in the Conversion of NO

ARTICLE pubs.acs.org/JPCA

The Importance of NOþ(H2O)4 in the Conversion of NOþ(H2O)n to H3Oþ(H2O)n: I. Kinetics Measurements and Statistical Rate Modeling Nicole Eyet,†,‡ Nicholas S. Shuman,‡ Albert A. Viggiano,*,‡ J€urgen Troe,§ Rachael A. Relph,^ Ryan P. Steele,^ and Mark A. Johnson^ †

Department of Chemistry, St. Anselm College, 100 Saint Anselm Drive, Manchester, New Hampshire 03102, United States Space Vehicles Directorate, Air Force Research Laboratory, 29 Randolph Road, Hanscom Air Force Base, Massachusetts 01731-3010, United States § Institut f€ur Physikalische Chemie, Universit€at G€ottingen, Tammannstrasse 6, and Max-Planck-Institut f€ur Biophysikalische Chemie, D-37077 G€ottingen, Germany ^ Department of Chemistry, Yale University, P.O. Box 208107, New Haven, Connecticut 06520, United States ‡

ABSTRACT: The kinetics for conversion of NOþ(H2O)n to H3Oþ(H2O)n has been investigated as a function of temperature from 150 to 400 K. In contrast to previous studies, which show that the conversion goes completely through a reaction of NOþ(H2O)3, the present results show that NOþ(H2O)4 plays an increasing role in the conversion as the temperature is lowered. Rate constants are derived for the clustering of H2O to NOþ(H2O)13 and the reactions of NOþ(H2O)3,4 with H2O to form H3Oþ(H2O)2,3, respectively. In addition, thermal dissociation of NOþ(H2O)4 to lose HNO2 was also found to be important. The rate constants for the clustering increase substantially with the lowering of the temperature. Flux calculations show that NOþ(H2O)4 accounts for over 99% of the conversion at 150 K and even 20% at 300 K, although it is too small to be detectable. The experimental data are complimented by modeling of the falloff curves for the clustering reactions. The modeling shows that, for many of the conditions, the data correspond to the falloff regime of third body association.

’ INTRODUCTION One of the great mysteries of lower atmospheric ion chemistry was discovered on the first successful flight of a mass spectrometer into the lower ionosphere on October 31, 1963.1 Positive ion profiles were measured from 64 to 112 km. At the highest altitudes, NOþ and O2þ were found to be the primary ions, as expected. However, below about 85 km, proton hydrates (H3Oþ(H2O)n) became important and then dominated the spectra below ∼80 km. Both O2þ and NOþ react with H2O only by clustering. Therefore, much effort over the next several years was expended to explain the presence of the proton hydrates. Numerous additional rocket flights proved that the observation of proton hydrates was not due to water vapor outgassing from the rocket, a common interpretation. The observations were only explained 67 years later when cluster-assisted routes to the proton hydrates were discovered by Ferguson and Fehsenfeld.2 For the O2þ system, they found that O2þ(H2O) reacted with H2O to form both H3Oþ and H3OþOH. Both product ions react with H2O to form H3Oþ(H2O) and then higher order proton hydrates. This was confirmed by the Kebarle group.3 The situation for NOþ is more complex and is described by reactions 14. NOþ þ H2 O þ M T NOþ ðH2 OÞ þ M

ð1Þ

NOþ ðH2 OÞ þ H2 O þ M T NOþ ðH2 OÞ2 þ M

ð2Þ

NOþ ðH2 OÞ2 þ H2 O þ M T NOþ ðH2 OÞ3 þ M

ð3Þ

NOþ ðH2 OÞ3 þ H2 O f H3 Oþ ðH2 OÞ2 þ HNO2

ð4Þ

r 2011 American Chemical Society

The rate constant for reaction 4 is small, prompting speculation that only one of several isomers of NOþ(H2O)3 reacted. These mechanisms were confirmed and details added by a number of further studies.47 Recently, there have been several studies of the infrared spectroscopy,8,9 metastable ion dissociation, and structure and energetics of the species involved. The theoretical work by Asada et al.10 shows that NOþ(H2O)4,5 each have 15 isomers. Previous theoretical work is summarized in that paper. The presence of isomers was also confirmed in the work from the Johnson and Okumura groups9,11 which showed that infrared dissociation of NOþ(H2O)4 produced NOþ(H2O)3, NOþ(H2O)2, and H3Oþ(H2O)2, with the first ion being the major species. For NOþ(H2O)5, infrared dissociation produced mainly H3Oþ(H2O)3 with lesser amounts of NOþ(H2O)4. Angel and Stace12 performed studies on both unimolecular dissociation of metastable cations and collision-induced dissociation which showed more equal amounts of products for NOþ(H2O)4 and a dominance of HNO2 loss for NOþ(H2O)5. A common thread in these studies is the observation that NOþ(H2O)5 may play a role in the kinetics—at least at low temperatures, since the ions are formed in supersonic beams. The theoretical and dynamical work cited above shows a more complex picture than suggested by past kinetics studies. In part that stems from the fact that only little work on the kinetics has Received: April 8, 2011 Revised: May 12, 2011 Published: May 19, 2011 7582

dx.doi.org/10.1021/jp2032803 | J. Phys. Chem. A 2011, 115, 7582–7590

The Journal of Physical Chemistry A been done since the early days of its discovery and none of the studies were performed as a function of temperature. Considering that the region of the atmosphere where these reactions are the most important is cold (100300 K),13 it is worthwhile to study the kinetics over a wide range of temperatures. In the present work the rate constants for H2O clustering to NOþ(H2O)13 and the conversion of NOþ(H2O)34 to proton hydrates from 150 to 400 K have been measured. Indeed the mechanism is more complex than previously thought; NOþ(H2O)4 plays a prominent role and the temperature dependences are steep. Here kinetics studies for the pertinent reactions in a helium buffer and complementary statistical rate modeling of the rate constants are presented so that the results can be transferred to a nitrogen buffer, as well as to broader ranges of temperature and pressure than assessed in laboratory work. Part II of this series will address the infrared spectroscopy of NOþ(H2O)4 and thermodynamics calculations of the various species involved.

’ EXPERIMENTAL SECTION The measurements were performed in the Air Force Research Laboratory's selected ion flow tube (SIFT). The instrument has been described in detail previously,14 and only a brief discussion is given here. Ions are created in a moderate pressure ion source, mass selected, and injected into a reaction flow tube through a Venturi injector. The ions are thermalized by a helium bath gas. Water vapor is added downstream, and a small amount of the gas is sampled into a quadrupole mass spectrometer through a pinhole in a nose cone. The ions are then mass analyzed by a second quadrupole mass filter and detected by a discrete dynode multiplier. The flow tube is heated or cooled in four zones by resistance heaters or pulsed liquid nitrogen, respectively. NOþ(H2O)n ions are created in the source from a mixture of NO and H2O. Our goal was to make and inject individual clusters into the flow tube so that individual reactions could be studied in isolation, particularly reaction 4. However, due to the fragile nature of the reactants, H2O ligands are lost during the injection process. Therefore, the kinetics had to be analyzed as a complete set. Details are described later. A heated neutral injector is used to introduce water vapor at low temperatures. A thin (1/8 in.) stainless steel tube is silver soldered to a thick 1/4 in. tube, both coaxial tubes otherwise being electrically insulated from each other and the flow tube. Passing a current through the inlet preferentially heats the thin inner tube. A thermocouple is placed near the tip to monitor the temperature. In this manner, H2O is prevented from freezing in the inlet while minimizing the amount of heat transferred to the buffer gas. Nevertheless, the actual temperature of the reaction zone is likely to be higher than the reported temperatures (which are measured at the wall) by an unknown but small (∼10 K) amount. This is taken into account in our error analysis, such as described in the analysis section. Water vapor is introduced by two methods. At 250 K and above, where the reaction is relatively slow, helium is bubbled through H2O to create large concentrations in the flow tube (∼1013 molecules cm3). The pressure over the reservoir is monitored, and the flow rate is derived by assuming that the helium becomes saturated with water vapor. At low temperatures, a different method is used since less H2O is needed as the reaction becomes faster. Mixtures of 510% H2O in He are made, and the mixture flows directly into the flow tube through an MKS flow controller. The latter method is simpler but does

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not deliver the larger concentrations of H2O necessary at higher temperatures. H2O was observed to be condensing on the flow tube walls at a low temperature. Normally, when kinetics are studied in our laboratory, flows are increased in equal steps and then decreased. This handles potential signal drift exceptionally well. However, we found that at low temperatures the time for signals to become stable after a H2O concentration decrease was many minutes, indicating that the H2O which had been sticking to the cold walls was outgassing. Therefore, the low temperature data presented were taken only by increasing H2O concentrations and allowing for several minute wait time between each data point. In addition, we found that at some point enough water adhered to the flow tube walls so that all ion signals decayed dramatically. Then the instrument had to be shut down and warmed up before data could again be taken. The time before shutdown could be as short as 2 h. To derive rate constants, an Eulerian equation solver was used to fit the data. Time steps were on the order of 110 μs (total reaction time ∼ ms). Rate constants were derived from a Monte Carlo procedure described in depth previously.15 For each reaction, rate constants were assumed for the forward rate constants. Physical constraints on the trial values were imposed; for example, no rate could occur above the collisional value or below the detection limit. Equilibrium constants were fixed using experimentally or theoretically determined thermodynamics. The ion concentrations were then predicted as a function of the H2O concentration; a weighted least-squares parameter, which we refer to as the goodness-of-fit, was calculated by comparing the predicted ion concentrations to the experimental data. Approximately 100 000 trial values were done for each data set. Plots of the individual rate constants versus the goodness-of-fit parameter were then used to derive the forward rates. The rate constants were fixed at the minima of such plots, when they existed. By eye, it was determined when the goodness-offit parameter was large enough (GOF(max)) to clearly lead to bad fits. The width of the rate constant versus goodness-of-fit curves at GOF(max) then determined the error limits. The system is complex enough that for many conditions only a few rate constants could be derived with certainty, with others having poorly defined minima. Varying the equilibrium constants (by varying the thermochemistry) of the reactions, while also varying the rate constants, did not provide better defined rate constants and/or little thermochemical information. Normally, we report errors as 25% for absolute rate constants and 15% for differences from one temperature to another. However, several conditions discussed above indicate that our errors here are probably much larger, even when the rate constants are well-defined by the goodness-of-fit plots: (1) the temperature is subject to error due to the heated inlet; (2) the results are very sensitive to the back reactions, which are in turn very sensitive to the thermochemistry and the uncertain temperature; (3) the freezing water vapor may have altered the sampling efficiency—certainly at 150 K after tens of minutes of H2O exposure. Estimating the effects of each potential problem is difficult. Individual errors are discussed in detail as all of the data are reported. The statistical modeling of the rate constants given below also suggests that individual errors must have been considerably larger than 25%.

’ RESULTS Determination of Rate Constants. Reactions 14 were previously considered as being the complete set important in the atmospheric conversion of NOþ(H2O)n to H3Oþ(H2O)n. In the 7583

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The Journal of Physical Chemistry A

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Figure 1. Modeled (lines) and measured (points) ion concentrations at 150 K (assuming k4 = 1.0 1010 cm3 molecule1 s1).

present study, NOþ(H2O)4 is observed but not predicted by the mechanism given above. Therefore, the additional clustering reaction NOþ ðH2 OÞ3 þ H2 O þ M T NOþ ðH2 OÞ4 þ M

ð5Þ

is assumed to compete with reaction 4 at low temperature. Once formed, NOþ(H2O)4 may undergo three competing reactions. It could collisionally dissociate, that is, undergo the reverse of reaction 5; it would form proton hydrates in a reaction similar to reaction 4, NOþ ðH2 OÞ4 þ H2 O f H3 Oþ ðH2 OÞ3 þ HNO2

ð6Þ

or release nitrous acid via thermal dissociation. NOþ ðH2 OÞ4 þ M f H3 Oþ ðH2 OÞ2 þ HNO2 þ M

ð7Þ

The complete mechanism only became visible when an extended temperature range was addressed. Normally in a SIFT experiment, one reaction is studied at a time. The complexity of the chemistry under study here made that impossible. In addition to reactions 17, clustering to form higher-order proton hydrates also occurs, H3 Oþ ðH2 OÞn þ H2 O þ He T H3 Oþ ðH2 OÞnþ1 þ He ð8Þ While NOþ(H2O)4 was observed at 150, 200, and 250 K, its importance being most clearly observed at 150 K, it probably still plays a role at 300 K although its concentration was below our detection limit. Data analysis at 150 K is simplified because thermal dissociation reactions (reactions 7 and the reverse of reactions 2, 3, 5, and 8) do not occur on the time scale of our experiment. In turn this eliminates uncertainties in the thermochemistry from impacting the modeling and also leads to a reduced set of reactions. The points in Figure 1 show ion concentrations as a function of H2O concentration at 150 K. At low [H2O], mainly NOþ and NOþ(H2O) are observed. NOþ(H2O)4 was the largest NOþ hydrate observed. The smallest H3Oþ-containing hydrate observed was H3Oþ(H2O)3. If reaction 4 was occurring, the observation of H3Oþ(H2O)2 would be expected, unless reaction 8 for n = 2 was much more rapid than reaction 7. However, the modeling shows that at 150 K, reaction 4 is significantly slower than reaction 6.

Figure 2. Goodness-of-fit plots at 150 K. (a) Variation of k4, leading to a poorly defined value; (b) variation of k6, leading to a well-defined optimum value (see text).

The solid lines in Figure 1 represent the modeling results described below. The modeling includes reaction 4 to produce H3Oþ(H2O)2. The orange line on the plot shows that this ion should be produced in quantities below our detection limit, consistent with our observations. Several facts lead to a switchover from reaction 4 to reaction 6 as the dominant conversion reaction. The clustering rates at 150 K are all fast, approaching gas kinetic limits, and leading to a fast chain of reactions required to produce NOþ(H2O)4.. Reaction 4 is endothermic by ∼6 kcal mol1; therefore it would be expected to become slower with decreasing temperature (as presented later, it is difficult to derive a true temperature dependence) and at 150 K should be unable to compete with reaction 5. Finally, reaction 6 is slightly exothermic and fast, providing a facile route to proton hydrates. The result is that, for NOþ(H2O)3, the clustering reaction 5 to produce NOþ(H2O)4 is significantly faster than conversion to proton hydrates (reaction 4). NOþ(H2O)4 then almost exclusively reacts by reaction 6 to produce H3Oþ(H2O)3. Flux calculations for the various reactions involving NOþ(H2O)3,4 are presented below. Figure 2 shows the goodness-of-fit parameter versus rate constants for the two reactions with H2O that lead to proton hydrates, reactions 4 and 6. Reaction 4 is poorly defined by this 7584



Figure 3. As for Figure 1 but for T = 200 K. The solid lines represent the model with the lowest goodness-of-fit parameters, see text. Removing the thermal decomposition of NOþ(H2O)4 (reaction 7) and redetermining the rate constants provides the fits given by the dashed lines.

modeling since so little H3Oþ(H2O)2 is observed. The rate constant was allowed to vary from 1 109 (approximately the collisional rate constant) to 1 1015 cm3 molecule1 s1 (the detection limit of the experiment). Reaction 6 is well-defined by the modeling, and the rate constant must be close to that of the collisional rate constant. The difference between the very shallow minimum in 2a and the more pronounced one in 2b is significant. Similar plots were used to derive all rate constants and their error limits. The most important clue for the importance of reaction 7, the thermal decomposition of NOþ(H2O)4 to lose HNO2, was observed at 200 K. The raw data are shown in Figure 3 along with model fits. Both H3Oþ(H2O)2 and H3Oþ(H2O)3 were observed as primary proton hydrates. Successive proton hydrates up to and including n = 7 (not shown for clarity) are formed from subsequent clustering at all low temperatures. These data were modeled with the method described above. The result of modeling reactions 18 is illustrated by the solid lines in Figure 3. Modeling of reactions 16 and 8 provides the fits shown by the dashed lines. It was not possible to obtain a good fit without reaction 7 unless reaction 4 occurred at nearly a collisional rate. The latter is not consistent with the temperature dependence of this rate constant predicted by the measured higher temperature rate constants for reaction 4 (discussed shortly.) Therefore, reaction 7 must be included in the reaction model, even if the rate constants determined become less defined. This reaction also has an effect on higher temperature data, even at 300 K where NOþ(H2O)4 is not observed. To derive rate data at temperatures above 150 K, the following simplifying assumptions were made in the data analysis. The 150 K data show that reaction 6 is fast. Exothermic reactions of this type usually remain rapid at all temperatures. When modeling this rate constant, it was constrained to be in the range 1.5 3 109 cm3 molecule1 s1, that is, near the collisional value. In addition, the reverse of reaction 5 should be faster than reaction 7, and that constraint was added in the Monte Carlo

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procedure. This is confirmed by photodissociation branching data presented in part II and by Okumura et al.16 The data were also modeled without these constraints; the resulting rate constants were less defined, and temperature trends were erratic, that is, they did not follow simple monotonic trends. The derived rate constants are listed in Table 1 for 150, 200, 250, and 300 K data; they are either represented as second-order or as third-order rate constants. For falloff effects see the statistical rate modeling given below. For each reaction, a best fit value is listed in bold, and upper and lower limits are listed in standard text, if they could be determined from the data. For equilibrium reactions, the thermodynamic values used to fix the equilibrium constant, and hence the back reactions are listed. Previous data taken at 300 K data are listed in the last column. No rate data are shown for reaction 2, the initial H2O clustering to NOþ, since the rate is slow and was only poorly measured for the amount of H2O added in the present experiments. Data were also taken at 400 K, but minimal clustering and no proton hydrates were observed. These data were not useful except to emphasize the steep temperature dependences of the clustering rates. Presumably at higher H2O concentrations, proton hydrates would be observed. The only rate constants that are well-defined at all temperatures are those for addition of a subsequent water to NOþ(H2O)1,2. The clustering reactions get much faster with decreasing temperature and roughly increase with n. There is good agreement between our value for NOþ(H2O) clustering and previous data at 300 K but poor agreement for n = 2 clustering. However, that disagreement is the result of the added complexity of the current model. If all rate constants are allowed to be variable over larger ranges, the value for that third-order rate constant increases to 2.9 1028 cm6 molecule2 s1, in good agreement with the published values, but with other rate constants making little physical sense (odd temperature dependences or unphysical rates). The derived values of the secondorder rate constants for reaction 4 vary from 0.83.6 1010 cm3 molecule1 s1. It was not possible to derive a value at 150 K. The derived values are in agreement with the 300 K values previously published within error.5,17 The rate constant for conversion of NOþ(H2O)4 to H3Oþ(H2O)3 could only be observed at 150 K, see Figure 2b, and as stated previously was assumed to be similar at all temperatures. The rate constant for thermal dissociation of NOþ(H2O)4 into H3Oþ(H2O)2 was not well determined, but it clearly increases with temperature as expected. The rate constant for production of NOþ(H2O)4 (reaction 5) could only be derived at the two lowest temperatures. The considerable uncertainty in many of the rate constants stems from potential trade-offs between the various channels. To determine the relative importance of NOþ(H2O)3 and NOþ(H2O)4 in the conversion to proton hydrates, flux calculations were performed. Several species in the reaction system are either produced or destroyed by two or more competing reactions. The relative contributions of each reaction are not necessarily immediately obvious and were deconvoluted from one another by calculating the total concentrations of each reactant consumed and each product yielded by each individual forward and reverse reaction throughout the full reaction time. In effect this calculation is simply tracking the progress of each reaction while iteratively solving the coupled differential equations as described above. The calculations are subject to uncertainty due to the large variations in rate constants possible, especially at 250 and 300 K. However, trends are reliable. The percentage of proton hydrates that are 7585



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Table 1. Rate Constants for Various Reactions of NOþ(H2O)n with H2O (Second-Order Rate Constants** in cm3 molecule1 s1; Third-Order Rate Constants* in cm6 molecule2 s1)a k at 150 K, 0.27 Torr

k at 200 K, 0.27 Torr

b

k at 300 K, 0.37 Torr

previous value (300 K)d

upper limit (top), best value (middle), lower limit (bottom)

reaction

a

k at 250 K, 0.34 Torr

(2)* NOþ(H2O) þ H2O þ He f NOþ(H2O)2 þ He ΔH = 65 kJ mol1 ΔS = 85 J mol1 K1

4.8 1026 4.1 1026 2.5 1026

1.2 1026 1.8 1026 2.4 1026

9 1028 7 1028 5 1028

4.8 1028 3.3 1028 2 1028

3.0 1028

(3)* NOþ(H2O)2 þ H2O þ He f NOþ(H2O)3 þ He ΔH = 64 kJ mol1 ΔS = 120 J mol1 K1

2.0 1025 (kcoll) 1.1 1025 7 1026

7.5 1026 4.3 1026 2.8 1026

1.8 1027 1.2 1027 8 1028

2 1028 9 1029 4 1029

3.7 1028

(5)* NOþ(H2O)3 þ H2O þ He f NOþ(H2O)4 þ He ΔH = 45 kJ mol1 ΔS = 80 J mol1 K1

2.0 1025 (kcoll) 1.3 1025 5 1026

2.3 1025 (kcoll) 1.1 1025 5 1026

2 1026 none none

2.5 1025 (kcoll) none none

not observed

(4)** NOþ(H2O)3 þ H2O f H3Oþ(H2O)2 þ HNO2

3.4 109 (kcoll) none none

3.0 109 (kcoll) 3.6 1010 2 1011

2.7 109 (kcoll) 8 1011c none

1.6 1010 1.1 1010 1.3 1012

8.5 1011

(6)** NOþ(H2O)4 þ H2O f H3Oþ(H2O)3 þ HNO2

3.4 109 (kcoll) 2.8 109 9 1010

fixed to be 1.53 109b



not observed

(7)** NOþ(H2O)4 þ He f H3Oþ(H2O)2 þ HNO2

none none none

1 1013 7.6 1014 none

6 1012 none none

1.2 1010 1.5 1011 none

not observed

Thermochemistry is at 298 K. kcoll is the collisional value. For termolecular kcoll, bimolecular collision rate constants are divided by the number density. Set to be small range near 150 K rate, which is relatively well-known. c Shallow minimum. d Average of refs 5 and 17.

Table 2. Modeled Yield of Proton Hydrates Formed from NOþ(H2O)3a H3Oþ(H2O)n yield T/K

[H2O]/molecules cm

3

þ

(in %) from NO (H2O)3

150

3 1011

200

6 1011

4

250 300

1 1013 3 1013

60 80

0.3

a

Errors are difficult to estimate given the uncertainties in the derived rate constants, particularly at 250 and 300 K.

produced by conversion of NOþ(H2O)3 (reaction 4) rather than NOþ(H2O)4 increases from essentially zero (0.3%) at 150 K to 80% at 300 K; see Table 2. The increase follows roughly an exponential dependence on 1/T. The 150 K flux is the most accurate since it is based on the fact that no H3Oþ(H2O)2 is observed. At 200 K, conversion of NOþ(H2O)4 (reaction 6) dominates proton hydrate production; at 250 K, NOþ(H2O)3 and NOþ(H2O)4 are equally important in producing proton hydrates within uncertainty (tens of percent), and at 300 K, the conversion to proton hydrates is dominated (80%) by conversion of NOþ(H2O)3 (reaction 4). This shows that the old mechanism is probably not exclusively responsible for proton hydrate production (due to the large errors in this calculation, a value nearer 100% is possible but less likely) but that roughly 20% of the conversion to proton hydrates occurs through NOþ(H2O)4 even though it is not observed. The flux calculations also show that, at 150 K, NOþ(H2O)4 is formed by reaction 6 almost exclusively (99.9%). This pathway's branching fraction drops to 20% at 200 K and 1% at 300 K. At the higher temperatures, the branching fraction for the reverse of reaction 5 increases.

Statistical Modeling of Rate Constants. The 300 K measurements of the rate constants for the formation of the hydrates NOþ(H2O)n with n = 135,6,17 have been extended toward lower temperatures for n = 2 and 3 and rate constants for n = 4 were added in the present experiments. It was assumed previously5,6,17 that at pressures near 1 Torr the reactions were at their limiting low-pressure third-order limit. By analogy, the rate data in Table 1 were represented as third-order rate constants. However, one cannot expect that third-order behavior prevails for all n at all temperatures studied and all pressures of interest for ionospheric applications. For this reason, modeling of the rate constants in terms of statistical unimolecular rate theory appears obligatory (see, e.g., the previously presented modeling of the hydration reaction of hydronium18 and ammonium cations19 as well as more recent work on hydration reactions in general).20 We consider a limiting low pressure range where the second-order rate constant k, here denoted by k0, is proportional to the bath gas concentration [M], that is, where the reaction is third order. At high pressure, there is a limiting range where the second-order rate constant k is independent of the bath gas concentration and reaches a value k¥ which possibly is given by ionmolecule capture theory.2023 Earlier in the paper this was termed “collisional” or “gas kinetic”. In the falloff range in between, the rate constant k gradually changes from k0 to k¥. k0 and k¥ generally have quite different temperature dependences such that the bath gas concentration at which the center of the falloff range is located moves with temperature. The limitations of the described experiments did not allow us to study details of the pressure dependences of the rate constants and to locate them along the respective falloff curves. In addition, the temperature dependences of k could only be specified to a limited 7586



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Table 3. Modeled Low Pressure Strong Collision Rate Coefficients k0sc from eq 9 for NOþ(H2O)n1 þ H2O þ He f NOþ(H2O)n þ He (see text) T/K n

150

1

1.5 1028

6.6 1029

3.5 1029

2.1 1029

1.3 1029

9.2 1028

26

26

27

27

28

2.8 1028

a

2

200

5.0 10

250

1.2 10

3.9 10

300

350

1.5 10

400

6.1 10

b

2 2c

27

1.3 10 6.5 1027

28

1.4 10 6.6 1028

29

2.0 10 9.5 1029

30

3.6 10 1.8 1029

31

8.2 10 4.1 1030

2.2 1031 1.1 1030

2d

1.5 1024

1.5 1025

2.3 1026

4.3 1027

1.0 1027

2.3 1028

e

27

28

23

30

31

1.0 1031

3

f

3

3.6 10

2.1 10

26

3.1 10

a

27

þ

3.3 10

2.1 10

5.4 10

2.8 10

28

4.7 10

28

29

1.1 10

2.7 10

þ

b

7.7 1030

þ

c

Low frequency oscillator model of NO (H2O)2. Low frequency oscillator model of NO (H2O)2 and ring structure of NO (H2O)3. Low frequency oscillator model of NOþ(H2O)2 and trigonal structure of NOþ(H2O)3; d Low frequency oscillator model of NOþ(H2O)2 and Cs structure of NOþ(H2O)3. e Cs structure of NOþ(H2O)3 and ring þ 1 front down structure of NOþ(H2O)4. f Ring structure of NOþ(H2O)3 and ring þ 1 front down structure of NOþ(H2O)4.

extent. In this situation, theoretical modeling of k(T,[M]) is helpful and extends the representation of the rate constants. In view of the uncertainty of the rate data we simplify the calculations to some extent by focusing attention on the limiting low and high pressure rate constants k0 and k¥. We then represent the falloff transition between the limiting ranges in analogy to the results for the H3Oþ(H2O) and NH4þ(NH3) systems.18,19 We express k0 in the form24 Fvib, h ðE0 ÞFE kTFanh Frot Qvib ðH2 OÞQvib ðNOþ ðH2 OÞn Þ !3=2 Qel, rot ðNOþ ðH2 OÞnþ1 Þ h2 Qel, rot ðH2 OÞQel, rot ðNOþ ðH2 OÞn Þ 2πμkT

Table 4. Modeled High Pressure Rate Coefficients k¥/cm3 molecule1 s1 from eq 10 for NOþ(H2O)n1 þ H2O f NOþ(H2O)n (see text) T/K n 1 2 3 4

150

200 9

3.80 10 3.5 109 3.4 109 3.2 109

250 9

3.3 10 3.1 109 3.01 109 2.9 109

300 9

3.0 10 2.8 109 2.7 109 2.6 109

350 9

2.8 10 2.6 109 2.5 109 2.4 109

400 9

2.6 10 2.5 109 2.4 109 2.3 109

2.5 109 2.3 109 2.2 109 2.2 109

k0 ½Mβc Z

ð9Þ

where βc is a collision efficiency related to the average energy transferred per collision through βc/(1 βc1/2) ≈ / FEkT. Z is represented by the Langevin collision frequency Z = kL = 2πe(R/μ)1/2 with the polarizability R of M and the reduced mass μ of the collision pair M-NOþ(H2O)n; Qvib,rot,el are vibrational, rotational, and electronic partition functions, respectively; Fvib,h(E0) is the harmonic vibrational density of states of the adduct NOþ(H2O)n at its dissociation energy E0, FE accounts for the energy dependence of Fvib,h, Fanh accounts for anharmonicity,25,26 and Frot is a rotational factor (here represented by its maximum value Frot,max, i.e., the value determined in absence of centrifugal barriers). The molecular parameters required for the calculation of k0 are given in the Appendix. Large additional anharmonicity effects may put the frequencies of the low frequency modes in question. For this reason, we put the lowest frequencies (below 25 cm1) arbitrarily at 25 cm1. In addition, there is the problem of the partication of several isomers of the hydrates, that is, three stable isomers (trigonal, ring, and Cs) for NOþ(H2O)3 and 15 stable isomers10 for NOþ(H2O)4. At this stage it remains uncertain how to account for the isomer formation in a proper way. We nevertheless show in Table 3 a series of calculations of strong collision rate constants k0sc (i.e., values for k0 putting βc = 1 and for M = He) for the formation of specific isomers. One should also note that the isomers after their formation may interchange by thermal isomerization processes and establish isomer equilibria. We assume that k¥ is represented sufficiently well by iondipole capture theory in the form of the SuChesnavich equation,21 modi-

Table 5. Strong Collision Broadening Factors F(y) Used in This Work (y = k0/k¥ and Modeling Results for H3Oþ þ H2O þ He f H3Oþ(H2O) þ He (from Hamon et al.18) y F(y)

104 0.90

103 0.75

102 0.53

101 0.31

1 0.20

101 0.27

102 0.40

103 0.53

104 0.65

fied for the asymmetric top character of H2O, in the form k¥ =kL 0:4767x þ 0:6200 for x g 2 ðx þ 0:5090Þ2 =10:526 þ 0:9754 for x e 2

ð10Þ

with x = μ0,eff/(2RkT) . Here, R is the polarizability of H2O (with R = 1.45 1024 cm3) and μD,b = 1.07 μD,b is an effective dipole moment20 of water (with μD,b = 1.857 D). Table 4 summarizes values for k¥ calculated with eq 10. Because of the uncertainties in the modeling of the limiting rate constants and in the experimental data, we refrain from doing a complete rate calculation in the intermediate falloff range. Instead, we employ the same strong collision “broadening factors” F(y) in the falloff representation24 1/2

k=k¥ ¼ ½y=ð1 þ yÞFðyÞ

ð11Þ

(where y = k0/k¥) as for the formation of H2O5þ and N2H7þ treated previously,18,19 see Table 5. For example, one has F(y = 1) = Fcenter ≈ 0.2, F(y = 102) ≈ 0.6, and F(y = 102) ≈ 0.4 In the following we compare the measured rate constants with modeled falloff curves. This is problematic for reaction 1, which was experimentally not accessible in the present work but for which previous data5,17 are available near 300 K and 1 Torr; see Figure 4. One observes that reaction 1 near 300 K and 1 Torr 7587



Figure 4. Modeled strong collision falloff curves for (1) NOþ þ H2O þ He f NOþ(H2O) þ He at T = 150 and 300 K [experimental results for T = 300 K from refs 5 (3) and 16 (2)]; the straight lines indicate the low- and high-pressure limits; the curved lines are modeled falloff curves, see text).

indeed is close to the low-pressure limit, and it remains near to this limit when the temperature is lowered to 150 K. The modeling corresponds to a temperature coefficient in k0 Tm of the order of m ≈ 2.8. By increasing the pressure one observes gradual deviations from third-order behavior. However, in this system this effect is not very pronounced; for example, k/k0 only changes from 0.9 to 0.5 when the pressure increases from 1 to 100 Torr. The measured rate constants do not deviate far from the modeled limiting low pressure strong collision values before any fine-tuning of the modeling is performed. Considering next reaction 2 (for which optimum experimental results were obtained in the present work), falloff deviations from third-order behavior are more pronounced than for reaction 1. Figure 5 compares the experimental data from the present work and from refs 5 and 17 with modeled falloff curves. One realizes that pressures of 1 Torr at 150 K nearly correspond to the center of the falloff curve, whereas they are closer to the low-pressure limit at 300 K, k nevertheless being about a factor of 23 below the extrapolated low pressure value of k0. The presently measured values at 150 and 200 K are about a factor of 4 above, at 250 and 300 K a factor of 23 below the modeled values. In view of the above-described uncertainties (in particular the mode with the lowest frequency), the results should be considered as satisfactory. In any case, deviations from third-order behavior at pressures near 1 Torr are considerable, and Figure 5 should be used for extending the pressure and temperature ranges. One expects even larger deviations from third-order behavior for reactions 3 and 5. At this stage, the problem of the contribution of isomer formation to k0 needs to be addressed. As the formation of NOþ(H2O)3 in Cs-symmetry dominates because of the larger number of low frequency modes, only this pathway needs to be considered. Figure 6 shows the results. One concludes that the reaction is near to its high pressure limit at 1 Torr for temperatures below 200 K and falls below the high

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Figure 5. Modeled strong collision falloff curves for (2) NOþ(H2O) þ H2O þ He f NOþ(H2O)2 þ He at T = 150, 200, 250, and 300 K (from top to bottom; measurements from the present work at 150 (Y),200 (y), 250 (O), and 300 K (b); measurements at 300 K from ref 5 (3) and16 (2); for the meaning of the modeled curves, see Figure 4 and text).

Figure 6. As for Figure 5, for (3) NOþ(H2O)2 þ H2O þ He f NOþ(H2O)3 þ He.

pressure limit at higher temperatures. As we have arbitrarily put the lowest frequencies (calculated values below 25 cm1) to a value of 25 cm1, the discrepancy with the measurements at 250 and 300 K may be an artifact of the modeling. At this stage it appeared meaningless to model k0sc for all 15 individual isomers of NOþ(H2O)4 forming from the 3 isomers of NOþ(H2O)3 in reaction 3. Instead we only considered one representative example, the formation of a “ring þ 1 front down” structure for n = 4 (see Appendix for its properties) from the ring structure for n = 3. The result is included in Table 3. As the reaction can lead to 15 isomers of n = 4, the overall rate constant 7588



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comparison with the experimental results in Table 1 indicates that reaction 6 indeed is capture-controlled, as the measured and modeled rate constants agree. On the other hand, reaction 4 is about 1 order of magnitude slower than the capture-controlled process, which has to be attributed to its endothermicity of 6 kcal mol1. However, that value would suggest much lower values of k4 than fitted. Therefore, one cannot rule out that the observed concentrations of H3Oþ(H2O)2 could also stem from the thermal dissociation of H3Oþ(H2O)3 such as formed by the fast reaction 6.

Figure 7. As for Figure 5, for (5) NOþ(H2O)3 þ H2O þ He f NOþ(H2O)4 þ He.

is the sum of all 15 rate constants. Therefore, we tentatively multiplied the single value by 15 and then divided by 2 because it is probable that a medium value was selected. Figure 7 shows the corresponding falloff curves. As in Figure 6, the comparison with the measurements at 150 and 200 K suggests that these experiments again were done close to the high pressure limit. In principle, the present modeling allows one to estimate the change of the rate constants when the bath gas changes from He to the atmospherically pertinent N2. For that the collision efficiency βc and the Langevin collision frequency Z in the expression 9 for k0 are modified. As the related changes are smaller than the general uncertainties of the modeling, however, they are not specified here. Modeling of the thermal dissociation of NOþ(H2O)4 through reaction 7 was done by considering the reverse reaction first. As HNO2 has nearly the same dipole moment as H2O, this reaction behaves very similarly to reaction 8 with n = 0, and its modeling follows that elaborated previously.18 It turns out that reaction 7 and its reverse under the present conditions are close to the high pressure limit where eq 10 applies. Conversion of the recombination to dissociation rate constants by the use of the equilibrium constants then lead to k7. The result can be represented in the form k7 ≈ 1017(T/300 K)2 exp(E0/kT) s1 where E0/k is in the range 72009100 K (depending on the considered isomer of NOþ(H2O)4). For 300 K, this gives k7 in the range 104106 s1 in good agreement with our fitted value of about 105 s1 (after converting the second order into a first-order rate constant in Table 1). However, the fitted value of about 103 s1 at 200 K (converted in the same way) is orders of magnitude above the best fit modeled value and, therefore, must be an artifact. However, as seen in Table 1, the data are not well fit for this reaction. One may also compare the measured rate constants for the HNO2-forming bimolecular reactions 4 and 6 with results from ion-dipole capture theory, that is, with eq 10, see above. For reaction 6, for example, one obtains kcap/109 cm3 molecule1 s1 = 3.2, 2.9, 2.6, and 2.4 at T/K = 150, 200, 250, and 300, respectively; for reaction 4, the results are almost the same. A

’ CONCLUSIONS Previous work at 300 K on the conversion of NOþ(H2O)n to H3Oþ(H2O)m found that all of the conversion occurred for n = 3. Temperature dependence measurements of the complicated kinetics system where NOþ sequentially clusters to H2O and then forms proton hydrates were presented here. Several competing reactions are identified as the temperature is lowered. At 150 K, NOþ(H2O)3 rarely reacts with H2O to form H3Oþ(H2O)3 but over 99% of the time forms NOþ(H2O)4; this hydrate reacts rapidly with H2O to form H3Oþ(H2O)3. The temperature dependences of the clustering rates were found to be steep, although large errors in deriving rate constants in the complicated system prevented definitive values. Interestingly, flux calculations show that NOþ(H2O)4 plays a small but nonzero role in conversion to proton hydrates at 300 K, even though it is not observed. Thermal dissociation of NOþ(H2O)4 to lose HNO2 is also observed. This study emphasizes the importance of studying temperature dependences in ionmolecule clustering reactions since such studies often reveal hidden reactions.27 When NOþ(H2O)4 reacts with H2O to form proton hydrates, specifically H3Oþ(H2O)3, and nitrous acid, HNO2, the derived rate constant is found to agree with the results from iondipole capture theory. On the other hand, the clustering reactions NOþ(H2O)n þ H2O þ M f NOþ(H2O)nþ1 þ M require an analysis in terms of statistical unimolecular rate theory. Modeling results show that the clustering reactions at pressures around 1 Torr are not in the third-order low pressure limit but in the falloff transition range toward the second-order high pressures limit. This influences the temperature coefficients of the rate constants at constant pressure. The full picture of the temperature and pressure dependences of these reactions, therefore, requires modeling of the full falloff curves. The present results show that in the mesosphere, where the temperatures are low, conversion of NOþ(H2O)n clusters to proton hydrates will occur much rapidly than previously thought. Rate constants become fast enough that conversion through a sequential mechanism involving N2 and CO2 clusters and ligand switching may not be needed. Details await modeling calculations beyond the scope of the present article. ’ APPENDIX: MOLECULAR PARAMETERS USED IN THE MODELING OF RATE CONSTANTS Molecular frequencies (in cm1). NOþ: 2377; H2O: 3651, 1595, 3756; NOþ(H2O): 92, 230, 245, 406, 417, 1645, 2126, 3752, 3859; NOþ(H2O)2: [25], 70, 70, 91, 189, 217, 227, 349, 351, 353, 366, 1654, 1655, 2156, 3781, 3783, 3887, 3887 ([25] corresponds to the “low frequency oscillator model” (a) used in Table 3); NOþ(H2O)3: 51, 107, 130, 197, 207, 218, 225, 248, 257, 264, 313, 343, 406, 486, 536, 640, 710, 1632, 1636, 1675, 2090, 3603, 3642, 3780, 3853, 3856, 3885 (corresponds to the 7589


The Journal of Physical Chemistry A “ring structure model” (b) used in Table 3); NOþ(H2O)3: 33, 55, 62, 77, 108, 195, 208, 256, 290, 320, 365, 374, 407, 524, 811, 919, 1006, 1594, 1647, 1660, 2040, 3154, 3272, 3789, 3800, 3896, 3910 (corresponds to the “trigonal structure model” (c) used in Table 3); NOþ(H2O)3: 6.4 [25], 16.1 [25], 48, 67, 108, 116, 181, 184, 189, 210, 275, 321, 324, 326, 343, 347, 1648, 1651, 1653, 2150, 3791, 3792, 3793, 3899, 3899, 3900 (corresponds to the “Cs structure model” (d) used in Table 3); NOþ(H2O)4: 25, 45, 64, 96, 110, 144, 201, 209, 211, 232, 240, 267, 301, 304, 322, 368, 446, 526, 592, 660, 804, 913, 1635, 1637, 1658, 1687, 2070, 3417, 3472, 3552, 3810, 3850, 3852, 3859, 3922 (corresponds to the “ring þ 1 down front structure model” (e and f) used in Table 3). Rotational constants (in cm1). NOþ: 2.00; H2O: 27.9, 14.5, 9.28; NOþ(H2O): 1.98, 0.23, 0.21; NOþ(H2O)2: 0.22, 0.177, 0.107; NOþ(H2O)3 (trigonal): 0.122, 0.067, 0.045; NOþ(H2O)3 (ring): 0.167, 0.076, 0.055; NOþ(H2O)3 (Cs): 0.101, 0.079, 0.050; NOþ(H2O)4 (ring þ 1 down front): 0.115, 0.034, 0.031. Frequencies and rotational constants for NOþ and H2O are from the JANAF tables,28 for NOþ(H2O)n from CCSD calculations, for details see Relph et al.9 and part II; for NOþ(H2O)2, [25] signals nearly free rotation. Dissociation energies (at 0 K, in kcal mol1). NOþ(H2O): 19.2; NOþ(H2O)2: 15.1; NOþ(H2O)3 (ring); 14.4, NOþ(H2O)3 (trigonal): 13.3; NOþ(H2O)3 (Cs): 13.6, NOþ(H2O)4 (ringþ1 down front f NOþ(H2O)3 (Cs)): 12.2; NOþ(H2O)4 (ring þ 1 down front f NOþ(H2O)3 (trigonal)): 12.4; from RI-MP2/augcc-pVTZ calculations of this work, for details see Relph et al.9 and part II; the NOþ(H2O)4 value coincides with that from Asada et al.10 Anharmonicity factors: Fanh ≈ 1.66, estimated as for H2O2 and H2CO26 and for HNO3;25 see text. Langevin collision frequencies Z for M = He (in cm3 molecule1 s1): 5.5 1010 for n = 1 and 2; 5.4 1010 for n = 3 and 4.

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(9) Relph, R. A.; Guasco, T. L.; Elliott, B. M.; Kamrath, M. Z.; McCoy, A. B.; Steele, R. P.; Schofield, D. P.; Jordan, K. D.; Viggiano, A. A.; Ferguson, E. E.; Johnson, M. A. Science 2010, 327, 308. (10) Asada, T.; Nagaoka, M.; Koseki, S. Phys. Chem. Chem. Phys. 2011, 13, 1590. (11) Choi, J. H.; Kuwata, K. T.; Haas, B. M.; Cao, Y. B.; Johnson, M. S.; Okumura, M. J. Chem. Phys. 1994, 100, 7153. (12) Angel, L.; Stace, A. J. J. Chem. Phys. 1998, 109, 1713. (13) Handbook of Geophysics and the Space Environment; Jursa, A. S., Ed.; National Technical Information Service: Springfield, VA, 1985. (14) Viggiano, A. A.; Morris, R. A.; Dale, F.; Paulson, J. F.; Giles, K.; Smith, D.; Su, T. J. Chem. Phys. 1990, 93, 1149. (15) Shuman, N. S.; Miller, T. M.; Hazari, N.; Luzik, E. D.; Viggiano, A. A. J. Chem. Phys. 133, 11. (16) Okumura, M.; Yeh, L. I.; Normand, D.; Lee, Y. T. J. Chem. Phys. 1986, 85, 1971. (17) Howard, C. J.; Rundle, H. W.; Kaufman, F. J. Chem. Phys. 1971, 55, 4772. (18) Hamon, S.; Speck, T.; Mitchell, J. B. A.; Rowe, B. R.; Troe, J. J. Chem. Phys. 2005, 123, 054303. (19) Hamon, S.; Speck, T.; Mitchell, J. B. A.; Rowe, B. R.; Troe, J. J. Chem. Phys. 2002, 117, 2557. (20) Maergoiz, A. I.; Nikitin, E. E.; Troe, J. Int. J. Mass Spectrom. 2009, 280, 42. (21) Su, T.; Chesnavich, W. J. J. Chem. Phys. 1982, 76, 5183. (22) Troe, J. J. Chem. Phys. 1987, 87, 2773. (23) Troe, J. J. Chem. Phys. 1996, 105, 6249. (24) Troe, J. J. Phys. Chem. 1979, 83, 114. (25) Troe, J. Int. J. Chem. Kinet. 2001, 33, 878. (26) Troe, J.; Ushakov, V. G. J. Phys. Chem. A 2009, 113, 3940. (27) Viggiano, A. A.; Arnold, S. T.; Morris, R. A. Int. Rev. Phys. Chem. 1998, 17, 147. (28) JANAF Thermochemical Tables, 4th ed.; Chase, M. W., Ed.; National Institute of Standards and Technology: Gaithersburg, MD, 1998.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT The AFRL authors are grateful for the support by the Air Force Office of Scientific Research of this work. N.E. acknowledges funding from the Institute for Scientific Research of Boston College (FA8718-04-C-0055) and the Air Force Summer Faculty Fellowship Program. Financial support by the European Office of Aerospace Research and Development (Grant Award No FA 8655-10-1-3057) is also gratefully acknowledged. ’ REFERENCES (1) Narcisi, R. S.; Bailey, A. D. J. Geophys. Res. 1965, 70. (2) Fehsenfeld, F. C.; Ferguson, E. E. J. Geophys. Res. 1969, 74. (3) French, M. A.; Hills, L. P.; Kebarle, P. Can. J. Chem. 1973, 51, 456. (4) Howard, C. J.; Rundle, H. W.; Bierbaum, V. M.; Kaufman, F. J. Chem. Phys. 1972, 57, 3491. (5) Fehsenfeld, F. C.; Mosesman, M.; Ferguson, E. E. J. Chem. Phys. 1971, 55, 2120. (6) Good, A.; Durden, D. A.; Kebarle, P. J. Chem. Phys. 1970, 52, 222. (7) Lineberger, W. C.; Puckett, L. J. Phys. Rev. 1969, 187, 286. (8) Buck, U.; Huisken, F. Chem. Rev. 2000, 100, 3863. 7590


in the Conversion of NO

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