--Exam 3 Oct 3. are. absorbed. electrons. described by. Quantum Numbers. Core Electrons. Valence Electrons. basis for.

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1 Chapter 7 Announcements Quantum Theory and Atomic Structure --Exam 3 Oct 3 --Chapter 7/8/9/10 Chapter 7: Skip Spectral Analysis p Skip calculations for de Broglie and Heisenburg, conceptual understanding only. The good thing is the material is much more descriptive. Chapter 10 will be done as a part of the lab with me. --Quiz 6 Next Week Tuesday Chapter 6 Quantum Theory and Atomic Structure 7.1 The Nature of Light 7.2 Atomic Spectra 7.3 The Wave-Particle Duality of Matter and Energy 7.4 The Quantum-Mechanical Model of the Atom absorbed are emitted involve energy changes in electrons described by EM radiation units called Quanta atoms molecules is a wave and a particle having amplitude energy frequency wavelength related by related by E = hv c = v e Wave functions - filling e - configuration gives comprising having quantum uunmbers determined by Aufbau Rules described by Wave Function (Orbital) described by Quantum Numbers which are e - filling Oribital Energy spdf electronic configuration determined by Aufbau Rules which involve Pauli Exclusion comprising Hund s Rule Core Electrons Valence Electrons basis for Periodic Table which summarizes Periodic Properties Quantum Numbers Principal n = 1,2,3,.. Angular momentum, l Magnetic ml Orbital size & energy Orbital shape Orbital orientation Spin, ms Electron spin

2 In 1873, Maxwell found that light was an electromagnetic wave that travels at the at the speed of light, c. speed of light = c = " = 3.00 x 10 8 m/sec The wavelength, λ, is the crest-tocrest-distance in space of the waveform taken at an instant in time. Amplitude x distance The wavelength, λ, is the crest-tocrest-distance in space. The frequency ν, is the number of times per second that a crest passes a given point on the x-axis. Text Electric field is perpendicular to an oscillating magnetic field both perpendicular direction to both direction of propagation = wavelength The amplitude, A of the electric field is the maximum disturbance of the waveform. The frequency ν, is the number of times per second that a crest passes any given point on the x-axis. It is related to the period (sec) of the wave by f = 1/T The speed of light which is a constant relates the frequency ν, and wavelength: c = x " Amplitude time The wavelength () of electromagnetic radiation is expressed in many different units of length...know the conversions between them. Electromagnetic radiation exists over a broad range of frequencies. (nm) Angstroms = 1 nm " (Hz) Gamma Radiation X-rays UV IR Microwave Radio Waves 400nm 500nm 600nm 700nm The Visible Spectrum of Light (wavelength) Red Orange Interconverting Wavelength and Frequency A dental hygienist uses x-rays ( = 1.00A ) to take a series of dental radiographs while the patient listens to a radio station ( = 325 cm) and looks out the window at the blue sky (= 473 nm). What is the frequency (in s -1 ) of the electromagnetic radiation from each source? (Assume that the radiation travels at the speed of light, 3.00 x 10 8 m/s.) Yellow Green Blue Indigo Violet 700 nm 525 nm 430 nm 370 nm

3 Sample Problem 7.1 Interconverting Wavelength and Frequency A dental hygienist uses x-rays ( = 1.00A ) to take a series of dental radiographs while the patient listens to a radio station ( = 325 cm) and looks out the window at the blue sky (= 473 nm). What is the frequency (in s -1 ) of the electromagnetic radiation from each source? (Assume that the radiation travels at the speed of light, 3.00 x 10 8 m/s.) Experiments over 300 years show that electromagnetic radiation (light) exhibits both wave-like and particle-like properties. WAVE PARTICLE c = " 1.00 A X m 1 A = 1.00x10-10 m " = 3x108 m/s 1.00x10-10 m = 3x1018 s cm X 10-2 m 1 cm = 325 x 10-2 m 10-9 m 473 nm X 1 nm = 473 x 10-9 m " = 3x10 8 m/s = 9.23x10 7 s x 10-2 m " = 3x10 8 m/s 473 x 10-9 m = 6.34x1014 s -1 Waves bend Particles fly straight Experiments over 300 years show that electromagnetic radiation (light) exhibits both wave-like and particle-like properties. WAVE PARTICLE Experiments over 300 years show that electromagnetic radiation (light) exhibits both wave-like and particle-like properties. Light Intensity Light Intensity Slit Screen When light is passed through a tiny single slit one bright line appears. Two Slits Screen When two slits spaced closely together a pattern of bright and dark appears showing that light is an wave that interfers. It has been known since the 1600 s that light acts like a wave; it can be diffracted by passing light through small slits producing an interference pattern. In the early 1900 s there were mysteries involving light and matter that the physicists could not explain including: Known in 1600 s by Huygens Wall of Screen Projection of Wall Bright spot I. Black-body Radiation Problem - the frequencies and intensities of light emitted by heated solids at a given temperature. Light Waves Dark spot II. The Photo-Electric Effect - the ejection of electrons from certain metals upon exposure of light depended on the frequency of light, not its intensity (how bright). Film With Slit Interference Pattern III. Line Emission and Absorption Spectra - compounds, gases emitted only emitted or absorb very specific frequencies (colors) of light.

4 BLACKBODY PROBLEM: a black body is an idealized object that absorbs all electromagnetic radiation falling on it. Blackbodies absorb and incandescently re-emit radiation in a characteristic, continuous spectrum. 1000K 3000K smoldering coal BLACKBODY PROBLEM: All solids heated above T > 0 Kelvin emit electromagnetic radiation. Higher T = higher frequencies (lower wavelength) and higher intensities of light. Classical physics could not model or fit these experimental observations. 1000K 3000K smoldering coal Observed Classical physics predicts 5000K Observed Classical physics predicts 5000K 7000K electric heating elemen Intensity 7000K electric heating elemen Wavelength (nm) 9000K light bulb filament Wavelength (nm) 9000K light bulb filament Planck postulated that energy and matter exist in fixed quantities or quanta of energy and not continuous energy states. MYSTERY #2: When a photoelectric tube is bombarded with electromagnetic radiation (light), only specific frequencies eject electrons from the surface of the light sensitive plate to generate current. Light evacuated tube Al Einstein Energy (joules) E = n h " n= whole number "= frequency of light (Hz = sec -1 ) h= Plank s constant = 6.63 x J s Max Planck Freed e - + pole battery light sensitive metal plate Current meter Einstein finds that light acts like a particle. He calls the light particles photons and each photon has an energy = h". energy (joules) E = h " h= Plank s constant = 6.63 x J s "= frequency of light (sec -1 ) Albert Einstein explained the photoelectric effect in Light acts like a particle and are quantized energy packets called photons. 1 photon has energy = E = h". Photon has to have a certain minimum energy to eject an electron from the metal. If a photon does not have enough energy no electrons are ejected no matter how intense the light is (the number of photons per unit time) Greater intensity of the wrong frequency (more photons) does not overcome the electron binding energy. It is the energy (frequency) and not the intensity that is key. collector collector + battery + battery no e - ejected 0 e - ejected 50 incoming red light ammeter no current incoming blue light emitter current flows Learning Check: Photon-Energy Calculations 1. Calculate the energy, in joules per photon, of radiation that has a frequency of 4.77 x s If the energy of an infrared krypton laser is x joules per photon, determine the wavelength of the radiation. 3. A cook uses a microwave oven to heat a meal. The wavelength of the radiation is 1.20 cm. What is the energy of one photon of this microwave radiation? Note: The value of Planck s constant is 6.63 x J/s per photon.

5 MYSTERY #3: Narrow bands of colors of light are emitted when gases are excited by high voltage. The colors are characteristic and reproducible for different elements. High voltage tube containing hydrogen gas The light generated by the H2 gas is dispersed into its component wavelengths by a prism and projected on a screen Prism Screen All elements display a characteristic emission spectra that we can use to fingerprint all elements. Li Na K Ca Sr Ba Zn Cd Hg H He Ne Ar alkali metals alkaline earth Metals gases Flame color tests cause a similar electron excitement and are characteristic for elements. The color says something tells us electronic structure E = h" of the elements and can help identify elements. The beauty and intriguing color of fireworks arises from the characteristic color of elements in their salts. Hydrogen Helium Lithium Sodium Potassium Emission spectra of elements How could scientists account for the lines if atomic theory and classical physics were correct? How does the emission spectrum relate the atomic and electronic structure of the H atom? The first explanations of the emission spectra were empirical (i.e. found an equation that worked but without any explanation behind it). In 1885 Balmer developed an empirical mathematical relationship between " and n for the visible lines of hydrogen emission spectrum. Balmer equation λ (nm) = In 1885 Johann Rydberg developed a generalized but empirical mathematical relationship between " and n for the all lines of hydrogen emission spectrum This is freaky (nm) Rydberg equation 1 = R 1 n for the visible series, n 1 = 2 and = 3, 4, 5,...

6 In 1913, scientist Neils Bohr uses a planetary-quantum mathematical model that explains the emission spectrum of the hydrogen atom. 1. e - can only have specific (quantized) energy values (planetary like orbits) 2. light is emitted as e - moves from one energy level to a lower energy level 3. the energy of a level: ( E n = -R 1 ) H 4. the difference in energy between any two levels: n =6 n =5 n =4 n =3 n =2 The Bohr model explains the H atomic spectral lines as quantized jumps between the various n energy levels. 1 1 #E = R H( ) i f R H (Rydberg constant) = 2.18 x J n =6 n #E = R ( 1 1 ) H i f i = initial e - state and f = final e - state. R H (Rydberg constant) = 2.18 x J Bohr postulates that light is emitted when an e - goes from an orbit with a high n value to a lower n value The Bohr s model works for hydrogen, but could not explain the emission spectra of other atoms. Classical physics also says that electrons can not accelerate around a nucleus without eventually collapsing into the nucleus (i.e. simple planetary motion of e- around a nucleus fails). In 1924, Louis DeBroglie reasoned that if light wave could act like a particle then perhaps a particle could act like a wave. He connects Plank s Law to Einstein s energy-mass equivalence equation. E = hν = h c λ = mc2 Louis DeBroglie How and why is e - energy quantized and what physics can explain it? Plank s Law Einstein s Law of Mass equivalence Solving for wavelength, substituting v = velocity for c and watching our units...we get: λ = h mv = wavelength (m) v = velocity (m/s) h = Planck s constant (6.626 $ J-s) Particles with mass m and velocity v are also waves with wavelength The wave-like properties only manifest when mass is small and high velocity: e - that move at high speeds. The de Broglie Wavelengths of Selected Matter Soon after de Broglie s theory was put forth it was found that e - particles (which were thought to have only particle-like properties) could be made to diffract and interfere just like light waves. Substance Mass (g) Speed (m/s) de Broglie (m) slow electron 9 x x10-4 fast electron 9 x x x10-10 alpha particle 6.6 x x x10-15 one-gram mass x10-29 baseball x10-34 Earth 6.0 x x10 4 4x10-63 λ = h mv x-ray diffraction of aluminum foil electron diffraction of aluminum foil

7 What is the de Broglie wavelength (in meters) of: a) pitched baseball with a mass of 120. g and a speed of 100 mph or 44.7 m/s and b) of an electron with a speed of 1.00 x 10 6 m/s (electron mass = 9.11 x kg; h = x kg m 2 /s). Erwin Schrodinger, an Austrian physicist puts forth an equation that describes position and time dependence of an electron in hydrogen. We call this the Schrodinger wave equation. DeBroglie wavelength (m) λ = Plank s Constant = h = 6.63 x J s h mν wave function potential energy at x,y,z Erwin Schrodinger mass of object (kg) velocity of object (m/s) λ = h mν = kg m 2 /s (0.120 kg) (44.7 m/s) = m = x kg m 2 /s (9.11x10-31 kg)(1.00x10 6 m/s) = 7.27 x m h 2 2 Ψ(x) 8π 2 m e x 2 constants + 2 Ψ(y) y 2 how % changes in space + 2 Ψ(z) z 2 Ze2 Ψ(x, y, z) = EΨ(x, y, z) r total quantized energy of the atomic system Schrodinger equation is a difficult partial differential equation Schrodinger s equation is easier to solve if we transform the x,y,z coordinate system to spherical coordinates: r, #, $. "(x,y,z) 1) change of coordinates 2) solve the differential equation A solution to the Schrodinger wave equation, "(x,y,z) is called an a wavefunction or an atomic orbital. It s a mathematical function that describes an electron in an H atom. It has no physical meaning except when it is squared "(x,y,z) 2 (like the square of electric field in light). " () = 2L n where n = 1, 2, 3 "(r,#,$) = R(r) P(#) F($) n principal quantum number l orbital quantum number m l magnetic quantum number The wavefunctions are quantized allowed states of an electron like modes of a vibrational string of a guitar. The solution to this equation gives rise to 3 quantum numbers associated with an electron s energy levels called orbitals. Wave Motion Is Quantized In Space = 2L n A wavefunction is a solution to the Schrodinger equation and has 3-quantum numbers that describe electrons in space. where n = 1, 2, 3 n l m # n,m,l Guitar strings (a) can only vibrate at certain wavelengths (frequencies). If we apply this situation to to electrons around a nucleus (b) then only certain whole number wavelengths would be allowed--it is quantized. Nonallowed states can not be observed.

8 What does the Schrodinger Equation Tell Us? 1. Electrons in atoms, elements and molecules exist in discrete quantized states that are mathematically described by a wavefunction or orbital. The wavefunction can not be observed in a lab 2. The solutions of the Schrodinger equation for the hydrogen atom is a list of functions called wavefunctions or orbitals that describe the spatial location of an electron in a hydrogen atom. 3. Each wavefunction solution gives rise to a function having as variables called 3-quantum numbers: n, l, m. 4. The wavefunction "(r,#,$) by itself is not a physically measurable quantity, but the square of the wavefunction is.