Chemistry 232

Kinetics of Complex Reactions

The Pre-Equilibrium Approximation

Examine the following process1

1

2

1 1

1 1

2 2

v k A

v k B

v k B

k

k

k

A B

B C

2

d Ck B

dt

Pre-Equilibrium (II)

B is obviously an intermediate in the above mechanism. • Could use SSA.

What if the initial equilibrium is fast?• Step 2 is the rds!

1

1

B A Ak

Kk

Pre-Equilibrium (III)

We now have a simple expression for the [B]; hence

12 2

1

A ' Ad C k

k B k kdt k

Lindemann-Hinshelwood Mechanism

An early attempt to explain the kinetics of complex reactions.

PA

A2AA

AAAA

2

1

1

k

k

k

Akv

AAkv

Akv

22

11

211

Mechanism Rate Laws

The ‘Activated’ Intermediate

Formation of the product depends directly on the [A*].

Apply the SSA to the net rate of formation of the intermediate [A*]

0AkAAkAkdtAd

212

1

Is That Your ‘Final Answer’?

Substituting and rearranging

Akk

AkkdtPd

12

212

The ‘Apparent Rate Constant’ Depends on Pressure

The rate laws for the Lindemann-Hinshelwood Mechanism are pressure dependent.

High Pressure Case Low Pressure Case

Ak

kAkk

dtPd

1

12

/

2

21

Ak

AkdtPd

/

The Pressure Dependence of k’ In the Lindemann-Hinshelwoood

Mechanism, the rate constant is pressure dependent.

21

1

1 kkk

Ak1

k1 /

Catalysts

So far, we have considered one way of speeding up a reaction (i.e. increasing T usually increases k). Another way is by the use of a catalyst.

A catalyst - a substance that speeds up the rate of the reaction without being consumed in the overall reaction.

look at the following two reactions

A+B C rate constant k

A+B C rate constant with catalyst is kcat

NOTE: RATE WITH CATALYST > RATE WITHOUT CATALYST

Types of Catalyst

We will briefly discuss three types of catalysts. The type of catalyst depends on the phase of the catalyst and the reacting species. • Homogeneous

• Heterogeneous

• Enzyme

Homogeneous Catalysis

The catalyst and the reactants are in the same phase

e.g. Oxidation of SO2 (g) to SO3 (g)

2 SO2(g) + O2(g) 2 SO3 (g) SLOW

Presence of NO (g), the following occurs.NO (g) + O2 (g) NO2 (g)

NO2 (g) + SO2 (g) SO3 (g) + NO (g) FAST

SO3 (g) is a potent acid rain gas

H2O (l) + SO3 (g) H2SO4 (aq) Note the rate of NO2(g) oxidizing SO2(g)

to SO3(g) is faster than the direct oxidation.

NOx(g) are produced from burning fossil fuels such as gasoline, coal, oil!!

Heterogeneous Catalysis

The catalyst and the reactants are in different phases• adsorption the binding of molecules on a

surface.

Adsorption on the surface occurs on active sites• Places where reacting molecules are adsorbed

and physically bond to the metal surface.

The hydrogenation of ethene (C2H4 (g)) to ethane

C2H4 (g) + H2(g) C2H6 (g)Reaction is energetically favourablerxnH = -136.98 kJ/mole of ethane.

With a finely divided metal such as Ni (s), Pt (s), or Pd(s), the reaction goes very quickly .

There are four main steps in the process

• the molecules approach the surface;

• H2 (g) and C2H4 (g) adsorb on the surface;

• H2 dissociates to form H(g) on the surface; the adsorbed H atoms migrate to the adsorbed C2H4 and react to form the product (C2H6) on the surface

• the product desorbs from the surface and diffuses back to the gas phase

Simplified Model for Enzyme Catalysis

E enzyme; S substrate; P product

E + S ES ES P + E

rate = k [ES]The reaction rate depends directly on the

concentration of the substrate.

Enzyme Catalysis

Enzymes - proteins (M > 10000 g/mol)High degree of specificity (i.e., they will

react with one substance and one substance primarily

Living cell > 3000 different enzymes

The Lock and Key Hypothesis

Enzymes are large, usually floppy molecules. Being proteins, they are folded into fixed configuration.

According to Fischer, active site is rigid, the substrate’s molecular structure exactly fits the “lock” (hence, the “key”).

The Lock and Key (II)

The Michaelis-Menten Mechanism

Enzyme kinetics – use the SSA to examine the kinetics of this mechanism.

ES – the enzyme-substrate complex.

1 2

1

k k

kE S ES P E

Applying the SSA to the Mechanism

Note that the formation of the product depends directly on the [ES]

What is the net rate of formation of [ES]?

ESkESkSEkdtESd

21o1

ES – The Intermediate

Apply the SSA to the equation for d[ES]/dt = 0

21

o1

21o1

kkSEk

ES

ESkESkSEk

0dtESd

Working Out the Details

Let [E]o = [E] + [ES]

Initial enzyme concentration

Complex concentration

Free enzyme concentration

o121

oo1

SkkkSEk

ES

Note that [E] = [E]o - [ES]

The Final Equation

Substituting into the rate law vp.

ESkv 2p

Mo

oo2

o121

oo12p

KSSEk

SkkkSEk

kv

The Michaelis Constant and the Turnover Number

The Michaelis Constant is defined as

1

12M k

kkK

The rate constant for product formation, k2, is the turnover number for the catalyst.

Ratio of k2 / KM – indication of catalytic efficiency.

The Maximum Velocity

As [S]o gets very large.

maxlim vEkv o2

Sp

o

Note – Vmax is the maximum velocity for the reaction. The limiting value of the reaction rate high initial substrate concentrations.

Lineweaver-Burk Equation

Plot the inverse of the reaction rate vs. the inverse of the initial substrate concentration.

oM

o S1

vK

v1

v1

maxmax

Chain Reactions

Classifying steps in a chain reaction.• Initiation

• C2H6 (g) 2 CH3•

• Propagation Steps

• C2H6 + •CH3 •C2H5 + CH4

• Branching Steps

• H2O + •O• 2 •OH

Chain Reactions (Cont’d)

Retardation Step

• HBr + H• H2 + Br•

Terminations Steps

• 2 CH3CH2• CH3CH2CH2CH3

Inhibition Steps

• R• + CH3• RCH3

The H2 + Br2 Reaction

The overall rate for the reaction was established in 1906 by Bodenstein and Lind

HBrkBr

BrHk

dt

HBrd/

2

23

22

The Mechanism

The mechanism was proposed independently by Christiansen and Herzfeld and by Michael Polyani.

Mechanism

Rate Laws

BrBr 22

HHBrHBr 2

BrHBrBrH 2

BrHHBrH 2

2BrBrBr

211 Brkv 222 HBrkv HBrkv 222 HBrHkv 33

244 Brkv

Using the SSA

Using the SSA on the rates of formation of Br• and H•

HBrkkBr

BrHkkk

dt

HBrd

/2

32

23

224

122

Hydrogenation of Ethane

The Rice-Herzfeld Mechanism

Mechanism 362 2CHHC

423362 CHCHCHCHHC HCHCHCHCH 2223

22333 HCHCHHCHCH 6223 HCHCHCH

Rate Laws for the Rice-Herzfeld Mechanism

The rate laws for the elementary reactions are as follows.

6211 HCkv

36222 CHHCkv

2322 CHCHkv 3322 // CHCHHkv

2333 CHCHHkv

Explosions

Thermal explosions • Rapid increase in the reactions rate with

temperature.

Chain branching explosions• chain branching steps in the mechanism lead

to a rapid (exponential) increase in the number of chain carriers in the system.

Photochemical Reactions

Many reactions are initiated by the absorption of light.

Stark-Einstein Law – one photon is absorbed by each molecule responsible for the primary photochemical process.

I = Intensity of the absorbed radiation

Iv I

Primary Quantum Yield

Define the primary quantum yield,

absorbed photons of #

productsprimary of #

Define the overall quantum yield,

absorbed photons of #

react that molecules reactant of #

Photosensitization

Transfer of excitation energy from one molecule (the photosensitizer) to another nonabsorbing species during a collision..

HHgHHHg

HHgHHg

HgHgnm

2

2

254

2

Polymerization Kinetics

Chain polymerization• Activated monomer attacks another

monomer, chemically bonds to the monomer, and then the whole unit proceeds to attack another monomer.

Stepwise polymerization• A reaction in which a small molecule (e.g.,

H2O) is eliminated in each step.

Chain Polymerization

The overall polymerization rate is first order in monomer and ½ order in initiator.

The kinetic chain length, kcl

• Measure of the efficiency of the chain propagation reaction.

produced centres active of #

consumed units monomer of #

i

pkcl v

v

Mechanism

InitiationI 2 R•

Or

M + R• M1 • Propagation

M + M1• M2 •

M + M2• M3 •

M + M3• M4 •Etc.

Ikv ii

Rate Laws

1npp MMkv

Mechanism (Cont’d)

Termination

M + M3• M4 •

2 Mkv tt

Note – Not all the initiator molecules produce chainsDefine = fraction of initiator molecules that produce chains

Ikdt

Mdi2

Return to Kinetic Chain Length

We can express the kinetic chain length in terms of kt and kp

2

12

12

2

2

pkcl

t

p

i t

k M M

k M

k M I

k k

Stepwise Polymerization

A classic example of a stepwise polymerization – nylon production.NH2-(CH2)6-NH2 + HOOC-(CH2)4COOH

NH2-(CH2)6-NHOC-(CH2)4COOH + H2O

After many stepsH-(NH-(CH2)6-NHOC-(CH2)4CO)n-OH

The Reaction Rate Law

Consider the condensation of a generic hydroxyacid

OH-M-COOH

Expect the following rate law

COOHOHkv polypoly

The Reaction Rate Law (Cont’d)

Let [A] = [-COOH]A can be taken as any generic end

group for the polymer undergoing condensation.

Note 1 –OH for each –COOH

2Ak

AOHkv

poly

polypoly

The Reaction Rate Law (Cont’d)

If the rate constant is independent of the molar mass of the polymer

opoly

o

opoly

ot

Atk

A

COOHtk

COOHCOOH

1

1

The Fraction of Polymerization

Denote p = the fraction of end groups that have polymerized

o

to

A

AAp

opoly

opoly

Atk

Atkp

1

Statistics of Polymerization

Define Pn = total probability that a polymer is composed of n-monomers

ppP nn 11

The Degree of Polymerization

Define <n> as the average number of monomers in the chain

t

o

A

A

pn

1

1

Degree of Polymerization (cont’d)

The average polymer length in a stepwise polymerization increases as time increases.

opoly

opoly

opoly

Atk

Atk

Atk

pn

1

11

1

1

Molar Masses of Polymers

The average molar mass of the polymer also increases with time.

Two types of molar mass distributions.

• <M>n = the number averaged molar mass of the polymer.

• <M>w = the mass averaged molar mass of the polymer.

Definitions of <M>n

Two definitions!

11

1

on

J JJ

M Mp

n Mn

Mo = molar mass of monomer n = number of polymers of mass Mn

MJ = molar mass of polymer of length nJ

Definitions of <M>w

<M>w is defined as follows

J JJ

JJ

j

xnow

Mn

Mn

pxMpM n 1221

Note - xn the number of monomer units in a polymer molecule

The Dispersity of a Polymer Mixture

Polymers consists of many molecules of varying sizes.

Define the dispersity index () of the mass distribution.

n

w

M

M

Note – monodisperse sample ideally has <M>w=<M>n

The Dispersity Index in a Stepwise Polymerization

The dispersity index varies as follows in a condensation polymerization

n

w

M

M1

Note – as the polymerization proceeds, the ratio of <M>w/<M>n approaches 2!!!

Mass Distributions in Polymer Samples

For a random polymer sample

09 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41

Monodisperse Sample

Polydisperse Sample

Molar mass / (10000 g/mole)

Pn