Communicationsto the Editor
1773
(3)N. Hirota, J. fhys. Chem., 71,127 (1967). (4)G.Levin, J. Jaugar-Grodzinski, and M. Szwarc, J. Amer. Chem. SOC.,92, 2268 (1970). ( 5 ) G. R. Stevenson and L. Echegoyen, J. Phys. Chem., 77, 2339 (1973). (6) It has been previously observed that salts such a3 KI are fully dissociated in HMPA, see 1'. Bruno, M. D.Monica, and E. Righetti, J. fhys. Chem., 77,
1258 (1973). (7)The potassium ion Concentration was taken as that of the added KI, since this concentration is seveiral orders of magnitude larger that the concentration of potassium used for the reduction. (8) D. A. Deranleau, ,1. Arner. Chem. SOC.,91, 4044 (1969). (9)G. R. Stevenson, J. 6.Concepcion, and J. Castillo, J. fhys. Chem., 77,
to-'O
IO"" ,"'
611 (1973).
Department of Chemistry University of Puerto R i m Rio Piedras, Puerto Rjco 0083 1
Gerald R. Stevenson* Antonio E. Alegrfa
I
Received May 6, 1974
lo2
-
lo3
to4
Temperature
lo5
(OK)
Figure 1. Temperature dependence of the rate constant for the reaction O+ N:! NO+ 4- N. Experimental data (---) are taken from ref 13;(- - -) calculated using TST. Calculated rate constant arbitrarily scaled to equal experimental rate constant at T = 300°K.
+-
On the Negative Temperature Dependence of Slow Ion-Molecule Reactions fubhcation costs wssarfed by the National Science Foundation
Sir: With the equipment and techniques developed over the past few years it has become possible to study 'WOW" ion-molecule reactions under conditions which permit the determination of the order of the reaction and the dependence of the rate on temperature. For many of these reactions the temperature dependence is unequivocally negative, that is, the rate increases as the temperature is lowered. In most cases the data best fit not the Arrhenius equation, but the equation k , = ATn where n has been found experimentally to have values from 0 to -4.6. Two kinds of theoretical explanations have been proposed for these observations. ;In the first,l it is assumed that the reaction proceeds in two stages, the formation of an intermediate, followed by the unimolecular decay of that intermediate. The assumption is made that the longer the lifetime of the intermediate, the higher the probability of its dissociation into the observed products. The other theoretical explanation is quantum chemical and analyzes the reaction path along potential energy surfaces in terms of symmetry and selection rules.2 In the study of one ion-molecule reaction3 t-CdH:
z - C , H ~--+. ~ i-C,Hi, + t-CSHii'
(1)
for which n = -3.Q, we have found that ordinary transition state theory (7"ST) accounts satisfactorily for the observed temperature dependence. We have, therefore, applied TST to other known glow bn-molecule reactions. The results are reported herein. The basic equation of"TST* may be written as
where h, is the reaction rate constant, c is the transmission coefficient, the $' are the partition functions for the activated complex, and the QLare the partition functions for the reactants. The temperature dependence of the transmission coefficiexlt and the electronic and vibrational parti-
tion functions will be assumed to be sufficiently small to be . ~ partition functions neglected as a first a p p r ~ x i m a t i o nThe of any internal vibrations that remain unchanged in the complex will cancel out. Labeling the origins of the temperature dependent terms, eq 2 may be rewritten as
(-E,/% T ) (3) Here A is a constant with respect to temperature; r is the total number of new internal rotations created upon the formation of the complex, i.e., the number of new internal rotations present in the complex minus the nurnber of internal rotations that became hindered upon the formation of the complex; j is 3 for nonlinear molecules, 2 for linear molecules, and 0 for atoms. If Eo is negligible, eq 3 reduces to the form k , = AT". The only reaction of neutrals for which there is a large negative temperature coefficient, that of nitric oxide with oxygen, was treated by Gershinowitz and EyringS6In that case, a s i n the case of the ion-molecule reactions which we are here discussing, attractive potentials between the reactants balance the zero point energy of the newly formed vibrations, giving an effectively zero activation energy. Although we wish to reserve a discussion of absolute rates for a more detailed consideration at a later time, we would like to point out here that the same factors which introduce negative exponents for T also explain the slowness of such reactions. For the ion-molecule reactions, other than reaction 1 reported in the literature, those which have the most markedly negative temperature coefficients are those which have been found to be third order. Table I gives the observed temperature dependences and those predicted from TST for reactions for which experimental data are available. It will be seen that the agreement is very good, considering the accuracy of the experimental data quoted and the approximations involved in the present calculations. We have already mentioned the reaction of tert-butyl ion with isopentane, in which the temperature dependence of T-3 indicates that about two free rotations of the methyl The Journal of Physical Chemistry, Vol. 78, No. 17. 1974
Communicationsto the Editor
1774
TABLE 1: Calculated and Experimental Temperature Dependence of the R a t e C o n s t a n t s Expressed i*t+k r =: A T nfor Trimolecular Ion-Molecule Reactions Calcd temperature dependence* Exptl temperature dependenceC
-
Type of reaction"
Examples
+ +
A1LAi-A A + + L L A L++L + A or
A + + L + E L++L t L NL+ NL+
Linear complex
+ +
H e + He He Hez+ He N + + N 2 + H e + N 3 + He N Z + N:! He - N q + He
T-' T-2
+ N2 + Nz N3r + Nz + + + N2+ + N:! + Nz + N 4 + + NZ Ha+ + + H:! H j T + Hz H 3 0 L+ H 2 0 + CH4 Hj02+ + CH4
T-3
7'-2.5
T-4
T-3.5
T-4.5
T-4 T--5
+ +
-f
N+
0 2
Hp
0 2
-f
-
0 4+
*
0 2
a Here A denotes an atomic, L a linear, and NL a nonlinear species. Calculated from eq 3,with 7 = 0. ing references.
groups are restricted in the activated complex. The bimolecular reaction 0'
t
N?
--+
NO'
-C
T-1
7
T-1.6
T-1.7
8
T-1.6
8
T-2.5
9 10 9 11 12
N
(4 1
has been studied over a wide temperature range.13 It has been observed that the temperature coefficient changes from negatbe at low temperatures to positive at high temperatures. This can be accounted for on the basis that a t higher temperatures the effect of the vibrational partition functions cicn no longer be neglected and, consequently, there is no need t o introduce arbitrarily a change in the mechanism at higher temperatures. We have msumax? that the activated complex is a linear N&+. As I N first approximation, we take the vibration frequencies of the activated complex to be those of the stable NzOt inn,'4 and we have calculated the temperature dependence of the sate constant. The results, compared with the experimental data, are shown in Figure 1. The agreement iri again gratifying, but more detailed study will be needed to determine whether this explanation based on T S T or one 2 f the alternatives proposed by Fergusonl or by Kaufrnan2 ii: the correct one. The fact that WE' ~ Q have W on hand a large amount of experimental data for reactions of small molecules with little or no activation energy makes possible new attempts to calculate a b s ~ute l rates from first principles and suggests possibilitl es far experimental investigations of such phenomena a s 14 1net.c isotope effects. We think that the study of ion-molerule reactions will contribute to a better understanding 3f Ibe prt?exponential factor in chemical reactions in general.
Achnoivledgmer I This research was supported in part by a grant from the National Science Foundation.
The Journal of ,?hysicai Chemistry, Vol 78, No, 17, 7974
T-3 T-4.4
T-5 5
Ref
T-0.i
-+
0 2 +
+ L 1-EA + NL + NL
+
Nonlinear complex
T-4.F T-4.2
Calculated from the data given in the correspond-
References a n d Notes (1) E. E. Ferguson, "Ion-Molecule Reactions," Vol. 2, J. L. Franklin, Ed., Plenum Press, New York, N. Y., 1972, Chapter 8. (2) J. J. Kaufman, "Wave Mechanics, The First Fifty Years," W. C. Price, et a/., Ed., Wiley, New York, N. Y., 1973, Chapter 14. (3) J. J. Solomon, M. Meot-Ner, and F. H. Field, J. Amer. Chem. SOC.,96, 3727 (1974). (4) For rigorous derivation and comprehensive discussion see H. S. Johnston, "Gas Phase Reaction Rate Theory," Ronald Press, New York, N. Y., 1966, Chapters 8 and 9. (5)The partition function for a vibration is (1 e-hu'kv-', which is clearly a function of temperature. Most of the experimental data treated in our Table I were obtained between 80 and 30O'K. At 80'K the partition function is unity for practically all vibration frequencies. At 300'K it is equal to 1.097 for Y = 500 cm-', 1.008 for v = 1000 cm-', and 1.0000005 for Y = 3000 cm-l. It is only for frequencies lower than 500 cm-' that one would have an error of 10% or more due to this assumption. One would not expect many such low frequencies in molecules as small as those we are considering. At higher temperatures the value of the partition function begins to change more rapldly and we discuss later in this paper one example for which sufficient data are avallable. (6) H. Gershinowitz and H. Eyring, J. Amer. Chem. SOC.,57, 985 (1935). (7) F. E. Niles and W. W. Robertson, J. Chem. Phys., 42, 3277 (1965). (8) D. K. Bohme, D. B. Dunkin, F. C. Fehsenfeld, and E. E. Ferguson, J. Chem. Phys., 51, 863 (1969). (9) A. Good, D. A. Durden, and P. Kebarle, J. Chem. Phys., 52, 212 (1970). (IO) D. A. Durden, P. Kebarle, and A. Good, J. Chem. Phys, 50,805 (1969). (11) R. C. Pierce and R. F. Porter, Chem. Phys. Loft.,23, 608 (1973). (12) A. J. Cunningham, J. D. Paysant, and P. Keharle, J. Amer. Chem. Soc., 94, 7627 (1972). (13) M. McFarland, D. L. Albritton, F. C . Fehsenfeld, E. E. Ferguson, and A. C. Schmeltekypf, J. Chem. Phys., 5g96620 ('1973). (14) G. Herzberg, Electronic Spectra of Polyatomic Molecules," Van Nostrand, Princeton, N. J. 1967, p 593.
-
Department of Chemistry The Rockefeller University New York. New York 10021 Received April 11, 1974
M. Meot-Ner d. d. Solomon Ft H. Field H. Gershinowltz"
