Combustion Chemistry
Hai Wang
Stanford University
2015 Princeton-CEFRC Summer School On Combustion
Course Length: 3 hrs
June 22 – 26, 2015
Copyright ©2015 by Hai Wang
This material is not to be sold, reproduced or distributed without prior written
permission of the owner, Hai Wang.
Lecture 5
5. Unimolecular Reactions
In the last lecture, we learned the collision and transition state theory that govern
bimolecular reactions. In this lecture, we shall discuss the reaction rate theory of
unimolecular reactions. It will become clear that the unimolecular reaction theory is also
applicable to a large number of bimolecular reactions of importance to combustion analysis.
Common to the class of reactions to be discussed here is the dependence of the rate
coefficients on the presence of a third body and therefore pressure.
5.1 Types of Unimolecular Reactions
There are several types of unimolecular reactions. The first and second types involve the
fragmentation of the reactant molecule with the difference that the back, association reaction
has or does not have an energy barrier (see, Figure 5.1). If the back reaction does not have
an energy barrier, we usually call this a dissociation reaction. Examples include the
dissociation of a molecular species, e.g.,
CH4 → CH3• + H•
C2H5 → CH3• + CH3•
dissociation
Potential energy
Potential energy
If an energy barrier exists, the reaction is usually termed as the unimolecular elimination
reaction. Examples are the β-scission reactions,
isomerization
Reaction coordinate
Potential energy
Potential energy
Reaction coordinate
elimination
Reaction coordinate
two-channel elimination/
chemically activated reaction
Reaction coordinate
Figure 5.1 Potential energy characterizing several types of unimolecular reactions.
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C2H3• → C2H2 + H•
n-C3H7• → C2H4 + CH3•
The third type of unimolecular reaction is called the isomerization reaction. The potential
energy is characterized by a double well connected by a potential barrier. Examples include
H-atom shift in the n-propyl radical:
CH3–CH2–CH2• ↔ CH3–CH•–CH3
Any of the above three types of reactions can be combined to form multi-channel,
competing unimolecular reaction reactions, as seen in Figure 5.1. Examples include the
dissociation of ethane,
C2H6 → CH3• + CH3•
C2H6 → C2H5• + H•
5.2 Chemically Activated Reactions
As we discussed before, an important characteristic of unimolecular reactions is that they
require collision to activate the reactants above the potential energy barrier before the
reaction can proceed to products. This activation process is thermal in nature (i.e., exchange
of kinetic energy between the reactant and a third body), and is termed as thermal activation
hereafter. Activated species can also come from association of two reactants. For example,
the association of CH3• and H• forms a vibrationally excited CH4, since upon association the
combined translation energy of CH3• and H• has gone into the vibrational energy in CH4.
There are two ways that this (excess) energy can be removed. Either CH4 has to redissociation, or it has to collide with a third body which removes this excess energy.
Now suppose that the activated species formed from combination of two reactants can
undergo dissociation following two competing channels. Then the dissociation of the
activated species needs not to go back to the same reactants. In fact, it can dissociate into
different products. Take the reaction of CH3• + CH3• as an example. The association
reaction produces a ro-vibrationally excited or hot [C2H6]* adduct,
CH3• + CH3• → [C2H6]* .
(5.1)
In contrast to thermally activated complex, the above [C2H6]* adduct or complex is
chemically activated, which can go back out to the reactants,
[C2H6]* → CH3• + CH3•,
(5.2a)
or it can dissociate into C2H5• + H•,
[C2H6]* → C2H5• + H•,
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(5.2b)
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or it can be stabilized by colliding with a third body M to acquire the Boltzmann distribution
[C2H6]* + M → C2H6 + M
(5.2c)
The net observables are two reactions,
CH3• + CH3• → C2H6
CH3• + CH3• → C2H5• + H•.
(5.3)
(5.4)
Reaction (5.3) is the back reaction of ethane dissociation, whereas reaction (5.4) is a
bimolecular reaction. Since the rate of reaction (5.2c) increases with an increase in the
concentration of third bodies or pressure, the population of the activated complex is also
dependent on pressure. Since the net rate of reaction (5.4) is the rate constant of reaction
(5.2b) multiplied by this population, the rate coefficient of reaction (5.4) is also dependent.
This class of bimolecular reactions is called the chemically activated reactions. The nature of
these reactions is the same as the unimolecular reaction, as we will discuss in this lecture.
5.3 Unimolecular Reaction Theory
We learned in Lecture 3 that the Lindemann treatment of unimolecular reactions. This
analysis starts by writing three separate rate processes:
A + M → A* + M
A* + M → A + M
A* → products,
(5.5f)
(5.5b)
(5.6)
A steady-state analysis yielded the following equation:
1
kuni
=
1
kuni ,0
+
1
kuni ,∞
,
(3.36)
where kuni ,0 is the rate coefficient at the low-pressure limit
kuni ,0 = k5 f !" M #$ ,
(5.7)
and kuni ,∞ is the high-pressure limit rate coefficient, independent of pressure
kuni ,∞ = k6 k5 f k5b
(5.8)
Figure 3.2 illustrates that as the pressure is decreased, the value of kuni initially stays
constant, but it eventually falls off towards the high-pressure limit. The behavior is known
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as the rate coefficient fall-off. This behavior exhibits itself not only for unimolecular
reactions, it also applies to bimolecular combination reactions. Consider the following
reaction
AB → A + B .
(5.9)
Let the rate coefficient of the forward reaction be kuni and that of the back reaction be kbi.
Since the forward and back rate coefficients are related by the equilibrium constant,
kbi =
kuni
kuni
.
=
K c K p RuT
(5.10)
and Kp is a function of temperature only, kbi must exhibit the same pressure dependence as
kuni.
5.3.1. The Hinshelwood Revision of Lindemann Mechanism
While the Lindemann mechanism captures the essence of unimolecular reactions, namely the
pressure dependence of their rate coefficients, it fails to predict the rate coefficient in a
quantitative fashion. Figure 5.2 shows the discrepancy between the actual experimental rate
coefficient and the Lindemann prediction for
C2H2 + H• → C2H3• .
(5.11)
Clearly, the Lindemann prediction is substantially larger than the actual rate values.
k (cm3mol-1s-1)
1012
Lindemann
k0
1011
Actual
1010
10-1
100
101
102
103
104
p (Torr)
Figure 5.2 Rate coefficient of reaction
(5.11) at T = 400 K as a function of
pressure. Data are taken from Payne, W.
A. and Stief, L. J. J. Chem. Phys. 64, pp.
1150-1155 (1976). The theoretical highand low-pressure limit rate coefficients
are taken from Knyazev, V. D. and Slagle,
I. R. J. Phys. Chem. 100, 16899-16911
(1996).
In 1926, Hinshelwood realized that the thermally activated molecule can assume different
rates of dissociation depending on its energy. Therefore he proposed that process (5.6)
should be written as
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A*(E) → products,
(5.6a)
with an energy-dependent rate constant k(E). This rate constant is basically a frequency
factor of a molecule with vibrational energy above the zero-point to undergo chemical
transformation. We shall term this energy-dependent rate constant as the microcanonical
rate constant hereafter.
Consider the potential energy diagram of a unimolecular reaction shown in Figure 5.3, where
the reaction energy barrier is denoted by E0. As before, this energy barrier is equal to the
potential energy difference from the zero-point energy of the reactant to the zero-point
energy of the transition state. Let E be an internal vibrational energy above the zero-point
energy of the molecule. Obviously, for E < E0, no reaction is possible and k(E) = 0.
Conversely, k(E) > 0 for E ≥ E0. The Hinshelwood treatment requires that kuni be treated
by considering k(E) averaged over the populations at all energy levels above the energy
barrier, i.e.,
kuni ∝
∑i k ( Ei ) f ( Ei )
∑i f ( Ei )
.
(5.12)
where f(E) is the population at the ith energy level.
A†
k(E)
)
Potential energy
A*
E
E†
E0
D H ro,0 K
Reaction coordinate
Figure 5.3 Schematic diagram of the
Lindemann-Hinshelwood mechanism.
5.3.2 The RRKM Theory
Further development of the theory was independently achieved by Rice and Ramsperger,
and Kassel. It was not until 1952 that Marcus rigorously demonstrated how the
microcanonical rate constant and thus the thermally-averaged rate constant could be
computed from knowledge of reaction potential energy surfaces. The resulting RRKM
theory, named after Rice, Ramsperger, Kassel and Marcus, has become the widely accepted
and used theory today.
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There are two principal assumptions in the RRKM theory. The first assumption is known as
the ergodicity assumption, namely the rapid randomization of energy throughout vibrationally
degrees of freedom. The second assumption is the existence of a critical geometry for the
transition from a vibrationally excited molecule to the products, i.e., step (5.6a) is expanded
to include a critical geometry or activated complex A†,
k( E )
†
A * ( E ) ! !!
→ A † ( E ) !k!→ products .
(5.6a)
The microcanonical rate constant is therefore the frequency factor for attaining the critical
geometry from all possible configurations of the excited molecule at energy E. In other
words, step (5.6a) denotes the rate of the generally excited molecule A* into the specifically
excited molecule A†, which has sufficient energy localized in the vibrational degree of
freedom so that it is ready to undergo transformation to the products (see, Figure 5.3). The
k† in (5.6a) is the frequency factor for the activated complex to undergo chemical
transformation. In general, k† is of the order of a vibrational frequency and k†  k(E). An
application of the steady state analysis for A† gives
k ( E ) = k † !# A † ( E )$& !# A * ( E )$& .
"
% "
%
(5.14)
Consider the formyl (HCO•) radical. The energy barrier of dissociation
HCO• → CO + H•
(5.15)
is 68.2 kJ/mol,* which is equivalent to E0 = 5700 cm–1 vibrational energy. The HCO• is a
nonlinear species. It has 3 normal-mode vibrational frequencies: ν1 = 1145 cm–1 for H-C-O
bending, ν2 = 1900 cm–1 for C-O stretch, and ν3 = 2750 cm–1 C-H stretch. Clearly the C-O
stretch mode is responsible for the elimination reaction. Assuming the three frequency
values are invariant during the elimination reaction, we may assign possible vibrational
quantum numbers (n1, n2, and n3) to each of the three vibrational degrees of freedom and
obtain the total vibrational energy E in the molecule, as shown in Table 5.1 for the first 120
vibrational energy states. Clearly, the first 16 such energy states are non-reactive, since the
total vibrational energy E < 5700 cm–1. At and above state 17, the molecule has enough
energy to undergo elimination. Therefore, A*(E) refers to these reactive energy states and
[A*(E)] is the population of a reactive energy state. An examination of Table 5.1 also tells us
that not every A*(E) can dissociate, because to do so it requires the C-H stretch to have at
least 3 quanta to overcome the energy barrier, i.e., 2×2750 = 5500 cm–1 < E0, but 3×2750 =
8250 cm–1 > E0. It follows that relatively a few energy states are truly ready for Helimination. These specific energy states correspond to the activated complex A† and
include 38, 51, 61, 67 etc (marked in bold face letters in Table 5.1).
*
Wagner, A. F. and Bowman, J. M. J. Phys. Chem. 91, 5314-5324 (1987).
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Table 5.1 Possible vibrational energy states in the HCO• radical. The Energies are
expressed in wavenumbers. E is the total vibrational energy and E3 is the energy of C-H
stretch.
State
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
n1
0
1
0
2
0
1
3
0
1
2
4
0
1
2
3
0
0
5
1
2
3
4
0
1
1
6
2
3
4
0
0
5
1
2
2
7
3
0
4
0
n2
0
0
1
0
0
1
0
2
0
1
0
1
2
0
1
0
3
0
1
2
0
1
2
0
3
0
1
2
0
1
4
1
2
0
3
0
1
0
2
3
n3
0
0
0
0
1
0
0
0
1
0
0
1
0
1
0
2
0
0
1
0
1
0
1
2
0
0
1
0
1
2
0
0
1
2
0
0
1
3
0
1
E
0
1145
1900
2290
2750
3045
3435
3800
3895
4190
4580
4650
4945
5040
5335
5500
5700
5725
5795
6090
6185
6480
6550
6645
6845
6870
6940
7235
7330
7400
7600
7625
7695
7790
7990
8015
8085
8250
8380
8450
E3 State n1
0
41 5
0
42 1
0
43 1
0
44 6
2750 45 2
0
46 3
0
47 3
0
48 8
2750 49 4
0
50 0
0
51 1
2750 52 0
0
53 5
2750 54 1
0
55 6
5500 56 2
0
57 2
0
58 7
2750 59 3
0
60 4
2750 61 0
0
62 4
2750 63 9
5500 64 0
0
65 5
0
66 1
2750 67 2
0
68 1
2750 69 6
5500 70 2
0
71 7
0
72 3
2750 73 0
5500 74 3
0
75 8
0
76 4
2750 77 0
8250 78 5
0
79 1
2750 80 0
n2
0
1
4
1
2
0
3
0
1
2
0
5
2
3
0
1
4
1
2
0
1
3
0
4
1
2
0
5
2
3
0
1
0
4
1
2
3
0
1
6
n3
1
2
0
0
1
2
0
0
1
2
3
0
0
1
1
2
0
0
1
2
3
0
0
1
1
2
3
0
0
1
1
2
4
0
0
1
2
2
3
0
E
8475
8545
8745
8770
8840
8935
9135
9160
9230
9300
9395
9500
9525
9595
9620
9690
9890
9915
9985
10080
10150
10280
10305
10350
10375
10445
10540
10645
10670
10740
10765
10835
11000
11035
11060
11130
11200
11225
11295
11400
E3
2750
5500
0
0
2750
5500
0
0
2750
5500
8250
0
0
2750
2750
5500
0
0
2750
5500
8250
0
0
2750
2750
5500
8250
0
0
2750
2750
5500
11000
0
0
2750
5500
5500
8250
0
State
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
n1
5
10
1
6
2
3
2
7
3
8
4
0
1
4
9
0
5
1
6
2
1
6
2
7
3
4
0
3
8
4
9
0
5
1
2
0
5
10
1
6
n2
3
0
4
1
2
0
5
2
3
0
1
2
0
4
1
5
2
3
0
1
6
3
4
1
2
0
1
5
2
3
0
4
1
2
0
7
4
1
5
2
n3
0
0
1
1
2
3
0
0
1
1
2
3
4
0
0
1
1
2
2
3
0
0
1
1
2
3
4
0
0
1
1
2
2
3
4
0
0
0
1
1
E
11425
11450
11495
11520
11590
11685
11790
11815
11885
11910
11980
12050
12145
12180
12205
12250
12275
12345
12370
12440
12545
12570
12640
12665
12735
12830
12900
12935
12960
13030
13055
13100
13125
13195
13290
13300
13325
13350
13395
13420
E3
0
0
2750
2750
5500
8250
0
0
2750
2750
5500
8250
11000
0
0
2750
2750
5500
5500
8250
0
0
2750
2750
5500
8250
11000
0
0
2750
2750
5500
5500
8250
11000
0
0
0
2750
2750
Though statistically we can easily count the number of excited energy states and “ready-togo” states, there is one problem here. As the H• atom dissociates away from the CO group,
ν1 and ν2 do not stay constant. In particular, the departure of the H• atom makes the H-CO bending a little easier, and leads to a smaller ν1 value (400 cm–1) in the activated complex.
In addition, the C-O bond also becomes stronger. This causes the C-O stretch to assume a
larger frequency (ν2 = 2120 cm–1 in the activated complex). In other words, the potential
energy associated with the activated complex A† can be very different from A*.
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It may be shown that by considering the equilibrium of A† and A* and partitioning the
density of states of the activated complex into that of a one-dimensional translation motion
and that of the remaining vibrational degrees of freedom, equation (5.14) may be written as
k ( E) =
( ).
W E†
(5.20)
hρ ( E)
( )
where ρ E is the density of energy states of the reactant A, i.e., the number of possible
vibrational energy states in the energy range of E to E + dE, and W E † is the sum of
states of the activated complex, i.e.,
( )
( )
†
( )
W E † = ∫ 0E ρ E † dE .
(5.21)
This is the RRKM expression for the microcanonical rate constant. Here the density of
states may be easily understood by examining Table 5.1. For example, the number of energy
states between 8000 and 9000 cm–1 is 11 (i.e., energy states from 36 to 46). Figure 5.4 shows
the density of states of the HCO• radical. The curve may be smoothed.
In the above discussion, the energies E and E† are purely vibrational energies. During a
bond breaking (dissociation/elimination) process, the molecule inevitably elongates itself,
leading to an increase in the moment of inertia for at least two rotational degrees of freedom,
and thus a decrease in the total rotational energy (see, for example, equations 2.37 and 2.38).
If the changes in the rotational energy are ignored, the molecule would undergo reaction as if
it loses energy adiabatically. This, of course, violates the first law of thermodynamics. To
account for the rotational energy variations, we correct equation (5.20) by adding the ratio of
the rotational partition function,
0.20
ρ (1/cm-1)
0.15
0.10
0.05
0.00
0
10000
20000
30000
40000
50000
E (cm-1)
5-8
Figure 5.4 Density of states of the HCO•
radical (dotted line: energy grain size = 100
cm–1, solid line: 250 cm–1)
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( )
†
qr† W E
.
k ( E) =
qr h ρ ( E )
(5.22)
Since the rotational partition function increases with an increase in the moment of inertia,
qr† qr > 1 . In this way, the rotational energy loss is recovered and used to “enhance” the
rate of dissociation/elimination.
Finally, we need to consider the reaction path degeneracy la. Consider H-elimination from
the vinyl radical (i.e., the back reaction of 5.11). Since C2H3• has two equivalent H• atoms,
both of which can be eliminated from C2H3•, the microcanonical rate constant is twice of
that of equation (5.22). Therefore the complete expression for the microcanonical rate
constant is
( )
†
qr† W E
.
k ( E) = l a
qr h ρ ( E )
(5.23)
5.3.3 Application of the RRKM Theory for Unimolecular Dissociation/Elimination
Reactions – The Strong Collision Model
A simple, single-channel unimolecular dissociation/elimination reaction may be described by
the following processes:
dk ( E )
a
AB ! !!
→ AB * ( E )
(5.24)
AB * ( E ) !ω!
→ AB
(5.25)
k( E )
( )
AB * ( E ) ! !!
→ AB † E † ! !
→A+B .
(5.26)
Here ka is the rate of activation due to collision of AB with a third body, and ω is the
collision frequency,
P
.
ω = kcoll !" M #$ = kcoll
RuT
(5.27)
A steady-state analysis for AB*(E) yields
AB * ( E ) =
dka ( E ) ω
1+ k ( E ) ω
5-9
!" AB#$ .
(5.28)
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Here the ratio dka ω is basically an equilibrium constant that quantifies the probability of
finding a molecule in the energy range of E to E + dE in a Boltzmann distribution of energy,
dka ( E ) ω = P ( E ) dE = ρ ( E )
exp (− E kBT )
zv
dE .
(5.29)
In other words, we assume here that the consumption of the excited molecules from the
reaction step (5.26) is small compared to collision deactivation. Equation (5.29) is
sometimes referred to as the equation of microscopic, detailed balancing.
The observable rate of reaction, i.e., the formation of A+B, is k(E)AB*(E) at a given energy
level. Putting equation (5.29) into (5.28), and integrating over all energy levels, we obtain the
rate coefficient for a unimolecular dissociation/elimination reaction as
kuni (T , P ) =
∫
∞
k ( E ) P(E)dE
E0
1+ k ( E ) ω
.
(5.30)
The above rate coefficient is often termed the thermal rate coefficient, since it is averaged over
all energy levels. At the low-pressure limit, ω → 0 and the above expression gives
kuni ,0 (T , P ) =
∫
=ω
∞
E0
∫
ω P ( E ) dE
∞
E0
ρ ( E ) exp (− E kBT ) dE
,
(5.31)
zv
which shows that the low-pressure limit rate coefficient is independent of the nature of the
activated complex, but it simply depends on or is limited by the rate of activation to energy
levels above the energy barrier E0. Because a larger molecule has more vibrational modes,
and a larger number of vibrational modes translates into a higher probability to find a
specific energy state within E to E+dE, the density of energy states ρ E and thus kuni ,0
are usually larger for larger molecules. In addition, molecules having smaller vibrational
frequencies also have larger ρ E and thus kuni ,0 .
( )
( )
The high-pressure limit rate coefficient is obtained by letting ω → ∞ . We have
kuni ,∞ (T ) =
∫
1
=
zv
∞
E0
k ( E ) P(E)dE
∫
∞
E0
.
k ( E ) ρ ( E ) exp (− E kBT ) dE
5-10
(5.32)
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In other words, kuni ,∞ is dependent on the nature of the activated complex. And the rate of
a unimolecular elimination/dissociation reaction is limited by the rate of conversion from
the excited molecule to the activated complex.
Equation (5.30) can be readily extended to multi-channel dissociation and elimination
reactions. In such cases, one only needs to replace the term k(E) in the numerator by ki(E),
the microcanonical rate constant of the ith channel, and in the numerator by a sum of ki(E)
over all channels in the denominator, i.e.,
kuni ,i (T , P ) =
∫
∞
ki ( E ) P(E)dE
E0 1+ ∑i ki
( E)
ω
.
(5.33)
5.3.4 Comparison of RRKM and Transition State Theories
We wish to demonstrate that at the high-pressure limit, the RRKM theory is consistent with
the transition state theory. Combining equations (5.23) and (5.32), we obtain
†
kuni ,∞ (T ) = l a
1 qr 1
h qr z v
∫
∞
E0
( )
W E † exp (− E kBT ) dE
.
†
1q 1
= la r
exp (− E0 kBT )
h qr z v
∞
∫ W ( E ) exp (− E
†
0
†
)
kBT dE
(5.34)
†
The sum of states may be replaced by the density of states (equation 5.21), and the integral
of the above equation is written by
∫
∞
E0
( )
†
W E exp (− E kBT ) dE =
∞$
E†
0
0
∫ ∫
&
&%
'
ρ ( E ) dE)exp − E † kBT dE † .
)(
(
)
(5.35)
Inverting the order of integration, one obtained
∫
∞
E0
( )
W E † exp (− E kBT ) dE = kBT
∞
∫ ρ ( E ) exp (− E
†
0
†
)
kBT dE †
.
(5.36)
= kBTz v†
Thus equation (5.34) is simply
kuni ,∞ (T ) = l a
kBT qr†z v†
exp (− E0 kBT ) .
h qr z v
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(5.37)
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Compared to the Eyring equation (4.71), we see that the above expression is entirely
consistent with the transition state theory.
5.3.5 Weak Collisions
In the above discussion, we equated the rate of collision stabilization of an excited molecule
to the rate of collision itself. Here the excess energy in the molecule is removed by a single
collision. This collision model, commonly referred to as the strong collision model, is
usually inadequate, as collision energy removal usually requires several collisions. In contrast
to the strong collision model, a weak collision model is needed to predict the pressure
dependence of kuni(T,P).
For single-channel unimolecular reactions with an appreciable energy barrier, Troe*
introduced a modified-strong collision model, in which the collision frequency w in
equation (5.30) is replaced by βω , where β ( 0 < β ≤ 1 ) is a collision efficiency factor, and
equation (5.30) is replaced by
kuni (T , P ) =
∫
( E) P(E)dE
.
1+ k ( E ) βω
∞k
E0
(5.38)
Troe derived a formula approximating the collision efficiency factor as
β
1− β 1 2
≈
− E
(1< FE < 3) ,
FEkBT
(5.59)
where − E is the average internal vibrational energy transferred between the reactant
molecule and a third body in up and down directions, and FE is the energy dependent
density of states
∞
FE =
∫E P ( E)dE
0
P ( E ) kBT
.
(5.60)
Alternatively, equation (5.59) may be expressed as a function of average energy in down
transitions Edown , i.e., the average energy removed from a vibrational excited, reactant
molecule per collision,
− E ≈
*
Edown
2
Edown + kBT
Troe, J. J. Chem. Phys. 83, 114 (1979).
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,
(5.61)
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⎡
⎤2
E
down
⎥ .
β ≈ ⎢⎢
⎥
⎢⎣ Edown + FEkBT ⎥⎦
(5.62)
In general, − E and Edown are unknown quantities. They are often used as adjustable
variables to fit experimental data. Their values with typical third bodies are given in Table
5.2.
Table 5.2 Typical − E and Edown values
Third body
-­‐ E
Ar
130
CO
92
H2
160
N2
130
Edown
260
190
280
260
For large molecules at high temperatures, FE is usually greater than 3. In that case, a more
involved expression should be used to obtain the collision efficiency,
β≈
∫
⎡
⎤2
Edown
⎢
⎥
⎢ E
⎥
+
F
k
T
⎢⎣ down
E B ⎥⎦
.
E0
⎡
⎛ E − E ⎞⎟⎤
FEkBT
⎜
0
⎢
⎥
⎟ dE
P ( E ) ⎢1−
exp⎜⎜−
⎜⎝ FEkBT ⎟⎟⎠⎥⎥
Edown + FEkBT
0
⎢⎣
⎦
(5.63)
5.3.6 Unimolecular Isomerization Reactions
A unimolecular , mutual isomerization reaction may be expressed by
k f (T ,P )
⎯⎯
⎯⎯
⎯⎯
→B .
A←
⎯
⎯
(5.64)
kb (T ,P )
We can formulate the unimolecular processes in a more detailed form within the framework
of a strong collision model as follows
dk ( E)
k′ ( E)
ω
ω
kb′( E)
dk1′( E)
f
⎯⎯
⎯1 ⎯
⎯⎯
→ A * ( E )←
⎯⎯
⎯
⎯⎯
→ B * ( E )←
⎯⎯
⎯⎯
⎯⎯
→B .
A←
⎯
(5.65)
The rates for the disappearance of A and the formation of B is
d ⎡⎣ B ⎤⎦
dt
{
}
= k f ⎡⎣ A ⎤⎦ −kb ⎡⎣ B ⎤⎦  − dk1′ ( E ) ⎡⎣ B ⎤⎦ − ω ⎡⎢ B * ( E )⎤⎥ dE .
⎣
⎦
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d ⎡ A⎤
− ⎣ ⎦ = k f ⎡⎣ A ⎤⎦ −kb ⎡⎣ B ⎤⎦  dk1 ( E ) ⎡⎣ A ⎤⎦ − ω ⎡⎢ A * ( E )⎤⎥ dE .
⎣
⎦
dt
{
}
(5.67)
Mass conservation requires that
{
} {
}
− dk1′ ( E ) ⎡⎣ B ⎤⎦ − ω ⎡⎢ B * ( E )⎤⎥ = dk1 ( E ) ⎡⎣ A ⎤⎦ − ω ⎡⎢ A * ( E )⎤⎥ ,
⎣
⎦
⎣
⎦
(5.68)
and
⎡B* E ⎤ =
⎢⎣ ( )⎥⎦
dk1 ( E ) ⎡⎣ A ⎤⎦ − ω ⎡⎢ A * ( E )⎤⎥ + dk1′ ( E ) ⎡⎣ B ⎤⎦
⎣
⎦
.
ω
(5.69)
We now assume steady state for the excited molecules A* and B*,
d ⎡⎢ A * ( E )⎤⎥
⎣
⎦ = dk ( E ) ⎡ A ⎤ − ω ⎡ A * ( E )⎤ − k ′ ( E ) ⎡ A * ( E )⎤ + k ′ ( E ) ⎡ B * ( E )⎤ = 0 , (5.70)
⎢⎣
⎥⎦
⎢⎣
⎥⎦
⎢⎣
⎥⎦
1
f
b
⎣ ⎦
dt
d ⎡⎢ B * ( E )⎤⎥
⎣
⎦ = dk ′ ( E ) ⎡ B ⎤ − ω ⎡ B * ( E )⎤ − k ′ ( E ) ⎡ B * ( E )⎤ + k ′ ( E ) ⎡ A * ( E )⎤ = 0 (5.71)
⎢⎣
⎥⎦ b
⎢⎣
⎥⎦
⎢⎣
⎥⎦
1
f
⎣ ⎦
dt
Solving the above equations for ⎡⎢ A * ( E )⎤⎥ and ⎡⎢ B * ( E )⎤⎥ and substituting the results into
⎣
⎦
⎣
⎦
equation (5.66), it is possible to show that
k f ( E ) PA ( E )dE
k ( E ) PB ( E )dE
− dk1′ ( E ) ⎡⎣ B ⎤⎦ − ω ⎡⎢ B * ( E )⎤⎥ dE =
− b
,
⎣
⎦
k f ( E ) kb ( E )
k f ( E ) kb ( E )
1+
+
1+
+
ω
ω
ω
ω
{
}
(5.72)
or
k f (T , P ) =
∫
∞
E0
k f ( E ) PA ( E )dE
1+
k f ( E)
ω
+
kb ( E )
,
(5.73)
.
(5.74)
ω
and
kb (T , P ) =
∫
∞
E0
kb ( E ) PB ( E )dE
1+
k f ( E)
5-14
ω
+
kb ( E )
ω
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In principle, the collision frequency in the above two equations may be replaced by βω to
account for the effect of weak collision, e.g.,
k f (T , P ) =
∫
∞
E0
k f ( E ) PA ( E )dE
k f ( E)
k ( E)
1+
+ b
βω
β ′ω
.
(5.75)
Since the natures of the reactant A and product B are different, their collision efficiencies are
not equal. Presently there are no simple, analytical approximations available to obtain the
values of β and β ′ , and as such equation (5.75) serves only as an intuitive guide to the
nature of unimolecular isomerization reaction. Accurate solution for the thermal rate
constant is obtained from the solution of master equation of collision energy transfer, as will
be discussed in section 5.4.
5.3.7 Chemically Activated Reactions
We discussed earlier that reaction (5.4) is a chemically activated reaction. The rate
coefficient of such reactions is dependent on both pressure and temperature. Within the
framework of the modified strong collision model, we may write the simplest chemically
activated reaction in the following form:
k1 f
k
2
⎯⎯
⎯⎯
→ AB * ⎯ ⎯
A + B←
→C + D
k1b
↓ βω
AB
.
(5.76)
That is, following the addition of A and B, the vibrationally excited adduct AB* may
decompose back to A and B; it may decompose to C and D, or it may be stabilized by
colliding with third bodies. The potential energy of this type of reaction has already been
discussed (see, Figure 5.1). Figure 5.5 shows a schematic energy diagram of reaction (5.76).
AB *
E
A+B
C+D
D Hr ,0 K
E0,1
E0,2
Figure 5.5 Schematic energy diagram of a
chemically activated reaction.
AB
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The observable reactions are
k
bi
A + B ⎯⎯
→ AB ,
(5.77)
kca
A + B ⎯ ⎯→ C + D
(5.78)
where the rate coefficient of the chemically activated reaction is denoted by kca. Here we
shall develop the mathematical expression for kca.
An application of the steady state analysis for [AB*] yields
k1 f ( E )
⎡ AB * E ⎤ =
⎡ A⎤ ⎡B⎤ ,
( )⎥⎦
⎢⎣
k1b ( E ) + k2 ( E ) + βω ⎣ ⎦ ⎣ ⎦
(5.79)
The rate coefficient of the chemically activated reaction is
kca =
∫
k1 f ( E + ΔH r ,0K )k2 ( E ) PA+B ( E + ΔH r ,0K )dE
∞
k1b ( E ) + k2 ( E ) + βω
max( E0,1 ,E0,2 )
,
(5.79)
where ΔH r ,0K is the enthalpy of reaction of the entrance channel (5.77), PA+B is the
combined Boltzmann distribution of energy of separate A and B,
exp ⎡⎢−( E + ΔH r ,0K ) kBT ⎤⎥
⎣
⎦.
PA+B ( E + ΔH r ,0K ) = ρ A+B ( E + ΔH r ,0K )
z Az B
(5.80)
The microscopic, detailed balancing requires that
k1 f ( E + ΔH r ,0K )ρ A+B ( E + ΔH r ,0K ) = k1b ( E)ρ AB ( E) .
(5.81)
It follows that
exp ⎡⎢−( E + ΔH r ,0K ) kBT ⎤⎥
⎣
⎦ . (5.82)
PA+B ( E + ΔH r ,0K ) =
ρ AB ( E )
z
z
k1 f ( E + ΔH r ,0K )
A B
k1b ( E )
The equilibrium constant of reaction (5.77) may be expressed by (cf, equation 4.63)
Kc =
z AB
exp ⎡⎢−ΔH r ,0K kBT ⎤⎥ .
⎣
⎦
z Az B
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Combining equation (5.82) and (5.83), we obtain
kca = K c
∫
( E)k1b ( E) PAB ( E)dE
.
k1b ( E ) + k2 ( E ) + βω
∞k
2
E0,1
(5.84)
Thus the determination of the rate coefficient of chemically activated reaction uses
parameters identical to those of multi-channel unimolecular reaction. The bimolecular
combination rate constant is obtained in a similar fashion,
kbi = K c
∫
∞
βωk1b ( E ) PAB ( E )dE
E0,1 k1b
( E) + k2 ( E) + βω
,
(5.85)
which is essentially identical to equation (5.33), developed earlier for multi-channel
unimolecular reactions.
It is evident that from equation (5.84) at the low-pressure limit, kca is independent of
pressure, whereas at the high pressure limit, kca is inversely proportional to pressure, since
kca =
Kc
βω
∫
∞
E0,1
k2 ( E )k1b ( E ) PAB ( E )dE .
(5.86)
5.3.8 The Problem of Modified Strong Collision Model
In recent years, the modified strong collision model has been largely abandoned because of
its limited applicability. It is known that equations (5.59), (5.62) and (5.63) are applicable to
only single-channel unimolecular reactions with large energy barriers. For multi-channel
unimolecular reactions and chemically activated reactions, it is necessarily to consider the
rates of detailed collision energy transfer processes for each molecule considered in a
unimolecular reaction. Such a treatment is called the solution of master equation of collision
energy transfer, as will be discussed briefly in section 5.4.
5.3.9 Angular Momentum Conservation
We shall revisit the expression of the microcanonical rate constant in this section. In
equation (5.23), we accounted for rotational contributions to k(E) by considering the ratio of
the partition functions of the transition complex and the reactant molecule. This approach
is quite adequate for unimolecular elimination reactions, in which the back association
reaction has an appreciable energy barrier. For dissociation reactions, however, the
transition state is often not clearly defined, since in most cases the back association of two
free radicals does not have an energy barrier. Consider the dissociation of methane,
CH 4 → CH 3 i+H i .
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Potential energy
The back reaction is known to be barrierless, if only electronic and zero-point vibrational
energies are considered in our construction of the potential energy function for reaction.
This potential is equivalent to zero rotational energy (or rotational quantum number J = 0).
If the initial reactant molecule has J > 0, under the same J value the increase in the C-H
bond distance causes the molecule to increase its external rotational energy. Since the energy
transfer with the molecule is assumed to be adiabatic, the increase in the rotational energy
will have to be compensated by a decrease in the vibrational energy. This phenomenon
causes an apparent energy barrier in the association direction, so long as J > 0 (see, Figure
5.6).
J>0
J=0
Reaction coordinate
Figure 5.6 Comparison of potential energy with J = 0
and J > 0 for a dissociation reaction.
An alterative observation may be made by considering the back, association reaction. For
any off-center collision, the complex formed from the two reactants (CH3• and H•) will
undergo rotation. Since this rotation represents a centrifugal force that the two reactants
must overcome in order for them to associate, there exists a rotational energy barrier at each
quantum number J.
In general the combined electronic and zero-point vibrational energy may be expressed as a
function of the separation distance r of the two combining/dissociating species in the form
of the Morse potential,
2
⎡
−(r−re ) ⎤
V (r ) = ΔH r ,0K ⎢1−e
⎥ ,
⎢⎣
⎥⎦
(5.88)
where re is the equilibrium distance (i.e., the C-H distance in CH4). Obviously, the above
equation gives V (re ) = 0 and V (r → ∞) = ΔH r ,0K , as schematically illustrated in Figure
5.6 as J = 0. If we consider further the potential energy arising from the centrifugal force of
rotation, we have an effective potential function
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2
I
⎡
−(r−re ) ⎤
V (r , J ) = ΔH r ,0K ⎢1−e
⎥ + e 2 Erot
⎢⎣
⎥⎦
µr
= ΔH r ,0K
2
⎛ h 2 ⎞⎟ 1
⎡
−(r−re ) ⎤
⎜
⎢1−e
⎥ + J ( J +1)⎜⎜ 2 ⎟⎟⎟ 2
⎜⎝ 8π c ⎟⎠ µr
⎥⎦
⎣⎢
,
(5.89)
where Ie is the moment of inertia of the equilibrium geometry (i.e., the reactant), and Erot is
the rotational energy of the reactant, i.e.,
Erot = J ( J +1) Be .
(5.90)
Equation (5.89) states that for J > 0 the rotational contribution to V (r , J ) is zero when the
separation distance r → ∞. At a finite distance of separation, the rotational contribution is
greater than zero, thereby an effective rotational energy barrier is introduced, as depicted in
Figure 5.6 for J > 0.
The transition state is then J-dependent, and may be estimated from equation (5.89) by
deriving a separation distance r†(J) that corresponds to a maximum V (r , J ) . The larger the J
value, the higher the effective energy barrier, and the smaller the r† value.
The need of angular momentum conservation leads us to consider a microcanonical rate
constant, which may be expressed as functions of both internal vibrational energy and
external rotational energy or quantum number, i.e.,
k ( E, J ) = l a
(
W E† , J
).
hρ ( E, J )
(5.91)
In fact, equation (5.23) is a simplified version of the above equation.
The application of equation (5.91) in thermal rate coefficient calculation requires the
integration over internal vibrational energy as well as the rotational quantum number. For
example, the high pressure limit rate coefficient may be given by
kuni ,∞ (T ) =
1
zv
∞
∫ ∫
0
∞
E0
k ( E, J )ρ ( E, J ) exp(−E kBT )dE dJ .
(5.92)
5.4 Master Equation of Collision Energy Transfer
Rovibrationally excited molecules are subject to competitive processes of collision
activation/deactivation and unimolecular isomerization, dissociation or elimination. The
time evolution of such a system is described by the master equation of collision energy
transfer. In the discrete form the equation is given by
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d[ A ( Ei ) ]
dt
©Hai Wang
( )
= ∑ j kij ⎡⎣ M ⎤⎦ ⎡ A E j ⎤ − k ji ⎡⎣ M ⎤⎦ ⎡⎣ A ( Ei )⎤⎦ − ∑ m km ( Ei ) ⎡⎣ A ( Ei )⎤⎦
⎣
⎦
(5.93)
( )
In the above equation, A E j is a species at a specific energy state Ei, [] denotes the
concentration, M is a third body, kij is the second-order rate constant for the collisional
energy transfer of a molecule from state Ej to state Ei and km ( Ei ) is the microcanonical rate
constant for the mth channel of isomerization, dissociation or elimination. The first term on
the right hand side of the above equation is the rate at which the population at the energy
state Ei is created, by collision of molecules at energy state j with the third body M. The
second and third terms account for the disappearance of A E j as it collides with a third
body, and as it undergo unimolecular reaction, respectively.
( )
An energy transition probability Pji may be defined as
P ji =
k ji
kcoll
,
(5.94)
where kcoll is the collision rate constant. Since the total probability is unity for collision
transition of A E j into all possible energy states j = 1,…∞, we have ∑ j P ji = 1 , . The
master equation may be re-written in terms of Pij as
( )
d ⎡⎣ A ( Ei )⎤⎦
dt
= kcoll ⎡⎣ M ⎤⎦
{∑ P ⎡⎣ A ( E )⎤⎦ − ⎡⎣ A ( E )⎤⎦} − ∑ k ( E ) ⎡⎣ A ( E )⎤⎦ .
j
ij
j
i
m
m
i
i
(5.95)
Various models have been proposed to calculate the transition probabilities, such as the
exponential down model, the stepladder model, the Poisson model, the Gaussian model and
the biased random walk model. The most-used model has been the exponential down
model,
Pij = C j exp ⎡ −α E j − Ei ⎤
j ≥i
(5.96a)
⎣
⎦
(
)
for upward transition, and
Pij = C i
ρi
exp ⎡ − (α + β ) Ei − E j ⎤
⎣
⎦
ρj
(
)
j <i
(5.96b)
for down transition, where Ci is the normalization factors, α is the reciprocal of the average
energy transferred in the ‘down’ transitions,
α=
1
Edown
5-20
(5.97)
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and β = 1 kT . Equations (5.96a) and (5.96b) satisfy the principle of detailed balancing,
⎛ Ej ⎞
⎛ E ⎞
P ji ρ ( Ei ) exp ⎜ − i ⎟ = Pij ρ E j exp ⎜ − ⎟
⎝ kT ⎠
⎝ kT ⎠
( )
(5.98)
which ensures that the distribution of a non-reacting system (i.e., km(E) ≡ 0) is the
Boltzmann distribution.
5.5 Parameterization of Unimolecular Reaction Rate Coefficient
In this lecture we learned the basic theories that govern unimolecular reactions. In
application, it is important to accurately capture the pressure and temperature dependence of
unimolecular reaction rate coefficients. The form of parameterization, currently used in
combustion modeling, is largely due to the work of Troe. The starting point of this
parameterization is the Lindemann Theory, which is now re-written as
kuni (T , P )
kuni ,∞ (T )
kuni ,0 (T , P )
kuni ,∞ (T )
=
1+
kuni ,0 (T , P )
F (T , P ) ,
(5.99)
kuni ,∞ (T )
where the ratio of low- and high-pressure limit rate coefficient is sometime termed as the
reduced pressure Pr, i.e.,
Pr =
kuni ,0 (T , P )
kuni ,∞ (T )
,
(5.100)
and the function F(T,P) satisfies the limiting condition of F(T,P→0) = F(T,P→∞) = 1, and 0
< F(T,0<P<∞) < 1. In Troe’s form, F is given as a function of the central broadening factor Fc
x
F = Fc (T ) ,
(5.101)
where x is empirically correlated to the reduced pressure as
−1
⎧⎪ ⎡ log P + c
⎤ ⎫⎪
⎪ ⎢
⎪
r
⎥
x = ⎨1+ ⎢
⎬ .
⎪⎪ ⎢ n −d ( log Pr + c ) ⎥⎥ ⎪⎪
⎦ ⎪⎭
⎪⎩ ⎣
(5.102)
The constants in equation (5.102) are
c = −0.4 − 0.67 log Fc ,
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(5.103)
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n = 0.75−1.27 log Fc ,
d = 0.14 .
(5.104)
(5.105)
The central broadening factor is a function of temperature only and may be parameterized
by
Fc = (1−a )e−T
T ***
+ ae−T
T*
+ e−T
**
T
,
(5.106)
where a, T*, T**, and T*** are parameters specific to a given reaction.
In the form of above parameterization, the description of kuni(T,P) requires the expressions
of high- and low-pressure limit rate coefficients, and the four parameters appeared in
equation (5.106). For example, the ChemKin format for unimolecular/bimolecular
association reactions is
H+CH3(+M)<=>CH4(+M)
1.390E+16
-.534
536.00
LOW / 2.620E+33
-4.760
2440.00/
TROE / 0.7830
74.0 2941.0 6964.0/
H2/2.0/ H2O/6.0/ CH4/3.0/ CO/1.5/ CO2/2.0/ C2H6/3.0/ AR/ .70/
where the parameters in first line (A∞, n∞, and E∞) are those of the high-pressure limit rate
coefficient,
k∞ = A∞T
n∞
⎛ E ⎞
exp⎜⎜⎜− ∞ ⎟⎟⎟ ,
⎜⎝ R u T ⎟⎠
(5.107)
those in the second line A0, n0, and E0 are for the low-pressure limit rate coefficient,
⎛ E ⎞
n
k0 [ M ]′ = A0T 0 exp⎜⎜⎜− 0 ⎟⎟⎟ ,
⎜⎝ R u T ⎟⎠
(5.108)
and the third line gives the parameters (a, T***, T*, and T**) for Fc(T). In equation (5.108),
[ M ]′ is a revised molar concentration of the gas, which accounts for different third body
efficiency of collision energy transfer, i.e.,
[ M ]′ = ∑ i γi c i ,
(5.109)
where γi is the relative collision efficiency factor of the ith species (given in the fourth line,
relative to N2), and ci is its molar concentration. Species not specified in the fourth line have
a default γi value of unity.
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The above discussion also points to the fact that fitting the unimolecular reaction rate
coefficient essentially involves the determination of Fc value at a given temperature, followed
by fitting these Fc values as a function of T in the form of equation (5.106). The
determination of Fc is carried out for Pr = 1 by fitting the following equation,
⎡ k T , P = 1) ⎤
⎡ ⎛ −0.4 −0.67 log F ⎞⎤−1
r
⎥ .
c ⎟⎟⎥ log F T = log ⎢ 2 uni (
⎢1+ ⎜⎜
(
)
⎢
⎥
c
⎢ ⎜⎜ 0.806−1.176 log F ⎟⎟⎥
k
T
⎢⎣
⎢⎣ ⎝
c ⎠⎥⎦
uni ,∞ ( ) ⎥⎦
5-23
(5.99)