Low-temperature limit of rate constants

The temperature dependence of this rate constant was measured by Al-Soufi et al. [1991], and is shown in Figure 6.17. It exhibits a low-temperature limit of rate constant kc = 8x 105 s 1 and a crossover temperature 7 C = 80K. In accordance with the discussion in Section 2.5, the crossover temperature is approximately the same for hydrogen and deuterium transfer, showing that the low-temperature limit appears when the low-frequency vibrations, whose masses are independent of tunneling mass, become quantal at Tisotope effect increases with decreasing temperature in the Arrhenius region by about two orders of magnitude and approaches a constant value kH/kD = 1.5 x 103 at T[Pg.174]

Macrokinetie Peculiarities of Solid-State Chemical Reactions in the Region of Low-temperature Limit of Rate Constant... 

Figure 4. Examples of low-temperature limit of rate constant of solid-state chamical reactions obtained in different laboratories of the USSR, United States, Canada, and Japan (1) formaldehyde polymerization chain growth (USSR, 1973 [56]) (2) reduction of coordination Fe-CO bond in heme group of mioglobin broken by laser pulse (United States, 1975 [65]) (3) H-atom transfer between neighboring radical pairs in y-irradiated dimethylglyoxime crystal (Japan, 1977, [72], (4, 5) H-atom abstraction by methyl radicals from neighboring molecules of glassy methanol matrix (4) and ethanol matrix (5) (Canada, United States, 1977 [11, 78]) (6) H-atom transfer under sterically hampered isomerization of aryl radicals (United States, 1978 [73]) (7) C-C bond formation in cyclopentadienyl biradicals (United States, 1979 [111]) (8) chain hydrobromination of ethylene (USSR, 1978 [119]) (9) chain chlorination of ethylene (USSR, 1986 [122]) (10) organic radical chlorination by molecular chlorine (USSR, 1980 [124,125]) (11) photochemical transfer of H atoms in doped monocrystals of fluorene (B. Prass, Y. P. Colpa, and D. Stehlik, J. Chem. Phys., in press.).

MACROKINETIC PECULIARITIES OF SOLID-STATE CHEMICAL REACTIONS IN THE REGION OF LOW-TEMPERATURE LIMIT OF RATE CONSTANT... 

Cryochemical research in the past 15 years has established the existence of the low-temperature limit of rate constants of various solid-state chemical reactions with transfer of atoms and molecular fragments of different masses over distances comparable with intermolecular ones, from H-atoms transfer under intramolecular rearrangement to organic radicals and halogen atom... 

Adiabatic reactions, occurring on a single-sheet PES correspond to B = 1, and the adiabatic barrier height occurs instead of E. The low-temperature limit of a nonadiabatic-reaction rate constant equals... 

The low-temperature limit of the rate constant for the isomerization of the biradical... 

The chain polymerization of formaldehyde CH2O was the first example of a chemical conversion for which the low-temperature limit of the rate constant was discovered (see reviews by Goldanskii [1976, 1979]). As found by Mansueto et al. [1989] and Mansueto and Wight [1989], the chain growth is driven by proton transfer at each step of adding a new link... 

Figure 1.1 Examples of temperature dependences of rate constants for the reactions in which the low-temperature rate constant limit has been observed 1, hydrogen transfer in excited singlet state of molecule (6.14) 2, molecular reorientation in methane crystal 3, internal rotation of CH3 group in radical (7.42) 4, inversion of oxyranyl radical (8.18) 5, hydrogen transfer in the excited triplet state of molecule (6.20) 6, isomerization in the excited triplet state of molecule (6.22) 7, tautomerization in the ground state of 7-azoindole dimer (6.15) 8, polymerization of formaldehyde 9, limiting stage of chain (a) hydrobromi-nation, (b) chlorination, and (c) bromination of ethylene 10, isomerization of sterically hindered aryl radical (6.44) 11, abstraction of a hydrogen atom by methyl radical from a methanol matrix in reaction (6.41) 12, radical pair isomerization in dimethylglyoxime crystal (Figure 6.25).

Chemical dynamics at low temperatures is connected with elementary reactions that surmount potential energy barriers separating reactants from products in the absence of thermal activation. The first experimental evidence of this type of reactions was obtained in the early 1970s in studies of solid-state conversion of free radicals. These investigations clearly demonstrated that there is a sufficiently sharp transition from Arrhenius-like exponential temperature dependence, characteristic of thermal activation, to much weaker power-like temperature dependence down to the low-temperature limit of the rate constant. 

The main significance of the works [8] was in revealing the existence, irrespective of the barrier shape, of the finite low-temperature limit of the rate constant K(0). Even for Eckart barrier V x)= V /ch (2x/d), having an infinite width at = 0, the tunneling probability remains finite due to the existence of zero-point vibrations. 

Since issues of Doklady AN SSSR were not regularly translated into English in 1959, the papers [8] were not considered in Western scientific literature. Only after a number of experimental papers were published confirming the existence of the low-temperature limit of the rate constants of... 

Studies of chain reactions of hydrobromination [119-121] and hali-dization [122, 123] of ethylene and chlorination of various saturated hydrocarbons and their derivatives [124-128] confirmed the existence of the low-temperature limit of the rate constant. The pattern of chain growth under hydrobromination of ethylene after y-irradiation-induced HBr decomposition is as follows ... 

Thus, the data of refs. 119-128 confirmed the existence of the low-temperature limit of the rate constant of solid-state chemical reactions found in refs. 57 and 58 and demonstrated an even stronger dependence of reaction rate on the properties of the medium than in the earlier discussed reactions with H-atom transfer. 

The low-temperature limit of the rate constant is obtained from Eqn. (60) when coth l hCO2) is replaced by unity. It is clear that the low-temperature limit exists only for nonendothermal reactions. This condition is satisfied by the first term in the exponent, which at A > 0 turns into nought and at A < 0 (endothermal process) leads to an activation dependence of the rate constant on temperature. The next term is responsible for the effective decrease of the potential barrier due to lattice vibrations. The last term corresponds to tunneling rearrangement of the medium accompanying particle transfer. 

Equation (72) enables us to consider the low-temperature limit of the chemical reaction, which can be obtained setting 7=0. The first term of the exponent ensures an increase in the rate constant resulting from inter-molecular vibrations the second term describes the reorganization of the medium. It is clearly seen that the contribution of the latter depends on medium properties as well as on potential barrier parameters. 

Further development of the notions about the nonthermal critical phenomena in the region of the low-temperature limit of the rate constant should be based on quantum-chemical computations of local deformation of the solid lattice and its effect on the consequent reaction act rate. 

On the low-temperature limit of the capture rate constants for inverse power poterA a[s,J.Chem.Phys. 118, 7313-7321. 

The transition is fully classical and it proceeds over the barrier which is lower than the static one, Vo = ntoColQl- Below but above the second cross-over temperature T 2 = hcoi/2k, the tunneling transition along Q is modulated by the classical low-frequency q vibration. The apparent activation energy is smaller than V. The rate constant levels off to its low-temperature limit k only at 7 < Tc2, when tunneling starts out from the ground state of the initial parabolic term. The effective barrier in this case is neither V nor Vo,... 

On the other hand, the low temperature dependance of the rate constants with activation energies around 5 kcal/mole indicates a diffusion limited reaction rate which could refer to diffusion of oxygene into the fibers of the board, i.e. into the fiberwalls. The corresponding negative activation energy for the groundwood based hardboard and the effect of fire retardants there upon are difficult to understand. 

The isomerizations of n-butenes and n-pentenes over a purified Na-Y-zeolite are first-order reactions in conversion as well as time. Arrhenius plots for the absolute values of the rate constants are linear (Figure 2). Similar plots for the ratio of rate constants (Figure 1), however, are linear at low temperatures but in all cases except one became curved at higher temperatures. This problem has been investigated before (4), and it was concluded that there were no diffusion limitations involved. The curvature could be the result of redistribution of the Ca2+ ions between the Si and Sn positions, or it could be caused by an increase in the number of de-cationated sites by hydrolysis (6). In any case the process appears to be reversible, and it is affected by the nature of the olefin involved. In view of this, the following discussion concerning the mechanism is limited to the low temperature region where the behavior is completely consistent with the Arrhenius law. 

Involvement of intramolecular high-frequency vibrational modes in electron transfer was considered (Efrima and Bixon, 1974 Nitzan et al., 1972 Neil et al., 1974, Jortner and Bixon, 1999b Hopfield, 1974 Grigorov and Chernyavsky, 1972 Miyashita et al. 2000). As an example, when the high-frequency mode (hvv) is in the low-temperature limit and solvent dynamic behavior can be treated classically (Jortner and Bixon, 1999 and references therein), the rate constant for non-adiabatic ET in the case of parabolic terms is given by... 

In this expression, k is the first-order rate constant for leaving one site and W is the broadening in excess of the natural line-width (in Hz). Thus, if one assumes that the low-temperature limit is reached when the excess broadening is sal Hz, one can make a fairly accurate estimate of AG from this temperature (see Table 1). One might note for a rule of... 

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