Empirical rate equation for association reactions and ion–molecule reactions

Chemical Physics 469-470 (2016) 49–59

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Empirical rate equation for association reactions and ion–molecule reactions Shin Sato ⇑ Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8550, Japan

a r t i c l e

i n f o

Article history: Received 11 October 2015 In final form 8 February 2016 Available online 2 March 2016 Keywords: Empirical rate equation Association reaction van der Waals potential Ion–molecule reaction

a b s t r a c t Temperature dependence of the rate constants of many association reactions is now available. In order to express the rate constants at temperatures from very low to high, we tried to use the sum of new empirical rate equations for association reactions and Arrhenius equations. Temperature dependence of a number of radical–molecule and some ion–molecule reactions could be successfully demonstrated. A new procedure to analyze ion–molecule reactions was proposed. This might suggest a new viewpoint to understanding chemical reactions. Ó 2016 Elsevier B.V. All rights reserved.

1. Introduction Arrhenius equation has been widely used to express the experimentally obtained temperature dependence of rate constants of chemical reactions [1].

kðTÞ ¼ A expðEa =RTÞ;

ð1Þ

where, A is preexponential factor, Ea is activation energy, R is gas constant, and T is absolute temperature. This equation can be applied to not only elemental reactions but also somewhat complex ones. Consequently, a chemical reaction theory has been developed to explain this equation experimentally obtained. However, it is regrettable to say that in spite of more than 80 years’ efforts, the theory which is based on classical mechanics with the help of correction of quantum mechanics such as ‘‘transition state theory” could not reproduce the Arrhenius equation obtained for any reaction within experimental errors. Pure quantum mechanical calculation was carried out in 2003 for the reactions representative to the Arrhenius-type, hydrogenatom-molecule reactions: D þ H2 ! DH þ H and H þ D2 ! HD þ D, and it was found that a complete agreement between theory and experiment. This is an epock-making success in the history of chemical kinetics [2]. This calculation, however, cannot be easily applied to other reactions containing many electrons in the reactant, though several calculations have been reported [3]. In other words, any chemical reaction is a quantum ⇑ Present address: 7 Karasawa, Minami-ku, Yokohama 232-0034, Japan. Tel./fax: +81 45 251 4174. E-mail address: [email protected] http://dx.doi.org/10.1016/j.chemphys.2016.02.013 0301-0104/Ó 2016 Elsevier B.V. All rights reserved.

mechanical event, which cannot be imitated by classical mechanics. If experimentally obtained data should be expressed with an equation, empirical rate equation has to be contrived. Though the chemical reaction theory is now up against the wall, experimental technique has been advanced significantly and the temperature range was broadened. As a result, Arrhenius equation was modified as follows:

kðTÞ ¼ AT n expðEa =RTÞ:

ð2Þ

Recent experimental skill reaches the measurement of rate constant at temperatures lower than 1 K [4] and it comes to be argued to mechanism of production of the interstellar matter [5]. Chemical reactions observed at low temperatures are not Arrhenius-type reactions accompanied with high energy barrier, so that the rate constant cannot be expressed by Arrhenius Eq. (1). As the empirical equation, Eq. (2) is used in which n is often put to be negative [6]. However, this equation has 3 parameters. Determining their values is not straightforward. Moreover, this equation contains an essential defect. Since all reactions observable at low temperatures are exothermic, according to Wigner’s threshold law [7], the rate constant should approach a finite value as the temperature approaches 0 K. Eq. (2) as well as Eq. (1) does not approach a finite value as the temperature approaches 0 K. Fig. 1 shows the rate constants as a function of temperature for the reactions of CN radicals with 3 hydrocarbons: ethane, ethene, and allene. In case of ethane, the rate constant clearly shows 2 parts: at low temperatures it decreases with increasing temperature, while at high temperatures it increases with increasing temperature. The latter one is an Arrhenius-type reaction, and the

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S. Sato / Chemical Physics 469-470 (2016) 49–59

As is stated with Fig. 1, in the most reactive system, the rate constant remains at a constant value from low to high temperatures. Let us put this value to be B and substitute it for kðTÞ in Eq. (3) to calculate NðEÞ.

Z

1

B¼

1

ð6Þ

NðEÞ expðE=kB TÞdE 0 hMðkB TÞ Z sn=2 1 1 NðEÞ expðsEÞdE s ¼ ðkB TÞ ¼ hM 0

1 sn=2

n=2

¼

Z

1 BhM

FðEÞ ¼ L1

1

NðEÞ esE dE ¼

0



Z

1

ð7Þ

FðEÞ esE dE

ð8Þ

0

1



sn=2

¼

En=21 Cðn=2Þ

ð9Þ

The result is shown in the following table as a function of n, where, NðEÞ ¼ ðBhMÞFðEÞ. Fig. 1. Rate constants of the reactions of CN radicals with ethane (), ethene () and allene (M). The curves are calculated by using parameters in Table 2.

n FðEÞ

former one must be an association reaction. All range of rate constant may be expressed by the sum of Arrhenius equation and a presumable association rate equation. In case of allene, the rate constant keeps a constant value from 20 to about 400 K and then starts to decrease. Probably this temperature dependence can be expressed by a simple association rate equation. In case of ethene, the temperature dependence is not simple but it is assumed that this experimental curve can be expressed as the sum of two equations: Arrhenius equation and a presumable association rate equation. These reactions of three types seem to be representative of many radical–molecule reactions. In the following sections, we will demonstrate many examples of temperature dependence of radical–molecule reactions and ion– molecule reactions as the sum of Arrhenius equation and presumable association rate equations. 2. Derivation of empirical rate equation Bimolecular thermal rate constant kðTÞ is connected with the cumulative reaction probability NðEÞ by Laplace transformation as follows [8]:

kðTÞ ¼

1 hq

Z

1

NðEÞ expðE=kB TÞdE;

ð3Þ

ð2plkB TÞ h

1=2

p1ffiffiffi E

p

2

3

1

p2ffiffiffi E

p

4 1=2

E

5 3

6

4 ffiffiffi p E

p

3=2

7 2

1 2E

8pffiffiffi E 15 p

8 5=2

1 3 6E

Here let us consider as follows: the excited associative complex AB formed between A and B may be more unstable if its energy is higher. The unstable complex shall go back to the original reactant. Let us assume its boundary energy to be Eb . Complex containing the energy above this value goes back to the reactant, while that with the energy less than this value proceeds to the product. Then, the rate constant of this reaction will be written as follows:

kðTÞ ¼

1 hq

Z

Eb

NðEÞ expðE=kB TÞdE

ð10Þ

0

By substituting the functions shown in the above table to NðEÞ, we can obtain the rate constants as a function of n as shown in Table 1. The equations for numbers above 9 are not presented to avoid confusion. Compact expression is shown in Appendix A. If the experiment for determination of rate constant was carried out at very low temperature, the value of B is not a parameter any more as far as the proposed empirical rate equation(s) is used. The value of B is the rate constant observed at very low temperature. In the case of modified Arrhenius Eq. (2), three parameters, A; n, and Ea interact with each other at any temperature.

0

where, h is Planck’s constant, q is partition function of reactant, kB is Boltzmann’s constant, and E is the energy of reactant. Let us assume that bimolecular reaction is represented as A þ B ! AB ! product, where, AB is the excited associative complex. The reactant partition function may be written as follows:

q¼

1

3=2

 qvib  qrot ;

3

ð4Þ

where, l is the reduced mass of A and B. The terms qvib and qrot are the partition functions of vibration and rotation of reactant. We assume, qvib ¼ 1 and qrot is proportional to the power of a half of the number of rotational freedom on the temperature of reactant. Then, the partition function q may be written as follows:

q ¼ MðkB TÞ

n=2

:

ð5Þ

Here, M stands for a constant value which does not depend on temperature and n is the sum of the number of relative translational freedom 3 between A and B and the sum of rotational freedoms of A and B.

3. Application to radical–molecule reactions Excited associative complex formed between a radical and a molecule is the excited state of van der Waals complex. Klippenstein and his collaborators have noticed that this complex has two transition states, one leading to the product and another to the original reactant [9–11]. By comparing the state densities of both transition states, they successfully calculated the temperature dependence of rate constants of radical–molecule reactions. In this paper, we use the sum of an empirical rate equation and an Arrhenius one. For many radical–molecule reactions, the empirical equation of n ¼ 2 seems to give good fit with experimental data.

kðTÞ ¼ k1 ðTÞ þ k2 ðTÞ

ð11Þ

k1 ðTÞ ¼ Bð1  expðEb =RTÞÞ

ð12Þ

k2 ðTÞ ¼ A expðEa =RTÞ

ð13Þ

At low temperatures Eq. (12) is dominant.

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S. Sato / Chemical Physics 469-470 (2016) 49–59 Table 1 Empirical rate equations for association reactions. n 1 3 5 7

k(T)/B pffiffiffi erfð yÞ pffiffiffiffiffiffiffiffiffi pffiffiffi erfð yÞ  2 y=p expðyÞ pffiffiffiffiffiffiffiffiffi pffiffiffi erfð yÞ  2 y=pð1 þ 23 yÞ expðyÞ pffiffiffiffiffiffiffiffiffi pffiffiffi 4 2 erfð yÞ  2 y=pð1 þ 23 y + 15 y Þ expðyÞ

n

k(T)/B

2 4

1  expðyÞ 1  ð1 þ yÞ expðyÞ

6

1  ð1 þ y þ 12 y2 Þ expðyÞ

8

1  ð1 þ y þ 12 y2 þ 16 y3 Þ expðyÞ

y ¼ Eb =ðRTÞ and erfðÞ is error function.

3.1. CN radicals with saturated hydrocarbons The rate constants of CN radical reactions with many hydrocarbons have been measured at temperatures above 200 K, whereas ethane is the only saturated hydrocarbon with which the reaction of CN radicals has been investigated by the CRESU method at low temperatures [12–16]. Figs. 2 and 3, respectively, show the temperature dependence of the rate constants of the reactions of CN radicals with two groups of hydrocarbons: ethane, propane and isobutane, and ethane, 2,2dimethylpropane and 2,2,3,3-tetramethylbutane. These figures suggest that every reaction seems to be accompanied by the association reactions at low temperatures. Fig. 4 shows the result of the CN þ CH4 reaction. Three points observed at temperatures lower than 250 K deviated from the Arrhenius equation, suggesting the contribution of the association reaction, even in the reaction of the CN radical with methane.

3.2. CN radicals with unsaturated hydrocarbons

Fig. 3. Rate constants of the reactions of CN radicals with ethane (), 2,2dimethylpropane () and 2,2,3,3-tetramethylbutane (}). The curves are calculated by using parameters in Table 2.

Fig. 5 shows the temperature dependence of CN radical reactions with ethene and acetylene. Both experimental data can be approximated on the same curve calculated by using Eq. (11) with parameters given in Table 2. The contributions from Eqs. (12) and (13) cannot be separately recognized from the curve shown in the figure. Fig. 6 shows the temperature dependence of the CN radical reactions with allene, methylacetylene, 1,3-butadiene and vinylacetylene. All data can be expressed by the curve calculated for the reaction with allene. The units of B; A; Eb and Ea used in Table 2 are also used up to Table 7.

Fig. 4. Rate constants of the reactions of CN radicals with ethane () and methane (). The curves are calculated by using parameters in Table 2.

3.3. CN radicals with ammonia and oxygen Among molecules other than hydrocarbons, ammonia and oxygen were used in the reactions with CN radicals in a relatively wide range of temperatures [17–19]. The reactions of vibrationally excited CN radicals (v = 1) were also measured and compared with those of the radicals (v = 0). Fig. 7 shows the result obtained with ammonia, and Fig. 8 for oxygen. The fitting curves in Fig. 7 were drawn by using the empirical equation of n ¼ 3 : Fig. 2. Rate constants of the reactions of CN radicals with ethane (), propane (M) and 2-methylpropane (). The curves are calculated by using parameters in Table 2.

pffiffiffiffiffiffiffiffiffi pffiffiffi k1 ¼ Bðerfð yÞ  2 y=p expðyÞÞ;

ð14Þ

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S. Sato / Chemical Physics 469-470 (2016) 49–59

Fig. 5. Rate constants of the reactions of CN radicals with ethene () and acetylene (). The curve is calculated by using parameters in Table 2.

Fig. 7. Rate constants of the reactions of CN radicals (v = 0) () and (v = 1) () with ammonia. The curves are calculated by using parameters in Table 3.

Table 2 Parameter values for the CN radical reactions with hydrocarbons. Hydrocarbon

B

Eb

A

Ea

Equation used for k1

Methane Ethane Propane Isobutane Neopentane Hexamethylethane

30 30 30 30 30 30

0.00025 0.075 0.48 0.60 0.68 1.15

2.4 10.0 15.0 15.0 12.0 12.0

9.00 4.20 4.20 4.20 4.20 4.20

(12) (12) (12) (12) (12) (12)

Ethylene, acetylene Allene, 1,3 - butadiene et al.

50 45

1.4 8.0

18.0

5.00

(12) (12)

B and A: 1011 cm3 s1 ; Eb and Ea : kJ mol

1

.

Fig. 8. Rate constants of the reactions of CN radicals (v = 0) () and (v = 1) () with oxygen. The curves are calculated by using parameters in Table 3.

Table 3 Parameters for the CN radical reactions with ammonia and oxygen. Reaction CN CN CN CN

Fig. 6. Rate constants of the reactions of CN radicals with allene (M), methylacetylene (), vinylacetylene (}), and 1,3-butadiene (). The curve is calculated by using parameters in Table 2.

while, for the reaction of oxygen, Eq. (12) was used to keep better fittings. As is shown in Table 3, the difference of vibrational states, v = 0 and v = 1, could be expressed by the values of Eb . Comparison of the values of B between the reactions of oxygen and ammonia shows that the value for oxygen is 1/3 of that for

(v = 0) + ammonia (v = 1) + ammonia (v = 0) + oxygen (v = 1) + oxygen

B

Eb

A

Ea

Equation used for k1

50

0.45 0.64 0.25 0.38

6.0

9.0

2.0

3.0

(14) (14) (12) (12)

17

ammonia. This difference might be due to that reaction with oxygen occurs on the lowest doublet surface which correlates with CN (2 Rþ ) and O2(3 R g ) [20]. 3.4. OH radicals with hydrogen halids The reaction of the OH radical with HBr has been studied by many researchers, since the rate constant shows a negative temperature dependence at room temperature [21–26]. Fig. 9 shows the temperature dependence of the rate constants of the OH radical reactions with HBr together with those of the reactions of HCl and

53

S. Sato / Chemical Physics 469-470 (2016) 49–59

Fig. 9. Rate constants of the reactions of OH radicals with HCl (), HBr () and HI (M). The curves are calculated by using parameters in Table 4.

Table 4 Parameters for the OH radical reactions with 3 hydrogen halides. Reaction

B

Eb

A

OH + HI OH + HBr OH + HCl

30 30 30

0.6 0.08 0.002

10.0 0.85

kp ¼ 30 ðs1 Þ kr ¼ 3  1013 expðEr =RTÞ ðs1 Þ;

Ea

Equation used for k1

10.0 7.60

(12) (12) (12)

HI collected from the literature [27–29]. The curves were drawn using Eq. (11) based on the assumption that B is 30; this value was obtained in a fitting for the reaction with HBr. The other parameter values are listed in Table 4. The curve shown for the reaction of HCl suggests the participation of the association reaction between OH and HCl at low temperatures, which has already been mentioned in the literature [28]. The curve for the reaction of HI may be explained in the same manner, which occurs at high temperatures. 3.5. OH radicals with ethene OH radical becomes larger adduct by adding ethene molecule. This adduct is stable at low temperatures, but starts to decompose near 450 K and dissociates back to reactants. Consequently, the decreasing rate of the concentration of OH radicals shows an interesting temperature dependence [30,31]. Fig. 10 shows the temperature dependence of the rate constants for the decay rate of OH radicals at different total pressures. At very high pressures of the third body (the top group of the data), no return can be observed [32], while at 1 atm of Ar, the decrease of the rate constant of OH radicals can be clearly observed (the middle group) [33]. In order to draw fitting curves for these data, the following reaction mechanism was considered. k1 kr

k2

OH þ C2 H4 ! products Here, ðC2 H4 OHÞ is the adduct. Then, the rate constant of the decay of OH radicals may be written as follows:

k ¼ k1

kp kp þ kr

ð16Þ

The value of Er has to be adjusted to each reaction. As Fig. 10 shows, the rate constants are strongly dependent upon the pressures of the third body (He, Ar, and N2). The experimental data of the bottom group in the figure are cited from a few literatures [34–36]. There are more data reported using different pressures, but we do not show them in Fig. 10 to avoid confusion. Table 5 shows the parameter values used for the reactions of hydroxide radicals with ethene, in which the reactions of deuterated compounds are included [37], though the graphs are not shown. 3.6. OH radicals with propene and 1-butene, and with 1,3-butadiene Propene and 1-butene react with OH radicals in the same manner as ethene [33,34,38–40], but 1,3-butadiene differently reacts with OH radicals [41,42]. Fig. 11 shows the result, where the curves are drawn using Eq. (14) instead of Eq. (12) to get better fitting. Although the data at low temperatures for 1,3 -butadiene are not available, the back-dissociation to the reactants from the adduct of 1,3-butadiene with OH radical does not seem to occur. Parameter values used for drawing the curves are shown in Table 6. 3.7. OH radicals with acetylene The back-dissociation of adducts similar to that described above in the reaction of OH radical with ethene has also been observed in Table 5 Parameters used for the reactions of hydroxide radicals with ethylene. Reaction

Pressure

B

Eb

A

Ea

Er

OH + C2H4

10 atm ðN2 Þ 1 atm (Ar) 1 atm (Ar) 1 atm (Ar) 1 atm (Ar)

60

0.05

60 60 60 60

0.03 0.025 0.03 0.025

4.0 4.0 4.0 4.0

22 23 23 26.5

150 150 150 150

(12): n ¼ 2 (12) (12) (12)

60

0.03

3.36

24.9

125

(12)

 kp

OH þ C2 H4 ¢ ðC2 H4 OHÞ ! products



Fig. 10. Rate constants of the reactions of OH radicals with ethene at total third body pressures of 10 atms (), 1 atm () and less than 1 bar (). The curves are calculated by using parameters in Table 5.

 þ k2

We assume the following values for kp and kr .

ð15Þ

OH + C2H4 OD + C2H4 OH + C2D4 OD + C2D4 OH + C2H4

4 mbar (Ar) for k1 0.6 bar (He) for k2

The unit of Er is kJ mol1.

Equation used for k1 n¼1

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S. Sato / Chemical Physics 469-470 (2016) 49–59

Fig. 11. Rate constants of the reactions of OH radicals with propene (), 1-butene () and 1,3-butadiene (). The curves are calculated by using parameters in Table 6.

Fig. 12. Rate constants of the reactions of O(3 P) with 8 mono-olefins: ethene (), propene (), 1-butene (), 2-methylpropene (M), trans-2-butene (/), cis-2-butene (5), 2-methyl-2-butene (.) and 2,3-dimethyl-2-butene (}). The curves are calculated by using parameters in Table 7.

Table 6 Parameters used for the reactions of OH radicals with 3 olefins. Olefin

B

Eb

A

Ea

Er

kp

Equation used for k1

Propene 1-butene 1,3-butadiene

50 50 50

0.4 0.5 0.9

2.30 3.74

10.4 9.3

116 108

50 50

(14) (14) (14)

3.9. Other radical–molecule reactions Many radical–molecule reactions other than those using CN or OH radical have been studied using CRESU method [50]. Each reaction observed with C2 H and CH radicals (v = 0) and (v = 1), C, S, Al, and B atoms showed their own characteristic behaviors, but the temperature dependence of their rate constants can roughly be classified by the reaction of three types of CN radicals with 3 hydrocarbon molecules as discussed in Introduction.

The unit of kp is s1.

Table 7 Parameters used for the reactions of O(3 P). Mono-olefin

B

Eb

A

Ea

Ethene Propene 1-butene 2-methylpropene cis-2-butene trans-2-butene 2-methyl-2-butene 2; 3-dimethyl-2-butene

30 30 30 30 30 30 30 30

0.003 0.053 0.070 0.32 0.32 0.32 1.01 1.42

1.063 1.60 1.63 3.00 3.00 3.00 3.00 3.00

6.65 4.16 4.16 3.33 3.33 3.33 3.33 3.33

the reaction with acetylene [43,44]. Unfortunately, however, the rate constants at temperatures lower than 200 K have not been reported. Moreover, according to ab initio calculations carried out by Senosian et al., a barrier following the van der Waals complex is higher than the energy of the reactant; for OH + C2 H4 , the barrier lies below the reactant energy [45,46]. The association reaction discussed above, therefore, will occur at more low temperatures in OH + C2 H2 [47]. 3.8. O(3 P) with mono-olefins

4. Application to ion–molecule reactions The rate of ion–molecule reaction is expressed as the decay rate of parent ions and the rate constants are usually represented by two kinds of units.



d½Aþ  ¼ kb ½Aþ ½B ¼ kt ½Aþ ½B½M; dt

where, Aþ is parent ion, B is component molecule, and M is the third body. The units of the rate constants, kb and kt , are 1

2

cm3 molecule s1 and cm6 molecule s1 , respectively. Therefore, if the concentration of the third body M (molecule cm3 ) is available, the rate constants kb and kt can be converted each other. The rate constant of kt is used to express for the reactions of atoms and simple molecules, while kb is used for the reactions of complex molecules. In general, the mechanism of associative ion–molecule reaction may be written as follows: kf

By using CRESU method, the problem on the negative temperature dependence of the rate constants observed in the reactions of Oð3 PÞ atoms with several mono-olefins, such as cis-2-butene and dimethyl-2-butene, at room temperature has been completely solved in 2007 [48]. Fig. 12 shows the result in which the temperature dependence is shown as a function of temperature. Eq. (14) was used for drawing the curves. The present author discussed this problem a few years ago, and demonstrated a similar graph, in which Eq. (12) was used for drawing the curves [49]. Because of the scattering of experimental data, we cannot determine which is better.

ð17Þ

 kc ½M

Aþ þ B ¢ ðABþ Þ ! ABþ þ M; kf



where, ðABþ Þ is the excited associative complex ion and AB+ is the product complex ion. By assuming the steady state of the excited complex ion, the decay rate of parent ions is expressed by the following equation:



kf kc ½M d½Aþ  ¼ ½Aþ ½B dt kf þ kc ½M

ð18Þ

When the concentration of the third body is small (kf kc ½M), kt can be written as follows:

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S. Sato / Chemical Physics 469-470 (2016) 49–59

kb ¼ Bð1  expðyÞÞ

ð21Þ

kb ¼ Bð1  ð1 þ yÞ expðyÞÞ

ð22Þ

The B values were assumed to be the Langevin rate constants which are shown in Table 8. Several combinations of rare gas ion–molecule reactions have also been calculated, although the graphs are not shown. The parameter n is related with the degree-of-freedom of reactants, but the value is not simple. For example, the degree-of-freedom of the reactants are 3 of relative translational motion in the ion–molecule-reactions of rare atoms (Table 8), but the values of n estimated from calculations extend from 2 to 8. The heavy rare atoms seem to show larger value of n. The boundary energy Eb has a trend to increase with the increase in degree-of-freedom of reactants. þ þ þ 4.2. Reactions of Nþ 2 þ 2N2 ! N4 þ N2 and O2 þ 2O2 ! O4 þ O2 þ

Arþ 2

Fig. 13. Rate constants of bimolecular ion–molecule reactions: Ar þ Ar ! (; ) and Arþ þ Ne ! ArNeþ (M). The curves are calculated by using parameters in Table 8.

kt ¼

kf kc kf

ð19Þ 2

The units of the rate constant is cm6 molecule s1 . On the other hand, if kf and kc ½M are comparable, kb is the rate coefficient of the right side of Eq. (18) and the units is 1

cm3 molecule s1 . At very low temperatures, especially for complex compounds, kf becomes small (kf kc ½M), then kb approaches kf . If the component molecule B has no polarity, kf may be approximated by the Langevin rate constant kL [51]:

 1=2 a k L ¼ 2p e ;

ð20Þ

l

where, a is the polarizability of the molecule B, e is elementary charge, and l is the reduced mass of A+ and B. In case that the molecule B has a polarity, many theorists discussed on this problem [52,53]. If, in the reaction mechanism stated above, the excited complex ion is replaced by the activated van der Waals complex which has been discussed in the reactions of radical and molecule, both situations are formally similar with each other. So, we tried to apply the empirical rate equations discussed above to ion–molecule reactions. (Rate constant kb will be used in the same way as k1 in radical–molecule reaction).

The association reaction of Nþ 2 ions was studied at temperatures from 4 to 500 K and the reaction of Oþ 2 ions was from 3.8 to 400 K [56–58]. Both rate constants kt ’s were expressed by inverse temperature dependence. From these data, kb ’s were calculated by assuming the concentration of third body to be 5  1014 molecule cm3 . The result is shown in Fig. 14. 4.3. Reactions of Cþ þ H2 þ He and Cþ þ D2 þ He In the measurement of these reactions, the pressure of third body He was reported to be 0.2 to 1.4 Torr [59]. We assumed 0.2 Torr of He to calculate kb ’s of these reactions. Fig. 15 shows the result. þ 4.4. Ion–molecule reactions of CHþ 3 and CD3 with simple molecules

The third body is He which is same as that of the previous section. In the calculations of kb ’s, we assumed 0.2 Torr of He as the third body. Fig. 16 shows the calculated result of the reactions of þ CHþ 3 ions. Table 11 includes the result of CD3 ions, which is very ions. similar to that of CHþ 3 þ For the reactions of CHþ 3 þ CO and CD3 þ CO, the B values are not Langevin rate constants (1:06  109 and 0:99  109 cm3 s1 , respectively). By adjusting the rate equations of kb ’s to the reported data, we obtained the B values shown in Table 11. The values of (40 or 70) 109 cm3 s1 are too large, although CO molecule has a dipole moment. The origin of these large values is not apparent. þ 4.5. Reaction of s-C3 Hþ 7 þ i-C4 H10 ! t-C4 H9 þ C3 H8

4.1. Reaction of Arþ þ 2Ar ! Arþ 2 þ Ar Ion–molecule reactions between rare gas atoms were studied by M. Hawley and D.A. Smith using the free-jet method [54]. The rate constants were measured down to temperatures lower than 1 K. All data were reported by kt . Unfortunately, the concentrations of the third body were not reported. In order to calculate the values 16

3

of kb , we assumed the concentration to be 5  10 molecule cm . Only for the title reaction of this section, the CRESU method was applied to measure the rate constants and the concentration of third body was reported for each measurement [55]. Fig. 13 shows the temperature dependence of the bimolecular rate constants kb calculated from the reported results of kt of the title reaction and Arþ þ 2Ne ! ArNeþ þ Ne reaction. For the title reaction, the empirical rate Eq. (21) (n ¼ 2) was found to give a good fitting to kb , while for Arþ þ 2Ne ! ArNeþ þ Ne reaction the Eq. (22) (n ¼ 4) gave a good fitting.

The title reaction is a representative reaction of H ion transfer [60]. Many hydrocarbons containing tertiary C–H bond were used as the component molecule. At low temperatures, the rate con-

Table 8 Ion–molecule reactions of rare gas atoms.



Reaction

Used equation for kb

B

Eb =J mol

Ar+ + Ar Ar+ + Ne Kr+ + Kr Kr+ + Ar Xe+ + Xe Xe+ + Kr Xe+ + Ar

n=2 n=4 n=5 n=3 n=5 n=8 n=6

0.67 0.40 0.57 0.58 0.58 0.52 0.54

0.062 0.271 3.00 0.80 2.84 8.00 4.00

Third body is the component atom. The units of B is 109 cm3 s1.



1

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S. Sato / Chemical Physics 469-470 (2016) 49–59

þ Fig. 14. Rate constants of bimolecular ion–molecule reactions: Nþ 2 þ N2 ! N4 (5; ) þ and Oþ þ O ! O (; M; ). The curves are calculated by using parameters in 2 2 4 Table 9.

Fig. 16. Rate constants of bimolecular ion–molecule reactions of CHþ 3 with simple molecules: H2 (M), N2 (5), O2 (), CO () and CO2 (). The curves are calculated by using parameters in Table 11.

Table 11 þ Ion–molecule reactions of CHþ 3 and CD3 ions. Table 9 Ion–molecule reactions of N2 and O2 .



Reaction

Used equation for kb

B

Eb =J mol

Nþ 2 Oþ 2

n¼4 n¼4

0.83 0.74

20.0 7.5

þ N2 þ O2

1

Third body is the component molecule. The units of B is 109 cm3 s1.





Reaction

Used equation for kb

B

Eb =kJ mol

CHþ 3 CHþ 3 CHþ 3 CHþ 3 CHþ 3 þ CD3 CDþ 3 CDþ 3 CDþ 3 CDþ 3

þ H2 þ N2

n¼8 n ¼ 10

1.56 0.99

1.25 2.1

þ O2 þ CO

n¼9 n¼8

0.93 (40)

1.0 1.25

þ CO2 þ H2

n ¼ 13 n¼8

1.14 1.16

6.5 3.2

þ N2 þ O2

n ¼ 10 n¼9 n¼8

0.94 0.87 (70)

3.5 1.12 1.26

n ¼ 13

1.06

7.2

þ CO þ CO2

1

Third body is He. The units of B is 109 cm3 s1.



Fig. 15. Rate constants of bimolecular ion–molecule reactions: Cþ þ H2 ! CHþ þ H () and Cþ þ D2 ! CDþ þ D (). The curves are calculated by using parameters in Table 10. Fig. 17. Rate constants of H ion transfer reactions from hydrocarbons containing tertiary C–H bond to s-C3 Hþ 7 ion. The curves are calculated by using parameters in Table 12. Table 10 Ion–molecule reactions of C+ ion with hydrogen. Reaction þ

C þ H2 C þ þ D2 

Used equation for kb

B

Eb =J mol

n¼5 n¼5

1.58 1.21

125 185

Third body is He. The units of B is 109 cm3 s1.



1

stants are close to Langevin rate constants, while at high temperatures, they are approximated by the inverse power of temperature. We have used the empirical rate equations described above to draw the curves shown in Fig. 17. Table 12 summarizes the parameter values.

57

S. Sato / Chemical Physics 469-470 (2016) 49–59 Table 12 H ion transfer reaction of s  C3 Hþ 7 ion.



Component molecule

Used equation for kb

B

Eb =kJ mol

(a) ðCH3 Þ3 CH (b) ðCH3 Þ2 CHC2 H5 (c) ðCH3 Þ2 CHC4 H9 (d) ðCH3 Þ2 CHC5 H11 (e) ðCH3 Þ2 CHCH2 CHðCH3 Þ2

n ¼ 12 n ¼ 12 n ¼ 20 n ¼ 20 n ¼ 22

1.10 1.15 1.30 1.45 1.30

16 19 44 46 47

1

The units of B is 109 cm3 s1.

Table 13 Other H ion transfer reactions. Reaction

Used equation for kb

B

Eb =kJ mol

C2 Hþ 5 þ ðCH3 Þ3 CH C3 Hþ 7 þ ðCH3 Þ2 CHCHðCH3 Þ2 C4 Hþ 9 þ ðCH3 Þ2 CHCHðCH3 Þ2 C5 Hþ 11 þ ðCH3 Þ2 CHCHðCH3 Þ2

n ¼ 20 n ¼ 22 n ¼ 10 n ¼ 12

1.3 1.3 1.3 1.3

58 47 4.00 4.02

t-C4 Hþ 9 þ ðCH3 Þ2 CHCHðCH3 Þ2 t-C4 Hþ 9 þ ðCH3 Þ2 CHCH2 CHðCH3 Þ2 t-C4 Hþ 9 þ ðCH3 Þ2 CHCH2 CH3

n ¼ 20 n ¼ 14 n¼6

1.3 1.3 1.3

14.15 7.1 1.19

1

Fig. 19. Rate constants of two ion–molecule reactions: Arþ þ N2 () and Oþ 2 þ CH4 (). The curves are calculated by using parameters in Table 15.

ture. Two reactions in the title of this section shows interesting contrast: the former reaction shows that the rate increases with increasing temperature, while the latter one shows that the rate decreases with increasing temperature. Together with these reactions, an odd-looking temperature dependence has been reported þ with the reaction of C2 Hþ 4 þ C2 H6 ! sec-C3 H7 þ CH3 [61,62]. Fig. 18 demonstrates them. Table 14 summarizes the parameters in which those for Arrhenius equation are included.

kðTÞ ¼ kb þ k2

k2 ¼ A expðEa =RTÞ

ð23Þ

4.7. Reactions of Arþ þ N2 and Oþ 2 þ CH4 Charge transfer reactions between atoms or molecules which have different ionization potentials are most common ion–molecule reactions, such as Arþ þ O2 ; Nþ þ CO; Cþ þ NO; COþ þO2 ; and Nþ 2 þ O2 . Among them, interesting behaviors can be observed in the title reactions [63,64]. Fig. 19 shows the temperature dependence of the rate constants of two reactions. The parameters used to draw the curves are summarized in Table 15. þ Fig. 18. Rate constants of 3 ion–molecule reactions: C2 Hþ 5 þ CH3 (), C2 H5 þ C2 H6 () and C2 Hþ þ C H (). The curves are calculated by using parameters in Table 14. 2 6 4

Many other H ion transfer reactions have been studied together with those shown in Table 12. Although we will not show them as figures, parameter values are summarized in Table 13, in which Langevin rate constants are assumed to be 1:3  109 cm3 s1 for all reactions. þ 4.6. Reactions of C2 Hþ 5 þ CH4 ! C3 H7 þ CH3 and þ þ C2 H5 þ C2 H6 ! C4 H9 þ H2

þ 4.8. Reaction of NHþ 3 þ H2 ! NH4 þ H

This reaction is believed to be the last step of the formation of NH3 in the interstellar cloud. Neutralization of NHþ 4 with an electron produces NH3. The title reaction is very slow even at room temperature, and several groups of investigators have measured the rate constants [65–67]. Fig. 20 shows the result. Two curves in the figure were drawn by using the parameters in Table 16. 5. Conclusion

Ion–molecule reactions discussed in previous sections showed that all of their rate constants decrease with increasing tempera-

As shown in the previous section, the sum of the empirical rate Eq. (12) or (14), whose n numbers are 2 and 3, and Arrhenius equations for each reaction has given good fittings to the rate constants

Table 14 Ion–molecule reactions accompanied with Arrhenius-type reaction.



Reaction

Equation for kb

B

Eb =kJ mol

C2 Hþ 5 þ CH4 C2 Hþ 5 þ C2 H6 C2 Hþ 4 þ C2 H6

n¼4 n¼4

1.13 1.14

1.10 0.40

9

The units of B is 10

3

cm s

1

.

1

A=109 cm3 s1

Ea =kJ mol

0.0006

10.46

3.64

25.0

1

58

S. Sato / Chemical Physics 469-470 (2016) 49–59

Table 15 Ion–molecule reactions which show interesting temperature dependence. Reaction þ

Ar þ N2 Oþ 2 þ CH4 

Equation for kb

B

Eb =kJ mol

n¼2 n¼6

0.765 1.15

0.01 0.45

1

A=1011 cm3 s1

Ea =kJ mol

10 3.0

5.5 5.5

1

The units of B is 109 cm3 s1.

n ¼ 2m; m P 1 kðTÞ ¼ B ½1  S0 ðyÞ expðyÞ; where; S0 ðyÞ ¼

m X yi1 ði  1Þ! i¼1

Appendix B The Eq. (11) in the present manuscript has similarity with Eq. (16) in the paper by Glosik et al. [68]. Eq. (16) in the Glosik’s paper þ shows the rate coefficients for the reactions of the ions H+, Hþ 2 , H3 , þ þ þ þ þ + + + + N , N2 , Ar , C , CH , CH2 , CH3 , CH4 , and CH5 with HCl as a function of reactant ion/reactant neutral average center-of-mass kinetic energy (KEc.m.) measured at 300 K.

kðKEc:m: Þ ¼ kI þ kII ¼ k1 ½1 þ ðKEc:m: =KEc:m:1 Þm  Fig. 20. Rate constants reported by different research groups on the þ NHþ 3 þ H2 ! NH4 þ H reaction. The curves are calculated by using parameters in Table 16.

Table 16 The reaction of ammonia ion and hydrogen molecule.



Reaction

Equation for kb

B

Eb =J mol

NHþ 3 þ H2 ! NHþ 4 þH

n¼2 n¼2

1.55

0.054 0.0065

1

A=1013 cm3 s1

Ea =kJ mol

32.4 130.9

4.4 9.63

1

The units of B is 109 cm3 s1.

experimentally obtained for the radical–molecule reactions in the temperature range between 20 and 500 K. This was one of the purposes of this paper (see Appendix B). On the other hand, for the ion–molecule association reactions, the n number of empirical rate equation was extended from 2 to 22, in order to get good fittings. In this treatment, several termolecular rate constants kt ’s were converted to the bimolecular rate constants kb ’s by assuming the concentration of third body, since the bimolecular rate constant is theoretically easier to understand than the termolecular rate constant. For example, Langevin’s rate constant is bimolecular. This procedure might open the new viewpoint to the understanding of ion-molecular reactions. Appendix A. Empirical rate equations for association reactions

pffiffiffi n ¼ 1 kðTÞ ¼ B erfð yÞ y ¼ Eb =ðRTÞ rffiffiffiffi   pffiffiffi y n ¼ 2m þ 1; m P 1 kðTÞ ¼ B erfð yÞ  2 SðyÞ expðyÞ ;

p

where; SðyÞ ¼

m X i¼1

ð2yÞi1 : 1  3  5    ð2i  1Þ

1

þ kII0 exp½DEA =ð2=3 KEc:m: Þ;

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