Quantum Theory and Atomic Models

Transcription

1 Slide 1 / 83 Quantum Theory and tomic Models The Electron Slide 2 / 83 Streams of negatively charged particles were found to emanate from cathode tubes. J. J. Thompson is credited with their discovery (1897). iscovery and Properties of the Electron Slide 3 / 83 It was found that these rays could be deflected by electric or magnetic fields. y adjusting those fields the charge to mass ratio of the unknown "ray" was found.

2 iscovery and Properties of the Electron First, they found the velocity of the particle by adjusting the magnetic field electric forces so that they cancelled out; the "ray" traveled in a straight line. Slide 4 / 83 ΣF= ma F - F E = 0 qv = qe FE F v v = E/ Since they could measure E and, they could calculate v. iscovery and Properties of the Electron Slide 5 / 83 Then they turned off the electric field and the particle moved in a circular path. They measured the radius of the circle, by seeing where the particle struck the tube, and then determined the charge to mass ratio: q/m. ΣF= ma F = ma qv = mv 2 /r v F q = mv/r q/m = v/r r q/m = 1.76 x /kg 1 Which one of the following is not true concerning cathode rays? Slide 6 / 83 E They originate from the negative electrode. They travel in straight lines in the absence of electric or magnetic fields. They impart a negative charge to metals exposed to them. They are made up of electrons. The characteristics of cathode rays depend on the material from which they are emitted.

3 Millikan Oil rop Experiment Slide 7 / 83 Once the charge/mass ratio of the electron was known, determination of either the charge or the mass of an electron would yield the other. iscovery and Properties of the Electron The mass of each droplet was estimated by its size. The electric field was adjusted so the drop fell with constant velocity. The data showed that the charge was always an integral multiple of a smallest charge, e. That must be the charge of one electron. Slide 8 / 83 ΣF= ma F E - mg = 0 qe = mg q = mg/e FE q was always an integer multiple of the same number, which was given the symbol "e" mg 2 Which of these could be the charge of an object? Slide 9 / x x x x 10-19

4 3 The charge on an electron was determined in the. Slide 10 / 83 E cathode ray tube, by J. J. Thompson Rutherford gold foil experiment Millikan oil drop experiment alton atomic theory atomic theory of matter lackbody Radiation Slide 11 / 83 ll objects emit electromagnetic radiation which depends on their temperature: thermal radiation. black body absorbs all electromagnetic radiation (light) that falls on it. ecause no light is reflected or transmitted, the object appears black when it is cold. However, black bodies emit a temperature-dependent spectrum termed blackbody radiation. lackbody Radiation Slide 12 / 83 t normal temperatures, we are not aware of this radiation. ut as objects become hotter, we can feel the infrared radiation or heat. t even hotter temperatures, objects glow red and at still hotter temperatures, object can glow white hot such at the filament in a light bulb.

5 lackbody Radiation Slide 13 / 83 This figure shows blackbody radiation curves for three different temperatures. The wavelength at the peak, # p, is related to the temperature by: # pt = 2.90 x 10-3 m-k lassical physics couldn't explain the shape of these spectra. 4 Which of the following colors would indicate the hotest temperature? Slide 14 / 83 lack Red Yellow lue Planck s Quantum Hypothesis Slide 15 / 83 The wave nature of light could not explain the way an object glows depending on its temperature: its spectrum. Max Planck explained it by assuming that atoms only emit radiation in quantum amounts...in steps given by the formula: E = hf where h is Planck s constant and f is the frequency of the light

6 Planck s Quantum Hypothesis Slide 16 / 83 Planck didn't believe this was a real...it just worked. It was like working from the answers in the book...getting something that works, but having no idea why. It didn't make sense that atoms could only have steps of energy. Why couldn't they have any energy? Planck thought a "real" solution would eventually be found...but this one worked for some reason. Which brings us to our next mystery... The Photoelectric Effect Slide 17 / 83 When light strikes a metal, electrons sometimes fly off. lassical physics couldn't explain some specific features about how the effect works. So Einstein used Planck's idea to solve it. The Photoelectric Effect Slide 18 / 83 If atoms can only emit light in packets of specific sizes; maybe light itself travels as packets of energy given by Planck's formula. E = hf where h is Planck s constant or 4.14 x ev-s He called these tiny packets, or particles, of light, photons.

7 5 What is the energy (in nj) of a photon with a frequency of 5 x Hz? Slide 19 / 83 The Photoelectric Effect Slide 20 / 83 The maximum kinetic energy of these photons can be measured using a variable voltage source and reversing the terminals so that the electrode is negative and P is positive. If the voltage is increased, there is a point when the current reaches zero. This is called the stopping voltage, V 0, and it is given by: KE max = ev 0 The Photoelectric Effect Slide 21 / 83 We said earlier that when light strikes a metal, electrons sometimes fly off. Since electrons are held in the metal by attractive forces, some minimum energy, W 0, called the work function, is required just to get an electron free from the metal. The input energy of the photon will equal the kinetic energy of the ejected electron plus the energy required to free the electron. hf = KE + W 0

8 The ompton Effect Slide 22 / 83. H. ompton scattered short-wavelength light from various materials and discovered that the scattered light had a slightly lower frequency than the incident light, which indicated a loss of energy. He applied the laws of conservation of momentum and energy and found that the predicted results corresponded with the experimental results. single photon wavelength, #, strikes an electron in some material, knocking it out of its atom. The scattered photon has less energy since it gave some to the electron and thus has a wavelength of, #'. Incident photon (#) Electron at rest Electron after collision # # Scattered photon (#') The ompton Effect Slide 23 / 83 The momentum of a photon is given by: p = E/c Since E = hf, p = hf/c = h/# Using conservation of momentum: Incident photon (#) Electron at rest Electron after collision # # Scattered photon (#') Where m 0 is the rest mass of the electron. Photon Interactions Slide 24 / The ompton Effect - the photon can be scattered by an electron and lose energy in the process. 2. The photoelectric effect - a photon can knock an electron out of an atom and disappear in the process. 3. The photon can knock and atomic electron to a higher energy state if the energy of the photon is not sufficient to knock it out of the atom. 4. Pair production - a photon can produce an electron and a positron and disappear in the process. (The inverse of pair production can occur if an electron collides with a positron. This is called annihilation.)

9 Photon Theory of Light Slide 25 / 83 This particle theory of light assumes that an electron absorbs a single photon and made specific predictions that proved true. For instance, the kinetic energy of escaping electrons vs. frequency of light shown below: This shows clear agreement with the photon theory, and not with wave theory. This shows that light is made of particles, photons; light is not a wave. Wave-Particle uality; the Principle of omplementarity Slide 26 / 83 Earlier we proved that light is a wave. Now we've proven that light is a particle. Which is it? This question has no answer; we must accept the dual wave-particle nature of light. While we cannot imagine something that is a wave and is a particle at the same time; that turns out to be the case for light. 6 The ratio of energy to frequency for a given photon gives Slide 27 / 83 its amplitude. its velocity. Planck's constant. its work function. E = hf c = lf c = m/s

10 7 What is a photon? Slide 28 / 83 an electron in an excited state a small packet of electromagnetic energy that has particle-like properties one form of a nucleon, one of the particles that makes up the nucleus an electron that has been made electrically neutral E = hf c = lf c = m/s 8 The energy of a photon depends on Slide 29 / 83 its amplitude. its velocity. its frequency. none of the given answers E = hf c = lf c = m/s 9 Which color of light has the lowest energy photons? Slide 30 / 83 red yellow green blue E = hf c = lf c = m/s

11 10 The photoelectric effect is explainable assuming Slide 31 / 83 that light has a wave nature. that light has a particle nature. that light has a wave nature and a particle nature. none of the above E = hf c = lf c = m/s 11 The energy of a photon that has a frequency 110 GHz is Slide 32 / J J J J E = hf c = lf c = m/s 12 The frequency of a photon that has an energy of 3.7 x J is Slide 33 / 83 E Hz Hz J J J E = hf c = lf c = m/s

12 13 The energy of a photon that has a wavelength of 12.3 nm is Slide 34 / 83 E J J J J J E = hf c = lf c = m/s 14 If the wavelength of a photon is halved, by what factor does its energy change? Slide 35 / /4 1/2 E = hf c = lf c = m/s Energy, Mass, and Momentum of a Photon Slide 36 / 83 learly, a photon must travel at the speed of light, since it is light. Special Relativity tell us two things from this: The mass of a photon is zero. The momentum of a photon depends on its wavelength. m = 0 p = hf / c p = h/l and since c = lf This last equation turned out to have huge implications.

13 Wave Nature of Matter de roglie asked, "If light can behave like a wave or a particle, can matter also behave like a wave?" Slide 37 / 83 mazingly, it does! de roglie combined p = h/l with p = mv to get The wavelength of matter l = h/(mv) This wavelength is really small for normal objects, so it had never been noticed before. ut it has a dramatic impact on the structure of atoms. Wave Nature of Matter Slide 38 / 83 Electron wavelengths are often about m, about the size of an atom, so the wave character of electrons is important. In fact, the two-slit experiment that showed that light was a wave, has been replicated with electrons with the same result...electrons are particles and waves. Electrons fired one at a time towards two slits show the same interference pattern when they land on a distant screen. The "electron wave" must go through both slits at the same time...which is something we can't imagine a single particle doing...but it does. The most amazing experiment ever!!! Slide 39 / 83 These photos show electrons being fired one at a time through two slits. Each exposure was made after a slightly longer time. The same pattern emerges as was found by light. Each individual electron must behave like a wave and pass through both slits. ut each electron must be a particle when it strikes the film, or its wouldn't make one dot on the film, it would be spread out. This one picture shows that matter acts like both a wave and a particle.

14 15 What is the wavelength of a 0.25 kg ball traveling at 20 m/s? Slide 40 / 83 l = h/(mv) 16 What is the wavelength of a 80 kg person running 4.0 m/s? Slide 41 / 83 l = h/(mv) 17 What is the wavelength of the matter wave associated with an electron (m e = 9.1 x kg) moving with a speed of m/s? Slide 42 / nm 0.36 nm 0.48 nm 0.56 nm l = h/(mv)

15 18 What is the wavelength of the matter wave associated with an electron (m e = 9.1 x kg) moving with a speed of m/s? Slide 43 / nm 0.36 nm 0.48 nm 0.56 nm l = h/(mv) The tom, circa 1900 Slide 44 / 83 The prevailing theory was that of the plum pudding model, put forward by Thompson. It featured a positive sphere of matter with negative electrons imbedded in it. iscovery of the Nucleus Slide 45 / 83 Ernest Rutherford shot a particles at a thin sheet of gold foil and observed the pattern of scatter of the particles.

16 iscovery of the Nucleus Slide 46 / 83 While most particles went straight through, as expected, some bounced back...which was totally unexpected. The only way to account for that was to assume all the positive charge was contained within a tiny volume. Early Models of the tom Slide 47 / 83 small very dense nucleus must lie within a mostly empty atom. Now we know that the radius of the nucleus is 1 / 10,000 that of the atom. The Nuclear tom Slide 48 / 83 Since some particles were deflected at large angles, Thompson s model could not be correct.

17 Early Models of the tom Rutherford's experiment showed that the positively charged nucleus must be very small compared to the rest of the atom. Slide 49 / 83 Then I remember two or three days later Geiger coming to me in great excitement and saying "We have been able to get some of the alpha-particles coming backward " It was quite the most incredible event that ever happened to me in my life. It was almost as incredible as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you. - Rutherford The Nuclear tom Slide 50 / 83 Rutherford postulated a very small, dense nucleus with the electrons around the outside of the atom. Most of the volume of the atom is empty space. The Nuclear tom Slide 51 / 83 If an atom were magnified to be the size of a gymnasium (about 100 m across), the proton would be about the size of a ping pong ball (1 cm across), the electrons would be too small to see, and all the rest would be just emptiness...not filled with air (like a gym), but nothing.

18 19 The gold foil experiment performed in Rutherford's lab. Slide 52 / 83 E confirmed the plum-pudding model of the atom led to the discovery of the atomic nucleus was the basis for Thomson's model of the atom utilized the deflection of beta particles by gold foil proved the law of multiple proportions 20 In the Rutherford nuclearatom model,. Slide 53 / 83 the heavy subatomic particles reside in the nucleus the principal subatomic particles all have essentially the same mass the light subatomic particles reside in the nucleus mass is spread essentially uniformly throughout the atom Subatomic Particles Slide 54 / 83 Protons were discovered by Rutherford in Neutrons were discovered by James hadwick in Protons and electrons are the only particles that have charge. Protons and neutrons have essentially the same mass. The mass of an electron is extremely small.

19 The Problem with the Nuclear tom Slide 55 / 83 The nucleus of an atom is small, 1/10,000 the size of the atom. The electrons are outside the nucleus, moving freely within the vast empty atom The nucleus is positive; the electron is negative There is an electric force, F E = kq 1q 2/r 2, pulling the electron towards the nucleus There is no other force acting on the electron; there is nothing to support it; it experiences a net force towards the nucleus Why don't the electrons fall in...why doesn't the atom collapse into its nucleus? The Problem with the Nuclear tom Slide 56 / 83 ased on the equations #F = ma and x = x 0 + v ot +1/2at 2 ll atoms would collapse in about s Earth would collapse to less than a mile across in less than a billionth of a second. The universe as we know it would end. The Problem with the Nuclear tom Slide 57 / 83 Perhaps electrons orbit the nucleus...like planets orbit the sun. ut then they would constantly be accelerating as they travel in a circle: a = v 2 /r ut it was known that an accelerating charge radiates electromagnetic energy...light. ll the kinetic energy would radiate away in about that same billionth of a second...then it would fall into the nucleus. ll the atoms in the universe would still collapse.

20 The Problem with the Nuclear tom Slide 58 / 83 lassical physics failed to explain how atoms could exist. new approach was needed. The next step led to the ohr model of the atom, which was a semi-classical explanation of atoms. It would be an important transition to modern quantum theory. n important clue was found in the spectra of gas discharge tubes. tomic Spectra Slide 59 / 83 very thin gas heated in a discharge tube emits light only at characteristic frequencies. tomic Spectra Slide 60 / 83 n atomic spectrum is a line spectrum only certain frequencies appear. If white light passes through such a gas, it absorbs at those same frequencies.

21 tomic Spectra Slide 61 / 83 Why don't atoms radiate, or absorb, all frequencies of light? Why do they radiate light at only very specific frequencies, and not at others? tomic Spectra The wavelengths of electrons emitted from hydrogen have a regular pattern: Slide 62 / 83 almer series Lyman series Paschen series tomic Spectra: Key to the Structure of the tom Slide 63 / 83 portion of the complete spectrum of hydrogen is shown here. The lines cannot be explained by the Rutherford theory.

22 The ohr tom Slide 64 / 83 ohr proposed that electrons could orbit the nucleus, like planets orbit the sun...but only in certain specific orbits. He then said that in these orbits, they wouldn't radiate energy, as would be expected normally of an accelerating charge. These stable orbits would somehow violate that rule. Each orbit would correspond to a different energy level for the electron. The ohr tom Slide 65 / 83 These possible energy states for atomic electrons were quantized only certain values were possible. The spectrum could be explained as transitions from one level to another. Electrons would only radiate when they moved between orbits, not when they stayed in one orbit. The ohr tom Slide 66 / 83 s long as an electron was in an orbit given by the below formula, it would not emit electromagnetic radiation. The observed spectrum of the hydrogen atom is predicted successfully by transitions between these orbits.

23 The ohr tom Slide 67 / 83 n electron is held in orbit by the oulomb force: The ohr tom Using the oulomb force, we can calculate the radii of the orbits. These matched the sizes of known atoms very well. Slide 68 / 83 The ohr tom Slide 69 / 83 The radii of the orbits of a hydrogen atom are given by the below formula, with the smallest orbit, r n = n 2 r 1, (for hydrogen, Z = 1) Z r 1 = 0.53 x m. n = 1, 2, 3, 4,... Notice that the orbits grow in size as the square of n, so they get much larger as n increases.

24 21 The radius of the orbit for the third excited state (n=4) of hydrogen is r 1. Slide 70 / The radius of the orbit for the fifth excited state (n=6) of hydrogen is x m. r 1 = 0.50 x m Slide 71 / 83 The ohr tom Slide 72 / 83 Using the oulomb force, he calculated the energy of each orbit. For hydrogen he arrived at this result: E = ev n 2 n = 1, 2, 3, 4,... Notice that the energy levels are all negative, otherwise the electron would be free of the atom. The levels get closer together, and closer to zero, as n increases.

25 The Electron Volt (ev) Slide 73 / 83 In atomic physics, energies are so low that it's awkward to use Joules (J). smaller unit of energy is the electron-volt (ev). It's value equals the potenial energy of an electron in a region of space whose voltage (V) is 1.0 volt. U E = qv U E = (1.6 x )(1.0V) U E = 1.6 x J # 1.0 ev 1.0 ev = 1.6 x J The Electron Volt (ev) Slide 74 / 83 It's also convenient to convert Plank's constant to units of ev-s rather than J-s. h = 6.63 x J-s h = (6.63 x J-s) h = 4.14 x ev-s ( 1.0 ev 1.6 x J ) h = 4.14 x ev-s 1.0 ev = 1.6 x J 23 What is the energy of the second excited state (n=3) of hydrogen? Slide 75 / 83

26 24 What is the energy of the fifth excited state (n=6) of hydrogen? Slide 76 / 83 The ohr tom Slide 77 / 83 The lowest energy level is called the ground state; the others are excited states. de roglie s Hypothesis pplied to toms Slide 78 / 83 Scientists didn't like the lack of an explanation for why electrons didn't radiate when in those orbits. ut de roglie's wave theory of matter explained it very well. s long as the wavelength on an electron in orbit was the same as the circumference of the orbit, it would not radiate. This approach yields the same relation that ohr 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.

27 de roglie s Hypothesis pplied to toms Slide 79 / 83 These are circular standing waves for n = 2, 3, and 5. Quantum Physics Slide 80 / 83 While a big step forward, ohr's model only worked for atoms that had one electron, like hydrogen or certain ionized atoms. It failed for all atoms other than hydrogen. The idea that the electron was a particle in orbit around the nucleus, but with wavelike properties that only allowed certain orbits, worked only for hydrogen. Semi-classical explanations failed except for hydrogen. It turned out that only a lucky chance let it work even in that case. Quantum Mechanics Slide 81 / 83 Our goal was to explain why electrons in an atom don't fall into the nucleus. n electron, as a charged particle, would fall in because of Newton's Second Law. #F = ma ut electrons, in atoms, aren't particles, they're waves. Waves don't follow Newton's Second Law. Schrodinger had to invent a new equation for wave mechanics. H# = E#

28 Quantum Mechanics Slide 82 / 83 H# = E# The simplicity of this equation is deceptive. Here's what it looks like when expanded for one type of problem. It's only solved for general cases in advanced university courses. However, computers have been used to exactly solve it for many specific cases: atoms, molecules, etc. Quantum Mechanics Those solutions will allow us to understand how the microscopic world works: atoms, the periodic table, molecules, chemical bonds, etc. Slide 83 / 83 Quantum mechanics is very different from classical physics you can predict what a lot of electrons will do on average, but have no idea what any individual electron will do. In hemistry, you'll be using the solutions to Schrodinger's Equation equations, and the physics you've learned this year, to explored the nature of matter.