Arrhenius equation, deviations from

In one dimension the truncation of the equations of motion has been worked out in detail [59]. This has allowed an accurate examination of the role of diffusion in desorption, and implications for the Arrhenius analysis in nonequilibrium situations. The largest deviations from the desorption kinetics of a mobile adsorbate obviously occur for an immobile adsorbate... 

The reader may now wish to verify that the activation energy calculated by logarithmic differentiation contains a contribution Sk T/l in addition to A , whereas the pre-exponential needs to be multiplied by the factor e in order to properly compare Eq. (139) with the Arrhenius equation. Although the prefactor turns out to have a rather strong temperature dependence, the deviation of a In k versus 1/T Arrhenius plot from a straight line will be small if the activation energy is not too small. 

The temperature dependence of the reaction rate constant closely (but not exactly) obeys the Arrhenius equation. Both theories, however, predict non-Arrhenius behavior. The deviation from Arrhenius behavior can usually be ignored over a small temperature range. However, non-Arrhenius behavior is common (Steinfeld et al., 1989, p. 321). As a consequence, rate constants are often fitted to the more general expression k = BTnexp( —E/RT), where B, n, and E are empirical constants. 

The nature of the neutral or acidic hydrolysis of CH2CI2 has been examined from ambient temperature to supercritical conditions (600 °C at 246 bar). Rate measurements were made and the results show major deviations from the simple behaviour expressed by the Arrhenius equation. The rate decreases at higher temperatures and relatively little hydrolysis occurs under supercritical conditions. The observed behaviour is explained by a combination of Kirkwood dielectric theory and ab initio modelling. 

Hence, the exponential Arrhenius equation has been transformed to a linear equation. Figure 1-2 shows kinetic data in k versus T (Figure l-2a) and in In k versus 1/T (Figure l-2b). Actually, 1000/Tinstead of 1/T is often used so that the numbers on the horizontal axis are of order 1, which is the same relation except now the slope is 0.001 / . Because the linear relation is so much simpler and more visual, geochemists and many other scientists love linear equations because data can be visually examined for any deviation or scatter from a linear trend. Hence, they take extra effort to transform a relation to a linear equation. As will be seen later, many other equations encountered in geochemistry are also transformed into linear equations. 

A more rigorous treatment of adsorption equilibria would include due allowance for the total number of surface sites available, which sets an upper limit on the surface concentrations to be used in the above equations, the variation in the heat of adsorption with coverage, and surface heterogeneity. The significant feature of Eq. (11), relevant to the present discussion of compensation behavior, is that this predicted temperature dependence of variations of c i and c2 results in no deviation from obedience to the Arrhenius equation. If a given set of kinetic results obey Eq. (1) the condition for a fit to Eq. (9) is... 

VFT behavior is obtained by equating z = T/(T — Tv ft) and noting that increment of chemical potential Ap — Vft [65], Fitting the VFT model to the experimental results of iH in the paraelectric phase gives Tv ft = 228 and Ap 0.02 eV. Identifying the temperature at which xB deviates from the Arrhenius model with the onset of cooperativity yields a minimum cluster size given by... 

The activation parameters for the acid-catalyzed hydrolysis of long chain alkyl sulfates compared to those for non-micellar ethyl sulfate calculated from potentiometric data indicate that the rate acceleration accompanying micellization is primarily a consequence of a decrease in the enthalpy of activation rather than an increase in the entropy (Kurz, 1962). However, the activation energies for the acid-catalyzed hydrolysis of sodium dodecyl sulfate calculated from spectrophotometric data have been reported to be identical (Table 8) for micellar and non-micellar solutions, but the entropy of activation for the hydrolysis of the micellar sulfate was found to be 6 9 e.u. greater than that for the non-micellar system (Motsavage and Kostenbauder, 1963). This apparent discrepancy may be due to the choice of the non-micellar state as the basis of comparison, i.e. ethyl sulfate and non-micellar dodecyl sulfate, to temperature dependent errors in the values of the acid catalyzed rate constant determined potentiometrically, or to deviations in the rate constants from the Arrhenius equation. 

Figure 5.3 depicts the Arrhenius plots of the apparent self-diffusion coefficient of the cation (Dcation) and anion (Oanion) for EMIBF4 and EMITFSI (Figure 5.3a) and for BPBF4 and BPTFSI (Figure 5.3b). The Arrhenius plots of the summation (Dcation + f anion) of the cationic and anionic diffusion coefficients are also shown in Figure 5.4. The fact that the temperature dependency of each set of the self-diffusion coefficients shows convex curved profiles implies that the ionic liquids of interest to us deviate from ideal Arrhenius behavior. Each result of the self-diffusion coefficient has therefore been fitted with VFT equation [6].

In contrast to the formally analogous van t Hoff equation [10] for the temperature dependence of equilibrium constants, the Arrhenius equation 1.3 is empirical and not exact The pre-exponential factor A is not entirely independent of temperature. Slight deviations from straight-line behavior must therefore be expected. In terms of collision theory, the exponential factor stems from Boltzmann s law and reflects the fact that a collision will only be successful if the energy of the molecules exceeds a critical value. In addition, however, the frequency of collisions, reflected by the pre-exponential factor A, increases in proportion to the square root of temperature (at least in gases). This relatively small contribution to the temperature dependence is not correctly accounted for in eqns 2.2 and 2.3. [For more detail, see general references at end of chapter.]... 

The factor i only occurs in solutions which are good conductors of electricity, and in 1887 Arrhenius succeeded in explaining these apparent deviations from the simple laws by his electrolytic dissociation theory. The molecules of an electrolyte are broken up to a greater or less extent into their free ions, even when the solution is not conducting a current of electricity. Thus we have the equation HCl H - - CL... 

For C-H bond cleavage, Equation (4) predicts a KIE equal to kH/kD 7 at room temperature. In the limit where the semiclassical theory is valid, experimentalists measure the Schaad-Swain exponent, ln(kH/A T)/ln(kD/kT). In the special case that the pre-Arrhenius factor A is the same for all isotopes (which is not true in most cases) then semiclassical theory predicts for this exponent a value 3.26. Deviations from this value are often interpreted as signs of increased tunneling, but in our opinion this line of argument is based on an oversimplified model of quantum transfer in condensed phases. Note that in tunneling reactions where the ratio Au/AD l, the semiclassical theory predicts an exponent that is not equal to 3.26 and is temperature dependent. 

E0 and the infinite temperature relaxation time To are independent of temperature, and (ii) in the isotropic phase near the I-N transition, the temperature dependence of ts2(T) shows marked deviation from Arrhenius behavior and can be well-described by the Vogel-Fulcher-Tammann (VFT) equation ts2(T) = TyFrQxp[B/(T — TVFF), where tvff, B, and tvft are constants, independent of temperature. Again these features bear remarkable similarity with... 

Figure 5 is a reproduction of the data referred to in the quotation from Aicher el al. (80). The straight line for 205° C. fits the data reasonably well, and one may conclude that the rate is retarded by the adsorbed products of the reaction. The data for 190° C. (Fig. 5) are inadequate to determine a straight line, and those for 186 C. indicate that at low conversions there is some deviation from the rule of inverse proportionality to the partial pressure of the reactants. Some of the data of Aicher and coworkers (80) (Fig. 5) are suitable for calculating the temperature coefficient for relatively short contact times (that is, conversions of about 10-50%). The coefficient for the temperature ranges 197-207° C. and 191-207° C. are 1.4 and 1.67/10° C., respectively. These coefficients correspond to an activation energy per mole of about 20 kcal. as calculated from the Arrhenius equation. This energy value is reasonable for a desorption process. 

The conductivity of ionic liquids often exhibits classical linear Arrhenius behavior above room-temperature. However, as the temperature of these ionic liquids approaches their glass transition temperatures (Tg) the conductivity displays significant negative deviation from linear behavior. The observed temperature-dependent conductivity behavior is consistent with glass-forming liquids, and is often best described using the empirical Vogel-Tammann-Fulcher (VTF) equation. 

A short but very instructive discussion of the subject has been given by Hulett (74) in 1964 under the title Deviations from the Arrhenius Equation more recent reviews are primarily concerned with the special aspects considered in Sections V (92, 136) and VI (23), respectively. The temperature-dependence of Ea in enzymatic reactions hat recently been discussed in detail (64). 

Our treatment, based on both the collision and the statistical formulations of reaction rate theory, shows that there exist two possibilities for an interpretation of the experimental facts concerning the Arrhenius parameter K for unimolecular reactions. These possibilities correspond to either an adiabatic or a non-adiabatic separation of the overall rotation from the internal molecular motions. The adiabatic separability is accepted in the usual treatment of unimolecular reactions /136/ which rests on transition state theory. To all appearances this assumption is, however, not adequate to the real situation in most unimolecular reactions.The nonadiabatic separation of the reaction coordinate from the overall rotation presents a new, perhaps more reasonable approach to this problem which avoids all unnecessary assumptions concerning the definition of the activated complex and its properties. Thus, for instance, it yields in a simple way the rate equations (7.IV), corresponding to the "normal Arrhenius parameters (6.IV), which are both direct consequences of the general rate equation (2.IV). It also predicts deviations from the normal values of the apparent frequency factor K without any additional assumptions, such that the transition state (AB)" (if there is one) differs more or less from the initial state of the activated molecule (AB). ... 

According to (101.IV), both E and K decrease when the temperature is lowered, in agreement with the observed deviations from the Arrhenius equation for both the gas reaction H2 + H (Sec.2.2.4.IV) and some proton-transfer reactions in solution /152,153/. 

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