Chem. 140B Dr. J.A. Mak Introdution to Quantum Chemistry Without Quantum Mehanis, how would you explain: Periodi trends in properties of the elements Struture of ompounds e.g. Tetrahedral arbon in ethane, planar ethylene, et. Bond lengths/strengths Disrete spetral lines (IR, NMR, Atomi Absorption, et.) letron Mirosopy & surfae siene Why as a hemist, do you need to learn this material? Without Quantum Mehanis, hemistry would be a purely empirial siene. (We would be no better than biologists ) 140B Dr. Mak 1 140B Dr. Mak 2 Classial Physis The Failures of Classial Mehanis On the basis of experiments, in partiular those performed by Galileo, Newton ame up with his laws of motion: 1. A body moves with a onstant veloity (possibly zero) unless it is ated upon by a fore. 2. The rate of hange of motion, i.e. the rate of hange of momentum, is proportional to the impressed fore and ours in the diretion of the applied fore. 3. To every ation there is an equal and opposite reation. 4. The gravitational fore of attration between two bodies is proportional to the produt of their masses and inversely proportional to the square of the distane between them. m1m2 F = G 2 r 140B Dr. Mak 3 1. Blak Body Radiation: The Ultraviolet Catastrophe 2. The Photoeletri ffet: instein's belt bukle 3. The de Broglie relationship: Dude you have a wavelength! 4. The double-slit experiment: More wave/partile duality 5. Atomi Line Spetra: The 1 st observation of quantum levels 140B Dr. Mak 4
Blak Body Radiation Light Waves: letromagneti Radiation Light is haraterized by a wavelength and frequeny: The wavelength is given the symbol ( lambda ) an be measured from rest to rest or trough to trough. Light is omposed of two perpendiular osillating vetors waves: a node is a point of zero amplitude A magneti field & an eletri field As the light wave passes through a substane, the osillating The amplitude fields an stimulate the movement of eletrons in a substane. measures the rest The frequeny of light ν (nu) measures the number height of a wave. of rests that pass through a point in spae per (m) ν(s 1 ) = (m s 1 ) ν= seond. 140B Dr. Mak 6 140B Dr. Mak 5 Wavelength has units of length: m m µm nm pm The letromagneti Spetrum: Frequeny has units of inverse time: 1 s or Hz (hertz) (m) ν(s 1 ) = (m s 1 ), the speed of eletromagneti radiation (light) moving through a vauum is: 2.99792458 10 8 m/s ν= = ν 140B Dr. Mak 7 140B Dr. Mak 8
Quantized nergy and Photons 1900: Max Plank explained the phenomenon Blak Body Radiation by onluding that light must be quantized. Cavity with a small opening When light enters the avity it is refleted throughout the internal surfae. (measured) xperimentally, the wavelength of maximum intensity shifts to the blue as temperature inreases for a BBR. Classially, the intensity (spetral density) of the light emitted by a blak body radiator is predited to inrease infinity as the temperature inreases (as dereases). The light that esapes is representative of the internal temperature of the the avity. Classial theory predits that the intensity of the light that esapes inreases with the frequeny of the light. This leads to what is alled the Ultraviolet Catastrophe. 140B Dr. Mak 9 Rayleigh-Jeans Law 2πkT f( ) = 4 Based on this lassial interpretation, for a given temperature, as approahes zero (more to the UV) the intensity approahes infinity. 140B Dr. Mak 10 Plank s equation orreted for lassial failure by stating that energy an only be transferred in a finite minimum quantity. h = Plank s onstant (6.626 10 34 J s) As dereases: 2πkT h e 1 = 5 kt f( ) Plank s orretion avoids the Ultraviolet Catastrophe by reduing the intensity as dereases: Plank s Law = h ν As the frequeny of light inreases, the energy of the inreases ombining: ν = yields: h = As the wavelength of light inreases, the energy of the dereases Blue Light, (higher frequeny) has more energy than Red Light, with a longer wavelength. 140B Dr. Mak 11 140B Dr. Mak 12
As the frequeny of light inreases, the energy inreases. As the wavelength of light inreases, the energy dereases. = h ν = Red Light (650 nm) = = 3.00 10 8 m s 1 m 650 nm 9 10 nm -34 6.626 10 Js -19 J = 3.06 10 In 1905 Albert instein used Plank s Law to explain the Photoeletri ffet. When light strikes the surfae of ertain metals, eletrons are ejeted. The ejeted eletrons produe a urrent that proportional to their number. This doesn t seem like muh, but when you onsider a mole of s 184 kj/mol Now that s what I m talkin about! 140B Dr. Mak 13 It is found that the urrent produed follows the light intensity. And there is a frequeny dependene, below whih, no eletros are ejeted. 140B Dr. Mak 14 1905 instein: Osillators in light soure an only have quantized energies nhν (n = 0,1,2,3, ). As osillators hange their energy from nhν to (n 1)hν they emit radiation of frequeny ν and energy hν (). Therefore, if an osillator is to absorb a, the s energy must be greater then or equal to a minimum threshold energy ø to stimulate ejetion of an eletron. Below the threshold energy, no ejetion of eletrons our. Above that, any exess energy supplied by the is manifested in the kineti energy of the ejeted eletron. ø At the threshold, the eletrons are ejeted with zero K. K (eletron) = () Φ = ½ m e v 2 () = hν= 140B Dr. Mak 15 140B Dr. Mak 16
xample: A ertain metal has a threshold nergy (ø) of 146.0 kj/mol. Will 532 nm light ause eletrons to be ejeted? if () > ø then eletrons are ejeted i.e. the energy of the must be greater then the threshold. xample: A ertain metal has a threshold nergy (Φ) of 146.0 kj/mol. Will 532 nm light ause eletrons to be ejeted? if () > Φ then eletrons are ejeted i.e. the energy of the must be greater then the threshold. Solution: Calulate the energy of a mole of 532 nm s = hν= = 3.74 10 19 J kj 3 10 J 23 6.022 10 s = 225 kj/mol mol = 34 6.626 10 Js 532 nm 8 m 3.00 10 s 140B Dr. Mak 17 m 9 10 nm = 3.74 10-19 J () > Φ letron are ejeted 140B Dr. Mak 18 The Wave-like Nature of a Partile Louis de Broglie in response to Plank & instein s assertion that light was partile-like () stated that small partiles moving fast ould exhibit a harateristi wavelength. = m 2 h ν = m h ν = m = p (momentum) 2 sine Conlusion: Light waves have mass, partiles have a wavelength. ν 1 h h = = p or = p What is the de Broglie wavelength of a 1 gram marble traveling at 10 m/s h = 6.63 10-34 J s = 6.6 10-30 m = 6.6 10-20 Å (insignifiant) What is the de Broglie wavelength of an eletron traveling at 0.1 (=speed of light)? = 3.00 10 8 m/s m e = 9.1 10-31 kg = 2.4 10-11 m = 0.24 Å (on the order of atomi dimensions) 140B Dr. Mak 20 140B Dr. Mak 21
The partile and the wave: The eletron looks like a wave superimposed on a partile: The Double-slit experiment When light waves impinge upon a single slit, they may pass suh that those inident lear with no destrutive interferene (a). When light waves at aute angles, they do so with interferene that is related to the angle of inidene. (b) and () The eletron appears as a build up of amplitude in the wave at position x: x Unertainty arises beause of the width at x. x 140B Dr. Mak 22 140B Dr. Mak 24 The result of wave transmission amplitude build up is a diffration pattern. A soure of eletrons are direted toward the slits. With one slit losed, we see the expeted build up of intensity. With both slits open, we see a diffration pattern that is fits wavelike harateristis! 140B Dr. Mak 25 140B Dr. Mak 26
Suh a pattern an only our if the partile passes through both slits simultaneously! Line Spetra and the Bohr Model 1860: Robert Wilhelm Bunsen and Gustav Kirhoff noted the presene of dark lines arising from absorption of light when observing the spetrum of a bright light soure through the flame seeded with alkali metals. The partile must have wavelike properties to do so. 140B Dr. Mak 27 Normal spetrum of white light. Gaps due to absorption by atoms in the flame 140B Dr. Mak 28 Atomi Line Spetra When a disharge lamp of He is passed through a prism, Zap! Atomi Line Spetra: Pre-1900 Numerous researhers produed atomi spetra by heating up atoms of a material to high temperature and olleting the emitted energy in the form of an atomi spetrum. He(g) + He*(g) 1911 Rutherford proposes model of the atom. Positive entral nuleus surrounded by many eletrons. The He(g) is exited. The exited state relaxes through ollisions, produing light only at ertain frequenies (olors). He*(g) He(g) + hν 140B Dr. Mak 29 1913 Bohr s laws of the Hydrogen atom struture: 1. letron orbits nuleus (like a planet around the sun) 2. Of the possible orbits only those for whih the orbital angular momentum of the eletron is an integral multiple of h/2π are allowed. 3. letrons in these orbits don t radiate energy. 4. When an eletron hanges its orbit a quantum of energy () is emitted with energy = hv, where is the energy differene between the two orbits. 140B Dr. Mak 30
mission of Light: letrons move from a higher level (state) to a lower level (state) Absorption of Light: letrons move from a lower level (state) to a higher level (state) 140B Dr. Mak 31 nergy Levels in the Bohr Atom: The spaing between adjaent levels is given by: = n+ 1 between n = 1 and 2: 3R = = 4 H 0.75 RH between n = 2 and 3: 5R = = 36 H 0.139 R H n nergy R = n n = H n 2 n = 4 n = 3 n = 2 n = 1 (as n inreases, the levels get loser together) virtual ontinuum of levels 140B Dr. Mak 34 The energies of different transitions: Sine the gaps between states gets loser and loser together with inreasing n, the frequeny of the light emitted hanges. n = 2 1 > 3 2 > 4 3 Determine the wavelength (in nm) assoiated with an eletron jumping from n = 2 to n = 5 in a hydrogen atom. 1 1 = R n 2 2 final n initial n final = 5 n initial = 2 And 3 1 > 4 2 nergy n = 4 n = 3 n = 2 = 18 1 1 2.179 10 J 2 2 5 2 19 = 4.576 10 J n =5 n =2 140B Dr. Mak 35 n = 1 The value of is positive beause this is an absorption. 140B Dr. Mak 36
Determine the wavelength (in nm) assoiated with an eletron jumping from n = 2 to n = 5 in a hydrogen atom. h = (meters) = h 19 = 4.576 10 J 34 8 m 6.626 10 Js 2.997 10 = 19 4.576 10 J s Shrödinger Wave quations and the Origins of Orbitals Taking on the ideas of Bohr, de Broglie and Heisenberg, Irwin Shrödinger proposed that matter an be desribed as a wave. In this theory, the eletron is treated as both a wave and a partile. 7 = 4.340 10 m 9 10 nm 1m = 434.0 nm An eletron is desribed by a Wave Funtion Ψ that ompletely defines a system of matter. 140B Dr. Mak 37 140B Dr. Mak 38 The mental piture of an eletron orresponds to a wave superimposed upon the radial trajetory of a partile orbiting the nuleus. The position of an eletron is best desribed by the image of a dart board: As the Shrödinger Wave quation is solved time and time again, the position of an eletron is found suh that eah hit builds up a pattern. Probability Distributions 140B Dr. Mak 39