Combustion Chemistry - Princeton University · In the last lecture, we learned the collision and transition state theory that govern bimolecular reactions. In this lecture, we shall

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.

5-1

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•

If an energy barrier exists, the reaction is usually termed as the unimolecular elimination reaction. Examples are the β-scission reactions,

Figure 5.1 Potential energy characterizing several types of unimolecular reactions.

Reaction coordinate

Pot

enti

al e

nerg

y

Reaction coordinate

Pot

enti

al e

nerg

y

Reaction coordinate

Pot

enti

al e

nerg

y

Reaction coordinate

Pot

enti

al e

nerg

y

dissociation elimination

isomerization two-channel elimination/chemically activated reaction

Reaction coordinate

Pot

enti

al e

nerg

y

Reaction coordinate

Pot

enti

al e

nerg

y

Reaction coordinate

Pot

enti

al e

nerg

y

Reaction coordinate

Pot

enti

al e

nerg

y

dissociation elimination

isomerization two-channel elimination/chemically activated reaction

Stanford University ©Hai Wang Version 1.2

5-2

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 re-dissociation, 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•, (5.2b)


5-3

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 (5.3) CH3• + CH3• → C2H5• + H•. (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 (5.5f) A* + M → A + M (5.5b) A* → products, (5.6) A steady-state analysis yielded the following equation:

1kuni

=1

kuni ,0+

1kuni ,∞

, (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 unik initially stays constant, but it eventually falls off towards the high-pressure limit. The behavior is known


5-4

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 =kuniKc

=kuni

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.

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 high- and 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

1010

1011

1012

10-1 100 101 102 103 104

k (c

m3 m

ol-1

s-1 )

p (Torr)

0k Lindemann

Actual


5-5

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 ∝k Ei( ) f Ei( )i∑f Ei( )i∑

. (5.12)

where f(E) is the population at the ith energy level.

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.

,0r KHDo

0EE

k(E)) †E

Reaction coordinate

Pot

enti

al e

nerg

y

A† A*


5-6

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†, A* E( ) k 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 H-elimination. 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).


5-7

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 n1 n2 n3 E E3 State n1 n2 n3 E E3 State n1 n2 n3 E E3

1 0 0 0 0 0 41 5 0 1 8475 2750 81 5 3 0 11425 0 2 1 0 0 1145 0 42 1 1 2 8545 5500 82 10 0 0 11450 0 3 0 1 0 1900 0 43 1 4 0 8745 0 83 1 4 1 11495 2750 4 2 0 0 2290 0 44 6 1 0 8770 0 84 6 1 1 11520 2750 5 0 0 1 2750 2750 45 2 2 1 8840 2750 85 2 2 2 11590 5500 6 1 1 0 3045 0 46 3 0 2 8935 5500 86 3 0 3 11685 8250 7 3 0 0 3435 0 47 3 3 0 9135 0 87 2 5 0 11790 0 8 0 2 0 3800 0 48 8 0 0 9160 0 88 7 2 0 11815 0 9 1 0 1 3895 2750 49 4 1 1 9230 2750 89 3 3 1 11885 2750 10 2 1 0 4190 0 50 0 2 2 9300 5500 90 8 0 1 11910 2750 11 4 0 0 4580 0 51 1 0 3 9395 8250 91 4 1 2 11980 5500 12 0 1 1 4650 2750 52 0 5 0 9500 0 92 0 2 3 12050 8250 13 1 2 0 4945 0 53 5 2 0 9525 0 93 1 0 4 12145 11000 14 2 0 1 5040 2750 54 1 3 1 9595 2750 94 4 4 0 12180 0 15 3 1 0 5335 0 55 6 0 1 9620 2750 95 9 1 0 12205 0 16 0 0 2 5500 5500 56 2 1 2 9690 5500 96 0 5 1 12250 2750 17 0 3 0 5700 0 57 2 4 0 9890 0 97 5 2 1 12275 2750 18 5 0 0 5725 0 58 7 1 0 9915 0 98 1 3 2 12345 5500 19 1 1 1 5795 2750 59 3 2 1 9985 2750 99 6 0 2 12370 5500 20 2 2 0 6090 0 60 4 0 2 10080 5500 100 2 1 3 12440 8250 21 3 0 1 6185 2750 61 0 1 3 10150 8250 101 1 6 0 12545 0 22 4 1 0 6480 0 62 4 3 0 10280 0 102 6 3 0 12570 0 23 0 2 1 6550 2750 63 9 0 0 10305 0 103 2 4 1 12640 2750 24 1 0 2 6645 5500 64 0 4 1 10350 2750 104 7 1 1 12665 2750 25 1 3 0 6845 0 65 5 1 1 10375 2750 105 3 2 2 12735 5500 26 6 0 0 6870 0 66 1 2 2 10445 5500 106 4 0 3 12830 8250 27 2 1 1 6940 2750 67 2 0 3 10540 8250 107 0 1 4 12900 11000 28 3 2 0 7235 0 68 1 5 0 10645 0 108 3 5 0 12935 0 29 4 0 1 7330 2750 69 6 2 0 10670 0 109 8 2 0 12960 0 30 0 1 2 7400 5500 70 2 3 1 10740 2750 110 4 3 1 13030 2750 31 0 4 0 7600 0 71 7 0 1 10765 2750 111 9 0 1 13055 2750 32 5 1 0 7625 0 72 3 1 2 10835 5500 112 0 4 2 13100 5500 33 1 2 1 7695 2750 73 0 0 4 11000 11000 113 5 1 2 13125 5500 34 2 0 2 7790 5500 74 3 4 0 11035 0 114 1 2 3 13195 8250 35 2 3 0 7990 0 75 8 1 0 11060 0 115 2 0 4 13290 11000 36 7 0 0 8015 0 76 4 2 1 11130 2750 116 0 7 0 13300 0 37 3 1 1 8085 2750 77 0 3 2 11200 5500 117 5 4 0 13325 0 38 0 0 3 8250 8250 78 5 0 2 11225 5500 118 10 1 0 13350 0 39 4 2 0 8380 0 79 1 1 3 11295 8250 119 1 5 1 13395 2750 40 0 3 1 8450 2750 80 0 6 0 11400 0 120 6 2 1 13420 2750

Though statistically we can easily count the number of excited energy states and “ready-to-go” 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-C-O 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*.


5-8

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†( )hρ E( )

. (5.20)

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†( ) = ρ E†( )0E†∫ 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,

Figure 5.4 Density of states of the HCO• radical (dotted line: energy grain size = 100 cm–1, solid line: 250 cm–1)

0.00

0.05

0.10

0.15

0.20

0 10000 20000 30000 40000 50000

ρ (1/cm-1)

E (cm-1)


5-9

k E( ) = qr†

qr

W E†( )hρ 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

k E( ) = l aqr†

qr

W E†( )hρ 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:

ABdka E( )! →!! AB* E( ) (5.24)

AB* E( ) ω! →! AB (5.25)

AB* E( ) k 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,

ω = kcoll M!" #$= kcollPRuT

. (5.27)

A steady-state analysis for AB*(E) yields

AB* E( ) =dka E( ) ω1+k E( ) ω

AB!" #$ . (5.28)


5-10

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( )

zvdE . (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)dE1+k E( ) ωE0

∞

∫ . (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( ) = ωP E( )dEE0∞

∫

=ωρ E( )exp −E kBT( )dEE0

∞

∫zv

, (5.31)

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( ) = k E( )P(E)dEE0∞

∫=1zv

k E( )ρ E( )exp −E kBT( )dEE0∞

∫. (5.32)


5-11

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)dE1+ ki E( )i∑ ωE0

∞

∫ . (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 a1h

qr†

qr

1zv

W E†( )exp −E kBT( )dEE0∞

∫

= l a1h

qr†

qr

1zvexp −E0 kBT( ) W E†( )exp −E† kBT( )dE†0

∞

∫. (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

W E†( )exp −E kBT( )dEE0∞

∫ = ρ E( )dE0E†

∫$

%&&

'

())exp −E† kBT( )dE†0

∞

∫ . (5.35) Inverting the order of integration, one obtained

W E†( )exp −E kBT( )dEE0

∞

∫ = kBT ρ E†( )exp −E† kBT( )dE†0∞

∫= kBTzv

†

. (5.36)

Thus equation (5.34) is simply

kuni ,∞ T( ) = l akBT

h

qr†zv†

qrzvexp −E0 kBT( ) . (5.37)


5-12

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


5-13

β ≈Edown

Edown +FEkBT

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

2

. (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-‐ downE Ar 130 260 CO 92 190 H2 160 280 N2 130 260

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 ≈

EdownEdown +FEkBT

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

2

P E( ) 1− FEkBTEdown +FEkBT

exp −E0−EFEkBT

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥dE

0

E0

∫. (5.63)

5.3.6 Unimolecular Isomerization Reactions A unimolecular , mutual isomerization reaction may be expressed by

Akf T ,P( )kb T ,P( )⎯ →⎯⎯⎯← ⎯⎯⎯⎯ B . (5.64)

We can formulate the unimolecular processes in a more detailed form within the framework of a strong collision model as follows

Adk1 E( )ω

⎯ →⎯⎯← ⎯⎯⎯ A* E( )′k f E( )′kb E( )

⎯ →⎯⎯← ⎯⎯⎯ B* E( ) ωd ′k1 E( )⎯ →⎯⎯← ⎯⎯⎯ B . (5.65)

The rates for the disappearance of A and the formation of B is

d B⎡⎣⎤⎦

dt= k f A⎡⎣

⎤⎦−kb B

⎡⎣⎤⎦− d ′k1 E( ) B⎡⎣ ⎤⎦−ω B

* E( )⎡⎣⎢

⎤⎦⎥{ }dE . (5.66)


5-14

−d A⎡⎣⎤⎦

dt= k f A⎡⎣

⎤⎦−kb B

⎡⎣⎤⎦ dk1 E( ) A⎡⎣ ⎤⎦−ω A

* E( )⎡⎣⎢

⎤⎦⎥{ }dE . (5.67)

Mass conservation requires that

− d ′k1 E( ) B⎡⎣ ⎤⎦−ω B* E( )⎡⎣⎢

⎤⎦⎥{ }= dk1 E( ) A⎡⎣ ⎤⎦−ω A* E( )⎡⎣⎢ ⎤⎦⎥{ } , (5.68)

and

B* E( )⎡⎣⎢

⎤⎦⎥=dk1 E( ) A⎡⎣ ⎤⎦−ω A

* E( )⎡⎣⎢

⎤⎦⎥+d ′k1 E( ) B⎡⎣ ⎤⎦

ω. (5.69)

We now assume steady state for the excited molecules A* and B*,

d A* E( )⎡⎣⎢

⎤⎦⎥

dt= dk1 E( ) A⎡⎣ ⎤⎦−ω A

* E( )⎡⎣⎢

⎤⎦⎥− ′k f E( ) A* E( )⎡⎣⎢

⎤⎦⎥+ ′kb E( ) B* E( )⎡⎣⎢

⎤⎦⎥= 0 , (5.70)

d B* E( )⎡⎣⎢

⎤⎦⎥

dt= d ′k1 E( ) B⎡⎣ ⎤⎦−ω B

* E( )⎡⎣⎢

⎤⎦⎥− ′kb E( ) B* E( )⎡⎣⎢

⎤⎦⎥+ ′k f E( ) A* E( )⎡⎣⎢

⎤⎦⎥= 0 (5.71)

Solving the above equations for A* E( )⎡

⎣⎢⎤⎦⎥ and B* E( )⎡

⎣⎢⎤⎦⎥ and substituting the results into

equation (5.66), it is possible to show that

− d ′k1 E( ) B⎡⎣ ⎤⎦−ω B* E( )⎡⎣⎢

⎤⎦⎥{ }dE=

k f E( )PA E( )dE

1+k f E( )ω+kb E( )ω

−kb E( )PB E( )dE


, (5.72)

or

k f T ,P( )=k f E( )PA E( )dE


E0

∞

∫ , (5.73) and

kb T ,P( )=kb E( )PB E( )dE


E0

∞

∫ . (5.74)


5-15

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( )=k f E( )PA E( )dE

1+k f E( )βω

+kb E( )′β ω

E0

∞

∫ . (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:

A+Bk1 f

k1b⎯ →⎯← ⎯⎯ AB*

k2⎯ →⎯ C +D

↓ 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).

Figure 5.5 Schematic energy diagram of a chemically activated reaction.

A+B

AB

C+D

,0r KHD

E

E0,1

E0,2

AB *

A+B

AB

C+D

,0r KHD

E

E0,1

E0,2

AB *


5-16

The observable reactions are

A+B kbi⎯ →⎯ AB , (5.77)

A+B kca⎯ →⎯ 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

AB* E( )⎡⎣⎢

⎤⎦⎥=

k1 f E( )k1b E( )+k2 E( )+βω

A⎡⎣⎤⎦ B⎡⎣⎤⎦ , (5.79)

The rate coefficient of the chemically activated reaction is

kca =k1 f E+ΔHr ,0K( )k2 E( )PA+B E+ΔHr ,0K( )dE

k1b E( )+k2 E( )+βωmax E0,1,E0,2( )

∞

∫ , (5.79) where ΔHr ,0K is the enthalpy of reaction of the entrance channel (5.77), A BP + is the combined Boltzmann distribution of energy of separate A and B,

PA+B E+ΔHr ,0K( )= ρA+B E+ΔHr ,0K( )exp − E+ΔHr ,0K( ) kBT⎡⎣⎢

⎤⎦⎥

z AzB. (5.80)

The microscopic, detailed balancing requires that

k1 f E+ΔHr ,0K( )ρA+B E+ΔHr ,0K( )= k1b E( )ρAB E( ) . (5.81) It follows that

PA+B E+ΔHr ,0K( )=k1b E( )

k1 f E+ΔHr ,0K( )ρAB E( )

exp − E+ΔHr ,0K( ) kBT⎡⎣⎢⎤⎦⎥

z AzB. (5.82)

The equilibrium constant of reaction (5.77) may be expressed by (cf, equation 4.63)

Kc =z ABz AzB

exp −ΔHr ,0K kBT⎡⎣⎢

⎤⎦⎥ . (5.83)


5-17

Combining equation (5.82) and (5.83), we obtain

kca = Kck2 E( )k1b E( )PAB E( )dEk1b E( )+k2 E( )+βωE0,1

∞

∫ . (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 = Kcβωk1b E( )PAB E( )dEk1b E( )+k2 E( )+βωE0,1

∞

∫ , (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 =Kcβω

k2 E( )k1b E( )PAB E( )dEE0,1

∞

∫ . (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, CH4→CH3i+H i . (5.87)


5-18

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).

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,

V r( )=ΔHr ,0K 1−e− r−re( )⎡

⎣⎢⎢

⎤

⎦⎥⎥

2, (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 →∞( )=ΔHr ,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

Reaction coordinate

Pot

enti

al e

nerg

y

J = 0

J > 0


5-19

V r , J( )=ΔHr ,0K 1−e− r−re( )⎡

⎣⎢⎢

⎤

⎦⎥⎥

2+Ieµr 2

Erot

=ΔHr ,0K 1−e− r−re( )⎡

⎣⎢⎢

⎤

⎦⎥⎥

2+ J J+1( ) h

2

8π2c

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟1

µr 2

, (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 aW 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( )=1zv

k E, J( )ρ E, J( )exp −E kBT( )dEE0

∞

∫ dJ0∞

∫ . (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


5-20

d[A Ei( )]dt

= kij M⎡⎣ ⎤⎦ A E j( )⎡⎣ ⎤⎦j∑ − k ji M⎡⎣ ⎤⎦ A Ei( )⎡⎣ ⎤⎦ − km Ei( ) A Ei( )⎡⎣ ⎤⎦m∑ (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, ijk 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

Pji =k jikcoll

, (5.94)

where collk 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 Pjij∑ = 1 , . The master equation may be re-written in terms of Pij as

d A Ei( )⎡⎣ ⎤⎦

dt= kcoll M⎡⎣ ⎤⎦ Pij A E j( )⎡⎣ ⎤⎦ − A Ei( )⎡⎣ ⎤⎦j∑{ }− km Ei( ) A Ei( )⎡⎣ ⎤⎦m∑ . (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 =Ciρiρ jexp − α + β( ) Ei − E j( )⎡⎣ ⎤⎦ 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 = 1Edown

(5.97)


5-21

and β = 1 kT . Equations (5.96a) and (5.96b) satisfy the principle of detailed balancing,

Pjiρ Ei( )exp − EikT⎛⎝⎜

⎞⎠⎟= Pijρ E j( )exp − E jkT

⎛

⎝⎜

⎞

⎠⎟ (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( )kuni ,∞ T( )

F T ,P( ) , (5.99)

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


5-22

n= 0.75−1.27 logFc , (5.104) d = 0.14 . (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∞Tn∞ exp −

E∞RuT

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟

, (5.107)

those in the second line A0, n0, and E0 are for the low-pressure limit rate coefficient,

k0 [M ′] = A0Tn0 exp −

E0RuT

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟ , (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 ′] = γii∑ 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.


5-23

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,

1+−0.4−0.67 logFc0.806−1.176logFc

⎛

⎝⎜⎜⎜⎜

⎞

⎠⎟⎟⎟⎟⎟

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

−1

logFc T( )= log 2kuni T ,Pr =1( )kuni ,∞ T( )

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥ . (5.99)

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Combustion Chemistry - Princeton University · In the last lecture, we learned the collision and transition state theory that govern bimolecular reactions. In this lecture, we shall