### Chapter 21: The Rates of Chemical Reactions

```Atkins & de Paula:
Atkins’ Physical Chemistry 9e
Chapter 21: The Rates of
Chemical Reactions
Chapter 21: The Rates of Chemical Reactions
 chemical kinetics, the study of reaction rates.
 mechanism of reaction, the sequence of elementary steps involved in a reaction.
CHEMICAL KINETICS
21.1 Experimental techniques
 real-time analysis, a procedure in which the composition of a system is analysed
while the reaction is in progress.
(1)flow method, a procedure in which the composition of a system is analysed as the
reactants flow into a mixing chamber.
(2)stopped-flow technique, a procedure in which the reagents are mixed very quickly in
a small chamber fitted with a syringe instead of an outlet tube.
Chapter 21: The Rates of Chemical Reactions
(3) flash photolysis, a procedure in which the reaction is initiated by a brief flash of light.
 quenching methods, techniques based on stopping the reaction after it has been allowed
to proceed for a certain time.
(1)chemical quench flow method, a technique in which the reactants are mixed as in the
flow method but the reaction is quenched by another reagent.
(2)freeze quench method, a technique in which the reaction is quenched by cooling the
mixture.
Chapter 21: The Rates of Chemical Reactions
21.2 The rates of reactions
21.2(a) The definition of rate
 rate of consumption of a reactant R, –d[R]/dt.
 rate of formation of a product P, d[P]/dt.
 rate of reaction, v = (1/V)dξ/dt where ξ is the extent of reaction.
 rate of homogeneous reaction, v = (1/vJ)d[J]/dt.
 rate of heterogeneous reaction, v = (1/vJ)dσJ/dt.
21.2(b) Rate laws and rate constants
 rate law, the rate as a function of concentration, v = f([A],[B], ...).
 rate constant, the constant k in a rate law.
 hydrogen–bromine reaction: the observed rate law is d[HBr]/dt = kr[H2][Br2]3/2/([Br2] +
kr[HBr]).
21.2(c) Reaction order
 reaction order, the power to which the concentration of a species is raised in a rate law of
the form v = [A]a[B]b... .
 first-order reaction, a reaction with a rate law of the form v = kr[A].
 second-order reaction, a reaction with a rate law of the form v = kr[A]2.
 overall order, the sum of the orders a + b +..., in a rate law of the form v = kr[A]a[B]b....
 zero-order rate law, a rate law of the form v = kr.
Chapter 21: The Rates of Chemical Reactions
21.2(d) The determination of the rate law
 isolation method, a procedure in which the concentrations of all the reactants except one
are in large excess.
 Pseudo first-order rate law, v = kr[A] with kr = kr[B]0 by maintaining B in large excess.
N O
2
NaBH4
N H
2
Nanoparticle Catalyst
AgNPs
O H
O H
N O
-
2
O H
O H
E (A g ) -1 .8 0
0.0
ln A 400
N H
+
2
-0.5
V
-
E E D (B H 4 )
e -rela y
E (A g n )
-1.0
+
-1.5
0
30
60
90
120 150 180
E E A (4 N P )
E (A g b u lk ) + 0 .7 9
Time (sec)
S. W. Han et al., Chem. Lett., 2007, 36, 1350.
(4 A P )
V
Chapter 21: The Rates of Chemical Reactions
 method of initial rates, a procedure in which the rate is measured at the beginning of
the reaction for several different initial concentrations of reactants; v0 = kr [A]0a
log v0 = log kr + a log [A]0.
Example 21.2
Chapter 21: The Rates of Chemical Reactions
21.3 Integrated rate laws
 integrated rate law, the integrated form of a rate law for concentration as a function
of time.
21.3(a) First-order reactions
 first-order integrated rate law, -d[A]/dt= kr[A] ln([A]/[A]0) = –krt, [A] = [A]0e–krt.
 half life, t1/2 = (ln 2)/kr.
 time constant, the time required for the concentration of a reactant to fall to 1/e of its
initial value,τ = 1/kr.
Example 21.3
Chapter 21: The Rates of Chemical Reactions
21.3(c) Second-order reactions
 second-order integrated rate law, -d[A]/dt= kr[A]2  1/[A] – 1/[A]0 = krt
[A] = [A]0/(1 + krt[A]0).
 half life, t1/2 = 1/kr[A]0.
 half life for nth-order reaction (n>1), t1/2 = 2n-1-1/(n-1)kr[A]0n-1.
A  B  P;
d [ A]
dt
d [ A]
dt

[ A ]  [ A ]  x  d [ A ] / dt   dx / dt
  k r ([ A ] 0  x )([ B ] 0  x )    0     
dx
x
0
( a  x )( b  x )


0

dt
 k r ([ A ] 0  x )([ B ] 0  x )
 k r  dt  k r t
0
1 
 1



baax bx
dx
( a  x )( b  x )
1

 dx


b  a  a  x
1
dx
([ A ] 0  x )([ B ] 0  x )
 [ B ] /[ B ] 0
ln 
 [ A ] /[ A ] 0
dx
t
([ A ] 0  x )([ B ] 0  x )
1
x
  k r [ A ][ B ]

dx 
 b  x  
1
1 

ln
 ln
 constant

ba  ax
b  x 
1
  [ A ]0 
 [ B ]0  




ln

ln

 [ B ]  x 
[ B ] 0  [ A ] 0   [ A ] 0  x 
0


1

  ([ B ] 0  [ A ] 0 ) k r t


Chapter 21: The Rates of Chemical Reactions
Check this out!
Chapter 21: The Rates of Chemical Reactions
21.4 Reactions approaching equilibrium
21.4(a) First-order reactions close to equilibrium
A  B v  k r [ A]
B  A v  k r [ B ]
d [ A]
dt
d [ A]
dt
[ A] 
  k r [ A ]  k r [ B ]
  k r [ A ]  k r ([ A ] 0  [ A ])   ( k r  k r )[ A ]  k r [ A ] 0
k r  k r e
 ( k r  k r ) t
k r  k r
t   ; [ A ] eq 
K 
[ B ] eq

[ A ] eq
[ A ]0
k r [ A ] 0
k r  k r
k r [ A ]0
k r  k r
kr
k r
at equilibriu m forward
For general
[ B ] eq  [ A ] 0  [ A ] eq 
& reverse rates must be same ; k r [ A ] eq  k r [ B ] eq
reaction , K 
ka
k a

kb
k b

Chapter 21: The Rates of Chemical Reactions
21.4(b) Relaxation methods
 temperature jump, a procedure in which a sudden temperature rise is imposed and
the system returns to equilibrium.
 pressure-jump techniques, as for temperature jump, but with a sudden change in
pressure.
After temp . jump , system readjusts
to the new equilibriu m
k r [ A ] eq  k r [ B ] eq ( k r & k r are new rate const .)
[ A ]  x  [ A ] eq , [ B ]  [ B ] eq  x
d [ A]
dt
d [ A]
dt
d [ A]
dt
  k r [ A ]  k r [ B ]
  k r ( x  [ A ] eq )  k r ([ B ] eq  x )   ( k r  k r ) x

dx
 above is 1 st  order differenti al equation
dt
 x  x0e
 t /
1 /   k r  k r
Chapter 21: The Rates of Chemical Reactions
21.5 The temperature dependence of reaction rates
 Arrhenius equation, ln k = ln A – Ea/RT.
 pre-exponential factor (frequency factor), the parameter A in the Arrhenius equation.
 activation energy, the parameter Ea in the Arrhenius equation; the minimum kinetic
energy for reaction during a molecular encounter.
 Arrhenius parameters, the parameters A and Ea.
 generalized activation energy, Ea = RT2(d ln k/dT).
 activated complex, the cluster of atoms that corresponds to the region close to the
maximum potential energy along the reaction coordinate.
 transition state, a configuration of atoms in the activated complex which, if attained,
Chapter 21: The Rates of Chemical Reactions
ACCOUNTING FOR THE RATE LAWS
21.6 Elementary reactions
 elementary reaction, a single step in a reaction mechanism.
H + Br2  HBr + Br
 molecularity, the number of molecules coming together to react in an elementary
reaction.
 reaction order, the power to which the concentration of a species is raised in a rate
law of the form v = [A]a[B]b... ; an empirical quantity, and obtained from the
experimental rate law.
 unimolecular reaction, an elementary reaction involving a single reactant molecule.
 bimolecular reaction, an elementary reaction involving the encounter of two reactant
molecules.
CH3I(alc) + CH3CH2O-(alc)  CH3OCH2CH3(alc) + I-(alc)
Mechanism: CH3I + CH3CH2O-  CH3OCH2CH3 + I-, a single elementary step
Rate law: v=kr[CH3I][CH3CH2O-]
Chapter 21: The Rates of Chemical Reactions
21.7 Consecutive elementary reactions
 consecutive first-order reactions, a sequence of first-order reactions.
a
b
A 
I 
P
k
d [ A]
dt
d[I ]
dt
d[P]
dt
d[I ]
dt
[I ] 
k
  k a [ A ], [ A ]  [ A ] 0 e
 kat
 k a [ A]  kb[ I ]
 kb[I ]
 k b [ I ]  k a [ A ]0 e
ka
kb  ka
(e
 kat
e
 kat
 kbt
)[ A ] 0
k t
k t

kae b  kbe a
[ A ]  [ I ]  [ P ]  [ A ] 0  [ P ]  1 
kb  ka


[ A ] 0

Chapter 21: The Rates of Chemical Reactions
of change of concentrations of all reaction intermediates are negligibly small: d[I]/dt
 0 and their concentrations are low.
 induction period, the initial stage of a reaction during which reaction intermediates
are formed.
a
b
A 
I 
P
k
d[I ]
dt
k
 k a [ A]  kb[ I ]  0
ka
k  k
b
[ I ]  ( k a / k b )[ A ]  
a  [ I ] 
d[P]
dt
kb  ka
(e
 kat
e
 kbt
a
 k b [ I ]  k a [ A ] ; A 
I is the rate determinin
k
t
[P]  k a [ A ] 0  e
0
 kat
dt  (1  e
 kat
)[ A ] 0
k t
k t

kae b  kbe a
    [ P ]  1 
kb  ka

k b  k a

[ A ] 0

)[ A ] 0
g step!
Chapter 21: The Rates of Chemical Reactions
 validation of steady-state approximation (QSSA)
a
b
A 
I 
P
k
k
Exact & QSSA ; [ A ]  [ A ] 0 e
Exact ; [ I ] 
ka
kb  ka
(e
 kat
 kat
e
 kbt
)[ A ] 0
QSSA ; [ I ]  ( k a / k b )[ A ]  ( k a / k b )[ A ] 0 e
k t
k t

kae b  kbe a
Exact ; [ P ]  1 
kb  ka

t
QSSA ; [P]  k a [ A ] 0  e
0
 kat
 kat

[ A ] 0

dt  (1  e
 kat
)[ A ] 0
k b  20 k a
Chapter 21: The Rates of Chemical Reactions
 An example of steady-state approximation
2 N 2 O 5 ( g )  4 NO 2 ( g )  O 2 ( g )
a
 N 2 O 5 
NO 2  NO 3
k
k
a
 NO 2  NO 3 
N 2O5
b
 NO 2  NO 3 
NO 2  O 2  NO
k
c
 NO  N 2 O 5 
NO 2  NO 2  NO 2
k
intermedia
d [ NO ]
dt
 k b [ NO 2 ][ NO 3 ]  k c [ NO ][ N 2 O 5 ]  0  [ NO ] 
d [ NO 3 ]
dt
tes; NO and NO 3
k c[ N 2O5 ]
 k a [ N 2 O 5 ]  k a [ NO 2 ][ NO 3 ]  k b [ NO 2 ][ NO 3 ]  0  [ NO 3 ] 
d [ N 2O5 ]
dt
k b [ NO 2 ][ NO 3 ]
  k a [ N 2 O 5 ]  k a [ NO 2 ][ NO 3 ]  k c [ NO ][ N 2 O 5 ]  
k a [ N 2O5 ]
( k a  k b )[ NO 2 ]
2 k a k b [ N 2O5 ]
k a  k b
Chapter 21: The Rates of Chemical Reactions
 rate-determining step, the step in a mechanism that controls the overall rate of the
reaction; commonly but not necessarily the slowest step.
Chapter 21: The Rates of Chemical Reactions
 pre-equilibrium, a state in which an intermediate is in equilibrium with the reactants
and which arises when the rates of formation of the intermediate and its decay back
into reactants are much faster than its rate of formation of products.
A  B  I  P ; when k a  k b
[I ]
K 

[ A ][ B ]
d[P]
dt
dt
d[I ]
dt
d[P]
dt
k a
 k b [ I ]  k b K [ A ][ B ]  k r [ A ][ B ], k r  k b K 
not assuming
d[P]
ka
kakb
k a
k a  k b
 kb[I ]
 k a [ A ][ B ]  k a [ I ]  k b [ I ]  0  [ I ] 
 k r [ A ][ B ], k r 
kakb
k a  k b
k   k
k a [ A ][ B ]
a

b  k r 
k a  k b
kakb
k a
Chapter 21: The Rates of Chemical Reactions
Examples of reaction mechanisms
21.8 Unimolecular reactions
 Lindemann–Hinshelwood mechanism, a theory of ‘unimolecular’ reactions.

d[A ]

A A A  A
dt
 k a [ A]
2

d[A ]

A A  A A
dt

  k a [ A ][ A ]

d[A ]

A  P
dt

 kb[ A ]

d[A ]
dt

[A ] 


 k a [ A ]  k a [ A ][ A ]  k b [ A ]  0
2
k a [ A]
2
k b  k a [ A ]


d[P]
dt

k a [ A ][ A ]  k b [ A ];
d[P]
dt

 kb[ A ] 
k a kb [ A]
2
k b  k a [ A ]
 k r [ A ], k r 
kakb
k a
Chapter 21: The Rates of Chemical Reactions
 Test of Lindemann–Hinshelwood mechanism


Low conc . of A ; k a [ A ][ A ]  k b [ A ]  k a [ A ]  k b
d[P]

dt
k a kb[ A]
2
 k a [ A ] ; 2 nd order reaction !
2
k b  k a [ A ]
At low concentrat ion of A, rate - determinin
d[P]
dt
 k r [ A ], k r 
Test of theory ; plot
k a kb[ A]

k b  k a [ A ]
1
1
k a

kr
against
kr
1
g step is the bimolecula
1

kakb
k a [ A]
 straight line
[ A]
 activation energies of composite reactions
kr 
kakb
k a

( Aa e
 E a ( a ) / RT
( A a e
)( Ab e
 E a ( b ) / RT
 E a ( a ) / RT
E a  E a ( a )  E a ( b )  E a ( a )
)
)

A a Ab
A a
e
 { E a ( a )  E a ( b )  E a ( a )} / RT
r formaiton
of A

Chapter 21: The Rates of Chemical Reactions
21.9 POLYMERIZATION KINETICS
 stepwise polymerization, a polymerization reaction in which any two monomers
present in the reaction mixture can link together at any time and the growth of the
polymer is not confined to chains that are already forming.
 chain polymerization, a polymerization reaction in which an activated monomer attacks
another monomer, links to it, then that unit attacks another monomer, and so on.
stepwise
polymerization
chain
polymerization
Chapter 21: The Rates of Chemical Reactions
21.9(a) Stepwise polymerization
 degree of polymerization, the average number of monomer residues per polymer
molecule, n = 1/(1 – p), where p is the average number of monomers per polymer
molecule; n = 1 + krt[A]0.
For HO - R - COOH  HO - R - COOH
d[A]
  k r [ OH ][ A ]   k r [ A ]
dt
[ A] 
2
([ A ]  [  COOH ])
[ A ]0
1  k r t[ A ]0
fraction, p, of  COOH condensed
p
[ A ]0  [ A ]
[ A ]0

k r t[ A ]0
1  k r t[ A ]0
degree of polymeriza tion;  N  
[ A ]0
[ A]

1
1 p
 1  k r t[ A ]0
Chapter 21: The Rates of Chemical Reactions
21.9(b) Chain polymerization
 CH 2 CH 2 X    CH
2
 CHX   CH 2 CHXCH
2
CHX 
(a) Initiation
I  R R 
v i  k i [ I ] ; rate  determinin g step
M  R    M 1 ( fast )
(b) Propagatio n
M  M 1  M
M  M
2
(c) Terminatio n
2
 M 3

M  M
 d [M

 dt
n 1
 M
n
v p  k p [ M ][  M ]
m
 M
 M n  M
m
 M n  M m (dispropor tionation)
M  M
  M  M n (chain tra nsfer)
 d [M

 dt
]
 2 fk i [ I ]

 production
d [M ]
rate of chain
polymerization
 M n  M
dt
n
nm
(mutual terminati on)
]
2
  2 k t [M ]

 depletion
 fk  1 / 2
2
 2 fk i [ I ]  2 k t [  M ]  0  [  M ]   i [ I ]
 kt 
 fk  1 / 2
1/ 2
v p  k p [  M ][ M ]  k p  i [ I ] [ M ]  k r [ I ] [ M ]
 kt 
v t  k t [M ]
2
Chapter 21: The Rates of Chemical Reactions
 kinetic chain length, v, the ratio of the number of monomer units consumed per
activated centre produced in the initiation step; v = k[M][I]–½.
v
v
v
# of monomer
units consumed
# of activated
centers produced
rate of propagatio
n of chains
rate of production
k p [  M ][ M ]
2 k t [M ]
2

k p[M ]
2 k t [M ]
 N   2 v  2 k r [ M ][ I ]
1 / 2
 fk i 

[  M ]  

 kt 
1/ 2
1/ 2
[I ]
       k r [ M ][ I ]
1 / 2
(k r 
1
2
k p ( fk i k t )
1 / 2
)
Chapter 21: The Rates of Chemical Reactions
21.10 PHOTOCHEMISTRY
 primary process, a process in which products are formed directly from the excited
state of a reactant.
 secondary process, a process in which products originate from intermediates formed
directly from the excited state of a reactant.
Chapter 21: The Rates of Chemical Reactions
21.10(a) The primary quantum yield
 primary quantum yield, ϕ, the number of photophysical or photochemical events
that lead to primary products divided by the number of photons absorbed by the
molecule in the same interval, ϕ = v/Iabs.
# of events
 
# of photons absorbed
  
i
i
i
vi
rate of process

intensity
of light absorbed
1
I abs
 f   IC   P  1 or  f   IC   P   r  1

i
vi
I abs
1  I abs 

i
vi   
v
v
i
i

v
I abs
Chapter 21: The Rates of Chemical Reactions
21.10(b) Mechanism of decay of excited singlet states
Absorption
S  hv i  S
:

v abs  I abs

ce : S  S  hv
Fluorescen
Internal conversion
:


S  S

: S T
Intersyste m crossing
rate of formation

v f  k f [S ]
f
v IC  k IC [ S ]


v ISC  k ISC [ S ]

of [ S ] : I abs





rate of decay of [ S ]   k f [ S ]  k ISC [ S ]  k IC [ S ]   ( k f  k ISC  k IC )[ S ]


[ S ]t  [ S ]0 e
 t / 0

1
 
 0 k k
 k IC
f
ISC






d [S ]
dt




 I abs  k f [ S ]  k ISC [ S ]  k IC [ S ]  I abs  ( k f  k ISC  k IC )[ S ]  0

 I abs  ( k f  k ISC  k IC )[ S ]   f 
vf
I abs


k f [S ]

( k f  k ISC  k IC )[ S ]

kf
k f  k ISC  k IC
Chapter 21: The Rates of Chemical Reactions
21.10(c) Quenching
 quenching, shortening of the lifetime of an excited state.
 Stern–Volmer equation, φf,0/φf = 1 + τ0kQ[Q].
Quenching


S Q  S Q
:
v Q  k Q [ Q ][ S ]

d [S ]
dt
f 
kf
k f  k ISC  k IC  k Q [ Q ]
 f ,0 
 f ,0
f

kf
k f  k ISC  k IC
Stern–Volmer plot
; with quencher
; without

kf

k k
 k IC
ISC
 f
 f ,0
f

 I abs  ( k f  k ISC  k IC  k Q [ Q ])[ S ]  0
quencher
  k f  k ISC  k IC  k Q [ Q ]

 
kf
 
 1   0 k Q [ Q ]; SternVolme
r equation

kQ
  1
[Q ]

k

k

k
f
ISC
IC

Chapter 21: The Rates of Chemical Reactions
 Modified Stern–Volmer equation
0 
 
1
k f  k ISC  k IC

kf

k k
 k IC
ISC
 f
f
kf
 f ,0
f

1



0

1
0
 1   0 k Q [Q ]
 k Q [Q ]
Example 21.9
Q: Fe(OH2)63+
  1   f ,0


 k 
kf
  f 
Chapter 21: The Rates of Chemical Reactions
 Three common mechanism of quenching
Collisiona
Resonance
l deactivati on :
energy tra nsfer :
Electron t ransfer :


S Q  S Q

S Q  S Q




S  Q  S  Q or S  Q

21.10(d) Resonance energy transfer
 Förster theory, a theory of resonance energy transfer,
- efficiency (ηT =1-φf,0/φf); ηT  1/R6 [ηT = R06/(R06 + R6)].
- hdonor  h acceptor
Chapter 21: The Rates of Chemical Reactions
 fluorescence resonance energy transfer (FRET), a technique used to measure
distances in biological systems.
ηT =1-φf,0/φf
ηT = R06/(R06 + R6)
7.9 nm
Protein rhodopsin
```