一日一句 - Shandong University · 2018-09-18 · A molecule of A cannot react with a molecule of B unless the two reactant molecules can somehow interact. This interaction can

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一日一句

§9.6 Rate Theories of elementary reaction

for bimolecular reaction—simple collision theory SCT

Extensive reading: Levine, pp. 879-881

Two important empirical rules:

Rate equation (law of mass action)

Arrhenius equation

RT

EAk aexp

Type of

reaction

Unimolecular

reaction

Bimolecular

reaction

Termolecular

reaction

A 1013

s

1011

mol-1dm3s-1

109

mol-2dm6s-1

A seems related to collision frequency.

RT

EaexpBoltzmann distribution term

[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 the reaction may be predicted by calculation of the

collision frequency of the reactants.

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

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


Basic consideration and brief history

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


9.6.1 Fundamental assumptions of SCT

For gaseous bimolecular reaction

1) The reaction rate of reaction is proportional to the collision frequency (Z), which can

be solved by kinetic theory of molecule;

ABr Z q

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

the portion of effective collision.

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).


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 BB AB

2 2

8 8

8[A][B]

AB A

AB

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

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.


Only the molecules posses energy excess to

a critical value (Ec) can lead to reactive

collision.

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

RT

EBTk c

SCT exp2

1

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.

2NOCl2NO+Cl2 2.95 3.23 0.91

H+Br2 HBr+Br 46 6.76 6.76

NO+O3NO2+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

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


9.6.6 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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一日一句 - Shandong University · 2018-09-18 · A molecule of A cannot react with a molecule of B unless the two reactant molecules can somehow interact. This interaction can