Arrhenius shift

However, as mentioned in Chapter 1, a variation in temperature corresponds to a shift in the time scale. A shift commonly used for semi-crystalline polymers is the Arrhenius shift, which is written as... 

Fig. 58. Lower left-hand scale Arrhenius plot of the linewidths (lower and upper curves, for inelastic and quasielastic peaks, respectively). Upper right-hand scale Arrhenius plot of the shift of the inelastic peaks.

The idea that /3 continuously shifts with the temperature employed and thus remains experimentally inaccessible would be plausible and could remove many theoretical problems. However, there are few reaction series where the reversal of reactivity has been observed directly. Unambiguous examples are known, particularly in heterogeneous catalysis (4, 5, 189), as in Figure 5, and also from solution kinetics, even when in restricted reaction series (187, 190). There is the principal difficulty that reactions in solution cannot be followed in a sufficiently broad range of temperature, of course. It also seems that near the isokinetic temperature, even the Arrhenius law is fulfilled less accurately, making the determination of difficult. Nevertheless, we probably have to accept that reversal of reactivity is a possible, even though rare, phenomenon. The mechanism of such reaction series may be more complex than anticipated and a straightforward discussion in terms of, say, substituent effects may not be admissible. 

At low temperatures the reaction is negatively affected by the lack of oxygen on the surface, while at higher temperatures the adsorption/desorption equilibrium of CO shifts towards the gas phase side, resulting in low coverages of CO. As discussed in Chapter 2, this type of non-Arrhenius-like behavior with temperature is generally the case for catalytic reactions. 

The maximum operating temperature may be evaluated from DTA/DSC curves if the shift of the peak due to reduction of the heating rate to 0.5-1 K min does not exceed 40 K. This corresponds to moderate activation energies of the secondary reactions, i.e. less than 50 kJ/mol for values obtained from the Arrhenius plots . 

Fig. 24. Arrhenius plots of InilciJ ( ) and Inlfc J (A) versus l/T where 1c,l and Icli are the rate constants for the spin-state transitions S = 3/2 - S = 1/2, and S = 1/2 - S = 3/2, respectively. For clarity of presentation, the data points have been shifted in temperature by one degree up for ln(fc,L) and by one degree down for In(lcLi). According to Ref. [164]...

Figure 10.11 Arrhenius plots of the ORR rate constants obtained at various electrodes. The symbols are the same as those in Fig. 10.10. Each solid line is the least squares fit of all the data at the constant applied potential. Since the standard potential E° and [RHE(r)] shift to less positive values in a different maimer, the corrected potential E is applied so as to keep a constant overpotential for the ORR at each temperature. The applied potentials of -0.485, -0.525, and -0.585 V vs. E° correspond to 0.80, 0.76, and 0.70 V vs. RHE, respectively, at 30 °C. (From Yano et al. [2006b], reproduced by permission of the PCCP Owner Societies.)...

The standard-potential, E°, shows a temperature dependence called the "zero shift , according to its direct relationship with the free enthalpy for the standard conditions chosen, - AG° = RTIn K (eqn. 2.37), and the Arrhenius equation for the reaction rate,... 

This behavior is in between that of a liquid and a solid. As an example, PDMS properties obey an Arrhenius-type temperature dependence because PDMS is far above its glass transition temperature (about — 125°C). The temperature shift factors are... 

Several attempts have been made to superimpose creep and stress-relaxation data obtained at different temperatures on styrcne-butadiene-styrene block polymers. Shen and Kaelble (258) found that Williams-Landel-Ferry (WLF) (27) shift factors held around each of the glass transition temperatures of the polystyrene and the poly butadiene, but at intermediate temperatures a different type of shift factor had to be used to make a master curve. However, on very similar block polymers, Lim et ai. (25 )) found that a WLF shift factor held only below 15°C in the region between the glass transitions, and at higher temperatures an Arrhenius type of shift factor held. The reason for this difference in the shift factors is not known. Master curves have been made from creep and stress-relaxation data on partially miscible graft polymers of poly(ethyl acrylate) and poly(mcthyl methacrylate) (260). WLF shift factors held approximately, but the master curves covered 20 to 25 decades of time rather than the 10 to 15 decades for normal one-phase polymers. 

Fig. 8 Temperature dependence of din f>/d(T 1), i.e., slope of the Arrhenius plot as a function of temperature for (a) (EDT-TTFBr2)FeBr4 at various pressures - the data for 0, 5.8 and 10.1 kbar are vertically shifted up by 60, 40 and 20 K, respectively, for clarity (b) (EDO-TTFBr2)2GaCl4 and (EDO-TTFBr2)2FeCl4 at 11 kbar. TMl and TN are the metal-insulator transition temperature and the Neel temperature, respectively, hi (b) the metal-insulator transition is observed as two separate peaks...

The final question we shall consider here has to do with the extrapolation of the solubility of hydrogen in silicon to lower temperatures. Extrapolation of a high-temperature Arrhenius line, e.g., from Fig. 11, would at best give an estimate of the equilibrium concentration of H°, or perhaps of all monatomic species, in intrinsic material the concentration of H2 complexes would not be properly allowed for, nor would the effects of Fermi-level shifts. Obviously the temperature dependence of the total dissolved hydrogen concentration in equilibrium with, say, H2 gas at one atmosphere, will depend on a number of parameters whose values are not yet adequately known the binding energy AE2 of two H° into H2 in the crystal, the locations of the hydrogen donor and acceptor levels eD, eA, respectively, etc. However, the uncertainties in such quantities are not so... 

Three possibilities were considered to account for the curved Arrhenius plots and unusual KIEs (a) the 1,2-H shift might feature a variational transition state due to the low activation energy (4.9 kcal/mol60) and quite negative activation entropy (b) MeCCl could react by two or more competing pathways, each with a different activation energy (e.g., 1,2-H shift and azine formation by reaction with the diazirine precursor) (c) QMT could occur.60 The first possibility was discounted because calculations by Storer and Houk indicated that the 1,2-H shift was adequately described by conventional transition state theory.63 Option (b) was excluded because the Arrhenius curvature persisted after correction of the 1,2-H shift rate constants for the formation of minor side products (azine).60... 

The studies of MeCCl refocused attention on benzylchlorocarbene (10a). Earlier studies of 10a, over a temperature range of 0-31°C, afforded linear Arrhenius correlations for the 1,2-H shift, with Ea = 4.5-4.8 kcal/mol and log A 11.2 s-1.36 Additionally, LFP studies of p-CF3 and p-Cl substituted benzylchlorocarbenes (lOf and lOg) in isooctane over a temperature range of —3 to 47°C gave linear Arrhenius correlations with a (4.9 and 4.5 kcal/mol) and log A (10.9 s-1) values comparable to those found for parent carbene 10a.64... 

In the more polar solvent, chloroform, linear Arrhenius correlations were observed for the rearrangements of both 10a and 10b from —55 to 60°C. Arrhenius parameters were a = 3.6 kcal/mol and log A = 10.4 s-1 for 10a, and Ea = 4.05 kcal/mol and log A = 10.3 s 1 for 10b.66 These results accord with the idea that polar solvents stabilize the polar 1,2-H shift, hydride-like transition state (51), accelerating this reaction at the expense of potential competitors.4,22... 

Of course carbene C-H insertion reactions are well known absolute kinetics have been reported for the insertions of ArCCl into isooctane, cyclohexane, and n-hexane,67 and of PhCCl into Si-H, Sn-H, and C-H bonds.68 More recently, detailed studies have appeared of PhCCl insertions into a variety of substrates bearing tertiary C-H bonds, especially adamantane derivatives.69 Nevertheless, because QMT is considered important in the low temperature solution reactions of MeCCl,60,63 and is almost certainly involved in the cryogenic matrix reactions of benzylchlorocarbene,59 its possible intervention in the low temperature solution reactions of the latter is a real possibility. We are therefore faced with two alternative explanations for the Arrhenius curvature exhibited by benzylchlorocarbene in solution at temperatures < 0°C either other classical reactions (besides 1,2-H shift) become competitive (e.g., solvent insertion, azine formation), or QMT becomes significant.7,59,66... 

The possible intervention of classical, competitive reactions in the low temperature solution chemistry of benzylchlorocarbene (10a) requires careful investigation. There are reasons to suspect azine (48) formation Goodman reported minor yields of azine in analogous MeCCl experiments,60 and Liu et al. found 40% of 48 in the photolysis of neat diazirine 9a.65 Perhaps azine formation is also significant at low temperature in hydrocarbon solvents. If so, the intervention of bimolecular azine formation, in competition with the unimolecular carbene 1,2-H shift, could lead to a nonlinear temperature dependence for the disappearance of 10a. Arrhenius curvature could then be explained without invoking QMT. 

The significant incursion of intermolecular products implies that the kinetic data previously obtained for the disappearance of 10a at low temperatures66 is biased and should not be used in Arrhenius treatments of the 1,2-H shift reaction. Therefore, the curved Arrhenius correlations do not require a QMT rationalization. 

Photolytic decomposition of diazirine 9a in methylcyclohexane led to substantial C-H insertion of PI1CH2CCI into the solvent, although azine was a minor product. At 25°C, there were 74% of 1,2-H shift products and 14% of C-H insertion. Insertion increased to 44% at —75°C. Here too, a curved Arrhenius correlation reflected the competition of two classical reactions, not the incursion of QMT.71... 

These results support the idea that Arrhenius curvature in the rearrangements of MeCCl60 (and MeCBr61) may be associated with QMT, although the theoretical analysis found that QMT dominated the 1,2-H(D) shift only below —73°C at higher temperatures, the classical process became more important.63 The benzylchlorocarbene case is less clear. QMT is clearly important in matrices at 10-34 K, where the KIE for 1,2-H(D) shift is 2000 59 cf. Section IV.A. However, the nonlinear Arrhenius behavior exhibited by 10a or 10b in solution is largely due to the intervention of intermolecular reactions (Section IV.C) which obscure any contribution of QMT.71... 

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