G. G. MEISELSAND H. F. TIBBALS
3746 rupted. Because of the very long radiative lifetime of Nz(Aa2),2zreaction 5 would be the rate-determining step of the decay if excitation of Hg(3PJ occurred solely via N2(A3Z). Under favorable circumstances the simultaneous operation of the latter mechanism
Higher Order Ion-Molecule Reactions. I.
along with one or both of the other two could also be identified. Such experiments are in progress. (22) W. Brennen, J . Chem. Phys., 44, 1793 (1966), reports a lifetime of 12 6 2.4 sec.
Theoretical Basis1
by G. G. Meisels and H. F. Tibbals Department of Chemistry, WnCversity of Houston, Houston, Texas
77001 (Received M a y 7, 1968)
Relative ion abundances are calculated on the basis of a collision model for multiple order ion-molecule reactions. Allowance is made for the energy dependency of reaction cross sections and the existence of long-lived intermediates whose dissociation is assumed to follow unimolecular decay kinetics. Cross seetion and rate constant values severely affect calculated abundance curves and permit the assignment of reaction mechanisms.
I. Introduction
tinuous, steady source of reactive intermediates, most of which are translationally in thermal equilibrium with their surroundings. Rates of disappearance and metathesis are determined by the usual competition kinetics, and the steady-state assumption can be applied to a good approximation, provided that allowance for possible spatial inhomogeneity of the primary species formation is made. An entirely different situation may exist in the ion source of the mass spectrometer. When ions are sampled from high-pressure CY radiolysis, they will typically have survived as long or longer than they would under normal radiolysis and photolysis conditions. Therefore, such experiments yield information princi-
A number of investigations of ion-molecule reactions at pressures up to a few torr2-s and even several hundred torrg in the ion source of a mass spectrometer have been reported in the last few years. Interest in the results of such investigations stems largely from their obvious relevance to an understanding of gas phase radiation chemical systems where more than one half of the product formation is usually ascribed to reactions of intermediate ions.lO,ll Moreover, product formation in the far vacuum ultraviolet may also involve ions as intermediates.12 At least two difficulties arise in the application of information obtained by mass spectrometry directly to radiation and photochemical systems. Mass spectro(1) Presented a t the Symposium on Photochemistry and Radiation Chemistry sponsored by the National Academy of Sciences at Natick, metric investigations can only report on the nature and Mass., April 1968. abundance of ionic species, and it is often difficult to (2) (a) C. E. Melton and P. S. Rudolph, J . Chem. Phys., 32, 1128 assign reaction sequences in complex systems where (1960); (b) F. H. Field, J . Amer. Chem. SOC.,83, 1523 (1961). several primary ions undergo ion-molecule reactions.2b)3 (3) S. Wexler and N. Jesse, ibid., 84, 3425 (1962). The nature of the neutral entity concomitant with an (4) M. S. B. Munson, J. L. Franklin, and F. H. Field, ibid., 85, 3576 (1963). ionic product can only be inferred if the precursor ion is (5) G. A. W. Derwish, A. Galli, A. Giardini-Guidoni, and G. G. Volpi, known, and mechanistic uncertainties can sometimes be J.Chem. Phys., 39,1599 (1963). resolved only with considerable difficulties. In radi(6) M. S. B. Munson, J . Amer. Chem. SOC.,87,5313(1965). ation and photochemical experiments, however, the (7) M. Henchman and C. H. Ogle, Discussions Faraday Soc., 39, 63 (1965). precise reverse is the case: the nature of the ionic spe(8) J. H. Futrell and T. 0. Tiernan, J . Phys. Chem., 72, 158 (1968). cies responsible for and products of a neutral forming (9) P. Kebarle and E. W. Godbole, J . Chem. Phys., 39, 1131 (1963). step can only be assessed indirectly. The two tech(10) P. .Ausloos and S. G. Lias, “Gas Phase Radiolysis of Hydrocarniques, although complementary, provide no direct bons,” in “Actions Chimiques et Biologiques des Radiations,” M. Haissinsky, Ed., Masson et Cie, Paris, 1967, Chapter 5. overlap for mutual calibration. (11) G. G. Meisels, “Organic Gases” in “Fundamental Processes in A second difficulty, perhaps somewhat more subtle, Radiation Chemistry,” P. Ausloos, Ed., John Wiley and Sons, Inc., is the difference in the kinetic character of the systems. New York, N. Y., 1968, Chapter 6. I n radiation and photochemistry one provides a con(12) R. D. Doepker and P. Ausloos, J . Chem. Phys., 43,3814 (1966). The Journal of Physical Chemistry
3747
HIGHERORDEF~ ION-MOLECULE REACTIONS pally on the nature of the ion which may be expected to undergo eventual neutralization in the normal radiolytic system. Such experiments can also assist in the assignment of ions resulting from fewer collisions but then pressure must be reduced significantly, or “inert” rare gases are added whose efficiency for deactivation of excited intermediates differs from that of the parent molecule. Most information on the lower order ionmolecule reactions, which presumably give rise to some of the more important relatively simple neutral products, is therefore derived from studies where ion sources are operated in a more or less conventional manner, but at pressures up to ca. 1 torr. Thus, the primary ions are formed in an electron beam ribbon and are continually accelerated by an electrostatic field on their way toward expulsion at the exit slit. Their translational energy therefore is continually increased, and the usual steady-state approximation of thermal homogeneous kinetics is a priori inapplicable because rate constants and residence times are energy dependent. Problems of particular interest in an assignment of mechanisms are the collision cross sections or rate constants for the ion-molecule reactions, the verification of the existence of an intermediate complex, and the assessment of its unimolecular dissociation rate constant. An analysis of this sort was first attempted by Field for the ethylene system,zb and included an estimate of rate constants based on the premise that the reaction sequence in the ion source could be analyzed using the conventional steady-state a~sumption.’~He postulated that all third- and higher order ions were formed by reactions of long-lived intermediate addition complexes such as C4H8+. The rate constant for dissociation of intermediate butene ion derived on this basis was also observed in radiation chemical experim e n t ~vacuum , ~ ~ ~ultraviolet ~ ~ photochemistry,1e and in ion-sampling mass spectrometry, l7 lending credulence to this approach. However, WexlerI8bas pointed out that Field’s data are also consistent with a mechanism where all tertiary ion formation proceeds only via the reactions of the dissociation products of intermediate C4Hs+ such as CaHS+. Wexler was able to account for ion abundance variation using an attenuated beam approach inserting energy independent reaction cross sections. Theoretical consideration^,^^ isotopic labeling,zO~zl the ability to affect branching ratios in the sequence ascribed to reactions of a long-lived intermediate butene ion,21and ion cyclotron double resonance experimentszz have recently provided definitive evidence that at low pressures the mechanism leading to third- and fourthorder ions included reactions of C3H5+only. It is clear that the agreement of the rate constant calculated by the steady-state assumption with that obtained by other essentially non-mass spectrometric methods is specious, and other theoretical methods must be developed. Wexler’s ion beam approach has been modified by Derwish, et aZ.,23to allow for relatively
long-lived intermediate complexes in acetylene ionmolecule reactions. However, these authors assumed energy-independent reaction cross sections and again found it necessary to take recourse to a steady-state assumption. ?tloreover, they treated the disappearance of complex ions by dissociation in terms of cross sections whose relationship with unimolecular dissociation rate constants depends on ion currents. A model based on energy averaging was developed by Lorquet and Hami1lZ4but does not allow for unimolecular dissociation. Hyatt, Dodman, and Henchman developed a model based on residence times in the ion s0urce,~5 but these authors did not make allowance for finite complex lifetimes, and their kinematic treatment does not readily lend itself to such an extension. I n this paper we report the development of a collision theory of higher order ion-molecule reactions based on an examination of simplified classical trajectories. It makes explicit allowance for energy dependency of cross sections and for finite complex lifetimes using firstorder decay kinetics. The sensitivity of calculated ion abundances to unknown reaction parameters is examined.
11. Development of Model A . Physical Model. Our approach is simply one of following the classical, one-dimensional trajectories of individual positive charges and their identity changes in the mass spectrometer ion source. Parent and fragment primary ions are assumed to be formed in an infinitely thin electron beam at Lo (Figure 1). These and product ions are accelerated by a repeller or drawout field toward the exit slit at L, at which there is a sharp boundary beyond which no further reactions occur. The distance L, is subdivided at points where collisions and unimolecular dissociations take place. Reactions a,re assumed to involve the formation of an intermediate complex and therefore to occur as a result (13) F. W. Lampe, J. L. Franklip, and F. H. Field, “Kinetics of Reactions of Ions with Neutral Molecules” in “Progress in Reaction Kinetics,” Vol. 1, G. Porter, Ed., Pergamon Press, London, 1961, p. 67. (14) G. G. Meisels, J . Chem. Phys., 42, 3237 (1965) (15) G. G. Meisels, Advances in Chemistry Series, No. 58, American Chemical Society, Washington, D. C., 1966, p 243. (16) R . Gorden, Jr., and P. Ausloos, J . Chem. Phys., 47, 1799 (1967). (17) P. Kebarle, R. M. Haynes, and S. Searles, ref 15, p 210. (18) S. Wexler and R. Marshall, J . Amer. Chem. Soc., 86,781 (1964). (19) G. G. Meisels and F. H. Tibbals, paper presented a t the 15th Annual Meeting on Mass Spectrometry at Denver Colo., May 1967. (20) J. 0 . Tiernan and J. H. Futrell, J . Phys. Chem., in press. (21) J. J. Myher and A. G. Harrison, Can. J . Chem., 46, 101 (1968). (22) M.T. Bowers, D. D. Elleman, and J. 1,.Beauchamp, J . Chem. Phys., 72,3599 (1968). (23) G. A. W. Derwish, A. Galli, A. Giardini-Guidoni, and G. G. Volpi, J . Amer. Chem. SOC.,87, 1159 (1965). (24) A. J. Lorquet and W. H. Hamill, J . Phys. Chem., 67, 1709 (1963). (25) D. H . Hyatt, E. A. Dodman, and M. J. Henchman, ref 15, p 131.
Volume 79, Number 11
October 1068
3748 PLANE OF FORATION
G. G. RIEISELS AND H. F. TIBBALS COMPLEX FORMATION
COMPLEX DISSOCIATION
FRAGMENT COLLISION
(electron
PLANE OF TERMINATION (exit I
I
‘
I
Fourth order ions
A. General Expressions
I
)I
Third order ions
Second order ions
Sllt)
beam)
I
Primary Ion (First Order)
IFF+
I
!
’IFM
IF+
1
I
+M
f
--+
IFFM+
/-R
7IFM:
L
IFMM+
OIFM
\
’ +M7 IM’
’ I M M IMF++ ~
&IMM+’-R
IMFM+
\
Figure 1. Schematic of collision model and identification of reaction positions.
\
:EM
*
IMMM’
B: Example For Ethylene
of capturing collisions. While a stripping26or pickup27 mechanism can easily be accommodated in this treatment, it need not be considered here since the treatment of Hyatt, et aZ.,25is satisfactory for such cases. We will be dealing with condensation type reactions which must involve an addition complex persisting more than one or two vibrational periods. The effect of the lifetime of such addition complexes on the calculated higher order ion abundances is our primary concern. Consider a primary ion (I+) formed at Loand accelerated a distance L1 where it forms a complex [I-14]+with a neutral species. The sequence may now be distinguished with respect to formation of tertiary ions. In the first, the complex reacts at Lzo before it has an opportunity to dissociate, and forms a tertiary “complex” ion [IMRII]+. In the second, the complex dissociates at L2p to yield a secondary fragment ion [IF+]which reacts at LBto form a tertiary ion [IFM+]. Dissociation of IM+to yield I + will change the energy of I + and secondary I+ is therefore treated as and named IF+. Possible sequences for ions up to fourth order in pressure dependence and the nomenclature for their identification are illustrated in Figure 2. B. Mathematical Formulation. We consider first the fate of the primary ion which can react by collision only. We define an unnormalized probability function Pl(L1, el) as the probability density that a primary ion I + passing through L1 will form a collision complex. €1 is the energy of the ion relative to a neutral reactant in the center of mass system. If one neglects thermal contributions, one can express the relative energy of the reactant pair in terms of the energy E1 of the ion I + in the laboratory coordinate system. Thus, since E1 = FeLl, where F is the field strength and e is the electronic charge, Pl(L1, € 1 ) can be expressed as a function of L1 only
Pl(L1, € 1 ) = Pl(L1) (1) The number of primary ions reacting between L1 and L1 dL1 is
+
-dio(L1) = io(LdPl(LddL1 The Journal of Physical Chemistry
(2)
--
Cal,+.(a)
L Ca,:
Figure 2. Nomenclature for identification of ions, rate constants, and reaction cross sections. (a) These species apparently predominantly redissociate to starting components as shown explicitly for C,H12+.
The dependence of the primary ion current io on L1 arises from the attenuation of the initial ion current lo= io(Lo)formed in the plane of the electron beam. The intensity of ioat L1 can be expressed as the product of loand an attenuation factor AI(L1) by integrating eq 2 between the limits Lo and L1 to give
The translational energies of all ions other than the primary are not only affected by acceleration but are also essentially instantaneously altered at every reaction point. Moreover, t,he identity of the ion may change by j channels of collision and unimolecular dissociation. For any subsequent process, the energy of the ion is therefore known only when the nature and location of each previous step is specified. I n general, if all ions formed at L, in a sequence of n precisely defined steps are considered, the number which pass some subsequent point L,+l will be given by an equation analogous to (3), or (26) A. Henglein, ref 15, p 63. (27) Z. Herman, J. D. Kerstetter, T. L. Rose, and R. Wolfgang, J . Chem. Phye., 46, 2844 (1967).
HIGHER ORDERION-MOLECULE REACTIONS
3749
n+l
dnin,j(L1, Lz,. . . >Ln+1)
=
IO n At(L1, L z ~. . ,, Lt) X i=l
n
n
II Pt,k(Ll, Lz,. . ., Lt) 2II dLi (4) 2=1 -1 where the Pi,k terms represent the specific processes leading to the ion of interest, while the A ifactors apply to all j possible exit channels for the ion. The ith attenuation factor, A,, is obtained by integrating an equation analogous to eq 2 with respect to the variable dLi+l’ over the limits Lt to Li+l, or At(L1, Lz,
,
,
., L , )
=
where the summation is again taken over all possible exit channels €or the ith ion. This expression for ion attenuation differs slightly from that of Wexler and Jesse3 by making explicit allowance for the energy dependence of the exponent and by allowing detailed consideration of unimolecular dissociation. When attenuation is by collision, the probability density P,,j is given by the product of the concentration of neutral reactant species and the cross section for reaction P n , o o l ~ ( L ,Lz,
..
. j
[Mlan(L~,Lz,.
., Ln, Et)
(6)
where E , is an energy not arising from acceleration and can be used to describe possible thermal contributions to the ion energy or excess kinetic energy in the dissociation process. We shall assume at this time that Et is negligible. The acceleration term is dependent on the reaction path and is shown for the first three processes in Table I. For convenience of expression, we have substituted the identity =
Pn,dissoo(Ll,LZ, . . . Ln)
L n - L,-1
=
where t, is the time required for the ion formed at L, to be accelerated to L,+l and k , is the unimolecular dissociation rate constant for the process of interest. t, is determined by the equations of motion of the complex ion formed at L,-I and hence can be readily expressed in terms of distance. For example, the time a secondary complex I M + of mass m, has existed is given by tz(L2 - L1)
4%
=
-
[e
+ -
L1
+ Lz - Ll -
m, Ll]
(9)
The total ion current Inof species n passing through the exit slit at L, can now be obtained by successively substituting attenuation factors and expressions for the precursor ion currents into expression 4 and integrating In(Lz)
Ln) =
I,
When attenuation is by unimolecular dissociation, the probability function Pn,jis given by
=
10J L x Li’=Lo
n+ 1
JLr
Lz’ n
-
L1’
...
x
JL* Ln’ = Ln-
I’
n
II A,(Li’, . . ., Lt’) II Pi,k(Li’, . . ., Lt’) II dL,’ (10) i=l i=l n=l
where L,+1
=
L,.
I,,
The identity
=
Io can be
nd
used to check calculations for self-consistency. Relationships obtained by the substitution into the precursor functions are given below for the formation of the ions In!M + and IFM+ when only one ion of each type is produced in each reaction step and when each ion is formed via a unique sequence of steps.
(7)
A critical distinction between tripping^^,^^ and complex formation is obvious here. For primary ions where dissociation is not in competition with reaction with
Table I : Ion Kinetic Energies Primary ion
El
=
F.11
+
Secondary complex ion E$,= (ml/m,)F.I1 F . 12 Secondary fragment ion E3 = [ma/m,][(ml/mc)F.ll
F.111
+ F.13
+
neutrals, the integral in expression 5 is essentially an energy averaged or macroscopic cross sectionz8which will. change with increasing pressure because the region over which it is averaged will put increased emphasis on lower energies.25
(28) D. A. Kubose and W. H . Hamill, J . Amer. Chem. Soc., 85, 125 (1963).
Volume 72, Number 11
October 1Q68
G. G. h!fEISELS AND H. F. TIBBALS
3750 When a species is the product of more than one reaction sequence, the ion current is calculated by summing all I,, which lead to that species.
111. Assumptions and Approximations Some assumptions and approximations are inherent even in this generalized form. First is the postulate of a collision complex, whose lifetime is of course of primary concern. Second is the neglect of excess kinetic energy in the dissociation coordinate after the fragmentation process. While allowance for this effect is possible in principle, the resultant complex relationship would require an excessive amount of computer time for evaluation since the one-dimensionality of the trajectory would have to be abandoned as an approximation. This should, however, introduce only a minor error since unimolecular reaction rate theory predicts only small if not negligible excess kinetic e n e r g ~ . ~ 9 The randomness of orientation of the dissociation would also tend to decrease a possible effect on further reactions. Third, we have neglected the thermal energies of all species, a common assumption in such treatments. However, the integration procedure requires that the lower limit not be Lo = 0 but have a finite value, and Lo was therefore chosen to correspond to an initial energy of O.O15kT/F. Fourth, we have taken the reaction boundary at the planes of formation and exit to be sharp, while in reality the electron beam has finite width and reactions are not quenched precisely at the exit slit. The effects of the last two approximations are now under investigation, and preliminary results for small variations of Lo have shown little effect on ion abundance. It should perhaps be mentioned that field penetration and inhomogeneity have also been neglected. Returning to the specific reactionships 11 and 12 and their higher order analogs, it is clear that they contain two unknown components: microscopic cross sections and unimolecular dissociation rate constants. I n this communication we use only cross sections of the simple ion-induced dipole sort30
a(E) = S . re(2am1/pE)”~
(13)
where S is an efficiency factor, e is the electronic charge, CY is the polarizability of the neutral molecule, and p is the reduced mass of the collision partners. The use of the efficiency factor is dictated by the known deviation of cross sections from those calculated theoretically. The value of S used in the calculation is the first adjustable parameter. Other forms of the cross section dependency on energy, even including discontinuities, can be substituted readily, but are of lesser interest at this time since we are primarily concerned with the low ion energy range of a few tenths of an electron volt where the approximations inherent in eq 13 should be best applicable. The other adjustable parameters are the rate conT h e Journal of Physical Chemistru
stants for dissociation of intermediate complexes. It is, of course, possible to estimate these using unimolecular dissociation rate theory,31but such an a priori approach does not yield values with sufficient reliability to satisfy our requirements since the final ion distribution curves are highly sensitive to the values of rate constants. There is a further difficulty. The assumption of a long-lived collision complex requires that the translational energy of the collision be tiansformed into internal energy modes, which should affect the value of the rate constant. Therefore, IC should really be expressed as a function of energy or L1, Lp, . . ., L , just like the cross section. At this point, however, it is adequate for a test of the general model to assume a fixed rate constant for the dissociation of a given complex, and thus the applicability of simple unimolecular dissociation kinetics.
IV. Numerical Evaluation Procedure The multiple integrations of equations such as (11) and (12) properly substituted with cross sections from eq 13 cannot be carried out in closed form and were therefore solved on an IBR‘T 7094 computer using FORTRAN IV language. Several numerical integration procedures, 32 including those according to the trapezoidal, Simpson’s and Wedell’s rule, and the RungeKutta method, were attempted. Difficulties were encountered with all these when parameters of interest were varied. Gaussian quadrature was finally adopted as the most appropriate technique for these calculations. In this procedure, the integral of a polynomial of the nth degree or lower can be evaluated exactly by calculation of the integrand at only n values of the variable of integration. The appropriate values of the variables to give such a fit are chosen by this method. An available routine was modified to suit our ~ y s t e m . ~ 3 Sixteen-point gaussian quadrature was found to give satisfactory results for pressures from 0 to 0.2 torr and kd sec-’, but rapid decrease of the exponential functions at higher pressures and larger rate constants required a larger number of points, and up to 32-point gaussian quadrature was used for the range from 0.2 to 1.0 torr and k d = lo8 sec-I. The accuracy of the integration was confirmed by several means. Different gaussian quadrature procedures were employed, the number of points was varied, and the integration was broken up into subintervals with individual evaluation of the regions and subsequent summation. This assured that the integra-
-
(29) C. E. Klots, J.Chem. Phys., 41, 117 (1964). (30) G. Gioumousis and D. P. Stevenson, ibid., 29,294 (1958). (31) M. Vestal, A. L.Wahrhaftig, and W. H. Johnston, ibid., 37, 1276 (1962). (32) J. Todd, Ed., “Survey of Numerical Analysis,” McGraw-Hill Book Co., Inc., New York, N. Y., 1962, p 66. (33) IBM System/360 Scientific Subroutine Package 360A-CM-O3X, Verson 11, 3rd ed, No. H20-0205-2, IBM Technical Publications Department, White Plains, N. Y., 1966, p 96.
HIGHERORDER ION-n/IOLECULE REACTIONS tions converged. Computations were then repeated using the double precision mode, which gave nearly identical results, indicating that the dynamic range of the normal precision mode was sufficient but that errors due to round-off or truncation were limitations. As a further check on such errors, several equivalent programs were written diff ering in the order of calculating factors in the integrand. The maximum differences found by any of the above comparison methods were less than 2% of calculated individual ion currents or 1% of the total ion current, whichever is larger. As a last check, the total ion current obtained by summing all individual ion current fractions was compared to the required sum of 1. The deviation of the calculated sum from unity increased from less than 1% at pressures up to 0.2 torr to a maximum of 5% at 1torr when 24-point gaussian quadrature was employed (Icd = lo7 sec-l). The error obviously increased when a smaller number of points was employed for the evaluation, but also increased with increasing fragmentation rate constant and decreasing field strengths. As indicated above, this can be ascribed to the more rapid decrease of the exponential expressions under these conditions. The total current was always calculated, and the curves were recomputed with a higher order quadrature until the maximus error in I(tota1) did not exceed 37,
V. Results and Discussion The most important aspect of these calculations is the evaluation of the effect of the adjustable parameters on the calculated curves. Our theoretical approach would be without value if the results are not sensitive to the assumed mechanism, to the values of the undefined parameters, if there are too many of them, or if their variation has identical effects on the calculated curves. Not more than four of these parameters enter directly into the evaluation of the variation of ion abundance for any given species. These are the collision cross section for and rate constant of dissociation of the precursor ion, and the rate constant for dissociation and the collision cross section for further reactions of the ion itself. The effect of these parameters is demonstrated best by assuming that a particular ion is not produced simultaneously by reactions of both fragment and precursor ions; that is, we assume that IMF+ and IFM+ are not of identical mle. This is not restrictive since abundances of ions arising by both processes can be obtained by summing the probabilities of alternate reaction paths. Calculated curves can be correlated with experimental results by successive assignment of cross sections and rate constants for first-, second-, third-, etc., order ions, but we shall not attempt such a curve-fitting procedure at this time. Instead we will systematically examine ion abundance curves by varying one factor at a time and keeping all other parameters constant.
3751 The experimental conditions for these calculations were chosen arbitrarily but correspond fairly well to those reported by W e ~ l e ra, ~reaction path length L, = 0.29 em, a field strength of 6 V cm-l (smaller than that of Wexler), and a gas temperature of 200”. The last factor of course enters only negligibly through the initial ion energy used in the integration. The characteristics of the reaction system closely parallel those of the ethylene system, with U I H = 7.6 X loz3E-’/a em2 where E is the energy of the reactant ion in ergs, (TIF = 6.9 X loz3E-”z cm2 and U I F ~=~ U I M M = 8.2 X lozaE-’’a cm2 corresponding to the theoretical cross sections of the ions C4H8+,C3H6+,and CaH9+. Only UI = 7.7 X loz3E-”z cm2 was taken as ca.25% larger than theoretical for ethylene ion as suggested by Wexlere3 No attempt was made to study variations in UI since the magnitude of this cross section is always easily accessible from experiment, and its influence on higher order ion distributions can be predicted qualitatively if the effect of the other factors is understood. One of the most interesting aspects of these calculations is the assessment of the extent to which secondorder intermediate complexes contribute to higher order reactions before they dissociate. The effect of changes in the adjustable parameters on the calculated ion distribution curves is therefore presented at first for the total of all third- and higher order ions arising by reactions of the complex and those formed by collisions of fragment secondary ions. Such ion currents are called unattenuated tertiary ion currents. Variation of the unimolecular dissociation rate constant between lo6 and lo8 sec-’ demonstrates that an almost complete change in mechanism occurs over this range (Figure 3). At the highest value of the rate constant reactions occur virtually exclusively via a fragment ion mechanism, although a small but real contribution of a few per cent by intermediate complex ions is still observable, and increases with increasing pressure, until at 1 torr 11% of the unattenuated tertiary ion current arises from reactions of the complex (Figure 4). At the other extreme, reactions of the fragment ion are still barely noticeable for a rate constant of lo5sec-l but the mechanism is dominated by I M + ion reactions. The decrease of IFM+ with increasing pressure above ca. 0.08 torr can be attributed to the removal of precursor complexes by reaction. It is noteworthy that the value of the rate constant reflects itself not only in the relative contributions of the two mechanisms, but also in the character of the curves for the unattenuated tertiary ion abundances. The current fraction of IMM+ arising from reactions of an undissociated complex shows no maximum but a steady increase with pressure, approaching unity asymptotically at a pressure depending on the rate constant. The current I F M +, however, has a pronounced maximum whose value depends on the rate constant, but whose position is apparently always at very nearly the same pressure. Volume 79, Number 11 October 1868
3752
G. G. MEISFLSAND H. F. TIBBALS
f , 8
f
J
D
Y C
Figure 5 . Dependence of secondary and unattenuated tertiary ion currents on the efficiency factor for reactions of IM+ ( ~ I X= lo7sec-l): A, S = I; B, S = 0.5; C, S = 0.1.
Figure 3. Dependence of secondary and unattenuated tertiary = lo5 sec-l; ion currents on the value of ,h:A, B, k13r = lo8 sec-'; C, ~ I J I= lo7 sec-'; D, k ~ =~ lo8 t sec-l. All cross sections and conditions are listed in the text. 1.0,
I
I
I
Torr
Figure 4. Variation with pressure of the fractional contribution of reactions of IF + and I M + to the higher order ion currents.
The maximum disappears when virtually all reactions occur via fragments (Ic~M= 108 sec-I). The rate constant for dissociation of the secondary complex also has a considerable effect on the relative abundances of the secondary ion currents. However, it is important to recall here that the ion abundances The Journal of Physical Chemistry
calculated are those emerging from the exit slit, and ions of the type I M f can undergo fragmentation also before entering the analyzing magnetic field, and thus give rise to metastable peaks. The calculated secondary complex ion current I M + therefore includes the metastable ion abundances corresponding to its dissociation. Both will be negligible for rate constants in excess of ca. lo7 sec-'. Variation of the efficiency of further reaction of the intermediate complex I M + by an order of magnitude does not show a considerable effect on the abundance of the secondary ion current for a rate constant of lo7 sec-l (Figure 5 ) . This is to be expected since at low pressures where the observable current IF+ is most abundant, IM + disappears chiefly by dissociation, and only at higher pressures does collisional removal from the system become significant. Lowering of the cross section VIM reflects itself much like an increase in the rate constant for these ions. It is important to note that translation into physical observabilities may be quite complicated in this example. Since we are only dealing with capturing collisions, complex formation must occur with collision efficiency and values of S less than unity correspond to return of the collision partners to their original states, almost certainly accompanied by some exchange of energy between internal and translational modes. It is therefore highly probable that collisions will at least reduce the rate constant for dissociation, or even lead to stabilization of the secondary complex. They would thus not change the probability of fragmentation from that calculated using the theoretical UIR.I since the intermediate I M f would be deactivated even when IMMf is produced with less than collision efficiency. Variation of u I F , the efficiency of reaction of the secondary fragment ion, has a pronounced effect on the abundance of both the secondary and tertiary fragment
3753
HIGHERORDERION-MOLECULE REACTIOXS
Figure 6. Dependence of secondary and unattenuated tertiary ion currents on the efficiency factor for reactions of IF+ ( ~ I X= lO7sec-l): A, S = 1; B, S = 0.5; C, S = 0.1.
Figure 7. Variation of attenuated tertiary ion currents with pressure for several efficiencies of their future reactions: A, S = 1.0; B, S = 0.5; C, S = 0.1
ions IF+ and I F M + as expected. A decrease in rate constant for this reaction increases the highest value the secondary current I I F can obtain and shifts the position of the maximum to higher pressures. At the same time the contribution of the fragment ion to tertiary ion formation is reduced, the maximum abundance of I F M + is lowered and shifted to higher values of pressure. Thus a decrease in UIF increases the pro+ all pressures as the behavior of the portion of I I M F at latter process remains constant and the removal of precursor complex ions increases (Figure 6). The tertiary ions can also undergo further reactions. Calculated tertiary ion currents allowing for fourthorder processes are shown in Figure 7. When such reactions occur with collision efficiency, the tertiary ion current vanishes at about 0.15 torr. The value of the maximum in the observed abundance of IFM+ will thus be indicative of both kIM and UIF. We have not at this time allowed for disappearance of I F M + by unimolecular dissociation. It is interesting to analyze now what one can deduce from the behavior of the primary and secondary ion currents alone. Observation of an ion current IF+ in the absence of IM+ and any metastable corresponding to its formation indicates a rate constant for dissociation of I M + greater than lo7 sec-'. From a semilogarithmic plot of the primary ion abundance vs. pressure one can assess uI, and from the initial portion of the secondary ion current one can obtain the fractional probability of formation of a given secondary. Assuming that the rate constant is lo7sec-' or greater, the efficiency of reactions of IF+ can be estimated by the value of the maximum in I I F and the pressure at which it occurs. The actual cross section can be deduced with any degree of precision only when kdiss is so great (ca. los sec-l or more) that attenuation of IF+ by reaction
of its precursor I M + is not in competition with fragmentation. A detailed comparison of calculated values with those reported for the ethylene,4,18t20,21 acetylene,23 and methanol6rZssystems will be given elsewhere.
VI. Significance to Radiation Chemistry The occasional practice of extrapolating mass spectrometric evidence over orders of magnitude in pressure to radiation chemical systems ignores the possible considerable lifetime of intermediate collision complexes. While direct evaluation of reactions of ion-molecule complexes by conventional mass spectrometry is limited to values of up to ca. lo7see-', only ca. 1% of the intermediates will dissociate before collision at a pressure of ca. 400 torr, typical of many radiation chemical investigations. Care must therefore be exercised in the correlation of mass spectrometric and radiation chemical results, and overlap of pressures between studies using charged particle and neutral analysis is highly desirable.
Acknowledgment. This investigation was supported in its initial phases by the United States Atomic Energy Commission under Contract AT(40-1)-3606, and subsequently by the National Science Foundation. The IBM 7094 computer employed in these calculations was the facility of the Texas Medical Center, which is supported by U.S.P.H.S. Grant No. FR 00254. It was made available to us as a grantee of the Robert A. Welch Foundation of Texas. We are deeply grateful to these agencies for support of this investigation. We are also indebted to Mr. Michael Raines, undergraduate research participant from Midwestern University, Wichita Falls, Texas, for programming some of the quadrature procedures, and to the referee for his exceptionally careful and constructive comments.
Volume 72. Number 11 Octobe'r 196'8
