J . Phys. Chem. 1989, 93, 3 145-3 151 react on the walls to form sufficient I, to double k'. This seems unlikely because the ratio of alkali-metal iodide to 0, entering the reactor is probably less than 0.2%. Furthermore, in our determination of k2 we also used LiOH and LiO, as photolytic precursors of Li atoms and measured k2 within 30% of the rate constant measured using LiI.9 Allowing for the possibility that 30% of k' in decays such as that pictured in Figure 1 is due to the reactions between akali-metal atoms and I, yields the marginally lower limits Do(Li-0,) L 306 kJ mol-' and Do(Na-02) 1 202 kJ mol-' which we adopt as our final lower limits.
Discussion We have thus obtained estimates of the bond energies of Li02 and NaO, from ab initio calculations and their lower limits from experiment. When attempting to compare between theory and experiment, we make reference to the work of Bauschlicher and L a n g h ~ f f ,who ~ ~ ,have ~ carried out an extensive intercomparison of the ab initio and experimental bond energies for most of the alkali-metal monoxides and hydroxides. Those a u t h o r ~con~~,~ clude that agreement within 5% can generally be achieved when the ab initio calculations use a sufficiently flexible basis set. In the case of our study of Li02, the ab initio value and the experimental lower limit differ by only 10 kJ mol-', or 3%. We believe that this indicates that the experimental lower limit must be close to the true value and recommend a value of Do(Li-0,) = 306 kJ mol-'. This value is substantially higher (40%) than the current value of 222 f 25 kJ mol-' obtained from flame measurementsI6 and theory.', These flame experiments by Dougherty et a1.l6 probably need to be reanalyzed now that k,(Li + 0, + N,) has been determined directlyg and found to be 2-3 orders of magnitude faster than was assumed16 by analogy with O2 + Nz. The semiempirical calculation of the reaction H Alexanderi2 is based on the ab initio study of LiO, by Grow and Pitzer.ls They15 employed a contracted gaussian basis set with
+
~~
(40) Bauschlicher, C. W.; Langhoff, S. R. J . Chem. Phys. 1988,88,6431.
3145
polarization functions which led to a caculated (0-0)- bond distance which was significantly smaller than experiment,' as found in the present study with the 6-31G* basis set. We consider that the excellent agreement in this work between the ab initio and experimental geometries (Table I), and the convergence in Do(Li-0,) between basis sets (Table IV), has produced a more accurate calculation of the bond energy. In the case of NaO,, the agreement between the ab initio and experimental determinations is less satisfactory, though with a difference of only 17 kJ mol-' (9%). This is illustrated in Figure 1, where a broken line indicates the decay that would have been observed if reaction 1 approached equilibrium with Do(Na-0,) = 185 kJ mol-' and assuming an absence of I2 in the reactor and no other removal of NaO, besides dissociation (reaction -1). The observed LIF decay in Figure 1 clearly does not support the theoretical value of Do(Na-0,) = 185 kJ mol-'. However, as discussed above, the theoretical bond energy decreases by 14 kJ mol-' on going from the 6-31G to the more flexible 6-31 1G basis set. A tight convergence between the basis sets, as seen in the case of LiO,, has not been reached and the theoretical Do(Na-02) can be expected to change further on increasing the flexibility of the basis set, although the direction is difficult to predict. We therefore recommend Do(Na-0,) = 202 kJ mol-'. This value is close to the molecular beam study of Figger et al.,l However, it is significantly higher than the values of the bond energy obtained in the recent flame studies of Jensenigand Hynes et al.,,O by 19% and 39%, respectively, indicating that the chemistry of the alkali metals in oxygen-rich flames is still not completely understood.
Acknowledgment. This work was supported under Grant ATM-8616338 from the National Science Foundation. We thank F. Miller0 for supporting the purchase and running costs of the Gaussian 86 program and S. Mroueh for assisting with the calculations. Registry No. Li02, 12136-56-0; N a 0 2 , 12034-12-7.
New Approach to Weak-Collision Factor ,8 in Thermal Unimolecular Reactions Wendell F o r d DPpartement de Chimie Physique des RPactions, UA 328 CNRS, INPL-ENSIC et UniuersitP de Nancy I , I Rue Granduille, 54042 Nancy Cedex, France (Received: July 28, 1988: In Final Form: October 10, 1988)
A new expression for the weak-collision correction factor @, suitable for rapid computation, is proposed, based on a matrix approach and related to the energy diffusion version of the Fokker-Planck equation (eq 19 in the text). It makes use of a transition matrix constructed for a specified transition probability model but does not require the actual solution of the master equation; thus the number of matrix operations is minimized, with consequent saving in machine time. This approach is useful for a closer identification of the weak-collision falloff with a particular transition probability model. The new approach is applied to the decomposition of ethane as test case and compared with results obtained by actual solution of the weak-collision master equation and with other definitions of @.
1. Introduction The pressure dependence of k , , ~the , unimolecular rate constant for dissociation at some finite pressure in a gas-phase thermal system, gives rise to the well-known "falloff" that generally serves for the comparison of theory with experiment. It is usually represented by the well-known RRKM form of kd [see eq 7 below and accompanying text] which incorporates the assumption that collisions are "strong", Le., that every collision deactivates an energized molecule with unit efficiency. It is agreed that the strong-collision version of kuniis apt to be poor in many instances,
but it has the virtue of making the falloff calculations relatively quick and easy. This is a nontrivial consideration if large amounts of data on many different molecular systems are to be treated, considering that full treatment of "weak" collisions, while feasible, is more demanding. It is therefore not surprising that over the years there have been many proposals for shortcuts that would conserve the simplicity of the strong-collision calculation and at the same time correct, at least approximately, for weak-collision effects. While they may differ in detail, in essence these proposals'-' boil down to mul-
Present address: Laboratoire de Physicochimie ThBorique, UA 503 CNRS, Universite de Bordeaux I, 33405 Talence Cedex, France.
(1) Tardy, D. C.; Rabinovitch, B. S. J . Chem. Phys. 1966, 45, 3720; J. Chem. Phys. 1968, 48, 1282.
0022-3654/89/2093-3 l45$01.50/0
0 1989 American Chemical Society
Forst
3146 The Journal of Physical Chemistry, Vol. 93, No. 8, 1989 tiplying the hard-sphere collision frequency (which serves as basis for the strong-collision assumption) by a weak-collision correction factor 0,also referred to as collision efficiency. The reason for adopting the expedient of a multiplicative correction factor has to do with the practical aspect of the nature of the usual experimental falloff curve: it is a log k,,, vs log pressure (or concentration) plot. This double log representation has the effect of making the experimental (or calculational) error in k,,, less apparent; in addition, kUniis often normalized to unity by dividing it by k,, the limiting high-pressure rate constant, so that only the log of the relative rate constant is represented. As a result, when it comes to calculating the falloff, only log (k,Jk,) is required, which means that, for the purpose of comparison with experiment, only modest accuracy in the calculated kuniand k , is needed; at the same time, the log pressure scale also places only modest premium on the accuracy of the pressure dependence. Thus recognizing that in principle @ should be an energy-dependent parameter, there is no compelling reason why a single constant properly chosen 0could not be used for calculating an approximate, but for the purpose of comparison with experimental results sufficiently accurate, falloff curve in the double logarithmic representation. The aim of this work is to propose a procedure for calculating the falloff of kuniin a weak-collision system that makes use of the concept of 0as a simple multiplicative correction factor, but at the same time preserves a realistic representation of the weakcollision transition probability. It does not produce an analytic formula for (3 and is therefore computationally more demanding than a simple RRKM formulation, but the effort remains quite small compared to a full weak-collision calculation. It will be assumed throughout that only vibrational energy is involved; Le., rotational effects are ignored. The paper is organized as follows: section 2 sets out the theory behind the proposed formula for 3( (eq 19); section 3 describes briefly the calculations used to produce exact values of k,,,, which are then used in section 4 to check on the performance of 0.The results and their limitation are discussed in section 5, and the conclusions are summarized in section 6.
2. Theory The time history of collisional events at some arbitrary pressure in a thermal system, assumed to be in contact with a constanttemperature heat bath, is commonly described by the Pauli master equation.8 Since rate constants in general, and k , in~ particular, are steady-state properties, we need be concerned only with the steady-state version of the master equation which can be written, assuming a quasi-continuum of energy levels, in the form
where kuniis the unimolecular rate constant for reaction defined as kuni= -[A(t)]-’ d[A(r)]/dt, where [A(t)] is the total concentration of reactant species A of all energies at time t , c(x) is the steady-state fractional population/unit energy of reactant molecules having (vibrational) energy x, w is the collision frequency (s-I), assumed to be energy-independent, q(x,y) is the probability, y in a per unit energy and per collision, of the transition x
collision, and k(x) is the microcanonical rate constant (s-l) for reaction of molecules having energy x. There will be a threshold energy x = Eo below which k(x) = 0. The transition probability q(x,y) in eq 1 is normalized Jmq(x,y) dx = 1
for all y
and satisfies detailed balance q(x,y) N(Y) exp(-y/kT) = q(Y,x) N(x) exp(-x/kT)
(3)
where N(x) is the density of states of reactant at energy x, k is Boltzmann’s constant, and T is heat bath temperature. The solution of eq 1 can be written formally as , k u n i = L0c(x)ss k(x) dx (4) where we have added the subscript “SS” to emphasize that c(x) is the steady-state distribution, normalized to unity. The explicit form of c(x), will depend on the nature of the transition probability dX,Y). A particularly simple form of q(x,y) is the strong-collision transition probability given by q k y ) = c(x),,
where c(xLq = N(x) exp(-x/kir)/Q
(5)
Q is the partition function and c(x), is the normalized equilibrium Boltzmann distribution. Equation 5 asserts that every strong collision leaves the molecule in equilibrium with the heat bath at T, regardless of initial energy; in other words, the strong-collision q(x,y) is given by the Boltzmann distribution of final energies appropriate to heat bath temperature. In order to distinguish rate constants based on the strong-collision transition probability, rate constants calculated from an arbitrary form of q(x,y) will be called henceforth “weak-collision” (subscript wc) rate constants; those calculated from q(x,y) of eq 5 will be called “strong-collision” (subscript sc) rate constants. Equation 1 is readily solved for the strong-collision probability of eq 5 with the result9
The traditional RRKM form of kuni,scconsists of assuming that kuni,scis “small”, Le., that kuni,sccan be neglected with respect to w k(x) in the denominator on the rhs of eq 6. This is justified in most cases except for large molecules with low activation energies at elevated temperature^;^ in particular, this neglect causes negligible error in the case of the decomposition of ethane at 2000 K discussed below. Thus in eq 4
+
c(x)s,(RRKM) = wc(x)eq/[o
+W)l
(7)
Equation 1 simplifies when pressure becomes so low that collisional energy transfer becomes rate determining; then every molecule reaching Eo is guaranteed to dissociate; Le., the energy level at Eo acts as the perfect sink. If we call the rate constant for dissociation under these conditions ko, eq 1 reads
+-
(2) Robinson, P. J.; Holbrook, K. Unimolecular Reactions; Wiley-Interscience: London, 1972; p 322. (3) Forst, W. Theory of Unimolecular~Reactions;Academic Press: New York, 4973; p 185 ff. (4) Troe, J . Ber. Bunsen-Ges. Phys. Chem. 1973, 77, 665. Quack, M.; Troe, J.; In Gas Kinetics and Energy Transfer; Ashmore, P. G., Donovan, R. J., Eds.; Specialist Periodical Report; Chemical Society: London, 1977; Vol. 2. Gilbert, R. G.; Luther, K.; Troe, J. Ber. Bunsen-Ges. Phys. Chem. 1983, 87, 169. (5) Snider, N . (a) J . Chem. Phys. 1982, 77, 789. (b) Snider, N . J . Chem. Phys. 1983, 78, 6030. (c) Snider, N . J . Phys. Chem. 1986, 90,4366. (6) Nordholm, S. Chem. Phys. 1978, 29, 5 5 . (7) Schranz, H. W.; Nordholm, S. Chem. Phys. 1983, 74, 365. (8) Oppenheim, I.; Shuler, K. E.; Weiss, G. H . Stochastic Processes in Chemical Physics: The Master Equation; MIT Press: Cambridge, MA, 1977. Van Kampen, N . G. Stochastic Processes in Chemistry and Physics; NorthHolland: Amsterdam, 1981.
which in the case of the strong-collision transition probability (eq 5) reduces to
= (w/Q)],N(x) Eo
exp(-x/kT) dx
(9)
The solution of eq 1 for an arbitrary form of q(x,y) is considerably more difficult. While numerical solutions are available?*I0in the present context we look for a simplified approximate (9) Nordholm, S.; Schranz, H. W . Chem. Phys. 1981, 6 2 , 459
P in Thermal Unimolecular Reactions
The Journal of Physical Chemistry, Vol. 93, No. 8, 1989 3147
solution. Comparing eq 6 and 9 we see that if we wish to replace w by fiw, it is only in ko that /3 appears directly as a simple multiplicative factor. It is therefore sufficient to seek a weakcollision solution for ko of eq 8. A useful starting point is the Kramers-Moyal expansion*of the time-dependent master equation in moments of q(x,y). If the resulting series is broken off after the second moment, such approximation is known as the Fokker-Planck equation**I'which, as shown by Keck and Carrier,12 can be transformed into an energy diffusion equation with the )), (aE2Cy))is the diffusion coefficient Ob) = 1 / 2 w ( A E 2 b where second moment of q(x,y) defined as
e
Here ( x - y) = AE is the energy transferred in a collision. This diffusion approach is based on the assumption that q(x,y) is a sharply peaked function at x = y, so that its first (or any odd) moment is small with respect to the second. On the diffusion model, the steady-state solution for ko,wcis given approximately byI3
which is essentially the inverse of a mean first-passage formula. Equation 1 1 applies to a quasi-diatomic [ N b ) = const] and anticipates that the diffusion coefficient [the second moment, eq 101 will be, in general, energy-dependent. A simplification of eq 1 1 therefore results if the second moment is replaced by an energy-independent quantity which is then taken outside the integral. Considering that only steady-state properties of the system are involved, this can be accomplished approximately by replacing the second moment by its steady-state bulk average, defined by
A further simplification can be made by observing that, for large Eo,the main contribution to the integral in eq 11 will come from energies in the neighborhood of Eo; thus EOexp(Y/kT) dy
k T exp(Eo/kT) %
NEO)
-32
h
I
I
I
0
0.5
1
1.5
1
x/E, Pressure dependence of the fractional deviation of the steady-statedistribution c ( x ) , from the equilibrium distribution c ( x ) , as a function of energy in units of Eo. Shown are "exact" results (section 3) using eigenvectors of transport matrix J with offset Gaussian weakcollision transition probability (eq 28 and 29, p = 500). (a) 0.1 Torr; (b) 10 Torr; (c) lo3 Torr; (d) lo5 Torr; (e) lo8 Torr. Line e corresponds to the high-pressure limit. The calculation is for the dissociation C2H6 2CH, at 2000 K (section 4). Figure 1.
-
In order to adapt eq 16 to polyatomics and to simplify it further, the numerator is approximated by
(13)
With these simplifications, eq 1 1 reduces toI4
If a simplification analogous to that made in eq 13 as regards the contribution of energies near Eo is made in the integral of eq 9, the result is w
kosC= -kTN(EO)e-EO/kT
Q
To obtain the weak-collision correction factor ratio of the two expressions for ko:
0,we now take the
P = ko,wc/ko,, = ((AE2))ss/2(kZJ2
(16)
Except for the appearance of the bulk average of (AE2b)), 16 is similar to that derived previously by NikitinI3 and Troe.el? In the above form, eq 16 was obtained before,I4 where it was pointed out that 2 ( k n 2 is in fact the classical equilibrium value of ( ( A E 2 )) for a harmonic oscillator. ( I O ) Gaynor, B. J.; Gilbert, R. G.: King, K. D. Chem. Phys. Leu. 1978, 55,40. Gilbert, R. G.; King, K. D. Chem. Phys. 1980,49,367. Schranz, H. W.; Nordholm, S.Chem. Phys. 1983, 74, 365. Vatsya, S. R. J . Phys. A 1982, 16, 201.
( I I ) Selected Papers on Noise and Stochastic Processes, Wax, N., Ed.; Dover: New York, 1954. (12) Keck, J.; Carrier, G. J . Chem. Phys. 1965, 43, 2284. (13) Nikitin, E. E. Theory of Atomic and Molecular Processes in Gases; Clarendon Press: Oxford, 1974; Chapter 6. See also: Borkovec, M.; Berne, B. J . J . Chem. Phys. 1985, 82, 794. (14) Forst, W. J . Phys. Chem. 1986, 90, 456. ( 1 5 ) Troe, J. J . Chem. Phys. 1977, 66, 4745.
In other words, compared with eq 12, (AE2b))is now averaged over the equilibrium, rather than steady-state, distribution of energies below threshold. Using the equilibrium distribution in eq 17 has the obvious advantage that the master equation does not have to be actually solved for the steady-state distribution. The justification for this is that at the low-pressure limit only the population of levels below Eo is involved, which is also where cb), and cb), are almost identical, as shown in Figure 1. The subscript wc is added in eq 17 to emphasize that ( AE2b)) refers to the second moment of the weak-collision transition probability. The denominator in eq 16, which is now understood to refer to polyatomics, is similarly modified to become ( ( AE2))sc,q =
L E O (
AE2W)S,C(Y),
= 2(( (Y2) )q - ( (Y) )2,)
dY (18)
where the subscripts on the Ihs are intended to emphasize that it refers to the equilibrium bulk average of the second moment of the strong-collision transition probability. At the same time eq 18 identifies ( ( A E Z ))=, as approximately twice the energy fluctuation at equilibrium? Introducing these modifications into eq 16, we obtain finally
P = ( (A@)
)wc,eq/
( (AE2))sc,eq
(19)
which is obviously correctly normalized to unity in the strongcollision limit. It is this expression for /3 that will be the object of an examination as a multiplicative weak-collision factor. To this end, w in eq 7 will be replaced by /3w calculated from eq 19
Forst
3148 The Journal of Physical Chemistry, Vol. 93, No. 8, 1989
for a specified weak-collision transition probability; thus
approximate solution can be obtained simply if this eigenvalue is well separated from the others. The eigenvalues of a N X N matrix M are given by the solution of the characteristic equation pN
and the resulting RRKM-type k,,,,,, of eq 20 will be compared with the “exact” result obtained for the same weak-collision transition probability as discussed below. Equation 20 shows that the introduction of constant p in k,,,,,, has the result that the weak-collision falloff curve is merely the strong-collision k,, falloff curve shifted to higher pressures by the amount p.
3. The “Exact” Formulation In order to generate exact results for an arbitrary weak-collision transition probability, as well as to evaluate from eq 19, it is convenient to cast all equations in matrix form. To do this, let Q be the matrix of transition probabilities, I the unit matrix, and K the diagonal matrix of the microcanonical rate constants k(x). All matrices are square of size N X N , meaning that the vibrational energy space of the reacting molecule has been subdivided into N levels (“grains”). The discrete analogue of the normalization condition of eq 2 is that all columns of Q add to unity; in addition, Q is constructed1 so that its elements obey detailed balance, eq 3. In matrix form, eq 1 states that -k,,, is the eigenvalue of the transport matrix J = w(Q - I) - K. It can be showni6 that k,,, is in fact the smallest eigenvalue (in absolute value) of J, and the corresponding eigenvector (a,, or its square if J is symmetrized firsti6), after normalization to unity, is the discrete equivalent of the steady-state distribution c(x),,. Alternatively, k,,, may be determined via eq 4 as
+ aN-1pN-‘ + ... + u p + a0 = 0
(22)
The assumption of eigenvalue separation amounts to linearization of eq 22, Le., to a l p + a, = 0 (23) Thus if the smallest eigenvalue is designated as M ~we, have pi = -ao/al. It can be shown18 that a. = -det M, a l = -sum of diagonal cofactors, and also that al/ao is in fact the trace of the inverse of M. Hence pi
= - l / T r M-l
(24)
For M = Q1’, eq 24 may be considered to be the matrix analogue of eq 1 1. If M happens to be tridiagonal, eq 24 can be evaluated anaIyticaIIy.13J9 For the purpose of calculating p from eq 19 we need vector ( AEz) and its bulk average as defined in eq 17. If E is the vector of the discrete energies of the system (“grains”), then (AEz) = E2(Q1
+ 11)
-
2(EQl)E
(25)
The weak- and strong-collision versions of ( AEz) are obtained by using for Qi in eq 25 the appropriate partitions of matrices Q,, and Q,,, respectively. As defined in eq 17 and 18, the equilibrium average of ( AEZ),whether weak or strong collision, then follows from ( ( AEz) ) eq =
EI ( AE2
(26)
where qll are the diagonal elements of the strong-collision matrix partition Q,,scand (AEZ), is the j t h element of vector (AEZ). where k,. are the diagonal elements of matrix K. This is equivalent to, but less direct than, -kuni via the eigenvalue inasmuch as the eigenvector is usually calculated only after the eigenvalue has been determined. Since kunicalculated as the negative of the smallest eigenvalue of J [or calculated from eq 211, for any given arbitrary w , Q and K, involves no approximations, it will be referred to as the “exact” kUniand will be used as a benchmark for evaluating the performance of k,ni,w, based on p of eq 19. The matrix K has elements that are zero below x = E,, so that the transition matrix Q can be usefully partitioned1’ at Eo into submatrices Q1, Q2, Q3, and Q4: Q
=
?.!I%
Q3 Q4
Of these, the odd-numbered submatrices are of particular interest. Q I involves only transitions below threshold (”inactive” transitions), while Q3 involves transitions from below threshold to above threshold (“activating” transitions). The matrix equivalent of eq 8 shows that the (“exact”) limiting low-pressure rate constant -ko is given by the eigenvalue (smallest in absolute value) of Q1’ = Q I - I i , where Il is the similarly partitioned unit matrix I. Alternatively, ko can be calculated as the “upward flow”, Le., the matrix product of Q3 and the eigenvector of QI’corresponding to the smallest eigenvalue. Both methods give the same result. The above relations are valid for any properly constituted stochastic matrix Q; thus weak-collision results are obtained if a weak-collision matrix Q,, is substituted for Q, and similarly Qs, a strong-collision version of matrix Q, will produce the corresponding strong-collision results. From the definition in eq 5 we see that Q, is a matrix where every column contains elements , to unity. of the T K Boltzmann distribution C ( X ) ~ ~normalized Since in the present context only the smallest eigenvalues of J or QI’ are of interest, it is useful to point out that a very good (16) Pritchard, H. 0. The Quantum Theory of Unimolecular Reactions; Cambridge University Press: Cambridge, 1984. (17) Hoare, M. J . Chem. Phys. 1963, 38, 1630.
4. Results We have chosen the dissociation of ethane into two methyl radicals at 2000 K as example of a fairly typical model system. Microcanonical rate constants for input as elements of matrix K were based on molecule and transition-state parameters as given by Smith and Gilbert.zo This is a rather primitive transition state but sufficient for the purpose at hand. Three weak-collision models were used, the exponential, offset Gaussian, and stepladder. The down form (y > x) of the exponential model is
dX,Y) = exp[-fj - XI/.]
(27)
where a is a constant that determines the amount of energy transferred. The up form follows by application of detailed balance, eq 3, followed by normalization of the matrix, as described by Tardy and Rabinovitch.’ The down form of the offset Gaussianz1is q(x,y) = exp[-fj - x - ~)’/411kT]
fj > x)
(28)
where M is a parameter that determines the offset and indirectly the amount of energy transferred. The up form is qfj,x) = [ W ) / N C v ) l exp[-(x - Y
+ c~)~/411kTl
(x > Y ) (29)
which already includes detailed balance; it then remains for the matrix to be normalized as above. Let t be the step size in the stepladder model; then the down form of the transition probability is
(18) See, for example: Demidovich, 9. P.; Maron, 1. A. Computational Mathematics; MIR Publishers: Moscow, 1981; p 376. (19) Montroll, E. W.; Shuler, K. E. Ado. Chem. Phys. 1958, 1 , 361, Appendix I . Andres, R. P.; Boudart, M. J . Chem. Phys. 1965, 42, 2057. Lewis, J. W. Numer. Math. 1982, 38, 333. (20) Smith, S. C.; Gilbert, R. G. Int. J . Chem. Kiner. 1988, 20, 307, Table 11.
(21) Bhattacharjee, R. C.; Forst, W. Chem. Phys. Left. 1974, 26, 395.
The Journal of Physical Chemistry, Vol. 93, No. 8,1989 3149
(3 in Thermal Unimolecular Reactions
r(
I
E
u
\
A
w
=I
-400
h 0
I
I
20
40
1
I
60
-2
2
0
e n e r g y / 10' cm-'
a
6
4
log P / torr
Figure 2. Energy dependence of ( A E ) for three transition probability
Figure 3. Comparison of "exact" falloff (continuous lines) and approx-
models discussed in the text. The molecule is ethane at 2000 K. Legend = -254 cm-I; E2, exponential, for models: E l , exponential, = -724 cm-'; GI, Gaussian, (AE)down= -607 cm-I; G2, Gaussian, ( AE),, = -995 cm-l; SL1, stepladder, ( AE)daw= -600 cm-l (dotted line); SL2, stepladder, (AE)down = -1000 cm-I.
imate falloff in a weak-collisionsystem using /3 from eq 19 (dashed lines) for ethane dissociation at 2000 K. Calculations are for the Gaussian, exponential, and stepladder transition probability models described in the text. The curves for the stepladder model are displaced one log unit to higher pressures for clarity.
where 6 is the delta function; the up form, including detailed balance, is
TABLE I: Comparison of (AE)do,,,, and Various Definitions of B at 2000 K" (AE)down,
Normalization then follows as above. All matrices were size 102 X 102, and eigenvalues and eigenvectors of transport matrix J were obtained by standard routines. Tests have shown that this size was sufficient for full convergence. For the purpose of determining (3 from eq 19, 25, and 26, a 50 X 50 matrix Q, was extracted from the full matrix Q. The energy-transfer properties of various models may be characterized either by ( AE), the average energy transferred in all collisions, or by ( AE)down,the normalized average energy lost in a collision. For (AE) we have (AE) = (EQ)
-E
(32)
It turns out that (AE) for the above models is energy-dependent (Figure 2), but (AE)do,, is a constant, and therefore a more suitable parameter, in agreement with the conclusions of Gilbert.22 If Qz represents the last (zth) column of matrix Q (which contains only down probabilities), and E, the last element of vector E, the is given simply by normalized ( (AE)down = (EQz) - Ez
(33)
Figure 3 shows typical falloff curves for ethane dissociation at 2000 K; continuous lines are "exact" calculations using weakcollision kUnias eigenvalues of matrix J, while dashed lines are approximations to kuni,wc from eq 20 using (3 calculated from eq 19. The transition probability parameter values that were used for these illustrative calculations correspond to ( A E ) curves identified in the legend of Figure 2 as E l , G1, and SLI, respectively. The listed values of ( AE)downrepresent roughly the (22) Gilbert, R. G.Chem. Phys. Lett. 1983, 96, 259.
model exponential (El) stepladder (SLI) Gaussian (Gl) exponential (E2) Gaussian ((32) stepladder (SL2)
eq 33 -254 -600 -607 -724 -995 -1000
/3*
eq 19 0.0132 0.0159 0.0264 0.0452 0.0499 0.0340
@tiad,
eq 34 0.0170 0.0173 0.0361 0.0595 0.0715 0.0448
8,
eq 35 0.0072 0.0323 0.0329 0.0437 0.071 1 0.0716
" Identification of models is that given in the legend to Figure 2. /3 of eq 35 is calculated from (AE)downin the second column. spread between an inefficient and fairly efficient collider. The (3 values obtained from eq 19 and used in the calculations are listed in Table I. 5. Discussion In order to better appreciate the performance of (3 of eq 19 as a multiplicative correction factor, it is useful to compare the data in Figure 3 with results obtained using in eq 20 the "traditional" definition of (3 as the ratio of the "exact" weak- and strong-collision limiting low-pressure rate constants: &rad
= k0,wdexact)/kO,sc(exact)
(34)
In the present context, ko(exact), whether weak or strong collision, is understood to mean the lowest eigenvalue of Q1 - 11, where Q1 is the appropriate partition of the weak- or strong-collision matrix, respectively. [More simply, ko,,(exact) can also be obtained as w Tr Q+; this is the matrix equivalent of eq 9.1 The results of such calculations, applied to the same falloff as in Figure 3 using the same transition probability models, are shown in Figure 4, with Bvadfor the Gaussian, exponential, and stepladder models listed in Table I. The "exact" falloff curves represent, as before, eigenvalues of the transport matrix J. As could be ex-
Forst
3150 The Journal of Physical Chemistry, Vol. 93, No. 8, 1989 0.6
EXP 0.5
0.4 ) .
c, .I+ I+
.I+
m
0.3
D 0
L
a 0.2
Gauss 0.1
c -2
0
2
4
6
8
l o g p / torr
-25
-5
-15
AE
5
15
25
/ I O 3 cm"
Figure 4. Comparison of "exact" falloff (continuous lines) and approximate falloff using "traditional" Ovadfrom eq 34 (dashed lines) for ethane dissociation at 2000 K calculated for the Gaussian, exponential, and stepladder transition probability models described in the text. The curves for the stepladder model are displaced one log unit to higher pressures for clarity.
Figure 5. Comparison of exponential (large peak) and Gaussian (small peak) transition probability models. Negative AE refers to probability of energy loss; positive AE refers to probability of energy gain. These transition probability models are the same as those used in Figures 3 and 4 and described in the text. Since the models are discrete, the curves represent actually the envelope of a string of delta functions.
pected, Ptradrenders correctly the low-pressure limit of the falloff but overestimates appreciably the rate constant in the intermediate pressure region. In this respect /?of eq 19 performs much better, with the exception of the stepladder model which yields mediocre results for either definition of 6. The poor results obtained with the stepladder model may be discounted by observing that this is not a very realistic model which normally would not be used in any but the crudest falloff calculation. It is more significant, however, as evident in Figure 3, that with p of eq 19 the Gaussian and to a large extent also the exponential model yield very good results in the intermediatepressure region: the curvature in the transition region from highto low-pressure limit is quite well rendered. Near the low-pressure limit the rate constant tends to be underestimated, which means that this approach is likely to work best with molecules that are not too small, Le., where experimental results are obtained in the intermediate-pressure regime. Since the energy diffusion approach, on which P of eq 19 is based, assumes that the transition probability q(x,y) is a sharply peaked function at x = y (Le., at A E = 0), it may appear curious that the offset Gaussian (eq 28 and 29), which seems to violate this condition, nevertheless produces results that are as good as, or even better, than the exponential function, which more nearly satisfies the condition of peakedness. This may be understood with reference to Figure 5, which compares the two transition probability models. It may be seen that both models assign negligible probability to energy transfers exceeding 5000 cm-' up or down. Since the energy scale for energy transfer goes, in principle, from -m to + m (or at least spans some large multiple of &5000), either probability will look essentially as a delta function on such an energy scale. The energy dependence of ( A E ) as shown in Figure 2 for various models deserves a comment. As energy increases, all (AE)'s change from positive to negative and pass with a steep slope through ( A E ) = 0 in the vicinity of 13 000 cm-l, which is the average vibrational energy (enthalpy) of ethane at 2000 K.
This is as it should be, since a molecule with energy less than average energy corresponding to heat bath temperature can, on the average, only gain energy in a collision, while a molecule with more than average energy will, on the average, suffer energy loss in a collision. Since none of the weak-collision transition probabilities considered here satisfies the so-called "linear sum rule", the energy dependence of ( a E ) is not linear;23instead at higher energies the energy dependence of (A,!?) tends to diminish considerably and in fact in the case of the smallest (negative) ( AE)'s (curves SL1 and E l ) appears to reach a constant value. Such a decrease in energy dependence of ( AE) at higher energies would explain why experiments measuring average energy loss by collision at different excitation energies often give different results. Recent experimental results seem to indicate just such a decreased energy dependence at high energies.24 The calculations done for the data in Figure 2 can be used to assess the performance of the often used model-independent relation between p and ( AE)down proposed by T r ~ e : ~ ~ (35) where FE is a factor that accounts for the energy dependence of density of states (eq 6.3 in ref 25). In the present case of ethane obtained for at 2000 K, FE = 1.9694. The values of ( the various models from eq 33 and listed in the second column of Table I, are used to calculate p from eq 35; also listed are the other definitions discussed in the text. On the whole, discounting the stepladder case, if @ of eq 35 is compared with @ of eq 19 used as reference, eq 35 seems to work best for the exponential model, which is not surprising since it was derived for that model; for (23) Forst, W.; Xu, G.-Y.; Gidiotis, G. Can. J . Chem. 1987, 65, 1639. (24) Shi, J.; Barker, J. R. J . Chem. Phys. 1988, 88, 6219. (25) Troe, J. J . Chem. Phys. 1977, 66, 4758.
J. Phys. Chem. 1989, 93, 3 15 1-3 158 the Gaussian model eq 35 yields results close to &ad. On account of the energy dependence of ( AI?) in most models, with of the analogous relation ( A E ) = -(AE)dow,@1/2, eq 35, obviously cannot apply, although it is sometimes used.
6. Conclusions The new expression for fl (eq 19) asserts, in fact, that only the second moment of the transition probability is important. This new (3 is seen to perform quite well with a minimum of computational labor, once the transition matrix Q is set up; in fact fl of eq 19 performs better than of eq 34 if weak-collision data near the low-pressure limit are not required. At or near the low-pressure limit, &rad can be obtained quickly and with reasonable accuracy using ko,,, calculated from eq 24. If short machine time is not the primary consideration, the actual k,,i,wcas the lowest eigenvalue of the transition matrix J can be also be obtained directly from eq 24, but the calculation has to be repeated for every value of w . Some caution is required here since the approximation involved in eq 24 is poor if the lowest eigenvalue is not well separated from the others; in that case eq 24 can be modified26 to obtain a better approximation p I = -1
/(Tr M-")'/"
(36)
However, for n larger than 3 or 4 not much is gained and an actual eigenvalue determination with an eigenvalue package is preferable. As an alternative to actual eigenvalue determination of J, an iterative scheme proposed by Malins and Tardy2' could be used. _ _ _ _ ~
~~
(26) Snider, N . S. J . Chem. Phys. 1976, 65, 1800
3151
Analytic formulas for the stepladder model based on eq 24 are a~ailable.~~,~~ The advantage of the proposed @ is that it, and consequently also the falloff of l~,,,,~,~,,can be calculated for any specified model of the transition probability, and therefore the falloff reflects the properties of that particular model (a conclusion arrived at previously in ref 1) rather than some average model-independent property, such4 as ( AE)down. This is of some importance since the data in Table I suggest that 0may be different for different models, even if they have similar ( AE)down.Conversely, curves El and S L l in Figure 2 show that almost identical ( A E ) ' s can give rise to different ( AE)down)s.The practical disadvantage, of course, is that the model of the transition probability model is not known for a specific collider system. The reasonable success of eq 20 with a constant p to account for the falloff of k,, as shown in Figure 3, suggests that the shape (or curvature) of the falloff curve is only very weakly dependent on the transition probability model and is in fact quite close to the strong-collision shape. Finally, there are limitations of the matrix approach itself. One is related to finite matrix size, Le., "graining", meaning that too coarse a discretization of the energy space can cause errors unless precautions are taken. The other is that round-off errors accumulate with each matrix operation. This second source of errors is minimized in the present approach by limiting the number of is numerically very operations; it can, however, play a role if kuni,wc large or very small. (27) Malins, R. J.; Tardy, D. C. Chem. Phys. Lett. 1978, 57, 289. (28) Pritchard, H. 0.; Vatsya, S. R. Can. J . Chem. 1984,62, 1867.
Periodic Trends in Chemical Reactivity: Reactions of Sc', Y', La', and Lu' with H,, D,, and HD J. L. Elkind, L. S. Sunderlin, Department of Chemistry, University of California, Berkeley, California 94720
and P. B. Armentrout*yt Department of Chemistry, University of Utah, Salt Lake City, Utah 84112 (Received: July 28, 1988; In Final Form: November 1 , 1988)
The reactions of Sc', Y', La+, and Lu+ with H2, D,,and HD are examined by use of guided ion beam mass spectrometry. Sc+, Y + ,and La' are found to react primarily via an insertion mechanism, while Lu+ reacts imputsively at threshold and in a direct manner at higher energies. A simple molecular orbital model coupled with adiabatic surface crossings is used to explain the reactivity seen. The results are analyzed to give the 0 K bond energies Do(Sc+-H) = 2.44 i 0.09 eV, Do(Y+-H) = 2.66 i 0.06 eV, Do(La+-H) = 2.48 & 0.09 eV, and a more tentative value of Do(Lu+-H) = 2.1 1 i 0.16 eV (48.6 f 3.7 kcal/mol). The results suggest that intrinsic M+-H bond dissociation energies for third-row metals are about 60 kcal/mol, similar to values for the first and second rows.
Introduction Over the past few years, our studies of the kinetic energy dependence of reactions of atomic transition-metal ions with H2 to form MH+ have been aimed at a comprehensive description of the effect of 4s and 3d orbital populations on reactivity.' To this end, we have altered the electron configuration of the metal ion in two ways: by moving smoothly across the periodic table and by producing various populations of ground and excited states
through several ion production techniques. The present report completes this work for the first-row transition metals by describing the reactions of atomic scandium ions with H2. Further, we extend the scope of this research down the periodic table by describing the reactions of H2 with scandium's isovalent analogues: yttrium, lanthanum, and lutetium. This comparison allows us to test whether our description of how electron configuration dictates the interactions of atomic first-row transition-metal ions with H2
'NSF Presidential Young Investigator, 1984-1989; Alfred P. Sloan Fellow; Camille and Henry Dreyfus Teacher-Scholar, 1988-1993.
(1) Elkind, J. L.; Armentrout, P. B. J . Phys. Chem. 1987,91,2037-2045, and references therein.
0022-365418912093-3151$01.50/0 0 1989 American Chemical Society
