! In 1873, Maxwell found that light was an electromagnetic wave that travels at the at the speed of light, c. ! = wavelength. Chapter 7. c =!

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1 Chapter 7 EM radiation units called is a wave and a particle Quantum Theory and Atomic Structure are Quanta having absorbed emitted amplitude energy frequency wavelength involve energy changes in electrons described by atoms molecules related by related by E = hv c =!v e Wave functions - filling e - 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 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 m l defines defines defines defines Orbital size & energy Orbital shape Orbital orientation Spin, m s defines Electron spin! In 1873, Maxwell found that light was an electromagnetic wave that travels at the at the speed of light, c. 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. speed of light = c =! "! = wavelength The wavelength, λ, is the crest-tocrest-distance in space of the waveform taken at an instant in time. 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 Amplitude Amplitude! x distance Electric field is perpendicular to an oscillating magnetic field both perpendicular direction to both direction of propagation The speed of light which is a constant relates the frequency ν, and wavelength: c =! x " time

2 The wavelength (!) of electromagnetic radiation is expressed in many different units of length...know the conversions between them. 10 Angstroms = 1 nm! Electromagnetic radiation exists over a broad range of frequencies.! (nm) " (Hz) Gamma X-rays UV IR Microwave Radio Waves Radiation nm 500nm 600nm 700nm The Visible Spectrum of Light (wavelength) Calculations using c =! v Red Orange Yellow Green Blue Indigo Violet 700 nm 525 nm 430 nm 370 nm 7.1 Your new cordless phone has a frequency of 2.4 GHz. What is the wavelength of the radiation used by the phone? In what region of the EM spectrum is this? 7.2 Yellow light has a wavelength of 598 nm. What is the frequency of this light in Hz? 7.3 A certain type of electro-magnetic radiation has a frequency of 3.5 # Hz. What is the wavelength of this radiation in nm and A? In what region of the spectrum is this? 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

3 Experiments over 300 years show that electromagnetic radiation (light) exhibits both wave-like and particle-like properties. 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. Light Intensity Light Intensity Known in 1600 s by Huygens Wall of Screen Projection of Wall Bright spot Dark spot Slit Screen Two Slits Screen Light Waves When light is passed through a tiny single slit one bright line appears. When two slits spaced closely together a pattern of bright and dark appears showing that light is an wave that interfers. Film With Slit Interference Pattern In the early 1900 s there were mysteries involving light and matter that the physicists could not explain including: I. Black-body Radiation Problem - the frequencies and intensities of light emitted by heated solids at a given temperature. 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). III. Line Emission and Absorption Spectra - compounds, gases emitted only emitted or absorb very specific frequencies (colors) of light. 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. Intensity Observed Classical physics predicts Wavelength (nm) 1000K 3000K smoldering coal 5000K electric heating elemen 7000K 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 )

4 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 x J/s per photon. 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

5 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 (which means they 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) = n 2 n 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 n 2 2 for the visible series, n 1 = 2 and n 2 = 3, 4, 5,... 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 n 2 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 n 2 $E = R H( ) n 2 i f R H (Rydberg constant) = 2.18 x J n =6 n $E = R ( 1 1 ) H n 2 i n 2 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). How and why is e - energy quantized and what physics can explain it? 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 = h! = hc/" = mc 2 Plank s Law! = h mv Einstein s Law of Mass equivalence! = wavelength (m) v = velocity (m/s) h = Planck s constant (6.626 # J-s) Louis DeBroglie Solving for wavelength and watching our units All particles can are waves! The wave-like properties only manifest when mass is small and high velocity: e - that move at high speeds.

6 The de Broglie Wavelengths of Selected Matter Substance Mass (g) Speed (m/s) de Broglie! (m) Soon after de Broglie s theory was put forth it was found that e - particles (which were thought to have only particlelike properties) could be made to diffract and interfere just like light waves. 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 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). DeBroglie wavelength (m) mass of object (kg) λ = Plank s Constant = h = 6.63 x J s h mν 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 The Heisenburg Uncertainty Principle The act of measuring where an electron is changes what we measure. We can never know precisely where an electron is only a probability of where it may be in space. "x "p # h 4$ $x = position uncertainty $p = momentum uncertainty (p = mv) h = Planck s constant An electron moving near an atomic nucleus has a speed 6x10 6 ± 1% m/s. What is the uncertainty in its position ($x)? 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. Erwin Schrodinger The uncertainty ($x) is given as ±1%(0.01) of 6x10 6 m/s. Once we calculate this, plug it into the uncertainty equation. $u = (0.01)(6x10 6 m/s) = 6x10 4 m/s $ x * m $ u # h 4% $x! 6.626x10-34 kg*m 2 /s = 9.52x10-9 m 4% (9.11x10-31 kg)(6x10 4 m/s) h 2 8π 2 m e constants wave function 2 Ψ(x) x Ψ(y) y 2 how & changes in space potential energy at x,y,z + 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

7 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. A wavefunction is a solution to the Schrodinger equation and has 3-quantum numbers that describe electrons in space. n l m # n,m,l What the Quantum Numbers Mean 1. PRINCIPAL QUATUM NUMBER (n): Defines the size and energy level of the orbital. Also called a shell. n = {1,2,3,4,.}. (or n = 1 = K shell; n = 2 = L shell; n = 3 = M shell ANGULAR MOMEMTUM QUANTUM NUMBER (l): Defines the shape of the orbital or volume in space where the electron is likely to be found. Also called a subshell. l = {0,1,2,3...up to a maximum of n-1} where (0 = s, 1 = p, 2 = d, 3 = f) 3. MAGNETIC QUANTUM NUMBER (ml): Defines the spatial orientation of an orbital of the same energy. ml = {-l, 0, +l} 4. MAGNETIC SPIN QUANTUM NUMBER (ms): Defines the orientation of electron spin. ms = {+1/2 or -1/2}. Summary of Quantum Numbers The Quantum Number Tree (3 numbers) Name principal angular momentum Symbol Permitted Values Property n positive integers (1,2,3, ) orbital energy (size) l integers from 0 to n-1 orbital shape (The l values 0, 1, 2, and 3 correspond to s, p, d, and f orbitals, respectively.) magnetic m l integers from -l to 0 to +l orbital orientation in space spin m s +1/2 or -1/2 direction of e - spin Quantum Number n l m l Allowed Values Positive integers 1,2,3, up to max of n-1 -l, l n => l => Orbitals s 2s 2p 3s 3p 3d

8 Putting all the information together principal angular momentum s angular momentum s p s magneticp d spin s p d f magnetic Where is the fourth quantum number? Determining Sublevel Names and Orbital Quantum Numbers Give the orbital name, possible magnetic quantum numbers, and number of orbitals for each sublevel with the following quantum numbers: SOLUTION: (a) (b) (c) (d) (a) n = 3, l = 2 (b) n = 2, l = 0 (c) n = 5, l = 1 (d) n = 4, l = 3 n l Orbital name possible m l values # of orbitals d 2s 5p 4f -2, -1, 0, 1, 2 0-1, 0, 1-3, -2, -1, 0, 1, 2, The 4th quantum number: Stern-Gerlach Experiment Hydrogen atoms are shot through a strong magnet. It is found that the atoms at the collection plate are split into two groups of atoms. The results show that electron spins must be quantized as if they were not one would expect a continuous rectangular patch extending dot to dot (continuous energy as in classical physics). The wavefunction with its 4-quantum numbers specify all the information we can know about electrons around a nucleus which we call the electronic structure of an atom. Continuous distribution if the result was classical Quantum result gives EITHER -1/2 or +1/2...only 2 states North South The order of filling of the orbitals can be remembered using a mnemonic device. Memorize how to write it out as it determines electronic structure. Determining Quantum Numbers for an Energy Level 1) Name the 4 quantum numbers and say what they mean? 2) How many quantum numbers are needed to specify and an atomic orbital for an electron? 3) What values of the angular momentum (l) and magnetic (m l ) quantum numbers are allowed for a principal quantum number n = 3 and n =4? 4) How many orbitals are allowed for n = 3 and n = 4? PLAN: SOLUTION: remember l values can be integers from 0 to n-1; m l can be integers from -l through 0 to + l. For n = 3, l = 0, 1, 2 For n = 4, l = 0, 1, 2,3 For l = 0 m For l = 0 m l = 0 l = 0 For l = 1 m l = -1, 0, or +1 For l = 1 m l = -1, 0, or +1 For l = 2 m l = -2,-1,0,+1,+2 For l = 2 m l = -2, -1, 0, +1, or +2 For l = 3 m l = -3,-2,-1,0,+1,+2+3 There are 9 m l values and therefore 9 orbitals with n = 3. There are 16 m l values and therefore 16 orbitals with n = 4.

9 $ has no physical meaning or reality (it s a math function). $ 2 represents the probability function for finding an electron. We can integrate the function to get the total probability of finding an electron. $ = Mathematical Construct $ 2 = Probability Density The l quantum number determines the orbital shape or the probability of where the electron is to be found. Name Symbol Permitted Values Property principal n positive integers (1,2,3, ) orbital energy (size) angular momentum l integers from 0 to n-1 orbital shape (The l values 0, 1, 2, and 3 correspond to s, p, d, and f orbitals, respectively.) magnetic m l integers from -l to 0 to +l orbital orientation in space spin m s +1/2 or -1/2 direction of e - spin How many 2p orbitals are there in an atom? n=2 2p l = 1 If l = 1, then m l = -1, 0, or +1 3 orbitals If two electrons can be placed in one orbital how many electrons can go exist in the 3d subshell? n=3 If l = 2, then m l = -2, -1, 0, +1, or +2 3d l = 2 5 orbitals which can each holding 2 electrons for a total of 10 e - in the subshell The shape of an orbital is given by the l quantum number (l = {0,1,2 up to n-1}. The number of orbitals and its orientation in space is given by the angular momentum quantum number ml {-l,..0..+l}. l = 0 s orbital l = 1 p orbital The shapes of orbitals are boundry surface plots showing 90% probability of finding an in that volume of space. Probability at point r in space for 1s orbital l = 2 d orbital l = 3 f orbital Probability of points added up in a circular strips of (r + rdr) for a 1s orbital Boundary Surface Plot ( orbital shape )

10 The shapes of orbitals are boundry surface plots showing 90% probability of finding an in that volume of space. Probability at point r in space Probability Density Plots nodes (0 probability) 1s 2s 3s s-orbital Boundary Surface Plot Probability of points added up in a circular strips of (r + rdr) Radial Probability Distribution Plots 1s 2s The Shapes of Orbitals 1s 2s 3s 1s-Orbital Shape is spherical and gets larger with the n- quantum number. The p Orbitals ( l = 1) 2-D Slice of probability at points in space 1s 2s 3s Notice the nodes or regions of 0 electron density in the plots. Boundary Surface Plot having 95% of all probability Radial Probability Distribution Plot for p-orbital. 2-D Slice of probability at points in space Boundary Surface Plot having 95% of all probability 1s 2s 3s The 3 p orbital Boundary Surface Plots (Shapes) The 5-d Boundry Surface Plots (Shapes) 2 p x 2 p y 2 p z d yz d xz d xy d 2 z Three p-orbitals superimposed d x 2 - d y 2 Five d-orbitals superimposed

11 Energy The seven f-orbitals (radial plots) 1of One 7 of f-orbitals the seven superimposed possible 4f orbitals. Quiz 5 Please place books on the floor Full sheet of paper and write your name Quiz 5 1) Calculate the energy of a single photon of light that has a wavelength of 525 nm. E = hν c = λν c E = h =( λ 34 Js) Quiz 6 2a) What are the n, l, ml for the 2p and 5f orbital m/s 525 nm 10 9 nm 1 m = J n = 2, l = 1, ml = -1, 0, +1 = 2p n = 5, l = 3, ml = -2, -1, 0, +1, +2 = 5f 2b) what do the values of n, l and ml stand for or designate? n = energy, l = shape, ml = orientation in space We can build up the electron of all the ground state elements (and their ions) by filling in the orbitals with electrons. Orbitals are degenerate or the same energy in Hydrogen! Electrons will fill lowest energy orbitals first! Hydrogen Atom Multi-electron atoms The lowest energy (ground state) electronic of all elements are constructed by filling lowest energy orbitals sequentially in what is called the Aufbau Process. 1. Lower energy (n-quantum number) orbitals fill first. 2. Hund s Rule-degenerate orbitals fill one at a time before electrons are paired. 3. Pauli Exclusion Principle: No two electrons can have same 4-quantum numbers) Electrons fill the lowest energy orbitals first, 2 at a time!

12 The order of filling of the orbitals can be remembered using a mnemonic device. Memorize how to write it out! Chemists use spdf notation and orbital box diagrams to denote or show the ground state electronic of elements. Element H spdf Notation 1s 1 orbital box diagram An arrow denotes an electron with spin up or spindown. He electron shell principal quantum # 1s 2 orbital type angular quantum # Remember, no two electrons can have the same 4 quantum numbers! # of electrons in orbital The Pauli Exclusion principle states: No two electrons can have the same 4-quantum numbers. each electron can be described by a unique set of quantum set of four quantum numbers. Building electronic using Aufbau and Hund Atomic Number/Element H Orbital Box Diagram Full-electronic 1s 1 Condensed-electronic 1s 1 Example: Atomic Number/Element Orbital Box Diagram Full-electronic Li 1s 2 2s 1 Condensed-electronic [He]2s 1 He Li 1s 2 1s 2 2s 1 1s 2 written with noble gas [He]2s 1 Be 1s 2 2s 2 [He]2s 2 Atomic Number/Element Orbital Box Diagram Full-electronic Condensed-electronic Exam II B 1s 2 2s 2 2p 1 [He]2s 2 2p 1 C 1s 2 2s 2 2p 2 [He]2s 2 2p 2 1s 2 2s 2 2p 3 [He]2s 2 2p 3 1s 2 2s 2 2p 4 [He]2s 2 2p 4 1s 2 2s 2 2p 5 [He]2s 2 2p 5 1s 2 2s 2 2p 6 [He]2s 2 2p 6