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Chapter One Atomic Structure Slide 1 of 58 The abundances of the elements in the universe Slide 2 of 58 Subatomic particle of relevance to chemistry Particle Symbol Mass/u* Mass number Charge/ e# Spin Electron e- 5.486 x 10-4 0 -1 ½ Proton p 1.0073 1 +1 ½ Neutron n 1.0087 1 0 ½ Neutrino ν c. 0 0 0 ½ Positron e+ 5.486 x 10-4 0 +1 ½ α particle α 4 2+ 2 He nucleus 4 +2 0 β particle β e- ejected from 0 nucleus -1 ½ γ photo γ Electromagnetic 0 radiation from nucleus 0 1 *Masses are expressed in atomic mass units, u, with 1 u = 1.6605 x 10-27 kg # Elementary charge e = 1.602 x 10-19 C. Slide 3 of 58 Nuclear Binding Energy for Helium Ebind = (∆m)C2 Slide 4 of 58 Average Binding Energies Slide 5 of 58 Rutherford’s Alpha Scattering Experiment Slide 6 of 58 Rutherford’s Model • Ernest Rutherford discovered the positive charge of an atom is concentrated in the center of an atom, the nucleus – An atom, can be visualized as a giant indoor football stadium – The nucleus can be represented by a pea in the center of the stadium, – The electrons are a few bees buzzing throughout. The roof of the stadium prevents the bees from leaving. Slide 7 of 58 Continuous Spectra Line Spectra Slide 8 of 58 Emission Spectrum of Hydrogen in Visible Light Region Slide 9 of 58 Line Spectra of Some Elements Slide 10 of 58 Planck’s Constant • Planck’s quantum hypothesis states that energy can be absorbed or emitted only as a quantum or as whole multiples of a quantum, thereby making variations discontinuous, changes can only occur in discrete amounts. • The smallest amount of energy, a quantum, is given by: E = hv as Planck’s constant, h = 6.626 X 10-34 J s. Slide 11 of 58 The Photoelectric Effect • Albert Einstein considered electromagnetic energy to be bundled in to little packets called photons. Energy of photon = E = hv – Photons of light hit surface electrons and transfer their energy hv = B.E. + K.E. hv e- (K.E.) – The energized electrons overcome their attraction and escape from the surface Slide 12 of 58 Bohr’s Hydrogen Atom Postulations • Rutherford’s nuclei model • The energy of an electron in a H atom is quantized • Planck & Einstein’s photon theory E = hv • Electron travels in a circle • Classical electromagnetic theory is not applied Z v r • e- Orbit Slide 13 of 58 (1) Classical physics centripetal force = Coulombic attraction mv2/r = Ze2/r2 (2) Total energy E = 1/2 mv2 - Ze2/r (3) Quantizing the angular momentum mvr = n (h/2π ) Quantum number E = - ( 2π2 mZ2e4)/(n2 h2) when n =1, E (1) = - (2π2 mZ2e4)/( h2) E = E (1) /n2 r = (n2 h2)/ (4π2 mZe2) Slide 14 of 58 Bohr’s Hydrogen Atom • Niels Bohr found that the electron energy (En) was quantized, that is, that it can have only certain specified values. • Each specified energy value is called an energy level of the atom En = - B/n2 – n is an integer, and B is a constant which equals 2.179 x 10-18 J – The energy is zero when the electron is located infinitely far from nucleus – The negative sign represents the forces of attraction Slide 15 of 58 The Bohr Model Slide 16 of 58 Bohr Explains Line Spectra • Bohr’s equation is most useful in determining the energy change (∆Elevel) that accompanies the leap of an electron from one energy level to another • For the final and initial levels: Ef = -B / nf2 Ei = -B / ni2 The energy difference between nf and ni is: ∆Elevel = Ef - Ei = ( -B / nf2 ) – (-B / ni2 ) = B(1/ni2 – 1/nf2) Slide 17 of 58 Energy Levels and Spectral Lines for Hydrogen IR Visible UV Slide 18 of 58 Ground States and Excited States • When an atom has its electrons in their lowest possible energy levels, it is in its ground state • When an electron has been promoted to a higher level, it is in an excited state – Electrons are promoted through an electric discharge, heat, or some other source of energy – An atom in an excited state eventually emits photons as the electron drops back down to the ground state Slide 19 of 58 Problems of Bohr’s Model of Atom • The energy levels of Bohr’s H atom cannot be applied to other atoms. • The orbit of electrons cannot be defined. Slide 20 of 58 The Uncertainty Principle • Werner Heisenberg’s uncertainty principle states that we can’t simultaneously know exactly where a tiny particle like an electron is and exactly how it is moving h h h= (∆Px) (∆x) = h/4π = Px = mvx 2 2π • The act of measuring the particle actually interferes with the particle • In light of the uncertainty principle, Bohr’s model of the hydrogen atom fails, in part, because it tells more than we can know with certainty. Slide 21 of 58 Uncertainty Principle Illustrated Slide 22 of 58 De Broglie’s Equation • Louis de Broglie speculated that matter can behave as both particles and waves, just like light • He proposed that a particle move with a mass m moving at a speed c will have a wave nature consistent with a wavelength given by the equation: p = mc E = mc2 = pc = hν p = hν /c = h/ λ λ = h/p = h/mc • De Broglie’s prediction of matter waves led to the development of the electron microscope Slide 23 of 58 What is the speed of an electron to have a wavelength of X-ray? Wavelength of X-ray ~ 0.1 nm = 1 x 10-10 m Mass of electron = 9.11 x 10-31 kg Planck constant h = 6.626 x 10-34 Js (kg m2/s) <Answer> λ= h/p = h/mv v = h/m λ = (6.626 x10-34)/[(9.11 x10-31)(1 x 10-10)] = 7 x106 m/s Slide 24 of 58 How to achieve the speed of an electron of 7 x106 m/s? V <Answer> E= eV= ½ mv2 V = ½ mv2 /e = ½ (9.11 x10-31)(7 x106)2 /(1.6022 x 10-19) V = 140 V (1 eV = 1.6022 x 10-19 J) Slide 25 of 58 Electron diffraction The experimental confirmation of de Broglie’s wave hypothesis was first made in 1927 by Davisson and Germer of the Bell Laboratories who investigated the scattering of electrons from various surfaces. Slide 26 of 58 Wave Functions • Quantum mechanics, or wave mechanics, is the treatment of atomic structure through the wavelike properties of the electron • Wave mechanics provides a probability of where an electron will be in certain regions of an atom Slide 27 of 58 Erwin Schrödinger developed a wave equation to describe the hydrogen atom • An acceptable solution to Schrödinger’s wave equation is called a wave function • A wave function (ψ) is characterized by an energy state of the atom ψ ( x, y , z ) ∫ +∞ −∞ 2 : the probability of finding an electron at (x, y, z) position in an atom ψ ( x, y, z) dτ = 1 2 The probability of finding an electron in the universe is equal to 1. Slide 28 of 58 y x Traveling wave y(x, t) = y0 sin(kx−ωt) y: amplitude of the wave If y is a function of x only, then, y(x) = y0 sinkx n=1 n=2 n=3 L x 0 L Standing wave y(x) = y0 sinkx Boundary condition: y = 0, when x = 0 y = 0, when x = L kL = nπ, k = nπ/L n = integers (quantum number) y(x) = y0 sin[(nπ/L) x] Slide 29 of 58 Standing Waves & Quantum Number Slide 30 of 58 Wave Mechanics Schrödinger equation one-dimensional P.E. d 2ψ 8π 2 m + [E − U (x )]ψ = 0 2 2 dx h Η̂ψ = Eψ Hamiltonian operator • ψ : amplitude of the wave • The probability that a particle will be detected is 2 proportional to ψ . h 2 d 2ψ − 2 + U (x )ψ = Eψ Operator for K.E. 2 P.E. 8π m dx h2 d 2 − 8π 2 m dx 2 + U ( x )ψ = Eψ 2 2 h d Ηˆ = − 2 + U (x ) 2 8π m dx Slide 31 of 58 three-dimensional Schrödinger equation ∂ 2ψ ∂ 2ψ ∂ 2ψ 8π 2 m + + 2 + [E − U ( x, y , z )]ψ = 0 2 2 2 ∂x ∂y ∂z h Η̂ ψ = Eψ 2 2 2 2 h ∂ ∂ ∂ ˆ Η = − 2 2 + 2 + 2 + U (x, y , z ) 8π m ∂x ∂y ∂z Slide 32 of 58 Particle in a Box One-dimensional Schrödinger equation d ψ 8π m + 2 [E − U (x )]ψ = 0 dx2 h 2 2 Η̂ ψ = Eψ 2 2 h d ˆ=− Η + U (x ) 2 2 8π m dx In the box, 0 < x < L and U = 0 2 2 2 2 h d h d ˆ=− Η =− 2 2 8π m dx 2m dx2 d 2ψ 8π 2m + 2 [E − U (x )]ψ = 0 dx2 h The probability that the particle will be detected is proportional to ψ 2 h= h 2π h 2 d 2ψ − = Eψ 2 2m dx Let ψ = A sin( kx) and A, k are constants h2 d 2 − [A sin( kx)] = E [A sin( kx)] 2 2m dx Slide 33 of 58 h2 d 2 − [A sin( kx)] = E [A sin( kx)] 2 2m dx Since d2 2 [ A sin( kx ) ] = − k [A sin( kx)] 2 dx h2 2 k [ A sin( kx) ] = E[ A sin( kx) ] 2m h2 k 2 E= 2m Boundary conditions for the particle in the box: 1. The particle cannot be outside the box. ψ ( 0) = 0 and ψ ( L) = 0 2. In a given state, the total probability of finding L the particle in the box must be 1. 3. The wave function must be continuous. ∫ 0 ψ ( x ) 2 dx = 1 Slide 34 of 58 ψ ( x ) = A sin( kx) Boundary condition ψ (0) = 0 and ψ ( L) = 0 ψ ( L) = A sin kL = 0 nπ kL = nπ , or k = , where n is intergers L nπ ψ ( x) = A sin x L Boundary condition ∫ L 0 0 ψ ( x ) 2 dx = 1 nπ 2 x ) dx = 1 ∫0 L L nπ 2 1 ∫0 sin( L x ) dx = A2 ψ ( x ) dx = 2 ∫ L L A 2 sin( Slide 35 of 58 nπ 2 L 1 ∫0 sin( L x) dx = 2 = A2 2 A= L L ψ ( x) = 2 nπ sin x L L h 2 k 2 h 2 nπ h 2 nπ E= = = 2 2m 2m L 8π m L 2 E= 2 2 nh 8mL2 2 ψ ψ2 Slide 36 of 58 Boundary condition for solving ψ in Schrödinger eq. for an electron in an atom: • ∫ +∞ −∞ ψ ( x, y , z ) d τ = 1 2 (normalization) • ψ(x,y,z) is a single valued function w.r.t. the coordinates • ψ(x,y,z) is a continuous function • ψ(x,y,z) is a finite function Slide 37 of 58 For H atom z θ Z= 1 x φ r ey − Ze 2 − e 2 U = = r r To solve the equation more easily, Cartesian coordinates x, y, z are transformed to polar coordinates r, θ, φ. ψ ( x , y , z ) = ψ (r , θ , φ ) = R (r )Θ(θ )Φ (φ ) = R (r )Υ (θ , φ ) Slide 38 of 58 Wavefunctions of Hydrogen Atom Slide 39 of 58 Slide 40 of 58 Quantum Numbers and Atomic Orbitals • The wave functions for the hydrogen atom contain three parameters that must have specific integral values called quantum numbers. • A wave function with a given set of these three quantum numbers is called an atomic orbital. • These orbitals allow us to visualize the region in which there is a probability of find an electron. Slide 41 of 58 Quantum Numbers When values are given to quantum numbers, a specific atomic orbital is defined The principal quantum number (n) – Can only be a positive integer n = 1, 2, 3 ···· – The size of an orbital and its electron energy depend on the n number – Orbitals with the same value of n are said to be in the same principle shell Value of n 1 2 3 4 5 Shell K L M N O Slide 42 of 58 Quantum Numbers (continued) • The orbital angular momentum quantum number (l) – Can have positive integral values 0≤ l ≤ n-1 – Determines the shape of the orbital – All orbitals having the same value of n and the same value of l are said to be in the same subshell – Orbitals and subshells are also designated by a letter: Value of l Orbital or subshell 0 s 1 2 3 p d f Slide 43 of 58 Quantum Numbers (continued) • The magnetic quantum number (ml): – Can be any integer from -l to +l -l ≤ ml ≤ l – Determines the orientation in space of the orbitals of any given type in a subshell – The number of possible value for ml = 2l + 1, and this determines the number of orbitals in a subshell Slide 44 of 58 The relationship between quantum numbers 0≤ l ≤ n-1 1s- orbital For example: n= 1, l= 0 2p, 2s- orbitals n= 2, l= 1, 0 n= 3, l= 2, 1, 0 3d, 3p, 3s- orbitals -l ≤ ml ≤ l For example: l= 1, ml = -1,0,1 px, py, pz- orbitals l= 2, ml = -2,-1,0,1,2 dxy, dyz, dzx, dz2, dx2-y2 orbitals Slide 45 of 58 Slide 46 of 58 Quantum Numbers Summary Slide 47 of 58 The 1s Orbital Υ0, 0(θ,φ) = 1/2π1/2 • The 1s orbital has spherical symmetry. • The electrons are more concentrated near the center Slide 48 of 58 The 2s Orbital Υ0,0(θ,φ) = 1/2π1/2 (+) r 0 (-) 0 node 0 Slide 49 of 58 The 2s Orbital • The 2s orbital has two regions of high electron probability, both being spherical • The region near the nucleus is separated from the outer region by a spherical node- a spherical shell in which the electron probability is zero node Slide 50 of 58 Slide 51 of 58 The Three p Orbitals Slide 52 of 58 The Five d Orbital Shapes Slide 53 of 58 The Seven f Orbital Shapes Slide 54 of 58 Electron Spin – the 4th Quantum Number • The electron spin quantum number (ms) explains some of the finer features of atomic emission spectra – The number can have two values: +1/2 and –1/2 ( ms= ½ , -½ ) – The spin refers to a magnetic field induced by the moving electric charge of the electron as it spins – The magnetic fields of two electrons with opposite spins cancel one another; there is no net magnetic field for the pair. Slide 55 of 58 The Stern-Gerlach Experiment Slide 56 of 58 Hydrogen atom and Schrö dinger equation • energy is quantized (n) • magnitude of angular momentum is quantized (l) • the orientation of angular momentum is quantized (ml) Electron has spin (ms) Slide 57 of 58 Z 2 me 4 E=− 2 2 2n h hydrogen atom Slide 58 of 58