Zumdahl’s Chapter 7Zumdahl’s Chapter 7

Atomic StructureAtomic Structure

Atomic PeriodicityAtomic Periodicity

Chapter ContentsChapter Contents

• EM RadiationEM Radiation• EM QuantizationEM Quantization• H SpectrumH Spectrum

– Niels BohrNiels Bohr– L. deBroglieL. deBroglie– W. HeisenbergW. Heisenberg– E. SchrödingerE. Schrödinger

• Quantum Nos.Quantum Nos.

• Orbital ShapesOrbital Shapes– DegeneraciesDegeneracies

• W. Pauli (spin)W. Pauli (spin)• MultielectronsMultielectrons• Periodic TablePeriodic Table

– AufbauAufbau– Property TrendsProperty Trends– GroupsGroups

Electromagnetic Electromagnetic RadiationRadiation

• Oscillating E & M fields forever.Oscillating E & M fields forever.• Wavelength, Wavelength, , distance, distance between between

successive peaks.successive peaks.• Period,Period, τ τ, time between peaks., time between peaks.• Speed, c = Speed, c = / / ττ = =

= frequency (cycles per second)= frequency (cycles per second) = c / = c / and and = c / = c /

Electromagnetic Electromagnetic QuantizationQuantization

• E = hE = h = hc / = hc / • Equipartition Theorem demands kT worth Equipartition Theorem demands kT worth

of thermal energy to all light and matter of thermal energy to all light and matter modes. Leads to modes. Leads to energy. energy.

• Vibration overtones in matter are Vibration overtones in matter are truncated by indivisible atoms.truncated by indivisible atoms.

in light energies overcome by Planck in light energies overcome by Planck with QUANTIZED energies. with QUANTIZED energies.

• h = Planck’s constant = 6.6x10h = Planck’s constant = 6.6x10–34–34 Js Js

Hydrogen Atom SpectrumHydrogen Atom Spectrum

• White light is all colors (all White light is all colors (all ); ); diffraction in prisms or raindrops diffraction in prisms or raindrops gives continuous spectra.gives continuous spectra.

• Atomic excitation gives instead Atomic excitation gives instead discrete colors (few discrete colors (few ).).

• In H atom, E In H atom, E lightlight = R (n = R (n22–2–2 – n – n11

–2–2))

• Why so simple?Why so simple?

Niels BohrNiels Bohr

• F F centripetalcentripetal = m r = m r ²²

• F F attractionattraction = – Z e² / r² = – Z e² / r²

• Balance: Z e² = m r³ Balance: Z e² = m r³ ²²• E = K+V = ½ m r² E = K+V = ½ m r² ² – Z e² / r² – Z e² / r• E = – ½ Z e² / r E = – ½ Z e² / r on substitutionon substitution

• E = – (½ Z e² / r) E = – (½ Z e² / r) (Z e² / m r³ (Z e² / m r³ ²)²)• E = – Z² eE = – Z² e44 m / (2 m² r m / (2 m² r44 ²)²)

Quantized MomentaQuantized Momenta

• E = – Z² eE = – Z² e44 m / (2 m² r m / (2 m² r44 ²)²)• ppθθ = m r² = m r² isis angular momentum angular momentum

• E = – Z² eE = – Z² e44 m / (2 p m / (2 pθθ²)²)

• ppθθ = n h / 2 = n h / 2 Bohr’s postulateBohr’s postulate

• EEnn = – (2 = – (2² Z² e² Z² e44 m / h²) n m / h²) n––²²

• EEnn = – R = – RHH Z² / n² Z² / n² YES!!YES!!

– But but but But but but WHYWHY quantize p quantize pθθ ?!? ?!?

Louis deBroglieLouis deBroglie

• Just as light moves as a wave but Just as light moves as a wave but lives and dies as a discrete energy lives and dies as a discrete energy packet (“photon”),packet (“photon”),

• Matter too has wave properties Matter too has wave properties only dominant for only dominant for lightlight masses: masses:

= h / p = h / p = h / mv= h / mv = h / (m r² = h / (m r² ))• And n And n = 2 = 2 or wave or wave killskills itself! itself!

Werner HeisenbergWerner Heisenberg

• Waves must S P A N Waves must S P A N ..• Attempts to narrow the wave, Attempts to narrow the wave,

reduce reduce , , increasinincreasing p = h / g p = h / • So a minimum uncertainty So a minimum uncertainty x in x in

position MUST exist, andposition MUST exist, and

• (( x) ( x) ( p pxx) ) ½ h / 2 ½ h / 2– Heisenberg’s Uncertainty PrincipleHeisenberg’s Uncertainty Principle

Edwin SchrödingerEdwin Schrödinger

• Bohr’s orbits are infinitesimally thin Bohr’s orbits are infinitesimally thin trajectories. Being trajectories. Being offoff them must them must be infinitely uncertain!be infinitely uncertain!

• Need a Need a fullfull 3-D wave, 3-D wave, , not 1-D., not 1-D.• Schrödinger’s Wave Equation Schrödinger’s Wave Equation

solves H energy in 3-d and finds:solves H energy in 3-d and finds:

• EEnn = – R = – RHH Z² / n² also Z² / n² also and more!and more!

Quantum NumbersQuantum Numbers

(n, (n, l l ,m,ml l ,m,mss))

• Principle Quantum Number, nPrinciple Quantum Number, n– n = 1, 2, 3, 4, 5, … , n = 1, 2, 3, 4, 5, … , – Governs the number of nodes in the Governs the number of nodes in the

electron’s matter wave, electron’s matter wave, !!– # of nodes (where # of nodes (where =0)=0) is is n – 1n – 1– For “hydrogenic” ions (and H itself) For “hydrogenic” ions (and H itself)

electron energy depends electron energy depends onlonly on n.y on n.– Nodes can be spherical Nodes can be spherical oror angular! angular!

Angular Momentum Angular Momentum Quantum Number, Quantum Number, ll

•ll = = 0, 1, 2, 3, 4, … , (n – 1)0, 1, 2, 3, 4, … , (n – 1)• s p d f g … s p d f g … chemist shorthandchemist shorthand

– ll measures # of nodes that are measures # of nodes that are angularangular; so it ; so it mustmust stop at n–1. stop at n–1.

– Increasing angular nodes squeezes Increasing angular nodes squeezes

waves, so E usually depends on waves, so E usually depends on ll– Z component of Z component of ll is also quantized! is also quantized!

Magnetic Quantum Magnetic Quantum

Number, mNumber, mll

• mmll is the component of is the component of ll along (up along (up

or down) the Z axis in space.or down) the Z axis in space.

– mmll = – = – ll, – , – l l + 1, … , –1, 0, 1, … , + 1, … , –1, 0, 1, … , ll–1, –1, ll– Because the component can’t Because the component can’t

exceed its vector.exceed its vector.

– E only depends upon E only depends upon mmll when a when a

magnetic field is applied.magnetic field is applied.

3-D Shapes of Orbit3-D Shapes of Orbitalalss

• Governed by n, Governed by n, ll, and m, and mll

• ll = 0 is spherical = 0 is spherical– n = 1 means n = 1 means nono nodes: nodes:– (n – 1) = # of nodes, all spherical.(n – 1) = # of nodes, all spherical.– n = 2 means n = 2 means oneone spherical node: spherical node:– Wavefunction, Wavefunction, , falls off in intensity , falls off in intensity

to zero at large distance from +to zero at large distance from +

Cross-section of 2sCross-section of 2s

SphericalNode

Angular OrbitalsAngular Orbitals

• ll = 1 implies one angular node = 1 implies one angular node– Cleave space with an x=0 planeCleave space with an x=0 plane– But y=0 or z=0 work as well, so there But y=0 or z=0 work as well, so there

are three or 2are three or 2ll+1 suborbitals.+1 suborbitals.

– The mThe mll sequence always gives 2 sequence always gives 2ll+1+1

– mmll differentiates directions in space differentiates directions in space

for chemical bonding!for chemical bonding!

DegeneraciesDegeneracies

• In the absence of applied magnetic In the absence of applied magnetic

field, all suborbitals of a given field, all suborbitals of a given ll have the same energy.have the same energy.

• This identity of energies is called This identity of energies is called “degenerate.”“degenerate.”

• Even nearly degenerate orbitals Even nearly degenerate orbitals may be mixed to give new ones.may be mixed to give new ones.

P.A.M. DiracP.A.M. DiracWolfgang PauliWolfgang Pauli

• Dirac applied Einstein’s fixed Dirac applied Einstein’s fixed cc to to Schrödinger’s Equation and found Schrödinger’s Equation and found newnew quantum number, m quantum number, mss..

– mmss is electron spin number and takes is electron spin number and takes on only two values, on only two values, ½.½.

• Pauli Principle says Pauli Principle says onlonly y twotwo electrons can occupy any orbital, electrons can occupy any orbital, and their mand their mss mustmust differ, differ,– without which without which NO CHEMISTRYNO CHEMISTRY..

Multielectronic AtomsMultielectronic Atoms

• Beyond H, repulsions BETWEEN Beyond H, repulsions BETWEEN electrons compete with nuclear electrons compete with nuclear attraction & complicate spectra.attraction & complicate spectra.– Hund’s RuleHund’s Rule: if electrons have the : if electrons have the

choice between degenerate orbitals, choice between degenerate orbitals, they choose NOT to double occupy they choose NOT to double occupy them.them.• It minimizes electronic repulsion.It minimizes electronic repulsion.

Repulsive ConsequencesRepulsive Consequences

• Energies are now a function of Energies are now a function of ll, the , the angular quantum number.angular quantum number.

• The “filling sequence” shows the The “filling sequence” shows the new energy order:new energy order:– 1s<2s<2p<3s<3p<1s<2s<2p<3s<3p<4s4s<<3d3d<4p<<4p<5s5s<<4d4d– <5p<<5p<6s6s<<4f4f<<5d5d<6p<<6p<7s7s<<5f5f<<6d6d etc. etc.– Periodic Table exemplifies it, but a Periodic Table exemplifies it, but a

simple pattern emerges:simple pattern emerges:

Filling Sequence Filling Sequence MnemonicMnemonic

1s1s

2s2s 2p2p

3s3s 3p3p 3d3d

4s4s 4p4p 4d4d 4f4f

5s5s 5p5p 5d5d 5f5f 5g5g

6s6s 6p6p 6d6d 6f6f 6g6g 6h6h

7s7s 7p7p 7d7d 7f7f 7g7g 7h7h 7i7i

8s8s 8p8p 8d8d 8f8f 8g8g 8h8h 8i8i 8k8k

And that’s as far as the known elements go.

Periodic TablePeriodic Table

• Aufbau (filling sequence) follows Aufbau (filling sequence) follows that table:that table:– 1s² 2s² 2p1s² 2s² 2p66 3s² 3p 3s² 3p66 4s² 3d 4s² 3d1010 4p 4p66 5s² 5s²– 4d4d1010 5p 5p66 6s² 4f 6s² 4f1414 5d 5d1010 6p 6p66 7s² 6d 7s² 6d1010 – and the latest elements among 7pand the latest elements among 7p66

• Irregularities occur where ½ filled Irregularities occur where ½ filled suborbitals acquire greater stability suborbitals acquire greater stability than a predecessor:than a predecessor:

Aufbau HiccupsAufbau Hiccups

• Vanadium: [Ar] 4s² 3dVanadium: [Ar] 4s² 3d33 suggests suggests• Chromium: [Ar] 4s² 3dChromium: [Ar] 4s² 3d44 is next, is next, BUT BUT

IT ISN’T SOIT ISN’T SO!!• Chromium: [Ar] 4sChromium: [Ar] 4s11 3d 3d55 lowers its lowers its

energy by borrowing a 4s to energy by borrowing a 4s to complete a complete a ½–filled½–filled d suborbital. d suborbital.

• Manganese: [Ar] 4sManganese: [Ar] 4s²² 3d 3d55 follows. follows.

Periodic PropertiesPeriodic Properties

• Rows are called “periods” on the Rows are called “periods” on the Periodic Table.Periodic Table.– Columns are called “groups.”Columns are called “groups.”

• Progression along rows implies Progression along rows implies adding new electrons & protons.adding new electrons & protons.– They get added 1:1 for neutrality.They get added 1:1 for neutrality.– But new repulsions keep pace with But new repulsions keep pace with

new attractions. Which wins?new attractions. Which wins?

Effective PotentialEffective Potential

• Protons exert attraction only Protons exert attraction only toward the atom’s center.toward the atom’s center.

• Electrons exert repulsion from all Electrons exert repulsion from all over their wavefunctions.over their wavefunctions.– ““Core” electrons are located very Core” electrons are located very

close to the nucleus where they repel close to the nucleus where they repel outer electrons as effectively as outer electrons as effectively as their their numbernumber of protons attract. of protons attract.

Effective Charge = Protons Effective Charge = Protons – – EffectiveEffective e e––

• So nucleus’s effective charge is Z – So nucleus’s effective charge is Z – (# of all core electrons) less the (# of all core electrons) less the effect of “outer” electrons.effect of “outer” electrons.

• While core are 100% effective, While core are 100% effective, “valence” electrons are LESS by “valence” electrons are LESS by virtue of spanning a greater virtue of spanning a greater fraction of the atom. Only their fraction of the atom. Only their inner portion is 100% effective.inner portion is 100% effective.

Inefficiency WinsInefficiency Wins

• Across a row, added electrons are Across a row, added electrons are valence, not core. So they repel valence, not core. So they repel one another less than the added one another less than the added protons attract them.protons attract them.

• Effective potential INCREASES Effective potential INCREASES across the row, binding the across the row, binding the subsequent electrons subsequent electrons ever moreever more titigghtlhtly!y!

Trends on PeriodsTrends on Periods

• Increased electron binding along Increased electron binding along rows (to the right), generally rows (to the right), generally results in:results in:– Increasing ionization potentialsIncreasing ionization potentials– Decreasing atomic sizesDecreasing atomic sizes– Growing electron affinitiesGrowing electron affinities– Increasing electronegativiesIncreasing electronegativies

GroupsGroups

• Elements in the same column have Elements in the same column have the same number and type of the same number and type of valencevalence electrons, differing only by electrons, differing only by n.n.

• Because increasing n by 1 puts one Because increasing n by 1 puts one more node in more node in , dimensions of , dimensions of increase, vaulting electrons outside increase, vaulting electrons outside their predecessors.their predecessors.

Group TrendsGroup Trends

• Dropping down a group (column) Dropping down a group (column) increases efficiency of core and increases efficiency of core and distance of valence from center. distance of valence from center. Both conspire to weaken the Both conspire to weaken the nucleus’s grasp.nucleus’s grasp.– Atomic size increases.Atomic size increases.– Ionization potential decreases.Ionization potential decreases.– Electronegativity decreases.Electronegativity decreases.