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CHEMICAL KINETICS 25-05-2015
Contact: By: Muhammad Asif
Phone: +92-303-7807073 Lecturer (Physical Chemistry)
E-mail: [email protected] Govt. College Sahiwal, Pakistan
4.1 Introduction 02
4.2 Rate of reaction ... 02
4.3 Rate equation and rate constant.. 03
4.4 Order and molecularity of reactions ... 04
4.5 Pseudo order reactions 05
4.6 Zero order reactions ... 06
4.7 First order reactions ... 07
4.8 Second order reactions ... 09
4.9 Third order reactions .. 13
4.10 Elementary and complex reactions 23
4.10.1 Opposing reactions . 23
4.10.2 Consecutive reactions . 34
4.10.3 Parallel reactions .... 36
4.10.4 Chain reactions 37
4.11 Thermal reactions .... 37
4.12 Photochemical reactions .. 40
4.13 Effect of temperature on rate of reaction .... 44
4.14 Theories of reaction rates 48
4.14.1 Collision theory ... 48
4.14.2 Transition state theory . 52
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4.1 INTRODUCTION
Chemical kinetics is the branch of chemistry that deals with the
rates of chemical
reactions. The chemical reactions can broadly be divided into
three classes:
1. Fast reactions e.g., the reaction between aqueous sodium
chloride and silver nitrate.
2. Moderate reactions e.g., hydrolysis of an ester.
3. Slow reactions e.g., rusting of iron.
However, the chemical reactions can occur at a variety of rates
from very fast to very slow.
For example, the paper on which this book is printed reacts
extremely slowly with the
atmosphere oxygen, whereas explosion is extremely fast reaction.
The two extreme classes of
reactions cannot be studied conveniently. Reactions which
proceed with measureable rates
are the subjects of study by physical chemists.
In 1864 C. M. Guldberg and P. Waage pioneered the development of
chemical
kinetics by formulating the law of mass action, which states
that the speed of a chemical
reaction is proportional to the quantity of the reacting
substances.
Experimentally it has been found that the rate of a chemical
reaction depends on the
nature of the reacting species, the temperature, the pressure,
the concentration of the reacting
species and the presence of catalyst.
The reactions may be classified kinetically as being either
homogenous, if they take
place in one phase only, or heterogeneous if two or more phases
are involved in the
processes. The reactions may be elementary or complex. An
elementary reaction occurs in a
single step while a complex reaction occurs in two or more
steps.
4.2 RATE OF REACTION
The rate of a chemical reaction is the change in the
concentration of reactants or
products in a unit time or a given time.
Change in concentration c
RateTime period of change t
This equation gives the average velocity of the reaction during
the time of observation. By
shortening the time of observation the average velocity
approaches more and more closely to
the actual velocity of the reaction. So in chemical kinetics the
rate of reaction is not
represented by the average velocity but by the instantaneously
velocity of the reaction that is
when the time of observation approaches zero then,
d[c]
Ratedt
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The brackets [ ] represent the concentration in moles per
decimeter cube. Thus rate of
reaction is expressed in units of moles per decimeter cube per
second ( 3 1mol dm sec ). For
gas phase reactions, pressure units are used in place of molar
concentration.
Consider the following general reaction,
A + B C + D
The rate at which reaction proceeds can be measured in terms of
the rate at which one of the
reactants disappears or one of the products appears.
dx d[A] d[B] d[C] d[D]
dt dt dt dt dt
Where [A], [B], [C] and [D] represent the concentrations of A,
B, C and D in moles per
decimeter cube at time t. The negative sign indicates that the
concentrations of A and B
decreases with increasing time. The plus sign indicates that the
concentrations of C and D
increase with increasing time.
4.3 RATE EQUATION AND RATE CONSTANT
According to law of mass action, the rate of a chemical reaction
is proportional to the
product of the molar concentrations of the reactants raised to
the power equal to the number
of molecules of each species taking part in the reaction.
Consider the reaction,
aA + bB cC + dD
a bdx
[A] [B]dt
a bdx
= k[A] [B]dt
Where, k is constant of proportionality known as the velocity
constant or rate constant or
specific rate constant. This expression shows how the reaction
rate is related to
concentration and is known as rate equation, rate expression or
rate law.
If the concentration of A and B are kept unity, then
a bdx
= k[1] [1] = kdt
Thus rate constant may be defined as the rate of reaction when
molar concentration of
each of the reactants is unity (1 mol dm3
). The value of k varies from reaction to reaction
and also varies with temperature for a given reaction. The value
of k for a reaction does not
change with time. The units of k depend on the order of
reaction.
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Anyhow it is not essential that the powers in the rate equation
are always equal to the
coefficients of balance chemical equations. So we can say that
the rate equation of above
reaction may be,
m ndx
= k[A] [B]dt
The power or exponent of a concentration term, m or n in the
rate equation is usually in small
whole number integers (0, 1, 2, 3) or may be fractional. Here n
is called the order of reaction
with respect to A, and m is the order of the reaction with
respect to B. The sum (n + m) is
called the overall order of the reaction.
Rate equation for an elementary reaction can be predicted from
balanced chemical
equation but it cannot be predicted for complex reactions
(involving more than one steps).
Therefore it is written from experimental facts for complex
reactions. Actually rate equation
is an experimental expression. The order of the reaction is
completely independent of the
reaction stoichiometry.
4.4 ORDER AND MOLECULARITY OF REACTIONS
The order of a reaction is defined as the total number of atoms,
ions or molecules
whose concentration changes during a chemical reaction. It can
be calculated from an
experimentally determined rate equation. It is the sum of
exponents of all the
concentration terms in the differential form of rate equation. A
reactant whose
concentration does not affect the rate of reaction is not
included in the rate equation. The
concentration of such a reactant has the power 0 e.g., [A]0 =1.
For a reaction maximum order
is three and the minimum is zero. However, the order of reaction
may be in minus with
respect to a particular species in the reaction; such a species
acts as an inhibitor and rate of
reaction is inversely proportional to its concentration. The
reactions may be classified
according to the order. If (m + n) in the rate equation is:
m + n = 0 the reaction is zero order reaction
m + n = 1 the reaction is first order reaction
m + n = 2 the reaction is second order reaction
The order of reaction provides valuable information about the
mechanism of a reaction and
the knowledge of mechanism of a given reaction allows us to
control that reaction.
The order of reaction should not be confused with molecularity
which is defined as
the total number of atoms, ions or molecules which take part in
a reaction as given by
the balanced chemical equation. Molecularity and order are
identical for elementary
reactions. Most chemical reactions are complex reactions which
occur in a series of steps.
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Each step is an elementary reaction. The stepwise sequence of
elementary reactions that
convert reactants to products is called mechanism of the
reaction. In any mechanism, some
of the steps will be fast, others will be slow. The rate of the
reaction is determined by the
slowest step known as the rate determining step. Hence,
molecularity of a reaction can
also be defined as the total number of atoms, ions or molecules
taking part in the rate
determining step. The term unimolecular, bimolecular and
termolecular (or trimolecular)
indicate the number of particles reacting in a single elementary
process.
No Order of Reaction Molecularity
1 It may be equal to zero. It can never be equal to zero.
2 It can be in fractions. It can never be in fractions.
3 It is at most equal to three. It can be more than three.
4 It can only be determined
experimentally.
It can be obtained from a simple balanced
chemical equation.
5 It is the sum of all the
exponents of the concentration
terms in the rate equation.
It is the sum of number of molecules of the
reactants taking part in a single step chemical
reaction.
6 It helps in determining the
mechanism of a reaction.
It gives no idea about the mechanism of a reaction.
7 In case of complex reactions
order is determined by the
slowest step of the reaction.
In case of complex reactions each step of the
reaction has its own molecularity i.e., molecularity
has no significance for complex reactions.
4.5 PSEUDO ORDER REACTIONS
A reaction in which one of the reactants is present in large
excess shows an order
different from the actual order. The experimental order which is
not the actual one is
referred as the pseudo order. Since for elementary reactions
molecularity and order are
identical, pseudo reactions may also be called pseudo molecular
reactions.
Let us consider a reaction,
A + B Products
A + B Products
In which the reactant B is present in a large excess. Since it
is an elementary reaction, its rate
equation can be written as;
Rate = k [A] [B]
As B is present in large excess its concentration remains
practically constant in the course of
reaction. Thus the rate law can be written as;
Rate = k [A]
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Where the new rate constant k = k [B]. Thus the actual order of
the reaction is second order
but in practice it will be first order. Therefore, the reaction
is said to have a pseudo first order.
Examples:
1. Hydrolysis of an ester: methyl acetate upon hydrolysis in
aqueous solution using a
mineral acid as a catalyst forms acetic acid and methyl
alcohol.
CH3COOCH3 + H2O CH3COOH + CH3OHH
Here a large excess of water is used and the rate law can be
written as
Rate = k [CH3COOCH3] [H2O]
Rate = k [CH3COOCH3]
The reaction is actually second order but in practice it is
found to be first order. Thus
it is a pseudo first order reaction.
2. Hydrolysis of sucrose: sucrose upon hydrolysis in the
presence of a dilute mineral
gives glucose and fructose.
4.6 ZERO ORDER REACTION
The reaction in which rate is independent of the concentration
of the reactants is
called zero order. Let us take a substance A which decomposes
into products. Its initial
concentration is a moles dm3. Let after time t seconds the
amount left behind is (a-x)
moles dm3
and that converted into product is x moles dm3. Then
A Products
A Products
When t = 0 a 0
When t = teq (a-x) x
Hence, the rate of reaction is given by,
dx
k(a x)dt
dx
kdt
dx kdt
Integrating,
dx k dt
x kt C
The value of integration constant C can be found by applying the
initial conditions of the
reaction when t = 0, x = 0. That is
x kt (1)
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This equation (1) is similar to the equation for a straight
line. Thus, if we plot a graph
between x on y-axis and t on x-axis, we should get a straight
line passing through the origin
with slope k.
Rearranging Equation (1) for k,
xk
t (2)
This equation (2) gives the rate constant for zero order
reaction. The units of k are mole
dm3
sec1
.
Half-Life
It is defined as the time required to reduce the concentration
of reactants to half of its
original value. Therefore, when t = t1/2 then x = a/2.
Substitute this value in equation (2),
a2
1/2
at
k 2k
Examples
1. Thermal decomposition of HI on gold surface,
Gold Surface
2HI
Gold Surface
Gold Surface
H2 + I
2
Gold Surface
2HI
Gold Surface
Gold Surface
H2 + I
2
2. Decomposition of ammonia in the presence of tungsten,
Tungston
2NH3
Tungston
Tungston
N2 + 3H
2
Tungston
2NH3
Tungston
Tungston
N2 + 3H
2
3. Photochemical reactions are usually zero order.
4. The reactions which are catalyzed by enzymes are also zero
order.
4.7 FIRST ORDER REACTIONS
The reaction in which the concentration of only one molecule is
changed is called
a first order reaction. Let us take a substance A which
decomposes into products. Its initial
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concentration is a moles dm3. Let after time t seconds the
amount left behind is (a-x)
moles dm3
and that converted into product is x moles dm3. Then
A Products
A Products
When t = 0 a 0
When t = teq (a-x) x
Hence, the rate of reaction is given by,
dx
k(a x)dt
dx
kdt(a x)
dx
k dt(a x)
ln(a x) kt C (1)
The value of integration constant C can be found by applying the
initial conditions of the
reaction when t = 0, x = 0. That is
ln(a 0) k(0) C
lna C
Therefore Eq.(1) becomes,
ln(a x) kt lna
lna ln(a x) kt
a kt
loga x 2.303
(2)
This equation is similar to the equation for a straight line.
Thus, if we plot a graph between
alog
(a x)on y-axis and t on x-axis, we should get a straight line
passing through the origin
with slope k/2.303.
Rearranging Equation (2) for k,
2.303 ak log
t a x
(3)
This equation (3) gives the rate constant k of first order
reaction. The units of k are sec1
.
Half-Life
When t = t1/2 then x = a/2. Substitute this value in equation
(3),
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1/2 a2
2.303 at log
k a ( )
1/2
2.303 0.693t log2
k k
Examples
1. Thermal decomposition of hydrogen peroxide is a first order
reaction,
H2O2 H2O + O
2. Decomposition of nitrogen pentoxide in CCl4 solution is a
first order reaction,
N2O5 2NO2 +1/2 O2
3. The hydrolysis of an ester in the presence of a mineral acid
as catalyst,
CH3COOC2H5 + H2O CH3COOH + C2H5OHH
4.8 SECOND ORDER REACTIONS
The reaction in which two molecules undergo a chemical change is
called a
second order reaction. Let us consider two substances A and B
which react to give the
products. The reaction can be carried out by;
a) By taking equal concentrations of A and B
b) By taking different concentrations of A and B
a) Second order reaction with equal concentrations of
reactants
Let the initial concentrations in moles dm3
are a moles dm3
and after time t the
concentrations left behind are (a-x) for both A and B. Then
A + B Products
A + B Products
When t = 0 a + a 0
When t = teq (a-x) + (a-x) x
Hence, the rate of reaction is given by,
dx
k(a x)(a x)dt
2dx
k(a x)dt
2
dxkdt
(a x)
2
dxk dt
(a x)
2(a x) dx k dt
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2 1(a x) ( 1)
kt C2 1
1
kt C(a x)
(1)
The value of integration constant C can be found by applying the
initial conditions of the
reaction when t = 0, x = 0. That is
1 1
k0 C C(a 0) a
Therefore equation (1) becomes,
1 1
kt(a x) a
1 1
kt(a x) a
x
kta(a x)
x
akt(a x)
(2)
This equation is similar to the equation for a straight line.
Thus, if we plot a graph between
x
(a x)on y-axis and t on x-axis, we should get a straight line
passing through the origin
with slope ak.
Rearranging Equation (2) for k,
1 x
kt a(a x)
(3)
This equation gives the rate constant k for second order
reaction with same initial
concentrations of the reactants. The units of k are dm3 mol
1 sec
1.
Half-Life
When t = t1/2 then x = a/2. Substitute this value in equation
(3),
a2
1/2 a2
1 1t
k a(a ) ka
Examples
Saponification of an ester with a strong base is a second order
reaction in solution,
CH3COOC
2H
5 + NaOH CH
3COONa + C
2H
5OH
CH3COOC
2H
5 + NaOH CH
3COONa + C
2H
5OH
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b) Second order reaction with different concentrations of
reactants
Let the initial concentrations of reactants A and B are a and b
moles dm3 respectively,
and x moles dm3 of the reactants are disappeared in time t.
Then
A + B Products
A + B Products
When t = 0 a + b 0
When t = teq (a-x) + (b-x) x
Hence, according to rate law,
dx
k(a x)(b x)dt
dx
kdt(a x)(b x)
dx
k dt(a x)(b x)
(1)
In order to integrate the L.H.S., we have to do the partial
fractions first by using partial
fraction method,
1 A B
(a x)(b x) (a x) (b x)
(2)
1 A(b x) B(a x) (3)
To get A, put (a-x) = 0 or x = a in equation (3),
1 A(b a)
1 1
Ab a (a b)
To get B, put (b-x) = 0 or x = b in equation (3),
1 B(a b)
1
B(a b)
Substituting the value of A and B into equation (2),
1 1 1
(a x)(b x) (a b)(a x) (a b)(b x)
Now substituting this value in equation (1),
1 1
dx dx k dt(a b)(a x) (a b)(b x)
1 dx 1 dx
k dt(a b) (a x) (a b) (b x)
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1 1
[ ln(a x)] [ ln(b x)] kt C(a b) (a b)
1 1
ln(a x) ln(b x) kt C(a b) (a b)
1
ln(a x) ln(b x) kt C(a b)
1 (a x)
ln kt C(a b) (b x)
(4)
The value of integration constant C can be found by applying the
initial conditions of the
reaction when t = 0, x = 0. That is
1 (a 0)
ln k0 C(a b) (b 0)
1 a
ln C(a b) b
Therefore equation (4) becomes,
1 (a x) 1 a
ln kt ln(a b) (b x) (a b) b
1 (a x) 1 aln ln kt
(a b) (b x) (a b) b
1 (a x) a
ln ln kt(a b) (b x) b
1 b(a x)
ln kt(a b) a(b x)
2.303 b(a x)
log kt(a b) a(b x)
b(a x) (a b)k
log ta(b x) 2.303
(5)
This equation is similar to the equation for a straight line.
Thus, if we plot a graph between
L.H.S. of this equation (5) on y-axis and time t on x-axis, we
should get a straight line
passing through the origin with slope (a b)k
2.303
.
Rearranging Equation (5) for k,
2.303 b(a x)
k logt(a b) a(b x)
(6)
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This equation (6) gives the rate constant k for a second order
reaction when the initial
concentrations of the reactants are different. The dimensions of
k are again dm3 mol
1 sec
1.
Half-Life
The half-life method cannot be used for reactions where the
concentrations of A and B are
different, since A and B will have different times for half
reaction.
Examples
1.
H2 + I
2 2HI
H2 + I
2 2HI
2.
2HI H2 + I
2
2HI H2 + I
2
3.
2O3 3O
2
2O3 3O
2
4.
2NO2 N
2 + 2O
2
2NO2 N
2 + 2O
2
4.9 THIRD ORDER REACTIONS
The reaction in which only three molecules undergo a chemical
change is called a
third order reaction. Four different cases of 3rd
order reaction are discussed here.
Case (I): when the initial concentration of all the reactants is
same
Consider the following reaction in which A, B and C reactants
are converting into
products in an elementary reaction. All reactants have same
initial concentration (a). After
time (t) the concentration of each reactant becomes equal to a x
as given below;
A + B + C Products
A + B + C Products
When t = 0 a a a 0
When t = t ax ax ax x
Rate equation for such a reaction can be written as
dx
k[A][B][C]dt
dx
k(a x)(a x)(a x)dt
3dx
k(a x)dt
This is the differential form of rate law for a third order
reaction in which concentration of
all the reactants is same.
Integrating the above equation after separating variables,
3
dxk dt
(a x)
3(a x) dx k dt
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3 1(a x)
kt c( 3 1)( 1)
2(a x)
kt c2
2
1kt c
2(a x)
(1)
When t = 0, x = 0, putting these values in equation (1), the
value of c can be calculated
2
1c
2a
Hence, equation (1) becomes
2 2
1 1kt
2(a x) 2a
2 2
1 12kt
(a x) a
(2)
This is the integrated form of rate law for third order reaction
with same initial
concentrations of reactants. This equation is similar to the
equation for a straight line.
Therefore if we plot a graph between 2
1
(a x)on y-axis and t on x-axis, we should get a
straight line with slope equal to 2k and intercept 21/ a .
Also from equation (2),
2 2
1 1kt
2(a x) 2a
2 2
2 2
a (a x)kt
2(a x) a
2 2 2
2 2
a (a x 2ax)kt
2(a x) a
2 2 2
2 2
a a x 2axkt
2(a x) a
2
2 2
2ax xkt
2(a x) a
2 2
x(2a x)kt
2(a x) a
2 2
1 x(2a x)k
2a t (a x)
(3)
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This equation (3) gives the rate constant k. Units of k can be
determined as follows,
3 3
3 2 3 2
1 (mol dm )(mol dm )k
(mol dm ) sec (mol dm )
2 6 1k mol dm sec
Half-Life
When t = t1/2 then x = a/2 = 0.5a. Substitute this value in
equation (3),
2 2
1 x(2a x)t
2a k (a x)
1/2 2 2 2 2
1 0.5a(2a 0.5a) 3 1.5t
2a k (a 0.5a) 2ka ka
(4)
It is evident from the above equation that the half-life period
is inversely proportional to the
square of the initial concentration of the reactant.
In general for a reaction of nth order, the half-life is found
to be,
1/2 n 1
1t
a (where n is the order of reaction)
Thus, half-life period of any order reaction is inversely
proportional to the initial
concentration raised to the power one less than the order of
that reaction.
Examples:
The reaction between NO and H2 to give N2 and H2O follows the
third order reaction
kinetics. The reaction is,
2NO + 2H2 N
2 + 2H
2O
2NO + 2H2 N
2 + 2H
2O
The rate law for this reaction is given by
Rate = k [H2][NO]2
Case (II): when the initial concentration of two reactants is
same and that of the third
one is different
Consider the following reaction in which the reactants A and B
have the same initial
concentrations equal to a but reactant C has different
concentration equal to c as given
below,
A + B + C Products
A + B + C Products
When t = 0 a a c 0
When t = t ax ax cx x
Rate equation for such a reaction is given by
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dx
k[A][B][C]dt
dx
k(a x)(a x)(c x)dt
2dx
k(a x) (c x)dt
This is the differential rate equation for a third order
reaction in which concentration of two
reactants are same and that of the third one is different.
Integrating the above equation after separating variables,
2
dxk dt
(a x) (c x)
(1)
Using partial fraction, we can write
2 2
1 A B C
(a x) (c x) (a x) (a x) (c x)
(2)
21 A(a x)(c x) B(c x) C(a x) (3)
For calculating the value of B, put (a x) = 0 or x = a that
is
21 A(a a)(c a) B(c a) C(a a)
1 B(c a) or 1
B(c a)
For calculating the value of C, put (c x) = 0 or x = c that
is
21 A(a c)(c c) B(c c) C(a c)
21 C(a c) or 2
1C
(a c)
For calculating the value of A, consider equation (3),
21 A(a x)(c x) B(c x) C(a x)
2 2 21 A(ac ax cx x ) B(c x) C(a x 2ax)
2 2 21 Aac Aax Acx Ax Bc Bx Ca Cx 2Cax
Comparing coefficients of x2, we get
0 A C
A C or 2
1A
(a c)
Putting the values of A, B and C in equation (2),
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2 2 2 2
1 1 1 1
(a x) (c x) (a c) (a x) (c a)(a x) (a c) (c x)
Hence equation (1) becomes
2 2 2
1 1 1dx k dt
(a c) (a x) (c a)(a x) (a c) (c x)
2 2 2
1 dx 1 dx 1 dxk dt
(a c) (a x) (c a) (a x) (a c) (c x)
2 1
2 2
1 ln(a x) 1 (a x) 1 ln(c x)kt c
(a c) 1 (c a) ( 2 1)( 1) (a c) 1
2 2
1 1 1 1ln(a x) ln(c x) kt c
(a c) (c a) (a x) (a c)
21 1 1
ln(a x) ln(c x) kt c(a c) (c a) (a x)
2
1 (a x) 1 1ln kt c
(a c) (c x) (c a) (a x)
(4)
When t =0, x = 0, putting these values in above equation, the
value of c can be calculated,
2
1 (a 0) 1 1ln k0 c
(a c) (c 0) (c a) (a 0)
2
1 a 1ln c
(a c) c a(c a)
Putting the value of c in equation (4),
2 2
1 (a x) 1 1 1 a 1ln kt ln
(a c) (c x) (c a) (a x) (a c) c a(c a)
2 2
1 (a x) 1 1 1 a 1ln ln kt
(a c) (c x) (c a) (a x) (a c) c a(c a)
2
1 (a x) a 1 1 1ln ln kt
(a c) (c x) c (c a) (a x) a(c a)
2
1 c(a x) a (a x)ln kt
(a c) a(c x) a(a x)(c a)
2
1 c(a x) a a xln kt
(a c) a(c x) a(a x)(c a)
2
1 c(a x) xln kt
(a c) a(c x) a(a x)(c a)
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2
1 c(a x) xln kt
(a c) a(c x) a(a c)(a x)
2c(a x) x(a c)
ln (a c) kta(c x) a(a x)
(5)
This is the integrated rate equation for a third order reaction
in which concentration of two
reactants are same and that of the third one is different. This
equation is similar to the
equation for a straight line passing through the origin, from
the slope of which k can be
calculated if initial concentrations are known that is,
2Slope (a c) k
2
Slopek
(a c)
Left hand side of equation (5) is a dimensionless quantity.
Hence units of k comes out to be,
231 mol / dm ksec 2 61 mol dm seck
2 6 1k mol dm sec
The half-life period cannot be determined for such type of a
reaction.
Case (III): when the initial concentration of all the reactants
is different
Consider the following reaction in which the initial
concentration of reactants A, B
and C are a, b and c respectively.
A + B + C Products
A + B + C Products
When t = 0 a b c 0
When t = t ax bx cx x
Rate equation for such a reaction can be written as
dx
k[A][B][C]dt
dx
k(a x)(b x)(c x)dt
Integrating the above equation after separating variables,
dx
k dt(a x)(b x)(c x)
(1)
Using partial fraction, we get
1 A B C
(a x)(b x)(c x) (a x) (b x) (c x)
(2)
-
19
1 A(b x)(c x) B(a x)(c x) C(a x)(b x)
For calculating the value of A, put (a x) = 0 or x = a that
is
1 A(b a)(c a) B(a a)(c a) C(a a)(b a)
1 A(b a)(c a)
1 1
A(b a)(c a) (a b)(c a)
For calculating the value of B, put (b x) = 0 or x = b that
is
1 A(b b)(c b) B(a b)(c b) C(a b)(b b)
1 B(a b)(c b)
1 1
B(a b)(c b) (a b)(b c)
For calculating the value of C, put (c x) = 0 or x = c that
is
1 A(b c)(c c) B(a c)(c c) C(a c)(b c)
1 C(a c)(b c)
1 1
C(a c)(b c) (c a)(b c)
Putting the values of A, B and C in equation (2),
1 1 1 1
(a x)(b x)(c x) (a b)(c a)(a x) (a b)(b c)(b x) (c a)(b c)(c
x)
Hence equation (1) becomes
1 1 1dx k dt
(a b)(c a)(a x) (a b)(b c)(b x) (c a)(b c)(c x)
1 dx 1 dx 1 dx
k dt(a b)(c a) (a x) (a b)(b c) (b x) (c a)(b c) (c x)
1 1 1
ln(a x) ln(b x) ln(c x) kt c(a b)(c a) (a b)(b c) (c a)(b c)
1
(b c) ln(a x) (c a) ln(b x) (a b) ln(c x) kt c(a b)(b c)(c
a)
(b c) ln(a x) (c a) ln(b x) (a b) ln(c x)
kt c(a b)(b c)(c a)
(3)
When t =0, x = 0, putting these values in above equation, the
value of c can be calculated,
(b c) ln(a 0) (c a) ln(b 0) (a b) ln(c 0)
k0 c(a b)(b c)(c a)
-
20
(b c) ln a (c a) ln b (a b) ln c
c(a b)(b c)(c a)
Putting the value of c in equation (3),
(b c) ln(a x) (c a) ln(b x) (a b) ln(c x) (b c) ln a (c a) ln b
(a b) ln ckt
(a b)(b c)(c a) (a b)(b c)(c a)
(b c) ln(a x) (c a) ln(b x) (a b) ln(c x) (b c) ln a (c a) ln b
(a b) ln ckt
(a b)(b c)(c a) (a b)(b c)(c a)
(b c) ln(a x) (c a) ln(b x) (a b) ln(c x) (b c) ln a (c a) ln b
(a b) ln ckt
(a b)(b c)(c a)
(a x) (b x) (c x)(b c) ln (c a) ln (a b) ln (a b)(b c)(c
a)kt
a b c
(4)
This is the integrated rate equation for a third order reaction
in which the concentrations of all
the reactants are different. This is the equation of a straight
line passing through the origin,
where the left hand side of the equation represents the
dependent variable and t is the
independent variable. The plot will give a straight line whose
slope is given by
Slope (a b)(b c)(c a)k
Slope
k(a b)(b c)(c a)
Hence, the value of k can be determined if the initial
concentrations of the three reactants are
known. Units of k can be determined by rearranging the equation
(4),
(a x) (b x) (c x)(b c) ln (c a) ln (a b) ln
a b ck(a b)(b c)(c a)t
3
33
mol dmk
mol dm sec
2 6 1k mol dm sec
Case (IV): when only two reactants are involved
Sometime the rate of a reaction has more dependence on the
concentration of one
reactant than that of the other. For example, 3rd
order reactions, rate of reaction may be
directly proportional to the square of concentration of one
reactant and concentration of the
other reactants. Consider a third order reaction of the type in
which two moles of reactant A
are reacting with one mole of reactant B to give products. The
initial concentration of reactant
A is a and that of B is b as given below,
-
21
2A + B Products
2A + B Products
When t = 0 a b 0
When t = t a2x bx x
Rate equation for such a reaction is given by
2dx
k[A] [B]dt
2dx
k(a 2x) (b x)dt
Integrating the above equation after separating variables,
2
dxk dt
(a 2x) (b x)
(1)
Using partial fraction, we get
2 2
1 A B C
(a 2x) (b x) (a 2x) (a 2x) (b x)
(2)
21 A(a 2x)(b x) B(b x) C(a 2x) (3)
For calculating the value of C, put (b x) = 0 or x = b that
is
21 A(a 2b)(b b) B(b b) C(a 2b)
21 C(a 2b)
2
1C
(a 2b)
For calculating the value of B, put (a 2x) = 0 or x = a/2 that
is
2a a a a
1 A(a 2 )(b ) B(b ) C(a 2 )2 2 2 2
a 2b a
1 B(b ) B( )2 2
2 2
B(2b a) (a 2b)
For calculating the value of A, consider equation (3),
21 A(a 2x)(b x) B(b x) C(a 2x)
2 2 21 A(ab ax 2bx 2x ) B(b x) C(a 4x 4ax)
2 2 21 Aab Aax 2Abx 2Ax Bb Bx Ca 4Cx 4Cax)
Comparing coefficients of x2, we get
1 2A 4C
-
22
2A 4C
2
2A 2C
(a 2b)
Putting the values of A, B and C in equation (2),
2 2
1 A B C
(a 2x) (b x) (a 2x) (a 2x) (b x)
2 2 2 2
1 2 2 1
(a 2x) (b x) (a 2b) (a 2x) (a 2b)(a 2x) (a 2b) (b x)
Hence equation (1) becomes
2 2 2
2 2 1dx k dt
(a 2b) (a 2x) (a 2b)(a 2x) (a 2b) (b x)
2 2 2
1 2dx 1 2dx 1 dxk dt
(a 2b) (a 2x) (a 2b) (a 2x) (a 2b) (b x)
2 1
2 2
1 1 2 (a 2x)ln(a 2x) ln(b x) k dt
(a 2b) (a 2b) (a 2b) ( 2 1)( 2)
2
1 (a 2x) 1ln kt c
(a 2b) (b x) (a 2b)(a 2x)
2
1 (a 2x) (a 2b)ln kt c
(a 2b) (b x) (a 2x)
(4)
When t =0, x = 0, putting these values in above equation, the
value of c can be calculated,
2
1 (a 2 0) (a 2b)ln k0 c
(a 2b) (b 0) (a 2 0)
2
1 a (a 2b)ln c
(a 2b) b a
Putting the value of c in equation (4),
2 2
1 (a 2x) (a 2b) 1 a (a 2b)ln ln kt
(a 2b) (b x) (a 2x) (a 2b) b a
2
1 (a 2x) (a 2b) a (a 2b)ln ln kt
(a 2b) (b x) (a 2x) b a
2b(a 2x) (a 2b) (a 2b)
ln (a 2b) kta(b x) (a 2x) a
2b(a 2x) 1 1
ln (a 2b) (a 2b) kta(b x) (a 2x) a
-
23
2b(a 2x) a (a 2x)
ln (a 2b) (a 2b) kta(b x) a(a 2x)
2b(a 2x) 2x
ln (a 2b) (a 2b) kta(b x) a(a 2x)
2b(a 2x) 2x(a 2b)
ln (a 2b) kta(b x) a(a 2x)
(5)
This is the equation of straight line passing through the
origin. The value of k can be found
from the slope if the initial concentrations are known.
2Slope (a 2b) k
2
Slopek
(a 2b)
Units of k can be found as follows,
1 n
1
1 n
moleUnits of k time
litre
2 2 1Units of k mol litre time
4.10 ELEMENTARY AND COMPLEX REACTIONS
An elementary reaction is a simple reaction which occurs in a
single step. A complex
reaction is that reaction which occurs in two or more steps.
Chemistry is comprised of
billions of reactions. The majority of reactions with which the
chemist deals are not
elementary instead they involve two or more elementary steps and
are complex. The complex
reactions are also known as composite reactions. There are
various types of complex
reactions which may involve:
1. Reversible or opposing reactions
2. Consecutive reactions
3. Simultaneous or parallel or side reactions
4. Chain reactions
4.10.1 REVERSIBLE OR OPPOSING REACTIONS
These reactions may be of the following types:
a) First order opposed by first order reaction
b) First order opposed by second order reaction
c) Second order opposed by first order reaction
d) Second order opposed by second order reaction
-
24
a) First order opposed by first order reaction
In such reactions, the forward as well as the backward reaction
is first order. This is
the simplest case of opposing reactions. Consider the following
reaction
A B
k1
k-1
When t = 0 a 0
When t = t ax x
When t = teq axe xe
Rate equation for such a reaction is given by
1 1
dxk (a x) k x
dt (1)
If xe is the equilibrium concentration then at the state of
dynamic equilibrium,
Rf = Rb and x = xe
1 e 1 ek (a x ) k x
1 e1
e
k (a x )k
x
(2)
Putting the value of k-1 from equation (2) in equation (1),
1 e1
e
k (a x )dxk (a x) x
dt x
e1e
(a x )xdxk (a x)
dt x
e e1e
x (a x) (a x )xdxk
dt x
e e e1e
ax xx ax x xdxk
dt x
e1e
a(x x)dxk
dt x
1e
e
k adx(x x)
dt x
Integrating the above equation after separating variables,
1
e e
k adxdt
(x x) x
-
25
1e
e
k aln(x x) t c
x (3)
When t =0, x = 0, putting these values in above equation, the
value of c comes out to be,
ec ln x
Putting the value of c in equation (3),
1e e
e
k aln(x x) t ln x
x
1e e
e
k aln x ln(x x) t
x
e 1
e e
x k aln t
(x x) x
e e1
e
x xk ln
at (x x)
(4)
This is the integrated rate equation for opposing reaction of
the type first order opposed by
first order.
b) First order opposed by Second order reaction
In such reactions, the forward reaction is first order and
backward reaction is second
order. Consider the following reaction,
A B + C
k1
k-1
When t = 0 a 0 0
When t = t ax x x
When t = teq axe xe xe
Rate equation for such a reaction is given by
21 1
dxk (a x) k x
dt (1)
If xe is the equilibrium concentration then at the state of
dynamic equilibrium,
Rf = Rb and x = xe
21 e 1 ek (a x ) k (x )
1 e1 2
e
k (a x )k
(x )
(2)
Putting the value of k-1 from equation (2) in equation (1),
-
26
21 e1 2
e
k (a x )dxk (a x) x
dt (x )
21 e1 2
e
k (a x )dxk (a x) x
dt (x )
2 2
1 e 1 e
2
e
k (a x)x k (a x )xdx
dt x
2 2 2e 1 e 1 e
dxx k (a x)x k (a x )x
dt
2 2 2 2 2e 1 e 1 e 1 1 e
dxx k ax k xx k ax k x x
dt
2 2 2 2 2e 1 e 1 1 e 1 e
dxx k ax k ax k x x k xx
dt
2 2 2e 1 e 1 e e
dxx k a(x x ) k xx (x x)
dt
2e 1 e e 1 e e
dxx k a(x x)(x x) k xx (x x)
dt
2e 1 e e edx
x k (x x) a(x x) xxdt
2e 1 e e edx
x k (x x) ax ax xxdt
12
e e e e
k dtdx
(x x)(ax ax xx ) x
Integrating the above equation,
12
e e e e
kdxdt
(x x)(ax ax xx ) x
(3)
Using partial fraction method, we get
e e e e e e
1 A B
(x x)(ax ax xx ) (x x) (ax ax xx )
(4)
e e e1 A(ax ax xx ) B(x x) (5)
For calculating the value of A, put (xe x) = 0 or x = xe that
is
e e e e e e1 A(ax ax x x ) B(x x )
2e e1 A(2ax x )
e e1 A x (2a x )
-
27
e e
1A
x (2a x )
For calculating the value of B, consider equation (5),
e e e1 A(ax ax xx ) B(x x)
e e e1 Aax Aax Axx Bx Bx
Comparing coefficients of x, we get
e0 Aa Ax B
eB Aa Ax
eB A(a x )
Putting the value of A,
e
e e
(a x )B
x (2a x )
Putting the values of A and B in equation (4),
e
e e e e e e e e e e
(a x )1 1
(x x)(ax ax xx ) x (2a x )(x x) x (2a x )(ax ax xx )
Hence equation (3) becomes
e 12
e e e e e e e e
(a x )dx kdxdt
x (2a x )(x x) x (2a x )(ax ax xx ) x
e 12
e e e e e e
(a x )dx k1 dxdt
x (2a x ) (x x) (ax ax xx ) x
1e e e 2e e e
k1ln(x x) ln(ax ax xx ) t c
x (2a x ) x
e e 12
e e e e
(ax ax xx ) k1ln t c
x (2a x ) (x x) x
(6)
When t =0, x = 0, putting these values in above equation, the
value of c comes out to be,
e e 12
e e e e
(ax a0 0x ) k1ln 0 c
x (2a x ) (x 0) x
e e
1c ln a
x (2a x )
Putting the value of c in equation (6),
e e 12
e e e e e e
(ax ax xx ) k1 1ln t ln a
x (2a x ) (x x) x x (2a x )
-
28
e e 12
e e e e e e
(ax ax xx ) k1 1ln ln a t
x (2a x ) (x x) x (2a x ) x
e e 12
e e e e
(ax ax xx ) k1ln t
x (2a x ) (x x)a x
e e e1
e e
x ax x(a x )k ln
t(2a x ) (x x)a
(7)
This is the integrated rate equation for opposing reaction of
the type first order opposed by
second order.
c) Second order opposed by first order reaction
In such reactions, the forward reaction is second order and
backward reaction is first
order. Consider the following reaction,
A + B Ck1
k-1
When t = 0 a a 0
When t = t ax ax x
When t = teq axe axe xe
Rate equation for such a reaction is given by
21 1
dxk (a x) k x
dt (1)
If xe is the equilibrium concentration then at the state of
dynamic equilibrium,
Rf = Rb and x = xe
21 e 1 ek (a x ) k x
2
1 e1
e
k (a x )k
x
(2)
Putting the value of k-1 from equation (2) in equation (1),
2
2 1 e1
e
k (a x )dxk (a x) x
dt x
2 2
1 e 1 e
e
k (a x) x k (a x ) xdx
dt x
2 21e e
e
kdx(a x) x (a x ) x
dt x
2 2 2 21e e e
e
kdx(a x 2ax)x (a x 2ax )x
dt x
-
29
2 2 2 21e e e e e
e
kdxa x x x 2axx (a x x x 2ax x)
dt x
2 2 2 21e e e e e
e
kdxa x x x 2axx a x x x 2ax x
dt x
2 2 2 21e e e
e
kdxa x x x a x x x
dt x
2 2 2 21e e e
e
kdxa x a x x x x x
dt x
21e e e
e
kdxa (x x) xx (x x )
dt x
21e e e
e
kdxa (x x) xx (x x)
dt x
21e e
e
kdx(x x)(a xx )
dt x
12
e e e
k dtdx
(x x)(a xx ) x
Integrating the above equation,
12
e e e
kdxdt
(x x)(a xx ) x
(3)
Using partial fraction method, we get
2 2
e e e e
1 A B
(x x)(a xx ) (x x) (a xx )
(4)
2e e1 A(a xx ) B(x x) (5)
For calculating the value of A, put (xe x) = 0 or x = xe that
is
2 e e e e1 A(a x x ) B(x x )
2 2e1 A(a x )
2 2
e
1A
(a x )
For calculating the value of B, consider equation (5),
2 e e1 A(a xx ) B(x x)
2 e e1 Aa Axx Bx Bx
Comparing coefficients of x, we get
-
30
e0 Ax B
eB Ax
Putting the value of A,
e2 2
e
xB
(a x )
Putting the values of A and B in equation (4),
e2 2 2 2 2 2
e e e e e e
x1 1
(x x)(a xx ) (a x )(x x) (a x )(a xx )
Hence equation (3) becomes
e 12 2 2 2 2
e e e e e
x dx kdxdt
(a x )(x x) (a x )(a xx ) x
e 12 2 2
e e e e
x dx k1 dxdt
(a x ) (x x) (a xx ) x
2 1e e2 2e e
k1ln(x x) ln(a xx ) t c
(a x ) x
2
e 1
2 2
e e e
(a xx ) k1ln t c
(a x ) (x x) x
(6)
When t =0, x = 0, therefore the value of c comes out to be,
2
e 1
2 2
e e e
(a 0x ) k1ln 0 c
(a x ) (x 0) x
2
2 2
e e
1 aln c
(a x ) x
Putting the value of c in equation (6),
2 2
e 1
2 2 2 2
e e e e e
(a xx ) k1 1 aln t ln
(a x ) (x x) x (a x ) x
2 2
e 1
2 2 2 2
e e e e e
(a xx ) k1 1 aln ln t
(a x ) (x x) (a x ) x x
2
e e 1
2 2 2
e e e
(a xx )x k1ln t
(a x ) (x x)a x
2
e e e1 2 2 2
e e
x (a xx )xk ln
t(a x ) (x x)a
(7)
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31
This is the integrated rate equation for opposing reaction of
the type second order opposed by
first order.
d) Second order opposed by second order reaction
In such reactions, the forward reaction is second order and
backward reaction is also
second order. Consider the following reaction,
A + Bk1
k-1
C + D
When t = 0 a a 0 0
When t = t ax ax x x
When t = teq axe axe xe xe
Rate equation for such a reaction is given by
2 21 1
dxk (a x) k x
dt (1)
If xe is the equilibrium concentration then at the state of
dynamic equilibrium,
Rf = Rb and x = xe
2 21 e 1 ek (a x ) k x
2
1 e1 2
e
k (a x )k
x
(2)
Putting the value of k-1 from equation (2) in equation (1),
2
2 21 e1 2
e
k (a x )dxk (a x) x
dt x
2 2 2 2
1 e 1 e
2
e
k (a x) x k (a x ) xdx
dt x
2 2 2 21e e2
e
kdx(a x) x (a x ) x
dt x
2 2 2 2 2 21e e e2
e
kdx(a x 2ax)x (a x 2ax )x
dt x
2 2 2 2 2 2 2 2 2 21 e e e e e2e
kdxa x x x 2axx a x x x 2ax x
dt x
2 2 2 2 2 21e e e2
e
kdxa x 2axx a x 2ax x
dt x
2 2 2 2 2 21e e e2
e
kdxa x a x 2axx 2ax x
dt x
-
32
2 2 21e e e2
e
kdxa (x x ) 2axx (x x)
dt x
21e e e e2
e
kdxa (x x)(x x) 2axx (x x)
dt x
21 e e e2e
kdx(x x) a (x x) 2axx
dt x
1
22ee e e
kdxdt
x(x x) a (x x) 2axx
Integrating the above equation,
1
22ee e e
kdxdt
x(x x) a (x x) 2axx
(3)
Using partial fraction method we get
22 e e ee e e
1 A B
(x x) a (x x) 2axx(x x) a (x x) 2axx
(4)
2 e e e1 A a (x x) 2axx B(x x) (5)
For calculating the value of A, put (xe x) = 0 or x = xe that
is
2 e e e e e e1 A a (x x ) 2ax x B(x x )
2 2e e1 A(2a x 2ax )
e e
1A
2ax (a x )
For calculating the value of B, consider equation (5),
2 e e e1 A a (x x) 2axx B(x x)
2 2
e e e1 Aa x Aa x 2Aaxx Bx Bx
Comparing coefficients of x, we get
2e0 Aa 2Aax B
2 eB A(a 2ax )
Putting the value of A,
2
e
e e
(a 2ax )B
2ax (a x )
Putting the values of A and B in equation (4),
-
33
2
e
22e e e e e e ee e e
(a 2ax )1 1
2ax (a x )(x x) [2ax (a x )][a (x x) 2axx ](x x) a (x x)
2axx
Hence equation (3) becomes
2
e 1
2 2 2
e e e e e e
(a 2ax )dx k1 dxdt
2ax (a x ) (x x) a x a x 2axx x
2 2 1e e e 2e e e
k1ln(x x) ln(a x a x 2axx ) t c
2ax (a x ) x
2 2
e e 1
2
e e e e
(a x a x 2axx ) k1ln t c
2ax (a x ) (x x) x
(6)
When t =0, x = 0, therefore the value of c comes out to be,
2 2
e e 1
2
e e e e
(a x a 0 2a0x ) k1ln 0 c
2ax (a x ) (x 0) x
2
e
e e e
a x1ln c
2ax (a x ) x
2
e e
1c ln a
2ax (a x )
Putting the value of c in equation (6),
2 2
2e e 1
2
e e e e e e
(a x a x 2axx ) k1 1ln t ln a
2ax (a x ) (x x) x 2ax (a x )
2 2
2e e 1
2
e e e e e e
(a x a x 2axx ) k1 1ln ln a t
2ax (a x ) (x x) 2ax (a x ) x
2 2
e e 1
2 2
e e e e
(a x a x 2axx ) k1ln t
2ax (a x ) (x x)a x
2
e e e1 2
e e e
x a(ax ax 2xx )k ln
2atx (a x ) (x x)a
e e e1e e
x ax ax 2xxk ln
2at(a x ) (x x)a
e e e1e e
x x(a 2x ) axk ln
2at(a x ) (x x)a
(7)
This is the integrated rate equation for opposing reaction of
the type second order opposed by
second order.
-
34
4.10.2 CONSECUTIVE OR SEQUENTIAL REACTIONS
The reactions which proceed from reactants to products in one or
more steps
through intermediates are known as consecutive reactions. Each
step involved here has its
own rate constant. Consider the simplest consecutive reaction as
follows,
k2A B C
k1
When t = 0 a 0 0
When t = t [A] [B] [C]
According to law of mass action, rate of decomposition of A is
given by,
1
d[A]k [A]
dt
Integrating this equation after separating variables,
1d[A]
k dt[A]
1ln[A] k t c (1)
When t =0, [A] = a, therefore the value of c comes out to
be,
1ln a k 0 c
ln a c
Putting the value of c in equation (1),
1ln[A] k t ln a
1ln[A] ln a k t
1
[A]ln k t
a
Taking antilog on both sides,
1k t[A]
ea
1k t[A] ae (2)
Now rate of formation of B,
1 2
d[B]k [A] k [B]
dt
2 1d[B]
k [B] k [A]dt
1k t2 1
d[B]k [B] k ae
dt
(By equation (2))
-
35
Multiplying both sides by 2k te
2 1 2k t k t k t2 1
d[B]k [B] e (k ae )e
dt
2 2 2 1k t k t (k k )t2 1
d[B]e [B]k e k ae
dt
2 2 1k t (k k )t1d
[B]e k aedt
2 2 1k t (k k )t1d [B]e k ae dt
2 2 1k t (k k )t1d [B]e k a e dt
2 1
2
(k k )tk t
1
2 1
e[B]e k a D
k k
(3)
When t =0, [B] = 0, therefore the value of D comes out to
be,
2 1
2
(k k )0k 0
1
2 1
e[0]e k a D
k k
1
2 1
k aD
k k
Putting the value of D in equation (3),
2 1
2
(k k )tk t 1
1
2 1 2 1
k ae[B]e k a
k k k k
2 2 1k t (k k )t12 1
k a[B]e e 1
k k
2 2 1k t k t k t12 1
k a[B]e e e 1
k k
2
1
2 2
k tk t1
k t k t
2 1
k a e 1[B] e
k k e e
1 2k t k t12 1
k a[B] e e
k k
(4)
The concentration of C can be found as,
a [A] [B] [C]
[C] a [A] [B]
1 1 2k t k t k t12 1
k a[C] a ae e e
k k
-
36
1 2
1
k t k tk t 1 1
2 1
(k ae k ae )[C] a ae
k k
1 1 2k t k t k t
2 1 2 1 1 1
2 1
a(k k ) ae (k k ) k ae k ae[C]
k k
1 1 1 2k t k t k t k t
2 1 2 1 1 1
2 1
a(k k ) k ae k ae k ae k ae[C]
k k
1 2k t k t
2 1 2 1
2 1
a(k k ) a(k e k e )[C]
k k
1 2k t k t
2 1
2 1
(k e k e )[C] a 1
k k
(5)
Radioactive decay is an example of such a process. Other
examples include chemical
processes such as polymerization, thermal cracking and
chlorination of hydrocarbons.
4.10.3 SIMULTANEOUS OR PARALLEL OR SIDE REACTIONS
The reactions in which reactants undergo two or more independent
reactions
simultaneously are called as parallel reactions. These are also
known as side reactions.
The reaction in which maximum yield of the product is obtained
is called main or major
reaction while the others are called side reactions. Consider a
general parallel reaction,
A
B
C
k1
k2when t = 0 awhen t = t a - x
0 at t = 0x at t = t
0 at t = 0x at t = t
Rate equation for such a reaction is given by,
1 2
dxk (a x) k (a x)
dt
1 2
dx(k k )(a x)
dt
Integrating the above equation after separating variables,
1 2dx
(k k ) dt(a x)
1 2ln(a x) (k k )t c (1)
When t =0, x =0, therefore the value of c comes out to be,
1 2ln(a 0) (k k )0 c
-
37
ln a c
Putting the value of c in equation (1),
1 2ln(a x) (k k )t ln a
1 2(k k )t ln a ln(a x)
1 2
1 a(k k ) ln
t a x
(2)
The equation (2) is the rate constant expression for a parallel
reaction. It is similar to the
equation of a first order reaction, (k1 + k2) being the sum of
the specific rates of the two
simultaneous reactions. If k1 > k2 then AB is main reaction
and AC is side reaction.
Graphically this can be represented as shown in figure (4.1).
For example, nitration of
phenols gives ortho-nitrophenol and para-nitrophenol in two
simultaneous side reactions
Figure (4.2).
Fig(4.1): Graphical representation Fig(4.2): Simultaneous side
reactions
4.10.4 CHAIN REACTIONS
The chemical reactions which take place in a series of
successive processes
involving the formation of free atoms and radicals are known as
chain reactions. The
kinetic laws for such reactions are considerably complex. A
well-known example of these
types of reactions is the hydrogenbromine reaction. Chain
reactions usually proceed very
rapidly. Many explosive reactions occur by a chain mechanism;
atomic fission and atomic
fusion are of this type of reactions.
4.11 THERMAL REACTIONS
A well known example of a thermal reaction is the hydrogen
bromine reaction. The
stoichiometry for this reaction is,
H2 + Br2 2HBr
But it does not follow the simple rate expression
2 2d[HBr]
k[H ][Br ]dt
-
38
It follows the following empirical rate equation given by
Bodenstein and Lind in 1906:
1/2
2 2
2
k[Br ] [H ]d[HBr]
[HBr]dt1 k
[Br ]
The complexity of this equation could not be explained until
1919 when Christiansen,
Herzfeld and Polanyi independently and almost simultaneously
solved the problem. They
proposed a chain of reactions with the following steps,
(1) k1
BrBr2 2 Chain initiation
(2) k2
Br + H2 HHBr + Chain propagation
(3) k3
+ Br2 HBr +H Br Chain propagation
(4) k4
BrH2 +H + HBr Chain inhibition
(5) k5
Br Br22 Chain termination
According to this mechanism HBr is formed in reactions (2) and
(3) and removed in reaction
(4). Consequently, the rate of formation of HBr is given by
2 2 3 2 4d[HBr]
k [Br ][H ] k [H ][Br ] k [H ][HBr]dt
2 2 3 2 4d[HBr]
k [Br ][H ] k [Br ] k [HBr] [H ]dt
(1)
The concentration of [Br ] and [H ] can be determined using
steady state approximation.
The steady state approximation is also occasionally called
stationary state approximation.
This approximation assumes that the concentration of reaction
intermediates remains constant
throughout the reaction after an initial buildup. This
approximation can only be applied to
short lived or very reactive species. Therefore mathematically
we can write,
21 2 2 2 3 2 4 5d[Br ]
k [Br ] k [Br ][H ] k [H ][Br ] k [H ][HBr] k [Br ] 0dt
(2)
2 2 3 2 4d[H ]
k [Br ][H ] k [H ][Br ] k [H ][HBr] 0dt
(3)
Solving equation (3),
2 2 3 2 4k [Br ][H ] k [H ][Br ] k [H ][HBr] 0
-
39
2 2 3 2 4k [Br ][H ] k [Br ] k [HBr] [H ] 0
3 2 4 2 2k [Br ] k [HBr] [H ] k [Br ][H ] (4)
2 2
3 2 4
k [Br ][H ][H ]
k [Br ] k [HBr]
(5)
Now solve equation (2),
21 2 2 2 3 2 4 5k [Br ] k [Br ][H ] k [H ][Br ] k [H ][HBr] k
[Br ] 0
21 2 2 2 3 2 4 5k [Br ] k [Br ][H ] k [Br ] k [HBr] [H ] k [Br ]
0
21 2 2 2 2 2 5k [Br ] k [Br ][H ] k [Br ][H ] k [Br ] 0
(By equation (4))
21 2 5k [Br ] k [Br ] 0
25 1 2k [Br ] k [Br ]
1/2
1 2
5
k [Br ][Br ]
k
(6)
Now putting the value of [Br ] from equation (6) in equation
(5),
1/2
1 22 2
5
3 2 4
k [Br ]k [H ]
k[H ]
k [Br ] k [HBr]
(7)
Putting the values of [Br ] and [H ] from Eq.(6) and Eq.(7) in
Eq.(1) we get
2 2 3 2 4d[HBr]
k [Br ][H ] k [Br ] k [HBr] [H ]dt
1/2
1 21/2 2 2
51 22 2 3 2 4
5 3 2 4
k [Br ]k [H ]
kk [Br ]d[HBr]k [H ] k [Br ] k [HBr]
dt k k [Br ] k [HBr]
1/2
3 2 41 22 2
5 3 2 4
k [Br ] k [HBr]k [Br ]d[HBr]k [H ] 1
dt k k [Br ] k [HBr]
1/2
3 2 4 3 2 41 22 2
5 3 2 4
k [Br ] k [HBr] k [Br ] k [HBr]k [Br ]d[HBr]k [H ]
dt k k [Br ] k [HBr]
1
5
1/2k
2 2 2 3 2k
3 2 4
k [Br ] [H ]2k [Br ]d[HBr]
dt k [Br ] k [HBr]
-
40
1
5
1/2k 1/2
2 2 2k
4
3 2
2k [Br ] [H ]d[HBr]
k [HBr]dt1
k [Br ]
1/2
2 2
2
k[Br ] [H ]d[HBr]
[HBr]dt1 k
[Br ]
(8)
Where 15
k
2 kk 2k and 4
3
k
kk are constants. This equation (8) is in agreement with the
rate
equation given by Bodenstein and Lind. The appearance of the
term [HBr] in the
denominator implies that the velocity of the reaction is
decreased by the product HBr and this
product acts an inhibitor of the reaction. This is an example of
self-inhibition.
4.12 PHOTOCHEMICAL REACTIONS
Photochemical reaction is a chemical reaction initiated by the
absorption of
energy in the form of light. In photochemical reactions free
radicals are produced which
initiate the chain reactions. An example is the photosynthesis
of HCl gas. All photochemical
reactions are of zero order. These reactions are independent of
concentration of reactants.
Change in concentration of reactants has no effect on rate of
photochemical reactions.
a) HydrogenBromine Reaction
Consider the following photochemical reaction in which H2 and
Br2 are going to
produce HBr,
H2 + Br2 2HBrh
The mechanism of this reaction consists of following five
elementary steps,
(1) Br2 + h 2k1
Br Chain initiation
(2) k2
Br + H2 HHBr + Chain propagation
(3) k3
+ Br2 HBr +H Br Chain propagation
(4) k4
BrH2 +H + HBr Chain inhibition
(5) k5
Br Br22 Chain termination
According to this mechanism HBr is formed in reactions (2) and
(3) and removed in reaction
(4). Consequently, the rate of formation of HBr is given by
-
41
2 2 3 2 4d[HBr]
k [Br ][H ] k [H ][Br ] k [H ][HBr]dt
2 2 3 2 4d[HBr]
k [Br ][H ] k [Br ] k [HBr] [H ]dt
(1)
The concentration of [Br ] and [H ] can be determined using
steady state approximation. This
approximation assumed that at steady state the rate of formation
of reaction intermediates can
be considered to be equal to their rate of disappearance.
Mathematically we can write,
21 a 2 2 3 2 4 5d[Br ]
k I k [Br ][H ] k [H ][Br ] k [H ][HBr] k [Br ] 0dt
(2)
2 2 3 2 4d[H ]
k [Br ][H ] k [H ][Br ] k [H ][HBr] 0dt
(3)
Solving equation (3),
2 2 3 2 4k [Br ][H ] k [H ][Br ] k [H ][HBr] 0
2 2 3 2 4k [Br ][H ] k [Br ] k [HBr] [H ] 0
3 2 4 2 2k [Br ] k [HBr] [H ] k [Br ][H ] (4)
2 2
3 2 4
k [Br ][H ][H ]
k [Br ] k [HBr]
(5)
Now solve equation (2),
21 a 2 2 3 2 4 5k I k [Br ][H ] k [H ][Br ] k [H ][HBr] k [Br ]
0
21 a 2 2 3 2 4 5k I k [Br ][H ] k [Br ] k [HBr] [H ] k [Br ]
0
21 a 2 2 2 2 5k I k [Br ][H ] k [Br ][H ] k [Br ] 0
(By equation (4))
21 a 5k I k [Br ] 0
25 1 ak [Br ] k I
1/2
1a
5
k[Br ] I
k
(6)
Now putting the value of [Br ] from equation (6) in equation
(5),
1/2
12 a 2
5
3 2 4
kk I [H ]
k[H ]
k [Br ] k [HBr]
(7)
Putting the values of [Br ] and [H ] from Eq.(6) and Eq.(7) in
Eq.(1) we get
-
42
2 2 3 2 4d[HBr]
k [Br ][H ] k [Br ] k [HBr] [H ]dt
1/2
11/2 2 a 2
512 a 2 3 2 4
5 3 2 4
kk I [H ]
kkd[HBr]k I [H ] k [Br ] k [HBr]
dt k k [Br ] k [HBr]
1/2
3 2 412 a 2
5 3 2 4
k [Br ] k [HBr]kd[HBr]k I [H ] 1
dt k k [Br ] k [HBr]
1/2
3 2 4 3 2 412 a 2
5 3 2 4
k [Br ] k [HBr] k [Br ] k [HBr]kd[HBr]k I [H ]
dt k k [Br ] k [HBr]
1
5
1/2k
2 a 2 3 2k
3 2 4
k I [H ]2k [Br ]d[HBr]
dt k [Br ] k [HBr]
1
5
1/2k 1/2
2 a 2k
4
3 2
2k I [H ]d[HBr]
k [HBr]dt1
k [Br ]
1/2
a 2
2
kI [H ]d[HBr]
[HBr]dt1 k
[Br ]
(8)
Where 15
k
2 kk 2k and 4
3
k
kk are constants. The equation (8) is the final expression.
b) HydrogenChlorine Reaction
The chemical reaction between hydrogen and chlorine in the
presence of light
produces hydrogen chloride with explosion. In 1930 Bodenstein
and Hanger investigated the
kinetics of this photochemical reaction. The rate law expression
for the reaction on the basis
of their experimental observations is given below,
H2 + Cl2 2HClh
a 2d[HCl]
kI [H ]dt
Where k is constant and Ia is the intensity of absorbed light
(expressed in Einstein dm-3
sec-1
).
They proposed the following mechanism for the reaction,
(1) Cl2 + h 2k1
Cl Initiation step
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43
(2) k2
Cl + H2 HHCl + Propagation step
(3) k3
+ Cl2 HCl +H Cl Propagation step
(4) k4
Cl Cl21/2 Termination step
HCl is more stable than HBr. Bond between HCl is strong so [H ]
radical do not react with
HCl molecule as in case of HBr. The net rate of reaction in
terms of rate of formation of HCl
according to above mechanism can be written as,
2 2 3 2d[HCl]
k [Cl ][H ] k [H ][Cl ]dt
(1)
The concentration of [Cl ] and [H ] can be determined applying
steady state approximation.
This approximation assumed that at steady state the net change
in concentration of reaction
intermediates is zero. Mathematically,
1 a 2 2 3 2 4d[Cl ]
k I k [Cl ][H ] k [H ][Cl ] k [Cl ] 0dt
(2)
2 2 3 2d[H ]
k [Cl ][H ] k [H ][Cl ] 0dt
(3)
Solving equation (3),
2 2 3 2k [Cl ][H ] k [H ][Cl ] 0
2 2 3 2k [Cl ][H ] k [H ][Cl ]
(4)
2 2
3 2
k [Cl ][H ][H ]
k [Cl ]
(5)
Now solve equation (2),
1 a 2 2 3 2 4k I k [Cl ][H ] k [H ][Cl ] k [Cl ] 0
1 a 3 2 3 2 4k I k [H ][Cl ] k [H ][Cl ] k [Cl ] 0
(By equation (4))
1 a 4k I k [Cl ] 0
1 a
4
k I[Cl ]
k
(6)
Now putting the value of [Cl ] from equation (6) in equation
(5),
2 1 a 2
3 4 2
k k I [H ][H ]
k k [Cl ]
(7)
-
44
Putting the values of [Cl ] and [H ] from Eq.(6) and Eq.(7) in
Eq.(1) we get
2 2 3 2
d[HCl]k [Cl ][H ] k [H ][Cl ]
dt
1 a 2 1 a 22 2 3 2
4 3 4 2
k I k k I [H ]d[HCl]k [H ] k [Cl ]
dt k k k [Cl ]
2 1 a 2 1 a 22
4 4
k k I k k I [H ]d[HCl][H ]
dt k k
2 1a 2
4
2k kd[HCl]I [H ]
dt k
a 2
d[HCl]kI [H ]
dt (8)
Where 2 1 4k 2k k / k is a constant. The expression (8) is same
as was given by Bodenstein
and Hanger.
4.13 EFFECT OF TEMPERATURE ON RATE OF REACTION
Arrhenius Equation (1889)
It is well established that the rate of reaction increases to an
appreciable extent with
the rise in temperature. Experimentally, it has been found that
for a 10 C rise in temperature,
the velocity of the reaction is doubled or trebled.
The first empirical equation to show the variation of rate
constant k of a reaction with
temperature T was suggested by Hood. His equation is,
B
log k AT
(1)
Where A and B are positive empirical constants and T is the
absolute temperature. The
verification of this equation lies in the fact that a plot of
log k versus 1/T is nearly a straight
line with negative slope for most of the reactions. This
equation was theoretically justified by
Vant Hoff in 1884. His arguments were based on the variation of
equilibrium constant with
temperature.
Svante Arrhenius extended his idea and suggested a similar
equation to show the
variation of rate constant with temperature in the year 1889.
The differential form of
Arrhenius equation is given by,
a2
Ed ln k
dT RT (2)
Where, Ea = Energy of activation, T = Absolute temperature and R
= Gas constant.
Integrating equation (2) we get,
-
45
a2
E dTd ln k
R T
aE
ln k ln A(constant)RT
aE
ln k ln ART
(3)
aEln k ln ART
aEk
lnA RT
Taking antilog on both sides,
aE
RTk
eA
aE
RTk Ae
(4)
The constants A and Ea are related to Aand B in equation (1).
The factor A is known as
Arrhenius pre-exponential factor, also called frequency factor,
have same units as that of
specific rate constant. The frequency factor is related to
collision frequency and steric factor.
The steric (or orientation) effect, also called probability
factor, is the fraction of collisions in
which the molecules have proper orientation favourable for
reaction. Equation (3) and
equation (4) are the alternate forms of the Arrhenius equation.
Arrhenius equation shows that
the value of k for a reaction is directly related to A. The
value of A increases if collision
frequency or steric factor increases.
The second factor in Arrhenius equation Ea/RT depends on the
value of Ea. This term
usually has a positive value because Ea is usually positive.
Since this term is subtracted from
lnA in equation (3), we see that the value of k decreases as the
value of Ea/RT increases. In
other words, an increase in the value of Ea decreases the rate
of the reaction. Conversely, an
increase in temperature (which is in the denominator) reduces
the value of the term Ea/RT
and thus increases the value of k. This increase is consistent
with the observation that the
rates of almost all reactions increase as the temperature rises
at the same concentration.
Rewriting equation (3) as follows,
aE
log k log A2.303RT
(5)
This equation is identical with the empirical equation (1). To
test the validity of this equation;
a plot of log k against 1/T should be a straight line having
slope Ea/2.303R and intercept logA
-
46
as shown in figure. Knowing the slope, the value of constant Ea
which is the characteristic
of the reaction can be calculated.
Alternatively, Ea can also be determined if the rate constants
are known at two
different temperatures. For this purpose Arrhenius equation (2)
can be integrated between the
limits k1 (the rate constant at T1) and k2 (the rate constant at
T2),
2 2
1 1
k T
a
2
k T
E dTd ln k
R T
22
1 1
Tk a 1Tk T
Eln k
R
a2 12 1
E 1 1ln k ln k
R T T
a2 2 1
1 1 2
Ek T Tln
k R T T
a2 2 1
1 1 2
Ek T Tln
k 2.303R T T
(6)
Equation (6) is another form of Arrhenius equation that is
useful for calculating the value of
activation energy of a reaction when the value of rate constant
at two different temperatures
is known for that reaction. The Arrhenius equation correlates
rate constant with temperature.
It gives the idea of energy of activation.
Energy of Activation and Activated Complex
The concept of energy of activation was developed by Arrhenius
in 1888 which
constitute the backbone of all the modern theories. According to
Arrhenius, the molecules
must acquire a discrete minimum amount of energy before the end
products are formed.
These activated molecules will then collide and lead to the
reaction. Collisions between
molecules which are not activated will be of no use and no
reaction will take place. Thus the
reactants must pass through an energy rich or activated state
before they can react. The
-
47
minimum amount of energy required by the molecules to overcome
the activated state
or energy barrier before the reaction takes place is known as
energy of activation.
It follows from the concept of activation that the reactants are
not directly converted
into products. The molecules first acquire the energy to form an
activated complex and this
activated complex is then decomposed into products.
Reactants Activated Complex Products
In other words, there exists an energy barrier between the
reactants and the products. If the
reactant molecules can cross this energy barrier, they will be
converted into products.
The energy of activation is usually expressed in J mol-1
and kJ mol-1
and is denoted by
the symbol Ea. The relationship between enthalpy of reaction and
energy of activation for an
exothermic reaction and an endothermic reaction is shown in
figure (4.3) and (4.4).
Fig (4.3): Exothermic Reaction Fig (4.4): Endothermic
Reaction
In an exothermic reaction products are at lower energy level
than the reactants and in
an endothermic reaction the products are at higher energy level
than the reactants. With both
types of reactions the activation energy is an energy barrier
which must be overcome before
products are formed. Surmounting this barrier is similar to
carrying a ball to the top of a hill
and then rolling down the either side. But if the ball cannot be
carried to the top, it will roll
back. Similarly if Ea is not supplied, the reaction will not
start and the reactants will never get
converted into the products.
According to the arguments given above, a general reaction of
the type
A2 + B
2 2AB
A2 + B
2 2AB
takes place as follows;
It must be kept in mind that the energy of activation is always
positive whether the reaction is
exothermic or endothermic.
-
48
4.14 THEORIES OF REACTION RATES
There are two main theories of reaction rates,
1) Collision theory
2) Transition state theory or activated complex theory
4.14.1 Collision Theory
This theory is based upon the kinetic theory of gases according
to which the
molecules of a gas are continuously moving and hence colliding
with each other. Since for a
collision to takes place, at least two molecules must be
involved, therefore the simple case to
consider is that of bimolecular reactions. Hence in discussion
that follows, we shall first
discuss the collision theory for bimolecular reactions and then
for unimolecular reactions.
1) Collision Theory for Bimolecular Reactions
Bimolecular collision theory for the reactions consisting of two
reactant molecules
was proposed independently by Max Trautz in 1916 and William
Lewis in 1918. This
theory is based on following assumptions:
1. For a chemical reaction to takes place reactants must collide
with each other.
2. Not all the collisions occurring lead to reaction but only
those which form activated
molecules.
3. The molecules must be suitably oriented at the time of
collisions.
Consider a gasphase bimolecular elementary reaction,
A + A Products
A + A Products e.g.,
2HI H2 + I
2
2HI H2 + I
2
A + B Products
A + B Products e.g.,
H2 + I
2 2HI
H2 + I
2 2HI
From reaction kinetics, the rate of reaction is given by,
2 2ARate k[A] kn (1)
A BRate k[A][B] kn n (2)
Where nA and nB are the number of molecules per dm3 of the
reactants.
The rate of reaction can also be written according to collision
theory as follows,
Rate Z q (3)
Where Z is number of binary collisions per second per dm3 of the
reaction mixture and q is
fraction of activated molecules.
From kinetic theory of gases, number of collisions per second
per dm3 between like
molecules of a gas in the reaction mixture is given by,
2 2AA A A1
Z n u2
(4)
-
49
Where A is the collision diameter of the reactant molecules, nA
is the number of molecules
per dm3 of reactants, u is the average velocity of molecules.
Now the value of u is equal to,
8RT
uM
Where R is the general gas constant, T is absolute temperature
and M is the molar mass.
Putting this value in equation (4),
2 2AA A A
1 8RTZ n
M2
2 2
AA A A
RTZ 2 n
M
(5)
When the two molecules are not alike then collision frequency
(collision number) is given by,
2AB A B AB
8 RTZ n n
(6)
Where A BAB
2
is the mean collision diameter of the molecules; A B
A B
M M
M M
is the
reduced mass; nA and nB are the number of molecules per dm3 of
the reactants; MA and MB
are the molecular weights of two gaseous reactants.
The fraction of molecules q having the necessary energy of
activation is given by
Boltzmann law as follows,
aE /RTn
q en
(7)
Where n is the number of activated molecules having energy equal
to or greater than the
energy of activation Ea, and n is the total number of molecules
per dm3.
Thus the expression for the rate of reaction involving identical
molecules is given by,
aE /RT2 2
AA A A
RTRate Z q 2 n e
M
(8) (By Eq. (3), (5) & (7))
And for different reactant molecules
aE /RT2AB A B AB8 RT
Rate Z q n n e
(9) (By Eq. (3), (6) & (7))
Hence the rate constant expressions for bimolecular reactions
comes out to be,
For like molecules, from equation (1) and (8)
aE /RT2 2 2
A A A
RTkn 2 n e
M
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50
aE /RT2
A
RTk 2 e
M
(10)
For unlike molecules, from equation (2) and (9)
aE /RT2A B A B AB
8 RTkn n n n e
aE /RT2AB
8 RTk e
(11)
Now the rate constant estimated by Arrhenius equation is equal
to,
aE
RTk Ae
(12)
Comparing Eq.(10) and Eq.(11) with Eq.(12) we find that the
frequency factor A in
Arrhenius equation is proportional to ZAA or ZAB and depends on
the square root of
temperature. Hence the expression for the rate constant
becomes,
aE
RTk Ze
(13)
Where 2
AZ 2 RT / M (for like molecules) and 2
ABZ 8 RT / (for unlike
molecules). It has been found experimentally that for reactions
between simple molecules Z
agrees well with the Arrhenius factor A. Therefore, collision
theory successfully accounts for
the rates of simple reactions but in many cases the rates differ
considerably. For this reason
the equation is further modified as
aE
RTk PZe
(14)
Where, P is called the probability factor or steric factor. The
factor P is a measure of the
geometrical requirements that must be met when activated
colliding molecules are
interacting. This factor, the steric factor, limits the
successful collisions to the ones in which
molecules are favourably oriented. In other words, P is the
fraction of collisions in which the
molecules have an orientation favourable for reaction. Equation
(14) is the equation of the
collision theory.
Limitations of Collision theory
The collision theory in the form of equation (14) is not
applicable to the following cases,
1. When the complex molecules are involved in the chemical
reactions.
2. When the chain mechanism is to be obeyed by the chemical
reactions.
3. When the reaction is surface catalyzed.
It is applicable only to gas phase reactions and cannot be
applied to kinetic studies in
solutions. It does not give any explanation for abnormally high
rates of reactions.
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51
2) Collision Theory for Unimolecular Reactions (Lindemanns
Mechanism)
In unimolecular reactions, only one molecule takes part in the
reaction so
consequently the question arises: How do molecules in
unimolecular reactions attain their
energy of activation? If the energy of activation is attained in
a bimolecular collision, then the
reaction should be of second order. But the fact is that the
unimolecular reactions are first
order except in gas phase at low pressure. Explanation of how a
bimolecular collision can
give rise to first order kinetics was given by F. A. Lindemann
in 1922.
Lindemann pointed out that the reacting molecules acquire
activation energy through
collisions with other molecules and as a result some molecules
are activated. Further he
postulated that the activated molecules do not decompose
immediately, but remain in the
activated form for a definite period. Meanwhile the added excess
energy may increase the
amplitude of vibration and thus can lead to rupture the bond, or
may be robbed by a less
energetic molecule in the ensuing collision. These possibilities
can be represented by a
mechanism consisting of following series of elementary
steps,
A + Ak1 A + A*(1) (Activation)
A* + Ak2 A + A(2) (Dectivation)
A*k3
Products(3) (Decomposition)
Where, A represents a normal molecule and A
* an activated molecule. k1, k2 and k3 are the
rate constants of three steps. According to this mechanism rate
of formation of product is
given by,
*3
dxk [A ]
dt (1)
The concentration of A*, the reaction intermediate, can be
determined by using steady state
approximation. That is
*2 * *
1 2 3
d[A ]k [A] k [A ][A] k [A ] 0
dt
* * 2
2 3 1k [A ][A] k [A ] k [A]
* 2
2 3 1[A ](k [A] k ) k [A]
2
* 1
2 3
k [A][A ]
k [A] k
(2)
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52
Purring the value of A* from equation (2) in equation (1), we
get
2
3 1
2 3
k k [A]dx
dt k [A] k
(3)
Equation (3) predicts two limiting possibilities for gaseous
reactions.
Case (1): At high pressure the concentration of reacting
molecules is relatively high. The
activated molecules will have a greater chance of colliding with
inactive molecules and thus
become deactivated. Under such conditions, the rate of
bimolecular deactivation will be much
faster than that of unimolecular decomposition then k2 [A]
>> k3 and equation (3) reduces to
2
3 1 3 1
2 2
k k [A] k k [A]dxk[A]
dt k [A] k
Where, 3 1 2k k k / k . The equation predicts a first order
reaction.
Case (2): At low pressure the concentration of A decreases and
hence the number of
collisions decreases. Consequently, the rate of decomposition
will be much faster than its rate
of deactivation. Under such conditions k3 >> k2 [A] and
equation (3) reduces to
223 1
1
3
k k [A]dxk [A]
dt k
i.e., the reaction should be second order. Therefore it is the
pressure which governs the order
of reaction. This conclusion was experimentally verified by
Hinshelwood.
Examples:
Such changes from first order kinetics in gaseous reactions at
higher pressures to
second order at lower pressures actually have been observed in
many gaseous reactions. For
example, the thermal decomposition of nitrogen pentoxide and of
azomethane at high
pressure are the first order reactions,
N2O5 N2O4 +
1/2 O2
(CH3)2N2 C2H6 + N2
whereas at very low pressures these reactions become second
order. Such observations
confirm the Lindemanns mechanism.
4.14.2 Transition State Theory
This theory was first formulated in 1935 by Henry Eyring who was
awarded Nobel
prize for this valuable work. This theory is also known as
absolute rate theory because with
the help of it is possible to get the absolute value of the rate
constant. The main postulates of
this theory are:
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53
1. All reactions proceed through an activated or transition
state which is higher in energy
than both reactants and products.
2. In the transition state the reactants are combined in a
species called the activated
complex.
3. In activated complex the bonds between the atoms are in the
process of being formed
and broken.
4. A true thermodynamic equilibrium exists between the reactant
molecules and the
activated complex species even though the overall chemical
reaction is irreversible.
5. The rate of the reaction is equal to the concentration of
activated complex species
times the frequency at which the complex intermediates
dissociate into products.
In the light of above mentioned postulates, the reaction between
two reactants A and B can be
represented as follows
A + B [AB]* Products
k
Where [AB]* is the activated complex and k is the rate
constant.
On the basis of these ideas, Eyring derived an equation for the
rate constant k of any reaction,
given by
*
A
RTk K
N h (1)
In this equation, R is the gas constant, NA is Avogadros number,
h is Planks constant, T is
Kelvin temperature and K* is equilibrium constant for the
formation of activated complex
from the reactants.
Now from thermodynamics equilibrium constant for the first step
reversible process is related
with the change in Gibbs free energy according to following
expression,
* *G RTln K
*
* Gln KRT
*G
* RTK e
(2) (Taking antilog on both sides)
Where G* is the change in free energy of activation and is
obtained by subtracting the free
energy of the activated complex from that of reactants.
Putting this value of K* from equation (2) in equation (1) we
get
*G
RT
A
RTk e
N h
(3)
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54
Also from fundamental relations of thermodynamics we know
that
* * *G H T S
Where G*, H* and S* represents the free energy, enthalpy and
entropy of activation
respectively and T is the absolute temperature. Putting this
value of G* equation (3)
becomes,
* *( H T S )
RT
A
RTk e
N h
* *S H
R RT
A
RTk e e
N h
*aES
R RT
A
RTk e e
N h
(4) ( H* = Ea)
Equation (4) is the equation of transition state theory and is
known as Eyring equation. This
equation gives valuable information about the reaction.
A comparison of equation (4) with Arrhenius equation shows that
the pre-exponential
factor A is related to the entropy of activation by the
expression,
*S
R
A
RTA e
N h
The entropy of activation introduced by the transition state
theory is analogous to the steric
factor from collision theory.
Transition state theory is applicable to all types of reactions.
This is due to the
presence of S* factor in the theory. It has been found that the
value of S* is negative. The
negative value of S* indicates that the formation of activated
complex (which is more
ordered than the molecules of the reactants) from the reactants
is accompanied by a large
decrease of entropy. Larger the complexity of the reacting
molecules, greater will be the
decrease in the value of entropy. This results in the smaller
value of the quantity *S /Re and
since ART / N h is same for all reactions at same temperature,
therefore, PZ value in collision
theory would be much less than the interpreted value.