1Figure 12.23 - Zumdahl - Chemical Principles (4/e)

Energy Level Scheme of 1-electron (Hydrogenic) Atoms:En ~ 1/n2 (n- only)

2

HANDOUT: “QM & PT”:QUANTUM MECHANICS & PERIODIC TABLE (page 1)

• Schrödinger Wave Equation (SWE) for MULTI-ELECTRON ATOMS … contains potential energy terms which include interactions BETWEEN electrons…

Nuclear attraction = ! Z • e2

r1

! Z • e2

r2

, where Z = 2 for He

Electron!Electron Re pulsion = Electron Correlation = + e2

| r1 ! r2 |

This last term correlates (couples) electron behavior…SWE can NOT be solved exactly … APPROXIMATIONS…

3

“QM & PT”: (page 1)

Nuclear attraction = ! Z • e2

r1

! Z • e2

r2

, where Z = 2 for He

Electron!Electron Re pulsion = Electron Correlation = + e2

| r1 ! r2 |

2p+

e1-

e2-

e-e repulsion(correlation)

nuclearattraction

nuclearattraction

4

“QM & PT”: (page 2)

• PRESUME that e!ect of e-e repulsion was “INDIRECT”, I.e, Each electron “separately” a"racts the nucleus…BUT…

• The specific amount of nuclear charge “felt” varied with EACH electron’s SUBSHELL designation (n & l).

• This ATTENUATED NUCLEAR CHARGE is termed EFFECTIVE NUCLEAR CHARGE (Ze! = (Z - s) < Z) - subshell dep. Energy of subshell = En l = - Ze!,nl

2•Ry/n2 , n = 1, 2, 3, …

• Otherwise, each electron occupies a hydrogenic atomic orbital - !nlml(r,",#) = Rnl(r)•Ylml(",#) …same pictures from before.

5

“QM & PT”: (pages 2-3)

• !nlml(r,",#) = Rnl(r)•Ylml(",#). Rnl(r) - expresses the radial behavior [RDF’s].

• Radial behavior (pictures) RATIONALIZE SUBSHELL ENERGY DEPENDENCE.

• Ze! = Z - s ; where “s” is the SCREENING or SHIELDING.

• Energy of subshell = En l = - Ze!,nl2•Ry/n2 , n = 1, 2, 3, …

• Ze! - for a given “n” - DECREASES as l INCREASES. “s” feels “greatest” & “f” the “least”…. SEE RDF’s …

6

Figure 12.33 - Zumdahl - Chemical Principles (4/e)

Radial Distributions {4#r2•[Rnl(r)]2}: n = 3 orbitalsFor the same n value, lower l value has

greater degree of “penetration”[ns (greatest) > np > nd > nf (least)]

7Figure 8.10 - Hill & Petrucci - General Chemistry (3/e)

E!ective Nuclear Charge (Ze!) & Multi-electron Atoms:Ze!, n,l = Z - sn,l depends on (n,l) - subshell:

• ENERGY of e- SUBSHELL: En,l = -Ze!2•Ry/n2 .

• IONIZATION ENERGY of e- in subshell:IEn,l = Ef - Ei = 0 -En,l = +Ze!

2•Ry/n2

8Figure 8.1 - Hill & Petrucci - General Chemistry (3/e)

Energy Level Scheme of Multi-electron Atoms:Depends on (n,l) - subshell - En,l = -Ze!

2•Ry/n2 .

9

Figure 12.33 - Zumdahl - Chemical Principles (4/e)

Radial Distributions {4#r2•[Rnl(r)]2}: 3d vs 4s subshellsE4s (more stable) < E3d (less stable) :

10Figure 7.29 - Hill & Petrucci - General Chemistry (3/e)

“QM & PT”: (page 4)How do we pile e-’s in orbitals?

First … one more quantum # …“Spin” Quantum # (ms) - 4th Quantum #:

ms = +1/2 or -1/2

11

“QM & PT”: (page 4)“Spin” Quantum # (ms) = +1/2 or -1/2

RESTRICTS Orbital Population:Pauli Exclusion Principle (PEP):

(Di!erent ways of saying the same thing):

• NO 2 e-’s are allowed (zero probability) to have the SAME set of 4 quantum #’s (n, l, ml , ms).

• IF 2 e-’s are in the SAME ORBITAL (same n, l, ml ), each MUST have DIFFERENT ms values, i.e., they MUST have “OPPOSITE” SPINS”.

• Any orbital can have, at MOST, ONLY 2 e-’s; 1 e- with ms = +1/2 ($) & 1 e- with ms = -1/2 (%), i.e., $ % .

12

Aufbau Ordering ofEnergies of Subshells of Multi-electron Atoms:

[1s subshell through 6s subshell]

En,l = - (Zeffective)2•Ry/n21s2s

2p3s

3p4s

3d4p

5s

6s

4d5p

Aufbau orderingof subshell energies.

Energy

En,l

Not to scale.

13

GROUND STATE (Most Stable) electron configurations:[H, He, Li, Be, & B] - Orbital Box Diagram

Lowest energy ----> Highest energy (Obey PEP)

En,l = - (Zeffective)2•Ry/n2

Energy

En,l

1s

2s

2p

{ H , He }

{ Li , Be }

Not to scale.

Aufbau ordering:(Ground StateConfigurations)

{ B}

14

GROUND STATE (Most Stable) electron configurations:

Populating subshells containing energy-degenerate orbitals…

Do NOT “bunch up” … “spread out” …

En,l = - (Zeffective)2•Ry/n2

Energy

En,l

1s

2s

2p

{ H , He }

{ Li , Be }

Not to scale.

Carbon? ... NOT a Lowest EnergyGround State Config. ... actuallyan Excited State Config.!!

15

GROUND STATE (Most Stable) electron configurations:Populating subshells containing energy-degenerate orbitals…

Ground State configuration of Carbon:Hund’s Rule (preference) for Ground State Configurations:

“Singly occupy all orbitals [$ $ ] of same energy -with SAME spin … BEFORE filling any of them [$% ]).”

En,l = - (Zeffective)2•Ry/n2

Energy

En,l

1s

2s

2p

{ H , He }

{ Li , Be }

Not to scale.

Aufbau ordering:(Ground StateConfigurations)

Carbon: Hund'sRule

16

GROUND STATE (Most Stable) electron configurations:Populating subshells containing energy-degenerate orbitals…

Ground State configuration of Nitrogen:Hund’s Rule (preference) for Ground State Configurations:

En,l = - (Zeffective)2•Ry/n2

Energy

En,l

1s

2s

2p

{ H , He }

{ Li , Be }

Not to scale.

Aufbau ordering:(Ground StateConfigurations)

Nitrogen: Hund'sRule

17

GROUND STATE (Most Stable) electron configurations:

2nd Row: [Li, Be, B, C, N, O, F, Ne]Lowest energy ----> Highest energy (Obey PEP, & Hund’s Rule)

En,l = - (Zeffective)2•Ry/n2

Energy

En,l

1s

2s

2p

{ H , He }

{ Li , Be }

{ B,C,O,N,F,Ne}

Not to scale.

Aufbau ordering:(Ground StateConfigurations)

18

GROUND STATE (Most Stable) electron configurations:

3rd Row: [Na, Mg, Al, Si, P, S, Cl, Ar]Lowest energy ----> Highest energy (Obey PEP, & Hund’s Rule)

En,l = - (Zeffective)2•Ry/n2

Energy

En,l

1s

2s

2p

3s

3p

{ H , He }

{ Li , Be }

{ B,C,O,N,F,Ne}

{ Na , Mg}

{Al,Si,P,S,Cl,Ar}

Not to scale.

Aufbau ordering:(Ground StateConfigurations)

19

Figure 12.26 - Zumdahl - Chemical Principles (4/e)

Electron Configuration Correlated with Periodic Table:[Ground State (Lowest Energy) Configurations]

20

GROUND STATE (Most Stable) electron configurations:4th Row - Includes 1st period of Transition Metals:

[K, Ca, Sc, Ti, V, Cr*, Mn, Fe, Co, Ni, Cu*, Zn, Ga, Ge, As, Se, Br, Kr]Lowest energy ----> Highest energy (Obey PEP, & Hund’s Rule)

En,l = - (Zeffective)2•Ry/n2

Energy

En,l

1s

2s

2p

3s

3p

4s

3d

4p

{ H , He }

{ Li , Be }

{ B,C,O,N,F,Ne}

{ K , Ca }

{ Na , Mg}

{Al,Si,P,S,Cl,Ar}

{Ga,Ge,As,Se,Br,Kr}

{Sc,Ti,V,Cr*,Mn,Fe,Co,Ni,Cu*,Zn}

Not to scale.

Aufbau ordering:(Ground StateConfigurations)

21

Figure 12.27 - Zumdahl - Chemical Principles (4/e)

Electron Configuration Correlated with Periodic Table:[4th Row (Period) - Ground State (note 24Cr & 29Cu)]:

22

“Condensed” Formats to Represent Electron Configurations:[page 7 of “QM & PT” handout]

Example: Sulfur (S) [For Orbital Box Diagram - see below]• Spectroscopic: 1s2 2s2 2p6 3s2 3p4 [in order of $ energy]• Noble gas core abbreviation : [Ne] 3s2 3p4

(3s & 3p e-’s are the VALENCE electrons)

En,l = - (Zeffective)2•Ry/n2

Energy

En,l

1s

2s

2p

3s

3p

Not to scale.

Aufbau ordering:(Ground StateConfigurations)

SULFUR (S)

23

Figure 12.28 - Zumdahl - Chemical Principles (4/e)

Electron Configuration Correlated with Periodic Table:

24

“Follow” the Periodic Table to Determine Ground StateElectron Configurations:

[page 5 of “QM & PT” handout]

• Transition metals: [Noble gas core] ns2 (n-1)d1 --> 10 (n = row #)[Some exceptions … Cr = [Ar] 4s1 3d5 , Cu = [Ar] 4s1 3d10]

• Lanthanides (Between 57La & 72Hf - 6th row): [Xe] 6s2 5d1 4f1 ---> 14 [58Ce through 71Lu]. [Some exceptions … Don’t worry about exceptions.]

• Actinides (Between 89Ac & 104Rf - 7th row): [Rn] 7s2 6d1 5f1 ---> 14 [90Th through 103Lr]. [Some exceptions … Don’t worry about exceptions.]

25

Figure 12.29 - Zumdahl - Chemical Principles (5/e)

Electron Configuration Correlated with Periodic Table:

26

Main Group Elements & Valence Shell Electron Configuration[page 8 of “QM & PT” handout]

Group Valence Shell e- Config*. # of Valence e-’s1A ns1 12A ns2 23A ns2 np1 34A ns2 np2 45A ns2 np3 56A ns2 np4 67A ns2 np5 78A ns2 np6 8

*[n is the row # (period) of the element in the group.]

27

Ground State Electron Configuration of Ions[page 8 & page 9 of “QM & PT” handout]

Cations:• Construct ground state electron configuration of NEUTRAL atom first.• Remove e-’s from populated shell of HIGHEST principal quantum # (n) until proper cation charge is achieved*. * If more than one subshell of highest principal quantum # is populated, BEGIN with subshell of HIGHEST l quantum #. (i.e., subshell that is MOST [<--- corrected] shielded.) *[n is the row # (period) of the element in the group.]Examples: Fe+2 & Pb+4:1st Fe: [Ar] 4s2 3d6 , then, Fe+2 : [Ar] 3d6 (NOT [Ar] 4s2 3d4).1st Pb: [Xe] 6s2 4f14 5d106p2 , then, Pb+4 : [Xe] 4f14 5d10 (NOT [Xe] 6s2 4f14 5d8).

28

Ground State Electron Configuration of Ions[page 8 & page 9 of “QM & PT” handout]

Anions (page 9):• Typically involve non-metals in the p-block.

• Construct ground state electron configuration of NEUTRAL atom first.

• Continue to add electrons consistent with the Aufbau ordering scheme.

Example: O-2 (oxide anion)1st O: [He] 2s2 2p4 , then O-2 : [He] 2s2 2p6

= [Ne] (isolectronic with Ne).

29

Paramagnetism & Diamagnetism(unpaired e-’s vs all e-’s paired )

[page 9 & page 10 (diagram) of “QM & PT” handout]

Weights

Sample - suspendedfrom one side of balance.

Magnet

Balance

SN

30

Paramagnetism & Diamagnetism(unpaired e-’s vs all e-’s paired )

[page 9 & page 10 (diagram) of “QM & PT” handout]

• Construct electron configuration of species & determine if there are any unpaired e-’s [orbital box diagram.]• If ANY unpaired e-’s & PARAMAGNETIC. Species is ATTRACTED by a magnetic field. # of unpaired electrons determines “degree of paramagnetism”. As # of unpaired e-’s $, degree of paramagnetism $’s. Species is a"racted more strongly by magnetic field.• If NO unpaired e-’s, i.e., all are paired & DIAMAGNETIC. Species is NOT a"racted by a magnetic field.

31

Figure 12.38 - Zumdahl - Chemical Principles (4/e)

Atomic Radii (in pm) (Main Group Elements): [pp. 11 - 16.]

32

Figure 13.7 - Zumdahl - Chemical Principles (4/e)

Ionic Radii (in pm) (Main Group Elements): [pp. 11 - 16.]

33

Figure 12.35 - Zumdahl - Chemical Principles (4/e)

1st IE’s (Main Group Elements): [pp. 11 - 16.]M(gas) -----> M+(gas) + e- ; $E =IE1 (+)

34

Figure 12.36 - Zumdahl - Chemical Principles (4/e)

Electron A%nities (EA) (Main Group Elements): [pp. 11 - 16.]A(gas) + e- -----> A-(gas) ; $E = EA (+ or -)

35

Figure 13.3 - Zumdahl - Chemical Principles (4/e)

Electronegativities (Pauling Values):

36

PERIODIC TRENDS - SUMMARY[See QM & Periodic Table - pages 11-14]

Radius, Diameter, Size [Fig. 12.38, Zumdahl] :• DECREASES L ---> R across a row (period) - generally. [Ze! increases].

• INCREASES Top ---> Bo"om down a column (group). [Ze! “decreases” … outermost e-’s “farther away” from nucleus.]

Ionic Radius, Diameter, Size [Fig. 13.7, Zumdahl] :• Radius of CATION < corresponding NEUTRAL ATOM.

• Radius of ANION > corresponding NEUTRAL ATOM.

37

PERIODIC TRENDS - SUMMARY[See QM & Periodic Table - pages 11-14]

IONIZATION ENERGY (1st IE) :A(gas) ' A+(gas) + e- IE of A = $E of reaction (+).• INCREASES L ---> R across a row (period) - generally. [Ze! increases]. “Glitches”. Look at (ground state) e- configs. IE1 values [3rd row (period)] (in kJ/mol): Na Mg Al Si P S Cl Ar 500 740 580 790 1060 1000 1260 1520 3s1 3s2 3p1 3p2 3p3 3p4 3p5 3p6 [before] 3s0 3s1 3p0 3p1 3p2 3p3 3p4 3p5 [after]

• DECREASES Top ---> Bo"om down a column (group). IE1 values [Group 2A)] (in kJ/mol): Be (900) ; Mg (740) ; Ca (590) ; Sr (550) ; Ba (500)

38

PERIODIC TRENDS - SUMMARY[See QM & Periodic Table - page 13]

SUCCESSIVE IONIZATION ENERGIES (IE1 … IEn). • Example - Aluminum (Al) :

Al(gas) ' Al+(gas) + e- IE1 = 580 kJ/mol [3p1 ----> 3p0].

Al+(gas) ' Al+2(gas) + e- IE2 = 1,820 kJ/mol [3s2 ----> 3s1].

Al+2(gas) ' Al+3(gas) + e- IE3 = 2,740 kJ/mol [3s1 ----> 3s0].

Al+3(gas) ' Al+4(gas) + e- IE4 = 11,580 kJ/mol [2p6 ---->2p5].

39

PERIODIC TRENDS - SUMMARY[See QM & Periodic Table - page 15]

ELECTRON AFFINITY (EA) - energy change of reaction: A(gas) + e- ' A-(gas) EA of A = $E (+ or -).• More negative - more exothermic acquisition.• More negative (generally) L ---> R . across a row (period). “Quirky”.• Less negative (generally) Top ---> Bo"om down a column (group). “Quirky”. Li Be B C N O F-59.6 (+) -26.7 -122 (+) -141 -328 Na Mg Al C P S Cl-59.6 (+) -43 -134 -72 -200 -349 Br -325

40

PERIODIC TRENDS - SUMMARY[See QM & Periodic Table - page 16]

ELECTRONEGATIVITY (x) - measure of ability for atom to “draw” electron density from an atom to which it ischemically bonded.

• F = greatest = 4.0 , H = 2.1 , Group 1A metals - “least”.

• INCREASES L ---> R across a row (period). [“closer to F”]

• DECREASES Top ---> Bo"om down a column (group). [“farther away from F”] continued …

41

PERIODIC TRENDS - SUMMARYElectronegativity Di!erence ($x) &• Covalent versus Ionic Bonds:xA & xB = electronegativity of atoms A & B, respectively.(x = | xA - xB | = magnitude of the electronegativity di!erence between atoms A & B.If (x = 0 (Exactly) AB bond is purely covalent (or non-polar).If 0 < (x & 1.7 AB bond is polar covalent, i.e., the electrons are “polarized” toward with larger x.If (x > 1.7 AB bond is ionic, i.e., the electrons in the bond are“totally owned” by atom with larger x (anion) & “totallylost” by atom with smaller x (cation).