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Physical Chemistry
Reaction Kinetics (6)
Xuan Cheng
Xiamen University
1
Physical Chemistry Reaction
Theories of Reaction Rates Kinetics
Hard-Sphere Collision Theory of Gas-Phase Reactions
Assumptions
(1) The molecules are hard spheres
(2) For a reaction to occur between B and C, the two
molecules must collide
(3) Not all collisions produce reaction. Reaction occurs if
and only if the reactive translational kinetic energy along
the line of centers of the colliding molecules exceeds a
threshold energy thr
(4) The Maxwell-Boltzmann equilibrium distribution of
molecular velocities is maintained during the reaction
2
Physical Chemistry Reaction
Kinetics
简单碰撞理论的基本假设
该理论的基本假设(即理论模型）：
(i)反应物分子可看作简单的硬球，无内部结构和相互作用；
(ii)反应分子必须通过碰撞才可能发生反应；
(iii)并非所有碰撞都能发生反应，相互碰撞的两个分子—碰撞
分子对的能量达到或超过某一定值 thr—称为阈能时，反应才能
发生，这样的碰撞叫活化碰撞；
(iV)在反应过程中，反应分子的速率分布始终遵守麦克斯韦—
玻耳兹曼(Maxwell-Boltzmann)分布。
3
Physical Chemistry Reaction
Theories of Reaction Rates Kinetics
Hard-Sphere Collision Theory of Gas-Phase Reactions
The number of B reacting in a bimolecular reaction
B + C Products Z BC e Ethr / RT Ethr NA thr
J 1 1 dn A
Ethr / RT
r (17.3) NA
V V a dt r Z BC e
Z BC e Ethr / RT
The predict rate constant k r k[ B][C ]
N A[ B][C ]
Z BC e Ethr / RT
k (23.2)
N A[ B][C ]
The use of (15.62) for ZBC
1/ 2
8RT 1
for B C
1
k N A (rB rC ) 2
e Ethr / RT (23.3)
M B MC
4
Physical Chemistry Reaction
Theories of Reaction Rates Kinetics
Hard-Sphere Collision Theory of Gas-Phase Reactions
For the bimolecular reaction 2B Products
1 d [ B]
r k[ B]2
2 dt
Ethr / RT
The rate of disappearance of B
d [ B]
2 Z BC e NA
dt
1 d [ B] / dt Z BB e Ethr / RT
k r k[B]2
2 [ B]2 N A[ B]2
The use of (15.63) for ZBB
1/ 2
1 2 8 RT
k N Ad B
M
e Ethr / RT for B = C (23.4)
21 / 2 B
5
Physical Chemistry Reaction
Theories of Reaction Rates Kinetics
Hard-Sphere Collision Theory of Gas-Phase Reactions
1/ 2
8RT 1
for B C
1
k N A (rB rC ) 2
e Ethr / RT (23.3)
M B MC
1/ 2
1 2 8 RT
k N Ad B
M e Ethr / RT for B = C (23.4)
1/ 2
2 B
1 Ethr
ln k const ln T
2 RT
1 E d ln k
Ea RT 2 T thr Ea RT 2 (17.68)
2 RT 2 dT
1
Ea Ethr RT (23.5)
2
6
Physical Chemistry Reaction
Theories of Reaction Rates Kinetics
Hard-Sphere Collision Theory of Gas-Phase Reactions
1/ 2
2 8RT 1 1
k N A (rB rC ) M M
e Ethr / RT for B C (23.3)
B
C
1 1
1/2RT is small Ea RT Ea Ethr RT (23.5)
2 2
Ea Ethr Ea
A ke RT
(17.69)
1/ 2
8RT 1
for B C
1
A N A (rB rC ) 2
e 1 / 2 (23.6)
M B MC
The hard-sphere threshold energy is nearly the same as the
activation energy. The simple collision theory gives only the
pre-exponential factor A (but not for the calculation of Ethr)
7
Physical Chemistry Reaction
Kinetics
A comparison of theoretic calculation and experimental measurement
k0
T E k0 (theo)
Reaction 1011dm3 · mol-1 · s-1
K kJ· mol-1 k0 (cal)
measured cal.
K + Br2 KBr + Br 600 0 10 2.1 4.8
CH3 + CH3 C2H6 300 0 0.24 1.1 0.22
2NOCl 2NO + Cl2 470 102 0.094 0.59 0.16
CHO
CHO
+ 500 83 1.5×10 -5 3.0 5×10- 6
H2 + C2H4 C2H6 800 180 1.24×10 -5 7.3 1.7×10- 6
8
Physical Chemistry Reaction
Potential-Energy Surfaces Kinetics
The hard-sphere collision theory does not give accurate rate constants.
In chemical reactions, bonds are being formed and broken.
Intramolecular forces
Forces acting on atoms in the molecules
Intermolecular forces
Consider two molecules to form a
supermolecule
single quantum-mechanical entity
Only exists during collision
9
Physical Chemistry Reaction
Potential-Energy Surfaces Kinetics
Morse potential Energy
Ep (r ) De [exp{2a (r r0 )} 2 exp{ a(r r0 )}]
当r>r0时,有引力,即化学键力。
当r<r0时，有斥力。
=0时的能级为振动基态能级，
E0为零点能。
D0为把基态分子离解为孤立
原子所需的能量，它的值可
从光谱数据得到。
10
Physical Chemistry Reaction
Potential-Energy Surfaces Kinetics
For a reaction
A BC ABC AB C
A
Rab
C
B Rbc
If < 180o
Potential is a function
of Rab and Rbc only.
11
Physical Chemistry Reaction
Kinetics
Potential-Energy Surfaces
图中R点是反应物BC分子的基态,随着A原子的靠近,势能
沿着RT线升高，到达T点形成活化络合物。
随着C原子的离去，势能沿
着TP线下降，到P点是生成 A-----B-----C
物AB分子的稳态。
D点是完全离解为A,B,C原子
时的势能；OEP一侧,是原子
间的相斥能,也很高。 A-------B---C A---B-------C
12
Physical Chemistry Reaction
Transition-State Theory Kinetics
Transition-State Theory (TST)
Activation-Complex Theory (ACT)
The potential-energy surface for a reaction has a reaction region and
a product region that are separated by a barrier.
TST chooses a boundary surface located between the reactant and
product regions and assumes that all supermolecules that cross this
boundary surface (critical dividing surface) become products.
The critical dividing surface (Fig. 23.7) is
taken to pass through the saddle point of
the potential-energy surface.
saddle point The maximum point on the
minimum-energy path
13
Physical Chemistry Reaction
Transition-State Theory Kinetics
Transition-State Theory (TST)
Assumptions
(1) all supermolecules that cross the critical dividing surface from the
reactant side become products.
Once a supermolecule crosses the critical surface it is a downhill
journey to products.
(2) during the reaction the Boltzmann
distribution of energy is maintained for the
reactant molecules.
(3) the supermolecules crossing the
critical surface from the reactant side have
a Boltzmann distribution of energy
corresponding to the temperature of the
reacting system.
14
Physical Chemistry Reaction
Transition-State Theory Kinetics
Transition-State Theory (TST)
Assumptions
Not all supermolecules cross the dividing surface at precisely the
saddle point of the potential-energy surface.
Activated complex
Potential-energy
Any supermolecule whose nuclear
configuration corresponds to any point on
the dividing surface or to any point within
Minimum-energy path
a short distance beyond the dividing
surface.
15
Physical Chemistry Reaction
碰撞理论的优缺点 Kinetics
优点： 碰撞理论为我们描述了一幅虽然粗糙但十分
明确的反应图像，在反应速率理论的发展中起了很
大作用。
对阿仑尼乌斯公式中的指数项、指前因子和阈
能都提出了较明确的物理意义，认为指数项相当于
有效碰撞分数，指前因子A相当于碰撞频率。
它解释了一部分实验事实，理论所计算的速率系
数k值与较简单的反应的实验值相符。
缺点：但模型过于简单，所以要引入概率因子，且
概率因子的值很难具体计算。阈能还必须从实验活
化能求得，所以碰撞理论还是半经验的。
16
Physical Chemistry Reaction
Kinetics
过渡态理论的优缺点
优点：
1.形象地描绘了基元反应进展的过程；
2.原则上可以从原子结构的光谱数据和势能面计算宏
观反应的速率常数；
3.对阿仑尼乌斯的指前因子作了理论说明，认为它与
反应的活化熵有关；
4.形象地说明了反应为什么需要活化能以及反应遵循
的能量最低原理。
缺点：引进的平衡假设和速决步假设并不能符合所有
的实验事实；对复杂的多原子反应，绘制势能面有困
难，使理论的应用受到一定的限制。
17
Physical Chemistry Reaction
Transition-State Theory Kinetics
For ideal-gas reactions
Denoting an activated complex by X
f (forward direction)
B C X E F
f (23.8)
0
Equation (22.129) gives N
f z
e kT (23.9)
N B NC z B zC
0 0 ( X ) 0 ( B) 0 (C )
f
The quantity 0 differs from the classical
barrier b, because of the zero-point
vibrational species of X , B, C, …
f
18
Physical Chemistry Reaction
Transition-State Theory Kinetics
For ideal-gas reactions
Division of each N in (23.9) by NAV converts it to a molar concentration
Kf
X
f
z / N AV
0
e kT
(23.10)
[ B][C ] ( z B / N AV )(zC / N AV )
U 0
o
v
zi i
K c (ci )vi e RT VN
Ideal gas (22.128)
i i A
Kf
X
f equilibrium constant The activated
[ B][C ] complexes are in equilibrium with reactants
The activated complexes are not in a true chemical-reaction
equilibrium with the reacting system, are assumed to be in thermal
equilibrium with the reacting system populated according to
Boltzmann distribution appropriate to the system temperature
19
Physical Chemistry Reaction
Transition-State Theory Kinetics
For ideal-gas reactions
The partition function z of the activated complex is given by
z ztr zrot zvib zel (22.110)
'
zvib zrc zvib '
z zrc z (23.12)
' '
z ztr zrot zvib zel (23.13)
1 ( 2mrc kT )1 / 2
z rc (23.14)
2 h
1 (2mrc kT )1 / 2 '
z z (23.15)
2 h
e 0 / kT [ B][C ]
vrc z / N A N
r (23.16)
z B / N A N zC / N A N
20
Physical Chemistry Reaction
Transition-State Theory Kinetics
For ideal-gas reactions
The probability density of g(vrc) for vrc is
2
1 / 2 mrc vrc
m (23.17)
g (vrc ) 2 rc e 2kT
2kT
vrc 0 vrc g (vrc )dvrc 1/ 2
2kT
vrc
m
(23.18)
rc
1/ 2
1 2kT (2mrc kT )1 / 2 '
e( 0 / kT )[ B][C ]
z / N AV
r
mrc
2h ' '
z / N AV z / N AV
'
kT z / N AV ( 0 / kT )
kr
'
'
h z / N AV z / N AV
e
Ideal gas (23.19)
TST expression for the rate constant of an ideal-gas elementary reaction
21
Physical Chemistry Reaction
Transition-State Theory Kinetics
For ideal-gas reactions
Eyring kT '
z / N AV
( 0 / kT )
equation kr
'
'
h z / N AV z / N AV
e
Ideal gas (23.19’)
is transmission coefficient (0 < < 1), in many cases 1
Relation between TST and Hard-Sphere Collision Theory
For the bimolecular reaction B + C Products (B C)
3/ 2 3/ 2
2mB kT 2mC kT
zB / V
zC / V
2 2
h h
3/ 2
z ztr 2 (mB mC )kT mB mC kT
zrot
8 2 (rB rC ) 2
V V h 2
mB mC h2
22
Physical Chemistry Reaction
Transition-State Theory Kinetics
Relation between TST and Hard-Sphere Collision Theory
'
e( 0 / kT )
kT z / N AV
kr
'
'
h z / N AV z / N AV Ideal gas (23.19)
Substitution in equation (23.19) (for B C)
1/ 2
8kT mB mC
kr N A (rB rC ) 2
e( 0 / kT ) for B C (23.20)
mB mC
1/ 2
8RT 1 1
k N A (rB rC ) 2
M
e Ethr / RT for B C (23.3)
B M C
1/ 2
1 2 8 RT
k N Ad B
M e Ethr / RT for B = C (23.4)
1/ 2
2 B
TST reduces to the hard-sphere collision theory when the structure of
the molecules is ignored.
23
Physical Chemistry Reaction
Transition-State Theory Kinetics
Temperature dependence of the rate constant
ztr T 3 / 2 zrot ,lin T zrot , nonlin T 3 / 2 zel T 0
kr CT e
C and m are constants m E0 / RT (23.21)
E0 N A 0
Ea RT 2d ln kr / dT Taking the log of (23.21) and differentiating
Ea E0 mRT (23.22)
A kr e Ea / RT
kTem '
z / N AV
A Ce mT m (23.23)
h z B / N AV zC / N AV
24
