18.3 Atomic Physics Atomic SpectraBohr ModelAtomic spectra and atomic structure.The spectra of atoms provide information about the energies of the electron in the atom. Sharp peaks at discrete wavelengths indicate that only specified energies are allowed in the atom.For the Hydrogen atom the Bohr theory explains the energies in a simple manner based on a quantization of angular momentum.The quantization is explained by the de Broglietheory in terms of standing waves for the electron.Atomic structureThe scattering of alpha particles (He2+) nuclei from a thingold foil. The back scattering of a few alpha particles showed that the nucleus is a small compact object.Ernest Rutherford 1911Geiger and MarsdenPlanetary model of the atomAlphaparticleScattering from a small compact nucleusAtomic spectraEmissionhigh voltagegas of atom Ae-AhfA -> e-(slow) + A*e-(fast) +A* -> A + hfExcitationEmissionspectrometerAtomic SpectraAbsorptionAlight source(white light)gas of atomslight minus absorbed wavelengthsspectrometer2Atomic SpectraEmissionAbsorption(dark lines)Discrete spectral lines are observed.Balmer series for Hydrogenultraviolet visibleLowest λ Highest λA series of peaks closer together (continuum) at low λRydberg ConstantThe Balmer series could be analyzed mathematically interms of an empirical equation. H22111R2n⎛⎞=−⎜⎟λ⎝⎠Rydberg Constant RH = 1.0973732x107m-1n = 3,4, 5 ..........Integers larger than 2.Disagreement with classical theoryClassical physics predicts that the energy of the electroncan have any value- not discrete values observe.The classical theory could not explain why the electron did not fall into the nucleus. Like a satellite falling into the earth.Bohr Theory1. Electrons move in circular orbits.2. Only specified atomic energy levels are allowed.3. Energy is emitted when electron go from one energy level to another.4. The orbital angular momentum of the electron is “quantized” in units of h/2π ==(called h bar)L = mvr = n=mvrn=1, 2, 3 .......Angular momentum of a tennis ballrmvr= 0.5 mm = 0.1 kgv= 2 m/sL =mvr2(0.1 )(2 / )(0.5 ) 0.1 0.1kgmkg m s m J ss===⋅34346.6 101.05 1022hxJsxJsππ−−⋅===⋅h33340.1101.0 10 /JsLxJs−⋅==⋅hhn=1033Ln=hWhat is n?is quantized.n is so large that quantization is not apparentnL3Angular momentum of an electronrmvm=9.1x10-31kgr =0.1 x10-9mv=?L is much smaller. Quantization is apparentLn ->1 2 3 4 5Classical dynamicsFor central forceF22e2kemvFmarr== =22ekemvr=v and r are not independent variablesBohr theory for hydrogen atomE=KE+PE22e12 ekqq ke1Emv2r2r=− =−mvr n= =Quantum conditionn= 1, 2, 3, ........integersnhmv2r=πangular momentum is quantizedmomentum increases as 1/rClassical energiesany value of r is allowedBohr restricted the values of rResults from Bohr theoryOnly specific values of r are allowed22n2eenrmke==n=1, 2, 3, ....... integersradius increases as n2n=1n=2n=3For n=123411131 9 2 19 2(1) (1.05 10 )5.3 10(9.1 10 )(8.9 10 )(1.6 10 )xJsrxmxkgxNmC xC−−−−−==0.053 nmSize of the Hydrogen atom in the ground stateExcited state energy levelsEnergy levels are quantized24een22 2mke1 13.6EeV2n n⎛⎞=− =−⎜⎟⎝⎠=Emission energies initial final22final initial11E E E 13.6nn⎛⎞∆= − = −⎜⎟⎝⎠Predicts spectral lines in the ultraviolet (Lyman series)and infrared (Paschen series), maximum energies, continuum.(proportional to 1/n2)hfmax=13.6 eV∆EmaxAgreement with Rydberg equation24ee32 2final initialmke1E 1 1hc 4 c n n⎛⎞∆== −⎜⎟λπ⎝⎠=24ee331 9 2 19 4718343mkeR4c9.109x10 (8.987x10 ) (1.602x10 )R1.099x10m4 (2.997x10 )(1.054x10 )−−−−=π==π=In excellent agreement with the experimentalvalue 1.097x107 m-14ExampleFind the wavelength of the transition from n=3 to n=2.722 22final initial7617111 11R 1.097x10nn 23111.097x10 1.52x10 m496.56x10 m 656nm−−⎛⎞⎛⎞=−= −⎜⎟⎜⎟λ⎝⎠⎝⎠⎛⎞=−=⎜⎟⎝⎠λ= =red line in Balmer seriesde Broglie wavelength explanation of Bohr theoryhmvr n2=πquantization of angular momentumh2r n nmv⎛⎞π==λ⎜⎟⎝⎠circumference = nλQuantization of angular momentum is equivalent to formingcircular standing waves.Bohr theoryShows that the energy levels in the hydrogen atom are quantized.Correctly predicts the energies of the hydrogen atom (and hydrogen like atoms.)The Bohr theory is incorrect in that it does not obey the uncertainty principle. It shows electrons in well defined orbits.Quantum mechanical theories are used to calculate the energies of electrons in atoms. (i.e. Shrödinger equation)Extension of the Bohr TheoryBohr theory can only be used to predict energies ofHydrogen-like atoms. (i.e. atoms with only one electron)This includes H, He+, Li2+ ....For example He+ ( singly ionized helium has 1 electron and a nucleus with a charge of Z = +2)For this case the energy for each state is multiplied byZ2 =4224een2222n222mkze1E2n111E 13.6(Z ) 13.6(2 ) 54.4 eVnnn⎛⎞=−⎜⎟⎝⎠⎛⎞ ⎛⎞=− =− =−⎜⎟ ⎜⎟⎝⎠ ⎝⎠=for He+Characteristic X-rays are due to emission from heavy atoms excited by electronsCharacteristic x-raysThe wavelength of characteristicx-ray peaks due to emission fromhigh energy states of heavy atoms (high Z).e-dislodgeK shell e-fast metal atom MoK shellUppershellhfKα5X-ray emissionBohr model with ShieldingEffective Z = 42-3Calculate the wavelength for Kαx-ray emission of Mo (Z=+42) The electron in the L shell must be in a l=1 state()223(Lshell)211E 13.6 Z 3 13.6(42 3) 5.17x10 eV24⎛⎞ ⎛⎞=− − =− − =−⎜⎟ ⎜⎟⎝⎠ ⎝⎠K shellL shellhfKα24Kshell21E 13.6(Z 1) 2.28x10 eV1⎛⎞=− − =−⎜⎟⎝⎠434E 2.28x10 5.17x10 1.76x10 eV∆= − =hcEhf∆= =λ34 81119 4hc (6.63x10 J)(3.0x10 m/s)7.1x10 mE (1.6x10 J/ eV)(1.76x10 eV)−−−λ= = =∆71 pm1s2s2pEffective Z =42-1Good agreementCalculatedvalue71