SF Chem 2201.
Chemical Kinetics
2011/2012.
Dr Mike Lyons
Room 3.2 Chemistry Building
School of Chemistry
Trinity College Dublin.
Email : melyons@tcd.ie
Course Summary.
• Contact short but sweet. 5 Lectures in total (4 this week, 1 next
week, 3 tutorials next week).
• We revise quantitative aspects of JF kinetics and discuss some
new more advanced topics and introduce the mathematical
theory of chemical kinetics.
• Topics include:
– Lecture 1-2. Quantitative chemical kinetics, integration of rate
equations, zero, first, second order cases, rate constant . Graphical
analysis of rate data for rate constant and half life determination
for each case . Dependence of rate on temperature. Arrhenius
equation and activation energy. Kinetics of complex multistep
reactions. Parallel and consecutive reactions. Concept of rate
determining step and reaction intermediate.
– Lecture 3,4. Enzyme kinetics (Michaelis-Menten case) and surface
reactions involving adsorbed reactants (Langmuir adsorption
isotherm).
– Lecture 5. Theory of chemical reaction rates : bimolecular
reactions. Simple Collision Theory & Activated Complex Theory.
1
Recommended reading.
•
Burrows et al Chemistry3, OUP Chapter 8. pp.339-403.
•
P.W. Atkins J. de Paula.The elements of physical chemistry. 4th edition.
OUP (2005). Chapter 10, pp.229-256; Chapter 11, pp.257-284.
•
P.W. Atkins and J. de Paula. Physical Chemistry for the Life Sciences. 1st
edition. OUP (2005). Part II entitled The kinetics of life processes (Chapters
6,7,8) is especially good.
•
•
P.W. Atkins, J. de Paula. Physical Chemistry. 8th Edition. OUP (2006).
Chapter 22, pp.791-829 ; Chapter 23, pp.830-868;
Chapter 24, pp.869-908.
•
•
A more advanced and complete account of the course material. Much of chapter 24 is
JS material.
M.J. Pilling and P/W. Seakins. Reaction Kinetics. OUP (1995).
•
•
Both of these books by well established authors are clearly written with an excellent
style and both provide an excellent basic treatment of reaction kinetics with emphasis
on biological examples. These books are set at just the right level for the course and
you should make every effort to read the recommended chapters in detail. Also the
problem sheets will be based on problems at the end of these chapters!
Modern textbook providing a complete account of modern
chemical reaction kinetics. Good on experimental methods and theory.
M. Robson Wright An introduction to chemical kinetics. Wiley (2005)
•
Another modern kinetics textbook which does as it states in the title, i.e. provide a
readable introduction to the subject ! Well worth browsing through.
SF Chemical
Kinetics.
Lecture 1-2.
Quantitative Reaction
Kinetics.
2
Reaction Rate: The Central Focus of
Chemical Kinetics
The wide range of reaction rates.
Reaction rates vary from very fast to very slow :
from femtoseconds to centuries !
1 femtosecond (fs)
= 10-15 s = 1/1015 s !
3
Picosecond (10-12s)
techniques
Femtosecond
(10-15 s)techniques
Atkins de Paula, Elements Phys. Chem.
5th edition, Chapter 10, section 10.2, pp.221-222
Reactions studies under constant temperature conditions.
Mixing of reactants must occur more rapidly than reaction occurs.
Start of reaction pinpointed accurately.
Method of analysis must be much faster than reaction itself.
Chemical reaction kinetics.
•
•
•
Chemical reactions involve the
forming and breaking of
chemical bonds.
Reactant molecules (H2, I2)
approach one another and
collide and interact with
appropriate energy and
orientation. Bonds are
stretched, broken and formed
and finally product molecules
(HI) move away from one
another.
How can we describe the rate
at which such a chemical
transformation takes place?
reactants
products
H 2 ( g )  I 2 ( g )  2HI ( g )
• Thermodynamics tells us all
about the energetic feasibility
of a reaction : we measure the
Gibbs energy DG for the chemical
Reaction.
• Thermodynamics does not tell us
how quickly the reaction will
proceed : it does not provide
kinetic information.
4
Basic ideas in reaction kinetics.
•
Chemical reaction kinetics deals with the rate of velocity of chemical
reactions.
•
We wish to quantify
•
These objectives are accomplished using experimental measurements.
•
We are also interested in developing theoretical models by which the
underlying basis of chemical reactions can be understood at a
microscopic molecular level.
•
Chemical reactions are said to be activated processes : energy (usually
thermal (heat) energy) must be introduced into the system so that
chemical transformation can occur. Hence chemical reactions occur
more rapidly when the temperature of the system is increased.
•
In simple terms an activation energy barrier must be overcome before
reactants can be transformed into products.
•
– The velocity at which reactants are transformed to products
– The detailed molecular pathway by which a reaction proceeds (the reaction
mechanism).
Reaction Rate.
What do we mean by the term
reaction rate?
–
•
How may reaction rates be
determined ?
–
–
•
•
•
The term rate implies that something
changes with respect to something
else.
The reaction rate is quantified in
terms of the change in concentration
of a reactant or product species with
respect to time.
This requires an experimental
measurement of the manner in which
the concentration changes with time
of reaction. We can monitor either
the concentration change directly, or
monitor changes in some physical
quantity which is directly proportional
to the concentration.
The reactant concentration
decreases with increasing time, and
the product concentration increases
with increasing time.
The rate of a chemical reaction
depends on the concentration of each
of the participating reactant species.
The manner in which the rate
changes in magnitude with changes in
the magnitude of each of the
participating reactants is termed the
reaction order.
Reactant
concentration
R  
Product
concentration
d R d P

dt
dt
Net reaction rate
Units : mol dm-3 s-1
[P]t
[R]t
time
5
Geometric definition of
reaction rate.
Rate expressed as tangent line
To concentration/time curve at a
Particular time in the reaction.
R
R
d [ R]
dt
d  P
dt
Reaction Rates and Reaction
Stoichiometry
O3(g) + NO(g)  NO2(g) + O2(g)
rate = -
dO3
d[NO]
d[ NO2]
d[O2]
==+
=+
dt
dt
dt
dt
Reaction rate can be quantified by monitoring changes in either reactant
concentration or product concentration as a function of time.
6
2 H2O2 (aq)  2 H2O (l) + O2 (g)
The general case.
• Why do we define
our rate in this way?
 removes ambiguity in
the measurement of
reaction rates in that
we now obtain a
single rate for the
entire equation, not
just for the change in
a single reactant or
product.
aA  bB  pP  qQ
Rate
1 d A
1 d B 

a dt
b dt
1 d Q 
1 d P 


q dt
p dt
R
7
Rate, rate equation and reaction order : formal
definitions.
• The reaction rate (reaction velocity) R is quantified in terms of
changes in concentration [J] of reactant or product species J
with respect to changes in time. The magnitude of the reaction rate
changes as the reaction proceeds.
RJ 
DJ  1 d J 

D
t

0
J
Dt
 J dt
1
lim
2 H 2 ( g )  O2 ( g )  2 H 2O( g )
R
1 d H 2 
d O2  1 d H 2O


2 dt
dt
2 dt
• Note : Units of rate :- concentration/time , hence RJ has units mol dm-3s-1 .
J denotes the stoichiometric coefficient of species J. If J is a reactant J
is negative and it will be positive if J is a product species.
• Rate of reaction is often found to be proportional to the molar
concentration of the reactants raised to a simple power (which
need not be integral). This relationship is called the rate equation.
The manner in which the reaction rate changes in magnitude with
changes in the magnitude of the concentration of each participating
reactant species is called the reaction order.
Reaction rate and reaction order.
• The reaction rate (reaction velocity) R is quantified in terms
ofchanges in concentration [J] of reactant or product species J
with respect to changes in time.
• The magnitude of the reaction rate changes (decreases) as the
reaction proceeds.
• Rate of reaction is often found to be proportional to the molar
concentration of the reactants raised to a simple power (which
need not be integral). This relationship is called the rate
equation.
• The manner in which the reaction rate changes in magnitude with
changes in the magnitude of the concentration of each
participating reactant species is called the reaction order.
• Hence in other words:
– the reaction order is a measure of the sensitivity of the reaction
rate to changes in the concentration of the reactants.
8
Working out a rate equation.
k = 5.2 x 10-3 s-1
2 N 2O5 ( g )  4 NO2 ( g )  O2 ( g )
T = 338 K
Initial rate is proportional to initial
concentration of reactant.
(rate) 0  N 2O5 0
Initial rate determined by
evaluating tangent to
concentration versus time
curve at a given time t0.
(rate) 0  k N 2O5 0
Rate
Equation.
rate  
k
xA  yB 

Products
empirical rate
equation (obtained
from experiment)
stoichiometric
coefficients
Reaction order determination.
Vary [A] , keeping [B] constant and
measure rate R.
Vary [B] , keeping [A] constant and
measure rate R.
k = rate constant
d N 2O5 
 k N 2O5 
dt
rate constant k
R
1 d A
1 d B



 k A B
x dt
y dt
,  = reaction
orders for the
reactants (got
experimentally)
Rate equation can not in
general be inferred from
the stoichiometric equation
for the reaction.
Log R
Log R
Slope = 
Slope = 
Log [A]
Log [B]
9
Different rate equations
imply different mechanisms.
H 2  X 2  2 HX
X  I , Br , Cl
• The rate law provides an important guide
to reaction mechanism, since any proposed
mechanism must be consistent with the
observed rate law.
• A complex rate equation will imply a complex
multistep reaction mechanism.
• Once we know the rate law and the rate
constant for a reaction, we can predict the
rate of the reaction for any given composition
of the reaction mixture.
• We can also use a rate law to predict the
concentrations of reactants and products at
any time after the start of the reaction.
H 2  I 2  2 HI
d HI 
 k H 2 I 2 
dt
H 2  Br2  2 HBr
R
d HBr  k H 2 Br2 

k HBr 
dt
1
Br2 
1/ 2
R
H 2  Cl2  2 HCl
R
d HCl 
1/ 2
 k H 2 Cl2 
dt
Integrated rate equation.
Burrows et al Chemistry3, Chapter 8, pp.349-356.
Atkins de Paula 5th ed. Section 10.7,10.8, pp.227-232
•
•
Many rate laws can be cast as differential equations which may then be
solved (integrated) using standard methods to finally yield an
expression for the reactant or product concentration as a function of
time.
We can write the general rate equation for the process A  Products
as
dc

•
dt
 kF (c )
where F(c) represents some distinct function of the reactant
concentration c. One common situation is to set F(c) = cn where n =
0,1,2,… and the exponent n defines the reaction order wrt the reactant
concentration c.
The differential rate equation may be integrated once to yield the
solution c = c(t) provided that the initial condition at zero time which is
c = c0 is introduced.
10
Zero order kinetics. The reaction proceeds at the same rate R
regardless of concentration.
R  c0
R
Rate equation :
dc
R k
dt
c  c0 when t  0
units of rate constant k :
mol dm-3 s-1
c
integrate
using initial
half life
condition
c(t )  kt  c0
c
t   1/ 2 when c 
 1/ 2 
slope = -k
c0
 1/ 2  c0
2k
 1/ 2
c0
slope 
diagnostic
plot
k
A

products
First order kinetics.
Initial
concentration c0
First order differential
rate equation.

dc
dt
1
2k
c0
t
rate  
c0
2
dc
 kc
dt
Initial condition
t  0 c  c0
Solve differential
equation
Via separation
of variables
First order reaction
(rate)t  c
(rate)t  kc
k = first order
rate constant, units: s-1
c(t )  c0e kt  c0 exp  kt 
Reactant concentration
as function of time.
11
Half life 1/2
First order kinetics.
t   1/ 2
c0
2
u  1/ 2
c
  1/ 2
k s
 1/ 2 
1
c(t )  c0e  kt  c0 exp  kt 
c(t )
u
 exp   
c0
ln 2 0.693

k
k
Mean lifetime of reactant molecule

  kt
First order kinetics: half life.
u
1
c0
a
a0


0
0
1
 c  t  dt  c 
0
c0e kt dt 
u 1
t  1/ 2
In each successive period
of duration t1/2 the concentration
of a reactant in a first order reaction
decays to half its value at the start
of that period. After n such periods,
the concentration is (1/2)n of its
initial value.
1
k
c  c0 / 2
u  0.5
u  0.25
u  0.125
 1/ 2
 1/ 2 
c0
ln 2 0.693

k
k
half life independent of
initial reactant concentration
12
Second order kinetics: equal
reactant concentrations.
dm3mol-1s-1
2A 
 P
k
slope = k
dc
  kc 2
dt
t  0 c  c0
half life
separate variables
integrate
1
1
 kt 
c
c0
t   1/ 2
 1/ 2 
1
kc0
 1/ 2 
1
c0
c
c0
2
t
c t  
 1/ 2
 1/ 2  as c0 
1st order kinetics
u ( )
1
c
c(t )  c0e  kt
u ( )  e
  kt
c0
1  kc0t
c0
rate varies as
square of reactant
concentration
2nd order kinetics
u ( )
c(t ) 
c0
1  kc0t
u ( ) 
1
1
  kc0t
13
1st and 2nd order kinetics : Summary .
Reaction
k1
A 
Products
k2
2 A 
Products
ln c(t)
Differential
rate equation


dc
 k1c
dt
dc
 k2 c 2
dt
Concentration
variation with
time
c(t ) 
c0 exp  k1t 
c(t ) 
c0
1  k2 c0t
Slope = - k1 1/c(t)
ln c(t )  k1t  ln c0
1
1
 k2t 
c(t )
c0
1/2
Slope = k2
t
 1/ 2
c
 1/ 2  c01 n
n  1  1/ 2  as c0 
n  1  1/ 2  as c0 
ln  1/ 2
 1/ 2 
1
k2 c0
Diagnostic
Plots .
k
nA 

P
1
c n 1
n 1
  n  1 kt 
1
c0 n1
rate constant k
obtained from slope
Half life
2n 1  1

 n  1 kc0n1
ln 2
k1
c0
1
n = 0, 2,3,…..
 1/ 2 
2nd order
separate variables
integrate
n 1
Half
Life
1st order
t
n th order kinetics: equal reactant
concentrations.
dc
  kc n
dt
t  0 c  c0
Diagnostic
Equation
 2n 1  1 


 ln 
   n  1 ln c0
n

1
k






slope  n  1k
t
ln  1/ 2
slope  n  1
reaction order n determined
from slope
ln c0
14
Summary of kinetic results.
t  0 c  c0
k
nA 

P
t   1/ 2
c
Rate equation
Reaction
Order
R
Integrated Units of k Half life
expression
1/2
c  t   kt  c0
mol dm-3s-1
c0
2k
dc
dt
0
k
1
kc
 c 


ln  0   kt
 c t  


s-1
ln 2
k
2
kc 2
1
1
 kt 
c t 
c0
dm3mol-1s-1
1
kc0
3
kc3
1
1
 2kt  2
c2 t 
c0
dm6mol-2s-1
3
2kc0 2
n
kc n
1
1
  n  1 kt  n1
c n1
c0
Second order kinetics:
Unequal reactant concentrations.
1  2n 1  1


n  1  kc0 n 1 
k
A B 

P
Pseudo first
order kinetics
when b0 >>a0
half life
rate equation

1
ln 2  


 
ka0   1
da
db dp
R


 kab
dt
dt dt
 1/ 2
initial conditions
b
  0
a0
t  0 a  a0
b  b0
  1
 1/ 2 
dm3mol-1s-1
F a, b  
1
b0  a0
  b b0 
  kt
ln 
a
a
0



F a, b 
1

0
ln 2 ln 2

kb0
k
k   kb0
a0  b0
integrate using
partial fractions
c0
2
slope = k
pseudo 1st order
rate constant
t
15
Temp Effects in Chemical
Kinetics.
Atkins de Paula
Elements P Chem 5th edition
Chapter 10, pp.232-234
Burrows et al Chemistry3,
Section 8.7,
pp.383-389.
16
Van’t Hoff expression:
Energy
TS
DU
 d ln K c 

 
2
 dT  P RT
0
E
Standard change in internal
energy:
E’
DU 0  E  E
R
k
k
R
DU0
P
P
Reaction coordinate
k
Kc 
k
 d  k 
d ln k d ln k  DU
 dT ln  k     dT  dT  RT 2
  P

0
d ln k
E

dT
RT 2
d ln k 
E

dT
RT 2
This leads to formal
definition of Activation
Energy.
Temperature effects in chemical kinetics.
• Chemical reactions are activated processes : they require an
energy input in order to occur.
• Many chemical reactions are activated via thermal means.
• The relationship between rate constant k and temperature T
is given by the empirical Arrhenius equation.
• The activation energy EA is determined from experiment, by
measuring the rate constant k at a number of different
temperatures. The Arrhenius equation
 E 
k  A exp  A 
is used to construct an Arrhenius plot
 RT 
of ln k versus 1/T. The activation energy
Pre-exponential
is determined from the slope of this plot.
factor
 d ln k 
 d ln k 
  RT 2 
E A   R



d
1
/
T
 dT 


ln k
Slope  
EA
R
1
T
17
18
In some circumstances the Arrhenius Plot is curved which implies that
the Activation energy is a function of temperature.
Hence the rate constant may be expected to vary with temperature
according to the following expression.
 E 
k  aT m exp  
 RT 
We can relate the latter expression to the Arrhenius parameters A and EA
as follows.
ln k  ln a  m ln T 
E
RT
E 
 d ln k 
2 m
E A  RT 2 
 E  mRT
  RT  
2
 dT 
 T RT 
E  E A  mRT
Hence
 E 
 E 
k  aT m em exp   A   A exp   A 
 RT 
 RT 
m m
A  aT e
Consecutive Reactions .
•Mother / daughter radioactive
decay.
Po  Pb  Bi
218
214
Svante August
Arrhenius
k1
k2
A 
X 
P
Mass balance requirement:
214
k1  5 103 s 1
k2  6 104 s 1
p  a0  a  x
The solutions to the coupled
equations are :
3 coupled LDE’s define system :
a(t )  a0 exp  k1t 
da
  k1a
dt
dx
 k1a  k 2 x
dt
dp
 k2 x
dt
x(t ) 
k1a0
 exp  k1t   exp  k2t  
k 2  k1
p(t )  a0  a0 exp  k1t  
k1a0
 exp  k1t   exp  k2t  
k 2  k1
We get different kinetic behaviour depending
on the ratio of the rate constants k1 and k2
19
Consecutive reaction : Case I.
Intermediate formation fast, intermediate decomposition slow.
Case I .

k2
 1
k1
TS II
k 2  k1
TS I
I : fast
DGI‡ <<
energy
k1
k2
A 
X 
P
II : slow
rds
DGI‡
DGII‡
A
DGII‡
X
P
reaction co-ordinate
Step II is rate determining
since it has the highest
activation energy barrier. The reactant species A will be
more reactive than the intermediate X.

Case I .
k2
 1
k1
k 2  k1
k1
k2
A 
X 
P
a
u
a0
x
v
a0
p
w
a0
Normalised concentration
1.2
k2/ k1 = 0.1
1.0
u = a/a0
v = x/a0
Intermediate X
0.8
w = p/a0
0.6
Reactant A
0.4
Product P
0.2
0.0
0
2
4
6
8
10
k1t
Initial reactant A more
reactive than intermediate X .
Concentration of intermediate
significant over time course of
reaction.
20
Consecutive reactions Case II:
Intermediate formation slow, intermediate decomposition fast.

k2
k1
k1
k2
A 
X 
P
Case II .
k
  2  1
k1
key parameter
Intermediate X fairly reactive.
[X] will be small at all times.
k 2  k1
k1
k2
A 
X 
P
DGI‡ >>
II : fast
DGII‡
TS II
DGI‡
energy
I : slow rds
TS I
DGII‡
A
Step I rate determining
since it has the highest
activation energy barrier.
k1
k2
A 
X 
P
P
reaction co-ordinate
1.2
normalised concentration
Case II .
X
k
  2  1
k1
k 2  k1
Intermediate X
is fairly reactive.
Concentration of
intermediate X
will be small at
all times.
1.0
Product P
0.8
u=a/a0
v=x/a0
0.6
w=p/a0
Reactant A
0.4
0.2
 = k 2/k1 = 10
Intermediate X
0.0
0
2
4
6
8
10
 = k1t
normalised concentration
1.2
u=a/a0
1.0
v =x/a0
Intermediate concentration
is approximately constant
after initial induction period.
w=p/a0
0.8
P
A
0.6
0.4
X
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
 = k1t
21
Rate Determining Step
k1  k2
k1
k2
A 
X 
P
Fast
Slow
• Reactant A decays rapidly, concentration of intermediate species X
is high for much of the reaction and product P concentration rises
gradually since X--> P transformation is slow .
k2  k1
Rate Determining
Step
k1
k2
A 
X 
P
Slow
Fast
• Reactant A decays slowly, concentration of intermediate species X
will be low for the duration of the reaction and to a good approximation
the net rate of change of intermediate concentration with time is zero.
Hence the intermediate will be formed as quickly as it is removed.
This is the quasi steady state approximation (QSSA).
Parallel reaction mechanism.
k1
A 
X
k2
• We consider the kinetic analysis of a concurrent
A 
Y
or parallel reaction scheme which is often met in
st
k1, k2 = 1 order rate constants
real situations.
• A single reactant species can form two
We can also obtain expressions
distinct products.
for the product concentrations
We assume that each reaction exhibits 1st order x(t) and y(t).
kinetics.
• Initial condition : t= 0, a = a0 ; x = 0, y = 0
.• Rate equation:
R
da
 k1a  k2 a  k1  k2 a  k a
dt
a(t )  a0 exp  kt   a0 exp  k1  k2 t 
• Half life:  1/ 2  ln 2  ln 2
k
k1  k2
dx
 k1a  k1a0 exp  k1  k 2 t 
dt
x(t )  k1a0
x(t ) 
 exp  k
t
1
0
 k 2 t  dt
k1a0
1  exp  k1  k2 t 
k1  k 2
dy
 k 2 a  k 2 a0 exp  k1  k 2 t 
dt
y (t )  k 2 a0
y (t ) 
 exp  k
t
0
1
 k 2 t  dt
k 2 a0
1  exp  k1  k2 t 
k1  k 2
x(t ) k1
• All of this is just an extension of simple Final product analysis
Lim

yields rate constant ratio. t  y (t ) k 2
1st order kinetics.
22
Parallel Mechanism: k1 >> k2
normalised concentration
1.0
Theory

  0.1
v  0.9079
a(t)
0.8
x(t)
u()
v()
w()
0.6
0.4
 = k2 /k1
/
0.2
w  0.0908
y(t)
0.0
0
1
2
3
4
5
6
 = k1t
k2
 0.1
k1
k1
v 
 10 
k2
w()
Parallel Mechanism: k2 >> k1
normalised concentration
1.0
Theory
u()
v()
w()
0.6
0.4
a(t)
0.2
k2
 10
k1
v()  0.0909
x(t)
0.0
0

w()  0.9091
y(t)
  10
0.8
1
2
3
4
5
6
 = k1t
k1
v 
 0.1 
k2
w()
23
Reaching Equilibrium on the
Macroscopic and Molecular Level
N2O4
NO2
N2O4 (g)
2 NO2 (g)
colourless
brown
Chemical Equilibrium :
a kinetic definition.
•
•
Countless experiments with chemical
systems have shown that in a state of
equilibrium, the concentrations of
reactants and products no longer change
with time.
This apparent cessation of activity occurs
because under such conditions, all
reactions are microscopically reversible.
We look at the dinitrogen tetraoxide/
nitrogen oxide equilibrium which
occurs in the gas phase.
N2O4 (g)
2 NO2 (g)
colourless
Kinetic analysis.

R  k N 2O4 

2
R  k NO2 
Valid for any time t
brown
Equilibrium:
t 
Concentrations vary
with time
NO2  t 
N 2O4  t
concentration
•
 
RR
k N 2O4 eq  k NO2 eq
NO2 eq2 k
 K
N 2O4  k 
2

Kinetic
regime
Concentrations time
invariant
NO2  eq
N 2O4  eq
Equilibrium
state
NO2
t 
N2O4
time
Equilibrium
constant
24
First order reversible reactions :
understanding the approach to chemical equilibrium.
Rate equation
k
A
B
k'
u  v 1
  0 u 1 v  0
da
 ka  k b
dt
Rate equation in normalised form
Initial condition
t  0 a  a0
b0
du
1
u 
d
1
Mass balance condition
Solution produces the concentration expressions
 t a  b  a0
u   
v  
Introduce normalised variables.
u
a
a0
v
b
a0
  k  k t  
k
k
1
1   exp   
1

1
1  exp   
Q  
Reaction quotient Q
v 

u  
 1  exp    


1   exp   
First order reversible reactions: approach to equilibrium.
Kinetic
regime
1.0
concentration
0.8
Equilibrium
Product B
K  Q   
v 

u  
0.6
u ()
v ()
0.4
Reactant A
0.2
0.0
0
2
4
6
8
(k+k')t
25
Understanding the difference between reaction quotient Q and
Equilibrium constant K.
12
Approach to
Equilibrium
Q<
10
  10
Q()
8
Equilibrium
Q=K=
6
4
2
0
0
2
4
6
8
 = (k+k')t
Q  
v 

u  
 1  exp    


1   exp   
t   Q  K  
k
k
K
v 
u  
Kinetic versus Thermodynamic
control.
• In many chemical reactions
the competitive formation of
side products is a common
and often unwanted
phenomenon.
• It is often desirable to
optimize the reaction
conditions to maximize
selectivity and hence
optimize product formation.
• Temperature is often a
parameter used to modify
selectivity.
• Operating at low
temperature is generally
associated with the idea of
high selectivity (this is
termed kinetic control).
Conversely, operating at high
temperature is associated
with low selectivity and
corresponds to
Thermodynamic control.
• Time is also an important
parameter. At a given
temperature, although the
kinetically controlled product
predominates at short times,
the thermodynamically
Refer to JCE papers dealing with this
controlled product will
topic given as extra handout.
predominate if the reaction
time is long enough.
26
Short reaction
times
Assume that reaction
product P1 is less stable
than the product P2.
Also its formation is
assumed to involve
a lower activation
energy EA.
Temperature effect.
•
Kinetic control.
–
–
–
•
Assume that energy of products
P1and P2 are much lower than that of
the reactant R then EA,1<<EA,-1 and
EA,2 << EA,-2.
At low temperature one neglects the
fraction of molecules having an
energy high enough to re-cross the
energy barrier from products to
reactants.
Under such conditions the product
ratio [P1]/[P2] is determined only by
the activation barriers for the
forward R  P reaction steps.
 DG*j 
k j  A exp  

 RT 
Thermodynamic control.
–
At high temperature the available
thermal energy is considered large
enough to assume that energy
barriers are easily crossed.
Thermodynamic equilibrium is
reached and the product ratio
[P1]/[P2] is now determined by the
relative stability of the products P1
and P2.
 P1   k1  exp   DG1*  DG2*  
RT
 P2  k2


 P1   K1  exp   DG10  DG20  
RT
 P2  K 2


27

d  R
dt
d  P1 
dt
d  P2 
dt
  k1  k2   R   k1  P1   k2  P2 
 k1  R   k1  P1 
 k2  R   k2  P2 
Short time
Approximation.
Neglect k-1, k-2
terms.
Long time approximation.
 P1  
 R    R 0 exp   k1  k2  t 
k  R
 P1   1 0 1  exp   k1  k2  t 
k2  R 0
k1  k2
1  exp   k  k  t 
1
2
 P1  
 P2  
t


 P1 
 P2 
k1  R 0
k1  k2
k2  R 0
k1  k2
 P1   k1
 P2  k2
t
 1
 k1  k2 
K1  R 0
1  K1  K 2
 P2  
k1  k2
 P2  
d  R  d  P1  d  P2 


0
dt
dt
dt
 R 0
 R  
1  K1  K 2
Kinetic control
Limit.

K 2  R 0
1  K1  K 2
K1
K2
Thermodynamic
control limit.
General solution valid for intermediate times.
 1  k2  1  k1  exp  t   2  k1  2  k2  exp  t 
 1
 2 
1  1  2 
2  2  1 
 12

k k

k   k 
k   k 
 P1    R 0  1 2  1 1 2 exp 1t   1 2 2 exp 2t 
2  2  1 
 12 1  1  2 

 k k k   k 

k   k 
 P2    R 0  1 2  2 1 1 exp 1t   2 2 1 exp 2t 
2  2  1 
 12 1  1  2 

 k1k2
 R    R 0 
1,2  

   2  4
2

  k1  k2  k1  k2
  k1k2  k1k 2  k1k2
 
4 
1  1  2 
2 
 
These expressions reproduce
the correct limiting forms
corresponding to kinetic and
Thermodynamic control in the
limits of short and long time
respectively.
28
Quasi-Steady State Approximation.
• Detailed mathematical analysis of complex
QSSA
reaction mechanisms is difficult.
Some useful methods for solving sets of
coupled linear differential rate equations
include matrix methods and Laplace Transforms.
• In many cases utilisation of the quasi steady
state approximation
induction
leads to a considerable simplification in the
period
kinetic analysis.
k1
k2
A 
X 
P
Consecutive reactions k1 = 0.1 k2 .
The QSSA assumes that after an initial
induction period (during which the
concentration x of intermediates X rise
from zero), and during the major part
of the reaction, the rate of change of
concentrations of all reaction
intermediates are negligibly small.
Mathematically , QSSA implies
dx
 RX formation  RX removal  0
dt
RX formation  RX removal
Normalised concentration
1.0
u()
A
P
0.5
w()
v()
0.0
0.0
0.5
1.0
 = k1t
intermediate X
concentration
approx. constant
QSSA: a fluid flow analogy.
• QSSA illustrated via analogy
with fluid flow.
• If fluid level in tank is to
remain constant then rate of
inflow of fluid from pipe 1
must balance rate of outflow
from pipe 2.
• Reaction intermediate
concentration equivalent to
fluid level. Inflow rate
equivalent to rate of
formation of intermediate
and outflow rate analogous to
rate of removal of
intermediate.
P1
Fluid level
P2
29
Consecutive reaction mechanisms.
k1
A
X
k2
P
k-1
Rate
equations
du
 u   v
d
dv
 u      v
d
dw
 v
d
a
u
a0
x
v
a0
p
w
a0
 u  v  w  1
  0 u 1 v  w  0

k 1
k1

k2
k1
  k1 t
Step I is reversible, step II is
Irreversible.
Coupled LDE’s can be solved via Laplace
Transform or other methods.
u   
v  
1
 
1
 
w   1 
      exp           exp    
exp      exp    
1
 
Note that  and  are composite quantities containing
the individual rate constants.
  
    1   
Definition of normalised variables
and initial condition.
QSSA assumes that
dv
0
d
u ss     vss  0
 exp       exp    
Using the QSSA we can develop more
simple rate equations which may be
integrated to produce approximate
expressions for the pertinent concentration
profiles as a function of time.
The QSSA will only hold provided that:
u ss
 
• the concentration of intermediate is small
and effectively constant,
and so :
du ss


u ss
d
 
• the net rate of change in intermediate
concentration wrt time can be set equal to
zero.
vss 



u ss  exp 






    
exp  
  
    
vss 
 
    
dwss

  vss 
exp  
  
d
 
    
wss 


  
 
   d
exp  
  
    
0
    
  
 1  exp  
    
30
normalised concentration
1.2
1.0
0.4
X
0.0
0.01
0.1
1
10
k2
P
Concentration versus log time curves
for reactant A, intermediate X and
product P when full set of coupled
rate equations are solved without
any approximation.
k-1 >> k1, k2>>k1 and k-1 = k2 = 50.
The concentration of intermediate X is
very small and approximately constant
throughout the time course of the
experiment.
u()
v()
w()
0.2
X
k-1
0.8
0.6
k1
A
P
A
100
log 
Concentration versus log time curves
for reactant A, intermediate X, and
product P when the rate equations
are solved using the QSSA.
Values used for the rate constants
are the same as those used above.
QSSA reproduces the concentration
profiles well and is valid.
normalised concentration
1.2
QSSA will hold when concentration
of intermediate is small and constant.
Hence the rate constants for getting
rid of the intermediate (k-1 and k2)
must be much larger than that for
intermediate generation (k1).
0.8
uss ()
0.6
v ss ()
wss ()
0.4
0.2
X
0.0
0.01
normalised concentration
1
u()
v()
w()
0.2
0.0
10
k2
P
Concentration versus log time curves
for reactant A, intermediate X and
product P when full set of coupled
rate equations are solved without
any approximation.
k-1 << k1, k2,,k1 and k-1 = k2 = 0.1
The concentration of intermediate is
high and it is present throughout much
of the duration of the experiment.
0.4
1
X
P
X
0.1
100
k-1
A
0.6
10
k1
A
1.0
0.01
0.1
log 
1.2
0.8
P
A
1.0
100
log 
The QSSA is not good in predicting
how the intermediate concentration
varies with time, and so it does not
apply under the condition where the
concentration of intermediate will be
high and the intermediate is long lived.
1.2
normalised concentration
Concentration versus log time curves
for reactant A, intermediate X and
product P when the Coupled rate
equations are solved using
the quasi steady state approximation.
The same values for the rate constants
were adopted as above.
A
1.0
0.8
P
uss ()
0.6
v ss ()
wss ()
0.4
X
0.2
0.0
0.01
0.1
1
10
100
log 
31