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Empirical rate formula for ion-dipolar molecule reactions
Accepted Manuscript Empirical rate formula for ion-dipolar molecule reactions Shin Sato PII: DOI: Reference:
S0301-0104(16)31016-3 http://dx.doi.org/10.1016/j.chemphys.2017.02.006 CHEMPH 9750
To appear in:
Chemical Physics
Received Date: Accepted Date:
11 December 2016 22 February 2017
Please cite this article as: S. Sato, Empirical rate formula for ion-dipolar molecule reactions, Chemical Physics (2017), doi: http://dx.doi.org/10.1016/j.chemphys.2017.02.006
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Empirical rate formula for ion-dipolar molecule reactions Shin Sato∗ Tokyo Institute of Technology, 2−12−1, Ookayama, Meguro-ku, Tokyo 152−8550 Japan
New empirical rate formula for the low temperature reactions is proposed. The formula proposed previously has been simplified by using incomplete gamma function. A few examples of temperature dependence of rate constants for the reactions of ions with dipolar molecules and that for the reaction F + H2 → HF + H are demonstrated by using new rate formula.
1. Introduction According to Wigner’s threshold law, the rate constant of bimolecular exothermic reaction approaches a finite value with decreasing temperature to 0 K [1]. lim k(T ) = constant (1) T →0
This value may be called Wigner’s limit† in the low temperature reaction. On the other hand, the rate constant of the reaction of ion with nonpolar molecule is well known to become a temperature independent constant value, which is called Langevin rate constant [2]. kL = 2πe
1/2 α , µ
(2)
where, e is elementary charge, α is polarizability of the component molecule, and µ is the reduced mass between ion and molecule. In fact, however, the rate constants observed in experiments decrease with increasing temperature. For the reaction of ion with dipolar molecule, Troe proposed the following equation as the rate constant at 0 K [3]. q k(T → 0) = kL 1 + µ2D /3αBe , (3) where, µD is the dipole moment and Be is the rotational constant of dipolar molecule. The units of both molecular constants have to be changed to the MKSA system. These rate constants observed at low temperatures (10−10 ∼ 10−13 cm3 s−1 for radical reactions and 10−7 ∼ 10−9 cm3 s−1 for ion-molecule reactions) decrease with increasing temperature. 2. New empirical rate formula ∗ present address: 7 Karasawa, Minami-ku, Yokohama, 232-0034, Japan Tel./fax: +81 45 251 4174, E-mail address : [email protected] † We learnt this name from Professor Guo of New Mexico University.
1
In order to contrive an empirical rate formula for low temperature reactions with negative activation energies, A + B → AB∗ → product, where AB∗ is the intermediate associative complex, we start from the following equation which connects the rate constant with the cumulative reaction probability [4]. Z ∞ 1 k(T ) = N (E)exp(−E/kB T )dE, (4) hq 0 where, h is Planck’s constant, q is the partition function of reactant, kB is Boltzmann’s constant, and E is the energy of reactant. The reactant partition function may be written as follows: q=
(2πµkB T )3/2 · qvib · qrot , h3
(5)
where, µ 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 = 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 = M (kB T )n/2 , (6) where, M stands for a constant value for the reactant which does not depend on temperature and n is the sum of the degrees of relative translational freedom 3 between A and B and the sum of the degrees of rotational freedom of A and B. If the reaction A + B is most effective to complete temperature independently, the rate constant k(T ) in Eq. (4) may be replaced by the value observed at 0 K, which we name B. When k(T ) is replaced by the constant value B, Eq. (4) can be solved, and N (E) is expressed as follows: N (E) = (BhM )
E n/2−1 = (BhM )F (E) Γ n2
(7)
Next, we assume that the decrease of the rate constant with increasing temperature is due to the excited intermediate complex formed in the reaction which contains higher energy cannot proceed to the product but goes back to the original reactant. We tentatively assume the boundary energy to be Eb . So, the reaction proceeds through the intermediate complex containing the energy lower than Eb . As a result, the rate constant of this reaction can be expressed as follows: Z Eb B F (E)exp(−E/kB T )dE (8) k(T ) = (kB T )n/2 0 In the previous paper [5], we showed the integrated results as functions of each value of n. But we found that this integration can simply be expressed by using incomplete gamma function as follows: " , # n n k(T ) = B 1 − Γ ,y Γ , (9) 2 2 2
where, y = Eb /kB T . As a matter of course, n becomes a parameter in the same as Eb . In the following a few examples of the calculated results by using Eq. (9) are demonstrated. 3. Results Numerous ion-molecule reactions have been studied over the last fifty years and their rate constants are summarized [5]. However, the data on their temperature dependence are quite limited, especially for the reactions of polar molecules at the temperatures lower than 200 K. (Polar compounds easily condense on the reaction vessel.) The CRESU (Cin´etique de R´eactions en Ecoulement Supersonique Uniforme) method is overcoming this difficulty [6]. 3.1. He+ , C+ and N+ ions with CO, NH3 and H2 O [7,8] Figure 1 shows the temperature dependence of the reactions of nitrogen ions with NH3 , H2 O, CO, and two non-polar molecules, CH4 and O2 for comparison. Table 1 summarizes the parameter values used in Fig. 1 with those for the reactions of He+ and C+ ions, although they are not shown as figures. Table 1 Parameters for the reactions of He+ , C+ and N+ ions a) a) b) Reaction kL k(T →0) n Eb Reaction + He + N2 1.66 0.5 0.65 He+ + NH3 + He + O2 1.57 0.5 0.35 He+ + H2 O + He + CO 1.75 4.49 1.0 0.2 C+ + NH3 + N + O2 0.95 0.5 0.2 C+ + H 2 O + N + CH4 1.38 0.5 0.4 N+ + NH3 + N + CO 1.07 2.74 0.5 0.0036 N+ + H 2 O a) −9 3 −1 −1 b) −1 10 cm molecule s , kJ mol
kL k(T →0) 1.95 31.4 1.57 32.3 1.33 21.4 1.06 21.8 1.27 20.5 1.01 20.7
n 1.0 2.0 1.0 1.25 1.1 1.1
+ 3.2. NH+ 4 + NH3 (+ M) and H3 O + H2 O (+M) [9]
Figures 2 and 3 show the temperature dependence of the rate constants of two title reactions. The parameters are summarized in Table 2. As it is noticed, two calculated curves in both figures nearly overlap with each other, indicating the stabilizing effects of third bodies, He and N2 , are similar in both reactions. At the temperatures lower than 100 K, N2 is more effective than He, while above 100 K, the collision frequency seems to be predominant. He atom’s collision with the excited intermediate complex ion may be more frequent than that of N2 at the same temperature.
3
Eb 0.005 0.035 0.01 0.07 0.016 0.054
Table 2 Parameters for the reactions of titlereactions a) a) b) Reaction kL k(T →0) n Eb NH+ 1.25 20.33 2.5 0.053 4 + NH3 (+He) NH+ + NH (+N ) 7.0 1.25 3 2 4 H3 O+ + H2 O (+He) 0.925 19.17 2.5 0.050 H3 O+ + H2 O (+N2 ) 7.0 0.825 a) 10−9 cm3 molecule−1 s−1 , b) kJ mol−1 The relation between two parmeters n and Eb in Eq. (9) is very sensitive. In order to discuss the energy-transfer mechanism included in Eq. (9), more accurate and extensive measurements of the rate constants are highly desired. 3.3. F + H2 → HF + H As the last demonstration, a splendid experimental result recently reported [10] is shown in Fig. 4: temperature dependence of the rate constant of the reaction F + H2 → HF + H, where two curves are drawn by using two equations, Arrhenius equation: k(T ) = A exp(−Ea /kB T ) and Eq. (9). 4. Conclusion As stated above, a single equation Eq. (9) can express the rate constants at low temperatures. Consequently, in all range of temperatures from very low to high, the rate constants of many reactions can be expressed by the sum of Arrhenius equation and Eq. (9), although small modifications might be necessary for each reaction. For example, Arrhenius equation is very useful for the short range of temperatures but when it is applied to the wide range of temperatures, modification is needed. For Eq. (9), if the component reactant is hydrogen molecule, para-ortho conversion occurs in the range of low temperatures.
References 1. E.P. Wigner, Phys. Rev. 73 (1948) 174; D.W. Schwenke, D.G. Trular, J. Chem. Phys. 83 (1985) 3454; T. Takayanagi, N. Masaki, K. Nakamura, M. Okamoto, S. Sato, G.C. Schatz, J. Chem. Phys. 86 (1987) 6133. 2. P. Langevin, Ann. Chim. Phys. 5 (1905) 245; G. Gioumousis, D.P. Stevenson, J. Chem. Phys. 29 (1958) 294. 3. J. Troe, J. Chem. Phys. 87 (1987) 2773; 105 (1996) 6249. 4. S. Sato, Chem. Phys. 469-470 (2016) 49. 5. Y. Ikezoe, S. Matsuoka, M. Takebe, A. Viggiano, “Gas Phase Ion-Molecule Reaction Rate Constants through 1986,” Maruzen, Tokyo (1987).
4
6. I.W.M. Smith, Angew. Chem. Int. Ed., 45 (2006) 2842. 7. B. Rowe, J.B. Marquette, G. Dupeyrat, Chem. Phys. Lett., 113 (1985) 403. 8. J.B. Marquette, B.R. Rowe, G. Dupeyrat, G. Poissant, C. Rebrion, Chem. Phys. Lett., 122 (1985) 431. 9. S. Hamon, T. Speck, J.B.A. Mitchell, B.R. Rowe, J. Troe, J. Chem. Phys. 117 (2002) 2557; 123 (2005) 054303. 10. M. Tizniti, S.D. Le Picard, F. Lique, C. Berteloite, A. Canosa, M.H. Alexander, I.R. Sims, Nat. Chem. 6 (2014) 141.
5
Figure captions Fig. 1. Rate constants of the reactions of N+ ions with H2 O (), NH3 (O), CO (4), CH4 (◦), and O2 (). Fig. 2. Rate constants of the reactions of NH+ 4 + NH3 (+ M). M = He (◦) and M = N2 (•). Fig. 3. Rate constants of the reactions of H3 O+ + H2 O (+ M). M = He (◦) and M = N2 (•). Fig. 4. Rate constants of the reaction of F + H2 →HF + H (The values used for drawhing: A = 5.76 × 10−11 cm3 s−1 , Ea = 2.056 kJ mol−1 , B = 3.0 × 10−13 cm3 s−1 , n = 1.0, and Eb = 0.095 kJ mol−1 ).
6
F ig .1
1 0 0
3s -1 k/10-9cm
1 0
1
0 .1 1
1 0
T/K
1 0 0
1 0 0 0
F ig .2
1 0 0
3s -1 k/10-9cm
1 0 1
0 .1 0 .0 1 1
1 0
T/K
1 0 0
1 0 0 0
F ig .3
1 0 0
3s -1 k/10-9cm
1 0 1
0 .1 0 .0 1 1
1 0
T/K
1 0 0
1 0 0 0
F ig .4
1 0 0 0
3s -1 k/10-13cm
1 0 0 1 0 1 0 .1 1
1 0
T/K
1 0 0
1 0 0 0
S0301-0104(16)31016-3 http://dx.doi.org/10.1016/j.chemphys.2017.02.006 CHEMPH 9750
To appear in:
Chemical Physics
Received Date: Accepted Date:
11 December 2016 22 February 2017
Please cite this article as: S. Sato, Empirical rate formula for ion-dipolar molecule reactions, Chemical Physics (2017), doi: http://dx.doi.org/10.1016/j.chemphys.2017.02.006
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Empirical rate formula for ion-dipolar molecule reactions Shin Sato∗ Tokyo Institute of Technology, 2−12−1, Ookayama, Meguro-ku, Tokyo 152−8550 Japan
New empirical rate formula for the low temperature reactions is proposed. The formula proposed previously has been simplified by using incomplete gamma function. A few examples of temperature dependence of rate constants for the reactions of ions with dipolar molecules and that for the reaction F + H2 → HF + H are demonstrated by using new rate formula.
1. Introduction According to Wigner’s threshold law, the rate constant of bimolecular exothermic reaction approaches a finite value with decreasing temperature to 0 K [1]. lim k(T ) = constant (1) T →0
This value may be called Wigner’s limit† in the low temperature reaction. On the other hand, the rate constant of the reaction of ion with nonpolar molecule is well known to become a temperature independent constant value, which is called Langevin rate constant [2]. kL = 2πe
1/2 α , µ
(2)
where, e is elementary charge, α is polarizability of the component molecule, and µ is the reduced mass between ion and molecule. In fact, however, the rate constants observed in experiments decrease with increasing temperature. For the reaction of ion with dipolar molecule, Troe proposed the following equation as the rate constant at 0 K [3]. q k(T → 0) = kL 1 + µ2D /3αBe , (3) where, µD is the dipole moment and Be is the rotational constant of dipolar molecule. The units of both molecular constants have to be changed to the MKSA system. These rate constants observed at low temperatures (10−10 ∼ 10−13 cm3 s−1 for radical reactions and 10−7 ∼ 10−9 cm3 s−1 for ion-molecule reactions) decrease with increasing temperature. 2. New empirical rate formula ∗ present address: 7 Karasawa, Minami-ku, Yokohama, 232-0034, Japan Tel./fax: +81 45 251 4174, E-mail address : [email protected] † We learnt this name from Professor Guo of New Mexico University.
1
In order to contrive an empirical rate formula for low temperature reactions with negative activation energies, A + B → AB∗ → product, where AB∗ is the intermediate associative complex, we start from the following equation which connects the rate constant with the cumulative reaction probability [4]. Z ∞ 1 k(T ) = N (E)exp(−E/kB T )dE, (4) hq 0 where, h is Planck’s constant, q is the partition function of reactant, kB is Boltzmann’s constant, and E is the energy of reactant. The reactant partition function may be written as follows: q=
(2πµkB T )3/2 · qvib · qrot , h3
(5)
where, µ 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 = 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 = M (kB T )n/2 , (6) where, M stands for a constant value for the reactant which does not depend on temperature and n is the sum of the degrees of relative translational freedom 3 between A and B and the sum of the degrees of rotational freedom of A and B. If the reaction A + B is most effective to complete temperature independently, the rate constant k(T ) in Eq. (4) may be replaced by the value observed at 0 K, which we name B. When k(T ) is replaced by the constant value B, Eq. (4) can be solved, and N (E) is expressed as follows: N (E) = (BhM )
E n/2−1 = (BhM )F (E) Γ n2
(7)
Next, we assume that the decrease of the rate constant with increasing temperature is due to the excited intermediate complex formed in the reaction which contains higher energy cannot proceed to the product but goes back to the original reactant. We tentatively assume the boundary energy to be Eb . So, the reaction proceeds through the intermediate complex containing the energy lower than Eb . As a result, the rate constant of this reaction can be expressed as follows: Z Eb B F (E)exp(−E/kB T )dE (8) k(T ) = (kB T )n/2 0 In the previous paper [5], we showed the integrated results as functions of each value of n. But we found that this integration can simply be expressed by using incomplete gamma function as follows: " , # n n k(T ) = B 1 − Γ ,y Γ , (9) 2 2 2
where, y = Eb /kB T . As a matter of course, n becomes a parameter in the same as Eb . In the following a few examples of the calculated results by using Eq. (9) are demonstrated. 3. Results Numerous ion-molecule reactions have been studied over the last fifty years and their rate constants are summarized [5]. However, the data on their temperature dependence are quite limited, especially for the reactions of polar molecules at the temperatures lower than 200 K. (Polar compounds easily condense on the reaction vessel.) The CRESU (Cin´etique de R´eactions en Ecoulement Supersonique Uniforme) method is overcoming this difficulty [6]. 3.1. He+ , C+ and N+ ions with CO, NH3 and H2 O [7,8] Figure 1 shows the temperature dependence of the reactions of nitrogen ions with NH3 , H2 O, CO, and two non-polar molecules, CH4 and O2 for comparison. Table 1 summarizes the parameter values used in Fig. 1 with those for the reactions of He+ and C+ ions, although they are not shown as figures. Table 1 Parameters for the reactions of He+ , C+ and N+ ions a) a) b) Reaction kL k(T →0) n Eb Reaction + He + N2 1.66 0.5 0.65 He+ + NH3 + He + O2 1.57 0.5 0.35 He+ + H2 O + He + CO 1.75 4.49 1.0 0.2 C+ + NH3 + N + O2 0.95 0.5 0.2 C+ + H 2 O + N + CH4 1.38 0.5 0.4 N+ + NH3 + N + CO 1.07 2.74 0.5 0.0036 N+ + H 2 O a) −9 3 −1 −1 b) −1 10 cm molecule s , kJ mol
kL k(T →0) 1.95 31.4 1.57 32.3 1.33 21.4 1.06 21.8 1.27 20.5 1.01 20.7
n 1.0 2.0 1.0 1.25 1.1 1.1
+ 3.2. NH+ 4 + NH3 (+ M) and H3 O + H2 O (+M) [9]
Figures 2 and 3 show the temperature dependence of the rate constants of two title reactions. The parameters are summarized in Table 2. As it is noticed, two calculated curves in both figures nearly overlap with each other, indicating the stabilizing effects of third bodies, He and N2 , are similar in both reactions. At the temperatures lower than 100 K, N2 is more effective than He, while above 100 K, the collision frequency seems to be predominant. He atom’s collision with the excited intermediate complex ion may be more frequent than that of N2 at the same temperature.
3
Eb 0.005 0.035 0.01 0.07 0.016 0.054
Table 2 Parameters for the reactions of titlereactions a) a) b) Reaction kL k(T →0) n Eb NH+ 1.25 20.33 2.5 0.053 4 + NH3 (+He) NH+ + NH (+N ) 7.0 1.25 3 2 4 H3 O+ + H2 O (+He) 0.925 19.17 2.5 0.050 H3 O+ + H2 O (+N2 ) 7.0 0.825 a) 10−9 cm3 molecule−1 s−1 , b) kJ mol−1 The relation between two parmeters n and Eb in Eq. (9) is very sensitive. In order to discuss the energy-transfer mechanism included in Eq. (9), more accurate and extensive measurements of the rate constants are highly desired. 3.3. F + H2 → HF + H As the last demonstration, a splendid experimental result recently reported [10] is shown in Fig. 4: temperature dependence of the rate constant of the reaction F + H2 → HF + H, where two curves are drawn by using two equations, Arrhenius equation: k(T ) = A exp(−Ea /kB T ) and Eq. (9). 4. Conclusion As stated above, a single equation Eq. (9) can express the rate constants at low temperatures. Consequently, in all range of temperatures from very low to high, the rate constants of many reactions can be expressed by the sum of Arrhenius equation and Eq. (9), although small modifications might be necessary for each reaction. For example, Arrhenius equation is very useful for the short range of temperatures but when it is applied to the wide range of temperatures, modification is needed. For Eq. (9), if the component reactant is hydrogen molecule, para-ortho conversion occurs in the range of low temperatures.
References 1. E.P. Wigner, Phys. Rev. 73 (1948) 174; D.W. Schwenke, D.G. Trular, J. Chem. Phys. 83 (1985) 3454; T. Takayanagi, N. Masaki, K. Nakamura, M. Okamoto, S. Sato, G.C. Schatz, J. Chem. Phys. 86 (1987) 6133. 2. P. Langevin, Ann. Chim. Phys. 5 (1905) 245; G. Gioumousis, D.P. Stevenson, J. Chem. Phys. 29 (1958) 294. 3. J. Troe, J. Chem. Phys. 87 (1987) 2773; 105 (1996) 6249. 4. S. Sato, Chem. Phys. 469-470 (2016) 49. 5. Y. Ikezoe, S. Matsuoka, M. Takebe, A. Viggiano, “Gas Phase Ion-Molecule Reaction Rate Constants through 1986,” Maruzen, Tokyo (1987).
4
6. I.W.M. Smith, Angew. Chem. Int. Ed., 45 (2006) 2842. 7. B. Rowe, J.B. Marquette, G. Dupeyrat, Chem. Phys. Lett., 113 (1985) 403. 8. J.B. Marquette, B.R. Rowe, G. Dupeyrat, G. Poissant, C. Rebrion, Chem. Phys. Lett., 122 (1985) 431. 9. S. Hamon, T. Speck, J.B.A. Mitchell, B.R. Rowe, J. Troe, J. Chem. Phys. 117 (2002) 2557; 123 (2005) 054303. 10. M. Tizniti, S.D. Le Picard, F. Lique, C. Berteloite, A. Canosa, M.H. Alexander, I.R. Sims, Nat. Chem. 6 (2014) 141.
5
Figure captions Fig. 1. Rate constants of the reactions of N+ ions with H2 O (), NH3 (O), CO (4), CH4 (◦), and O2 (). Fig. 2. Rate constants of the reactions of NH+ 4 + NH3 (+ M). M = He (◦) and M = N2 (•). Fig. 3. Rate constants of the reactions of H3 O+ + H2 O (+ M). M = He (◦) and M = N2 (•). Fig. 4. Rate constants of the reaction of F + H2 →HF + H (The values used for drawhing: A = 5.76 × 10−11 cm3 s−1 , Ea = 2.056 kJ mol−1 , B = 3.0 × 10−13 cm3 s−1 , n = 1.0, and Eb = 0.095 kJ mol−1 ).
6
F ig .1
1 0 0
3s -1 k/10-9cm
1 0
1
0 .1 1
1 0
T/K
1 0 0
1 0 0 0
F ig .2
1 0 0
3s -1 k/10-9cm
1 0 1
0 .1 0 .0 1 1
1 0
T/K
1 0 0
1 0 0 0
F ig .3
1 0 0
3s -1 k/10-9cm
1 0 1
0 .1 0 .0 1 1
1 0
T/K
1 0 0
1 0 0 0
F ig .4
1 0 0 0
3s -1 k/10-13cm
1 0 0 1 0 1 0 .1 1
1 0
T/K
1 0 0
1 0 0 0