Lecture PowerPoints. Chapter 27 Physics: Principles with Applications, 7th edition Giancoli
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1 Lecture PowerPoints Chapter 27 Physics: Principles with Applications, 7th edition Giancoli This work is protected by United States copyright laws and is provided solely for the use of instructors in teaching their courses and assessing student learning. Dissemination or sale of any part of this work (including on the World Wide Web) will destroy the integrity of the work and is not permitted. The work and materials from it should never be made available to students except by instructors using the accompanying text in their classes. All recipients of this work are expected to abide by these restrictions and to honor the intended pedagogical purposes and the needs of other instructors who rely on these materials.
2 Chapter 27 Early Quantum Theory and Models of the Atom
3 Contents of Chapter 27 Discovery and Properties of the Electron Blackbody Radiation; Planck s Quantum Hypothesis Photon Theory of Light and the Photoelectric Effect Energy, Mass, and Momentum of a Photon Compton Effect Photon Interactions; Pair Production
4 Contents of Chapter 27 Wave-Particle Duality; the Principle of Complementarity Wave Nature of Matter Electron Microscopes Early Models of the Atom Atomic Spectra: Key to the Structure of the Atom The Bohr Model de Broglie s Hypothesis Applied to Atoms
5 27-1 Discovery and Properties of the Electron In the late 19 th century, discharge tubes were made that emitted cathode rays. These rays are emitted at the cathode, or negative terminal. In 1879 Crookes had proposed that the cathode rays were 'radiant matter', negatively charged particles that were repelled from the negatively charged cathode and attracted to the positively charged anode. The nature of the cathode rays was controversial. Although Thomson thought the rays must be particles, many Europeans thought they were waves. In Germany Hertz had observed the rays passing through thin sheets of gold. It seemed impossible that particles could pass through solid matter.
6 27-1 Discovery and Properties of the Electron J.J. Thomson realized that he could deflect the cathode rays in an electric field produced by a pair of metal plates. One of the plates was negatively charged, while the other was positively charged. The cathode rays moved towards the positively charged plate so the rays must be negatively charged. A current in a coil of wire produces a magnetic field. Two coils will produce a uniform magnetic field. A beam of charged particles passing through the magnetic field will be bent. The magnetic field produces a force which deflects the cathode rays. Thomson positioned the coils so that the deflection was in the opposite direction to the deflection produced by the electric field. By adjusting the strengths of the fields the rays could be deflected, in one direction by the electric field, and back an equal amount by the magnetic field. The forces were balanced.
7 27-1 Discovery and Properties of the Electron By accelerating the rays through a known potential and then measuring the radius of their path in a known magnetic field, the charge to mass ratio could be measured: The result is e/m = C/kg. (27-1)
8 27-1 Discovery and Properties of the Electron Cathode rays were soon called electrons. Millikan devised an experiment to measure the charge on the electron by measuring the electric field needed to suspend an oil droplet of known mass between parallel plates.
9 27-1 Discovery and Properties of the Electron The mass and charge of each droplet were measured; careful analysis of the data showed that the charge was always an integral multiple of a smallest charge, e.
10 27-1 Discovery and Properties of the Electron The currently accepted value of e is: e = C Knowing e allows the electron mass to be calculated: m e = kg
11 27-2 Blackbody Radiation; Planck s Quantum Hypothesis We shall now turn to another puzzle confronting physicists at the turn of the century (1900): just how do heated bodies radiate? There was a general understanding of the mechanism involved heat was known to cause the molecules and atoms of a solid to vibrate. What is meant by the phrase black body radiation? The point is that the radiation from a heated body depends to some extent on the body being heated. All objects emit radiation whose total intensity is proportional to the fourth power of their temperature. This is called thermal radiation; a blackbody is an object that emits thermal radiation only. The spectrum of blackbody radiation has been measured; it is found that the frequency of peak intensity increases linearly with temperature.
12 27-2 Blackbody Radiation; Planck s Quantum Hypothesis This figure shows blackbody radiation curves for three different temperatures. Note that frequency increases to the left.
13 27-2 Blackbody Radiation; Planck s Quantum Hypothesis This spectrum could not be reproduced using19 th -century physics. A solution was proposed by Max Planck in 1900: The energy of atomic oscillations within atoms cannot have an arbitrary value; it is related to the frequency: E = hf The constant h is now called Planck s constant. The smallest amount of energy possible hf is called the quantum of energy
14 27-2 Blackbody Radiation; Planck s Quantum Hypothesis Planck found the value of his constant by fitting blackbody curves: h = J s Planck s proposal was that the energy of an oscillation had to be an integral multiple of hf. This is called the quantization of energy.
15 Example Estimate the peal wavelength of light emitted from the pupil of the human eye assuming normal body temperature (~37 C). Solution: T = = 310 K. λ P T = m K λ P = m K 310K = m
16 27-3 Photon Theory of Light and the Photoelectric Effect Einstein suggested that, given the success of Planck s theory, light must exist as a set of small energy packets: (27-4) These tiny packets, or particles, are called photons.
17 27-3 Photon Theory of Light and the Photoelectric Effect The photoelectric effect: If light strikes a metal, electrons are emitted. The effect does not occur if the frequency of the light is too low; the kinetic energy of the electrons increases with frequency.
18 27-3 Photon Theory of Light and the Photoelectric Effect If light is a wave, theory predicts for the photoelectric effect: 1. Number of electrons and their energy should increase with intensity 2. Frequency should not matter
19 27-3 Photon Theory of Light and the Photoelectric Effect If light is particles (photons), theory predicts for the photoelectric effect: Increasing intensity increases number of electrons but not energy Above a minimum energy required to break atomic bond, kinetic energy will increase linearly with frequency There is a cutoff frequency below which no electrons will be emitted, regardless of intensity
20 27-3 Photon Theory of Light and the Photoelectric Effect The particle theory assumes that an electron absorbs a single photon. Plotting the kinetic energy vs. frequency: This shows clear agreement with the particle theory of light, not wave theory
21 27-3 Photon Theory of Light and the Photoelectric Effect The photoelectric effect is how electric eye detectors work. It is also used for movie film soundtracks.
22 27-4 Energy, Mass, and Momentum of a Photon Photons must travel at the speed of light. Looking at the relativistic equation for momentum, it is clear that this can only happen if its rest mass is zero. We already know that the energy is hf; we can put this in the relativistic energy-momentum relation and find the momentum: (27-6)
23 Example About 0.1 ev is necessary to break a hydrogen bond in a protein molecule. Calculate the minimum frequency and maximum wavelength of a photon that can do this. Solution: E min = hf min f min = E min h λ max = c f min = 0.1 ev J / ev J s = m / s Hz = m = Hz;
24 27-5 Compton Effect Compton did experiments in which he scattered X-rays from different materials. He found that the scattered X-rays had a slightly longer wavelength than the incident ones, and that the wavelength depended on the scattering angle: (27-7)
25 27-6 Photon Interactions; Pair Production Photons passing through matter can undergo the following interactions: 1. Photoelectric effect: photon is completely absorbed, electron is ejected 2. Photon may be totally absorbed by electron, but not have enough energy to eject it; the electron moves into an excited state 3. The photon can scatter from an atom and lose some energy 4. The photon can produce an electron-positron pair.
26 27-6 Photon Interactions; Pair Production In pair production, energy, electric charge, and momentum must all be conserved. Energy will be conserved through the mass and kinetic energy of the electron and positron; their opposite charges conserve charge; and the interaction must take place in the electromagnetic field of a nucleus, which can contribute momentum.
27 We have phenomena such as diffraction and interference that show that light is a wave, and phenomena such as the photoelectric effect and the Compton effect that show that it is a particle. Which is it? 27-7 Wave-Particle Duality; the Principle of Complementarity This question has no answer; we must accept the dual waveparticle nature of light. The principle of complementarity states that both the wave and particle aspects of light are fundamental to its nature. Indeed, waves and particles are just our interpretation of how light behaves.
28 27-8 Wave Nature of Matter Just as light sometimes behaves as a particle, matter sometimes behaves like a wave. The wavelength of a particle of matter is: (27-8) This wavelength is extraordinarily small for most objects. The wave nature of matter becomes more important for very light particles such as the electron.
29 Example 2 Visible light incident on a diffraction grating with slot spacing of 0.01mm has the first maximum at an angle of 3.6 from the central peak. If electrons could be diffracted by the same grating, what electron velocity would produce the same diffraction pattern as the visible light? Solution: First maximum means that we have constructive interference here: d sinθ = nλ λ = d sinθ n v = hn md sinθ = = h p = h mv ( J s) kg ( )( m) sin 3.6 ( ) =1159 m / s
30 27-8 Wave Nature of Matter Electron wavelengths can easily be on the order of m; electrons can be diffracted by crystals just as X-rays can.
31 Examples (1) Calculate the wavelength of a 0.21-kg ball traveling at 0.1 m/s. Solution: λ = h p = h mv = J s 0.21kg ( ) 0.1m / s ( ) = m Extremely small! Cannot detect it at all in everyday s life. (2) What is the wavelength of an electron (m= kg) traveling at m/s? λ = h p = h mv = J s ( kg) m / s ( ) = m Waves with such wavelength can be observed in diffraction experiments
32 27-9 Electron Microscopes The wavelength of electrons will vary with energy, but is still quite short. This makes electrons useful for imaging remember that the smallest object that can be resolved is about one wavelength. Electrons used in electron microscopes have wavelengths of about nm. Example: What voltage is needed to produce electron wavelength of 0.06 nm? Solution: The kinetic energy is acquired by electrostatic potential energy. KE = mv 2 2 = p2 2m ; p = h λ KE = h 2 2mλ 2 KE = ev electron is accerated by electric field ( ) h 2 2mλ 2 = ev V = h 2 2meλ 2 = kg ( J s) 2 ( )( C) ( m) = 419 V 2
33 27-9 Electron Microscopes Transmission electron microscope the electrons are focused by magnetic coils
34 27-9 Electron Microscopes Scanning electron microscope the electron beam is scanned back and forth across the object to be imaged
35 27-9 Electron Microscopes Scanning tunneling microscope up and down motion of the probe keeps the current constant. surface. Plotting that motion produces an image of the
36 27-10 Early Models of the Atom It was known in the late 19 th century that atoms were electrically neutral, but that they could become charged, implying that there were positive and negative charges and that some of them could be removed. One popular atomic model was the plum-pudding model. This model had the atom consisting of a bulk positive charge, with negative electrons buried throughout.
37 27-10 Early Models of the Atom Around 1911, Rutherford did an experiment that showed that the positively charged nucleus must be extremely small compared to the rest of the atom. He scattered alpha particles helium nuclei from a metal foil and observed the scattering angle. He found that some of the angles were far larger than the plum-pudding model would allow.
38 27-10 Early Models of the Atom The only way to account for the large angles was to assume that all the positive charge was contained within a tiny volume. Therefore, Rutherford s model of the atom is mostly empty space: Now we know that the radius of the nucleus is 1/10000 that of the atom.
39 27-11 Atomic Spectra: Key to the Structure of the Atom A very thin gas heated in a discharge tube emits light only at characteristic frequencies.
40 27-11 Atomic Spectra: Key to the Structure of the Atom An atomic spectrum is a line spectrum only certain frequencies appear. If white light passes through such a gas, it absorbs at those same frequencies.
41 27-11 Atomic Spectra: Key to the Structure of the Atom The wavelengths of electrons emitted from hydrogen have a regular pattern: (27-9) This is called the Balmer series. R is the Rydberg constant: R = m 1
42 27-11 Atomic Spectra: Key to the Structure of the Atom Other series include the Lyman series: And the Paschen series:
43 27-11 Atomic Spectra: Key to the Structure of the Atom A portion of the complete spectrum of hydrogen is shown here. The lines cannot be explained by the Rutherford theory.
44 27-12 The Bohr Atom Bohr proposed that the possible energy states for atomic electrons were quantized only certain values were possible. Then the spectrum could be explained as transitions from one level to another.
45 27-12 The Bohr Atom Bohr found that the angular momentum was quantized: (27-11)
46 27-12 The Bohr Atom An electron is held in orbit by the Coulomb force:
47 27-12 The Bohr Atom Using the Coulomb force, we can calculate the radii of the orbits: (27-12)
48 27-12 The Bohr Atom The lowest energy level is called the ground state; the others are excited states. Bohr assumed that electrons in fixed orbits do not radiate light. He was not able to say how an electron moved when it made a transition from one energy level to another. The idea of electron orbits was rejected. Today electrons are thought as forming probability clouds.
49 Electrons accelerated from rest by a potential difference of 12.3 V pass through a gas of hydrogen atoms at room temperature. What wavelengths of light will be emitted? Solution: the potential difference gives the electrons a kinetic energy of 12.3 ev, so it is possible to provide this much energy to the hydrogen atom through collisions. From the ground state, the maximum Example energy of the atom is ev ev = 1.3 ev. From the energy level diagram, we see that this means that the atom could be excited to the n = 3 state, so the possible transitions when the atom returns to the ground state are n=3 to n=2, n=3 to n=1, and n=2 to n=1.
50 Example (continued) So we will calculate the wavelengths using equation 1 λ = 1 ( hc E E ) n n 1 h = ev s; c = m / s λ 3 1 = λ 2 1 = λ 3 2 = hc E 3 E 1 = hc E 2 E 1 = hc E 3 E 2 = 1240eV nm [ 1.5eV ( 13.6eV )] 1240eV nm [ 3.4eV ( 13.6eV )] =102 nm =122 nm 1240eV nm = 650 nm [ 1.5eV ( 3.4eV )]
51 27-12 The Bohr Atom The correspondence principle applies here as well when the differences between quantum levels are small compared to the energies, they should be imperceptible. Correspondence principle means that when you move from quantum world to macro-world, the theory must be able to predict classical results.
52 27-13 de Broglie s Hypothesis Applied to Atoms De Broglie s hypothesis is the one associating a wavelength with the momentum of a particle. He proposed that only those orbits where the wave would be a circular standing wave will occur. This yields the same relation that Bohr had proposed. In addition, it makes more reasonable the fact that the electrons do not radiate, as one would otherwise expect from an accelerating charge..
53 27-13 de Broglie s Hypothesis Applied to Atoms De Broglie argued that the electron wave was a circular standing wave that closes on itself. If this does not happen, destructive interference takes place as the wave travels around the loop, and the wave quickly dies out. For constructive interference we need 2πr n =nλ; λ=h/mv mvr n =nh/(2π) exactly what Bohr proposed! These are circular standing waves for n = 2, 3, and 5.
54 Example 3 Construct the energy-level diagram for doubly ionized lithium, Li 2+. Solution: Lithium nucleus has 3 protons (positively charged particles), so Z=3. Energy levels are given as E n = ( 13.6 ev ) Z 2 n 2 = ( 13.6 ev ) 32 ev 2 = n n 2 E 1 = ev ; E 2 = 30.6 ev ; E 3 = 13.6 ev ;
55 Summary of Chapter 27 Planck s hypothesis: molecular oscillation energies are quantized Light can be considered to consist of photons, each of energy Photoelectric effect: incident photons knock electrons out of material
56 Summary of Chapter 27 Compton effect and pair production also support photon theory Wave-particle duality both light and matter have both wave and particle properties Wavelength of an object with mass:
57 Summary of Chapter 27 Principle of complementarity: both wave and particle properties are necessary for complete understanding Rutherford showed that atom has tiny nucleus Line spectra are explained by electrons having only certain specific orbits Ground state has the lowest energy; the others are called excited states
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