Chapter 7 Quantum Theory and Atomic Structure 7-1 7-2 제 7장 양자론과 원자 구조 7.1 빛의 파동성, 빛의 입자성 고전물리학에서 양자론까지 광전효과 7.2 원자 스펙트럼, 수소 원자의 보어 모형 Bohr 수소 7.3 물질과 에너지의 파동-입자 이중성 하이젠베르크의 불확정성 원리 전자의 이중성 양자역학 7.4 원자의 양자역학적 모형, 원자 궤도함수 양자수 원자궤도함수 7-3 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 7-4 Maxwell (1873), proposed that visible light consists of electromagnetic waves. Electric field electromagnetic waves. Amplitude Magnetic field Direction of wave Speed of light (c) in vacuum = 299 792 458 m/s (exact) = 3.00* × 108 m/s 7-5 All electromagnetic radiation l×n=c 7.1 Human km m MHz radio Atom Bacteria mm GHz microwave μm THz THz nm PHz IR Ǻ EHz UV X-ray pm ZHz Gamma FM Radio 88-108 MHz AM Radio 0.6-1.6 MHz microwave oven 2.4 GHz 7-6 Medical X 10-0.1Ǻ PET Imaging 0.1-0.01Ǻ Convert frequency to wavelength A photon has a frequency of 6.0× 104 Hz. Convert this frequency into wavelength (nm). Does this frequency fall in the visible region? λ×n=c λ = c/n λ = 3.00 × 108 m/s / 6.0 × 104 Hz λ = 5.0 × 103 m AM Radio wave λ = 5.0 × 1012 nm 7-7 7.1 Different behaviors of waves and particles. 7-8 from Silberberg Figure 7.4 The diffraction pattern caused by light passing through two adjacent slits. 7-9 Mystery #1, “Black Body Problem” Stefan–Boltzmann law Wien’s law Ultraviolet catastrophe! Rayleigh-Jeans formula F(l ) = 7-10 2ckT l4 7.1 Mystery #1, “Black Body Problem” Solved by Planck in 1900 Energy (light) is emitted or absorbed in discrete units (quantum). F(l ) = 2c l 5 2 h e hc lkT 1 E=h×n Planck’s constant (h) h = 6.626×10-34 J•s 7-11 7.1 When (hc/lkT) << 1 (long wavelength range) h e = hc lkT 1 h 2 hc hc 1 ... 1 lkT lkT h 2 hc hc hc 1 ... lkT lkT lkT = lkT hc hc 2 c 1 ... lkT lkT lkT c F(l ) = = 7-12 = 2c l 5 2 h e hc lkT 2c lkT = 5 l c 1 2 2ckT l 4 7.1 When (hc/lkT) >> 1 (short wavelength range) h hc lkT e F(l ) = = 7-13 2c 2c h 2 l 5 1 e 2 l 5 e h lhc kT hc lkT = he h e hc lkT 1 2c l 5 2 lhc kT he lhc kT 7.1 Mystery #2, “Photoelectric Effect” 7-14 from:http://www.mtmi.vu.lt/pfk/funkc_dariniai/quant_mech/wave_f.htm 7.2 Mystery #2, “Photoelectric Effect” Photoelectric Effect on Sodium Plate hn KE e- Reference: http://hyperphysics.phy-astr.gsu.edu/hbase/mod2.html 7-15 7.2 Mystery #2, “Photoelectric Effect” KE = hn - BE KE Photoelectric Effect on Sodium Plate = 1.25 eV = 6.6 × 10-34 Js × 1.60217646 × 10-19 J/eV = 2.0025 × 10-19 J n 0 = 1.82 eV -BE 7-16 = 1.82 × 1.60217646 × 10-19 J/eV = 2.91 × 10-19 J 7.2 Work Functions for Photoelectric Effect Work function (Binding Energy) CRC handbook on Chemistry and Physics version 2008, p. 12-114 7-17 Element Aluminum Beryllium Cadmium Calcium Carbon Cesium Cobalt Copper Gold Iron Lead Magnesium Mercury Nickel Niobium Potassium Platinum Selenium Silver Sodium Uranium Zinc Work Function(eV) 4.06-4.26 4.98 4.08 2.87 ~5 2.14 5.0 4.53-4.95 5.1 -5.47 4.67-4.81 4.25 3.66 4.475 5.014-5.35 3.95-4.87 2.29 5.12-5.93 5.9 4.52-4.74 2.36 3.63-3.90 3.63-4.9 Mystery #2, “Photoelectric Effect” Solved by Einstein in 1905 Light has both: 1. wave nature 2. particle nature hn KE e- Photon is a “particle” of light hn = KE + BE KE = hn - BE 7-18 7.2 Convert wavelength to energy When copper is bombarded with high-energy electrons, X rays are emitted. Calculate the energy (in joules) associated with the photons if the wavelength of the X rays is 0.154 nm. E=h×n E=h×c/l E = 6.63×10-34 (J•s) × 3.00×10 8 (m/s) / 0.154×10-9 (m) E = 1.29×10 -15 J 7-19 7.2 Comparing the diffraction patterns of x-rays and electrons. x-ray diffraction of aluminum foil electron diffraction of aluminum foil from Silberberg Figure 7.14 7-20 The diffraction pattern caused by electron passing through two adjacent slits. 7-21 The line spectra of several elements. 7-22 from Silberberg Figure 7.7 Line Emission Spectrum of Hydrogen Atoms Rydberg equation 1 1 = RH 2 2 l n2 n1 1 RH is the Rydberg constant = 1.096776×107 m-1 Lyman Balmer n1 = 1 and n2 = 2, 3, 4, ... n1 = 2 and n2 = 3, 4, 5, ... 7-23 Paschen n1 = 3 and n2 = 4, 5, 6, ... 7-24 7.3 Chemistry Mystery: Discovery of Helium In 1868, Pierre Janssen detected a new dark line in the solar emission spectrum that did not match known emission lines Mystery element was named Helium (from helios) In 1895, William Ramsey discovered helium in a mineral of uranium (from alpha decay). Hydrogen Waveleng th(nm) Color 435 purple 486 blue 657 red 7-25 587.49 nm Helium Wavelength (nm) Color 447 purple 469 blue 472 blue 493 blue-green 501 blue-green 505 blue-green 587 yellow 669 red Bohr’s Model of the Atom (1913) 1. e- can only have specific (quantized) energy values 2. light is emitted as e- moves from one energy level to a lower energy level +Ze Z2 En = RH 2 n 2 n rn = a0 Z n (principal quantum number) = 1,2,3,… RH (Rydberg constant) = 2.17987209 ×10-18J = 1.096776×107 m-1 7-26 7.3 1 En = RH 2 n 1 𝐸𝑓 = −𝑅H ( 2 ) 𝑛𝑓 1 𝐸𝑖 = −𝑅H ( 2 ) 𝑛𝑖 ni = 3 nf = 2 7-27 656.3 nm ∆𝐸 = 𝐸𝑓 − 𝐸𝑖 = 𝑅H ( 1 𝑛𝑖 2 − 1 𝑛𝑓 2) 7.3 Calculate a hydrogen atomic emission line wavelength 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. 1 1 Ephoton =DE = RH ( 2 ) 2 ni nf Ephoton = - 2.18×10-18 J × (1/25 - 1/9) = 1.55 ×10-19 J 1/ l = -1.10×107 m-1 × (1/25 - 1/9) Ephoton = h×c / l = 7.82 × 10 5 m-1 l = 1.28×10-6 m = 1280 nm l = h×c/ Ephoton l = 6.63×10-34 (J•s) × 3.00×108 (m/s)/1.55×10-19J = 1.28×10-6 m l = 1280 nm 7-28 7.3 De Broglie (1924) reasoned that e- is both particle and wave. 2 r = nl , n = 1, 2,3,... h l= mu u = velocity of e- m = mass of e- 7-29 7.4 Calculate the de Broglie wavelength What is the de Broglie wavelength (in nm) associated with an electron traveling at 1.00× 106m/s? λ = h/mu h in J•s m in kg u in (m/s) λ = 6.63×10-34 / (9.11×10-31 × 1.00×106) λ = 7.27×10-10 m = 0.727 nm 7-30 7.4 Bohr 수소 원자와 양자역학 Bohr 이론 양자역학 전자는 원운동을 함 전자 궤도 상에 존재함 s 전자는 원운동 하지 않음 전자의 위치는 정할 수 없고 확률 분포함수 s 전자는 핵에서 발견될 확률이 최대 전자는 핵에 가까이 갈 수 없음 다전자 원자 적용 불가능 7-31 다전자 원자 및 분자에 적용 가능 Electron Imaging Electron Microscopy le = 0.004 nm STM's ability to image the wave patterns of electrons on the surface of a metal 1993 - IBM's Quantum Corral 7-32 Classical Theory to QUANTUM THEORY Observation Explanation by when blackbody radiation Energy is quantized; only certain values allowed Planck 1900 photoelectric effect Light has particulate behavior (photons) Einstein 1905 atomic line spectra Energy of atoms is quantized; photon emitted when electron changes orbit. Bohr 1913 Davisson/Germer: electron diffraction by metal crystal All matter travels in waves; energy de Broglie of atom is quantized due to wave motion of electrons Compton scattering(1923) photon wavelength increases (momentum decreases) after colliding with electron Mass and energy are equivalent; particles have wavelength and photons have momentum 7-33 Einstein/ de Broglie 1924 Heisenberg Uncertainty Principle 위치와 운동량과의 관계 Dx Dp h , where = 2 2 위치와 운동량을 동시에 정확히 알 수 없다. 7-34 Estimate uncertainty of a position 양성자의 위치를 1×10-11m의 정확도로 측정하였다. 1초 후의 양성자 위치의 불확정성은? Dx Dp = 2 Dx = D v t 7-35 , Dp 2mDx0 t 2Dx0 , Dv = 3.15 103 m Dp m 2mDx0 Estimate minimum uncertainty of momentum and speed 원자핵의 반경은 5×10-15m이다. 전자가 원자핵의 일부가 되기 위해 필요한 운동에너지는? Dx Dp = 2 Dp 1.110 20 kg m / s 2Dx Dp = mDv = 1.11020 kg m / s Dv = 1.11020 kg m / s /(9.11031 kg) = 1.2 1010 m / s c KE = p 2c 2 m 2c 4 m c2 pc KE = pc 1.11020 kgm / s 3 108 m / s = 3.31012 J 20 MeV 너무 큰 에너지로 전자가 핵 안에 있을 확률이 거의 없다. 7-36 Estimate minimum uncertainty of momentum and speed 수소원자의 반경이 5.3×10-11m이다. 전자의 최소 운동에너지는? Dx = 5.31011 m 9.9 10 25 kgm / s 2Dx p2 KE = = 5.4 1019 J = 3.4 eV 2m Dp = 7-37 Schrodinger Wave Equation In 1926 Schrodinger wrote an equation that described both the particle and wave nature of the eWave function (Y) describes: 1. energy of e- with a given Y 2. probability of finding e- in a volume of space Schrodinger’s equation can only be solved exactly for the hydrogen atom. Must approximate its solution for multi-electron systems. 7-38 7.5 The Schrödinger Equation wave function Hamiltonian Operator kinetic energy HY = EY potential energy total quantized energy of the atomic system H = T V 2 2 2 2 = ( 2 2 2 ) V ( x, y, z ) 2me x y z mass of electron 2 2Y 2Y 2Y ( 2 2 2 ) (V ( x, y, z) E)Y = 0 2me x y z 7-39 Schrodinger Wave Equation Ψ = f(n, l, ml, ms) principal quantum number n n = 1, 2, 3, 4, …. distance of e- from the nucleus n=1 7-40 n=2 n=3 7.6 Where 90% of the e- density is found for the 1s orbital 7-41 7.6 Schrodinger Wave Equation Ψ = f(n, l, ml, ms) angular momentum quantum number l for a given value of n, l = 0, 1, 2, 3, … n-1 n = 1, l = 0 n = 2, l = 0 or 1 n = 3, l = 0, 1, or 2 l=0 l=1 l=2 l=3 s orbital p orbital d orbital f orbital Shape of the “volume” of space that the e- occupies 7-42 7.6 l = 0 (s orbitals) l = 1 (p orbitals) 7-43 7.6 l = 2 (d orbitals) 7-44 7.6 Schrodinger Wave Equation Ψ = f(n, l, ml, ms) magnetic quantum number ml for a given value of l ml = -l, …., 0, …. +l if l = 1 (p orbital), ml = -1, 0, or 1 if l = 2 (d orbital), ml = -2, -1, 0, 1, or 2 orientation of the orbital in space 7-45 7.6 ml = -1 ml = -2 7-46 ml = 0 ml = -1 ml = 0 ml = 1 ml = 1 ml = 2 7.6 Schrodinger Wave Equation Ψ = f(n, l, ml, ms) spin quantum number ms ms = +½ or -½ ms = +½ 7-47 ms = -½ 7.6 Schrodinger Wave Equation Ψ = f(n, l, ml, ms) Existence (and energy) of electron in atom is described by its unique wave function Ψ. Pauli exclusion principle - no two electrons in an atom can have the same four quantum numbers. Electron is a fermion. Only one fermion can occupy a quantum state at a given time; this is the Pauli Exclusion Principle. 7-48 7.6 Schrodinger Wave Equation Ψ = f (n, l, ml, ms) Shell – electrons with the same value of n Subshell – electrons with the same values of n and l Orbital – electrons with the same values of n, l, and ml How many electrons can an orbital hold? If n, l, and ml are fixed, then ms = ½ or - ½ Ψ = f (n, l, ml, ½) 7-49 or Ψ = f (n, l, ml, -½) An orbital can hold 2 electrons 7.6 How many 2p orbitals are there in an atom? n=2 If l = 1, then ml = -1, 0, or +1 2p 3 orbitals l=1 How many electrons can be placed in the 3d subshell? n=3 3d l=2 7-50 If l = 2, then ml = -2, -1, 0, +1, or +2 5 orbitals which can hold a total of 10 e- 7.6 Figure 7.16 Electron probability in the ground-state H atom. 7-51 Determining Quantum Numbers for an Energy Level What values of the angular momentum (l) and magnetic (ml) quantum numbers are allowed for a principal quantum number (n) of 3? How many orbitals are allowed for n = 3? For n = 3, l = 0, 1, 2 For l = 0 ml = 0 For l = 1 ml = -1, 0, or +1 For l = 2 ml = -2, -1, 0, +1, or +2 There are 9 ml values and therefore 9 orbitals with n = 3. 7-52 Determining Sublevel Names and Orbital Quantum Numbers Give the name, magnetic quantum numbers, and number of orbitals for each sublevel with the following quantum numbers: (a) n = 3, l = 2 (b) n = 2, l = 0 (c) n = 5, l = 1 (d) n = 4, l = 3 n l (a) 3 2 3d -2, -1, 0, 1, 2 5 (b) 2 0 2s 0 1 (c) 5 1 5p -1, 0, 1 3 (d) 4 3 4f -3, -2, -1, 0, 1, 2, 3 7 7-53 sublevel name possible ml values # of orbitals Figure 7.18 7-54 The 2p orbitals. 7-55 Figure 7.20 7-56 The 3d orbitals. The 3d orbitals along z-axis on xy-plane, betw. xy-axis 7-57 on xy-plane, along xy-axis on yz-plane, betw. yz-axis on zx-plane, betw. zx-axis 4f orbitals. 7-58