J. Phys. Chem. 1982, 86, 1771-1775
TABLE I: Intermolecular Vibrational Frequencies of Hydrogen-Bonded Phenols in the First Excited Electronic State and the Spectral Red Shifts from the 0,O Band of Free Phenol (in cm-') proton acceptor
v,ja
water methanol ethanol 1-propanol
27 24 23 29 26 3o 31 23 21 20 16
dimethyl ether diethyl ether dioxane
THF
cy clohexene benzene a
Bending.
Stretching.
VU b
A uC
121 175 150 131
355 41 5 409 395 447 455 56 2 431 411 477 235 147
137 101 50
Spectral shift.
- A H (Kcal/mol)
't 0
100
200 red shift
300 dV
400
500
(cm-1)
Flgun, 7. Plot of the spectral red shift vs. the heat of formation of the hydrogen bond In the ground state taken from ref 13.
also supported by the difference of the frequency of the
bending vibration between the two spectral regions. It is concluded from these observations that there exist two types of 7~ hydrogen-bonded complexes with quite different
1771
bond strengths. As seen from Figure 6, each group also has several complexes, suggesting the existence of complexes with slightly different structures. Table I summarizes the excited-state bending and stretching vibrational frequencies and the spectral red shifts of the complexes from the 0,Oband of free phenol. In the cases where several complexes are involved, those giving the strongest bands are included in the table. It can be seen from the table that there is a qualitative correlation among the bending and stretching frequencies and the spectral shifts. Therefore, the observed spectral shift gives a good measure of the strength of hydrogen bonding. In Figure 7, the spectral shift is plotted against the heat of formation of the hydrogen bond in the ground state.13 A good correlation between the two quantities is seen from the figure. In all of the cases except phenol-water, the progressions of the excited-state bending vibration were observed in the spectra. The appearance of the progression seems to indicate appreciable change of the angle 0-H ...A (A is a proton-accepting molecule) in the electronic transition. The change might be ascribed to the steric hindrance arising from the bulkiness of the accepting molecule. Finally, it is concluded that the supersonic free jet technique is a very powerful way to investigate hydrogen-bonded complexes and gives detailed information about these complexes. Identification and structure of the hydrogen-bonded complexes are now under study in our laboratory by means of fluorescence spectra and multiphoton ionization-mass spectra.
Acknowledgment. We thank Dr. Jun-ichi Murakami for his experimental assistance and useful suggestions. (13) Data taken from: (a) A. S. N. Murthy and C. N. R. b o , Appl. Spectrosc. Reu., 2, 69 (1968), Table 2; (b) G. C. Pimentel and A. L. McClellan, Annu. Reu. Phys. Chem., 22,347 (1971); (c) G. C. Pimentel and A. L. McClellan, 'The Hydrogen Bond", W. H. Freeman, San Francisco, CA, 1960.
Test of Laplace Transform Inversion of Unlmoiecular Rate Constant Wendell Forst Department of Chem&fty and Centre de Recherches sur les Atomes et les M&cules (CRAM), Universit6 Lava/, Q&ec, Canade G1K 7P4 (Received: June 17, 1081; I n Final Form: November 16, 1081)
"Exact" quantum-mechanical rate constants k(E) of specified energy E for the nonadiabatic decomposition of N20are used to calculate, by Boltzmann averaging, the exact limiting high-pressureunimolecular rate constant k,. The temperature dependence of this k , is non-Arrhenius. Inversion leads to the recovery of a good smooth-function approximation to the exact k(E) if the activation energy E,, is expressed at least as a fourth-degree polynomial in temperature. This is because k(E)in the example chosen has an unusual energy dependence, and so provides a severe test. The pressure dependence of E,, (but not of kUi) is shown to reflect sensitively the temperature dependence of k,, and through it the energy dependence of k(E).
Introduction It is well-hown that at some temperature the high-pressure unimolecular rate constant k, is by definition the Boltzmann average of k(E), the microcanonical rate constant for decomposition of molecules having specified energy E. This average can be considered operationally' as the Laplace transform of k(E)N ( E ) / Q ,i.e. km
@'L{k(E) N ( E ) J
(1)
0022-3654/82/2086-177 1$01.25/0
where N ( E ) is the density of states of the decomposing molecule at E , and Q is its partition function at T. It is now also well-known2-6that one can recover k(E) from k , by inversion (Le., by taking the inverse Laplace (1) Slater, N. B. h o c . Leeds Philoe. Lit. SOC.,Sci. Sect. 1955, 6, 259. (2) Forat, W. J . Phys. Chem. 1972, 76, 342. (3) Forat, W. In "Reaction Transition States"; Dubois, J. E., Ed.; Gordon and Breach New York, 1972; p 75. (4) Forat, W. J . Phys. Chem. 1979, 83, 100. (5) Forat, W.; Turrell, S. Znt. J. Chem. Kinet. 1981, 13, 283.
0 1982 Amerlcan Chemical Society
1772
The Journal of Physical Chemistry, Vol. 86, No. 70, 1982
transform), provided the temperature dependence of k, is known,and the k(E) so recovered can be substituted into the usual expression for kh, the general-pressure unimolecular rate constant, from which the pressure dependence of kd can be reconstructed. While the inversion procedure rests on solid mathematical ground, in practice it can only yield an approximation to k(E) since (1)the temperature dependence of k, is not usually known over a wide enough temperature range and therefore is mostly assumed to be of the simple Arrhenius (exponential) form at all temperatures
k, = A, exp(-E,,/kr)
Inversion of Arrhenius Form of k, With reaction threshold calculated to be at Eo = 14135 cm-’, Figure 1 shows the energy dependence of the exact discrete rate constants k(E) for the decomposition of N20. Note that the rate constant is exceedingly small at threshold (s-l) but increases to 10l2s-l (i.e., by 19 orders of magnitude) over an energy range of just --8ooo cm-l. Remarkable as this may seem, one must not forget that the decomposition of N20 is a spin-forbidden (i.e., nonadiabatic) reaction which proceeds by a process akin to tunneling, hence the strong energy dependence of k(E) near threshold. In a spin-allowed (i.e., adiabatic) reaction, such strong energy dependence of k(E) would be characteristic of a polyatomic molecule very much larger than N20. Least-squares fit of the k(E)’s in Figure 1gives their energy dependence as -0 - E o / E P .
-
H. 0. Can. J. Chem. 1979, 57, 2458.
10075-
--
50-
I
25-
W
r
0-
I
(2)
and (2) several approximations are involved in the actual numerical application. Up to now, the practical reliability of the inversion procedure had to be taken more or less on faith since no sufficiently accurate kinetic data exist over a wide enough temperature range that could provide a test. A direct test would consist of calculating an exact k(E) from first principles (i.e., a k(E) not making use of any kinetic information) and then obtain a k , by Boltzmann averaging. The result would be an exact k, since Boltzmann averaging is a straightforward procedure that can be accomplished without any approximations. If the inversion procedure really “works”, inverting the exact k, should then lead to the recovery of the exact k(E), or at least of a very good approximation to it. This sort of test has been applied by Yau and Pritchard6 (YP), who calculated a two-dimensional quantum-mechanical k(E) for the decomposition of N 2 0 and C02. Using a somewhat different theory, we have recently7 likewise calculated the exact quantum-mechanical (this time multidimensional) k(E) for the decomposition of N20, which, although it yields a similar k,, differs from YP’s k(E) in one important aspect: it shows a different behavior at reaction threshold Eo. We do not propose to discuss here whether our k(E) is “better” or “worse” than YP’s, but merely wish to use ours for the direct test of the inversion procedure outlined above; hence, for the purpose of the test, our k(E) and the corresponding k, will be taken as exact by definition. It turns out that the test is a very severe one: mainly because of the threshold behavior of our exact k(E),the inversion procedure is no longer quite so straightforward as in the cases (including YP’s) that have been discussed so far, but nevertheless it is capable of giving back a very good approximation to the exact k(E), albeit under conditions unlikely to be satisfied in practice.
(6) Yau, A. W.; Pritchard,
Forst
(7) Lorquet, A. J.; Lorquet, J. C.; Forst, W.Chem. Phys. 1980,51,253.
s
c’
- 7 5 1
14
/
I
/
1
I
I
I
1
’ ‘
’
I
I
18
16
E
I
I
M
I
I
I
I
I
22
, cm-’
Flgure 1. Microcanonical rate constant k ( E ) (s-’)for N20decompositkn as a function of energy: (circles) exact: (dashed line) by inversion from eq 3: (full line) by Inversion from eq 13.
We have shown previously7that (discrete) Boltzmann averaging of the discrete exact k(E)’s of Figure 1leads, at 900 K, to k, having the simple form of eq 2, with A, = 1.64 X 10l2s-l, E,, = 20484 cm-’ (=58.6 kcallmol). Roughly the same limiting high-pressure activation e n e r d (E,, 59 kcal/mol) has been obtained both in bulb experiments near 900 K and in shock tube experiments near -2000 K. One is therefore tempted to suppose that eq 2 is valid at all temperatures and proceed to invert k,, thus obtaining the standard result
-
E > E,, k(E) = A,N(E - E,-) k(E) = 0 E < E,,
(3)
Since this k(E) is a continuous-function approximation, it is clear that it cannot reproduce the discontinuous character of the exact k(E),but it should give correctly its mean energy dependence. The dashed line in Figure 1shows k(E) of eq 3. There are two things wrong with it: (1)because of the large difference between Eo (14 135 cm-’) and E,, (20484 cm-l), the inversion procedure appears to have led to the loss of all information about k(E) between Eoand E,,, Le., over a range of -6500 cm-l above threshold; (2) since N(E) for N20 depends on energy as E2, the energy dependence of the inverted k(E) is too small, inasmuch as we have seen already that the exact k(E) depends on energy as -El8. Let us now reconstruct k~ and its pressure falloff. The exact kmi is given by
-
(4)
where k(E) is the exact discrete k(E), W(E)is the number of states at E (obtained by direct count), p is pressure, and the denominator is the quantum-mechanical form of the partition function. Inasmuch as the inversion procedure does not establish the absolute position of the falloff curves on the pressure axis, we assume for simplicity strong collisions under Lennard-Jones potential. Thus Z = 213 2/ 3)*d2(8kT/up) l12(t/ k V1l3in concentration units, where d is the mean collision diameter, e = (ele2)’I2is the mean Lennard- Jones well depth of the two colliding (8) Baulch, D. L.; Drysdale, D. D.; Home, D. G. ‘Evaluated Kinetic Data for High Temperature Reactions”; Butterworths: London, 1973; VOl. 2.
The Journal of Physical Chemistty, Vol. 86, No. 10, 1982 1773
Laplace Transform Inversion of Rate Constant
t
17
-6
1
-2
1
1
0
1
1
,
2 log,,
1
4
I
I
I
6
I
8
p (torr)
Flgure 2. Pressure dependence of k,/k,
for N20decomposition at
901 K (a) exact, from eq 4 and 1 0 (b) by inversion (eq 5),using k ( E ) from eq 3; this curve was shifted to h w r pressures by 1.5 log units, slnce the calculation does not establish absolute pressure fit.
16
t
/
L:
L
l
l
200
l
l
r
l
r
r
T
r
l
r
600
400
l
r
r
r
800
O K
Flgure 4. Temperature dependence of exact limiting high-pressure activation energy E,, calculated from eq 7.
log, p (torr1
Fbwe 3. Pressure dependence of E, for N,O decompositlon at 901 K (a) exact, from eq 4 and 6;the low-pressure limit is at E, = 13617 cm-‘ and is reached at -lo-’’ torr; (b) by inversion from eq 3, 5, and 8.
partners, and p is their reduced mass. The approximate “inverted” kmi is k,i(inverted) =
where k(E) is given by eq 3 and the other symbols have their usual meaning. Taking the logarithmic derivative of each k~ with respect to T,we can calculate the falloff with pressure of activation energy, both exact and inverted: E, = k P ( d In kmi/dT)
(6)
Figure 2 shows the pressure dependence of the exact and inverted kUi. It is remarkable that the inverted kmi has a pressure dependence that is not much different from that of the exact kmi, although the former makes use of k(E) of eq 3 that is clearly wrong. This illustrates the extent to which the pressure falloff of kUi is insensitive to the quality of k(E).
Figure 3 compares the pressure falloff of the exact and inverted activation energies E,, and the difference with respect to kh and its falloff is startling. The inverted E, falls off in a way characteristic of a “small” molecule whose k(E)has a weak energy dependence, whereas the exact E, falls off dramatically in a way reminiscent of a “large” molecule whose k(E) has a strong energy dependence. Thus, it turns out that, in a thermal system, the pressure dependence of activation energy is clearly much more sensitive to the quality of k(E) than the pressure dependence of kmi. It should be pointed out that the results shown in Figures 2 and 3 are not substantially changed if the inversion procedure is modified4to take into account the difference between E,, and E,,. The reason is that the difference is much too large for the modification (which is essentially an interpolation) to be applicable.
Inversion of General Non-Arrhenius Form of k, If k(E) of eq 3 obviously incorrect, what went wrong? It turns out that eq 2, contrary to assumption, is not valid at all temperatures, which means that in the present instance we must have a temperature-dependent E,”. This is shown in Figure 4, which gives, as a function of temperature, the exact limiting high-pressure activation energy
where (E) = k P ( d In Q/dT). Thus, the bulk of the temperature dependence of E,, is seen to be at low temperature, Le., below -600 K. The calculation in Figure 4 does not extend above 900 K since at higher temperatures rate constants larger than those that were actually calculated start contributing. It is nevertheless quite clear that the activation energy reaches a limit near 900 K, so that the calculation c o n f i i the experimental result* that between -900 and -2000 K there is little change in activation energy.
1774
The Journal of Physical Chemktry, Vol. 86, No. 10, 1982
In view of the complicated temperature dependence of E,,, let us write it in terms of an mth order polynomial in kT ( k = Boltzmann's constant): m
E,, = C ai(kT)' i=O
(8)
Since E,, is, by definition, k P ( d In k,/dT), integration of E , , / k P yields
k , = (eC/kal)(kT)aleQO/kTeR
(9)
where
and C is a constant of integration that is determined by comparing k , of eq 9 with the exact k, given by C k ( E ) W(E)emEIkT k,(exact) = (10) C W(E)e-E/kT Equation 9 is clearly not of simple Arrhenius form; it shows a temperature-dependent preexponential factor that combines a simple power dependence with a complicated exponential dependence. Although preexponential factors with simple power or exponential temperature dependence have been discussed ~eparately?~ the two combined represent a new complication not considered before. For the purpose of inversion, let us write s = l/kT for the inversion parameter; substitution in eq 9 and inversion of eq 1 gives N ( E ) k(E) = Bo-C-'{Q(s)e+@eR(') / sal1 (11)
Forst
be checked by noting, from eq 12, that we must have
L(F,JE))= Q(kT)"1eR (16) the Laplace transform being evaluated numerically, e.g., by Gauss-Laguerre integration, and the partition function directly from the quantum-mechanical expression Q = ?Iirl - exp(-hvi/kT)]-' where vi are the frequencies of the oscillators involved (in the case of N20 these are 2224, 1285, and 589 cm-l, the last doubly degenerate). Independent harmonic oscillators are assumed throughout. In the case of the data shown in Figure 4, it turns out that a reasonable fit of the temperature dependence of E,, below 900 K requires at least a fourth-degree polynomial in kT. The five coefficients ai for this fit (eq 8), as well as the constant C, are as follows: a. = 11102.32 cm-', al = 52.37074, a2 = -0.095481 (cm-')-', u3 = 4.84886 X (cm-1)-2,a4 = 1.56888 X lo4 (~m-')-~, and C = -275.08 s-'. Numerical overflow problems in evaluating the inverse transform in eq 12 necessitate working with In Fm(E)rather than with Fm(E).
Results The full line in Figure 1 compares k(E) calculated from eq 13 with the exact kQ, and it is quite apparent that now the inversion procedure does reproduce a reasonable average k(E) at a given energy, and with the correct energy dependence. Although the threshold energy is not given correctly, since u0 = 11102 cm-' instead of 14135 cm-l, the consequences are minimal because the inversion produces negligibly small rate constants below 14 135 cm-'. Thus, for example, at 2300 cm-' above ao,i.e., -735 cm-' below the exact threshold, the inverted k(E) is still only s-l, and many orders of magnitude smaller at lower enerwhere Bo = eC/koland R(s) = Ci=2m[ai/(i - 1)s' - '1. The gies. As a result, relative to the correct threshold, the above term e- in eq 11 merely causes a zero shift of the inverse a. has little effect on falloff not too far from the hightransform of f(s) = Q(s)eR(s)/sal.If we let pressure limit, but obviously it will not yield the correct limiting low-pressure parameters (activation energy and F,(E) = LWsN (12) rate constant), in keeping with the known fact that the then eq 11 becomes inversion procedure, based as it is on high-pressure parameters k , and E,,, cannot yield the correct low-pressure k(E) = BoFm(E- aO)/N(E) E > (13) parameters unless the actual threshold Eo is known from k(E) = 0 E< other information. Presumably in the limit as the polynomial in eq 8 approaches an infinite series, the threshold The coefficient a. (which has the dimension of energy) thus obtained by inversion will approach the true threshold. plays the role of threshold energy. Figure 5 compares the pressure dependence of exact The inverse transform in eq 12 is best evaluated by the / k ,~ (eq 4 and 10) and of k ~ / k obtained , by inversion method of steepest descentss carried to second ~ r d e r , ~ k~e ~ (eq 5), and similarly Figure 6 compares the corresponding which requires the derivatives of 4(s) = In V(s)e"]. We activation energies. The curves based on the inversion have procedure contain a few kinks which can be traced to the $(s) = In Q(s) + sE - al In s + R(s) fact that eq 8 with m = 4 is not a perfect representation of the exact E,, (within mi50 cm-' near 200 K, within 4%) = d In Q(s)/ds + E - al/s - & / s i (14) i -A25 cm-' near 800 K). Figure 6 is remarkable in two respects: first, by comparison with Figure 3, thus showing The subsequent derivatives are clearly the sensitivity of activation energy falloff with 4'"'(s) = pressure to the quality of the k(E) used as input; secondly, by noting that the activation energy continues to decrease well below the lowest pressure shown, to reach ultimately (at 901 K) 13 618 cm-' (this is the exact value based on eq Then 4). The continued decrease of the activation energy over such wide pressure range is a reflection of the very strong energy dependence of k(E),on the one hand, and of the very small value of k(E) at threshold on the other. where s* is the solution of 4'(s) = 0. The calculation can (9) Forst, W. 'Theory of Unimolecular Reactions"; Academic Press: New York, 1973; Appendix. (10) Hoare, M.R. J. Chem. Phya. 1970,52,5695; 1971,55, 3058.
-
Discussion There are four main conclusions that emerge from the present test: (1) The results presented here show that, if the temperature dependence of k , is known accurately enough, the inversion procedure "works" even if k , is non-Ar-
The Journal of phvsical Chemlstry, Vol. 86, No. 10, 1982 1775
Laplace Transform Inversion of Rate Constant
-T
1
1
1
-2
1
,
1
1
1
2
0
1
1
1
1
S
6
4
logH) p (torr)
Figure 5. Pressure dependence of k d k , for N20decomposition at 901 K (a, fun line) exact, eq 4 and 1 0 (b, dashed Hne) by inversion, from eq 5 and 13; this curve was shifted to hi@w pressures by 2 log Units Since the calculation does not establish absolute pressure fit.
I/
/-
0 85 ,
I
-2
/,
I
0
I
I
2
l
4
,
I
6
,
I
8
loglo p (torr)
Figue 6. Pressure dependence of relative activation energy E,/€,, for N20decomposltkn at 901 K (a, MI #ne)exact, eq 4 and 6 (same as in Flgure 3); (b, dashed line) by inversion, from eq 5, 6, and 13. The bw-pressure limit of curve a conesponds to E,/€,, = 0.665 and is reached at -io-” torr.
rhenius, albeit at the price of eome numerical complication. Therefore, in actual applications of the inversion procedure to experimentally determined k,, the main issue is making sure that the temperature dependence of k, is known in sufficient detail. In the present instance, “in sufficient detail” means principally at temperatures below 600 K, where, from the point of view of experiment, the dissociation of NzO is unfortunately too slow to be measured. Presumably, around 300 K the rate of the reverse association Nz(’Z) + O(3P) Nz0(lZ) should be measurable, which could then be converted into the forward rate by means of the
-
equilibrium constant. It is doubtful, however, that k, could be determined over the entire temperature interval of interest with enough accuracy, particularly since the curvature in the usual log k, w. 1/T plot is noticeable only below 300 K. Thus “in sufficient detail” means, at least in the case of NzO, a requirement unlikely to be satisfied in practice insofar as actual experimental measurements of NzO decomposition are concerned. (2) The pressure dependence of k~ is quite insensitive to the energy dependence of k(E),and through it to the temperature dependence of k,. Hence, even if the temperature dependence of k, is not known accurately or is simply assumed to be Arrhenius (eq 2) because of insufficient information, the inversion procedure will always , in rather produce a fairly good falloff curve for k ~except extreme cases of which N20 is one example. (There are strong indications that such extreme cases are represented mainly by nonadiabatic reactions or reactions proceeding by tunneling. For example, tunneling in the decomposition HzCO H2 + CO gives rise“ to k(E)’s having an extreme energy dependence similar to that shown in Figure 1). (3) The pressure dependence of activation energy, in contrast, is quite sensitive to the energy dependence of k(E)and therefore reflects the temperature dependence of k, much more strongly than the falloff of k ~ Hence, . the pressure dependence of activation energy, which need be determined at only one temperature, offers a useful experimental check on the temperature dependence of k,, for sometimes it may be experimentally more feasible to work over a wide range of pressures rather than a wide range of temperatures. As Figures 3 and 6 show, any important discrepancy between the pressure dependence of activation energy as determined from experiment and by inversion immediately raises doubts about the functional dependence of k, on temperature. Conversely, substantial agreement between pressure falloff of activation energy determined by inversion and by experiment confiims that the temperature dependence of the k, used as a basis of the inversion is indeed the correct one. Such is the case of the dissociation of ethane into two methyl radicals, where inversion of an Arrhenius-form k, leads to a calculated activation energy falloff that is in good agreement with e ~ p e r i m e n t . ~ (4) The second-order region of a unimolecular reaction is reached when kmi is proportional to pressure, which means unit slope on the log k~ vs. log p plot? The present (exact) calculations show, not surprisingly, that the approach to unit slope parallels the activation energy falloff, which in the case of NzO is exceedingly slow: at 901 K, it takes something like a drop of 4 orders of magnitude in pressure for the slope to increase from 0.9 to 0.95. Given the inevitable scatter in experimental data, a slope of 0.95, or even of 0.9, could easily be mistaken for slope of unity and hence lead to incorrect assignment of the limiting low-pressure region.
-
Acknowledgment. This work has received financial support from the National Sciences and Engineering Council of Canada. (11) Miller, W. H., J. Am. Chem. SOC.1979, 101, 6810, Figure 2.
