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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