emits a photon of energy equal to the difference in the states:

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Ch Bohr’s Assumptions Radiation is only emitted when an electron changes from one stationary state to another He found he could derive the Balmer formula if he assumed that the electrons moved in circular orbits with angular momentum (L) satisfied the following quantum condition: and thus

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Ch Bohr Radius Z is the number of protons so Q nucleus = Ze The electrical force equals the centripetal force

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Ch Bohr Energy Levels The electric potential of the nucleus is With potential energy Substituting Bohr radius equation and values gives

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Ch Summary of Bohr Model Electrons obit in stationary states that are characterized by a quantum number n and a discrete energy E n. Sometimes this is called a energy level. E n is negative indicating a bound electron At room temperature, most H atoms have their electron in the n=1 energy level When electron changes to a lower n it emits a photon of energy equal to the energy difference. Electron must be given energy to move to a higher n This formula can be used for any single electron atom or ion such as a singly-ionized He ion in which case Z=2. The radius of the orbit is given by

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Ch Energy Level Diagram Red arrows indicate transitions where electron emits a photon and moves to a lower state Vertical scale is energy.

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Ch Example 27-2A. A hydrogen atom initially in its ground state (n=1) absorbs a photon and ends up in the n=3 state. Calculate the energy and wavelength of the absorbed photon. First calculate the energy of the first three states.

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Ch Example 27-2B. A hydrogen atom initially in its ground state (n=1) absorbs a photon and ends up in the n=3 state. Calculate the energy and wavelength of the absorbed photon. When the atom returns to the ground state, what possible energy photons could be emitted?

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Ch Example Singly ionized 4 He consists of a nucleus with two protons and two neutrons with a single electron in orbit around this nucleus. Use the Bohr model to calculate the energy of a photon that is emitted when the electron goes from the first excited state to the ground state of singly ionized 4 He. Calculate the ionization energy of singly ionized 4 He. Ionization energy = Could you use the Bohr model for atomic 4 He?

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Ch Wave Nature of Particles Louis de Broglie proposed that if light had a wave-particle duality, then perhaps particles, such as electrons, also had a wave nature. He assumed that the following equation for photons also applied to electrons

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Ch Diffraction of Electrons Later experiments showed that a beam of electrons was diffracted just like light.

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Ch De Broglie Hypothesis and Hydrogen De Broglie was able to give a reason for the Bohr quantum hypothesis by assuming that allowed electron orbits had to be in standing waves around orbit. Circumference =2 r n = n where n = 1, 2, 3... and we get the following equation which was Bohr’s original quantum assumption. Combine with

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