9.6 Rate Theories of elementary reaction for bimolecular ...

§9.6 Rate Theories of elementary reaction for bimolecular reaction

—simple collision theory SCT

Extensive reading: Levine, pp. 879-881

(1) Recite some terms: collision, collision diameter, collision cross-section;

effective collision; critical energy.

Teaching goal:

(1) Make comments on the discrepancy between kSCT and kexp.

(2) Explain the necessity for oversimplification.

(1) Be able to calculate the collision frequency and fraction of effective collision.

(2) Describe the reaction process based o SCT.

(3) Explain the physical meaning of pre-exponential factor.

(4) Explain the difference between Ec and Ea.

Two important empirical rules:

Rate equation (law of mass action)

Arrhenius equation

−=

RT

EAk aexp

Type of

reaction

Unimolecular

reaction

Bimolecular

reaction

Termolecular

reaction

A1013

s

1011

mol-1dm3s-1

109

mol-2dm6s-1

A seems related to possibility of encounter.

−

RT

Eaexp Boltzmann distribution term

[A][B]r k=

§9.6 Rate Theories of elementary reaction


−=

RT

EAk aexp

[A][B]r k=

A molecule of A cannot react with a molecule of B unless the two reactant molecules

can somehow interact.

This interaction can only take place if they come within a certain distance of each

other, i.e., collides with each other.

Therefore, the rate constant of a reaction may be predicted by calculation of the

collision frequency of the reactants.

Basic considerations and brief history

http://en.wikipedia.org/wiki/Collision_theory


Collision theory is proposed independently by Max Trautz in 1916 and William

Lewis in 1918. Thereafter, C. Hinshelwood made modification on it.

Basic consideration and brief history

Z. anorg. Chem. 94 (1916), 79


9.6.1 Fundamental assumptions of SCT

For gaseous bimolecular reaction

where ZAB is the collision frequency of A with B per unit cubic meter per second, q is

the portion of effective collision.

ABr Z q=reaction rate can be expressed as:

2) The collision can be either non-reactive (elastic) collision or reactive collision. Only the

molecules posses energy excess to a critical value (Ec) can lead to reactive collision.

The reaction rate should be in proportion to the fraction of reactive collision (q).


1) The reaction rate is proportional to the collision frequency (Z) which can be solved by

kinetic theory of molecule;


Kinetic theory of gasesIn ca. 50 BCE, the Roman

philosopher Lucretius proposed that

static macroscopic bodies were

composed on a small scale of rapidly

moving atoms all bouncing off each

other.

In 1738 Daniel Bernoulli published

“Hydrodynamica”, which laid the basis

for the kinetic theory of gases.

•The gas consists of very small particles. The volume of molecule is negligible.

•The number of molecules is so large that statistical treatment is applicable.

•The rapidly moving particles constantly collide among themselves. All these

collisions are perfectly elastic.

•The interactions among molecules are negligible

9.6.2 Calculation of ZAB

SCT assumes that molecules can be taken as rigid ball without inner structure.

dAdB

Definition:

mean collision diameter: dAB

ABBA d

dd=

+

2


Definition: collision cross-section

2

ABdS =

2

ABAB dZ =V

NB

A

AV

NBmotionless



When the concentration of A is NA/V (molecm-3):

2

ABAB dZ =V

N

V

N BAA

When both A and B moves, the relative velocity AB should be used.

( )22

BAAB +=



i

iM

RT

8=

according to the kinetic theory of gases

A BAB

A B A B

8 8 8 M MRT RT RT

M M M M

+= + =

AB

8RT

=A B

A B

M M

M M =

+(reduced mass)

2 2A B A BAB AB AB

2 2

AB

8 8

8[A][B]

N N Ln LnRT RTZ d d

V V V V

RTL d

= =

=



Decomposition of HI: 2HI = H2 + I2

2 2 2

AA AA

A

2 8[A]

2

RTZ L d

M

=

2 2

AB AB

8[A][B]

RTZ L d

=

At 1.0 105 Pa and 700 K, d = 3.50 10-10 m, Z HI-HI = ?

53

1 1

1.0 10 Pa[HI] 17.41mol m

8.314J K mol 700K

p

RT

−

− −

= = =

23 2 10 2 2 34 3 1

AA 3

2 8 8.314 7003.1416(6.02 10 ) (3.50 10 ) (17.41) 1.017 10 m s

2 3.1416 128 10Z − − −

−

= =

Generally, ZAB of gaseous reactions at ambient temperature and pressure is of the

magnitude of 1035 m-3s-1.



If reaction takes place whenever the molecules collides:

2 2

AB AB

8[A][B]

RTZ L d

=

( ) ( )A A

AB

d d [A]

d d d d

N n L dZ L

V t V t dt= − = − = −

2ABAB

[A] 8[A][B]

Zd RTr d L

dt L

= − = = [A][B]r k=

2

AB

8RTk d L

=

k = 7.88 104 mol-1dm3s-1


Not all the collisions is effective/reactive: Only the molecules posses energy excess to a

critical value (Ec) can lead to reactive collision.

When c0 = 1.00 mol dm-3, the half-life of HI is 1.27 10-5 s. This result differs greatly

from the experimental fact.

9.6.3 Calculation of q

It is apparent that E of translational energy of motion is

related to the relative motion of two molecules. And Ec is thus

the minimum translational energy of motion


If the energy exchange between

colliding molecules is much rapid than

reaction, the energy distribution of

molecules may still obey the Maxwell-

Boltzmann distribution equation.

(critical / threshold energy) along the connecting line between the mass-point of the two molecules

which are to collide.

−==

RT

E

n

nq cexp

*

Boltzmann factor

If Ec = 120 kJmol-1, T = 300 K, then

q = 1.27 10-21

This suggest than among 7.8 1020

collision only one collision is effective.The fraction of the collision with the

energy equal to or greater than Ec is:

9.6.3 Calculation of q

In 1909, Max Trautz had introduced

fraction of reactive collision (q) to solve

this great discrepancy.


9.6.4 Calculation of k

2 2 A BAB AB

A B

8[A][B]

M MRTZ L d

M M

+=

2

AB

8exp [A][B]cERT

r d LRT

= −

2

SCT AB

8exp cERT

k d LRT

= −

1

2SCT exp cE

k BTRT

= −

B is a constant independent of T.

−==

RT

E

n

nq cexp

*


−=

RT

EAk aexp

ca ERTE +=2

1

−=

RT

EBTk c

SCT exp2

1

The experimental activation energy (Ea) depends on temperature.

Using Ea for substitution of Ec,

2

AB

8exp c

SCT

ERTk d L

RT

= −

2

AB

8exp a

SCT

ERTek d L

RT

= −

The pre-exponential factor corresponds to the collision frequency. This is the reason why A

is also named as frequency factor.

9.6.4 Calculation of k


9.6.5 Comment on SCT

1) The expression for the rate coefficient

given by SCT conforms qualitatively to

the Arrhenius equation observed

experimentally. This suggests that SCT

reveal the principal features of the

reaction, i.e., in order to react, molecules

have to collide (the pre-exponential term)

and the collision should be sufficiently

energetic (the exponential term)

(1) Success SCT gives a vivid physical image of the

reaction process:


2) As pointed out by SCT, the pre-

exponential factor, dependent only on

the masses of the species involved in the

collision, can be calculated easily.

ca ERTE +=2

1

SCT reveals the physical meaning of

the pre-exponential factor, i.e., the

collision frequency.



3) SCT demonstrated theoretically that

experimental activation energy depends on

temperature.

2

AB

8RTA L e d

=

(2) Shortcomings

1) For calculating k, Ec is needed.

However, SCT can not give Ec.

Calculation of k depends on the

experimental determination of Ea.

2) The quantitative agreement between

SCT and experiments is poor.

Reaction 10-9Acal 10-9Aexp Acal./Aexp.

2NOCl→2NO+Cl2 2.95 3.23 0.91

H+Br2 →HBr+Br 46 6.76 6.76

NO+O3→NO2+O2 7.94 0.063 1.25102

CH3+CHCl3 →CH4+CCl3 15 1.2610-3 1.19104

2-cyclopentadiene →dimer 8.13 2.4510-6 3.32106



In some cases, the agreement between

experimental and calculated A values can be

quite good. However, in many cases, the

observed rate is definitely too small: the more

complex of the reactant molecules, the greater

the discrepancy between Acal and Aexp.



OH¯+ CH3Br → CH3OH + Br¯

图片1.gif

The great discrepancies between

experimental and calculated A were

recognized around 1925. The equation

−=

RT

EAk a

SCT exp

was then modified by introduction of an

empirical factor P called the steric factor /

probability factor.

−=

RT

EPAk a

SCT exp

.

.exp

calA

AP =

Steric factor (P), ranging between 1~10-9,

represents the fraction of energetically

suitable collisions for which the

orientation is also favorable, can be only

determined experimentally.

SCT can not give any clue to calculate P.

9.6.6 Modification of SCT


9.6.7 further application of SCT


,

AB expcERT

k d LRT

= −

12 8

,

AB expcERT

k L dRT

= −

22 8

( )1, ,

2

log log log a a

kE E

k RT

= + − −1 1

1 2

2 2

1

9.6.7 further application of SCT


（a）碰撞频率。大量溶剂分子的存在会导致反应

物分子间的碰撞几率降低，指前因子（A）下降；

（b）分子间的阻滞作用。溶质分子与溶剂分子之

间存在的作用力致使反应物分子的运动速率（v）

降低；

（c）反应物活度。由于溶剂—溶质分子间存在作

用力，导致反应物的活度系数（）降低--“原盐

效应（primary salt effect）”；

（d）溶剂化。包围在反应物、中间产物或者过渡

态、反应产物周围的溶剂分子会构成一个静电场，

影响这些物质的结构、尺寸、电子分布 [2, 3, 10-16]，

进而影响反应的活化能（Ea）；

（e）溶剂催化。溶剂分子的作用可能使某些化学

键被活化，从而改变反应途径和活化能（Ea）；

（f）溶剂分子的运动速率。溶剂分子量小，反应

物分子的运动会被加速，呈现“负黏度”

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