chapter 9 alkynes dr. wolf's chm 201 & 202 1

Dr. Wolf's CHM 201 & 202 9-1

Chapter 9Chapter 9AlkynesAlkynes

Dr. Wolf's CHM 201 & 202 9-2

Sources of AlkynesSources of Alkynes

Dr. Wolf's CHM 201 & 202 9-3

AcetyleneAcetylene

Industrial preparation of acetylene isIndustrial preparation of acetylene isby dehydrogenation of ethyleneby dehydrogenation of ethylene

CHCH33CHCH33800°C800°C

1150°C1150°C

cost of energy makes acetylene a morecost of energy makes acetylene a moreexpensive industrial chemical than ethyleneexpensive industrial chemical than ethylene

HH22CC CHCH22

HH22CC CHCH22 HCHC CHCH

HH22++

HH22++

Dr. Wolf's CHM 201 & 202 9-4

NomenclatureNomenclature

Dr. Wolf's CHM 201 & 202 9-5

HCHC CHCHAcetylene and ethyne are both acceptableAcetylene and ethyne are both acceptableIUPAC names forIUPAC names for

NomenclatureNomenclature

Higher alkynes are named in much the sameHigher alkynes are named in much the sameway as alkenes except using an -way as alkenes except using an -yneyne suffix suffixinstead of -instead of -eneene..

HCHC CCHCCH33

PropynePropyne

HCHC CCHCCH22CHCH33

1-Butyne1-Butyne

(CH(CH33))33CCCC CCHCCH33

4,4-Dimethyl-2-pentyne4,4-Dimethyl-2-pentyne

Dr. Wolf's CHM 201 & 202 9-6

Physical Properties of AlkynesPhysical Properties of Alkynes

The physical properties of alkynes are The physical properties of alkynes are similar to those of alkanes and alkenes.similar to those of alkanes and alkenes.

Dr. Wolf's CHM 201 & 202 9-7

Structure and Bonding in Alkynes:Structure and Bonding in Alkynes:spsp Hybridization Hybridization

Dr. Wolf's CHM 201 & 202 9-8

StructureStructure

linear geometry for acetylenelinear geometry for acetylene

CC CCHH HH120 pm120 pm

106 pm106 pm 106 pm106 pm

CC CCCHCH33 HH121 pm121 pm

146 pm146 pm 106 pm106 pm

Dr. Wolf's CHM 201 & 202 9-9

Cyclononyne is the Cyclononyne is the smallest cycloalkyne smallest cycloalkyne stable enough to be stable enough to be stored at room temperaturestored at room temperaturefor a reasonable length for a reasonable length of time. of time.

Cyclooctyne polymerizesCyclooctyne polymerizeson standing.on standing.

Dr. Wolf's CHM 201 & 202 9-10

22ss

22pp

22spsp

Mix together (hybridize) the 2s orbital Mix together (hybridize) the 2s orbital and and oneone of the three 2p orbitals of the three 2p orbitals

22pp

Bonding in acetylene is based on Bonding in acetylene is based on spsp-hybridization-hybridizationfor each carbonfor each carbon

Dr. Wolf's CHM 201 & 202 9-11

22spsp

Mix together (hybridize) the 2s orbital Mix together (hybridize) the 2s orbital and and oneone of the three 2p orbitals of the three 2p orbitals

22pp

Bonding in acetylene is based on Bonding in acetylene is based on spsp-hybridization-hybridizationfor each carbonfor each carbon

Each carbon has two Each carbon has two half-filled half-filled spsp orbitals orbitalsavailable to form available to form bonds. bonds.

Dr. Wolf's CHM 201 & 202 9-12

Bonds in AcetyleneBonds in Acetylene

Each carbon isconnected to ahydrogen by a bond. The twocarbons are connectedto each other by a bond and two bonds.

Dr. Wolf's CHM 201 & 202 9-13

Bonds in AcetyleneBonds in Acetylene

One of the two bonds in acetylene isshown here.The second bond is at rightangles to the first.

Dr. Wolf's CHM 201 & 202 9-14

Bonds in AcetyleneBonds in Acetylene

This is the secondof the two bonds in acetylene.

Dr. Wolf's CHM 201 & 202 9-15

The region of highest negative charge lies aboveand below the molecular plane in ethylene.

The region of highest negative charge encirclesthe molecule around itscenter in acetylene..

Dr. Wolf's CHM 201 & 202 9-16

C—C distanceC—C distance

C—H distanceC—H distance

H—C—C anglesH—C—C angles

C—C BDEC—C BDE

C—H BDEC—H BDE

% % ss character character

ppKKaa

153 pm153 pm

111 pm111 pm

111.0°111.0°

368 kJ/mol368 kJ/mol

410 kJ/mol410 kJ/mol

spsp33

25%25%

6262

134 pm134 pm

110 pm110 pm

121.4°121.4°

611 kJ/mol611 kJ/mol

452 kJ/mol452 kJ/mol

spsp22

33%33%

4545

120 pm120 pm

106 pm106 pm

180°180°

820 kJ/mol820 kJ/mol

536 kJ/mol536 kJ/mol

spsp

50%50%

2626

hybridization of Chybridization of C

Table 9.1 Comparison of ethane, ethylene, and acetyleneTable 9.1 Comparison of ethane, ethylene, and acetylene

Ethane Ethylene Ethane Ethylene AcetyleneAcetylene

Dr. Wolf's CHM 201 & 202 9-17

Acidity of Acetylene and Terminal AlkynesAcidity of Acetylene and Terminal Alkynes

HH

CC CC

Dr. Wolf's CHM 201 & 202 9-18

In general, hydrocarbons are In general, hydrocarbons are exceedingly weak acidsexceedingly weak acids

CompoundCompound ppKKaa

HFHF 3.23.2HH22OO 1616

NHNH33 3636

4545CHCH44 6060

HH22CC CHCH22

Dr. Wolf's CHM 201 & 202 9-19

Acetylene is a weak acid, but not nearlyAcetylene is a weak acid, but not nearlyas weak as alkanes or alkenes.as weak as alkanes or alkenes.

CompoundCompound ppKKaa

HFHF 3.23.2HH22OO 1616

NHNH33 3636

4545CHCH44 6060

HH22CC CHCH22

HCHC CHCH 2626

Dr. Wolf's CHM 201 & 202 9-20

Electronegativity of carbon Electronegativity of carbon increases with its increases with its ss character character

CC HH HH++ ++

HH++ ++

HH++ ++

1010-60-60

1010-45-45

1010-26-26

sp3CC :

sp2

sp

HH

CC CC

CC CC HH

CC CC

CC CC :

:

Electrons in an orbital with more s character are closer to thenucleus and more strongly held.

Dr. Wolf's CHM 201 & 202 9-21

Objective:Objective:Prepare a solution containing sodium acetylidePrepare a solution containing sodium acetylide

Will treatment of acetylene with NaOH be effective?Will treatment of acetylene with NaOH be effective?

NaCNaC CHCH

HH22OONaOHNaOH ++ HCHC CHCH NaCNaC CHCH ++

Dr. Wolf's CHM 201 & 202 9-22

No. Hydroxide is not a strong enough No. Hydroxide is not a strong enough base to deprotonate acetylene.base to deprotonate acetylene.

HH22OONaOHNaOH ++ HCHC CHCH NaCNaC CHCH ++

––HOHO

....

.... ::

HH CC CHCH HOHO HH........++ ++ CC CHCH::

––

weaker acidweaker acidppKKaa = 26 = 26

stronger acidstronger acidppKKaa = 16 = 16

In acid-base reactions, the equilibrium lies toIn acid-base reactions, the equilibrium lies tothe side of the weaker acid.the side of the weaker acid.

Dr. Wolf's CHM 201 & 202 9-23

Solution: Use a stronger base. Sodium amideSolution: Use a stronger base. Sodium amideis a stronger base than sodium hydroxide.is a stronger base than sodium hydroxide.

NHNH33NaNaNHNH22 ++ HCHC CHCH NaCNaC CHCH ++

––HH22NN

....::

HH CC CHCH HH....

++ ++ CC CHCH::––

stronger acidstronger acidppKKaa = 26 = 26

weaker acidweaker acidppKKaa = 36 = 36

Ammonia is a weaker acid than acetylene.Ammonia is a weaker acid than acetylene.The position of equilibrium lies to the right.The position of equilibrium lies to the right.

HH22NN

Dr. Wolf's CHM 201 & 202 9-24

Preparation of Alkynes Preparation of Alkynes byby

Alkylation of Acetylene and Terminal AlkynesAlkylation of Acetylene and Terminal Alkynes

Dr. Wolf's CHM 201 & 202 9-25

Preparation of AlkynesPreparation of Alkynes

Carbon-carbon bond formationalkylation of acetylene and terminal alkynes

Functional-group transformationselimination

There are two main methods for the preparationThere are two main methods for the preparationof alkynes:of alkynes:

Dr. Wolf's CHM 201 & 202 9-26

Alkylation of acetylene and terminal alkynesAlkylation of acetylene and terminal alkynes

H—C H—C C—HC—H

RR—C —C C—HC—H

RR—C —C C—C—RR

Dr. Wolf's CHM 201 & 202 9-27

RR XXSSNN22

Alkylation of acetylene and terminal alkynesAlkylation of acetylene and terminal alkynes

XX––::++CC––::H—C H—C C—RC—RH—C H—C ++

• The alkylating agent is an alkyl halide, andThe alkylating agent is an alkyl halide, andthe reaction is nucleophilic substitution.the reaction is nucleophilic substitution.

• The nucleophile is sodium acetylide or the The nucleophile is sodium acetylide or the sodium salt of a terminal (monosubstituted) sodium salt of a terminal (monosubstituted) alkyne.alkyne.

Dr. Wolf's CHM 201 & 202 9-28

Example: Example: Alkylation of acetylene Alkylation of acetylene

NaNHNaNH22

NHNH33

CHCH33CHCH22CHCH22CHCH22BrBr

(70-77%)(70-77%)

HCHC CHCH HCHC CCNaNa

HCHC CC CHCH22CHCH22CHCH22CHCH33

Dr. Wolf's CHM 201 & 202 9-29

NaNHNaNH22, NH, NH33

CHCH33BrBr

Example: Example: Alkylation of a terminal alkyne Alkylation of a terminal alkyne

CCHH(CH(CH33))22CHCHCHCH22CC

CCNaNa(CH(CH33))22CHCHCHCH22CC

(81%)(81%)

C—CHC—CH33(CH(CH33))22CHCHCHCH22CC

Dr. Wolf's CHM 201 & 202 9-30

1. NaNH1. NaNH22, NH, NH33

2. 2. CHCH33CHCH22BrBr

(81%)(81%)

Example: Example: Dialkylation of acetyleneDialkylation of acetylene

H—C H—C C—HC—H

1. NaNH1. NaNH22, NH, NH33

2. 2. CHCH33BrBr

C—HC—HCHCH33CHCH22—C—C

C—C—CHCH33CHCH33CHCH22—C—C

Dr. Wolf's CHM 201 & 202 9-31

LimitationLimitation

Effective only with primary alkyl halides

Secondary and tertiary alkyl halides undergo elimination

Dr. Wolf's CHM 201 & 202 9-32

E2 predominates over SE2 predominates over SNN2 when alkyl 2 when alkyl halide is secondary or tertiaryhalide is secondary or tertiary

HH CC

C— C— XX

E2E2

CC––::H—C H—C

++CCH—C H—C ——HH CC CC XX––::++

Dr. Wolf's CHM 201 & 202 9-33

Preparation of Alkynes byPreparation of Alkynes byElimination ReactionsElimination Reactions

Dr. Wolf's CHM 201 & 202 9-34

Preparation of Alkynes by Preparation of Alkynes by "Double" Dehydrohalogenation"Double" Dehydrohalogenation

Geminal dihalideGeminal dihalide Vicinal dihalideVicinal dihalide

XX

CC CC

XX

HH

HH

XX XX

CC CC

HHHH

The most frequent applications are in preparation The most frequent applications are in preparation of terminal alkynes.of terminal alkynes.

Dr. Wolf's CHM 201 & 202 9-35

Geminal dihalide Geminal dihalide Alkyne Alkyne

(CH(CH33))33CCCCHH22—CH—CHClCl22

1. 3NaNH1. 3NaNH22, NH, NH33

2. H2. H22OO

(56-60%)(56-60%)

(CH(CH33))33CCCC CHCH

Dr. Wolf's CHM 201 & 202 9-36

NaNHNaNH22, NH, NH33

NaNHNaNH22, NH, NH33

NaNHNaNH22, NH, NH33HH22OO

Geminal dihalide Geminal dihalide Alkyne Alkyne

(CH(CH33))33CCCCHH22—CH—CHClCl22

(CH(CH33))33CCCCHH CHCHClCl

(CH(CH33))33CCCC CHCH

(CH(CH33))33CCCC CNaCNa

(slow)(slow)

(slow)(slow)

(fast)(fast)

Dr. Wolf's CHM 201 & 202 9-37

CHCH33(CH(CH22))77CCHH—C—CHH22BrBr

BrBr

Vicinal dihalide Vicinal dihalide Alkyne Alkyne

1. 3NaNH1. 3NaNH22, NH, NH33

2. H2. H22OO

(54%)(54%)

CHCH33(CH(CH22))77CC CHCH

Dr. Wolf's CHM 201 & 202 9-38

Reactions of AlkynesReactions of Alkynes

Dr. Wolf's CHM 201 & 202 9-39

Reactions of AlkynesReactions of Alkynes

Acidity (Section 9.5)Hydrogenation (Section 9.9)Metal-Ammonia Reduction (Section 9.10)Addition of Hydrogen Halides (Section 9.11)Hydration (Section 9.12)Addition of Halogens (Section 9.13)Ozonolysis (Section 9.14)

Dr. Wolf's CHM 201 & 202 9-40

Hydrogenation of AlkynesHydrogenation of Alkynes

Dr. Wolf's CHM 201 & 202 9-41

Hydrogenation of AlkynesHydrogenation of Alkynes

RCRCHH22CCHH22R'R'catcat

catalyst = Pt, Pd, Ni, or Rhcatalyst = Pt, Pd, Ni, or Rh

alkene is an intermediatealkene is an intermediate

RCRC CR'CR' ++ 22HH22

Dr. Wolf's CHM 201 & 202 9-42

Heats of hydrogenationHeats of hydrogenation

292 kJ/mol292 kJ/mol 275 kJ/mol275 kJ/mol

Alkyl groups stabilize triple bonds in the Alkyl groups stabilize triple bonds in the same way that they stabilize doublesame way that they stabilize doublebonds. Internal triple bonds are more bonds. Internal triple bonds are more stable than terminal ones.stable than terminal ones.

CHCH33CHCH22CC CHCH CHCH33CC CCCHCH33

Dr. Wolf's CHM 201 & 202 9-43

Partial HydrogenationPartial Hydrogenation

RCHRCH22CHCH22R'R'

Alkynes could be used to prepare alkenes if acatalyst were available that is active enough to catalyze the hydrogenation of alkynes, but notactive enough for the hydrogenation of alkenes.

catcatHH22

RCRC CR'CR' catcatHH22

RCHRCH CHR'CHR'

Dr. Wolf's CHM 201 & 202 9-44

There is a catalyst that will catalyze the hydrogenationof alkynes to alkenes, but not that of alkenes to alkanes.

It is called the Lindlar catalyst and consists ofpalladium supported on CaCO3, which has been poisoned with lead acetate and quinoline.

syn-Hydrogenation occurs; cis alkenes are formed.

Lindlar PalladiumLindlar Palladium

RCHRCH22CHCH22R'R'catcatHH22

RCRC CR'CR' catcatHH22

RCHRCH CHR'CHR'

Dr. Wolf's CHM 201 & 202 9-45

ExampleExample

+ + H H22

Lindlar PdLindlar Pd

CHCH33(CH(CH22))33 (CH(CH22))33CHCH33

HH HH(87%)(87%)

CHCH33(CH(CH22))33CC C(CHC(CH22))33CHCH33

CCCC

Dr. Wolf's CHM 201 & 202 9-46

Metal-Ammonia Reduction of Metal-Ammonia Reduction of AlkynesAlkynes

Alkynes Alkynes transtrans-Alkenes-Alkenes

Dr. Wolf's CHM 201 & 202 9-47

Partial ReductionPartial Reduction

RCHRCH22CHCH22R'R'

Another way to convert alkynes to alkenes isby reduction with sodium (or lithium or potassium)in ammonia.

trans-Alkenes are formed.

RCRC CR'CR' RCHRCH CHR'CHR'

Dr. Wolf's CHM 201 & 202 9-48

ExampleExample

CHCH33CHCH22

CHCH22CHCH33HH

HH

(82%)(82%)

CHCH33CHCH22CC CCHCCH22CHCH33

CCCC

Na, NHNa, NH33

Dr. Wolf's CHM 201 & 202 9-49

MechanismMechanism

four steps(1) electron transfer(2) proton transfer(3) electron transfer(4) proton transfer

Metal (Li, Na, K) is reducing agent; Metal (Li, Na, K) is reducing agent; HH22 is not involved is not involved

Dr. Wolf's CHM 201 & 202 9-50

MechanismMechanism

Step (1): Transfer of an electron from the metalStep (1): Transfer of an electron from the metalto the alkyne to give an anion radical.to the alkyne to give an anion radical.

MM ..++RR R'R'CC CC RR R'R'CC.... ..––

CC

MM++

Dr. Wolf's CHM 201 & 202 9-51

Step (2) Transfer of a proton from the solvent Step (2) Transfer of a proton from the solvent (liquid ammonia) to the anion radical.(liquid ammonia) to the anion radical.

MechanismMechanism

HH NHNH22

....

RR R'R'CC......––CC

..R'R'

RR

CC CC

HHNHNH22

....

––::

Dr. Wolf's CHM 201 & 202 9-52

Step (3): Transfer of an electron from the metalStep (3): Transfer of an electron from the metalto the alkenyl radical to give a carbanion.to the alkenyl radical to give a carbanion.

MechanismMechanism

MM ..++..

R'R'

RR

CC CC

HH

MM++

R'R'

RR

CC CC

HH

....––

Dr. Wolf's CHM 201 & 202 9-53

Step (4) Transfer of a proton from the solventStep (4) Transfer of a proton from the solvent(liquid ammonia) to the carbanion .(liquid ammonia) to the carbanion .

MechanismMechanism

HH NHNH22

....

R'R'

RR

CC CC

HH

.... ––

R'R'HH

HH

CCCC

RR NHNH22

....

––::

Dr. Wolf's CHM 201 & 202 9-54

Suggest efficient syntheses of (Suggest efficient syntheses of (EE)- and ()- and (ZZ)-2-)-2-heptene from propyne and any necessary organicheptene from propyne and any necessary organic or inorganic reagents. or inorganic reagents.

Dr. Wolf's CHM 201 & 202 9-55

1. NaNH1. NaNH22

2. CH2. CH33CHCH22CHCH22CHCH22BrBr

Na, NHNa, NH33HH22, Lindlar Pd, Lindlar Pd

Dr. Wolf's CHM 201 & 202 9-56

Addition of Hydrogen Halides Addition of Hydrogen Halides to Alkynesto Alkynes

Dr. Wolf's CHM 201 & 202 9-57

Follows Markovnikov's RuleFollows Markovnikov's Rule

HBrHBr

BrBr

(60%)(60%)

Alkynes are slightly less reactive than alkenesAlkynes are slightly less reactive than alkenes

CHCH33(CH(CH22))33CC CHCH CHCH33(CH(CH22))33CC CHCH22

Dr. Wolf's CHM 201 & 202 9-58

CHCH

Termolecular transition stateTermolecular transition state

....BrBrHH ::....

RCRC

....BrBrHH ::....

Observed rate law: rate = Observed rate law: rate = kk[alkyne][HX][alkyne][HX]22

Dr. Wolf's CHM 201 & 202 9-59

Reaction with two moles of a hydrogenReaction with two moles of a hydrogenhalide yields a geminal dihalidehalide yields a geminal dihalide

(76%)(76%)

CHCH33CHCH22CC CCHCCH22CHCH33

2 H2 HFF

FF

FF

CC CC

HH

HH

CHCH33CHCH22 CHCH22CHCH33

Dr. Wolf's CHM 201 & 202 9-60

Free-radical addition of HBr occurs whenFree-radical addition of HBr occurs whenperoxides are presentperoxides are present

HBrHBr

(79%)(79%)

regioselectivity opposite to Markovnikov's ruleregioselectivity opposite to Markovnikov's rule

CHCH33(CH(CH22))33CC CHCH CHCH33(CH(CH22))33CCHH CHCHBrBrperoxidesperoxides

Dr. Wolf's CHM 201 & 202 9-61

Hydration of AlkynesHydration of Alkynes

Dr. Wolf's CHM 201 & 202 9-62

Hydration of AlkynesHydration of Alkynes

expected reaction:expected reaction:

observed reaction:observed reaction:

RCHRCH22CR'CR'

OO

HH++

RCRC CR'CR' HH22OO++

HH++

RCRC CR'CR' HH22OO++

OHOH

RCHRCH CR'CR'

Dr. Wolf's CHM 201 & 202 9-63

• enols are regioisomers of ketones, and exist in equilibrium with them

• keto-enol equilibration is rapid in acidic media

• ketones are more stable than enols andpredominate at equilibrium

enolenol

OHOH

RCHRCH CR'CR' RCHRCH22CR'CR'

OO

ketoneketone

Dr. Wolf's CHM 201 & 202 9-64

Mechanism of conversion of enol to ketoneMechanism of conversion of enol to ketone

OO HH

CC CC

HH++OO

HH

HH

::

....::

Dr. Wolf's CHM 201 & 202 9-65

Mechanism of conversion of enol to ketoneMechanism of conversion of enol to ketone

OO HH

CC CC

HH++OO

HH

HH

::

....::

Dr. Wolf's CHM 201 & 202 9-66

Mechanism of conversion of enol to ketoneMechanism of conversion of enol to ketone

OO HH

CC CCHH++

OO

HH

HH

::

....::

::

Dr. Wolf's CHM 201 & 202 9-67

Mechanism of conversion of enol to ketoneMechanism of conversion of enol to ketone

OO HH

CC CC

HH

HH

OO:: ::

HH++

....::

Dr. Wolf's CHM 201 & 202 9-68

Mechanism of conversion of enol to ketoneMechanism of conversion of enol to ketone

OO HH

CC CC

HH

HH

OO:: ::

HH++

....::

Dr. Wolf's CHM 201 & 202 9-69

Mechanism of conversion of enol to ketoneMechanism of conversion of enol to ketone

OOHH

CC CC

HH

HH

OO::

HH

++....::

Dr. Wolf's CHM 201 & 202 9-70

Key carbocation intermediate is stabilized by Key carbocation intermediate is stabilized by electron delocalization (resonance)electron delocalization (resonance)

OO HH

CC CCHH++

....:: OO

CC CCHH

....HH++

Dr. Wolf's CHM 201 & 202 9-71

HH22O, HO, H++

(89%)(89%)

viavia

ExampleExample

CHCH33(CH(CH22))22CC C(CHC(CH22))22CHCH33

HgHg2+2+

OO

CHCH33(CH(CH22))22CHCH22C(CHC(CH22))22CHCH33

OHOH

CHCH33(CH(CH22))22CHCH C(CHC(CH22))22CHCH33

Dr. Wolf's CHM 201 & 202 9-72

HH22O, HO, H22SOSO44

HgSOHgSO44

CHCH33(CH(CH22))55CCHCCH33

(91%)(91%)

viavia

Markovnikov's rule followed in formation of enolMarkovnikov's rule followed in formation of enol

CHCH33(CH(CH22))55CC CHCH22

OHOH

CHCH33(CH(CH22))55CC CHCH

OO

Dr. Wolf's CHM 201 & 202 9-73

Addition of Halogens to Addition of Halogens to AlkynesAlkynes

Dr. Wolf's CHM 201 & 202 9-74

ExampleExample

+ 2 + 2 ClCl22

ClCl

ClCl

(63%)(63%)

CCClCl22CHCH CHCH33HCHC CCHCCH33

Dr. Wolf's CHM 201 & 202 9-75

Addition is antiAddition is anti

BrBr22

CHCH33CHCH22

CHCH22CHCH33BrBr

BrBr

(90%)(90%)

CHCH33CHCH22CC CCHCCH22CHCH33 CC CC

Dr. Wolf's CHM 201 & 202 9-76

Ozonolysis of AlkynesOzonolysis of Alkynes

gives two carboxylic acids by cleavage gives two carboxylic acids by cleavage of triple bondof triple bond

Dr. Wolf's CHM 201 & 202 9-77

ExampleExample

1. O1. O33

2. H2. H22OO

++

CHCH33(CH(CH22))33CC CCHH

CHCH33(CH(CH22))33CCOHOH

(51%)(51%)

OO

HOHOCCOHOH

OO

Dr. Wolf's CHM 201 & 202 9-78

End of Chapter 9