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# Principles of Chemistry I CHEM 111

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This 25 page Class Notes was uploaded by Gavin Harvey on Friday October 23, 2015. The Class Notes belongs to CHEM 111 at University of Idaho taught by Staff in Fall. Since its upload, it has received 32 views. For similar materials see /class/227962/chem-111-university-of-idaho in Chemistry at University of Idaho.

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Date Created: 10/23/15

Quantum Theory of the Atom hanter 7 Remember the description ofthe atom from Chapter 2 in the form ofthe Rudmerford model vm may mm mm Guld M Our picture up to this point is only mat electrons in a cloud surround a nucleus protons neutrons How do electrons arrange themselves interms ofenergies and geometry about the nucleus 0 How atoms do chi is NOTintuitive 0 Let s rst discuss how atoms DO NOT follow d1 satelliteplanet model Because of earth s gravitational eld we can see that moving the satellite from the lower orbit to the higher orbit will require energy Likewise energy will be released if the satellite assmes a lower orbit There are in nite variations in how this model can absorb and release energy 0 THIS MODEL DOES NOT WORK FOR ATOMS Atoms absorb and release discrete1 quantized energies The vaules of energy aborbedreleased by the atom is NOT in nitely variable To begin the discussion of the modern model of the atom we must rst consider the type of energy absorbed Electromagnetic waves or ghotons 1 discrete Mathematics Defined for a finite set of values not continuous The Wave Nature of Light A wave is a continuously repeating change or oscillation in matter or in a physical eld Light is also a wave It consists of oscillations in electric and magnetic elds that travel through space Visible light Xrays and radio waves are all forms of electromagnetic radiation Think of electromagnetic radiation as pure waves with no mass We characterize waves in terms of 2 wavelength distance from crest to crest v frequency number crests that pass a xed point per second L3939H v H HP II second I Origin Time Units of l wavelength are distance units e g nm V frequency are cyclessec or lsec Note s39l commonly referred to as Hertz Hz is de ned as cycles or waves per second The product of the frequency V 1 sec and the wavelength l m would give the speed of the wave in ms In a vacuum the speed oflight c is 300 X 108 ms Therefore c V The range of possible 9 and V of electromagnetic waves is called the electromagnetic spectrum Figure 75 The electromagnetic spectru m llllquot39 I ll Inquot In IIIquot III quot mquot In 1r39 l39Zl m39 Frequmcy n 39I I l l l I E Mquot u Iv M1 w lllllllh m quotNH II 3 quot Mwmn HJIIH 39 V 39 ma L InlIIml Irrmm kuhn mum 39 win IIIJul l I I l l l I I III 391 III 39 In In 39 n I ll 39 lll quot Ill 39 Inquot In 39 I quot lll39 anlunurlrlnn II mu H mm I Ill lvml ul IInI r l mu IIIYI III It I InII ll an ll lM39JJllH ll llmll all Inml I ll mml 39lenllulu sawmill IILII ur 1m 5 tsll Inquot I II TIZ I J 7 Visible light extends from the Violet end of the spectrum at about 400 nm to the red end With Wavelengths about 800 nm 7 Beyond these extremes electromagnetic radiation is not Visible to the human eye Example What is the Wavelength of yellow light With a frequency of509 x 10H squot Note squot commonly referred to as Hertz Hz is de ned as cycles or Waves per second 7 Ifc V7 then reamnging We obtain 7 cV 7 589X10 7m or 589 nm Quantum Effects and Photons Planck Max 185871947 eerrnan onvsrcrsl From nrs nvoolnesrs 1900 lnal atoms emlt and absorb energv onlv rn drscrele bundles quanta lnstead or contlnuously as assurned rn classrcal o vsros quantum mechanlcsvvas developed Planckrecelved lne 1918 Nobel Prlze rn onvsrcs ror nrs Work on blackbodv radlallon Hevvas ororessor 18894928 atme Unlv or Berlrn and oresrdenl 193045 or lne lltarser ernelrn Socrelv ror lne a cementofSclence Berlrn Planck sconstanMsee quantum rnecnanrcs rs narned ror nrrn Max Planck Culver Plduvesr rne 2w some Coumbla snowoperas rs lrcerrsed rrurn Culumbla Unlvelsllv Pless CDPVVlEh 1555 W Culumbla unrversnv Pless All llgms reserved Einstein in stin Albert 18791955 Germanborn American theoretical physicist whose special and general theories of relativity revolutionized modernt ought on the na ure 0 s ace an ime an energy He won a 1921 Nobel Prize for his explanation of the photoelectric effect Albert mm Culver Piduves inc Planck s Quantization of Energy Hypothesis 1900 amp Explanation of the Photoelectric Effect by Einstein 1905 helped explain how energy is absorbed and released by atoms The energy of the photons proposed by Einstein would be proportional to the observed frequency and the proportionality constant would be Planck s constant Ehv 7 In 1905 Einstein used this concept to explain the ghntnzlec c 2amp2 Planck s constant h 6626 x 103934 I s u m N Quantum Effects and Photons 39 Photoelectric Effect read page 284285 The ane and particle pictures of light should be regarded as complementary Views of the same physical entity This is called the waveparticle dualitv of light The equation E hv displays this duality E is the energy of the particle photon and V is the frequency of the associated wave Example What is the energy of a photon corresponding to radio waves of frequency 1255 X 10 6 3391 E hv 6626 x1034 Js1255 x106 s39l 8316x103928 J The Bohr Theory of the Hydrogen Atom 18851962A new model ofthe atom devised by Danish physicist Niels Bohr 28 violates class39ca electromagnetic theory but successfully accounts for the spectrum of hydrogen Bohr applies Max Planck s uantumtheory of1900 to Ernest Rutherford s nuclear atom of1911 see 1939 Schrodinger 19264 Niels aw Culver plumes inc Prior to the Work ofNiels Bohr the stability ofthe atom could not be explained using the thencurrent theories I In 1913 using the Work of Einstein and Planck he applied a new theon to the simplest atom hydrogen I Before looking at Bohr s theory We must rst examine the line spectra of atoms I The light emitted by a heated gas such as hydrogen results in a line spectruma spectrum showing only speci c wavelengths of light see Figure 72 The People s CoronaOwls licensed 1mm Henry Holt and cempaw inc CUWHEM 15551556 vaamesTvauev All nemsvesewee 39 Atomic Line Spectra In 1885 J J Balmer showed that the wavelengths M in the visible spectrum of hydrogen could be reproduced by a simple formula 1097gtlt 107m 1 A II The known wavelengths of the four visible lines for hydrogen correspond to values ofn 3 n 4 n 5 and n 6 see Figure 72 Figure 72 Emission line spectra of some elements EMISSION LINEII SPECIFYEA O 39 Bohr s Postulates Hypothesis Bohr set down postulates to account for l the stability of the hydrogen atom and 2 the line spectrum of the atom 1 Energy level postulate An electron can have only speci c energy levels in an atom 2 Transitions between energy levels An electron in an atom can change energy levels by undergoing a transition from one energy level to another see Figures 710 and 711 In order to account for Balmer s observations 7 1 1097x10 m i 22 Bohr derived the following formula from his postulates for the energy levels of the electron in the hydrogen atom Rh is a constant expressed in energy units with a value of218 x 103918 J E R n 1 2 3 for H atom 11 Bohr s Postulates When an electron undergoes a transition from a higher energy level to a lower one the energy is emitted as a photon Energy of emitted photon From Postulate 1 Now we have EhvRJ7 nf hvEi Ef 1 g 1 Does Bohr s equation converge with Balmer s observations 1 22 1097X107m1 YES EhvRh SNI H n3 Rh218XlO3918J 1 112 speed oflight c 300 X 108 ms Planck s const h 6626 X 103934 I s c V V c0 Eh 218gtlt10 18J iz iZ xi nf mi l be H i 218x1018J 1 2 f i 1 7 almost looks like the Blamer eqn 1 1 1 2 let Ilf2 1 7 l l l 1 1097X10 This is the Balmer eqn What was proven Theory Bohr s Eqn agrees with experimental observation Balmer s Eqn Why does Balmer s eqn start with n 2 and not 11 1 Example Bohr s Eqn Calculate the energy of a photon of light emitted from a hydrogen atom when an electron falls from level n 3 to level n 1 amt 12 nf ni E 218x10 18J112 312 E 194x10 18J Bohr s Eqn Represented in terms of an energy diagram 71 Baimer scrim mimev quot2 Finge 711 Transitions of the elech on in me hydrogen atom wag Lyman scncs munniulcn liner Quantum Mechanics exp am the 2 r E E a 3 E S o Bohr s Lheoxy established the concept ofatomic energy ughly behavior ofche electron Current ideas about atomic structure depend on the principles of quantum mechanics a theory that applies to subatomic particles such as electrons Broglie Louis Victor duc de br gle 1892 1987 French physicist From his hypothesis that particles should exhibit certain wavelike properties wave mechanics a form of quantum mechanics was developed Experiments proved 1927 the existence of these waves he was awarded the 1929 Nobel Prize in physics for his theory5 The rst clue in the development of quantum theory came with the discovery of the de Broglie relation it hmv v velocity in ms In 1923 Louis de Broglie reasoned that if light exhibits particle aspects perhaps particles of matter show characteristics of waves If matter has wave properties why are they not commonly observed The de Broglie relation shows that a baseball 0145 kg moving at about 60 mph 27 ms has a wavelength of about 17 X 103934 m 663x10 34kg39sm2 34 kW l7gtlt10 m This value is so incredibly small that such waves cannot be detected 5The Concise Columbia Encyclopedia is licensed from Columbia University Press Copyright 1995 by Columbia University Press All rights reserved However consider the 7x of an electron travelling at 10 the speed of light 1 663X10 34 243 pm 243x10 m 911gtlt10 31 k 300gtlt107 E g S The wavelength of the electron is signi cant relative to the size of an atom The waveparticle duality of the electron has implications for the electron moving about a nucleus To begin let s rst consider water waves VIIIWIIUl u We ve all seen the result of simultaneously tossing 2 pebbles in a pond and getting that pattern to the right FIGURE 37 4 The interference of water waves in a ripple tank There is destructive interference along the lines marked Line of nodes and constructive interference between these lines Courtesy Physical Science Study Committee That pattern can be explained by the addition of waves of the same wavelength but from 2 different sources v 613 V Waves in phase Waves partially out of phase Waves completely out of phase Constructive Destructive interference interference a Result b Result c Result Figure 21 9 Interference of adjacent waves that are a 0 17 90 and 0 180 out of phase Now consider the movement of a wave in a circular path a wave undergoes a constructive reinforcement b wave will destructive interference l6 We should be able to reason that for constructive reinforcement the circumference of the circle 2 71 is related to the wavelength of the wave through 0 For constructive reinforcement 27tr 11 Where n 1 2 3 o This is quantization only certain orbits are allowed In a very general sense this is why energy is quantized for the electron in an atom Caveat the wave function of an electron moving about a nucleus is not a simple sine wave That wave function is described by the Schrodinger Eqn 1 Schrodinger Erwin 1887 1961 Austrian theoretical physicist For his mathematical development 1926 of wave mechanics a form of quantum mechanics and his formulation of the wave equation that bears his name he shared with Paul Dirac the 1933 Nobel Prize in physics6 1 is a mathematical relationship that we will not discuss in Chem 1 l 1 In 1927 Werner Heisenberg showed from quantum mechanics that it is impossible to know both the exact position and velocity of an electron in an atom simultaneously 6The Concise Columbia Encyclopedia is licensed from Columbia University Press Copyright 1995 by Columbia University Press All rights reserved Heisenberg Werner 1901 76 German physicist A founder ofquantum mechanics he is famous for his uncertainty principle which states that it is impossible to determine both the position and momentum of a subatomic particle such as the electron with arbitrarily high accuracy The effect of this principle is to convert the laws of physics into statements about relative instead of absolute certainties Heisenberg39s matrix mechanics a form of quantum mechanics was shown to be equivalent to Erwin Schrodinger39s wave mechanics Heisenberg received the 1932 Nobel Prize in physics for his work in nuclear physics and quantum theory7 Heisenberg s uncertainty principle is a relation that states that the product of the uncertainty in position AX and the uncertainty in momentum mAVX of a particle can be no larger than h4Tt AxmAvx 2 L 4 When in is large for example a baseball the uncertainties are small but for electrons high uncertainties disallow de ning an exact orbit Although we cannot precisely de ne an electron s orbit we can obtain the probability of nding an electron at a given point around the nucleus 7The Concise Columbia Encyclopedia is licensed from Columbia University Press Copyright 1995 by Columbia University Press All rights reserved Erwin Schrodinger de ned this probability in a mathematical expression called a wave function denoted 111 psi The probability of nding a particle in a region of space is de ned by see Figures 718 and 719 Bottomline we can only describe the probability of locating that electron in an atom Figure 719 Probability of nding an electron in a spherical shell about the nucleus Hmllul pmlminlil it I ll I 5 Ill r Ij 393ij s A Ill That 99 contour for the probability of n l is shown Figure 724 amp 725 R E 7 n 1 2 3 for H atom 11 E39J39Zl ii mmnur x K a V I AI 39 139 x ll I n39 V Al I rnrbl39ltal 15 39IJI IJi39lill Quantum Numbers and Atomic Orbitals Further derivation of the Schrodinger Eqn 1 reveals that there are 4 quantum numbers Principal quantum number n Angular momentum guantum number I Magnetic guantum number m Spin guantum number ms The rst three de ne the wave function for a particular electron The fourth quantum number refers to the magnetic property of electrons 20 The principal quantum numbern represents the shell number in which an electron resides The smaller 11 is the smaller the orbital The smaller 11 is the lower the energy of the electron The angular momentum quantum number I distinguishes sub shells within a given shell that have different shapes Each main shell is subdivided into sub shells Within each shell of quantum number 11 there are 11 sub shells each with a distinctive shape I can have any integer value from O to n 1 example 11 1 l 0 1 The different subshells are denoted by letters Letter s p d f g l 0 1 2 3 4 The magnetic quantum number m1 distinguishes orbitals within a given subshell that have different shapes and orientations in space Each sub shell is subdivided into orbitalsz each capable of holding a pair of electrons 21 m can have any integer value from I to 1 example 11 2 l 0amp1 miaH Each orbital within a given sub shell has the same energy The spin quantum number ms refers to the two possible spin orientations of the electrons residing within a given orbital Each orbital can hold only two electrons whose spins must oppose one another The possible values of rnS are l2 and l2 see Table 71 and Figure 723 n 123 l0n 1 m1 ll ms JA NikkiJ cv IE mlsf gl lmluuijfft inhm Humben ful39fkfl lyhitgll mutual Silbill l Dmhin an n 1 mr39 Nutum nishIi U 0 1 2 I E 27 l 2 1 10 4 2p 3 3 0 9 33 l 3 a 1ni 3p 3 3 2 2 L0 4412 3d 5 4 U 0 it I 1 J Iltl I tip 3 4 2 ll l1 4 3d 5 4 3 3 quot2 1 1 3 3 if T Awlpw mm m mum 1mm may hi Imiaw him m n ind Inmlum nm mm nu IIquot In 22 Using calculated probabilities of electron position the shapes of the orbitals can be described The s sub shell orbital there is only one is spherical see Figures 724 and 725 The p sub shell orbitals there are three are dumbbell shape see Figure 726 The 0 sub shell orbitals there are ve are a miX of cloverleaf and dumbbell shapes see Figure 727 The s sub shell orbital there is only one is spherical see Figures 724 and 725 quotHie rumMquot t ls mhlnd 1i nrhiml n1 n2 0 0 23 Figure 725 another representation of orbitals 1 o 39 m p sub shell orbitals mm are mm are dumbbell shape see Figure 725 n 2 1 1

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