absorbed amplitude energy frequency wavelength 7.1 The Nature of Light electrons described by Quantum Numbers Core Electrons Valence Electrons

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1 Chapter 7 Quantum Theory and Atomic Structure are EM radiation units called Quanta is a wave and a particle having absorbed emitted amplitude energy frequency wavelength 7. The Nature of Light 7. Atomic Spectra 7.3 The Wave-Particle Duality of Matter and Energy involve energy changes in electrons described by atoms molecules related by related by E = hv c = v 7.4 The Quantum-Mechanical Model of the Atom e Wave functions - filling e - configuration gives comprising having quantum numbers determined by Aufbau Rules described by Wave Function (Orbital) described by Quantum Numbers which are e - filling Orbital 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 =,,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 873, Maxwell found that light is an electromagnetic wave that travels at the at the speed of light, c. speed of light = c = " = 3. x 8 m/sec The wavelength, λ, is the crest-to-crestdistance 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 are perpendicular direction to both direction of propagation. = 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,, is the number of times per second that a crest passes any given point on the x-axis (units of /sec called Hertz = Hz). It is related to the period (sec) of the wave by f = /T The frequency and wavelength of a wave are connected by the speed of light: c = " = 3. X 8 m/s Amplitude Amplitude x distance time

2 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 = nm 9 nm = m 6 um = m 3 mm = m cm = m " (Hz) Gamma Radiation X-rays UV IR Microwave Radio Waves 4nm 5nm 6nm 7nm Picture of amplitude of waveforms are shown vs space. which has the greatest frequency, which has the greatest wavelength, which has teh greatest frequency, and which has the greatest amplitude? longest wavelength = smallest frequency Sample Problem 7. Interconverting Wavelength and Frequency A dental hygienist uses x-rays ( =.A ) to take a series of dental radiographs while the patient listens to a radio station ( = 35 cm) and looks out the window at the blue sky (= 473 nm). What is the frequency (in s - ) of the electromagnetic radiation from each source? (Assume that the radiation travels at the speed of light, 3. x 8 m/s.) c = " largest amplitude shortest wavelength = largest frequency. A X - m A =.x - m 35 cm X - m cm = 35 x - m " = 3x8 m/s.x - m = 3x8 s - " = 3x 8 m/s = 9.3x 7 s - 35 x - m Note how these wave are Amplitude vs distance. They could be amplitude vs time -9 m 473 nm X nm = 473 x -9 m " = 3x 8 m/s 473 x -9 m = 6.34x4 s - Experiments over 3 years show that electromagnetic radiation (light) exhibits both wave-like and particle-like properties. Light can interact constructively and destructively interfere forming light and dark regions (fringes) on a wall. WAVE PARTICLE Light Intensity Light Intensity Waves bend Particles fly straight Slit Screen SINGLE SLIT: When light is passed through a tiny single slit one bright line appears. No interference of light waves (particle like) Two Slits Screen DOUBLE SLIT: two slits spaced closely together a pattern of bright and dark appears showing that light is an wave that interfers.

3 It has been known since the 6 s that light acts like a wave; it can be diffracted by passing light through small slits producing an interference pattern. In the early 9 s there were mysteries involving light and matter that the physicists could not explain including: Known in 6 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. BLACKBODY PROBLEM: All solids heated above T > 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 K 3K smoldering coal 5K electric heating elemen 7K Energy (joules) Planck postulated that energy and matter exist in fixed quantities or quanta of energy and not continuous energy states. E = n h " n= whole number "= frequency of light (Hz = sec - ) h= Plank s constant = 6.63 x -34 J s Matter exists in quantized states as opposed to continuous set of energy states. Max Planck Wavelength (nm) 9K light bulb filament Light can be absorbed from n = to any other higher n. Light can be emitted if electron falls from one level to another. It can also be emitted from one n to another. MYSTERY #: 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. Freed e - + pole battery Light evacuated tube light sensitive metal plate Current meter Al Einstein 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 -34 J s "= frequency of light (sec - ) Detailed Explanation of the Photoelectric Effect. Light acts like a particle and are quantized energy packets called photons. photon has energy = E = h". Photon has to have a certain minimum energy to eject an electron from the metal. E = hv = KE + Binding Energy 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 e - ejected 5 incoming red light ammeter no current incoming blue light emitter current flows

4 Learning Check: Photon-Energy and Photo Electric Effect Work- Function Calculations. Calculate the energy, in joules per photon, of radiation that has a frequency of 4.77 x 4 s -. E = h" = 6.63 x -34 J s X 4.77 x 4 s - = 3.6 x -9 J. If the energy of an infrared krypton laser is.484 x -9 joules per photon, determine the wavelength of the radiation. E = h" = hc = hc E = (6.63 x -34 J s) (3. x 8 m/s).484 x -9 joules per photon 3. Calculate the maximum kinetic energy of an electron ejected by the photon in part (b) from a metal with a binding energy of 3.7 ev. Binding Energy (Joules) = 3.7 ev x (.6 x -9 J / ev) = 5.9 x -9 J E = h" = KE + Binding Energy = 8. x -7 J KE = h" - Binding Energy = 8. x -7 J x -9 J = 8. x -7 J MYSTERY #3: Narrow bands of colors of light are emitted when gases are excited by high voltage in a tube. The colors are characteristic and reproducible for different elements. High voltage tube containing gaseous element Classical physics predicts a continuous spectrum (top). We observe discrete wavelengths, however, which correspond to discrete energy levels. These levels are quantized. The gas is excited by voltage generating light that 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 Wavelength (not scaled) alkali metals alkaline earth Metals gases MYSTERY #3: How could scientists account for the spectral lines if atomic theory and classical physics were correct? How does the emission spectrum relate the atomic and electronic structure of the H atom? (nm) for the visible series, n = and n = 3, 4, 5,... The first explanations of the emission spectra were empirical (i.e. some old dudes found an equation that worked, but without any explanation behind it). In 885 Balmer developed an empirical mathematical relationship between " and n for the visible lines of hydrogen emission spectrum. Balmer equation In 885 Johann Rydberg developed a generalized but empirical mathematical relationship between " and n for the all lines of hydrogen emission spectrum This is freaky Rydberg equation n λ (nm) = n = R n - n In 93, scientist Neils Bohr explains the emission spectrum of huydrogen using a quantized planetary model.. e - can only have specific (quantized) energy values (planetary like orbits around nucleus). light emission occurs when e - moves from higher energy level to a lower energy level (high n to low n) 3. the energy of a level: ( E n = -R ) H n 4. the difference in energy between any two levels: ( ) #E = h" = R H n i n f i = initial e - state and f = final e - state. R H (Rydberg constant) =.8 x -8 J n =6 n =5 n =4 n =3 n = Bohr postulates that light is emitted when an e - goes from an orbit with a high n value to a lower n value

5 The Bohr model explains all of the emission spectral lines of hydrogen as quantized jumps between the various n energy levels. E photon = hv = #E atom Calculate the wavelength (in nm) of a photon emitted by a hydrogen atom when its electron drops from the n = 5 state to the n = 3 state. n i = n f = E photon = E f - E i E f = -R H ( ) n f E i = -R H ( ) n i n #E = R H( ) i n f E photon = n #E = R H ( ) i n f E photon =.8 x -8 J x (/5 - /9) E photon = #E = -.55 x -9 J E photon = h x c / = h x c / E photon = 6.63 x -34 (J s) x 3. x 8 (m/s)/.55 x -9 J = 8 nm 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 94, 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 = mc Plank s Law Einstein s Law of Mass equivalence Solving for wavelength, substituting v = velocity for c and watching our units...we get the de Broglie equation: Louis DeBroglie λ = h mv = wavelength (m) v = velocity (m/s) h = Planck s constant (6.66 $ -34 J-s) It says that all particle of mass m, moving at velocity v have an associated wavelike wavelength We only observe wavelike properties of particles when the mass is tiny (electron). Macroscopic objects give de Broglie wavelengths that are too small to be observed. Substance Mass (g) Speed (m/s) de Broglie (m) slow electron fast electron alpha particle one-gram mass baseball Earth 9 x -8 9 x x -4. λ = 6. x 7 h mv. 5.9 x 6.5 x x 4 7x -4 x - 7x -5 7x -9.x -34 4x -63 Observable Not Observable The wave-like properties only manifest when mass is small and high velocity e - that move at high speeds. baseball What is the de Broglie wavelength (in meters) of: a) pitched baseball with a mass of. g and a speed of mph or 44.7 m/s and b) of an electron with a speed of. x 6 m/s (electron mass = 9. x -3 kg; h = 6.66 x -34 kg m /s). DeBroglie wavelength (m) mass of object (kg) λ = electron = 6.66 x -34 kg m /s (9.x -3 kg)(.x 6 m/s) Plank s Constant = h = 6.63 x -34 J s h mν velocity of object (m/s) λ = h mν = kg m /s (. kg) (44.7 m/s) =.4 34 m = 7.7 x - m

6 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. 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 x-ray diffraction of aluminum foil electron diffraction of aluminum foil A photon of light impinging on an electron would collide with an electron and alter it s position or momentum. Erwin Schrodinger, an Austrian physicist puts forth an equation that describes position and time dependence of an electron in hydrogen. It s called the Schrodinger wave equation. Schrodinger s equation is easier to solve if we transform the x,y,z coordinate system to spherical coordinates: r, ", #. The solution to this equation gives rise to function that specify 3 quantum numbers associated with an electron s energy levels that we call orbitals. Schrodinger s equation is a difficult partial differential equation. It is the framework for all of quantum mechanics Erwin Schrodinger %(x,y,z) ) change of coordinates ) solve the differential equation x,y,z to r, &, ' constants wave function h Ψ(x) 8π m e x + Ψ(y) y how % changes in space potential energy at x,y,z + Ψ(z) z Ze Ψ(x, y, z) = EΨ(x, y, z) r total quantized energy of the atomic system %(r,&,') = R(r) P(&) F(') n principal quantum number l orbital quantum number m l magnetic quantum number The solutions to the Schrodinger equation gives rise to functions that specify 3-quantum numbers that completely describe electrons in space. Here they are: n l m n,m, 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) (like the square of electric field in light). Allowed state = L n where n =,, 3 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. Non-allowed states can not be observed.

7 Energy What does the Schrodinger Equation Tell Us?. The solutions of the Schrodinger equation for the hydrogen atom is a list of functions called wavefunctions or orbitals that completely describe the spatial location and energy of an electron in a hydrogen atom.. Electrons in atoms, elements and molecules exist in discrete quantized states that are mathematically described by a wavefunction or orbital. The 4th quantum number: Stern-Gerlach Experiment-9. 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). Continuous distribution if the result was classical Quantum result gives EITHER -/ or +/...only states North 3. The wavefunctions or orbitals is specified by 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. South The wavefunction with its 4-quantum numbers specify all the information we can know about electrons around a nucleus. Chemists call this the electronic structure of an atom. Orbitals are degenerate or the same energy in Hydrogen Electrons will fill lowest energy orbitals first Hydrogen Atom Multi-electron atoms What the Quantum Numbers Mean. PRINCIPAL QUATUM NUMBER (n) or SHELL: Defines the size and energy level of the orbital. n = {,, 3, 4, etc}. (n = = K shell; n = = L shell; n = 3 = M shell. ANGULAR MOMEMTUM QUANTUM NUMBER (l) or SUBLEVEL Defines the shape of the orbital or volume in space where the electron is likely to be found. Also called a subshell. l = {,,,3...up to a maximum of n-} where ( = s, = p, = d, 3 = f) 3. MAGNETIC QUANTUM NUMBER (ml): Defines the spatial orientation of an orbital of the same energy. ml = {-l,, +l} 4. MAGNETIC SPIN QUANTUM NUMBER (ms): Defines the orientation of electron spin. ms = {+/ or -/}. Simple Summary of Quantum Numbers Name principal angular momentum Symbol Permitted Values Property n positive integers (,,3...) orbital energy (size) l integers from to n- orbital shape (,,, and 3 correspond to s, p, d, and f orbitals, respectively.) Allowed Values of Quantum Numbers n, l and m l magnetic m l integers from -l to to +l orbital orientation in space spin m s +/ or -/ direction of e - spin

8 Quantum Number n Allowed Values Positive integers,,3,4... Possible Orbitals 3 4 quantum numbers will describe each electron in an atom. The order of filling of the orbitals must be remembered using a mnemonic device. We are building toward the electronic configuration of atoms. l up to max of n- m l -l,...+l Orbital Name Shapes or Boundry Surface Plots s s p 3s 3p 3d # electrons 6 6 A shell describes the n quantum number (n = => K shell; n = => L shell and so on). A subshell refers to a specific n, l quantum number set: a s subshell or p subshell, or 3s subshell, or 3d subshell. Study the table. principal angular momentum s angular momentum s p s magneticp d spin s p d f magnetic Determining Quantum Numbers from Orbital Diagrams Write a set of quantum numbers for the third electron and a set for the eighth electron of the F atom. Use the orbital diagram to find the third and eighth electrons. SOLUTION: 9F s s p The third electron is in the s orbital. Its quantum numbers are: n = l = m l = m s = +/ The eighth electron is in a p orbital. Its quantum numbers are: n = l = m l = -,, or + m s = -/ Determining Sublevel Names and Orbital Quantum Numbers Give the orbital name, possible magnetic quantum numbers, and number of orbitals and number of electrons held in a sublevel with the following quantum numbers: (a) n = 3, l = (b) n =, l = (c) n = 5, l = (d) n = 4, l = 3 Determining Quantum Numbers for an Energy Level ) Name the 4 quantum numbers and say what they mean? ) 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? SOLUTION: n l Orbital name possible m l values # of orbitals (a) (b) (c) (d) d s 5p 4f -, -,,, -,, -3, -, -,,,, # e PLAN: SOLUTION: remember l values can be integers from to n-; m l can be integers from -l through to + l. For n = 3, l =,, For n = 4, l =,,,3 For l = m For l = m l = l = For l = m l = -,, or + For l = m l = -,, or + For l = m l = -,-,,+,+ For l = m l = -, -,, +, or + For l = 3 m l = -3,-,-,,+,++3 There are 9 m l values and therefore 9 orbitals with n = 3. There are 6 m l values and therefore 6 orbitals with n = 4.

9 How many p orbitals are there in an atom? n= p l = If l =, then m l = -,, or + 3 orbitals n = 4, l = p 3d n =, l =, m l = +/ If two electrons can be placed in one orbital how many electrons can go exist in the 3d subshell? n=3 3d l = If l =, then m l = -, -,, +, or + 5 orbitals which can each holding electrons for a total of e - in the subshell " has no physical meaning or reality (it s only a function). " represents the probability function for finding an electron. We can integrate the function to get the total probability of finding an electron. " = Mathematical Construct " = Probability Density The square of the wavefunction " is the probability density: a measure of the probability of finding an electron in a region of space. Probability density at point r in space s-orbital shape 3-D Boundary Surface Plot where electron spends 9% of its time Radial probability distribution of points added up in a circular strips of (r + rdr) Probability Density Plots nodes ( probability) Quantum Number n Allowed Values Positive integers,,3,4... Possible Orbitals 3 s s 3s l up to max of n- m l -l,...+l Orbital Name Shapes or Boundry Surface Plots s s p 3s 3p 3d Radial Probability Distribution Plots

10 The shape of an orbital is given by the l quantum number (l = {,, up to n-}. The number of orbitals and its orientation in space is given by the angular momentum quantum number ml {-l,...+l}. l = s orbital l = p orbital ml {-,,+} 3-p orbitals The 3 p (l = ) orbital Boundary Surface Plots (Shapes) l = d orbital ml {-,-,,+,+} 5-d orbitals l = 3 f orbital ml {-3,-,-,,+,+,+3} 7-f orbitals p x p y p z Three p-orbitals superimposed The Five d-orbital (l = ) Boundry Surface Plots (Shapes) The Seven f-orbital (l = 3) Boundry Surface Plots (Shapes) d yz d xz d xy d z Five d-orbitals superimposed d x - d y The Aufbau Process is used to generate the electronic configuration of elements filling the lowest energy orbitals sequentially.. Lower energy orbitals fill first.. Hund s Rule-degenerate (i.e. orbitals with the same energy) orbitals fill one at a time before electrons are paired in an orbital. 3. Pauli Exclusion Principle: No two electrons in an atom can have same 4- quantum numbers. Electrons fill the lowest energy orbitals first, at a time Chemists use spdf notation and orbital box diagrams to denote or show the ground state electronic configuration of elements. Element H He electron shell principal quantum # spdf Notation s s orbital type angular quantum # orbital box diagram An arrow denotes an electron with spin up or spindown. Remember, no two electrons can have the same 4 quantum numbers # of electrons in orbital

11 Building electronic configuration using Aufbau and Hund Atomic Number/Element H Orbital Box Diagram Full-electronic configuration s Condensed-electronic configuration s Atomic Number/Element B C Orbital Box Diagram Full-electronic configuration s s p s s p Condensed-electronic configuration [He]s p [He]s p He s s s s p 3 [He]s p 3 Li Be s s s s [He]s written with noble gas configuration [He]s s s p 4 s s p 5 s s p 6 [He]s p 4 [He]s p 5 [He]s p 6