Arrhenius curves

Similar plots have been obtained for the gas-phase rearrangement of 35 (A = CH3 Aik = ethyl Alk = methyl) and 36 (A = CH3 Aik = methyl Alk = ethyl) in 720 torr methyl chloride in the temperature range from 40 to 120 Regression analysis of the relevant Arrhenius curves leads to the activation parameters listed in Table 22. 

Fig. 29. The Arrhenius curve for the isomerization of the 2,4,6-tri-t-butylphenyl radical and its completely deuterated analogue [77],...

Fig. 30. The scheme of the transfer of the H atom in an irradiated crystal of dimethylglyoxime. (a) The arrow indicates the direction of the transfer leading to the transformation of the J radical pair into the radical pair K (b) the Arrhenius curve for the process [79],...

Figure 2.4 Arrhenius curve for monomolecular Langmuir-Hinshelwood kinetics.

Such a singularity is clearly visible on the Arrhenius curve of Aciass determined for a crystal of acetylsalicylic acid-iV, 8. In Fig. 17 this curve is compared with those for the values of and k, determined recently for the same system cf. Subsection 4.2). 

The linear portion of the Arrhenius curves in Figure 1 showed some significant differences among the resins. Within the temperature range of 120- to 160°C, resin X0 released about an order of magnitude more bromide ion than resin Q. At 150°C, 53 PPM bromide was extracted from resin X0 this could... 

A Kinetic Model. The shapes of the Arrhenius curves in Figure 1 are indicative of the mechanism of aqueous ion extraction. A closer examination of the shapes entails a detailed mathematical analysis of the kinetics of the elementary steps comprising the aqueous bromide extraction process. A rate equation will be derived here in order to interpret the experimental data in... 

The second type of reactions discussed refers mostly to liquid state solutions and involves major heavy atom reorganization. Here, equilibria between reactive and non-or less reactive molecular configurations may play a role. Several cases are discussed where the less reactive forms dominate at low or at high temperature, leading to unusual Arrhenius curves. These cases include examples from small molecule solution chemistry like the base-catalyzed intramolecular H-transfer in diaryltriazene,... 

Various quantum-mechanical theories have been proposed which allow one to calculate isotopic Arrhenius curves from first principles, where tunneling is included. These theories generally start with an ab initio calculation of the reaction surface and use either quantum or statistical rate theories in order to calculate rate constants and kinetic isotope effects. Among these are the variational transition state theory of Truhlar [15], the instanton approach of Smedarchina et al. 

The Bell-Limbach model is not designed to give definite interpretations of Arrhenius curves of hydrogen transfer reactions which have to come from more sophisticated methods. However, it provides an opportunity to check whether the number of parameters describing a given set of Arrhenius curves matches or exceeds the number of parameters necessary to describe the same set in terms of sums of single Arrhenius exponentials. This check also tells whether it is useful... 

The simplest tunnel model which allows one to calculate Arrhenius curves of H-transfer reactions is the Bell tunneling model [7] which has been modified in our laboratory [9]. The model has been reviewed recently by Limbach et al. [26]. It is visualized in Fig. 6.8 which will be explained in the following. 

A set of Arrhenius curves calculated using Eq. (6.22) depends then on the following parameters ... 

AH -r -r represents the minimum energy for tunneling to occur as described above and is assumed to be isotope independent. Note that a similar effect on the Arrhenius curves may be obtained by using more complex barrier shapes [10]. 

Let us discuss the effects of 2a and Am on the Arrhenius curves. In Fig. 6.9(a) and (b) the H curves are identical, as the product 2a +AmyP is the same. However, the introduction of an extra turmeling mass of 3 in Fig. 6.9(b) reduces the kinetic isotope effects in the low-temperature regime but not in the high-temperature regime. On the other hand, by comparison of Fig. 6.9(a) with Fig. 6.9(c), or of Fig. 6.9(b) with Fig. 6.9(d), it follows that when a barrier of constant height becomes narrower, the regime of temperature independent kinetic isotope effects is reached at higher temperatures. 

Figure 6.9 Arrhenius curves of H and Dtrans-fer calculated according to the Bell-Limbach tunneling model. Minimum energyfortunnel-ing to occur E/ = 12.55 kj moh, barrier heights = 20.9 kJ moh E° = 27.2 kJ mol", tunnel-...

In the case of the single-barrier mechanism, all four hydrons are in flight in the transition state. Subsequent replacement of H by D involves similar primary kinetic isotope effects P, leading to equally spaced Arrhenius curves of the isotopic reactions in Fig. 6.19(a), with an overall kinetic isotope effect of k nnn/ki DDD P. This result is analogous to the single-barrier HH and the HHH-transfer cases discussed above. Note that, generally, the transfer of n hydrons is expected to give rise to an overall kinetic isotope effect of P" [26]. 

One observes three groups of Arrhenius curves, i.e. the HHHH curve, the group of the HHHD and the HDHD curves, and the group of the HHDD, HDDD and DDDD curves. Within each group the differences are small. They are further attenuated by isotope fractionation (Fig. 6.19c). The overall kinetic isotope effect, given by feHHHH DDDD js typical for a concerted double proton transfer reaction. It is interesting to note that replacement of the first H by D already leads to a... 

In other words, the observed rate constants depend in a similar way on the equilibrium constants in the three cases discussed. In order to have an impression of the effects on the Arrhenius curves let us discuss the intramolecular case given by Eq. (6.43). 

Figure 6.20(b) depicts the case where the formation of the reactive state involves a positive entropy and enthalpy. Such a case could happen if the reaction partners AH and B are involved in strong interactions with other species. For example, AH could be hydrogen bonded to any proton acceptor, or B to any proton donor, which requires this interaction to be broken before the partners can react. Now, the reacting state predominates at high temperatures and the non-reactive state at low temperatures. Only at high temperatures is the true Arrhenius curve measured, exhibiting a normal pre-exponential factor of about 13. At low temperatures, the... 

In this section various hydrogen transfer systems are reviewed for which Arrhenius curves of the different isotopic reactions are available over a large temperature range. Mainly the systems are discussed exhibiting degenerate hydrogen transfers which could be studied by dynamic NMR. The main question is how the reaction properties are related to the molecular structure. 

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