kinetics

1 Chemical kinetics
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CHEMICAL KINETICS
Chemical kinetics deals with
1) Study of rates of reactions
2) Factors affecting rate of reaction
3) Study of mechanisms of reactions
Rate of reaction  : The rate of reaction is defined as the change in molar concentration of either
reactants or products in unit time.
 
change in concentration
rate r =
time interval
Let ‘dc’ is the decrease in concentration of reactants in a small interval of time ‘dt’ then
dc
r = -
dt
  ( negative sign indicates decrease in concentration )
If ‘dx’ is the increase in concentration of products in a small interval of time ‘dt’ then
dx
r =
dt
  ( positive sign indicates increase in concentration )
E.g., For a reaction, A B 
 
    d A d B
rate r
dt dt

 
In general for a reaction, pP + qQ  rR + sS
the relations between rates of reaction with respect to P,Q,R and S can be written as
        d P d Q d R d S 1 1 1 1
- = - = + = +
p dt q dt r dt s dt
Eg., for the reaction,
      2 g 2 g 3 g
N +3H 2NH  , the relations can be expressed as
      2 2 3 d N d H d NH 1 1
- -
dt 3 dt 2 dt
  
Factors affecting the rate of reaction
1) Nature of reactants:   Rate of a reaction depends on the nature of bonding in the reactants. Ionic
compounds react faster than covalent compounds.
The reactions between ionic compounds in water occur very fast as they involve only exchange of
ions.
Eg., AgCl is precipitated out immediately when AgNO3
 solution is added to NaCl solution.
3 3 AgNO +NaCl AgCl NaNO  
This reaction involves only exchange of ions as shown below and hence occurs very fast.
+ - + - + -
3 3 Ag +NO +Na +Cl AgCl Na NO  
Reactions between covalent compounds take place slowly because they require energy for the
cleavage of existing bonds.
Eg., Following esterification of acetic acid occurs slowly as the breaking of bonds require energy.
C H3 C
OH
O
O C2H5 H C H3 C
OC2H5
O
+ O H2 +
H2SO42 Chemical kinetics
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VAAGDEVI JUNIOR COLLEGE
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2) Concentration:  Rate of a reaction is directly proportional to the concentration of reactants.
rate (r)   cn
where c = concentration
n = order of the reaction
or r = kcn
where k = specific rate
The number of collisions increases with increase in concentration and hence the rate of the reaction
also increases.
The rate of a reaction decreases with time as the concentration of reactants is decreasing. This can
be shown graphically as follows.
concentration of reactants
Time
3) Temperature : The average kinetic energy and hence the number of collisions increase with abso-
lute temperature. Hence rate of reaction increases with increase in temperature.
Usually the rate of a reaction is doubled when the temperature is increased by 10o
C.
Temperature Coefficient : The ratio of rate constants of a reaction at two different temperatures
which differ by 10o
C is called temperature coefficient.
 
o
o
t +10 C
t C
k
Temperature coefficient = 2 to3
k

The relation between rate constant and temperature can be shown by Arrhenius equation.
k = A.e-Ea/RT
Where k = specific rate constant
      A = Frequency factor
      Ea
 = Activation energy
      R = Universal Gas constant
      T = Absolute Temperature
Multiplying by 'ln' (natural logarithm) on both sides,
ln k = ln a
E
A-
RT
or 2.303 log k = 2.303 log
a
E
A-
RT
or log k =
a
E
log A-
2.303 RT
When a graph is plotted by taking log k on y - axis and 1/T on x-axis, a straight line with negative slope
is obtained.
slope =
a
E
-
2.303 R =
a
E
-
4.576
where R = 1.987 cal / K / mole3 Chemical kinetics
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log A
1
T
Ea
slope =  -
2.303 R
log k
Relation between two rate constants at two different temperatures can be given as
a 2
1 1 2
E k 1 1
log = -
k 2.303R T T
 
 
 
or
  2 1 a 2
1 1 2
T -T E k
log =
k 2.303R TT
4) Catalyst: Catalyst is a substance which alters the rate of reaction without being consumed or without
undergoing any chemical change during the reaction.
A catalyst increases the rate of reaction by providing a new path with lower activation energy for the
reaction. In case of reversible reactions, catalyst lowers the activation energies of both forward and
backward  reaction to the same extent and helps in attaining the equilibrium quickly.

Reaction coordinate
Potential energy
without catalyst
in presence of catalyst
Problems :
1) Write the relations between the rates with respect to reactants & products in the following reaction.
2N2
O  2N2
 + O2
Rate Equation or Rate Expression or Rate Law: Equation that describes mathematical dependance
of rate of reaction on the concentration terms of the reactants is called Rate Equation.
For a general reaction, xA+yB+zC  products.
      r = k A B C x y z
where k= rate constant or specific rate
x,y and z are orders with respects to A,B and C respectively.
Order of reaction :Sum of the powers of the concentration terms in the rate equation is called order
of  reaction.
For a general reaction, xA + yB + zC  products.4 Chemical kinetics
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if the rate expression is r =         k A B C x y z
Then the order of this reaction is equal to x + y + z
Note: Order of a reaction is to be determined experimentally. It may have positive or negative or zero
or a fractional values.
In case of simple reactions, the order of reaction and hence the rate equation can be written
according to stoichiometric equation. But the actual rate law should be written by conducting experi -
ments.
Depending on the order of reaction, chemical reactions can be divided into zero, first, second &
third order reactions as follows.
Zero order reaction: Total order of the reaction is zero.
Example:
1)  Decomposition of NH3
 on metal surfaces such as gold and molybdenum.
      2 g 2 g 3 g
N +3H 2NH Au

r = k[N2
]
0
[H2
]
0
First order reaction: Total order of the reaction is one.
Examples:
1) Thermal decomposition of N2
O5
N2
O5
  N2
O4
 + 1/2 O2
r = k[N2
O5
]
2) Thermal decomposition of SO2
Cl
2
SO2
Cl
2SO2
+Cl
2
r  = k [ SO2
Cl
2
]
3) Thermal decomposition of H2
O2
H2
O2
  H2
O + 1/2 O2
r  = k[H2
O2
]
4) All radioactive disintegrations
Second order reaction : Total order of the reaction is two.
Examples:
1) Thermal decomposition of N2
O
2N2
O  2N2
 + O2
r= k [N2
O]
2
2) Decomposition of Cl
2
O
2Cl
2
O  2Cl
2
 + O2
r= k [Cl
2
O]
2
Third order reaction : Total order of the reaction is three.
Examples:
1) Reaction between NO and O2
to give NO2
2NO + O2
--------> 2NO2
r = k[NO]
2
[O2
]
2) Reaction between NO and Cl
2
 to give NOCl
2NO + Cl
2
  2NOCl
r = k[NO]
2
 [Cl
2
]5 Chemical kinetics
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VAAGDEVI JUNIOR COLLEGE
WARANGAL
Problems :
1) What is the order of the reaction if the rate law for the reaction A + 2B products is
r = k [A]
1/2
[B]
2
Rate constant (k): The rate of a reaction at the unit concentrations of all the reactants of the reaction is
called specific rate (or) rate constant.
rate = k when [reactants] = 1
Units of rate constant
For a general reaction, A  products
Rate expression can be written as
 
n dc
rate = k A
dt

where n = order of the reaction
 
n
dc 1
k = x
dt A
 Units of
 
n
mole / litre 1
k = x
sec mole / litre
 
1-n
mole / litre
=
sec
  - n-1 n-1 -1
= L .mole .sec
For 1st
 order reaction  sec-1
For 2nd
 order reaction  L.mole-1
sec-1
For 3rd
 order reaction  L2
.mole-2
sec-1
For zero order reaction  mole .L-1
sec-1
Molecularity
Elementary reactions: The chemical reactions which occur in one single step are called elementary
reactions.
But most of the reactions occur in more than one step. The sequence of elementary steps that occur
during a reaction is referred to as mechanism of that reaction.
Rate determining step: The elementary step or reaction with slowest rate is called rate determining step
or rate limiting step. The rate of overall reaction depends on this step.
Molecularity: Total number of atoms or molecules or ions taking part in an elementary reaction is called
molecularity of that elementary reaction (or step).
It is only possible to define the molecularity of an elementary step. But the molecularity of the overall
reaction which involves more than one step cannot be defined.
Molecularity is an integer. It cannot be zero or fractional. The maximum value observed is 3. It can be
deduced theoretically from the proposed mechanism6 Chemical kinetics
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Differences between Order and Molecularity
Order Molecularity
1) Sum of the powers of concentration terms
in rate expression is called order.
1) Total number of atoms or molecules or
ions taking part in an elementary reaction is
called molecularity
2) It may have zero or positive or negative or
integer or fractional values.
2) It can only have integral values (1,2 or 3).
Its value cannot be zero or fractional.
3) It is determined experimentally 3) It is deduced theoretically from the
mechanism of the reaction.
Integrated rate equations for zero and  first order reactions
Zero order rate equation:
Reactions in which the rate  is independent of the concentrations of reactants are called Zero order
reactions. The integrated rate equation for zero order reactions is derived as follows.
o dc dx
- = =kc
dt dt
or  
o dx
=k a-x k
dt

or dx = k dt
 
or x = kt
or k =
x
t
Where k = specific rate
x = amount of reactant consumed or reacted in time 't'.
First order reaction
For a first order reaction R  P
Let
The initial concentration of reactant = a mol. dm-3
.
Amount of reactant decomposed in 't' sec = x mol. dm-3
.
The concentration of reactant after 't' sec = (a - x ) mol. dm-3
.
 The rate
dx
dt
 
 
 
 at time 't' is proportional to the concentration of reactant (a - x) at that time. Since the
reaction is a first order reaction,
 
dx
a-x
dt

 
dx
k a-x
dt

 
dx
=k.dt
a-x [where k = specific rate]
On integration7 Chemical kinetics
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 
dx
k.dt
a-x
  
     -ln (a-x) = k.t + c
when t = 0 then x = 0
 -ln a = c
Hence -ln(a-x) = kt-ln a
or
 
a
kt =ln
a-x
or
 
1 a
k = ln
t a-x
or
 
2.303 a
k= log
t a-x
Where a = initial concentration of reactant
x = concentration reacted in an interval of time 't'
(a-x) = concentration of reactant after time 't'
Half life (t
1/2
) of a reaction :The time required for initial concentration of reactants to become half is
called half life.
Half life of first order reaction can be derived as given below.
x = a/2 when t = t
1/2
Then a-x =
a
a-
2
 = a/2
Therefore
1/2 1/2
2.303 a 2.303
k= log log2
t a/2 t
 
  
 
 =
1
2
2.303x 0.3010
t
 (  log2 = 03010)
or 1/2
0.693
t
k

Methods of determination of order of a reaction :
1) Integrated equation method or Trial and error method
* The initial concentration (a) and the concentrations at various intervals of time (a - x) are measured
during the progress of reaction by suitable analytical methods. These values are substituted in the  rate
equations of different rate orders. The order corresponding to the rate equations, which gives constant 'k'
values is taken as the order of reaction.
Rate equations
For  a  Zero order reaction R  P
x = kt
or
  o t
a- a-x R -R x
k = = =
t t t8 Chemical kinetics
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For  first order reaction     R  P
o
t
R 2.303 a 2.303
k= log = log
t a-x t R
For second order reaction     2R  P
o t
o t
R -R 1 x 1
k= = .
at a-x R t R
For second order reaction R1
 + R2
 P
2.303 ( )
k= log
t(a-b) ( )
b a x
a b x


The order can also be ascertained graphically as follows
For a zero order reaction, the graph of 'x' versus 't' must be a straight line parallel to time (t)  axis.
x
t
For a first order reaction, the graph of
a
log
a-x
 versus  't'   should be a straight line with positive slope
and should pass through the origin
t
a
a-x
log
For a second order reaction, the graph of
 
x
a-x a  vs 't'   must be a straight line passing through the
origin with a positive slope.
t
x
a(a-x)
t
b(a-x)
a(b-x)
log OR
Half life method
Half life   1
2
t  is inversely proportional to an-19 Chemical kinetics
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1 n -1
2
1
a
t 
where a = initial concentration
n = order
The order 'n' of a reaction can be calculated by comparing two half lives -  1
2
t
 &   1
2
t
 with initial
concentrations a& a  respectively. .
n-1 1
2
1
2
t
a
=
t a

  
 
   
3) Van't Hoff differential method
The rate
dc
-
dt
 for a reaction of nth
 order is given by the equation,
n dc
- =kc
dt
For two initial concentrations c1
,c2
, we have
n 1
1
dc
- =kc
dt
;
n 2
2
dc
- =kc
dt
1
1
dc
log - log k +nlogc
dt
 
   
 
2
2
dc
log - log k +nlogc
dt
 
  
 
or  
1 2
1 2 1 2
dc dc
log - - log - n log c nlogc n log c logc
dt dt
   
       
   
 
1 2
1 2
dc dc
log - - log -
dt dt
n
log c logc
   
   
   
 

For two different initial concentrations c1
 and c2
, the rates are determined at any given time from c-t graph.
These are substituted in the above equation and 'n' is calculated.
concentration of reactants
Time (t)
c1
concentration of reactants
c2
Time (t)
4) Ostwald's Isolation Method
In this method, order with respect to each reactant is determined separately by taking other reactants
in larger quantity.
For example, for the given reactions A + B  products, the order is determined as follows:
1) First the concentration of 'B' is taken in larger excess and the order (nA) with respect to 'A' is deter-
mined by any of the other three methods.10 Chemical kinetics
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2) In the second experiment the order (nB) with respect to 'B' is determined by taking 'A' in large excess.
The order of the overall reaction is taken as nA +  nB.
Kinetic Molecular Theory of Reaction Rates or Collision theory of reaction rates :
This theory was proposed by Arrhenius. The main postulates of this theory are
1) Chemical reactions occur due to collisions between reactant molecules.
2) But all the collisions do not lead to the formation of products.
3) Only those collisions taking place between molecules possessing certain minimum amount of energy
can form the products. This minimum amount of energy is called Threshold Energy (ET).
4) The energy possessed by the normal molecules under STP conditions is called 'Average Energy'. The
difference between threshold energy and average energy of the reacting molecules is called Activation
energy (Ea
).
Ea
 = Threshold energy (ET) - Average energy (ER)
5) Some molecules will get threshold energy during collisions and are called activated molecules and
denoted by asterick(*).
6) The collisions taking place between activated molecules are called activated collisions. The reaction
occurs only during the activated collisions.
Activated molecules constitute a small fraction of total molecules. Similarly activated collisions consti-
tute small fraction of total collisions. Hence all the chemical reactions do not occur in fraction of second.
A  +  A                 A* + A A
A* + A*              products
Where A = normal molecule
A* = Activated molecule
7) The number of binary collisions per unit time (Z) is given by
2
AB A B
8kT
Z .n .n  
 

where AB   = Collision diameter
     = reduced mass
A B n &n  are number of molecules of A & B
8) The relation between specific rate (k), temperature (T) and activation energy Ea
 can be given as follows
a -E /RT
k =A.e
     or
a -E /RT
k = p.Z.e
where p = probability factor
The concept of activation energies of both forward and backward reactions (Ea
 and Ea
') can be
shown graphically as follows.11 Chemical kinetics
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R E
a E
T E
T E
1
a
E
p E
H 
Energy barrier
Where ER = Energy of reactants
EP
 = Energy of products
ET = Threshold energy
Ea
 = Activation energy of  forward reaction
Ea
1
 = Activation energy of  backward reaction
H  = Enthalpy of reaction
Problems :
1) The rate constant of a first order reaction is 1.386 sec-1
. What is it's half life.
2) The initial concentration of  A  is 0.5 moles/litre. If the concentration becomes 0.25 mole/litre in 5 sec.
What is the value of rate constant.
TEST YOUR UNDERSTANDING
State whether the following statements are true or false.
1) The rate of a reaction increases with time as the concentration of reactants decreases.
2) The order of a reaction is always determined experimentally.
3) The units of rate constants for the following reaction are (L / mol /sec).
2NO(g)  + Cl
2
 (g)  2NOCl (g)
4) The half life of zero order reaction is proportional to the initial concentration.
5) The rate of consumption of hydrogen gas is thrice of formation of Ammonia in the following reaction.
N2
 (g) + 3H2
 (g)  2 NH3
 (g)
6) In Ostwald isolation method, the reactant for which the order is going to be determined is taken in
excess when compared to the concentrations of other reactants.
7) A positive catalyst increases the activation energy of both forward and backward reactions.
8) In exothermic reaction the activation energy of forward reaction is always greater than that of back
ward reaction.
9) For a second order reaction, a straight line is obtained when log
a
a x 
 is plotted against time (t)
10) Rate of reaction increases with increasing temperature as the threshold energy increases.