Chemical Kinetics1

ChemicalKinetics

Chapter I

Peter Atkins, Physical Chemistry, 7th editionJeffrey I. Steinfeld et al, Chemical Kinetics and Dynamics, 2nd edition

Basic concepts of kinetics

Thermodynamics• First law• Second law• Third law

Kinetics

Deal with changes of the system properties in time.

(Physical-Chemical)

Feasibility of any process or reacction to take place

(DSUniv. > 0, DG < 0)

a) Rate of a chemical reaction:

Chemical kinetics: Study chemical systems whose composition changes with time

!Gas, liquids, solid• Homogeneous: Single-phase

• Heterogeneous: Multi-phase

Stoichiometric representation

aA + bB + … cC + dD …

a and b: # of moles of A and B, i.e. the reactants

c and d: # of moles of C and D, i.e. the products

IrreversibleChemical reaction

ReversibleChemical reaction

2H2 +O2 2H2O H2 + I2 2HI

Type of reactions:• Elementary: One step

• Complexes: Multi-steps

Rate of reaction: Change in composition of the reaction mixtures

dtDd

ddtCd

cdtBd

bdtAd

aR 1111

Reac. [i] (-) Prod. [j] (+)

I2 + hn I2*I2* 2I

2I + M I2 + MH2 + 2I 2HI

b) Order and Molecularity of a chemical reaction:

• One or more reactants[i]’s

• One or more intermadiates[E]’s

• One or more species that do not appear in the reaction

[C]

Reaction rate

R = f([A], [B])

R a [A]m [B]n

• m and n: Integer, fractional or negative

R = k [A]m [B]n

Rate equation!

Rate constant(Proportionality constant)

• m: Order of reaction with respect to A.

• n: Order of reaction with respect to B

• p = m + n : Overall order of reaction

http://www.chem.uci.edu/education/undergrad_pgm/applets/sim/simulation.htm

General expression

k

i

ni

ickR1

• ni: Reaction order with respect to i’s component.

*Units of K : [i]-(p-1)t-1

k

iinp

1

Elementary reactions are described by their molecularity

Molecularity: # of reactants involved in the reaction step! (always an integer)

Spontaneousdecomposition

A and B reactwith each other

Three reactantsthat comes together

A Products Unimolecular

A + B Products Bimolecular

A + B + C Products Termolecular

http://www.chm.davidson.edu/ChemistryApplets/kinetics/ReactionRates.html

Elementary reaction rate lawsTime behavior Integrating the rate law

a) Zero-Order reaction:

ktAAktAA

t

dtkAdkdtAd

Aa

AkdtAd

aR

tt

t

t

A

A

t

00

0

][

][

0

0

][][][][

0

][][

1][,1

][][1

00

t

[A]t

[A]0

m = -K

a) A productsHeterogeneous reactions

b) First-Order reaction:

ktt

t

eAA

ktAA

0

0

][][

][][ln

303.2]log[]log[

]ln[]ln[][][

][][][][1

0

00

][

][

1

0

ktAA

ktAAdtkAAd

kdtAAdAk

dtAd

aR

t

t

tA

A

t

a = 1

t0=0

t

log[A]t

log[A]0

m = -k/2.303

t

[A]t

[A]0

t

e-1[A]0

k-1 = t (Decay time)[A]t = [A]0/ee=2.7183

b) A productsCH3NC CH3CN

(CH3)3CBr + H2O (CH3)3COH + HBr[90% acetone & 10% water]T (min) [C] (mol/L)

0 0.1056

9 0.0961

18 0.0856

24 0.0767

40 0.0645

54 0.0536

72 0.0432

105 0.0270

0 20 40 60 80 100 120

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0.11

[C] (

mol

/L)

t (min)

KtAA t 0][][

0 20 40 60 80 100 120

0.1

log[

C] (

mol

/L)

t (min)

303.2]log[]log[ 0

KtAA t

c) Second-Order reaction:

c1) A + A products

ktAA

dtkA

AdktAA

xxdx

kdtA

AdAkdtAdR

t

tA

At

t

2][

1][

1

2][

][2][

1][

1

1

2][

][][][21

0

0

][

][2

0

2

22

0

t

[A]t-1

[A]0-1

m = 2k

2CH3 C2H6b

a

b

a

c2) A + B products

kt

BABA

BA

baxabax

dx

xABAdx

xBBAdx

xBxAdx

xBxAkdtdx

BBAAx

BAkdtAdR

t

t

tt

][][][][ln

][][1

)ln(1

][][][][][][][][

][][

][][][][

][][][

0

0

00

00000000

00

00

11

H2 + O OH + H

b

a

b

a

d) Third-Order reaction:

d1) A + A + A products

ktAA

dtkA

AdktAA

xnxdx

kdtA

AdAkdtAdR

t

tA

At

n

b

an

t

6][1

][1

3][

][3][1

][1

21

1)1(

1

3][

][][][31

20

2

0

][

][32

02

1

33

0

t

[A]t-2

[A]0-2

m = 6k

b

a

I + I + M I2 + M

d2) A + A + B products

kt

BABA

BAAABA

kdtyByA

dy

yByAkdtdy

BAyBB

yAA

BAkdtAdR

t

t

t

t

t

][][][][ln

][2][1

][1

][1

][2][1

][2][

][2][

][,][][][

2][][

][][][21

0

02

00000

02

0

02

0

00

0

0

2

Partial fractions!

tkAA

orderpseudoBkk

AkdtAdR

t

nd

'2][

1][

1

2],['

]['][21

0

2

[B]>>[A][B] = const.

O + O2 + M O3 + M

d3) A + B + C products

]][][[][ CBAkdtAdR

Reaction rate for n order respect with only one reactant

ktnnAA

nktAAn

dtnkA

Ad

AkdtAd

nR

nnt

nnt

A

A

t

n

n

t

)1(][1

][1

][1

][1

)1(1

][][

][)(

1

)1(0

)1()1(0

)1(

][

][ 00

e) Reaction half-lives: Alternative method to determine the reaction order

2][

][ 0

21

21

AAt

2ln1

][2/][ln

2/][][

][][ln

2/1

2/1

0

0

0

0

kt

ktA

A

AA

ktAA

t

t

First-order reaction

For a reaction of order n>1 in a single reactant

Independent of [A]0

1

0

1

21 ])[1(

12

n

n

Ankt

This is function of [A]0

1st order reaction half-lives:

0 10 20 30 40 500

0.2

0.4

0.6

0.8

11

6.738 10 3

A t( )

500 t

T dependence of the rate constant, kThe Arhenius equation:

k(T):

• [i]• t• pH (only in solution) T! (Strongly)

TkE

TB

Act

Aek )(

1/T

ln[k(T)]lnA

m = -EAct/kB

Chemical coordinates

E

EAct(F)EAct(R)

Reac

Prod.

DH0Rxn

A :Frequency factor

0Re

0Pr

0

0 )()(

acodRxn

ActActRxn

HHH

FEREH

DDD

D

Ineffective Effective

Determinant Parameters

Catalysis

Complex reactionsa) Reversible reactions:

A1 A2kf

kr

12

2

211

AkAkdtAd

AkAkdtAd

fr

rf

If @ t=0[A1] = [A1]0

[A2] = [A2]0

[A1] + [A2] = [A1]0 + [A2]0[A2] = [A1]0 + [A2]0 - [A1]

020111

1020111

AAkkkAdtAd

AkAAkAkdtAd

rrf

rrf

(Mass conservation law)

ClCl

Cl

Cl

102011 A

kkAAk

kkdtAd

rf

rrf

Introducing the variable m

tkk

AkAkAkAk

dtkkAm

AdAmkk

dtAd

kkAAk

m

rfrf

rf

t

trf

A

Arf

rf

r

0201

21

1

11

1

0201

ln

0

1

01

tkkAk

AkAkrf

f

rf

01

21ln

If @ t=0[A2]0=0

tkk

rf

f

tkkfr

rf

rf

rf

ekk

AkA

AAA

ekkkk

AA

1012

1012

011

When the equilibrium is reached

eq

eq

eq

r

f

eqreqf

eqreqf

KA

A

kk

AkAk

AkAk

dtAd

dtAd

1

2

21

21

21

0

0

b) Consecutive reactions:

A1 A2 A3k1 k2

223

22112

111

AkdtAd

AkAkdtAd

AkdtAd

b1) First:

tkeAA 1011

b2) Second:

For [A2] Linear differential equation of first order

tktk

tktktktk

eekk

AkA

eekk

AkeAAeAkAk

dtAd

AkAkdtAd

21

2121

12

0112

12

01102201122

2

11222

Standardmethods

b3) Third:

If @ t=0[A2]0=0

[A1]0 = [A1] + [A2] + [A3] [A3] = [A1]0 - [A1] - [A2]

tktktktktk e

kkk

ekk

kAAee

kkk

eAA 21211

12

1

12

2013

12

1013 11

0 2 4 6 8 100

0.2

0.4

0.6

0.8

11

0

A t( )

A 0( )

B t( )

A 0( )

C t( )

A 0( )

100 t0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

11

0

A t( )

A 0( )

B t( )

A 0( )

C t( )

A 0( )

100 t

K1 = 1K2 = 100

K1 = 1K2 = 0.1

A B C CBA

0 2 4 6 8 100

0.2

0.4

0.6

0.8

11

0

A t( )

A 0( )

B t( )

A 0( )

C t( )

A 0( )

100 t

K1 = 1K2 = 0.01

0 2 4 6 8 100

0.2

0.4

0.6

0.8

11

0

A t( )

A 0( )

B t( )

A 0( )

C t( )

A 0( )

100 t

K1 = 1K2 = 0.1

0 2 4 6 8 100

0.2

0.4

0.6

0.8

11

0

A t( )

A 0( )

B t( )

A 0( )

C t( )

A 0( )

100 t

K1 = 1K2 = 10

0 2 4 6 8 100

0.2

0.4

0.6

0.8

11

0

A t( )

A 0( )

B t( )

A 0( )

C t( )

A 0( )

100 t

K1 = 1K2 = 100

c) Parallel reactions:(First order decay to different products)

A1 A2

A3

k2

k3

133

122

13121

AkdtAd

AkdtAd

AkAkdtAd

c1) First:

tkk

t

t

A

A

eAAdtkkAAd

AkkdtAd

32

0

1

01

011321

1

1321

32' kkk

c2) Second:

tkk

atat

ttkk

A

tkk

ekk

AkA

ea

dte

dteAkAd

eAkdtAd

32

32

2

32

1

1

32

0122

0012

02

0122

[A2]0 = 0

c3) Third:

tkk

atat

ttkk

A

tkk

ekk

AkA

ea

dte

dteAkAd

eAkdtAd

32

32

3

32

1

1

32

0133

0013

03

0133

[A3]0 = 0

b

a

b

a

b

a

b

a

0 0.5 1 1.5 20

0.5

11

0

A1 t( )

A2 t( )

A3 t( )

20 t

k2= 1, k3=10

0 0.5 1 1.5 20

0.5

11

0

A1 t( )

A2 t( )

A3 t( )

20 t

k2= 1, k3=1

0 0.5 1 1.5 20

0.5

11

0

A1 t( )

A2 t( )

A3 t( )

20 t

k2= 1, k3=0.1

A1 A2

A3

k2

k3

d) Parallel reactions:

A1 A2

A3

k1

k3

333

33112

111

AkdtAd

AkAkdtAd

AkdtAd

0

0

2

033

011

A

AAAA

t

tktk

tk

tk

eAkeAkdtAd

eAA

eAA

31

3

1

0330112

033

011

tktk

ttk

ttk

A

eAAeAAA

dteAkdteAkAd

31

31

2

030301012

0033

0011

02

0 1 2 3 40

0.5

1

1.51.311

0

A1 t( )

A2 t( )

A3 t( )

40 t0 1 2 3 4

0

0.5

1

1.5

21.963

0

A1 t( )

A2 t( )

A3 t( )

40 t0 1 2 3 4

0

0.5

1

1.5

21.982

0

A1 t( )

A2 t( )

A3 t( )

40 t

k1= 1, k3=10k1= 1, k3=1k1= 1, k3=0.1

A1 A2

A3

k1

k3

Steady-State approximation

This method is very usefulwhen intermediates are presentin small amount [Ai]

0

dtAd i

If we take the case b2) of Consecutive reactions(k1 << k2)

12

12

11222 0

Akk

A

AkAkdtAd

SS

tkSS eA

kk

A 101

2

12

tkSS eAA 11013

A1 A2 A3

k1 k2

A1 A2 A3

k1

K-1

k2

Home work!0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

11

0

A t( )

A 0( )

B t( )

A 0( )

C t( )

A 0( )

100 t

0

0

0302

011

AAAA

t

)1( 1013

tkSS eAA

The Michaelis-Menten mechanism(Enzyme action)

+ +

S (Substrate)E (Enzyme) X(Enzyme-Substrate

complex)

E (Enzyme)

P1 & P2

(Products)

True chemical Intermediate

[ES] << 1K2 > k1

0dtESd

Another Initial condition is:

[E]0 = [E] + [ES][S]0 = [S] + [P] SEES

E + S ES E + Pk1

K-1 K-2

K2

12

2

12

1

2112

11

0

kkPEk

kkSEk

ES

PEkSEkESkkdtESd

ESkSEkdtPd

dtSd

SS

SS

2121

02121

kkPkSkEPkkSkk

dtSd

E + S ES E + Pk1

K-1 K-2

K2

Considering[S] >> [P] @ t = 0

SEES

SK

vvEkv

kkk

K

Skkk

EkdtSdv

Sk

kkSkESkk

dtSd

M

SS

M

1

1

02

1

21

1

21

02

1

211

021

Michaelis-Mentenconstant

KM

vS

[S]

v

vS

2

t 0

t 0

Galaxies