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# 12.4, 12.7-12.8 CHEM 1B

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This 14 page Class Notes was uploaded by Stacy Vargas on Saturday October 10, 2015. The Class Notes belongs to CHEM 1B at University of California - Santa Cruz taught by Roberto Bogomolni in Fall 2015. Since its upload, it has received 39 views. For similar materials see CHEM 1B in Chemistry at University of California - Santa Cruz.

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Date Created: 10/10/15

124 The Quantum Mechanical of the Atom an electron bound to the nucleus was similar to a standing wave 0 An example is standing waves in musical instruments such as guitars or violins The image below explains the concept U npl LlC ktl stri n g If lts l linese e n gllt s Figure 1211 The standing wese predLl eee by the vlbretien ef e gelter string festenee st beth ends Eeeln Clef represents e nettle Figure s peint efzere tlispleeernent 1212 Only certain circular orbits have a circumference into which a whole of wavelengths of the standing electron wave will t other orbits would produce destructive interference of the standing electron wave Schrodinger developed a mathematical equation in which describes the hydrogen electron as a wave based on the classical description of a wave phenomena Jim Es 0 Refer to equation above the wave function W is a function of coordinates x y and z of the electron s position in a 3D space Fl Hamiltonian represents a set of mathematical instruction called an operator 0 Operator a mathematical tool that acts on a function to produce another function 0 In Schrodinger s equation the H acts to give back the wave function multiplied by the constant E which represents the total energy of the atom sum of potential energy due to attraction between the proton and the electron amp the kinetic energy KE of the moving electron o Orbital a speci c wave function for a given electron Mi emefeh ti l it i a W w Figure 1212 The hydregen eleetren iiieuellieecl eel e etendiing wave ereunci the nucleus The eireumferenee elf e lpertieuller eireulleitr erbit nee tie eerreepnd te a wlhele number elf weeelengthe ale ehewn in lei and lb er melee deetruetiv interier einee eeeulre ae ehewn in C This medel ie C39DvlfiSlSltEl39iT with the feet that enly certain eleetren energiee ere eileweci the etern is quantized elltheugh thie idea bee eneeuraged eelentlete tel uee e we39ve thleew it deee net meen that the eleetren reeiiy treeelle in elireuler erbite 1St orbital the wave function corresponding to the lowest energy 0 The meaning of orbital is the rst point of interest 0 Wave function does not give information about the movements of the electrons we can only predict their trajectories o there is a fundamental limitation to just how precisely we can know both position and momentum of a particle at a given time o This principle is stated mathematically as fig 0 Where Ax is the uncertainty in a particle s position Ap is the uncertainty in a particle s momentum and h is Planck s constant divided by 275h275 o The minimum uncertainty is h47c This relationship shows that the more precisely a particle s position is known the less precisely its momentum is known and vice versa 0 Limitation is so small for large particles but limitation becomes important when it comes to 5 small particle such as an electron 0 Since an electron s exact path cannot be known it is not appropriate to assume that the electron is moving around the nucleus 127 The Wave Equation for the Hydrogen Amm The electron of the hydrogen atom moves in 3D and has potential energy because of its attraction to the positive nucleus of the atom Differences can be accounted for by including the 2nOI derivatives with respect to wrt all 3 of the Cartesian coordinates and by inserting a term that speci es the dependence of the electron s potential energy For the convenience of math the coordinate system is changed from Cartesian to spherical polar coordinates as demonstrated in the image below E all Figure 1215 The spherical pelar eeerdiinate system 0 Spherical polar coordinates a give point in space speci ed by speci ed by values of the Cartesian coordinates x y and z are described by values r G and 45 Spherical polar coordinate system the wave function KIJr G 45 can be written as a product of one function depending on r one depending on G and one depending on 45 ilflra fl l5 Renewals Separation of variables allows an exact solution to the Schrodinger equation for the hydrogen atom In spherical polar coordinates the potential energy in cgs units of the electron is 2MB 3quot Vi Refer to equation above where Ze represents the nuclear charge 2 1 for the hydrogen atom When the Schrodinger equation for Hydrogen is solved and the boundary conditions are applied a series of wave functions is obtained each corresponding to a particular energy 0 In contrast to the particle in a 1D box the three dimensional hydrogen atom gives rise to 3 quantum numbers Conventional symbols for quantum numbers 0 m the principle quantum number 0 ll the angular momentum quantum number 0 m l the magnetic quantum number Wave mechanics expression of hydrogen s electron Z ml ZZ 39 39 I 21quot X Til 18 I Tl H g2 g nlhz 7 0 HE Where 21 for hydrogen and n is integer values 1 2 3 Characteristics of the equation above 0 Energy of an electron depends only on the principle quantum number this is only true for oneelectron species 0 Since n us restricted to integer values hydrogen s electron can assume only discrete energy values energy levels are quantized o It is exactly the same equation for energy as obtained in the Bohr model The picture bellow shows the rst few wave functions for hydrogen along with the 3 quantum numbers n I m Table 121 Solutions If the E hr tjlinger Wave Equation far a Cline Electan Elam r rm Drlljiiml Sul utiiun ll Z 4 I I Inil is if 3 t rr EU 3 I w 2 i gl h j l r E r b alibilavas I u 1 a if E 1 c 3quot w 2 i ll 2 tit3 4H are mquot ccis ill 7 quot IQT if 2 ii I ip ifij JE EWlIa sin El cos ii A quotJ 4 IE ill 1 in air 1 2 2p Elli 4 Ref 4 I In quott 5111 H 5111 il 39l39 1 3 39 1 j l l 35 lth 31 L 2 llEtr l Etrfllazf j ifquot quotIT HE quotiiti 7 i f i ll 3 aiml 317 lim Fiji 5 cos H it L g l V 1 a Elffjill39l E Eff fir jezm 51in C39 if 1 Fr 39quotJ Lil Ir r ip 335 81 xix I Err rr 1aquot39cquotquot39 quot sun fl sin ab I 3 1 if 52 1 j c 5 1 Li 3112 atria g v5 tries 3 cost ti 39 I 1 LI 7 152 Z 1 u r 339 l J39 3n 8 H M r39j 51111 if cos El cos ugh K HE Hi 1 J J 3rd 31f U2 if 31m Ell QUIE 5 511m ti 39 v 1 1 E quotif j 3 Sift illEffmaw 31 E ETDE if 5111 El tiIle39le lab I I any v39 ft 39l l 339 r l Bdrrr n ill Z i rtrfm efaiirirl El ns 2 f A 3 Il 8 1 L If Mute rr 2 Erie quotwhere E i l fur hydrogen a Eu gf meg f Sit 3 iii 139 m atom some of the solutions contain complex numbers i When solving Schrodinger s equation for the hydrogen l 1 0 Even though it is more convenient to deal with orbitals that contain real numbers complex orbitals are usually combined added or subtracted to remove the complex portions 128 The Physical Meaning of a Wave Function Warning there is always danger in taking a mathematical description of nature and using our human experiences to interpret it Simple pictorial models of natural phenomenon always oversimplify the phenomenon and shouldn t be taken too literally Note that the uncertainty principle indicates that there is no way of knowing the detailed movements of the electron The square of the function evaluated at a particular point in space indicates the probability of nding an electron near that point Relative probability of nding the electron near positions 1 and 2 is determined by substituting the values of r G and l for the two positions into the wave function squaring the function value and computing the following ratio l l llfiirlln 6391 1 g tilte 923 152 6331 N2 N1N2 is the ratio of the probabilities of nding the electron in the in nitesimally small volume elements dv The model gives no information concerning when the electron will be at either position or how it moves between positions square of the function in which the intensity of color used to indicate the probability value at a given point in space The image below depicts the probability distribution for hydrogen ls orbital at e J2 E m D39i litltlUE f39rern heelette r lb Figure 1215 The more at The prebebilliity dietributiein fer the times the electron visits hydrogen le erlbitel in thr39ee dirneneienel a partiCUaF IOOm E Ithe epeee let The prebelbility density et the da erl the negathe elleetreh at peiihte elehg a line drawn blecomles I the darkness eutwerdl fremi the nucleus in any diree mtlens39ty Of a 9039 t ttein fer tlhe hydregein te erbiteli 39 d 39Cates the probability of nding an electron at the position 0 This diagram is otherwise known as an electron density map electron density and electron probability equal the same thing Another way of to represent the electron probability for orbital ls is to calculate the probability at points along a line drawn outward in any direction from the nucleus refer to the image above letter b Need to know total probability of nding the electron at a particular distance from the nucleus 0 Radial probability distribution a plot of 47 2R2 versus r where R represent the radial part of the wave function Radial prebebiility elf EFF iljistanee freitt l lEii ll rt th Figu re 1 21 let Greee eeetien ef the hydregen te erbital pvrvebehility dietributien divided inte eueeeeeiee thin epheriieel shells in The radial preteebility dietnibutien iii plet elf the tetel prebalbillty ref finding the elect tren in eeeh thin epherieel ehell ae e funetien elf dietanee frem the nueleue t z 10 m the engetrem ie eften used ae the unit fer eternie rediue EC tlSE ef ite eeneenient eize edether eerwenien t unit ie the pieemeler fl pin 2 3910 19 mi Maximum in curve occurs because of two opposing effects Probability of nding an electron at a particular position is greatest near the nucleus but the volume of the spherical shell increases wdistance from nucleus 0 The farther away you move from the nucleus the probability of nding the electron at a certain position decreases However the total probability increases to a certain radius and then decreases as the electron probability at each position becomes very small The innermost orbit in the Bohr model is called Bohr radius and denoted as a0 0 Note In Bohr s model the electron is assumed to have a circular path and so is always found at this distance In the wave mechanical model the speci c electron motions are unknown so the most probable distance at which the electron is found Characteristic of hydrogen ls orbital that should be considered is its size because it cannot be precisely de ned 0 Therefore the hydrogen orbital has no distinct size 0 But it is useful to have a de nition of relative orbital size Hydrogen atom ls orbital gives a sphere with a radius 26 a or 14 x 103910 m 140pm 124 The Quantum Mechanical of the Atom an electron bound to the nucleus was similar to a standing wave 0 An example is standing waves in musical instruments such as guitars or violins The image below explains the concept U npl LlC ktl stri n g If lts l linese e n gllt s Figure 1211 The standing wese predLl eee by the vlbretien ef e gelter string festenee st beth ends Eeeln Clef represents e nettle Figure s peint efzere tlispleeernent 1212 Only certain circular orbits have a circumference into which a whole of wavelengths of the standing electron wave will t other orbits would produce destructive interference of the standing electron wave Schrodinger developed a mathematical equation in which describes the hydrogen electron as a wave based on the classical description of a wave phenomena Jim Es 0 Refer to equation above the wave function W is a function of coordinates x y and z of the electron s position in a 3D space Fl Hamiltonian represents a set of mathematical instruction called an operator 0 Operator a mathematical tool that acts on a function to produce another function 0 In Schrodinger s equation the H acts to give back the wave function multiplied by the constant E which represents the total energy of the atom sum of potential energy due to attraction between the proton and the electron amp the kinetic energy KE of the moving electron o Orbital a speci c wave function for a given electron Mi emefeh ti l it i a W w Figure 1212 The hydregen eleetren iiieuellieecl eel e etendiing wave ereunci the nucleus The eireumferenee elf e lpertieuller eireulleitr erbit nee tie eerreepnd te a wlhele number elf weeelengthe ale ehewn in lei and lb er melee deetruetiv interier einee eeeulre ae ehewn in C This medel ie C39DvlfiSlSltEl39iT with the feet that enly certain eleetren energiee ere eileweci the etern is quantized elltheugh thie idea bee eneeuraged eelentlete tel uee e we39ve thleew it deee net meen that the eleetren reeiiy treeelle in elireuler erbite 1St orbital the wave function corresponding to the lowest energy 0 The meaning of orbital is the rst point of interest 0 Wave function does not give information about the movements of the electrons we can only predict their trajectories o there is a fundamental limitation to just how precisely we can know both position and momentum of a particle at a given time o This principle is stated mathematically as fig 0 Where Ax is the uncertainty in a particle s position Ap is the uncertainty in a particle s momentum and h is Planck s constant divided by 275h275 o The minimum uncertainty is h47c This relationship shows that the more precisely a particle s position is known the less precisely its momentum is known and vice versa 0 Limitation is so small for large particles but limitation becomes important when it comes to 5 small particle such as an electron 0 Since an electron s exact path cannot be known it is not appropriate to assume that the electron is moving around the nucleus 127 The Wave Equation for the Hydrogen Amm The electron of the hydrogen atom moves in 3D and has potential energy because of its attraction to the positive nucleus of the atom Differences can be accounted for by including the 2nOI derivatives with respect to wrt all 3 of the Cartesian coordinates and by inserting a term that speci es the dependence of the electron s potential energy For the convenience of math the coordinate system is changed from Cartesian to spherical polar coordinates as demonstrated in the image below E all Figure 1215 The spherical pelar eeerdiinate system 0 Spherical polar coordinates a give point in space speci ed by speci ed by values of the Cartesian coordinates x y and z are described by values r G and 45 Spherical polar coordinate system the wave function KIJr G 45 can be written as a product of one function depending on r one depending on G and one depending on 45 ilflra fl l5 Renewals Separation of variables allows an exact solution to the Schrodinger equation for the hydrogen atom In spherical polar coordinates the potential energy in cgs units of the electron is 2MB 3quot Vi Refer to equation above where Ze represents the nuclear charge 2 1 for the hydrogen atom When the Schrodinger equation for Hydrogen is solved and the boundary conditions are applied a series of wave functions is obtained each corresponding to a particular energy 0 In contrast to the particle in a 1D box the three dimensional hydrogen atom gives rise to 3 quantum numbers Conventional symbols for quantum numbers 0 m the principle quantum number 0 ll the angular momentum quantum number 0 m l the magnetic quantum number Wave mechanics expression of hydrogen s electron Z ml ZZ 39 39 I 21quot X Til 18 I Tl H g2 g nlhz 7 0 HE Where 21 for hydrogen and n is integer values 1 2 3 Characteristics of the equation above 0 Energy of an electron depends only on the principle quantum number this is only true for oneelectron species 0 Since n us restricted to integer values hydrogen s electron can assume only discrete energy values energy levels are quantized o It is exactly the same equation for energy as obtained in the Bohr model The picture bellow shows the rst few wave functions for hydrogen along with the 3 quantum numbers n I m Table 121 Solutions If the E hr tjlinger Wave Equation far a Cline Electan Elam r rm Drlljiiml Sul utiiun ll Z 4 I I Inil is if 3 t rr EU 3 I w 2 i gl h j l r E r b alibilavas I u 1 a if E 1 c 3quot w 2 i ll 2 tit3 4H are mquot ccis ill 7 quot IQT if 2 ii I ip ifij JE EWlIa sin El cos ii A quotJ 4 IE ill 1 in air 1 2 2p Elli 4 Ref 4 I In quott 5111 H 5111 il 39l39 1 3 39 1 j l l 35 lth 31 L 2 llEtr l Etrfllazf j ifquot quotIT HE quotiiti 7 i f i ll 3 aiml 317 lim Fiji 5 cos H it L g l V 1 a Elffjill39l E Eff fir jezm 51in C39 if 1 Fr 39quotJ Lil Ir r ip 335 81 xix I Err rr 1aquot39cquotquot39 quot sun fl sin ab I 3 1 if 52 1 j c 5 1 Li 3112 atria g v5 tries 3 cost ti 39 I 1 LI 7 152 Z 1 u r 339 l J39 3n 8 H M r39j 51111 if cos El cos ugh K HE Hi 1 J J 3rd 31f U2 if 31m Ell QUIE 5 511m ti 39 v 1 1 E quotif j 3 Sift illEffmaw 31 E ETDE if 5111 El tiIle39le lab I I any v39 ft 39l l 339 r l Bdrrr n ill Z i rtrfm efaiirirl El ns 2 f A 3 Il 8 1 L If Mute rr 2 Erie quotwhere E i l fur hydrogen a Eu gf meg f Sit 3 iii 139 m atom some of the solutions contain complex numbers i When solving Schrodinger s equation for the hydrogen l 1 0 Even though it is more convenient to deal with orbitals that contain real numbers complex orbitals are usually combined added or subtracted to remove the complex portions 128 The Physical Meaning of a Wave Function Warning there is always danger in taking a mathematical description of nature and using our human experiences to interpret it Simple pictorial models of natural phenomenon always oversimplify the phenomenon and shouldn t be taken too literally Note that the uncertainty principle indicates that there is no way of knowing the detailed movements of the electron The square of the function evaluated at a particular point in space indicates the probability of nding an electron near that point Relative probability of nding the electron near positions 1 and 2 is determined by substituting the values of r G and l for the two positions into the wave function squaring the function value and computing the following ratio l l llfiirlln 6391 1 g tilte 923 152 6331 N2 N1N2 is the ratio of the probabilities of nding the electron in the in nitesimally small volume elements dv The model gives no information concerning when the electron will be at either position or how it moves between positions square of the function in which the intensity of color used to indicate the probability value at a given point in space The image below depicts the probability distribution for hydrogen ls orbital at e J2 E m D39i litltlUE f39rern heelette r lb Figure 1215 The more at The prebebilliity dietributiein fer the times the electron visits hydrogen le erlbitel in thr39ee dirneneienel a partiCUaF IOOm E Ithe epeee let The prebelbility density et the da erl the negathe elleetreh at peiihte elehg a line drawn blecomles I the darkness eutwerdl fremi the nucleus in any diree mtlens39ty Of a 9039 t ttein fer tlhe hydregein te erbiteli 39 d 39Cates the probability of nding an electron at the position 0 This diagram is otherwise known as an electron density map electron density and electron probability equal the same thing Another way of to represent the electron probability for orbital ls is to calculate the probability at points along a line drawn outward in any direction from the nucleus refer to the image above letter b Need to know total probability of nding the electron at a particular distance from the nucleus 0 Radial probability distribution a plot of 47 2R2 versus r where R represent the radial part of the wave function Radial prebebiility elf EFF iljistanee freitt l lEii ll rt th Figu re 1 21 let Greee eeetien ef the hydregen te erbital pvrvebehility dietributien divided inte eueeeeeiee thin epheriieel shells in The radial preteebility dietnibutien iii plet elf the tetel prebalbillty ref finding the elect tren in eeeh thin epherieel ehell ae e funetien elf dietanee frem the nueleue t z 10 m the engetrem ie eften used ae the unit fer eternie rediue EC tlSE ef ite eeneenient eize edether eerwenien t unit ie the pieemeler fl pin 2 3910 19 mi Maximum in curve occurs because of two opposing effects Probability of nding an electron at a particular position is greatest near the nucleus but the volume of the spherical shell increases wdistance from nucleus 0 The farther away you move from the nucleus the probability of nding the electron at a certain position decreases However the total probability increases to a certain radius and then decreases as the electron probability at each position becomes very small The innermost orbit in the Bohr model is called Bohr radius and denoted as a0 0 Note In Bohr s model the electron is assumed to have a circular path and so is always found at this distance In the wave mechanical model the speci c electron motions are unknown so the most probable distance at which the electron is found Characteristic of hydrogen ls orbital that should be considered is its size because it cannot be precisely de ned 0 Therefore the hydrogen orbital has no distinct size 0 But it is useful to have a de nition of relative orbital size Hydrogen atom ls orbital gives a sphere with a radius 26 a or 14 x 103910 m 140pm

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