§9.6 Rate Theories of elementary reaction

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

It is obvious that 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.

A reaction can take place only if the molecules of the reactants

collide. 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 its.

Basic consideration and brief history

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

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).

6.2 Calculation of ZAB

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

dA dB

dA and dB are the diameter of A and B molecule, respectively.

Definition: mean collision diameter: dAB

ABBA d

dd

2

The way to collide: 撞个满怀、擦肩而过，失之交臂

Definition:collision cross-section

2ABdS

2ABAB dZ V

NB

A

A V

NBmotionless

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

2ABAB dZ V

N

V

N BAA

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

22BAAB

ii M

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 2AA AA

A

2 8[A]

2

RTZ L d

M

2 2AB AB

8[A][B]

RTZ L d

?

For example

At 1.0 105 Pa and 700 K, d = 3.50 10-10 m, Z HI-HI = ?5

31 1

1.0 10 Pa[HI] 17.41mol m

8.314J K mol 700K

p

RT

23 2 10 2 2AA 3

34 3 1

2 8 8.314 7003.1416(6.02 10 ) (3.50 10 ) (17.41)

2 3.1416 128 10

1.017 10 m s

Z

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 2AB AB

8[A][B]

RTZ L d

A

AB

[A]N

ddV

r L Zdt dt

2ABAB

[A] 8[A][B]

Zd RTd L

dt L

[A][A][B]

dk

dt 2

AB

8RTk d L

because

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. In 1909, Max

Trantz introduced fraction of reactive collision (q) to solve this great

discrepancy.

6.3 Calculation of q

Only the molecules posses energy excess to a critical value (Ec) can

lead to reactive collision.

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 (critical / threshold energy) along the

connecting line between the mass-point of the two molecules which

are to collide.

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.

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:

6.4 Calculation of k

ABr Z q

2 2 A BAB AB

A B

8[A][B]

M MRTZ L d

M M

2AB

8exp [A][B]cERT

r d LRT

[A][B]r k

2SCT 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,

2AB

8exp c

SCT

ERTk d L

RT

2AB

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.

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) Successfulness

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) Shortcomings

1) For calculating k, Ec is needed. However, SCT can not give Ec.

Calculation of k depends on the experimental determination of Ea.

Therefore, SCT can not predict the kinetic features of the reaction

theoretically.

2) The quantitative agreement between SCT and experiments is poor.

Reaction Ea Acal Aexp Acal./Aexp.

2NOCl2NO+Cl2 107.8 2.95109 3.23109 0.91

H+Br2 HBr+Br 3.76 4.61010 6.76109 6.76

NO+O3NO2+O2 9.61 7.94109 6.31107 1.25102

CH3+CHCl3 CH4+CCl3 24.2 1.51010 1.26106 1.19104

2-cyclopentadiene dimer 60.6 8.13109 2.45103 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. It was found that the more complex of the reactant molecules, the greater the discrepancy between Acal and Aexp.

In fact, the reactant is of complex molecular structure. To take reactant molecules as rigid balls without inner structure will spontaneously result in systematic error.

?

2 ONBr Br2 + 2 NO

CH3+CHCl3 CH4+CCl3

The colliding molecules must be suitably oriented.

Substitution

OH¯ + CH3Br CH3OH + Br¯

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.