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Tunneling in bimolecular reactions
Chemical Physics 315 (2005) 65–75 www.elsevier.com/locate/chemphys
Tunneling in bimolecular reactions Shin Sato
*
Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-0033, Japan Received 15 December 2004; accepted 28 January 2005 Available online 27 April 2005
Abstract The estimation of the tunneling fraction of bimolecular reaction from the empirical rate equation has been made by using the Laplace transform relation between the rate equation and the cumulative reaction probability on the knowledge of the reaction barrier height. The tunneling fractions of the D + H2 ! HD + H reaction were found to be 95%, 65%, 24%, and 2.4% at the temperatures of 200, 300, 500, and 1000 K, respectively. Similar estimations were made for several reactions including the reactions of F-atoms with H2 and its isotopes, the reactions of O-atoms and OH-radicals with H2 and D2, the reactions of H-atoms and Mu (muonium) with F2, and the methoxy radical decomposition reaction. For most reactions containing H2 as the reactant, the tunneling fractions exceeded 50% at 300 K, while for the reactions containing D2 in place of H2, found were about a half of those for the reactions containing H2. Exception was found in the reactions of F-atoms with H2 and its isotopes. The tunneling fraction of the F + H2 reaction at 300 K was found to be 14%. The value is much smaller than those of other hydrogen atom transfer reactions calculated in this paper. The reason is probably the fact that the barriers of the F atom-reactions are flat and their heights are very low. Although there are some uncertainties on the accuracy of barrier heights, the Mu + F2 reaction seems to occur thoroughly through tunneling at 300 K, and the methoxy radical decomposition also seems to occur through tunneling at the temperatures up to 1000 K. Finally, an example of the application of the present method to the analysis other than the estimation of tunneling fraction was presented. Ó 2005 Elsevier B.V. All rights reserved.
1. Introduction In the previous Letter [1], I reported that approximate cumulative reaction probabilities for simple atom–diatom exchange reactions can be derived from the empirical rate equations by using Laplace transformation. As a result, it could be shown that almost all of the reaction of D + H2 ! HD + H occurs through tunneling at 200 K, and more than a half of that occurs at 300 K. For this reaction, the accurate reaction barrier height has been quantum mechanically calculated. It is well known that for the reaction in which the transfer of light species such as hydrogen atom is the rate-determining step, the tunnel effect plays an impor*
Present address: 7 Karasawa, Minami-ku, Yokohama 232-0034, Japan. Fax: +81 45 251 4174. E-mail address: [email protected]. 0301-0104/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2005.01.034
tant role [2]. However, if we want to estimate the quantitative extent of the tunnel effect, we have to perform the quantum mechanical calculation of rate constants on the accurate potential energy surface, or we have to be satisfied with simplified approximate calculations for tunneling attached to the transition-state theory [3]. As stated above, if the rate equation is experimentally known and the barrier height for this reaction is accurately calculated, we can obtain the cumulative reaction probability for the reaction, and can numerically estimate the extent of the tunnel effect. This paper will study further the reaction of D + H2 ! HD + H and will discuss the relation with the transition-state theory. Then, several bimolecular reactions with barriers will be analyzed by using the technique of Laplace inverse transformation, and the extent of tunnel effect will be discussed. For the last example of the calculations, the analysis of the reactions with
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S. Sato / Chemical Physics 315 (2005) 65–75
no barriers will be shown, and the application of this technique to other reactions will be discussed. If a reaction occurs thoroughly through tunneling, so-called transition-state will not be on the col of the potential energy surface, but somewhere under the barrier. This issue will be discussed in the last section.
By using Laplace transform operator, we obtain 3=2
N ðEÞ ¼
A ð2plÞ 8p2 I F ðEÞ k nB rh4
with F ðEÞ ¼ L1
2. Theory
1 snþ5=2
ð5Þ
eEa s . 1 ehmBC s
ð6Þ
In the previous Letter, I assumed ehmBC s ¼ 0. Here, I expand the vibrational term as follows:
2.1. Basic calculation Bimolecular thermal rate constant k(T) is connected with the cumulative reaction probability N(E) by Laplace transformation as follows [4]: Z 1 1 kðT Þ ¼ N ðEÞ expðE=k B T Þ dE; ð1Þ hq 0 where h is the PlanckÕs constant, q is partition function of reactant, kB is BoltzmannÕs constant, T is temperature and E is the energy of reactant. If the rate constant k(T) and the partition function q can be expressed as simple functions of T, then the cumulative reaction probability N(E) is obtained from k(T) by Laplace inverse transformation. By using the approximation of harmonic oscillator and rigid rotor, the partition function of reactant can be expressed as a function of 1/(kBT); for example, the reactant partition function of the A + BC ! AB + C reaction may be expressed as
1 ¼ 1 þ ehmBC s þ . 1 ehmBC s
ð7Þ
If we take until the second term, Eq. (6) can be expressed as F ðEÞ ¼ F 0 ðEÞ þ F 1 ðEÞ 1 1 ¼ L1 nþ5=2 eEa s þ L1 nþ5=2 eðEa þhmBC Þs . s s ð8Þ By performing Laplace inverse transformation, we can obtain the following equations: ( nþ3=2 1 ðE Ea Þ ; E > Ea ; ð9Þ F 0 ðEÞ ¼ Cðnþ5=2Þ 0; E 6 Ea ; ( F 1 ðEÞ ¼
1 ðE Cðnþ5=2Þ
Ea hmBC Þ
nþ3=2
; E > Ea þ hmBC ; E 6 Ea þ hmBC .
0;
ð10Þ
BC q ¼ qt qBC vib qrot
¼
ð2plk B T Þ h3
3=2
1 8p2 Ik B T ; 1 expðhmBC =k B T Þ rh2
ð2Þ
where l is the reduced mass of A and BC, and mBC, I and r are the vibrational frequency, the moment of inertia and the symmetry number of the molecule BC, respectively. The zero-point energy of vibration is taken into account in the barrier height. Experimentally obtained thermal rate constant is often expressed as kðT Þ ¼ AT n expðEa =k B T Þ;
ð3Þ
where n is a constant. When n = 0, Eq. (3) is the Arrhenius equation. Since both Eqs. (2) and (3) are simple functions of 1/(kBT), the cumulative reaction probability N(E) can be calculated as follows. Substitution of Eqs. (2) and (3) into Eq. (1) with s = 1/(kBT) gives 1 snþ5=2
eEa s k nB rh4 ¼ 1 ehmBC s A ð2plÞ3=2 8p2 I
Z
1
N ðEÞ e
Es
dE.
0
ð4Þ
2.2. Wigner’s threshold law Although Eq. (3) has been widely used to express the temperature dependence of experimentally obtained reaction rates, this equation cannot be extended to very low temperatures. According to WignerÕs threshold law [5], the bimolecular rate constant for any exothermic reaction satisfies the following relation: lim kðT Þ / T 0 .
ð11Þ
T !0
This relation contradicts Eq. (3). Based on this finding, we contrived an empirical equation that connects the rate constants measured above room temperature and those calculated at low temperatures [6] 0 1 B kðT Þ ¼ A0 exp @
E0 C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA; k B T 2 þ T 20
ð12Þ
where A 0 , E0 and T0 are parameters, although E0 is not a simple parameter but is considered to correspond to the reaction barrier height. In the harmonic oscillator approximation, E0 may be expressed by
S. Sato / Chemical Physics 315 (2005) 65–75
E0 ¼ Ec þ
X1 z X1 hm hmj ; 2 i 2 i j
ð13Þ
1
z 1 ehms s
67
2
1 1e
hmzb s
z
z
ð20Þ
mzi
where Ec is the classical barrier height. and mj are the vibrational frequencies of transition-state and reactant. The calculation of N(E) using Eq. (12) has been reported in detail in the previous Letter. I do not repeat it but show the result 3=2
A0 ð2plÞ 8p2 I N ðEÞ ¼ F ðEÞ rh4
z
¼ 1 þ ehms s þ 2ehmb s þ e2hmb s þ .
Finally, we can obtain the cumulative reaction probability, NTST(E), as the sum of a series of Laplace inverse transforms: N TST ðEÞ ¼ N 00 ðEÞ þ N 10 ðEÞ þ 2N 01 ðEÞ þ N 02 ðEÞ þ ; ð21Þ
ð14Þ
1 E > E0 ; 1 z z ðE E 0 Þ; 1 1 E0 s N 00 ðEÞ ¼ z z L e ¼ rB 2 s rB 0; E 6 E0 ;
with F ðEÞ
" (Z Z )# b E pffiffiffi 4 E3=2 1 abs dq ¼ pffiffiffi s cosqðE sÞds sin pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; p 0 0 p 3 b2 q2 q
ð15Þ
ð22Þ ( N 10 ðEÞ ¼
1 rz Bz
ðE E0 hmzs Þ;
0;
E > E0 þ hmzs ; E 6 E0 þ hmzs ;
ð23Þ
where a = E0 and b = 1/(kBT0). In this derivation, the vibrational partition function qvib is assumed to be unity.
where 1/Bà = 8p2Ià/h2. N01(E), N02(E) and the succeeding terms can be obtained in the same manner.
2.3. Cumulative reaction probability obtained by the transition-state theory, NTST(E)
2.4. Wigner’s tunneling correction
As an example, let us consider the following atom– diatom exchange reaction: A þ BC ABCz ! AB þ C
ð16Þ
The transition-state theory rate constant may be written as follows: k B T qzABC E0 kðT Þ ¼ j exp ; ð17Þ h q kBT where j is the transmission coefficient and qzABC is the internal partition function of the transition-state. By ignoring the tunnel effect and the so-called recrossing, I assume j = 1, and also assume that three atoms ABC in the transition-state are collinear. Then, the internal partition function may be written as follows: 2 8p2 I z k B T 1 1 z qABC ¼ ; ð18Þ z z rz h2 1 ehms =kB T 1 ehmb =kB T à
à
where I and r are the moment of inertia and the symmetry number of the transition-state complex, and mzs and mzb are its stretching and doubly degenerated bending vibrational frequencies. Substitution of Eqs. (17) and (18) into Eq. (1) gives 2 2 z 8p I 1 1 1 eE0 s z z rz h2 s2 1 ehms s 1 ehmb s Z 1 ¼ N TST ðEÞ eEs dE. ð19Þ 0
Here, I expand the vibrational term in the same manner as Eq. (7)
WignerÕs (first order) correction for tunneling has been widely used as a convenient method attached to the transition-state theory [7]. In this method, the transmission coefficient j appearing in Eq. (17) is expressed by 2 1 hmz j¼1 ; ð24Þ 24 k B T where mà is the imaginary vibrational frequency of the transition-state complex. When Eq. (17) containing WignerÕs correction shown in Eq. (24) is substituted into Eq. (1), the following equation can be obtained: Z 1 1 1 1 E0 s 2 1 ðhmz sÞ e ¼ N TST ðEÞ eEs dE; 24 rz Bz s2 0 ð25Þ where, for simplicity, all vibrational partition functions are put as one. The cumulative reaction probability in this case is expressed as follows: N TST ðEÞ ¼ N 00 ðEÞ þ N 00 ðEÞ;
ð26Þ
1 ðhmz Þ2 F ðEÞ 24 rz Bz 00
ð27Þ
N 00 ðEÞ ¼ with
F 00 ðEÞ ¼ L1 feE0 s g ¼ dðE E0 Þ.
ð28Þ
If the vibrational partition function is included, the contribution to the tunnel effect due to the stretching vibration can be derived as follows:
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S. Sato / Chemical Physics 315 (2005) 65–75 2
N 10 ðEÞ
1 ðhmz Þ ¼ dðE E0 hmzs Þ. 24 rz Bz
ð29Þ
To calculate the values of N 01 ðEÞ and N 02 ðEÞ, hmzb and 2hmzb are substituted, respectively, in place of hmzs in Eq. (29). 2.5. Electronic state of reactant: the definition of k(E; T) When the electronic state of the reactant changes with temperature; for example, in the atom–diatom exchange reaction, A + BC ! AB + C, if A is oxygen atom in the ground state, the experimentally obtained rate constants are the mixture of the rate constants obtained with the three different states of oxygen atom, 3 P2, 3P1 and 3P0. At present, the observation for the rate constants of the reaction of oxygen atoms in the different sublevels of the ground state has not been succeeded. In order to calculate the cumulative reaction probability from such mixture of rate constants by using Eq. (1), the temperature dependence of the electronic states of reactant has to be considered. Since the ground state of oxygen atom is accurately analyzed [8], the calculation of the temperature dependence may be possible, but it is not a simple function of temperature which can be used in the estimation of N(E) in Eq. (1). In this paper, I aim to clarify the tunnel effect in simple reactions, so that, for such reactions, we avoid the calculation of the cumulative reaction probability, and calculate the integrand in Eq. (1) on the assumption that the temperature dependence of the partition functions of reactants containing atoms in different sublevels is same kðE;T Þ ¼ that is kðT Þ ¼
Z
1 N ðEÞ expðE=k B T Þ; hq
:
1 ðE k nB Cðnþrþ3 2 Þ
rþ1
Ea Þnþ 2
0;
1 kB T
rþ3 2
eE=kB T ; E > Ea ; E 6 Ea . ð33Þ
With this equation, we can estimate the extent of the tunnel effect in the reaction at desired temperature by comparing with the reaction barrier height, if it has been accurately calculated. In this section, I have discussed the electronic degeneracy of atoms in the reactant. When we formulate the rate constant of the reaction involving homonuclear diatomic molecules such as hydrogen and deuterium, we have to consider the nuclear-spin degeneracy. Strictly speaking, ortho- and para-molecules have to be treated as different molecules. However, in this paper, the rate constants discussed in the following sections are those obtained at the temperatures higher than 167 K. The ortho–para ratio above this temperature does not severely change even for hydrogen molecule. Consequently, it will not be necessary to be concerned about the temperature dependence due to the nuclear-spin degeneracy. In the following sections, I will not consider the contribution of both electronic and nuclear-spin degeneracies of atoms in the reactant and in the transition-state. If I take these nuclear degeneracies into the calculation of partition functions, the absolute values of cumulative reaction probabilities, N(E) and NTST(E), will be changed. Therefore, when I discuss on the cumulative reaction probabilities, I will pay attention only on their relative values between the probabilities obtained by different methods.
3. Results and discussion ð30Þ 3.1. Rate equation and cumulative reaction probability for the D + H2 ! HD + H reaction
1
kðE;T Þ dE.
ð31Þ
0
By this procedure, we can skip the calculation of the partition functions of the reactants involving atoms in the different sublevels. For the calculation of k(E; T), I assume r=2
qvib ¼ 1
kðE;T Þ ¼
8
and
qrot ¼
ðk B T Þ B
;
ð32Þ
where r is the sum of the degree of freedom of reactant rotation. Most of gas-phase rate constants reported are expressed by Eq. (3). On the definition of Eqs. (30) and (31) with the assumption of Eq. (32), we can derive the following equation by using the above-described technique of Laplace inverse transformation:
In the previous Letter, I used the following two rate equations to estimate the cumulative reaction probability for the D + H2 ! HD + H reaction: 22.43 ½kJ=mol kðT Þ ¼ 2.72 1017 T 2 exp ; ð34Þ kBT ! 37.07 ½kJ=mol kðT Þ ¼ 2.35 1010 exp pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð35Þ k B T 2 þ 1302 where the unit of k(T) is cm3 molecule1 s1. As stated in the previous Letter, neither rate equation completely agrees with the experimental data. Recently, an epoch-making convergence of theory and experiment has been reported on the rate constants of thermal hydrogen atom–diatom exchange reactions
S. Sato / Chemical Physics 315 (2005) 65–75 Table 1 The temperature dependence of the rate constants for the D + H2 ! HD + H reaction
8000
T (K)
Ref. [9]
Eq. (34)
Eq. (35)
Eq. (36)
6000
170 180 200 250 300 350 400 450 500 600 800 1000 1200 1500 2000
2.16(19) 4.57(19) 1.86(18) 3.40(17) 2.96(16) 1.51(15) 5.38(15) 1.48(14) 3.38(14) 1.21(13) 6.46(13) 1.90(12) 4.12(12) 9.53(12) 2.45(11)
1.01(19) 2.73(19) 1.51(18) 3.50(17) 3.04(16) 1.50(15) 5.13(15) 1.37(14) 3.09(14) 1.09(13) 5.97(13) 1.83(12) 4.14(12) 1.01(11) 2.82(11)
2.11(19) 4.47(19) 1.79(18) 3.16(17) 2.81(16) 1.53(15) 5.85(15) 1.73(14) 4.20(14) 1.65(13) 9.59(13) 2.82(12) 5.85(12) 1.21(11) 2.54(11)
2.10(19) 4.61(19) 1.93(18) 3.38(17) 2.80(16) 1.41(15) 5.03(15) 1.40(14) 3.26(14) 1.21(13) 6.68(13) 1.97(12) 4.23(12) 9.59(12) 2.44(11)
Numbers in parentheses are powers of 10.
[9]. For the D + H2 ! HD + H reaction, they showed the rate constants at different temperatures between 170 and 2000 K, which are listed in Table 1 together with the values calculated by using Eqs. (34) and (35). The last column shows the values calculated with Eq. (36) kðT Þ ¼ k 1 ðT Þ þ k 2 ðT Þ;
ð36Þ
N = N 1+ N 2
4000
N(E)
N1
2000
N2
0
0
20
60
80
100
-1
Fig. 1. Cumulative reaction probability of D + H2 ! HD + H reaction.
By substituting the parameter values in Table 2 into Eq. (21), we can obtain the NTST(E) shown in Fig. 2, which may be compared with N(E) obtained in the previous section. Naturally, below 35.4 kJ/mol, the reaction Table 2 The properties of the transition-state and the reactant of the D + H2 ! HD + H reaction D H H
ð37Þ !
35.4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . k B T 2 þ 1262
ð38Þ
I tried to find a rate equation better than Eqs. (34) and (35) by repeated try and error, and reached Eq. (36) which consists of the sum of two equations similar to Eqs. (34) and (35). I failed to find a single equation which can express the accurate rate constants. Michael et al. [10] proposed an equation which accurately expresses the temperature dependence of the rate constants for the D + H2 ! HD + H reaction. However, their equation is not adequate for the technique of Laplace transformation used in this paper. Fig. 1 shows the cumulative reaction probability for the D + H2 ! HD + H reaction calculated from Eq. (36) by using the technique described in the previous section. N1(E) and N2(E) shown in Fig. 1 correspond to those calculated from k1(T) and k2(T) in Eq. (36), respectively. 3.2. NTST(E) for the D + H2 ! HD + H reaction
0.0930 0.0930
mzs =cm1 mzs =cm1 à 1
1720 870 (doubly degenerate) 1472i
0.0741
m /cm
4401
Ec = 41.0 kJ/mol (= 9.80 kcal/mol) E0 = 35.4 kJ/mola a
Numerical mistake was found in the calculation of E0 in the previous paper [6]. The value of 35.4 kJ/mol is used in this paper in place of 37.07 kJ/mol in the previous Letter.
10000
N TST(E) N(E) 1000
100
10 20
To calculate the NTST(E), we need the various parameters of the reactant and the transition-state.
H2
r(D–H)/nm r(H–H)/nm r(H–H)/nm
Cumulative reaction probabilities
20.0 20 2.7 k 1 ðT Þ ¼ 5.41 10 T exp kBT
k 2 ðT Þ ¼ 9.29 1011 exp
40
E / kJ mol
where
and
69
30
40
50
60
E / kJ mol
70
80
-1
Fig. 2. Comparison between N(E) and NTST(E).
90
100
70
S. Sato / Chemical Physics 315 (2005) 65–75
barrier height, the value of NTST(E) does not exist. At higher energies, the series for NTST(E) were calculated up to the term of N07(E), and up to 27th rotational levels for N00(E). The two curves of N(E) and NTST(E) cross with each other at 63 kJ/mol. This might suggest that so-called recrossing starts to occur from this energy. But, we cannot conclude it at present, because of many assumptions involved in the calculation, such as the omission of unharmonicity in vibration and of nuclear-spin degeneracy. Originally, classical cumulative reaction probability is defined as the number of the quantum levels between E0 and E of the energy of the transition-state. For the D + H2 ! HD + H reaction, the cumulative reaction probability Nà(E) may be expressed by (P z E P E0 ; z ns ;nb ;J z ð2J þ 1Þ; N ðEÞ ¼ ð39Þ 0; E < E0 ;
(30). The result is shown in Fig. 4. A linear plot for the k(E; T) at 300 K is shown in Fig. 5. The area surrounded by the curve and the energy axis corresponds to the rate constant at this temperature. The fractional area under the barrier height, 35.4 kJ/mol, is the tunneling fraction, the value of which has been calculated to be 65% from this figure. The values at other temperatures have been found to be 95% at 200 K, 24% at 500 K and 2.4% at 1000 K. Similar estimation from theoretically calculated rate constants was made by Schatz [3], when the values of transmission coefficient j for many reactions were reported. For the D + H2 ! HD + H reaction calculated on the potential energy surface of LSTH, j = 68 at 200 K and 6.7 at 300 K were presented. If the tunneling fraction mentioned above has been calculated from these values of j, we can obtain 98.5% at 200 K and 85.1% at 300 K. Our values are a little smaller than those by Schatz.
where Jà is the rotational quantum number of the transition-state which has to satisfy the following relation: 1E-13
E P E0 þ J z ðJ z þ 1ÞBz þ ns hms X þ nbi hmb ; ns ; nbi ¼ 0; 1; 2; . . .
1000 K 1E-14
ð40Þ
1E-15
i¼1;2
500 K 1E-16
à
1E-17
k(E;T)
Fig. 3 shows N (E) up to 55 kJ/mol of E. It is obvious that NTST(E) is the smoothed function of Nà(E). When we discuss the reaction probability for each rotational level of the transition-state, the step-wise Nà(E) has to be considered.
300 K
1E-18 1E-19
200 K
1E-20
3.3. k(E; T) for the D + H2 ! HD + H reaction and its tunneling fraction
1E-21 1E-22 10
In this case, k(E; T) at desired temperature can be obtained by substituting N(E) shown in Fig. 1 into Eq.
30
40
50
60
E / kJ mol
70
80
90
100
-1
Fig. 4. The k(E; T) for the D + H2 ! HD + H reaction.
600
16 14
500
12 400
10
k(E;T)/10 -18
Cumulative reaction probabilities
20
N(E)
300
+
N (E)
200
8 6 4
100
N TST(E)
2
0
0 30
35
40
E / kJ mol
45
50
55
-1
Fig. 3. Cumulative reaction probability obtained by direct count, Nà(E) compared with N(E) and NTST(E).
10
20
30
40
E / kJ mol
50
60
-1
Fig. 5. The linear plot of k(E; T) for the D + H2 ! HD + H reaction at 300 K.
S. Sato / Chemical Physics 315 (2005) 65–75
3.4. Case of the reaction of F-atoms with hydrogen molecule and its isotopes The rate constants of the reaction of F-atoms with H2 and its isotopes are reported for the wide range of temperatures from 190 to 7000 K as follows [11] F þ H2 ! HF þ H kðT Þ ¼ 7.74 1011 ðT =298Þ0.50 e2.652=kB T ; F þ D ! DF þ D kðT Þ ¼ 5.73 1011 ðT =298Þ0.50 e3.475=kB T ; 2
F þ HD ! HF þ D kðT Þ ¼ 3.44 1011 ðT =298Þ0.50 e3.076=kB T ; F þ HD ! DF þ H kðT Þ ¼ 3.16 1011 ðT =298Þ0.50 e3.658=kB T .
ð41Þ Several other rate constants have been reported, but their temperature ranges are narrow. The k(E; T)Õs for these reactions calculated at 200 and 300 K from the above k(T)Õs are shown in Fig. 6. The theoretical study of the F + H2 ! HF + H reaction might be ahead of that of the hydrogen atom– diatom exchange reaction [12]. It is now well known that the transition-states of the F atom-reactions are not collinear [13], and the barrier heights are very low, i.e, Ec = 6.07 ± 1.05 kJ/mol (= 1.45 ± 0.25 kcal/mol) [14]. The barrier heights E0Õs for these reactions can be calculated from the data reported by Takayanagi and Kurosaki [15], and found to be 5.9 ± 1.0 kJ/mol for four reactions. By comparing this value with Fig. 6, the tunneling fractions of the F + H2 and F + D2 reactions at 200 and 300 K were found to be 34% and 14%, and 19% and 7.4%, respectively. These fractions are much smaller than those of the D + H2 reaction and the reactions of O-atoms and OH-radicals with hydrogen and its isotopes described in the following sections. The reason may be simply the fact that the barrier height is low and the reaction path to the col of the potential energy surface is so flat that the quantum effect does not much participate, as a whole, in the reaction,
71
although several interesting quantum effects have been discussed [12,15]. The ground state of F atoms is in the 2P state. Therefore, the F atom-reaction is the mixture of the reactions of F-atoms in the two states, 2P3/2 and 2P1/2. The difference in energy between these two sublevels is reported to be about 4.8 kJ/mol. Since the accurate potential energy surface has been calculated, the partition functions of the transition-states containing the two different stateF atoms could be constructed on the assumption of the appropriate structures of the transition-states. Then, the NTST(E)Õs for these reactions may be calculated, and the mixed rate constants will be discussed much more in detail, including the difference in reactivity of the sublevels of F-atoms. 3.5. Case of the reaction of O-atoms with H2 and D2 Among many rate equations reported, I selected the following two equations. The applicable temperature range is from about 300 to 2500 K [16,17] O þ H2 ! OH þ H; kðT Þ ¼ 3.39 1013 ðT =298Þ2.67 e26.274=k B T ; O þ D ! OD þ D kðT Þ ¼ 3.92 1012 ðT =298Þ1.70 e40.824=kB T . 2
ð42Þ Fig. 7(a) shows the k(E; T)Õs at 300 K calculated for the two reactions. Since the barrier height of the O + H2 ! OH + H reaction is calculated to be 38.5 kJ/mol [8], the tunneling fraction at 300 K can be estimated to be 52%. If the rate equation was extended to 200 K as a try, the tunneling fraction was found to be 88%. Since the barrier height for the O + D2 ! OD + D reaction was not available, I calculated the vibrational frequencies from the data of the O + H2 reaction by using the usual technique [18]. The result is shown in Table 3.
25
20
15
2 3
5
4 0
0
(a)
1: F + H 2 = HF + H 2: F + D 2 = DF + D 3: F + HD = HF + D 4: F + HD = DF + H
1
10
k(E;T)/10 -13cm 3molecule -1s-1/kJ mol -1
k(E;T)/10 -13cm 3molecule -1s-1/kJ mol -1
30
5
10
15
E / kJ mol
20 -1
25
25
1 20
15
2 10
3
5
4
0 0
30
(b)
5
10
15
E / kJ mol
20 -1
Fig. 6. The k(E; T)Õs for the reactions of F-atoms with H2 and its isotopes at (a) 200 K and (b) 300 K.
25
30
72
S. Sato / Chemical Physics 315 (2005) 65–75 5
k(E;T)/10 -20cm 3molecule -1s-1/kJ mol -1
70
k(E;T)/10 -16cm 3molecule -1s-1/kJ mol -1
80
O + H 2 = OH + H
60 50 40
O + D 2 = OD + D
30
x 1/10
20 10
OH + H 2 = H 2O + H
4
OH + D 2 = HDO + D
H2
3
2
1
D2 0
0 20
30
40
50
(a)
E / kJ mol
60
70
80
10
(b)
-1
20
30
E / kJ mol
40
50
60
-1
Fig. 7. The k(E; T)Õs at 300 K for the reactions of (a) O-atoms and of (b) OH-radicals with H2 and D2.
Table 3 The vibrational frequencies of the transition-states for the reactions of O + H2 and O + D2 and of the reactants Modea
O H H 3
A
mzs =cm1 mzb à m
a
O D D 3
00
A
0
3
3
A00
1577.0 514.0
A0
1579.8 1052.0
365.0
747.1
1738.0i
914.7i
m(H2) = 4266
m(D2) = 3017
The electronic state of the bent structure.
The barrier height for O + D2 ! OD + D reaction thus obtained was 44.8 kJ/mol. By comparing this value with the curve shown in Fig. 7(a), the tunneling fraction of the O + D2 reaction at 300 K was estimated to be only 7%. The trial extension to 200 K showed 21%. By substituting the imaginary vibrational frequencies shown in Table 3 into Eq. (24), we can estimate the transmission coefficients jÕs at 300 K as follows: 7.51 for the O + H2 and 2.80 for the O + D2 reactions, the values which correspond to 88% and 74% as the tunneling fraction, respectively. The discrepancy in tunneling fraction between the value calculated by the present method and that by WignerÕs tunneling correction, especially for the O + D2 reaction (7–74%), is remarkable. At present, I have no reasonable interpretations on this. 3.6. Case of the reaction of OH radicals with H2 and D2 To calculate the k(E; T)Õs for the title reactions, I used the following rate equations [19]: OHþH2 !H2 OþH kðT Þ¼4.481013 ðT =298Þ2.44 e10.643=kB T ; 1.18 OHþD2 !HDOþD kðT Þ¼3.641012 ðT =298Þ e19.373=kB T .
ð43Þ
Fig. 7(b) shows the k(E; T)Õs at 300 K for both reactions. Since the classical barrier height Ec for these reactions is reported to be about 28 kJ/mol [20], the tunneling fractions are about 60% for the OH + H2 reaction and 30% for the OH + D2 reaction, respectively. Precise values cannot be calculated because of the lack of the data for the transition-states. The isotope effect of H and D for the rate constants are not so large compared with that for the O atom-reactions described in the previous section. 3.7. Case of the reactions of H + F2 ! HF + F and Mu + F2 ! MuF + F For the H + F2 ! HF + H reaction, I selected the following rate equation [21]: 1.40 kðT Þ ¼ 1.4 1011 ðT =298Þ e5.565=kB T . ð44Þ The applicable temperature range is from 225 to 2000 K. In another title reaction, Mu + F2 ! MuF + F, Mu stands for muonium, which is an isotope of hydrogen atom whose atomic weight is 1/9 of H-atom. No rate equation has been proposed for this reaction. Experimentally obtained rate constants at each temperature are reported in a table [22], probably because the appropriate function could not be found. Fig. 8 shows the Arrhenius plot for this reaction. The curve shown in the figure is calculated by the following equation: ! 12.5 10 kðT Þ ¼ 6.33 10 exp pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . ð45Þ k B T 2 þ 3352 The applicable temperature range is from 100 to 450 K. We used these two equations to calculate the k(E; T)Õs. Fig. 9(a) shows the k(E; T)Õs for the H + F2 ! HF + F reaction at 200 and 300 K. The classical barrier height has been reported to be 15.5 kJ/mol (= 3.7 kcal/ mol) [23]. The zero-point-inclusive barrier height E0 was
S. Sato / Chemical Physics 315 (2005) 65–75
CH3 O þ M ! CH2 O þ H þ M
4.5
ð46Þ
where M stands for inert atom. The rate constants at the temperatures from 300 to 1700 K are expressed by a simple Arrhenius equation [24] kðT Þ ¼ 9.0 1011 e56.44=kB T . ð47Þ
3.5 3.0 2.5 2.0 1.5 1.0 0.5 2
3
4
5
6
7
8
9
10
11
1000/T
Fig. 8. Arrhenius plot for the reaction of Mu + F2 ! MuF + F.
discussed. Although the final value was not reached, it will not be too far from 15.5 kJ/mol. If we take this value, the tunneling fractions are almost same as those for the reactions of OH radicals with H2 shown in the previous section, i.e., about 90% at 200 K and 60% at 300 K. Fig. 9(b) shows the k(E; T)Õs for the Mu + F2 ! MuF + F reaction at the temperatures from 100 to 450 K. Since the rate constants do not much change in this temperature range, the k(E; T)Õs calculated can be demonstrated in the figure of linear scale. It is obvious that almost all reactions occur through tunneling under 300 K. Even at 450 K, more than a half of the reaction occurs through tunneling.
The unit of rate constant is cm3 molecule1 s1, and that of activation energy is kJ/mol. The barrier height E0 of this reaction has been quantum mechanically calculated to be 107.1 kJ/mol (= 25.6 kcal/mol) [25]. By substituting Eq. (47) into Eq. (33), we can obtain the k(E; T) at desired temperature. Fig. 10 shows the logarithmic plot of k(E; T)Õs as a function of energy at the temperatures from 300 to 1700 K. Comparison with the barrier height (107.1 kJ/mol) suggests that almost all reactions under 1000 K occur through tunneling. A detailed calculation has shown that the tunneling fraction at 1700 K is 71%.
1E-13 1E-14
1700 K k(E;T)/cm 3molecule -1s-1/kJ mol -1
4.0
k(T)/10 -11cm 3molecule -1s-1
73
1E-15 1E-16
1000 K
1E-17 1E-18 1E-19
500 K
1E-20 1E-21
300 K
1E-22 1E-23
3.8. Case of the methoxy radical decomposition reaction
1E-24 40
Methoxy radicals are believed to decompose by collision with inert atoms such as Ar and He as follows:
60
80
100
120
140
E / kJ mol
160
180
200
220
-1
Fig. 10. The k(E; T)Õs for the methoxy radical decomposition reaction.
3.0
100 K
12 10
300 K
8 6 4
200 K 2 0 0
(a)
k(E;T)/10 -12cm 3molecule -1s-1/kJ mol -1
k(E;T)/10 -14cm 3molecule -1s-1/kJ mol -1
14
5
10
15
20
E / kJ mol
25 -1
30
35
2.5
2.0
450 K 1.5
1.0
300 K
0.5
200 K
0.0
0
40
(b)
10
20
30
E / kJ mol
40
50
-1
Fig. 9. The k(E; T)Õs for (a) the H + F2 ! HF + F reaction at 200 and 300 K and (b) the Mu + F2 ! MuF + F reaction at the temperatures from 100 to 450 K.
74
S. Sato / Chemical Physics 315 (2005) 65–75
tion for these reactions. The reactions and their rate equations are as follows [27]: OH þ C H ! C H OH kðT Þ ¼ 8.7 1012 ðT =300Þ0.85 ;
This reaction has been analyzed by using the RRKM theory with the tunneling correction; however, such a high tunneling fraction seems to suggest that classical RRKM theory has to be modified. The true structure of the transition-state complex under the barrier might be quite different from that classically estimated and may be dependent upon the energy and its quantum level of the reactant.
2
4
2
4
OH þ C3 H6 ! C3 H6 OH kðT Þ ¼ 2.95 1011 ðT =300Þ1.06 ; OH þ 1-C H ! C H OH kðT Þ ¼ 3.02 1011 ðT =300Þ1.44 . 4
8
4
8
ð48Þ
The applicable temperature range is from 96 to 296 K. Note that no exponential terms are contained in these rate equations. By substituting these equations into Eq. (33), we can obtain the k(E; T)Õs for these reactions, which are shown in Fig. 11(a) and (b). Although no abnormal trend can be found in three curves at 100 K, the curves at 300 K for the reactions with propene and 1-butene cross with each other at 6.5 kJ/mol of the reactant energy. This seems to suggest that an unknown reaction other than the association reaction between OH and propene occurs simultaneously at high
3.9. Case of the association reaction between OH radicals and olefins Special experimental techniques have been developed recently for gas-phase reactions at very low temperatures [26], and the temperature dependence of the rate constants was reported. Most of the reactions have no barriers, so that the tunnel effect does not participate in these reactions. However, when I calculated the k(E; T)Õs for the association reactions between OH radicals and olefins, I found a small but interesting sugges-
40
3
40
at 100 K
2
30
20
10
1
0
0
(a)
3 k(E;T)/10 -13cm 3molecule -1s-1/kJ mol -1
k(E;T)/10 -12cm 3molecule -1s-1/kJ mol -1
50
1: ethylene 2: propene 3: 1-butene
2
4
6
E / kJ mol
8
30
2
1
10
0
10
0
-1
at 300 K
20
5
(b)
10
15
E / kJ mol
20
25
30
-1
40
k(E;T)/10 -13cm 3molecule -1s-1/kJ mol -1
3 30
at 300 K 2'
20
10
2''
0
0
(c)
5
10
15
E / kJ mol
20
25
30
-1
Fig. 11. The k(E; T)Õs for the association reactions of OH-radicals with olefins (a) at 100 K, (b) at 300 K, and (c) the result of speculation.
S. Sato / Chemical Physics 315 (2005) 65–75
energy. On this suggestion, the curve for the OH + propene reaction is divided into two parts as shown in Fig. 11(c). I speculate the following reaction as a candidate: OH þ C3 H6 ! H2 O þ C3 H5
ð49Þ
Incidentally, the rate equation proposed for the low-temperature gas-phase reactions, k(T) / Tn, contradicts WignerÕs threshold law, Eq. (11); therefore, it is necessary to modify the rate equation when the experimental measurement is extended to lower temperatures.
4. Concluding remarks When I proposed a new method of drawing the potential energy surface which is now called LEPS method [28], it was vaguely believed that the reaction barrier height is equal to the activation energy in Arrhenius equation experimentally obtained. Consequently, so-called Sato parameter was adjusted to obtain the activation energy. It is now obvious that the barrier height, especially for the H-atom transfer reactions, is larger than the activation energy in Arrhenius equation experimentally obtained near room temperature. Although the quantitative extent is not certain, the activation energies for these reactions are 70–80% of their barrier heights [29]. For the intramolecular reactions such as the methoxy radical decomposition reaction discussed above, the percentage may go down further. Transmission coefficient was originally proposed as a correction factor in the transition-state theory, which involved the corrections of recrossing of the activated complex at the col of potential energy surface and of tunneling through the barrier. As discussed in Section 3.3, the transmission coefficient due to the tunneling for the reaction of D + H2 ! HD + H is quite large; 68 and 6.7 at 200 and 300 K, respectively. These values are no longer the correction factors. If possible, these corrections should be taken into account when building up the transition-state theory at the beginning. In order to construct such a theory, it is necessary to introduce the thermodynamical concept of the tunneling state which is not necessarily in reaction. Classical thermodynamical treatment has been widely used to discuss the details of the reactions involving the transfer of light species such as H-atom and H+ and H ions, especially in organic reactions. In these treatments, the transmission coefficient is considered to be close to one and is often ignored.
75
Acknowledgements I thank Professor Kazuhiko Shibuya for helpful discussions and Dr. Mitsuhiko Takasawa for programming of numerical calculations.
References [1] S. Sato, Chem. Phys. Lett. 387 (2004) 216. [2] R.P. Bell, The Tunnel Effect in Chemistry, Chapman & Hall, London, 1980. [3] G.C. Schatz, Chem. Rev. 87 (1987) 81. [4] W.H. Miller, Acc. Chem. Res. 26 (1993) 174. [5] E.P. Wigner, Phys. Rev. 73 (1948) 1002; T. Takayanagi, N. Masaki, K. Nakamura, M. Okamoto, S. Sato, G.C. Schatz, J. Chem. Phys. 86 (1987) 6133. [6] K. Nakamura, T. Takayanagi, S. Sato, Chem. Phys. Lett. 160 (1989) 295. [7] E.P. Wigner, Z. Phys. Chem. B 19 (1932) 203. [8] S.P. Walch, A.F. Wagner, T.H. Dunning Jr., G.C. Schatz, J. Chem. Phys. 72 (1980) 2894. [9] S.L. Mielke, K.A. Peterson, D.W. Schwenke, B.C. Garrett, D.G. Truhlar, J.V. Micael, M.-C. Su, J.W. Sutherland, Phys. Rev. Lett. 91 (2003) 63201. [10] J.V. Michael, M.-C. Su, J.W. Sutherland, J. Phys. Chem. A 108 (2004) 432. [11] N. Cohen, K.R. Westberg, J. Phys. Chem. Ref. Data 12 (1983) 531. [12] D.E. Manolopoulos, J. Chem. Soc., Faraday Trans. 93 (1997) 673. [13] D.W. Schwenke, R. Steckler, F.B. Brown, D.G. Truhlar, J. Chem. Phys. 86 (1987) 2443; T. Takayanagi, S. Sato, Chem. Phys. Lett. 144 (1988) 191. [14] K. Stark, H.-J. Werner, J. Chem. Phys. 104 (1996) 6515. [15] T. Takayanagi, Y. Kurosaki, Chem. Phys. Lett. 286 (1998) 35. [16] J.W. Sutherland, J.V. Michael, A.N. Pirraglia, F.L. Nesbitt, R.B. Klemm, Symp. Int. Combust. Proc. 21 (1988) 929. [17] J.V. Michael, J. Chem. Phys. 90 (1989) 189. [18] G. Herzberg, Molecular Spectra and Molecular Structure II. Infrared and Raman Spectra of Polyatomic Molecules, D. van Nostrand, Princeton, NJ, 1968, p. 173. [19] A.R. Ravishankara, J.M. Nicovich, R.L. Thompson, F.P. Tully, J. Phys. Chem. 85 (1981) 2498. [20] A.D. Isaacson, J. Chem. Phys. 107 (1997) 3832. [21] K.H. Homann, H. Schweinfurth, J. Warnatz, Ber. Bunsenges. Phys. Chem. 81 (1977) 724. [22] A.C. Gonzalez, I.D. Reid, D.M. Garner, M. Senba, D.G. Fleming, D.J. Arseneau, J.R. Kempton, J. Chem. Phys. 91 (1989) 6164. [23] T. Takayanagi, Y. Kurosaki, J. Phys. Chem. A 101 (1997) 7098. [24] D.L. Baulch, C.J. Cobos, R.A. Cox, P. Frank, G. Hayman, Th. Just, J.A. Kerr, T. Murrells, M.J. Pilling, J. Troe, R.W. Walker, J. Warnatz, J. Phys. Chem. Ref. Data 23 (1994) 847. [25] M. Page, M.C. Lin, Y. He, T.K. Choudhury, J. Phys. Chem. 93 (1989) 4404. [26] I.W.M. Smith, B.R. Rowe, Acc. Chem. Res. 33 (2000) 261. [27] A.B. Vakhtin, J.E. Murphy, S.R. Leone, J. Phys. Chem. A 107 (2003) 10055. [28] S. Sato, J. Chem. Phys. 23 (1955) 592, 2465. [29] S. Sato, in: T. Ichikawa, T. Miyazaki, Y. Aratono, M. Shiotani (Eds.), Proceedings of Meeting on Tunneling Reaction and Low Temperature Chemistry, Sapporo, Japan, 1998, p. 2.
Tunneling in bimolecular reactions Shin Sato
*
Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-0033, Japan Received 15 December 2004; accepted 28 January 2005 Available online 27 April 2005
Abstract The estimation of the tunneling fraction of bimolecular reaction from the empirical rate equation has been made by using the Laplace transform relation between the rate equation and the cumulative reaction probability on the knowledge of the reaction barrier height. The tunneling fractions of the D + H2 ! HD + H reaction were found to be 95%, 65%, 24%, and 2.4% at the temperatures of 200, 300, 500, and 1000 K, respectively. Similar estimations were made for several reactions including the reactions of F-atoms with H2 and its isotopes, the reactions of O-atoms and OH-radicals with H2 and D2, the reactions of H-atoms and Mu (muonium) with F2, and the methoxy radical decomposition reaction. For most reactions containing H2 as the reactant, the tunneling fractions exceeded 50% at 300 K, while for the reactions containing D2 in place of H2, found were about a half of those for the reactions containing H2. Exception was found in the reactions of F-atoms with H2 and its isotopes. The tunneling fraction of the F + H2 reaction at 300 K was found to be 14%. The value is much smaller than those of other hydrogen atom transfer reactions calculated in this paper. The reason is probably the fact that the barriers of the F atom-reactions are flat and their heights are very low. Although there are some uncertainties on the accuracy of barrier heights, the Mu + F2 reaction seems to occur thoroughly through tunneling at 300 K, and the methoxy radical decomposition also seems to occur through tunneling at the temperatures up to 1000 K. Finally, an example of the application of the present method to the analysis other than the estimation of tunneling fraction was presented. Ó 2005 Elsevier B.V. All rights reserved.
1. Introduction In the previous Letter [1], I reported that approximate cumulative reaction probabilities for simple atom–diatom exchange reactions can be derived from the empirical rate equations by using Laplace transformation. As a result, it could be shown that almost all of the reaction of D + H2 ! HD + H occurs through tunneling at 200 K, and more than a half of that occurs at 300 K. For this reaction, the accurate reaction barrier height has been quantum mechanically calculated. It is well known that for the reaction in which the transfer of light species such as hydrogen atom is the rate-determining step, the tunnel effect plays an impor*
Present address: 7 Karasawa, Minami-ku, Yokohama 232-0034, Japan. Fax: +81 45 251 4174. E-mail address: [email protected]. 0301-0104/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2005.01.034
tant role [2]. However, if we want to estimate the quantitative extent of the tunnel effect, we have to perform the quantum mechanical calculation of rate constants on the accurate potential energy surface, or we have to be satisfied with simplified approximate calculations for tunneling attached to the transition-state theory [3]. As stated above, if the rate equation is experimentally known and the barrier height for this reaction is accurately calculated, we can obtain the cumulative reaction probability for the reaction, and can numerically estimate the extent of the tunnel effect. This paper will study further the reaction of D + H2 ! HD + H and will discuss the relation with the transition-state theory. Then, several bimolecular reactions with barriers will be analyzed by using the technique of Laplace inverse transformation, and the extent of tunnel effect will be discussed. For the last example of the calculations, the analysis of the reactions with
66
S. Sato / Chemical Physics 315 (2005) 65–75
no barriers will be shown, and the application of this technique to other reactions will be discussed. If a reaction occurs thoroughly through tunneling, so-called transition-state will not be on the col of the potential energy surface, but somewhere under the barrier. This issue will be discussed in the last section.
By using Laplace transform operator, we obtain 3=2
N ðEÞ ¼
A ð2plÞ 8p2 I F ðEÞ k nB rh4
with F ðEÞ ¼ L1
2. Theory
1 snþ5=2
ð5Þ
eEa s . 1 ehmBC s
ð6Þ
In the previous Letter, I assumed ehmBC s ¼ 0. Here, I expand the vibrational term as follows:
2.1. Basic calculation Bimolecular thermal rate constant k(T) is connected with the cumulative reaction probability N(E) by Laplace transformation as follows [4]: Z 1 1 kðT Þ ¼ N ðEÞ expðE=k B T Þ dE; ð1Þ hq 0 where h is the PlanckÕs constant, q is partition function of reactant, kB is BoltzmannÕs constant, T is temperature and E is the energy of reactant. If the rate constant k(T) and the partition function q can be expressed as simple functions of T, then the cumulative reaction probability N(E) is obtained from k(T) by Laplace inverse transformation. By using the approximation of harmonic oscillator and rigid rotor, the partition function of reactant can be expressed as a function of 1/(kBT); for example, the reactant partition function of the A + BC ! AB + C reaction may be expressed as
1 ¼ 1 þ ehmBC s þ . 1 ehmBC s
ð7Þ
If we take until the second term, Eq. (6) can be expressed as F ðEÞ ¼ F 0 ðEÞ þ F 1 ðEÞ 1 1 ¼ L1 nþ5=2 eEa s þ L1 nþ5=2 eðEa þhmBC Þs . s s ð8Þ By performing Laplace inverse transformation, we can obtain the following equations: ( nþ3=2 1 ðE Ea Þ ; E > Ea ; ð9Þ F 0 ðEÞ ¼ Cðnþ5=2Þ 0; E 6 Ea ; ( F 1 ðEÞ ¼
1 ðE Cðnþ5=2Þ
Ea hmBC Þ
nþ3=2
; E > Ea þ hmBC ; E 6 Ea þ hmBC .
0;
ð10Þ
BC q ¼ qt qBC vib qrot
¼
ð2plk B T Þ h3
3=2
1 8p2 Ik B T ; 1 expðhmBC =k B T Þ rh2
ð2Þ
where l is the reduced mass of A and BC, and mBC, I and r are the vibrational frequency, the moment of inertia and the symmetry number of the molecule BC, respectively. The zero-point energy of vibration is taken into account in the barrier height. Experimentally obtained thermal rate constant is often expressed as kðT Þ ¼ AT n expðEa =k B T Þ;
ð3Þ
where n is a constant. When n = 0, Eq. (3) is the Arrhenius equation. Since both Eqs. (2) and (3) are simple functions of 1/(kBT), the cumulative reaction probability N(E) can be calculated as follows. Substitution of Eqs. (2) and (3) into Eq. (1) with s = 1/(kBT) gives 1 snþ5=2
eEa s k nB rh4 ¼ 1 ehmBC s A ð2plÞ3=2 8p2 I
Z
1
N ðEÞ e
Es
dE.
0
ð4Þ
2.2. Wigner’s threshold law Although Eq. (3) has been widely used to express the temperature dependence of experimentally obtained reaction rates, this equation cannot be extended to very low temperatures. According to WignerÕs threshold law [5], the bimolecular rate constant for any exothermic reaction satisfies the following relation: lim kðT Þ / T 0 .
ð11Þ
T !0
This relation contradicts Eq. (3). Based on this finding, we contrived an empirical equation that connects the rate constants measured above room temperature and those calculated at low temperatures [6] 0 1 B kðT Þ ¼ A0 exp @
E0 C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA; k B T 2 þ T 20
ð12Þ
where A 0 , E0 and T0 are parameters, although E0 is not a simple parameter but is considered to correspond to the reaction barrier height. In the harmonic oscillator approximation, E0 may be expressed by
S. Sato / Chemical Physics 315 (2005) 65–75
E0 ¼ Ec þ
X1 z X1 hm hmj ; 2 i 2 i j
ð13Þ
1
z 1 ehms s
67
2
1 1e
hmzb s
z
z
ð20Þ
mzi
where Ec is the classical barrier height. and mj are the vibrational frequencies of transition-state and reactant. The calculation of N(E) using Eq. (12) has been reported in detail in the previous Letter. I do not repeat it but show the result 3=2
A0 ð2plÞ 8p2 I N ðEÞ ¼ F ðEÞ rh4
z
¼ 1 þ ehms s þ 2ehmb s þ e2hmb s þ .
Finally, we can obtain the cumulative reaction probability, NTST(E), as the sum of a series of Laplace inverse transforms: N TST ðEÞ ¼ N 00 ðEÞ þ N 10 ðEÞ þ 2N 01 ðEÞ þ N 02 ðEÞ þ ; ð21Þ
ð14Þ
1 E > E0 ; 1 z z ðE E 0 Þ; 1 1 E0 s N 00 ðEÞ ¼ z z L e ¼ rB 2 s rB 0; E 6 E0 ;
with F ðEÞ
" (Z Z )# b E pffiffiffi 4 E3=2 1 abs dq ¼ pffiffiffi s cosqðE sÞds sin pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; p 0 0 p 3 b2 q2 q
ð15Þ
ð22Þ ( N 10 ðEÞ ¼
1 rz Bz
ðE E0 hmzs Þ;
0;
E > E0 þ hmzs ; E 6 E0 þ hmzs ;
ð23Þ
where a = E0 and b = 1/(kBT0). In this derivation, the vibrational partition function qvib is assumed to be unity.
where 1/Bà = 8p2Ià/h2. N01(E), N02(E) and the succeeding terms can be obtained in the same manner.
2.3. Cumulative reaction probability obtained by the transition-state theory, NTST(E)
2.4. Wigner’s tunneling correction
As an example, let us consider the following atom– diatom exchange reaction: A þ BC ABCz ! AB þ C
ð16Þ
The transition-state theory rate constant may be written as follows: k B T qzABC E0 kðT Þ ¼ j exp ; ð17Þ h q kBT where j is the transmission coefficient and qzABC is the internal partition function of the transition-state. By ignoring the tunnel effect and the so-called recrossing, I assume j = 1, and also assume that three atoms ABC in the transition-state are collinear. Then, the internal partition function may be written as follows: 2 8p2 I z k B T 1 1 z qABC ¼ ; ð18Þ z z rz h2 1 ehms =kB T 1 ehmb =kB T à
à
where I and r are the moment of inertia and the symmetry number of the transition-state complex, and mzs and mzb are its stretching and doubly degenerated bending vibrational frequencies. Substitution of Eqs. (17) and (18) into Eq. (1) gives 2 2 z 8p I 1 1 1 eE0 s z z rz h2 s2 1 ehms s 1 ehmb s Z 1 ¼ N TST ðEÞ eEs dE. ð19Þ 0
Here, I expand the vibrational term in the same manner as Eq. (7)
WignerÕs (first order) correction for tunneling has been widely used as a convenient method attached to the transition-state theory [7]. In this method, the transmission coefficient j appearing in Eq. (17) is expressed by 2 1 hmz j¼1 ; ð24Þ 24 k B T where mà is the imaginary vibrational frequency of the transition-state complex. When Eq. (17) containing WignerÕs correction shown in Eq. (24) is substituted into Eq. (1), the following equation can be obtained: Z 1 1 1 1 E0 s 2 1 ðhmz sÞ e ¼ N TST ðEÞ eEs dE; 24 rz Bz s2 0 ð25Þ where, for simplicity, all vibrational partition functions are put as one. The cumulative reaction probability in this case is expressed as follows: N TST ðEÞ ¼ N 00 ðEÞ þ N 00 ðEÞ;
ð26Þ
1 ðhmz Þ2 F ðEÞ 24 rz Bz 00
ð27Þ
N 00 ðEÞ ¼ with
F 00 ðEÞ ¼ L1 feE0 s g ¼ dðE E0 Þ.
ð28Þ
If the vibrational partition function is included, the contribution to the tunnel effect due to the stretching vibration can be derived as follows:
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S. Sato / Chemical Physics 315 (2005) 65–75 2
N 10 ðEÞ
1 ðhmz Þ ¼ dðE E0 hmzs Þ. 24 rz Bz
ð29Þ
To calculate the values of N 01 ðEÞ and N 02 ðEÞ, hmzb and 2hmzb are substituted, respectively, in place of hmzs in Eq. (29). 2.5. Electronic state of reactant: the definition of k(E; T) When the electronic state of the reactant changes with temperature; for example, in the atom–diatom exchange reaction, A + BC ! AB + C, if A is oxygen atom in the ground state, the experimentally obtained rate constants are the mixture of the rate constants obtained with the three different states of oxygen atom, 3 P2, 3P1 and 3P0. At present, the observation for the rate constants of the reaction of oxygen atoms in the different sublevels of the ground state has not been succeeded. In order to calculate the cumulative reaction probability from such mixture of rate constants by using Eq. (1), the temperature dependence of the electronic states of reactant has to be considered. Since the ground state of oxygen atom is accurately analyzed [8], the calculation of the temperature dependence may be possible, but it is not a simple function of temperature which can be used in the estimation of N(E) in Eq. (1). In this paper, I aim to clarify the tunnel effect in simple reactions, so that, for such reactions, we avoid the calculation of the cumulative reaction probability, and calculate the integrand in Eq. (1) on the assumption that the temperature dependence of the partition functions of reactants containing atoms in different sublevels is same kðE;T Þ ¼ that is kðT Þ ¼
Z
1 N ðEÞ expðE=k B T Þ; hq
:
1 ðE k nB Cðnþrþ3 2 Þ
rþ1
Ea Þnþ 2
0;
1 kB T
rþ3 2
eE=kB T ; E > Ea ; E 6 Ea . ð33Þ
With this equation, we can estimate the extent of the tunnel effect in the reaction at desired temperature by comparing with the reaction barrier height, if it has been accurately calculated. In this section, I have discussed the electronic degeneracy of atoms in the reactant. When we formulate the rate constant of the reaction involving homonuclear diatomic molecules such as hydrogen and deuterium, we have to consider the nuclear-spin degeneracy. Strictly speaking, ortho- and para-molecules have to be treated as different molecules. However, in this paper, the rate constants discussed in the following sections are those obtained at the temperatures higher than 167 K. The ortho–para ratio above this temperature does not severely change even for hydrogen molecule. Consequently, it will not be necessary to be concerned about the temperature dependence due to the nuclear-spin degeneracy. In the following sections, I will not consider the contribution of both electronic and nuclear-spin degeneracies of atoms in the reactant and in the transition-state. If I take these nuclear degeneracies into the calculation of partition functions, the absolute values of cumulative reaction probabilities, N(E) and NTST(E), will be changed. Therefore, when I discuss on the cumulative reaction probabilities, I will pay attention only on their relative values between the probabilities obtained by different methods.
3. Results and discussion ð30Þ 3.1. Rate equation and cumulative reaction probability for the D + H2 ! HD + H reaction
1
kðE;T Þ dE.
ð31Þ
0
By this procedure, we can skip the calculation of the partition functions of the reactants involving atoms in the different sublevels. For the calculation of k(E; T), I assume r=2
qvib ¼ 1
kðE;T Þ ¼
8
and
qrot ¼
ðk B T Þ B
;
ð32Þ
where r is the sum of the degree of freedom of reactant rotation. Most of gas-phase rate constants reported are expressed by Eq. (3). On the definition of Eqs. (30) and (31) with the assumption of Eq. (32), we can derive the following equation by using the above-described technique of Laplace inverse transformation:
In the previous Letter, I used the following two rate equations to estimate the cumulative reaction probability for the D + H2 ! HD + H reaction: 22.43 ½kJ=mol kðT Þ ¼ 2.72 1017 T 2 exp ; ð34Þ kBT ! 37.07 ½kJ=mol kðT Þ ¼ 2.35 1010 exp pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð35Þ k B T 2 þ 1302 where the unit of k(T) is cm3 molecule1 s1. As stated in the previous Letter, neither rate equation completely agrees with the experimental data. Recently, an epoch-making convergence of theory and experiment has been reported on the rate constants of thermal hydrogen atom–diatom exchange reactions
S. Sato / Chemical Physics 315 (2005) 65–75 Table 1 The temperature dependence of the rate constants for the D + H2 ! HD + H reaction
8000
T (K)
Ref. [9]
Eq. (34)
Eq. (35)
Eq. (36)
6000
170 180 200 250 300 350 400 450 500 600 800 1000 1200 1500 2000
2.16(19) 4.57(19) 1.86(18) 3.40(17) 2.96(16) 1.51(15) 5.38(15) 1.48(14) 3.38(14) 1.21(13) 6.46(13) 1.90(12) 4.12(12) 9.53(12) 2.45(11)
1.01(19) 2.73(19) 1.51(18) 3.50(17) 3.04(16) 1.50(15) 5.13(15) 1.37(14) 3.09(14) 1.09(13) 5.97(13) 1.83(12) 4.14(12) 1.01(11) 2.82(11)
2.11(19) 4.47(19) 1.79(18) 3.16(17) 2.81(16) 1.53(15) 5.85(15) 1.73(14) 4.20(14) 1.65(13) 9.59(13) 2.82(12) 5.85(12) 1.21(11) 2.54(11)
2.10(19) 4.61(19) 1.93(18) 3.38(17) 2.80(16) 1.41(15) 5.03(15) 1.40(14) 3.26(14) 1.21(13) 6.68(13) 1.97(12) 4.23(12) 9.59(12) 2.44(11)
Numbers in parentheses are powers of 10.
[9]. For the D + H2 ! HD + H reaction, they showed the rate constants at different temperatures between 170 and 2000 K, which are listed in Table 1 together with the values calculated by using Eqs. (34) and (35). The last column shows the values calculated with Eq. (36) kðT Þ ¼ k 1 ðT Þ þ k 2 ðT Þ;
ð36Þ
N = N 1+ N 2
4000
N(E)
N1
2000
N2
0
0
20
60
80
100
-1
Fig. 1. Cumulative reaction probability of D + H2 ! HD + H reaction.
By substituting the parameter values in Table 2 into Eq. (21), we can obtain the NTST(E) shown in Fig. 2, which may be compared with N(E) obtained in the previous section. Naturally, below 35.4 kJ/mol, the reaction Table 2 The properties of the transition-state and the reactant of the D + H2 ! HD + H reaction D H H
ð37Þ !
35.4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . k B T 2 þ 1262
ð38Þ
I tried to find a rate equation better than Eqs. (34) and (35) by repeated try and error, and reached Eq. (36) which consists of the sum of two equations similar to Eqs. (34) and (35). I failed to find a single equation which can express the accurate rate constants. Michael et al. [10] proposed an equation which accurately expresses the temperature dependence of the rate constants for the D + H2 ! HD + H reaction. However, their equation is not adequate for the technique of Laplace transformation used in this paper. Fig. 1 shows the cumulative reaction probability for the D + H2 ! HD + H reaction calculated from Eq. (36) by using the technique described in the previous section. N1(E) and N2(E) shown in Fig. 1 correspond to those calculated from k1(T) and k2(T) in Eq. (36), respectively. 3.2. NTST(E) for the D + H2 ! HD + H reaction
0.0930 0.0930
mzs =cm1 mzs =cm1 à 1
1720 870 (doubly degenerate) 1472i
0.0741
m /cm
4401
Ec = 41.0 kJ/mol (= 9.80 kcal/mol) E0 = 35.4 kJ/mola a
Numerical mistake was found in the calculation of E0 in the previous paper [6]. The value of 35.4 kJ/mol is used in this paper in place of 37.07 kJ/mol in the previous Letter.
10000
N TST(E) N(E) 1000
100
10 20
To calculate the NTST(E), we need the various parameters of the reactant and the transition-state.
H2
r(D–H)/nm r(H–H)/nm r(H–H)/nm
Cumulative reaction probabilities
20.0 20 2.7 k 1 ðT Þ ¼ 5.41 10 T exp kBT
k 2 ðT Þ ¼ 9.29 1011 exp
40
E / kJ mol
where
and
69
30
40
50
60
E / kJ mol
70
80
-1
Fig. 2. Comparison between N(E) and NTST(E).
90
100
70
S. Sato / Chemical Physics 315 (2005) 65–75
barrier height, the value of NTST(E) does not exist. At higher energies, the series for NTST(E) were calculated up to the term of N07(E), and up to 27th rotational levels for N00(E). The two curves of N(E) and NTST(E) cross with each other at 63 kJ/mol. This might suggest that so-called recrossing starts to occur from this energy. But, we cannot conclude it at present, because of many assumptions involved in the calculation, such as the omission of unharmonicity in vibration and of nuclear-spin degeneracy. Originally, classical cumulative reaction probability is defined as the number of the quantum levels between E0 and E of the energy of the transition-state. For the D + H2 ! HD + H reaction, the cumulative reaction probability Nà(E) may be expressed by (P z E P E0 ; z ns ;nb ;J z ð2J þ 1Þ; N ðEÞ ¼ ð39Þ 0; E < E0 ;
(30). The result is shown in Fig. 4. A linear plot for the k(E; T) at 300 K is shown in Fig. 5. The area surrounded by the curve and the energy axis corresponds to the rate constant at this temperature. The fractional area under the barrier height, 35.4 kJ/mol, is the tunneling fraction, the value of which has been calculated to be 65% from this figure. The values at other temperatures have been found to be 95% at 200 K, 24% at 500 K and 2.4% at 1000 K. Similar estimation from theoretically calculated rate constants was made by Schatz [3], when the values of transmission coefficient j for many reactions were reported. For the D + H2 ! HD + H reaction calculated on the potential energy surface of LSTH, j = 68 at 200 K and 6.7 at 300 K were presented. If the tunneling fraction mentioned above has been calculated from these values of j, we can obtain 98.5% at 200 K and 85.1% at 300 K. Our values are a little smaller than those by Schatz.
where Jà is the rotational quantum number of the transition-state which has to satisfy the following relation: 1E-13
E P E0 þ J z ðJ z þ 1ÞBz þ ns hms X þ nbi hmb ; ns ; nbi ¼ 0; 1; 2; . . .
1000 K 1E-14
ð40Þ
1E-15
i¼1;2
500 K 1E-16
à
1E-17
k(E;T)
Fig. 3 shows N (E) up to 55 kJ/mol of E. It is obvious that NTST(E) is the smoothed function of Nà(E). When we discuss the reaction probability for each rotational level of the transition-state, the step-wise Nà(E) has to be considered.
300 K
1E-18 1E-19
200 K
1E-20
3.3. k(E; T) for the D + H2 ! HD + H reaction and its tunneling fraction
1E-21 1E-22 10
In this case, k(E; T) at desired temperature can be obtained by substituting N(E) shown in Fig. 1 into Eq.
30
40
50
60
E / kJ mol
70
80
90
100
-1
Fig. 4. The k(E; T) for the D + H2 ! HD + H reaction.
600
16 14
500
12 400
10
k(E;T)/10 -18
Cumulative reaction probabilities
20
N(E)
300
+
N (E)
200
8 6 4
100
N TST(E)
2
0
0 30
35
40
E / kJ mol
45
50
55
-1
Fig. 3. Cumulative reaction probability obtained by direct count, Nà(E) compared with N(E) and NTST(E).
10
20
30
40
E / kJ mol
50
60
-1
Fig. 5. The linear plot of k(E; T) for the D + H2 ! HD + H reaction at 300 K.
S. Sato / Chemical Physics 315 (2005) 65–75
3.4. Case of the reaction of F-atoms with hydrogen molecule and its isotopes The rate constants of the reaction of F-atoms with H2 and its isotopes are reported for the wide range of temperatures from 190 to 7000 K as follows [11] F þ H2 ! HF þ H kðT Þ ¼ 7.74 1011 ðT =298Þ0.50 e2.652=kB T ; F þ D ! DF þ D kðT Þ ¼ 5.73 1011 ðT =298Þ0.50 e3.475=kB T ; 2
F þ HD ! HF þ D kðT Þ ¼ 3.44 1011 ðT =298Þ0.50 e3.076=kB T ; F þ HD ! DF þ H kðT Þ ¼ 3.16 1011 ðT =298Þ0.50 e3.658=kB T .
ð41Þ Several other rate constants have been reported, but their temperature ranges are narrow. The k(E; T)Õs for these reactions calculated at 200 and 300 K from the above k(T)Õs are shown in Fig. 6. The theoretical study of the F + H2 ! HF + H reaction might be ahead of that of the hydrogen atom– diatom exchange reaction [12]. It is now well known that the transition-states of the F atom-reactions are not collinear [13], and the barrier heights are very low, i.e, Ec = 6.07 ± 1.05 kJ/mol (= 1.45 ± 0.25 kcal/mol) [14]. The barrier heights E0Õs for these reactions can be calculated from the data reported by Takayanagi and Kurosaki [15], and found to be 5.9 ± 1.0 kJ/mol for four reactions. By comparing this value with Fig. 6, the tunneling fractions of the F + H2 and F + D2 reactions at 200 and 300 K were found to be 34% and 14%, and 19% and 7.4%, respectively. These fractions are much smaller than those of the D + H2 reaction and the reactions of O-atoms and OH-radicals with hydrogen and its isotopes described in the following sections. The reason may be simply the fact that the barrier height is low and the reaction path to the col of the potential energy surface is so flat that the quantum effect does not much participate, as a whole, in the reaction,
71
although several interesting quantum effects have been discussed [12,15]. The ground state of F atoms is in the 2P state. Therefore, the F atom-reaction is the mixture of the reactions of F-atoms in the two states, 2P3/2 and 2P1/2. The difference in energy between these two sublevels is reported to be about 4.8 kJ/mol. Since the accurate potential energy surface has been calculated, the partition functions of the transition-states containing the two different stateF atoms could be constructed on the assumption of the appropriate structures of the transition-states. Then, the NTST(E)Õs for these reactions may be calculated, and the mixed rate constants will be discussed much more in detail, including the difference in reactivity of the sublevels of F-atoms. 3.5. Case of the reaction of O-atoms with H2 and D2 Among many rate equations reported, I selected the following two equations. The applicable temperature range is from about 300 to 2500 K [16,17] O þ H2 ! OH þ H; kðT Þ ¼ 3.39 1013 ðT =298Þ2.67 e26.274=k B T ; O þ D ! OD þ D kðT Þ ¼ 3.92 1012 ðT =298Þ1.70 e40.824=kB T . 2
ð42Þ Fig. 7(a) shows the k(E; T)Õs at 300 K calculated for the two reactions. Since the barrier height of the O + H2 ! OH + H reaction is calculated to be 38.5 kJ/mol [8], the tunneling fraction at 300 K can be estimated to be 52%. If the rate equation was extended to 200 K as a try, the tunneling fraction was found to be 88%. Since the barrier height for the O + D2 ! OD + D reaction was not available, I calculated the vibrational frequencies from the data of the O + H2 reaction by using the usual technique [18]. The result is shown in Table 3.
25
20
15
2 3
5
4 0
0
(a)
1: F + H 2 = HF + H 2: F + D 2 = DF + D 3: F + HD = HF + D 4: F + HD = DF + H
1
10
k(E;T)/10 -13cm 3molecule -1s-1/kJ mol -1
k(E;T)/10 -13cm 3molecule -1s-1/kJ mol -1
30
5
10
15
E / kJ mol
20 -1
25
25
1 20
15
2 10
3
5
4
0 0
30
(b)
5
10
15
E / kJ mol
20 -1
Fig. 6. The k(E; T)Õs for the reactions of F-atoms with H2 and its isotopes at (a) 200 K and (b) 300 K.
25
30
72
S. Sato / Chemical Physics 315 (2005) 65–75 5
k(E;T)/10 -20cm 3molecule -1s-1/kJ mol -1
70
k(E;T)/10 -16cm 3molecule -1s-1/kJ mol -1
80
O + H 2 = OH + H
60 50 40
O + D 2 = OD + D
30
x 1/10
20 10
OH + H 2 = H 2O + H
4
OH + D 2 = HDO + D
H2
3
2
1
D2 0
0 20
30
40
50
(a)
E / kJ mol
60
70
80
10
(b)
-1
20
30
E / kJ mol
40
50
60
-1
Fig. 7. The k(E; T)Õs at 300 K for the reactions of (a) O-atoms and of (b) OH-radicals with H2 and D2.
Table 3 The vibrational frequencies of the transition-states for the reactions of O + H2 and O + D2 and of the reactants Modea
O H H 3
A
mzs =cm1 mzb à m
a
O D D 3
00
A
0
3
3
A00
1577.0 514.0
A0
1579.8 1052.0
365.0
747.1
1738.0i
914.7i
m(H2) = 4266
m(D2) = 3017
The electronic state of the bent structure.
The barrier height for O + D2 ! OD + D reaction thus obtained was 44.8 kJ/mol. By comparing this value with the curve shown in Fig. 7(a), the tunneling fraction of the O + D2 reaction at 300 K was estimated to be only 7%. The trial extension to 200 K showed 21%. By substituting the imaginary vibrational frequencies shown in Table 3 into Eq. (24), we can estimate the transmission coefficients jÕs at 300 K as follows: 7.51 for the O + H2 and 2.80 for the O + D2 reactions, the values which correspond to 88% and 74% as the tunneling fraction, respectively. The discrepancy in tunneling fraction between the value calculated by the present method and that by WignerÕs tunneling correction, especially for the O + D2 reaction (7–74%), is remarkable. At present, I have no reasonable interpretations on this. 3.6. Case of the reaction of OH radicals with H2 and D2 To calculate the k(E; T)Õs for the title reactions, I used the following rate equations [19]: OHþH2 !H2 OþH kðT Þ¼4.481013 ðT =298Þ2.44 e10.643=kB T ; 1.18 OHþD2 !HDOþD kðT Þ¼3.641012 ðT =298Þ e19.373=kB T .
ð43Þ
Fig. 7(b) shows the k(E; T)Õs at 300 K for both reactions. Since the classical barrier height Ec for these reactions is reported to be about 28 kJ/mol [20], the tunneling fractions are about 60% for the OH + H2 reaction and 30% for the OH + D2 reaction, respectively. Precise values cannot be calculated because of the lack of the data for the transition-states. The isotope effect of H and D for the rate constants are not so large compared with that for the O atom-reactions described in the previous section. 3.7. Case of the reactions of H + F2 ! HF + F and Mu + F2 ! MuF + F For the H + F2 ! HF + H reaction, I selected the following rate equation [21]: 1.40 kðT Þ ¼ 1.4 1011 ðT =298Þ e5.565=kB T . ð44Þ The applicable temperature range is from 225 to 2000 K. In another title reaction, Mu + F2 ! MuF + F, Mu stands for muonium, which is an isotope of hydrogen atom whose atomic weight is 1/9 of H-atom. No rate equation has been proposed for this reaction. Experimentally obtained rate constants at each temperature are reported in a table [22], probably because the appropriate function could not be found. Fig. 8 shows the Arrhenius plot for this reaction. The curve shown in the figure is calculated by the following equation: ! 12.5 10 kðT Þ ¼ 6.33 10 exp pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . ð45Þ k B T 2 þ 3352 The applicable temperature range is from 100 to 450 K. We used these two equations to calculate the k(E; T)Õs. Fig. 9(a) shows the k(E; T)Õs for the H + F2 ! HF + F reaction at 200 and 300 K. The classical barrier height has been reported to be 15.5 kJ/mol (= 3.7 kcal/ mol) [23]. The zero-point-inclusive barrier height E0 was
S. Sato / Chemical Physics 315 (2005) 65–75
CH3 O þ M ! CH2 O þ H þ M
4.5
ð46Þ
where M stands for inert atom. The rate constants at the temperatures from 300 to 1700 K are expressed by a simple Arrhenius equation [24] kðT Þ ¼ 9.0 1011 e56.44=kB T . ð47Þ
3.5 3.0 2.5 2.0 1.5 1.0 0.5 2
3
4
5
6
7
8
9
10
11
1000/T
Fig. 8. Arrhenius plot for the reaction of Mu + F2 ! MuF + F.
discussed. Although the final value was not reached, it will not be too far from 15.5 kJ/mol. If we take this value, the tunneling fractions are almost same as those for the reactions of OH radicals with H2 shown in the previous section, i.e., about 90% at 200 K and 60% at 300 K. Fig. 9(b) shows the k(E; T)Õs for the Mu + F2 ! MuF + F reaction at the temperatures from 100 to 450 K. Since the rate constants do not much change in this temperature range, the k(E; T)Õs calculated can be demonstrated in the figure of linear scale. It is obvious that almost all reactions occur through tunneling under 300 K. Even at 450 K, more than a half of the reaction occurs through tunneling.
The unit of rate constant is cm3 molecule1 s1, and that of activation energy is kJ/mol. The barrier height E0 of this reaction has been quantum mechanically calculated to be 107.1 kJ/mol (= 25.6 kcal/mol) [25]. By substituting Eq. (47) into Eq. (33), we can obtain the k(E; T) at desired temperature. Fig. 10 shows the logarithmic plot of k(E; T)Õs as a function of energy at the temperatures from 300 to 1700 K. Comparison with the barrier height (107.1 kJ/mol) suggests that almost all reactions under 1000 K occur through tunneling. A detailed calculation has shown that the tunneling fraction at 1700 K is 71%.
1E-13 1E-14
1700 K k(E;T)/cm 3molecule -1s-1/kJ mol -1
4.0
k(T)/10 -11cm 3molecule -1s-1
73
1E-15 1E-16
1000 K
1E-17 1E-18 1E-19
500 K
1E-20 1E-21
300 K
1E-22 1E-23
3.8. Case of the methoxy radical decomposition reaction
1E-24 40
Methoxy radicals are believed to decompose by collision with inert atoms such as Ar and He as follows:
60
80
100
120
140
E / kJ mol
160
180
200
220
-1
Fig. 10. The k(E; T)Õs for the methoxy radical decomposition reaction.
3.0
100 K
12 10
300 K
8 6 4
200 K 2 0 0
(a)
k(E;T)/10 -12cm 3molecule -1s-1/kJ mol -1
k(E;T)/10 -14cm 3molecule -1s-1/kJ mol -1
14
5
10
15
20
E / kJ mol
25 -1
30
35
2.5
2.0
450 K 1.5
1.0
300 K
0.5
200 K
0.0
0
40
(b)
10
20
30
E / kJ mol
40
50
-1
Fig. 9. The k(E; T)Õs for (a) the H + F2 ! HF + F reaction at 200 and 300 K and (b) the Mu + F2 ! MuF + F reaction at the temperatures from 100 to 450 K.
74
S. Sato / Chemical Physics 315 (2005) 65–75
tion for these reactions. The reactions and their rate equations are as follows [27]: OH þ C H ! C H OH kðT Þ ¼ 8.7 1012 ðT =300Þ0.85 ;
This reaction has been analyzed by using the RRKM theory with the tunneling correction; however, such a high tunneling fraction seems to suggest that classical RRKM theory has to be modified. The true structure of the transition-state complex under the barrier might be quite different from that classically estimated and may be dependent upon the energy and its quantum level of the reactant.
2
4
2
4
OH þ C3 H6 ! C3 H6 OH kðT Þ ¼ 2.95 1011 ðT =300Þ1.06 ; OH þ 1-C H ! C H OH kðT Þ ¼ 3.02 1011 ðT =300Þ1.44 . 4
8
4
8
ð48Þ
The applicable temperature range is from 96 to 296 K. Note that no exponential terms are contained in these rate equations. By substituting these equations into Eq. (33), we can obtain the k(E; T)Õs for these reactions, which are shown in Fig. 11(a) and (b). Although no abnormal trend can be found in three curves at 100 K, the curves at 300 K for the reactions with propene and 1-butene cross with each other at 6.5 kJ/mol of the reactant energy. This seems to suggest that an unknown reaction other than the association reaction between OH and propene occurs simultaneously at high
3.9. Case of the association reaction between OH radicals and olefins Special experimental techniques have been developed recently for gas-phase reactions at very low temperatures [26], and the temperature dependence of the rate constants was reported. Most of the reactions have no barriers, so that the tunnel effect does not participate in these reactions. However, when I calculated the k(E; T)Õs for the association reactions between OH radicals and olefins, I found a small but interesting sugges-
40
3
40
at 100 K
2
30
20
10
1
0
0
(a)
3 k(E;T)/10 -13cm 3molecule -1s-1/kJ mol -1
k(E;T)/10 -12cm 3molecule -1s-1/kJ mol -1
50
1: ethylene 2: propene 3: 1-butene
2
4
6
E / kJ mol
8
30
2
1
10
0
10
0
-1
at 300 K
20
5
(b)
10
15
E / kJ mol
20
25
30
-1
40
k(E;T)/10 -13cm 3molecule -1s-1/kJ mol -1
3 30
at 300 K 2'
20
10
2''
0
0
(c)
5
10
15
E / kJ mol
20
25
30
-1
Fig. 11. The k(E; T)Õs for the association reactions of OH-radicals with olefins (a) at 100 K, (b) at 300 K, and (c) the result of speculation.
S. Sato / Chemical Physics 315 (2005) 65–75
energy. On this suggestion, the curve for the OH + propene reaction is divided into two parts as shown in Fig. 11(c). I speculate the following reaction as a candidate: OH þ C3 H6 ! H2 O þ C3 H5
ð49Þ
Incidentally, the rate equation proposed for the low-temperature gas-phase reactions, k(T) / Tn, contradicts WignerÕs threshold law, Eq. (11); therefore, it is necessary to modify the rate equation when the experimental measurement is extended to lower temperatures.
4. Concluding remarks When I proposed a new method of drawing the potential energy surface which is now called LEPS method [28], it was vaguely believed that the reaction barrier height is equal to the activation energy in Arrhenius equation experimentally obtained. Consequently, so-called Sato parameter was adjusted to obtain the activation energy. It is now obvious that the barrier height, especially for the H-atom transfer reactions, is larger than the activation energy in Arrhenius equation experimentally obtained near room temperature. Although the quantitative extent is not certain, the activation energies for these reactions are 70–80% of their barrier heights [29]. For the intramolecular reactions such as the methoxy radical decomposition reaction discussed above, the percentage may go down further. Transmission coefficient was originally proposed as a correction factor in the transition-state theory, which involved the corrections of recrossing of the activated complex at the col of potential energy surface and of tunneling through the barrier. As discussed in Section 3.3, the transmission coefficient due to the tunneling for the reaction of D + H2 ! HD + H is quite large; 68 and 6.7 at 200 and 300 K, respectively. These values are no longer the correction factors. If possible, these corrections should be taken into account when building up the transition-state theory at the beginning. In order to construct such a theory, it is necessary to introduce the thermodynamical concept of the tunneling state which is not necessarily in reaction. Classical thermodynamical treatment has been widely used to discuss the details of the reactions involving the transfer of light species such as H-atom and H+ and H ions, especially in organic reactions. In these treatments, the transmission coefficient is considered to be close to one and is often ignored.
75
Acknowledgements I thank Professor Kazuhiko Shibuya for helpful discussions and Dr. Mitsuhiko Takasawa for programming of numerical calculations.
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