一日一句 - shandong university · 2018-09-18 · a molecule of a cannot react with a molecule...
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--《曾国藩家书》慎独主敬求仁习劳
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
§9.6 Rate Theories of elementary reaction
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
§9.6 Rate Theories of elementary reaction
Basic consideration and brief history
Z. anorg. Chem. 94 (1916), 79
§9.6 Rate Theories of elementary reaction
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 Rate Theories of elementary reaction
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
§9.6 Rate Theories of elementary reaction
Definition:collision cross-section
2
ABdS
2
ABAB dZ V
NB
A
AV
NBmotionless
9.6.2 Calculation of ZAB
§9.6 Rate Theories of elementary reaction
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
9.6.2 Calculation of ZAB
§9.6 Rate Theories of elementary reaction
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
9.6.2 Calculation of ZAB
§9.6 Rate Theories of elementary reaction
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.
9.6.2 Calculation of ZAB
§9.6 Rate Theories of elementary reaction
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.
§9.6 Rate Theories of elementary reaction
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
§9.6 Rate Theories of elementary reaction
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 Rate Theories of elementary reaction
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
*
§9.6 Rate Theories of elementary reaction
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 Rate Theories of elementary reaction
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:
§9.6 Rate Theories of elementary reaction
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.
9.6.5 Comment on SCT
§9.6 Rate Theories of elementary reaction
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
9.6.5 Comment on SCT
§9.6 Rate Theories of elementary reaction
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.
9.6.5 Comment on SCT
§9.6 Rate Theories of elementary reaction
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.
9.6.6 Modification of SCT
§9.6 Rate Theories of elementary reaction
,
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
§9.6 Rate Theories of elementary reaction
(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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