4338
RODNEY L. LEROY
Relationships between Arrhenius Activation Energies and Excitation Functions
by Rodney L. LeRoy’ Department of Chemistry, University of Colorado, Boulder, Colorado 80802
(Received J u n e 90,1969)
Analytical expressions are derived which relate the ilrrhenius activation energy to the energy dependent reaction cross section or excitationfunction. Results are presented for reactions which proceed with and without a threshold energy. It is shown that the activation energy for neutral-neutral reactions may display a strong temperature dependence, and that activation energies determined in thermal studies of ion-molecule reactions can commonly be expected to be either positive or negative, and strongly temperature dependent. Particular attention is given to recent rate measurements for the exchange reactions of atomic hydrogen with molecular hydrogen, and an attempt is made to derive excitationfunctionsfrom these results. Three general formulations of the excitation function are used. These reflect the forms which have been suggested by experiments and by theoretical calculations. In particular one class of functions is found to be able to accurately reproduce the available data on the energy dependence of the reaction cross section for neutral-neutral reactions, and representative data for ion-impact induced fragmentation reactions. A second class of functions is introduced to include the special case of hard spheres which require a critical energy along the line of centers to react, while a final class adequately reproduces data for many ion-molecule processes.
Introduction Much of the interesting information about an elementary chemical reaction can be summarized in its excitation function or energy dependent reaction cross section. Considerable progress is now being made towards devising “non-classical” experiments which will allow such excitation functions to be determined directly. -5 Also, calculations performed by Monte Carlo averaging over quasi-classical trajectories have made it possible to predict excitation functions for some simple systems.6” All of the results which have been obtained for neutral reactions suggest that the excitation function rises from zero a t some critical value of the relative translational energy of the reagents-the threshold energy. This function then passes through a maximum, and decreases towards zero again as the energy is further increased.436,’ Also it is well establisheds that the excitation function for most ion-molecule processes is a monotonically decreasing function of energy. Menzinger and Wolfgangghave recently examined the relationship between the Arrhenius activation energy, E,,,, and the more fundamental excitation function for reaction. I n the present paper we extend their results. Three general analytical formulations of the energy dependence of the reaction cross section are suggested, and E,,, and the bimolecular rate constant k ( T ) are derived for each of these.
Theory The Arrhenius activation energy is defined by the relation T h e Journal of Physical Chemistry
E,,,
=
RT2
d In k ( T ) dT
where T is the absolute temperature of the reaction mixture, R is the universal gas constant, and k(T) is the bimolecular rate constant, given by 9t1O
Here E and p are the relative translational energy and the reduced mass respectively of the collision partners, and a@) is the excitation function for reaction.“ (1) Address requests for reprints to Department of Chemistry, Yale University, New Haven, Connecticut 06520. (2) (a) M. Menringer and R. Wolfgang, J. Chem. Phya., 50, 2991 (1969): (b) E.F. Greene and J. Ross, Science, 159,587 (1968). (3) A. Kuppermann and J. M. White, J . Chem. Phys., 44, 4352 (1966). (4) R. Wolfgang, in “Progress in Reaction Kinetics,” Vol. 3, G. Porter, Ed., Pergamon Press, New York, 1965,p 97; Ann. Rev. Phys. Chem., 16,15 (1965). ( 5 ) “Ion-Molecule Reactions in the Gas Phase,” R. F. Gould, Ed., Vol. 58,American Chemical Society Publications, Washington, D. C., 1966. (8) LM.Karplus, R. N. Porter, and R. D. Sharma, J . Chem. Phys., 43, 3259 (1965). (7) XI. Karplus, R. N. Porter, and R. D. Sharma, J. Chem. Phyls., 45, 3871 (1966). (8) See, for example, C. F. Gieseand W. B. Maier, J. Chem. Phya., 39, 739 (1963). (9) M. Menzinger and R. Wolfgang, Angew. Chem., 8,438 (1969). (10) M. Eliason and J. 0. Hirschfelder, J . Chem. Phys., 30, 1426 (1959). (11) Note that we do not significantly restrict our considerations by treating only bimolecular processes. This is because the Arrhenius activation energy for a chemical reaction generally reflects the temperature dependence of the rate of an elementary collisional activation step, whatever the overall reaction order.$
4339
ARRHENIUS ACTIVATION ENERGIES AND EXCITATION FUNCTIONS
1 3 N
5
- 2 W
c
b
I
0
8
4
12
16
E(eV)Figure 2. Excitation functions for the displacement reaction of of atomic tritium with cyclohexane. The data is taken from Menzinger and Wolfgang,as who derived it by assuming that the total reaction probability is 3.5 times the probability of reaction to form c-CeHltT. The different curves correspond to different values of the energy loss parameter p. Parameters of the class I functions drawn through the data are recorded in Table I.
Figure 1. Excitation functions for the reactions of atomic tritium with molecular hydrogen and deuterium. The points were obtained by Monte Carlo averaging over quasi-classical trajectories,' i13 and the curves through the data correspond to the best least-squares fits of class I excitation functions. The lower three pointe, for the exchange reaction with hydrogen and the lower two points for the exchange reaction with deuterium (see insert), were omitted in the fitting procedure. The derived class I parameters are recorded in Table I.
Equations 1 and 2 are evaluated below for three general classes of functions. These can be used to describe most of the forins of the excitation function which have been postulated. Class I .
c(E -
(E
z Eo) (3)
and a(E) = 0
( E < Eo)
where m,n 3 0. These functions increase from0 at E = Eo. The rising portion is concave upward when n is greater than 1, and is convex upward when n has a value between 0 and 1. The exponential term causes the excitation function to pass through a maximurnl2 as the energy E increases, and then to decrease at a rate determined by the parameter m. I n the simple case where n = m = 0, u(E) becomes a step function. The best fits of class I functions to some experimental and theoretical excitation functions for neutral-neutral
t/ LA
0
I
2
I
4
I
6
I
a
E ( k cal /mole)--c Figure 3. Excitation function for the reaction of potassium with methyl iodide.2b The circled points were deduced from measurements of elastically scattered potassium, and the squared point from measurements of reactively scattered potassium iodide.
reactions are presented in Figures 1to 3. The excitation functions of Figure 1 for the reactions of atomic tritium with hydrogen and deuterium were obtained from trajectory calculation^.^^^^ The fit seems quite good, except for the exchange reactions in the vicinity of the reaction threshold (insert, Figure 1). Although the number of calculated trajectories in this region may be too small for the Monte Carlo averages to be reliable,14they indicate structure which class I functions can not reproduce. Such structure is negligible when compared with the (12) The maximum of class I excitation functions occurs at E =
+ n/m.
Eo
(13) R. N. Porter kindly provided the original quasi-classical results which were used in plotting u ( E ) for the exchange reactions treated in
ref 7. (14) R. N. Porter, private communication.
Volume 78,Number 12 December 1969
4340
RODNEY L. LEROY
Table I : Parameters of Class I Functions Plotted in Figures 1 and 2
I2
Figure
0, eV-n
n
m, eV-1
Eo, eV
1
2.55
0.57
0.21
0.33
1
1.90
0.58
0.22
0.40
1
0.263
0.59
0.018
6.07
1
0.291
1.60
0.074
4.59
2
1.14
0.92
0.085
1.88
2
0.74
0.80
0.102
1.94
2
0.13
1.28
0.188
1.51
can quite satisfactorily describe the energy dependence of the available excitation functions for neutral reactions. In Figure 4 (bottom curve) we show that this class of functions can give an equally good description of the excitation function for an ion-impact induced dissociation reaction. The illustrated data were obtained by Vance and Bailey15for the impact of molecular hydrogen ions on hydrogen molecules. I n addition eq 3 has been fitted with comparable success to the numerous excitation functions reported by 31aierl6 for analogous processes. Substituting u ( E ) from eq 3 into eq 1 and 2 gives analytical expressions for the bimolecular rate constant k ( T )and for Eexp
r(n overall excitation function, but it could have a significant effect on the low temperature thermal rate constants for these reactions. Figure 2 presents three of the excitation functions which Menzinger and Wolfgang derived from an analysis of their beam studies of the hot atom reaction of atomic tritium with solid cyclohexane.2a The curves correspond to three different values of the energy loss parameter p . Parameters of the excitation functions which were used in fitting curves to the data of Figures 1 and 2 are listed in Table I. The excitation function for the reaction of potassium with methyl iodide is plotted in Figure 3. These data were obtained by Greene and Rosszb in a molecular beam study of the angular distribution of elastically scattered potassium, and of reactively scattered potassium iodide. It is evident from Figures 1 to 3 that class I functions
+ 1)(1+ mRT)Eo/RT] (4)
and
+ RT(n +
E,,, = EO
- m(n
+
2 ) ( R T ) 21 f mRT
1/2)
Eo
(n
+
+ 1) + (1 + mRT)Eo/RT
(5)
+
where r ( n 1) and r ( n 2 ) are gamma function^.^' When m is set equal to zero, eq 5 becomes equivalent to the expression quoted by Menzinger and Wolfgang.9
E,,,
=
Eo
+ RT [I
+
(n Eol ) R T
I-')
(6)
Class I I
and
a(E) = 0 H i + H2 CHARGE TRANSFER
i 0
,
o
0
These functions are similar in form to those of class I. However they are introduced here in order to include the well known excitation function for the collision of hard spheres which require a critical relative translational energy EO,measured along their line of centers, in order to react.l0 The form of eq 7 for this special case is
40
(1 - Eo/E)
(E
3 Eo)
(8)
and 0
IO
20
E(eV)--.
30
40
so
Figure 4. Excitation functions for the inelastic scattering of H z + on molecular hydrogen16. T h e cross section for ion impact induced dissociation (bottom curve) is reproduced by a class I function, while that for charge exchange (top curve) is well represented by a class I11 function. The Journal of Physical Chemistry
a(E) = 0
( E < Eo)
(15) D.W.Vance and T. L. Bailey, J . Chem. Phys., 44,486 (1966). (16) W.B.Maier,ibid., 41,2174 (1964). (17) See for example the tables of r ( x ) in "Handbook of Mathematical Functions," M. Abramowita and I. A. Stegun, Ed,, Dover Publications, Inc., New York, N.Y.,1965,p 267.
4341
ARRHENIUS ACTIVATION ENERGIES AND EXCITATION FUNCTIONS Substitution of the general class 11 function into eq 2 yields
+ l)e-Eo/RT + mRT)"+'
'Iz (RT)"-lr(n
(1
For the hard sphere excitation function of eq 8 ( n = 1, = 0 ) ,eq 10 reduces to the well known result10~18
+
EO R T / 2
(11)
It is also interesting to note that if the excitation function has the unlikely form u ( E ) a ( E - Eo)"'/E
(E
> Eo)
and
m
=
u ( E ) = CE" ( E
(9)
and the activation energy assumes a particularly simple form
Eexp
ergy Eo. This latter observation was made by Karplus, Porter, and Sharma.6 Class KII
> Eo)
( E < Eo)
(13) The best known example of this type of function is the Langevin cross s e ~ t i o n , 'u(E) ~ a E-''a, which applies for close collisions between low energy ions and polarizable molecules. Typical fits of class 111 functions to measured cross sections for ion-molecule processes are illustrated in Figure 4 (top curve) and Figure 5. For these functions, solution of eq 1 and 2 gives a(E) = 0
k(T) = C
CPTY -
+
(RT)"[I'(n 2 ) -
and
( E < Eo)
u(E)= 0
and
(12)
Eexp
=
RT(n
+
'/z)
+
then the experimental activation energy calculated from eq 10 will be identically equal to the threshold en-
+
Hi+ H 2 4 H3+ + H o G I E S E - M A I E R ( R E F 8) o V A N C E - B A I L E Y ( R E F 15)
where r ( n 2,Eo/RT)is an incomplete gamma function.20,21 For integer and half-integer values of n, Eexpis expressed in terms of analytical and tabulated functions in the Appendix. We only record here the expression which applies when a(E) is the Langevin cross section ( n = --I/') for an endoergic process
Eexp = Eo
(Eo/RT )'/2e-Eo/RT (7r) 'I2
-[1 2 Q(
+
- erf(Eo/RT)'/'] (Eo/RT)'/ze-Eo/RT
E ) = 89.1
(16) Reaction with No Threshold Energy For reactions which occur with no threshold energy, the class I and I1 excitation functions become u ( E ) = CEne-mE
(17)
Examples of this form will be discussed below. When the excitation function is given by eq 17 E,,, is nonzero, having the form
where n and m are greater than zero.
Figure 5. Excitation function for the ion-molecule reaction of Data of both Giese and Maiers and Vance and Bailey16 are plotted, and all the data are well represented by a class I11 function.
Ht+ with molecular hydrogen.
(18) W. J. Moore, "Physical Chemistry," 3rd ed., Prentice-Hall, Inc., Englewood Cliffs, N. J., 1962, p 278. (19) P. Langevin, Ann. Chim. Phys., 5, 245 (1906); G. Gioumousis and D. P. Stevenson, J. Chem. Phgs., 29,294 (1958). (20) The incomplete gamma function is defined and plotted on pages 260 and 261 of ref 17, and is discussed more fully in ref 21. (21) F. G. Tricomi, Ann. Mat., 31,263 (1950). Volume 73. Number 12 December 1960
4342
RODNEY L. LEROY
Discussion The results presented above offer several formal relationships between the Arrhenius activation energy and the excitation function for reaction. It should be emphasized, however, that by making use of eq 2 we assume that the velocities of the reactant molecules are distributed according to the Maxwell-Boltzmann function. In fact this will only be true if u ( E ) (and thus the reaction rate) is small enough that collisions can maintain an equilibrium distribution of translational energies. Menzinger and Wolfgang9 have discussed some of the consequences of a departure from equilibrium. Secondly, it is important to realize that the excitation function is different for each vibrational and rotational state of the reagent molecules.6 Thus the overall excitation function for a reaction represents an appropriate temperature dependent average over all possible microscopic processes. Assuming functional forms such as those given in eq 3, 7, and 13 t o be temperature independent is an approximation. In spite of these limitations, it is instructive to examine the variation of E,,, with temperature. Esxp/ Eo is plotted against RT/Eo in Figure 6a for three class I functions which correspond to reasonable (see Figures '21
rn = 0.05/E0 n = 0.5
1 to 3) values of the parameters n and m. The activation energy shows a strong temperature dependence in each case. At very high temperatures, the activation energy calculated for class I and I1 functions will always become negative, provided rn is not equal to zero. This is because the average translational energy will, in the limit, exceed the average energy at which reaction occurs. This is particularly obvious if m is sufficiently large that the energy dependence of the reaction cross section approaches a delta function. Then the values of E*,, predicted by eq 5 and 10 both approach
E,,,
=
Eo - 3RT/2
(19) Most neutral reactions are studied at temperatures which are low enough that R T
