Arrhenius parameters for ionic fragmentations - ACS Publications

---

J. Phys. Chem. 1992,96, 1733-1737 overlapping precisions in the L Y values, ~ structural distinctions from the latter are often unclear. (2) Both scales demonstrate the influence of increasing halogen substitution on HBD acidity for the haloalkanols and haloalkanes. (3) The overall trends in the dissimilarity indices for the two HBD acidity scales are parallel among the members of all four functional classes, including the full range from weak-to-strong hydrogen bond donors. On the other hand, the AaKT values are consistently the superior discriminators with respect to the identification of smaller structural influences upon solvent donor acidity. (4) For the scaling intervals used in the three HBD acidity parameters, there are significant numerical differences as well. These may be demonstrated by comparing the parameter ratios for the strongest (st-HBD) and weakest (w-HBD) members in Tables I and 11, i.e. trifluoroethanol and acetone. As measures of the scale ranges, it is clear that the AN* and a? parameters are similar in spread with ratios (st-HBD:w-HBD) of 6.5:l and 6.3:1, respectively; however, the corresponding ratio for the LYKT scale is nearly 3 times larger (18.8:l) than the other two acidity scales and leads to the LYKT parameter being a more sensitive factor in linear free energy descriptions of kinetic solvent effects. The search for universal HBD acidity parameters for applications to electron-transfer kinetics in solution seems justified because of the evolving structural models for self-associated solvents. Even those strongly associated solvents having very diverse polar functionalities appear to have solvent characteristics

1733

in common. Rossky has concluded that packing effects are a major structural determinant and that for even the lower alkanols the unit structure can be depicted as a simple heteronuclear diatomic species with the alkyl and hydroxyl groups defining the dipolar sites in that molecule.29 Likewise, among such dissimilar HBSA liquids as alkylamides and alkanols, there appear to be common polar liquid structures (Le. “winding chains”) in which the monomeric unit is hydrogen bonded to two nearest neighbors. Computer models of solute solvation by such liquids indicate that there are only slightly stronger nearest-neighbor HBD interactions in solution rather than major population shifts favoring more linear hydrogen bonded combinations within the solvent structure.29 Registry No. Benzonitrile, 100-47-0; pyridine N-oxide, 694-59-7; 1-butanol, 71-36-3; ethanol, 64-17-5; 1-hexanol, 11 1-27-3; methanol, 67-56- 1; 2-methyl-2-propano1, 75-65-0; I-pentanol, 7 1-41-0;1-propanol, 7 1-23-8; 2-propanol, 67-63-0; 2-chloroethanol, 107-07-3; 2,2,2-trichloroethanol, 115-20-8; 2-fluoroethanol, 37 1-62-0; 2,2-difluoroethanol, 359-13-7; 2,2,2-trifluoroethanol,75-89-8; acetic acid, 64-19-7;butanoic acid, 107-92-6;formic acid, 64-18-6; propanoic acid, 79-09-4; acetone, 67-64-1; acetonitrile, 75-05-8; chloroform, 67-66-3; methylene chloride, 75-09-2; 1,2-dichIoroethane, 107-06-2; dimethylacetamide, 127-19-5; dimethylformamide, 68- 12-2; dimethyl sulfoxide, 67-68-5; ethyl acetate, 141-78-6; methyl acetate, 79-20-9; benzene, 71-43-2; carbon tetrachloride, 56-23-5; l,Cdioxane, 123-91-1;diethyl ether, 60-29-7;n-hexane, 110-54-3;tetrahydrofuran, 109-99-9;formamide, 75-12-7; nitromethane, 75-52-5. (29) Rossky, P. J. Annu. Reo. Phys. Chem. 1985, 36, 321-346.

Arrhenius Parameters for Ionic Fragmentations Cornelius E.Mots Chemical Physics Section, Health and Safety Research Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831 -61 25 (Received: June 1 1 , 1991; In Final Form: October 15, 1991)

Rate constants as a function of energy for the dissociation of more than 25 molecular ions, drawn from the literature, are analyzed in order to extract their attendant Arrhenius parameters. Subtleties of the extraction process are discussed, and uses to which the parameters might be put are indicated.

Introduction The rate constant for a unimolecular reaction is typically a strong function of pressure and temperature. The pressure dependence can be thought of as arising from an imperfect contact between the reacting species and its heat bath. The limiting, high-pressure rate constant is the parameter of greatest intrinsic interest. It may be obtained from an extrapolation of measurements obtained at finite pressures, guided by theory and with due attention to artifacts arising from such matters as the cage effect.’ Finally, the temperature dependence of this limiting rate constant may be studied. Using a standard Arrhenius plot one may extract an activation energy, M a , and then cast the rate constants in the form k ( T ) = A exP(-L\E,/k,T)

(1)

This representation contains two parameters, the activation energy and the frequency factor, A . Each is of interest. Nevertheless, the traditional experimental method just outlined for obtaining them is both problematical and tedious. An alternative method for obtaining the same information has been describedS2 It makes use of measurements of k ( E ) , rate (1) Troe, J. J . Phys. Chem. 1986, 90, 357.

0022-365419212096-1733$03.00/0

constants for the reaction under consideration as a function of the energy. The measurements are made on isolated molecules, Le., under conditions which are the antithesis of the high-pressure heat bath. Nevertheless, generalized Arrhenius plots can be constructed, and the parameters Ma,, and A can be extracted. In effect, data garnered in the energy domain can be transferred to the temperature domain. Such transformations require that one know the properties of the parent molecule, in order to calculate canonical energies and heat capacities. One does not need to know anything about transition states, or what the products of the reaction are, or even whether or not it involves chemistry. In that sense the transformation is model-free. In addition to the opportunity for data-smoothing which generalized Arrhenius plots offer, two uses of such a transformation will be evident: it provides a convenient two-parameter condensation of the data and it permits a direct comparison with data garnered in the temperature domain by conventional means. Because of the facility with which charged particles can be manipulated, most of the existing measurements of rate constants as a function of energy involve the fragmentation of ions.3 We (2) Klots, C . E.J . Chem. Phys. 1980, 90, 4470; 1990, 93, 2513. (3) Baer, T. Adu. Chem. Phys. 1986, 64, 111.

0 1992 American Chemical Society

Klots

1734 The Journal of Physical Chemistry, Vol. 96, No. 4, 1992 shall focus here on these data, and in particular on the extraction of Arrhenius parameters from them. The uses to which they might be put will be indicated, but only briefly. A more thorough examination of their magnitudes will be reserved for another place.

Background Molecular ions can be prepared with a well-defined internal energy in a number of ways. Ascertaining the magnitude of this energy contains two subtleties, which we shall discuss in the context of the single-photon ionization technique. We suppose that a photon of energy hv is used to ionize a molecule with an ionization potential indicated by IP. The energy of the molecular ion may then be written as E = hv - IP + &To) - ke where &To) is the (thermal) energy in the parent molecule prior to ionization and ke is the energy of the ejected electron. A recurrent question is whether or not the rotational energy of the neutral parent should be included within the term E(To). The question arises because of concem that the rotational energy might expedite the rate constant. It is an interesting and multifaceted question, but one which in the present context is something of a diversion. This may be seen by assuming for simplicity that the neutral parent molecule is a spherical top, with a rotational energy Bdo(Jo 1). Strictly speaking, this term should be included on the right-hand-side of eq 2. Recall, however, that the construction of a generalized Arrhenius plot2 calls for the definition of a temperature of the ion via the relation

+

TABLE I: Arrheniw Parameters for Ionic Fragmentations and Apparent Heats of Formation of hoduct Ions w

parent (product) ion benzene (C4H4)" benzene (C3H3)" pyrole (C2H3N)b phenyl bromide (C6H5)' phenol ( C S H ~ ) ~ phenyl chloride (C6HS)'

A, s-' 3.6 (16) 1.3 (16) 6 (15) 1.7 (15) 1.4 (15) 1.1 (15) 1(15) 8.5 (14) 8 (14) 7 (14) 6 (14) 2.3 (14) 1.7 (14) 1.5 (14) 9 (13) 5 (13) 5.0 (1 3) 2.3 (13) 2.2 (13) 9 (12) 5 (12) 3 (12) 5 (12) 4 (12) 2.1 (11) 1.1 (11) 2 (9) 2 (8) 2 (12)

MAMd

thiophene (C,H2S)J aniline (C5H6)8

benzene (C6HS)" m-nitrotoluene (C7H7)j p-nitrotoluene (C,H7)' benzene (C6H4)" phenyliodide (C6HS)' dibromobenzene (C&Br) benzonitrile (C6HJL n-butylbenzene (C7H7)i dichlorobenzene (C6H4C1) dioxane (C3H60)" styrene (C6H6)" nitrobenzene (C6H50)P nitrobenzene (NO)P naphthalene (C8D6)9 butadiene (C3H3)' p-nitrotoluene (C7H70)j n-butylbenzene (C7H8)' ethyl acetate (C4HsO)S o-nitrotoluene (C,H6NO)" J

(3)

where &(T) is the canonical energy of the parent ion. This energy must then include a term BJi(Jj l), where Bj is the rotational constant of the parent ion, and Ji indicates its rotational quantum number. In the experiments analyzed here, Bi N Bo and Ji N Joe Therefore, in deducing the temperature corresponding to a given experiment, it will scarcely matter whether these terms are included or not, so long as a consistent procedure is followed. On the other hand, in chemical activation experiments the angular momentum changes are large, and this cancellation cannot be assumed. One will then need to address these terms explicitly. In any case the question of whether or not rotational energy expedites a reaction does not enter into the discussion. This is as it should be in a model-free procedure. These considerations also serve as a reminder that a microcanonical rate constant k(E,Ji) is a function of the conserved angular momentum as well as the energy. Strictly speaking, the Arrhenius parameters which are deduced will refer to a curious sort of heat bath whose constituents all have the same angular momentum. In the experiments addressed here, for example, the rotational temperature is negligible compared with that of the other degrees of freedom. The parameters obtained then need not correspond to those which one might measure in a more orthodox setting. We will return to this matter below. Equation 2 prescribes an energy as measured from the ground-state of the parent ion. If (as is often the case) this ion isomerizes to a more stable structure prior to dissociation, the energy of isomerization must be added to the right-hand side of the equation. If this energy is not known, the usefulness of the present analysis is greatly restricted. We shall see an example of this below. If this is not a problem, then one may proceed to construct a generalized Arrhenius plot, consisting of the logarithms of the rate constants a5 a function of their respective reciprocal temperatures. The slope of such a plot is given by2

+

d In k(E)/d(kBr)-* = (a In k(E)/aE)(aE/a(kBT)-')

(4a)

= -[ (kgT*)-l - (k~r)-']c(?")(k,T)' where use has been made of the theorem2

(4b)

m

9

AE., eV

kJ/mol

4.18 4.18 3.42 2.73 3.25 3.24

1170 1054 1018 1140

3.07 3.66 3.72 1.61 1.65 3.72 2.14 2.76 3.09 1S O 3.14 1.40 2.43 1.15 1.20 3.38 2.08 1.10 0.91 0.67' 0.34 1.33"

1051 1008h 1136 1079 1096 1352 1118 1149 1337 919" 1078 849 994

1135

1059 1057 923

"Data from ref 4. *Reference 5. 'Reference 6. dReference 7. cEvaporation;see the text. /Reference 8. ZReference 9. *mmO for the neutral product, HNC, assumed to be 196 kT/mol. 'Reference 10. jReference 11. &Reference12. 'Reference 13. "eutral product assumed to be s-C,H,. "Reference 14. OReference 15. PReference 16. 9Reference 17. 'Reference 18. SReference 19. 'An isomerization energy of 0.62 eV, as proposed in ref 19, was assumed here. " Reference 20. "An isomerization energy of 1.5 eV, as proposed in ref 20, was assumed here. and C(Tj is the heat capacity, in units of kB, defined by the energy-temperature relation of eq 3; and where the temperature Tt is defined by

+

E = Ma Ej(T*) - k B P

(6) where &Tt) is again the canonical energy of the parent ion, but now evaluated at T). Since this temperature is the only parameter on the right-hand side of eq 4b which is not known, it may be obtained from the Arrhenius plot and then entered into eq 5 to obtain the activation energy. A frequency factor may then be obtained by focussing on some k(E), drawn preferably from the midrange of the Arrhenius plot. Its corresponding isokinetic temperature, Tb,defied by k(E) being equal to k(Tb), can be obtained from the relation2 where

is a dimensionless heat capacity defined by

e = AE,/kB(T

- TI)

(8)

The Arrhenius Parameters We have used this prescription to analyze most of the available rate constants for ionic fragmentations. The reactions surveyed and the sources of the data are given in Table I. The vibrational frequencies of the parent ions, which are needed to calculate canonical energies as prescribed by eqs 3 and 6, were also taken from these sources. The derived Arrhenius parameters for these reactions are given in the table. Recall, however, that activation energies and preexponential factors are never strictly independent of temperature. The listed parameters should thus be thought of as pertaining only in the temperature range where the rate constants

The Journal of Physical Chemistry, Vol. 96, No. 4, 1992 1735

Arrhenius Parameters for Ionic Fragmentations

Included in this table is the frequency factor A 2 X lOI5s-l for evaporation from liquidlike van der Waals molecules. This number derives from a recent resultz2 kcyap= 3 X 10'3n2/3e ~ p [ 6 / n ' / ~exp[-hEa/kBTb] ] (13) 108

r

p-

107

x

1OE

where n is the number of monomeric units in the aggregate. It will be seen that this result yields frequency factors which, as a function of cluster size, pass through a broad minimum at the cited value. Although superficially unlike the other reactions in the table, the process of evaporation offers a useful benchmark. Equation 13, while supported by experiment, is rooted in the traditional Langmuir model for the reverse process, that of condensation. The model assumes that every collision leads to reaction, Le., that the steric factor for condensation is unity. The discussion in a later section will indicate how an analogous quantity for the other reactions in the table may be obtained. We have indicated two sets of Arrhenius parameters for the reaction O-CH3C6H4NO2' C7H6NO' + OH (14) The extremely low frequency factor results when the experimental data are treated as described above. The larger value is obtained when it is assumed, as suggested by the original authors,*O that the parent ion isomerizes prior to dissociation to a structure which is 1.5 eV more stable. We show in Figure 2 the Arrhenius plots corresponding to these two cases. The discernible concavity in the plot which assumes isomerization suggests that 1S eV is an upper limit to the energy for this process. The parameters in the table thus probably constitute lower and upper limits to the correct values. In either case, the frequency factor for this reaction is clearly low compared with most of the entries in the table. Nevertheless, this example also illustrates that, when there is ambiguity in the energy, the present method is of limited utility. For this same reason, the parameters derived for the decomposition of the ethyl acetate ion must also be approached with some caution. Having transformed microcanonical rate constants into the temperature domain, one might now entertain their interpretation with any of the plethora of models available for this purpose. This prospect is especially inviting since such comparisons can surely be done more simply and with greater definition when temperature is the dependent variable instead of energy. We shall eschew this possibility in order to focus in the next two sections on more immediate applications of these data. +

105

8

10

12

[ksT(ev)].' Figure 1. Generalized Arrhenius plots for reactions 9 and 10; solid line,

extrapolation as in eq

12.

assume magnitudes of 105-106s-', this being the usual magnitude of the microcanonical rate constants from which these parameters derive. Several examples of the generalized Arrhenius plots used to obtain these parameters have already been given.2 There is an additional nuance to these plots which should be discussed here. We show in Figure 1 the datal3 for the reaction

C&C~HS'

-

CH3CbHS'

+ C3H6

(9)

from which the parameters in the table were obtained. Experiments a t much higher energies also yielded the ratios of rate constants for this reaction to those for

C ~ H ~ C ~ H SC' H ~ C ~ H + S ' C3H7

(10) If all these data had been obtained in an infinite heat bath, one might simply extrapolate the plot of the data for the first of these reactions and thus arrive at absolute rate constants for its competitor. Arrhenius plots of data obtained in a finite heat bath, however, are necessarily concaveS2Let us donate by Sothe slope of such a plot centered around the temperature Toand the rate constant k,,. From eqs 4b and 8 it is given by

so = -ua(C/O(T/T*)

(11)

Let us now assume that the activation energy and heat capacities are only weak functions of the temperature. It is then straightforward to show that, to second order, a generalized Arrhenius plot has the form In k/ ko = So[(kBT)-' - (k~To>-'l- ( s o 2 / 2 C ) [ ( k ~ T ) --' ( ~ B T O ) -(12) 'I~

As the molecule becomes very large, so does its heat capacity. The final term in this relation becomes then of no importance. For smaller molecules, the intrinsic concavity is still usually negligible over the range of typically accessible temperatures. When making a prolonged extrapolation, however, its effects may be discernible. The solid line in the figure contains this correction, whereas the dashed line does not. The former was used in obtaining the Arrhenius parameters for the reaction denoted by eq 10. Extrapolations to lower temperatures may also be called for. Thus when the cited data for the dissociation of the C8H8+ion are extended using eq 12, excellent agreement with the datum of Dunbar*' is obtained.

Activation Energies In a chemical reaction the activation energy often serves as a first approximation to the height of a potential barrier which must be surmounted. As the activation energy is measured at lower (4) Kiermeier, A.; Kiihlewind, H.; Neusser, H. J.; Schlag, E. W.; Lin, S. H . J . Chem. Phys. 1988, 88, 6182. ( 5 ) Willett, G. D.; Baer, T. J. Am. Chem. SOC.1980, 102, 6774. (6) Baer, T.; Tsai, B. P.; Smith, D.; Murray, P. T. J . Chem. Phys. 1976, 64, 2460. Baer, T.; Kury, R. Chem. Phys. Lett. 1982, 2, 659. (7) Fraser-Monteiro, M. L.; Monteiro, L. F.; de Witt, J.; Baer, T. J. Phys.

Chem. 1984,88, 3622. (8) Butler, J. J.; Baer, T. J. Am. Chem. Soc. 1980, 102, 6764. (9) Baer. T.: Carney. T. E. J. Chem. Phvs. 1982. 76. 1304. (IO) Baer, T.; Mor;ow, J. C.; Shao, J. D:; Olesik; S. >. Am. Chem. Soc. 1988, 110, 5633. (11) Olesik, S.; Baer, T.; Morrow, J. C. J. Phys. Chem. 1986, 90, 3563. (12) Eland, J. H. D.; Schulte, H. J. Chem. Phys. 1975, 62, 3835. (13) Baer, T.; Dutuit, 0.;Mestdagh, H.; Rolando, C. J. Phys. Chem. 1988, 92, 5674. (14) Fraser-Monteiro, M. L.; Fraser-Monteiro. L.; Butler, J. J.; Baer, T. J . Phvs. Chem. 1982. 86. 739. (15) Smith, D.; Baer,'T.; Willett, G. D.; Ormerod, R. C. Int. J. Mass Spectrom. Ion Phys. 1979, 30, 155. (16) Panczel, M.; Baer, T. Int. J. Mass Spectrom. Ion Phys. 1984,58,43. (17) Ruhl, E.; Price, S. D.; Leach, S. J. Phys. Chem. 1989, 93, 6312. (18) Werner, A. S.; Baer, T. J. Chem. Phys. 1975, 62, 2900. (19) Fraser-Monteiro, L.; Fraser-Monteiro, M. L.; Butler, J. J.; Baer, T. J . Phys. Chem. 1982,86, 752. (20) Shao, J.-D.; Baer, T. Int. J. Mass Spectrom. Ion Processes 1988,86, 357. (21) Dunbar, R. C. J. Phys. Chem. 1990, 94, 3283. (22) Klots. C. E. Z . Phys. D 1991, 20, 105.

1736 The Journal of Physical Chemistry, Vol. 96, No. 4, 1992

and lower temperatures, and if tunneling is not important, it will approach the magnitude of this barrier. Alternatively, one might use a model to deduce the latter from measurements at some high temperature. The simplest such model is to assume that the activation energy is independent of temperature. The magnitude of this barrier is especially interesting if one can also assume that there is no barrier for the reverse of the reaction under consideration. In that case the barrier height can be identified with the heat of reaction at absolute zero. If the heat of formation of the parent ion and that of one of the fragments are known, the heat of formation for the other fragment can be deduced. We give in Table I the heats of formation of the fragment ions that can be deduced from the present analysis on making these assumptions. No values are given for those reactions where some criterion, such as excess kinetic energy release, implicates an activation energy for the reverse reaction. Those reactions are also excluded where, for the reasons discussed above, an ambiguity in the energy of the parent ion precludes a meaningful determination of the Arrhenius parameters. Of particular interest are the four reactions which yield the C6H5+ion. Its apparent heat of formation equals 1132 (*6) kJ/mol and is in good agreement with the value previously deduced* when a model, the statistical adiabatic channel model, was used to extrapolate the observed activation energies to absolute zero. This agreement suggests that whatever is gained by the (tedious) application of such models is less than the errors which arise either from the experimental data or in their subsequent analysis. We will return to these errors in a later section.

Frequency Factors The activation energy is closely associated with a minimum energy requirement for a reaction. The frequency parameter may then be identified with those other factors involved in the harnessing of heat to produce directed motion. In chemistry its magnitude is often identified with the “looseness” of the transition state. In particular, some authors speak of an apparent entropy of activation, AS*,in order to quantify this idea. As usually defined, this quantity may be correlated with the frequency factor of the reaction via A = (keTt)e/h)exp(AS*/kB)

(15)

where e is the base of natural logarithms. Alternative definitions of this entropy of activation may include reaction path degeneracy factors or rotational partition function symmetry factors explicitly. In any case they constitute equivalent parametrizations of the reaction rate. We will indicate here another parametrization scheme. It is less general in that it is applicable only to dissociations and also in that it requires assumptions about the properties of the reaction products. Its conceptual advantages will be clear, however. We begin by noting that one may define a frequency factor corresponding to the loosest of all possible transition states. This is the phase-space rate constant, applicable when all reaction channels permitted by the conservation laws are open. When the rotational “temperature” of the parent species is much less than that of its other degrees of freedom, the frequency factor may be written as Aps(T,Ji=O) = (kBTe/h)(.O/.l.2)

exp(Mps/kB)

(16)

where uo,ul, and u2 are the rotational partition function symmetry factors of the parent ion and its two fragments, and where AS,, is defined by Here Siiband SLibare the entropies associated with the final and initial vibrational degrees of freedom, and Sg, is the entropy associated with the partition function where (2s

+ 1) equals the number of final rotational degrees of

Klots freedom and 6 is a function of the rotational constants of the reaction products. These functions have been given elsewhere.23 It will prove instructive to apply this to the reaction

C&&+ C.&+ C2H2 (19) which from Table I is seen to have an exceptionally large frequency factor. On using the vibrational frequencies and rotational constants assumed elsewhere for these species,23one obtains A , equal to 5 X lo’*s-l for this reaction. Clearly the experimentally derived rate constant, although large, could have been even more so. With this preliminary, we shall now turn to how one might extract an apparent hard-sphere reaction cross-section for the reverse of a dissociation reaction. When J , N 0, we may use the relationz2 -+

+

A/Ap, = p b 2 / ( p b 2 + ZI Zz) (20) where p is the reduced mass of the two fragments and ZI,Zz their moments of inertia, to obtain rb2, the apparent reaction cross section. In this way one obtains for the reaction denoted by eq 19, a value of 0.06 A2 for the cross section. This is very much less than a collision cross section. Again, but now even more clearly, it is not a fast reaction. Each such case must be considered separately. To illustrate this we focus on the reaction C&+ C&5+ H (21) which is seen from the table to have a significantly lower frequency parameter. Nevertheless one obtains the value 0.5 AZ for the apparent cross section for its reverse. While still not large by collision standards, on comparison with reaction 19 it illustrates nicely the enhancement of reaction probabilities when only one reagent needs to be ~ r i e n t e d . ~ ~ , ~ ~ Parametrization in terms of apparent reaction cross sections can thus be useful. It is therefore worth noting circumstances when eqs 16 and 20 can be condensed. When pb2 > Z2 is clearly fulfilled, and nearly so for (19).

Effect of Angular Momentum We now return to a question raised earlieI-the relation between a frequency factor obtained in an orthodox heat bath and that obtained when the rotational temperature is negligible. The corresponding ratio A( T ) / A ( T , J O y O )might be called the Rabinovitch factor in view of his prescient discussion26of its magnitude. It is the factor by which preexponential parameters such as those given in the table must be multiplied when comparing them with data obtained by traditional means. He noted that a number of authors had obtained the quantity (Z*/Z0)3/zfor this factor, where I* is a (geometric mean) moment of inertia of the transition state. This factor will tend to be greater than unity. It is of interest to compare this result with what obtains when a hard-sphere collision cross-section model for the reverse (23) Klots, C . E. Z . Naturforsch. 1972, 27a, 5 5 3 . (24) Johnston, H. S . Gas Phase Reaction Rare Theory; Ronald Press: New York, 1966. (25) Klots, C . E. Chem. Phys. Lett., submitted for publication. (26) Waage, E. V . ; Rabinovitch, B. S. Chem. Reu. 1970, 70, 377.

The Journal of Physical Chemistry, Vol. 96, No. 4, 1992 1737

Arrhenius Parameters for Ionic Fragmentations 10

16

18

20

I

I

I

in which by contrast the location of the energy maximum which conventionally defines the transition state can very from one exit channel to another. The effects of rotation on microcanonical rate constants can differ significantly from one model to another, and the magnitudes of the resulting Rabinovitch factors are accordingly uncertain. In view of this the earlier suggestion that these factors will not be large must be considered only tentative and in need of further scrutiny.

>\ 0

i

b \.

Error Analysis Experiments are subject to random errors. Let us suppose that some data have been acquired in an infinite heat bath, and an Arrhenius plot has been constructed. An incorrect assignment of its slope will lead to an incorrect activation energy. Any such error, say 6AEa, will result in an error in the frequency factor given by 6 In A = 6AE,/kBT

p- 101 x

\

o - NO? CeH&H3+

+ C,HeNO+ lo?

;

t

OH

When the data are acquired in a finite heat bath, the consequenm of such an error will be magnified. This occurs because the relevant temperature Tb cannot be read from a reliabie thermometer but must be obtained from eq 7, which also contains the error-prone activation energy. The net effect of an incorrect activation energy may nevertheless be written succintly as

I

10

and so may be estimated. For the experiments analyzed here, the second [size-dependent] factor typically will be about one-third greater than that arising in the infinite heat bath. Thus, while it is always useful to have a thermometer, the absence of one does not lead to prohibitively large uncertainties. Equation 28b has a consequent application. Let us suppose that one decides that some activation energy in Table I needs to be adjusted. This equation then prescribes the attending adjustment to the frequency factor. There will be no need to return to the original data. Summary Studies of rate constants as a function of energy are said to be more fundamental than those which yield rate constants as a function of temperature. A casting of the former in terms of the latter would then appear to be retrograde. Nevertheless, Arrhenius parameters constitute a convenient form for the representation of the data, as has been remarked before.29 The present contribution makes it clear that this can be accomplished in a model-free manner. Further, the use of generalized Arrhenius plots offers the particular opportunity for smoothing the data at an early stage in the analysis. As noted earlier, the extraction of an energy barrier from an activation energy requires a model, while the extraction of an effective reaction cross section requires assumptions about the thermodynamic properties of the products. To underscore the distinction between these two procedures and the model-free manipulations which precede them, no further discussion of the Arrhenius parameters in the table has been given here. Additionally, there exists a considerable sum of data for the dissociation of neutral systems, some acquired from single molecule studies and some in a conventional heat bath. It seems best to defer a more detailed analysis until this larger body of data has been assimilated. Acknowledgment. This research was sponsored by the Office of Health and Environmental Research, US. Department of Energy, under contract DE-AC05-840R21400 with Martin Marietta Energy Systems, Inc., and in part by the North Atlantic Treaty Organization via Grant CRG 900480. Stimulating conversations with Professor J. Troe are gratefully acknowledged. (29) Rosenstock, H. M.; Stockbauer, R.; Parr, A. C . J . Chem. Phys. 1979, 71, 3708.