The Arrhenius Plot Method

To determine the A and E parameters by the Arrhenius plot method, one usually invokes the Arrhenius equation in logarithmic form 

The pre-exponential factor. A, however, as shown before (Tables 3.1 and 3.2), depends on T, both directly and through the diffusion coefficient, D, which is proportional to Therefore, the values of E derived from the Arrhenius 

If these corrections are neglected, the difference between the In A parameters measured in vacuum and in a foreign gas would constitute at 1,000 K about 1.25, and between the E parameters, approximately 10.4 kJ moP The majority of researchers using the Arrhenius plot method fail to take these corrections into account, although the final figures for the E parameter are given frequently (see, e.g., [8]) to within a few kJ moP, or even a few tenths of kJ moP and the values of InA, to within an uncertainty of a few tenths of percent. 

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Fig. 4.1 The distribution of publications on the determination of the E parameter by the Arrhenius plot method as a function of Tmax/AT ratio, in steps of equal increments (0.3). 220 items are included...

This theoretical estimate is borne out by the data obtained in the high-vacuum decompositions of the carbonates, hydrates, sulphates, and hydroxides (Chapter 16). In full agreement with theoretical estimates (see Chapters), the difference in temperature between the temperature-stabilized heater, say, a crucible and a sample contained in this crucible is, under high-vacuum conditions, a few tens of degrees and may become, in extreme cases, as high as 3% of the heater temperature. In many cases where the second-law and the Arrhenius plot methods are employed, this systematic error would translate into an underestimation of the values of and E by 15-20%. 

One more typical systematic error that arises in determination of the E parameter with the Arrhenius plot method is the fairly arbitrary choice of kinetic model used to estimate the rate constant k from primary TA measurements. This statement is confirmed by numerous studies generalized, in... 

Such a scatter in results may be due to experimental errors, and primarily to the use of the Arrhenius plot method in conditions far from optimal when the ratio Tmax/AT achieves too high values (see Sect. 4.6). For example, the Tmax/AT ratio in work by Zawadski and Bretsznajder [10] was 30, 30, and 100 at CO2 pressures of 0.026, 0.039, and 0.059 bar, respectively. Even if the rate constants k2 and ki were measured to within an error of 2-3% only, the uncertainty in the determination of the E parameter could reach 200-300%. 

Parameters E had been calculated using the Arrhenius plot method. 

In fig. 26 the Arrhenius plot ln[k(r)/coo] versus TojT = Pl2n is shown for V /(Oo = 3, co = 0.1, C = 0.0357. The disconnected points are the data from Hontscha et al. [1990]. The solid line was obtained with the two-dimensional instanton method. One sees that the agreement between the instanton result and the exact quantal calculations is perfect. The low-temperature limit found with the use of the periodic-orbit theory expression for kio (dashed line) also excellently agrees with the exact result. Figure 27 presents the dependence ln(/Cc/( o) on the coupling strength defined as C fQ. The dashed line corresponds to the exact result from Hontscha et al. [1990], and the disconnected points are obtained with the instanton method. For most practical purposes the instanton results may be considered exact. 

AC is interpreted as the difference in heat capacities between the transition state and the reactants, and it may be a valuable mechanistic tool. Most reported ACp values are for reactions of neutral reactants to products, as in solvolysis reactions of neutral esters or aliphatic halides. " Because of the slight curvature seen in the Arrhenius plots, as exemplified by Fig. 6-2, the interpretation, and even the existence, of AC is a matter of debate. The subject is rather specialized, so we will not explore it deeply, but will outline methods for the estimation of ACp. 

Arrhenius plot. In an Arrhenius plot the ordinate is the log of the material life. The abscissa is the reciprocal of the absolute temperature. The linear curves obtained with the Arrhenius plot overcome the deficiency of most of the standard tests, which provide only one point and indicate no direction in which to extrapolate. Moreover, any change in any aspect of the material or the environment could alter the slopes of there curves. Therein lies the value of this method. 

FIGURE 1.37 The Arrhenius plot of Ce08Sm0i2Oli9 from different methods (A) solid-state reaction [95] (B) sol-gel process [116] (C) oxalate coprecipitation [91] (D) carbonate coprecipitation, our work and (E) glycine-nitrate process [157]. 

The best fit to the Arrhenius plot can be found by the least squares method (applied to In t or log t) and extrapolated to find the time to the threshold value (tu) at a temperature of interest (Tu). To obtain an estimate of the maximum temperature of use, extrapolate the line to a specified reaction rate or time to reach a threshold value. 20,000 or 100,000 hours duration and 50% change as the threshold value are commonly used for establishing a general maximum temperature of use. 

Another important factor is the speed of measurements. Here TDH usually comes out ahead of TSC and PITS, because the latter methods involve a separate temperature scan for each data point in the Arrhenius plot, since eni is determined at the spectral peak. However, it is also possible, with more sophisticated equipment, to actually measure eni from the current transient itself, at each temperature (Kirchner et al., 1981). Then only one temperature scan is necessary for the Arrhenius plot of eni. However, in this case it is necessary to know the precise position of the transient base line. 

To use the Roller model requires the measurement of cure curves at several different temperatures so that Arrhenius plots may be made for the rate constant, k, and the Isothermal melt viscosity, n. The four model constants are then obtained from the Arrhenius plots. This entails considerable experimental work, especially for a method one would like to use in a more or less routine fashion. In addition, it is invalid in principle to... 

Rudzinski W., Borowiecki T., Panczyk T., and Dominko A., On the Applicability of Arrhenius Plot Methods to Determine Surface Energetic Heterogeneity of Adsorbents and Catalysts Surfaces from Experimental TPD Spectra, Adv. Coll. Interface Sci., 84... 

The first-order rate coefficients and the Arrhenius parameters determined at higher pressures (generally above 100 torr) agree reasonably well, as can be seen from Table 6 and Fig. 5. The straight line drawn by the least-squares method through the points of the Arrhenius plot gives the first-order rate coefficient as... 

The discussion of interpolation and extrapolation methods has so far assumed that the reaction involved proceeds to one set of products over the whole temperature range. If more than one channel is open to the reaction, the branching ratio between the channels may also vary with temperature and this variation will lead to curvature of the Arrhenius plot... 

Boum describe their use of the line shape method in the temperature range —24° to — 82°C (deuterium decoupling was used so that only the unperturbed single proton resonance was observed) and of a double-resonance method in the temperature range —97° to — 116°C. This involved the observation of recovery of magnetization of one of the two lines in the spectrum after a saturating r.f. field applied to the other line was removed. Consistent rates of inversion were found from both methods as evidenced by linearity of the Arrhenius plot. The results do not agree with the spin-echo results of Allerhand et al In this type of work, while fairly consistent results of rate constants may be obtained, there is dispute as to how the thermodynamic parameters should be derived, even in the relatively simple case of cyclohexane. ... 

A comparison of the line-shape method with an equilibration method has shown good agreement in results (the Arrhenius plots for both methods are colinear). The example chosen was the interconversion of rotamers in A7 -methyl-A(-benzylformamide, which between 90° and 170°C shows averaged spectra. One of the rotamers can be obtained substantially pure at low temperatures, and the return to equilibrium was studied in the range —2 5° to 20°C. A similar study using AT-benzyl-A -2,4,6-tetramethylbenzamide and N,AT-2,4,6-penta-methylbenzamide has also been made. °... 

Another reason that can account for the systematic differences between the values of A H (II), measured by the second-law and Arrhenius plot methods, and those of A H (III) measured by the third-law method, is the systematic decrease of the contribution of the condensation energy to the reaction enthalpy with increasing temperature and the ensuing slight increase of A H (III) and a substantially larger decrease of A H (II). This effect will be considered in detail in Sect. 8.2. 

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