Determination of the Rate Constants of the Reactions Cr + O2 + M → CrO2 + M and Cr + O2 → CrO + O

The rate constants of the interactions of chromium atoms with molecular oxygen through recombination Cr + O₂ + M → CrO₂ + M (I) and exchange Cr + O₂ → CrO + O (II) were determined by a new method for treatment of experimental data. The results, together with the available literature data, led to the following equations for the rate constants of recombination in the low-pressure limit and of the exchange reaction: cm⁶ mol^–2 s^–1, , cm³ mol^‒1 s^‒1. An expression for the rate constant of the reverse reaction was obtained from k₂(T) and the equilibrium constant for reaction (II): cm³ mol^‒1 s^‒1. Modeling within the framework of the RRKM theory shows that calculation of the rate constant k_1,0(T) requires inclusion of not only the ground electronic state of the CrO₂ molecule, but also the low-lying excited electronic states up to the dissociation threshold. A comparison of the experimental and calculated temperature dependences shows that the best agreement between them is achieved at an average portion of energy transferred in deactivating collisions of the excited CrO₂ molecule with diluent gas molecules of ΔE = 2.8 kJ/mol.

Keywords: chromium atoms; molecular oxygen; recombination; exchange reaction; rate constants; RRKM model

Abbreviations and designations : RRKM, Rice–Ramsperger–Kassel–Marcus theory.

The interaction of chromium atoms with molecular oxygen in the gas phase is of considerable interest from the viewpoint of both fundamental reactivity studies [[1]] and applications [[3]–[8]], e.g., for analysis of the formation and transformations of chromium compounds during combustion of coal, industrial and domestic waste [[3]–[5]], explanation of the inhibitory effect of chromium on gas combustion [[6]], and for creating models of epitaxial growth of chromium dioxide films [[8]].

Like the reaction Fe + O2 [[9]], the interaction of a chromium atom with an oxygen molecule can occur by two channels: recombination and exchange:

I

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II

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As far as we know, the rate constant of reaction (I) was measured only in three works [[10]–[12]]. In [[10]], measurements were performed in a static reactor in the temperature and pressure ranges 298–323 K and 2–600 Torr, respectively. There were no signs of exchange reaction, which is quite understandable as reaction (II) is basically endothermic [[13]].

In [[11]], experiments on the interaction of chromium atoms with molecular oxygen were performed in a fast-flow reactor in the temperature and pressure ranges 290–1510 K and 11–460 Torr, respectively. The experimental data were interpreted based on the following assumptions: (1) At temperatures below 709 K, reaction (II) does not make a significant contribution to the observed rate constant of the consumption of chromium atoms. (2) Reaction (I) proceeds in the fall-off region. At higher temperatures, reaction (I) proceeds at low pressures; therefore, the observed rate constant was presented by the authors as the sum of the rate constant of reaction (I) multiplied by the total concentration and the rate constant of reaction (II) (formula (6) in [[11]]). We used this concept here for new interpretation of data from [[11]]; along with the data from [[10]–[12]], the result of this interpretation was used to compare with predictions based on the model described below for calculating the rate constant of reaction (I) in the low-pressure limit.

Table 1. Electronic states, their energies Ei, products of the moments of inertia (I1I2I3)0.5, and vibrational frequencies for isomers of CrO2

а, bCalculated in [[17]] and [[16]], respectively.

In [[12]], the interaction of Cr atoms with O2 was studied in a shock tube equipped with facilities for recording the atomic absorption of chromium atoms in the temperature range 720–3550 K at a total gas density of 2.8 × 10–6 mol/cm3 and in the range 720–980 K at 1.4 × 10–6 mol/cm3. The rate constants of reactions (I) and (II) were obtained by a rather complicated procedure based on the assumption that at ~700 K the contribution of exchange reaction (II) is negligible. The temperature dependences of the rate constants of reaction (I) were calculated for this temperature using the fall-off curves in the limit of low and high pressures and extrapolated to the temperature range of up to 4000 K using the Rice–Ramsperger–Kassel–Marcus (RRKM) theory. Then the temperature dependence of the rate constant of reaction (I) at a total density of 2.8 × 10–6 mol/cm3 was calculated from these temperature dependences in the range 720–4000 K using the fall-off curves method and subtracted from the temperature dependence of the observed (total) rate constant of the interaction of Cr with O2 to obtain the rate constant of exchange reaction (II).

An unexpected result of the above-mentioned studies was that the rate constant of the recombination of Cr atoms with O2 in the low-pressure limit proved significantly higher than for similar reactions of other atoms, including metal atoms (Fig. 3 in [[12]]). It was assumed [[12]] that this is due to a large number of low-lying electronic terms, which, when added together, create high density of vibrational levels near the dissociation barrier of the CrO2 molecule, the value of which, according to the RRKM theory, is proportional to the rate constant of dissociation (recombination) in the low-pressure limit [[15]]. However, there was no information about these electronic terms at that time, nor a suitable model for calculating their contribution. These data appeared afterwards [[16]], and a model was developed in [[9]] for calculating the rate constant in the low-pressure limit, which included the contribution of electronic terms.

Here we calculated the rate constant of recombination (I) in the low-pressure limit within the framework of the additive model proposed in [[9]] and compared the results with the available literature data. In addition, processing of the experimental data from [[12]] by the new method gave refined rate constants of reactions (I) and (II).

The rate constant of recombination (I) was determined by the equation

1

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where k–1,0(T) is the rate constant of the reverse reaction, and K1,eq(T) is the equilibrium constant. The expression for K1,eq(T) is of the form [[13]]

2

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where ΔrФ(T) and ΔrH(0) are the change in the reduced Gibbs energy at a temperature T and the change in enthalpy at 0 K for reaction (I), respectively; Rp is the gas constant (Rp = 82.057 cm3 atm mol–1 K–1); Δn is the change in the number of moles (Δn = –1 for (I)). ΔrФ○(T) and ΔrH○(0) for reaction (I) were calculated by the equations

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The values of Φ°(Cr, T) and Φ°(O2, T) were taken from [[13]].

The rate constant of dissociation of CrO2 in the low-pressure limit was determined by the equation [[15]]

3

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where βс is the collision efficiency during collisions, and k–1,sc is the rate constant of dissociation in the limit of strong collisions.

Calculation of the rate constant of dissociation of the CrO2 molecule in the limit of strong collisions is a rather difficult problem because it is necessary to take into account a large number of low-lying excited electronic levels. The experimental information on the molecular parameters of CrO2 is very limited: only some vibrational frequencies of OCrO and CrOO molecules in cryogenic matrices have been measured [[16]]. On the other hand, quantum-chemical calculations show that the CrO2 molecule exists in the form of three isomers: oxo (OCrO), peroxo (Cr(O2), cyclic structure), and superperoxo (CrOO) forms, each having a complex system of embedded electronic terms [[16]].

According to [[9]], k–1,sc can be represented as the sum of contributions from individual electronic states:

4

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5

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where i is the index of the electronic state, l is the number of included electronic states, R is the gas constant, T is the temperature, ρv,i (E0 – Ei) is the density of vibrational levels near the dissociation barrier, Fr,i(T) is the rotation factor, FE,i is the energy dependence factor, Fanh is the anharmonicity correction, E0 is the dissociation barrier, Ei is the energy, Qv,i(T) is the vibrational partition function, Mi is the multiplicity, and Qr,i(T) is the rotational partition function of the ith electronic state. The Qv,i(T) and Qr,i(T) values were determined by the standard technique [13]. The collision frequency factor of gas-kinetic collisions ZLJ was calculated by the equation [[15]]

6

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where NA is the Avogadro number; k is the Boltzmann constant; and μ is the reduced molecular mass of counterparts in collision, CrO2–Ar. The values of the parameters of the Lennard-Jones potential σ and ε were taken from [23]: σ(CrO2–Ar) = 5.0 Å, ε(CrO2–Ar)/k = 400 K.

The collision efficiency was calculated by the equation obtained from the expression presented in [[15]]:

7

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where 〈ΔE〉 is the average portion of energy transferred in all collision transitions (upward and downward); for the other symbols, see above. The energy dependence factor FE in (7) was represented as the sum of FE,i values weighted by the specific contributions of electronic states to the rate constant of dissociation in the limit of strong collisions:

8

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The correction for anharmonicity in (5) was calculated by the equation [[15]]

9

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where s is the total number of oscillators, and m is the number of Morse oscillators. As applied to the system under study, m = 3 for OCrO, and m = 2 for Cr(O2) and CrOO.

Note that this model should be regarded as a first approximation as it remains unclear how various isomers of CrO2 and their electronic states interact with each other at high excitation energies, in particular, near the dissociation barrier and what the resulting structure of energy levels is in this region. In addition, the majority of molecular parameters, namely, all parameters of electronically excited states, are determined by calculations, the result depending significantly both on the quantum-chemical method and on the basis set.

As noted in the introduction, the temperature dependence k1,0(T) presented in [[12]] was obtained in a rather complicated way. In addition, the data given in [[12]] for high pressures show a significant scatter. Therefore, we tried to find an alternative, simpler approach to determination of the rate constants of reactions (I) and (II). For this purpose, as in [[11]], the total (observed) rate constant for the consumption of chromium atoms in the reaction with molecular oxygen measured in [[12]] at low pressures ([M] = 2.8 × 10‒6 mol/cm3) was represented by the sum

10

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where k1,0(T) and k2(T) have the form

11

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12

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where A1, A2, n1, and n2 are the constants; and Δr2H°(0 K) is the change in enthalpy in reaction (II) at 0 K (37.385 kJ/mol [[13]]). Approximation of the experimental temperature dependence kt(T) shown in Fig. 1 by Eq. (10) gave the following values of parameters for the range 700‒4000 K: A1 = 4.3 × 1018 cm6 mol‒2 s‒1, n1 = ‒1.17; A2 = 3.9 × 1014 cm3 mol‒1 s‒1, n2 = ‒0.22.

Graph: Fig. 1. Temperature dependence of the total rate constant of consumption of Cr atoms in reactions (I) and (II), kt(T), constructed based on the data of [[12]]: (1) experimental points and (2) approximation by functional dependence (10).

Figure 2 shows the temperature dependence of the rate constant of recombination (I) in the low-pressure limit obtained in this work from the experimental data of [[12]] using the above-described method and the corresponding literature data. Also shown are the theoretical temperature dependences of this rate constant, calculated for different values ​​of the average portion of energy transferred in all collisional transitions. According to Fig. 2, the experimental temperature dependence k1,0(T) (curve 4) found in this work is in good agreement with the results of measurements from [[10]] and [[11]] (curves 1 and 2). At the same time, the temperature dependence k1,0(T) determined in [[12]] using a complex procedure (curve 3) differs significantly from other dependences, both experimental and theoretical.

Graph: Fig. 2. Temperature dependences of the rate constant of recombination of Cr atoms with O2 (reaction (I)) in the low-pressure limit. Experiment: (1), (2), (3) data from [[10]], [[11]], and [[12]], respectively; (4) dependence obtained in the present study from the experimental data of [[12]] (Eq. (11)); (5) general dependence for the results of this study and for data from [[10]] and [[11]]. Theory: (6), (7) calculation in the limit of strong collisions ((∆E → ∞, respectively; βс = 1) including only the (6) ground and (7) all electronic states of the CrO2 molecule; (8‒10) calculation including all electronic states at different values ​​of the average portion of transferred energy ∆E in the equation for βс (Eq. (7)): ∆E = (8) 1.0, (9) 3.0, and (10) 5.0 kJ/mol.

The general dependence for the results of this work and data from [[10]] and [[11]] (curve 5) is described by Eq. (11) with A1 = 3.7 × 1018 cm6 mol‒2 s‒1, n1 = ‒1.49.

According to Fig. 2, even in the limit of strong collisions, the values of k1,0(T) calculated neglecting the excited low-lying electronic states lie below the experimental values. For calculations including all the excited low-lying electronic states, the best agreement with experiment is achieved at an average portion of transferred energy ΔE = 2.8 kJ mol‒1. This value is close to the value obtained for recombination of iron atoms with molecular oxygen: ΔE = 3.3 kJ mol‒1 [[9]].

Figure 3 shows the temperature dependences of the rate constant of exchange reaction (II) found in this work (curve 3) and in [[11]] (curve 1) and [[12]] (curve 2). Unlike k1,0(T), they are all in good agreement with one another. In the temperature range where these dependences overlap, agreement is within 25%. In addition, the activation energy obtained in the present work is slightly higher than that measured in [[11]]. Approximation of the experimental data presented in Fig. 3 using Eq. (12) yields

13

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Graph: Fig. 3. Temperature dependence of the rate constant of exchange reaction (II): (1) and (2) data from [[11]] and [[12]], respectively; (3) data of this study; and (4) general approximation by Eq. (13).

In Fig. 3, this dependence is shown by the dotted line.

The rate constant of the reverse reaction calculated using the equilibrium constant has the form

14

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This reaction plays an important role in the mechanism of inhibition of hydrogen ignition by chromium atoms [[18]].

A new method has been proposed for determining the rate constants of the interaction of metal atoms with molecular oxygen occurring by recombination and exchange. Its applicability was demonstrated using the Cr + O2 reaction as an example. The rate constants of the reactions Cr + O2 + M = CrO2 + M (I) in the low-pressure limit and Cr + O2 = CrO + O (II) in the direct and reverse directions were obtained. The additive model developed in [[9]] was used to calculate the rate constant of reaction (I). It was shown that it is impossible to satisfactorily describe the kinetic parameters of reaction (I) without taking into account the low-lying excited electronic states. Calculations within the framework of the proposed model including all electronic states made it possible to evaluate the average portion of energy transferred by an excited CrO2 molecule in collisions with diluent gas molecules. It should be stated, however, that the obtained kinetic parameters need to be refined based on the new calculated and experimental data. In particular, it would be highly desirable to perform more detailed quantum-chemical calculations of the molecular parameters of all CrO2 isomers.

The authors declare that they have no conflicts of interest.

Translated by L. Smolina