§9.3 The rate equation of reaction
with simple order
H2 + I2 = 2 HI
r  k[H 2 ]1[I 2 ]1
H2 + Cl2 = 2 HCl
r  k[H 2 ]1[Cl2 ]0.5
H2 + Br2 = 2 HBr
[H 2 ][Br2 ]0.5
rk
[HBr]
1 k '
[Br2 ]
Overall
reactions
Reaction with
definite order
Reaction without
definite order
Reaction with
simple order
It was found that reactions with same reaction order are
usually of same kinetic characteristics, therefore, reactions
are usually classified on the basis of reaction order.
Reaction with simple order:
The reaction whose rate only depends on the concentration of
reactants, and both the partial order and the reaction order is zero
or plus integer is called reaction with simple order.
r = kcn
n
kinds
0
zeroth-order reaction
1
first-order reaction
2
second-order reaction
3
third-order reaction
Comparison between reactions with different reaction orders
order
First
Second
Third
Zeroth
Differential
rate equation
Integrated
rate equation
Linearity
Half-life
Unit of k
3.1 First-order reaction:
Reaction: A  P
at t = 0
c0
at t = t
c
Differential rate equation:
dc

 k1c
dt
can be rearranged into:
dc

 k1dt
c
Which can be integrated directly
c0
ln  k1t
c
c  c0 exp(k1t )
c  c0 exp(k1t )
1.0
C / mol dm-3
c~t curve of firstorder reaction
0.8
0.6
0.4
0.2
0.0
0
1000
2000
3000
4000
5000
t/s
Half-life
c0
c
2
c0
ln  k1t
c
ln 2 0.6932
t1 

k1
k1
2
Only when t  , can c
 0, which suggests that,
the first-order reaction
can not complete.
ln(C/mol dm-3)
0
lnc ~ t curve of the
first-order reaction
-1
-2
-3
-4
-5
0
1000
2000
3000
4000
5000
t/s
ln c  ln c0  k1t
The slope of the lnc ~ t curve is the k1
Characteristics of the first-order reaction
1) Unit of k is s-1
2) lnc is in linear proportion to t
3) can not complete
4) Half-life does not depend on c0
Example:
1) Decay of isotopes
226
88
Ra 
226
86
Rn  42 He
2) Decomposition
1
N 2 O5  N 2 O 4  O 2
2
3) Isomerization
Example:
The half-life of the first-order decay of
radioactive
14C
is about 5720 years. The
natural abundance of
14C
isotope is 1.1 
10-13 mol% in living matter.
Willard F. Libby
1960 Noble Prize
Radiochemical analysis of an object
obtained in an archeological excavation
USA
shows that the
1908/12/17 ~1980/09/08
 10-14 mol%.
Application of 14C for age
determinations
(radiocarbon dating)
14C
isotope content is 0.89
3.2 Second-order reaction
2A  P;
A + B  P
A +
B  P
a
b
cA= ax cB =bx
Differential rate equation:
dx
 k 2 (a  x)(b  x)
dt
dCA

 k 2 CA C B
dt
dx
 k2 dt
(a  x)(b  x)

x
0

x
0
t
dx
  k2 dt
(a  x)(b  x) 0
x
t
dx
dx

  k 2 dt
0
(a  b)(b  x)
(a  b)( a  x) 0
ln( b  x)
ln b
ln( a  x)
ln a




 k 2t
( a  b) ( a  b) ( a  b) ( a  b)
1
b( a  x )
ln
 k 2t
(a  b) a(b  x)
When a = b
dc

 k2 c 2
dt
dc
  2  k2 dt
C0 c
C
1
1 
 
  k2t
 c c0 
C / mol dm-3
1.0
0.8
0.6
0.4
0.2
0.0
0
1000
2000
3000
t/s
4000
5000
c~t curve of
second-order
reaction
When c  0, t  , which suggests that, the pure secondorder reaction can not complete, either.
Half-life
t1/ 2
1

k2 c0
1/C/ mol dm-3
1/c ~ t curve of
second-order
reaction
50
40
30
20
10
0
0
1000
2000
3000
4000
5000
t/s
For pure second-order reaction
1 dc
2

 k2 c
2 dt
1
1 
 
  2k2t
 c c0 
t1/ 2
1

2k2 c0
Characteristics of second-order reaction
1) Unit of k is mol-1dm3s-1
2) 1/c is in linear proportion to t
3) can not complete
4) Half-life
1
t1 
c0
2
Increasing the initial concentration of the reactant
will shorten the reaction time.
Example:
1) dimerization
2) decomposition
2HI = H2 + I2
3) recombination
2CH3 = C2 H6
4) esterification
CH 3 COOH+C 2 H 5 OH →
CH 3 COOC2 H 5 +H 2 O
5) hydrolysis
C12H22O11 + H2O 
C6H12O6 + C6H12O6
C12H22O11 + H2O  C6H12O6 + C6H12O6
In 1850, experiment done by Wilhelmy suggested that the
rate equation of the reaction is:
r  k[C12 H22O11 ]
r  k[C12 H 22O11 ][H 2O]v
Because the amount of water keeps nearly unchanged during the
reaction, [H2O] keeps nearly constant, and the rate equation can be
then simplified as
r  k '[C12 H 22O11 ]
Pseudo first-order reaction
r  k[C12 H 22O11 ][H 2O]6 [H 3O  ]
3.3 third-order reaction
3A  P
A + B + C  P
2A + B  P
3A  P
1 1 1 
 2  2   3k3t
2  c c0 
1 dc

 k3 c 3
3 dt
1 1
 2  6k3t
2
c c0
1
t1 
2
2
k
c
3 0
2
For A + B + C  P
with same initial concentration
Differential rate equation
dc

 k3 c 3
dt
Integrated rate equation
1 1
 2  2k3t
2
c c0
1 1 1 
 2  2   k3t
2  c c0 
3
t1 
2
2
k
c
3 0
2
Only five third-order gaseous reactions have been observed.
2NO + X2  N2O + X2O; X = H, D
2NO + O2  2NO2;
2NO + X2  2NOX; X = Br, Cl
Are these true third order reactions ?
r = k [C6H5CHO]2[CN-]
r = k [C2H4O][H+][Br-]
3.4 Zeroth-order reaction
A  P
Differential rate equation
c
t
c0
0
  dc   k0 dt
c0  c  k0t
dc
  k0
dt
t1/ 2
c0

2k0
When c = 0, the reaction completes, the reaction time is:
c0
t 
k0
The zero-order reaction can complete.
C / mol dm-3
1.0
0.8
0.6
0.4
0.2
0.0
0
1000
2000
3000
4000
5000
t/s
c ~ t curve for zero-order reaction
Characteristics of zeroth-order reaction
1) Unit of k is mol dm-3s-1
2) c is in linear proportion to t
3) can complete
4) When c increases, reaction time will
be prolonged.
C0
t 
k0
Examples:
Decomposition over catalysts:
1) 2N2O  2N2 + O2 over Pt wire
2) 2NH3  N2 + 3H2 over W wire
Photochemical reaction:
r=kI
I: intensity of radiation
5.5 for nth-order reaction
r  kc
n
1
1
1
{
 n1 }  k n t
n 1
n  1 (a  x)
a
t1/ 2
2n 1  1
A

 n 1
n 1
(n  1)kc0
c0
For n  1