§9.3 The rate equation of reaction

with simple order

Overall reactions

Reaction with definite order

Reaction without definite order

Reaction with simple order

1 12 2[H ] [I ]r kH2 + I2 = 2 HI

H2 + Cl2 = 2 HCl1 0.5

2 2[H ] [Cl ]r k

H2 + Br2 = 2 HBr0.5

2 2

2

[H ][Br ][HBr]

1 '[Br ]

r kk

It was found that reactions with same reaction order are usu

ally of same kinetic characteristics, therefore, reactions are us

ually 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

order First Second Third Zeroth

Differential rate equation

Integrated rate equation

Linearity

Half-life

Unit of k

Comparison between reactions with different reaction orders

3.1 First-order reaction:

Reaction: A P

at t = 0 c0

at t = t c

Differential rate equation: 1

dck c

dt

can be rearranged into: 1

dck dt

c

Which can be integrated directly

01ln

ck t

c 0 1exp( )c c k t

0 1000 2000 3000 4000 5000

0.0

0.2

0.4

0.6

0.8

1.0

C /

mol

dm

-3

t / s

c~t curve of first-order reaction0 1exp( )c c k t

Only when t , can c 0, which suggests that, the first-order reaction can not complete.

0

2

cc 0

1lnc

k tc

Half-life

112

1

6932.02ln

kkt

0 1000 2000 3000 4000 5000

-5

-4

-3

-2

-1

0

ln(C

/mol

dm

-3)

t / s

lnc ~ t curve of the first-order reaction

0 1ln lnc c k t

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

2) Decomposition

226 226 488 86 2Ra Rn He

2 5 2 4 2

1N O N O O

2

3) Isomerization

Willard F. Libby

1960 Noble Prize

USA

1908/12/17 ~1980/09/08

Application of 14C for age determinations (radiocarbon dating)

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.

Radiochemical analysis of an object

obtained in an archeological excavation

shows that the 14C isotope content is 0.89

10-14 mol%.

3.2 Second-order reaction

2A P; A + B P

A + B P a b

cA= ax cB =bx

A2 A B

dCk C C

dt Differential rate equation:

))((2 xbxakdt

dx dtk

xbxa

dx2))((

txdtk

xbxa

dx0 20 ))((

txxdtk

xaba

dx

xbba

dx0 200 ))(())((

tkba

a

ba

xa

ba

b

ba

xb2)(

ln

)(

)ln(

)(

ln

)(

)ln(

tkxba

xab

ba 2)(

)(ln

)(

1

22

dck c

dt

When a = b

022

C

C

dck dt

c 2

0

1 1k t

c c

0 1000 2000 3000 4000 5000

0.0

0.2

0.4

0.6

0.8

1.0

C / m

ol d

m-3

t / s

c~t curve of second-order reaction

When c 0, t , which suggests that, the pure second-order reaction can not complete, either.

1/ 22 0

1t

k cHalf-life

0 1000 2000 3000 4000 50000

10

20

30

40

50

1/C

/ mol

dm

-3

t / s

1/c ~ t curve of second-order reaction

For pure second-order reaction

22

1

2

dck c

dt 2

0

1 12k t

c c

1/ 2

2 0

1

2t

k c

Characteristics of second-order reaction

1) Unit of k is mol-1dm3s-1

2) 1/c is in linear proportion to t

3) can not complete

4) Half-life1

02

1t

c

Increasing the initial concentration of the reactant will shorten the reaction time.

Example:

1) dimerization

2) decomposition 2 22HI = H + I

3) recombination

4) esterification 3 2 5

3 2 5 2

CH COOH+C H OH

CH COOC H +H O

→

5) hydrolysis C12H22O11 + H2O

C6H12O6 + C6H12O6

3 2 62CH = C H

C12H22O11 + H2O C6H12O6 + C6H12O6

In 1850, experiment done by Wilhelmy suggested that the rate equation of the reaction is:

12 22 11[C H O ]r k

12 22 11 2[C H O ][H O]vr k

Because the amount of water keeps nearly unchanged during the reaction, [H2O] keeps nearly constant, and the rate equation can be

then simplified as

12 22 11'[C H O ]r k Pseudo first-order reaction

612 22 11 2 3[C H O ][H O] [H O ]r k

3.3 third-order reaction

3A P

A + B + C P

2A + B P

33

1

3

dck c

dt

32 20

1 1 13

2k t

c c

32 2

0

1 16k t

c c 1 2

3 02

1

2t

k c

3A P

For A + B + C P

with same initial concentration

33

dck c

dt Differential rate equation

32 20

1 1 1

2k t

c c

Integrated rate equation32 2

0

1 12k t

c c

1 23 02

3

2t

k c

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 0

dck

dt

000

c t

cdc k dt 0 0c c k t 0

1/ 202

ct

k

When c = 0, the reaction completes, the reaction time is:

0

0

ctk

The zero-order reaction can complete.

0 1000 2000 3000 4000 50000.0

0.2

0.4

0.6

0.8

1.0

C /

mo

l dm

-3

t / s

0 1000 2000 3000 4000 50000.0

0.2

0.4

0.6

0.8

1.0

C /

mo

l dm

-3

t / s

c ~ t curve for zero-order reaction

Characteristics of zeroth-order reaction

1) Unit of k is mol dm-3s-1

2) c is in linear proportion to t

3) can complete

4) When c increases, reaction time will be prolonged.

0

0

k

Ct

Examples:

Decomposition over catalysts:

1) 2N2O 2N2 + O2 over Pt wire

2) 2NH3 N2 + 3H2 over W wire

Photochemical reaction:

r = k I

I: intensity of radiation

5.5 for nth-order reaction

nr kc

tkaxan nnn

}

1

)(

1{

1

111

1

1/ 2 1 10 0

2 1

( 1)

n

n n

At

n kc c

For n 1