J. Phys. Chem. 1983, 87, 1283-1285
1283
Tunneling and the Temperature Dependence of Hydrogen Transfer Reactions W. R. McKinnon and C. M. Hurd" Chemistry Division, National Research Council of Canada, Ottawa, Canada, K1A OR9 (Received: September 27, 1982; In Final Form: January 25, 1983)
The temperature dependences of rate constants ( k ) of various hydrogen and deuterium transfer reactions, which conventionally are discussed in terms of curved Arrhenius plots, can alternatively be described by log & / K O ) = constant T. The dependence can arise in the conventional model of tunneling when, as in the Eckart case, the barrier has long exponential tails. The log (k/ko)= constant T behavior can arise alternatively in a model with a temperature-dependent tunneling length between reactant sites due to their thermal vibrations. The two types of model are contrasted and compared with experiment.
Introduction Transfer of hydrogen or deuterium between sites during a reaction is conventionallymodeled one-dimensionallyas a particle impinging on a potential barrier.l The rate constant ( k ) is described in terms of the transmission rate through a barrier of fixed shape. To obtain a temperature-dependent k ( T ) , it is asserted that the reaction involves a large number of systems covering a range of energies, and this is modeled as a stream of incident particles having a Boltzmann distribution. So that the observed curvature in an Arrhenius plot can be explained, it is assumed that k ( T ) at higher temperatures is dominated by classical over-the-barrier transfers, but that with decreasing temperature the transfers by quantum tunneling become increasingly important. This view of tunneling as a correction to classical behavior encourages authors tQ present their data in the form of log k vs. 1 / T , even when such a plot shows curvature over the entire range of temperature studied, showing that an Arrhenius law is not obeyed. Frequently in such cases, the data may be described alternatively by log (k/Ito) = constant T , where ko is a constant with the same dimensions as k . We discuss how this relation can arise in the conventional model, and point out that the theoretical fits to curved Arrhenius plots obtained by some authors are actually fits to log (k/It,) = constant T. We also show how this relation is predicted by a different model of the reaction in which the temperature dependence of k( T ) arises from the thermal oscillation of the potential barrier. Discussion The linear temperature dependence of log It can frequently be found in results for intramolecular transfers of hydrogen or deuterium. For example, the data points in Figures 5-7 of Le Roy et aL2show that the reactions of hydrogen abstraction by methyl radicals from acetonitrile (I and I1 in their notation) and methyl isocyanide follow log ( k / k o )= constant T over the range of measurement, and Trakhtenberg et al.3 have previously cited these data as support for such a relation. But results for the isomerization of aryl radicals (Brunton et aL4t6 shown in Figure 1) and those for the formation of hexahydrocarbazole (1) Bell, R. P. "Tunnel Effect in Chemistry"; Chapman and Hall: London, 1980. (2) Le Roy, R. J.; Murai, H.; Williams, F. J.Am. Chem. SOC.1980,102, 2325. (3) Trakhtenberg, L. I.; Klochikhin, V. L.; Pshezhetskii, S. Ya. Chem. Phys. 1982, 69, 121. (4) Brunton, G.; Griller, D.; Barclay, L. R. C.; Ingold, K. U. J. Am. Chem. SOC.1976,68,6803. (5) Grellmann, K.-H.;Schmitt, U.;Weller, H. Chem. Phys. Lett. 1982, 88, 40. (6) Brunton, G.; Gray, J. A.; Griller, D.; Barclay, L. R. C.; Ingold, K. U.J . Am. Chem. SOC.1978, 100, 4197.
(Grellmann et included in Figure 1 ) are more convincing since they cover wider temperature ranges. (There may be other examples, for we have not made an exhaustive search.) Consider how the linear temperature dependence of log k can arise in the conventional picture. If the density of energy states of the incident particle is constant, or cancels the frequency factor,' then the rate constant may be written'?'
k(T) = klfm 0 G ( E )e-EIkBTdE
(1)
where G(E) is the transmission ratio for an incident energy E , and k , is a factor with negligible temperature dependence com ared to the integral. Suppose G ( E ) depends on E as @&2 up to some energy El (where A is a constant), then
k(r) = kl~E1eAE1ize-E/kBT a + k'(T) = k l e A 2 * B T / 4 ~ E , ' i Z e - l - A k B T / 2 ) z / k ~ T ~dy y
+ k'(T)
(2)
where y Ell2,and k'(T) is the contribution to k for El 5 E 5 E = a. If the Gaussian in eq 2 is narrow and its peak lies within the integral's limits, the factor 2y can be replaced by its value at the peak, and k'(T) should be negligible. The integral becomes A ( k B T ) 3 / 2 ~and ' / 2 we can write approximately k ( T ) = kZeA2kBT/4
(3)
(where k2 = k l A ( k B T ) 3 / z ~ 1 / Whence 2). the linear temperature dependence of log k . The energy dependence G ( E ) = constant X e"" required to justify eq 3 arises from the exponential dependence of the barrier potential on distance at large distances. The required energy dependence therefore occurs for the Eckarts and inverted Morseg potential but not for the parabolic or Gaussian.' In fact, it can be shown with the WKB approximation (although we do not have space here) that, for any potential with such exponential tails, the dominant term in G ( E ) at small E will be For example, for the Eckart potential with the usual notation V(x) = Vo/cosh2 ( x / a )
(4)
and we find A = ~ a ( 2 m ) ' / ~ / hwhich , gives, using eq 3
k(T) = k2 e x p ( 2 m ~ ~ a ~ k ~ T / h ~ ) (5) Equation 5 is in complete agreement with the leading term (7) Cribb, P. H.; Nordholm, S.; Hush, N. S. Chem. Phys. 1978,29,43. (8)Eckart, C. Phys. Reu. 1930,35, 1303. (9) Bell, R. P. J . Chem. SOC.,Faraday Trans. 2 1978, 74, 688.
0022-3654f83f2087-l203$01.5010 0 1983 American Chemical Society
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The Journal of Physical Chemistry, Vol. 87,No. 8, 1983 /
5
2t
-1
t-
'
'
'
I
"
'
l
' 9
I+--; J
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135
l o
i
200
265
330
T/ K Flgure 1. Showing the temperature dependence of some hydrogen and (0)are the protonated and deuterium reaction rate constants. (0) and deuterated cases of the decay of transient 5 from Figure 3 of Grellmann et aL5 (El) and (a)are corresponding results for the isomerization reaction of Figure 5 of Brunton et SI.' (with low temperature results from Brunton et al.'). (0) are from Figure 2 of Brunton et al.' and (0) have been for the decay of their translent 7. The curves (0) displaced by 3 and 4 ordinate units, respectively. Reading from top to bottom, the slopes of the least-squares fits to the data are (in units X lo-' K-') 2.00,2.19, 2.70, 4.34, and 3.62.
in the earlier analysis of k(T) for the Eckart potential by Shinlo (see especially his eq 6), although Shin makes no comment about the simple temperature dependence of eq 5. To estimate the range of temperature where eq 3 holds, we require the peak of the Gaussian in eq 2 to be at least one half-width (kBT/2)'/2 within the limits 0 and E1112. This implies eq 3 should hold if T satisfies A ( k ~ T ) ~ / ' / 2 ( 2 )> ~ /1'
(6)
(E1/2k~T)'/' - A(kBT)1/2/2(2)1/2> 1
(7)
For the Eckart potential, G(E) = constant X eAE'" for E < O.SV,. For the parameters obtained by Brunton et ala4 for the uppermost data in Figure 1 (Vo= 60.7 kJ/mol, a = 0.330 A), and setting El = 0.8V0, their theoretical fit should obey eq 5 for 45 K ,< T S 380 K. Thus the excellent fit obtained by these authors arises because both theory and experiment follow a log (k/ko) = constant T law. (The breakdown of this linear T relation in the theory a t the lower limit can be seen in Figure l b of Le Roy et a1.2) In contrast to the conventional model, which has a fixed barrier width, we can consider tunneling of a particle between single energy levels through a potential barrier whose transmission ratio depends on the width of the barrier x by C ( x ) = e-2ax,where a is a constant. Let the barrier oscillate about its equilibrium width R at a frequency w to simulate the thermal vibration of the sites. Since the probability that x = R + y is proportional to e-mwy/2kBT for k T 2 hw/2 (where m is the mass of the site), the rate constant will be k ( T ) = k3$
me-2aRe2~Yye-mw2y2/2kBTdy
(8)
0
where k3 is a constant. The integral is essentially the same as that in eq 2 and so leads to the same temperature dependence k(T) =
k3e-2aRe2aZksT/mw2
(10) Shin, H. J. Chem. Phys. 1963, 39, 2934.
(9)
This model, or a more sophisticated version of it, has previously been proposed for hydrogen tunneling3J1J2and electron tunne1ing.l3-l6 The slopes in Figure 1 can be related to parameters in both models. For an Eckart potential in the conventional model the only unknown parameter in the exponential of eq 5 is a, assuming m is the mass of the hydrogen atom. (Note that the slope is independent of the barrier height.) For the hydrogen reactions in Figure 1, reading from top to bottom, we find a = 0.336, 0.350, and 0.389 A, respectively. The corresponding fitted values of Brunton et al.4,6 and of Grellmann et aL5 are 0.330,0.339, and 2.35 A. Our agreement with Brunton et al. is not fortuitous but follows necessarily from eq 5. In the fit by Grellmann et al.,5the value of a was inferred from intermolecular distances, and apparently Vowas varied to obtain the best fit at V , N 50 kJ/mol. According to eq 6 and 7, the fit by Grellmann et al. will be linear in a log k vs. T plot only over the range 1K 5 T S 85 K. In view of Figure 1,we would argue that a better fit is obtained for a = 0.389 8, and, from eq 7, for any Vo2 25 kJ/mol. In the vibrating barrier model, the slope in Figure 1 is determined by a2/mw2. For the straight-line region w is constrained2s5by kBT 2 hw/2. Putting w = 1.3 X 1013s-l (corresponding to hw/kB = 100 K), and taking m to be the mass of a hydrogen atom, we find a-l = 0.457,0.439, and 0.394 A for the three hydrogen reactions in Figure 1. However, it is difficult to judge whether these are reasonable results since estimates of CY depend in detail on the assumed barrier, where both the width and height change as the barrier oscillate^.^ The data in Figure 1for deuterium transfer show higher slopes than for the corresponding hydrogen reaction (by a factor of 1.80 and 1.61 for the upper and lower reactions, respectively). The conventional model with the Eckart potential (eq 5) predicts that the slopes in Figure 1 should scale as the mass of the tunneling atom, providing a is constant. The gradients for the proton and deuteron cases should therefore be in the ratio 1:2. The same conclusion follows for the oscillating barrier model if we assume a varies as m1l2(as is the case of a simple barrier of fixed height) and the "spring constant" (mw2)is constant. Thus both models are in qualitative agreement with the isotope effect on the slopes in Figure 1. At low temperatures some rate constants which show log (k/ko) = constant T a t higher temperatures become temperature i n d e ~ e n d e n t This . ~ occurs naturally in the vibrating barrier model, where the vibrations freeze out at low temperatures leaving only the zero-point motion.3J4J6 In the conventional model, the tendency to temperature independence can be accounted for if the lowest energy level is at E > 0, and the discreteness of the levels is considered.2 It should be pointed out that although log (k/ko) = constant T is a good description of the results shown in Figure 1, it is not the only possible one. Because of the limited precision of the data and the restricted temperature ranges, equally good linear plots are obtained for log (11) Klochikhin, V. L.; Pshezhetakii, S. Ya.; Trakhtenberg, L. I. Dokl. Akad. Nauk SSSR 1978,239,879. Engl. trans: Dokl. Phys. Chem. 1978, 239, 324. (12) Klochikhin, V. L.; Pshezhetakii, S. Ya.; Trakhtenberg, L. I. Zh. Fiz. Khim. 1980,54, 1324. Engl. trans: Russ. J . Phys. Chem. 1980,54, 761. (13) Tredgold, R. H. h o c . Phys. SOC.(London) 1962,80,807. (14) Shiozaki, I.; Hurd, C. M.; McAlister, S. P.; McKinnon, W. R.; Strobel, P. J . Phys. C 1981, 14, 4641. (15) McKinnon, W. R.; Hurd, C. M.; Shiozaki, I. J . Phys. C 1981,14, L877. (16) De Vault, D.; Chance, B. Biophys. J . 1966,6, 825.
Letters
k vs. T",where n is a small positive number such as 'Iz 5 n 5 1. The appearance of log (Iz/ko) = constant T behavior may thus be fortuitous; our point is that, where the behavior is a good description of the results, the interpretation in terms of eq 3 or 9 cannot be ruled out. Moreover, the observation of a similar linear variation in the electrical conductivity with temperature in some transition metal o ~ i d e s ~ ~gives J ~ J 'further support for a law like eq 3 or 9. We see that both models discussed can explain the experimental results in Figure 1. In our view, because of the special barrier shape required in the conventional model, the vibrating barrier model is more tenable. Furthermore, eq 9 predicts that log k should increase linearly with pressure through the term e-2aR,assuming dR/dp is constant. This dependence is seen in some reactions, although a slightly curved dependence is more c ~ m m o n . ~ *To J~ explain the pressure dependence in the conventional model requires further assumptions, such as a barrier with ex(17) Inglis, A. D.; Le Page, Y.; Strobel, P.; Hurd, C. M. J.Phys. C 1982, 16, 317.
(18)Palmer, D. A.; Kelm, H. Aust. J. Chem. 1977, 30, 1229. (19) Isaacs, N. S.; Javaid, K. J. Chem. SOC.,Perkin Trans. 2 1979, 1583.
The Journal of Physical Chemistry, Vol. 87, No. 8, 1983
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ponential tails (to give the linear T law) and with a central section which compresses under pressure. The common feature of both models, however, is the assumption that tunneling behavior dominates over the entire range of temperature studied (although, of course, both must give way at high enough temperature to an Arrhenius law when over-the-barrier transitions finally become important). In both models, tunneling is not a correction to the classical behavior in the temperature range of Figure 1, but dominates the behavior completely. The presentation of data in an Arrhenius plot in such cases is misleading, and we propose that a plot of log k vs. T i s more appropriate. Le Roy et a1.2 have commented on the unsuitability of Arrhenius plots and they employed log k vs. T to present their data, but they did not draw attention to the linearity of these plots nor to its implications. Le Roy20 has also discussed in detail the interpretation of the data from Brunton et al.4v6in terms of the conventional model.
Acknowledgment. It is a pleasure to acknowledge the help and encouragementgiven particularly by K. U. Ingold and D. Griller when studies of electron transport in oxides brought us into their field. (20) Le Roy, R. J. J. Phys. Chem. 1980,84, 3508.
