Diffusivity Arrhenius plot

Figure 2. Helium diffusion Arrhenius plot with Otway Basin borehole constraint. The laboratory-measured He diffusion from this sample of Durango apatite defines an extremely linear array (filled symbols), consistent with simple thermally activated volume-diffusion. He age measurements from the Otway Basin provide a completely independent estimate of He diffusivity (open symbol), under natural conditions, which is in excellent agreement with the extrapolated laboratory data. Laboratory data are from Farley (2000). The Otway Basin constraint is from House et al. (1999).

Fig. 47. Arrhenius plot of diffusion coefficient for (a) H and (b) D atoms on the (110) face of a tungsten crystal at coverage degree 0.1-0.9 as indicated. The cusps on the curves correspond to the phase transition.

Data was collected over a two-year period on the effect of water on DuPont s Zytel 101. In an Arrhenius plot of this data the failure point was the time when the elongation and impact strength started to decrease. This is not a chemical degradation, but rather a permeation or diffusion rate phenomenon. It shows that high temperature water tests can be used to predict normal temperature exposure results. 

Pai Vemeker and Kannan [1273] observe that data for the decomposition of BaN6 single crystals fit the Avrami—Erofe ev equation [eqn. (6), n = 3] for 0.05 < a < 0.90. Arrhenius plots (393—463 K) showed a discontinuous rise in E value from 96 to 154 kJ mole-1 at a temperature that varied with type and concentration of dopant present Na+ and CO2-impurities increased the transition temperature and sensitized the rate, whereas Al3+ caused the opposite effects. It is concluded, on the basis of these and other observations, that the rate-determining step in BaN6 decomposition is diffusion of Ba2+ interstitial ions rather than a process involving electron transfer. 

To summarize, the surface kinetics (or near surface kinetics) is the limiting step at lower temperature and diffusion is the rate limiting step at higher temperature. It is possible to switch from one rate-limiting step to the other by changing the temperature. This is illustrated in Fig. 2.9, where the Arrhenius plot (logarithm of the deposition rate vs. the reciprocal temperature) is shown for several reactions leading to the deposition of silicon,... 

R is the gas constant Dq and activation energy Eu are constants derived from an Arrhenius plot for diffusion coefficients applying at different temperatures, and solubility coefficient was obtained from a separate permeation test at TiK. Suitable testing using a specially constmcted permeation cell water-cooled at one end provided good validation data. 

Figure 5.37. Arrhenius plot illustrating the effect on the apparent activation energy of pore diffusion and transport limitations through the stagnation layer surrounding a catalyst...

As the working temperature of the substrate was increased, the induction period (the delay time) of increased conductivity decreased due to increased rate of lateral diffusion of hydrogen atoms towards the sensor. The activation energy for surface migration of particles along a Si02 substrate estimated from the tilt of the Arrhenius plot was found to be about 20 kj/mol. 

At high temperatures there is experimental evidence that the Arrhenius plot for some metals is curved, indicating an increased rate of diffusion over that obtained by linear extrapolation of the lower temperature data. This effect is interpreted to indicate enhanced diffusion via divacancies, rather than single vacancy-atom exchange. The diffusion coefficient must now be represented by an Arrhenius equation in the form... 

The activation energy can be determined from the gradient of a plot of In D versus 1 IT (Fig. 5.19). Such graphs are known as Arrhenius plots. Diffusion coefficients found in the literature are usually expressed in terms of the Arrhenius equation D0 and Ea values. Some representative values for self-diffusion coefficients are given in Table 5.2. 

Figure 5.19 Arrhenius plot of diffusion data, In D versus 1/T. The slope of the straight-line graph allows the activation energy of diffusion, a, to be determined, and the intercept at 1/T = 0 gives a value for the pre-exponential factor.

In most ordinary solids, bulk diffusion is dominated by the impurity content, the number of impurity defects present. Any variation in D0 from one sample of a material to another is accounted for by the variation of the impurity content. However, the impurity concentration does not affect the activation energy of migration, Ea, so that Arrhenius plots for such crystals will consist of a series of parallel lines (Fig. 5.21a). The value of the preexponential factor D0 increases as the impurity content increases, in accord with Eq. (5.13). 

Figure 7. Arrhenius plots of diffusion coefficients for Rb, Cs, and Sr. Solid lines are high temperature data (numbers are literature references). Dashed lines are extrapolated coefficients based on Equation 5. Open circles are from this study.

Figure 4.8 Arrhenius plot of elemental diffusivity in feldspars. From Yund (1983). Reprinted with permission of The Mineralogical Society of America.

Figure 11,22 (A) Saddle-shaped age spectra of calcic plagioclases from amphibolites, Broken Hill, Australia. (B) Arrhenius plots of reactor-produced isotopes for two of the hve samples, defining existence of three diffusion domains corresponding to albite-rich lamellae (domain 1) and anorthite-rich lamellae of different widths (domains 2 and 3). Reproduced with modifications from T. M. Harrison and I. McDougall (1981), with kind permission from Elsevier Science Publishers B.V, Amsterdam, The Netherlands.

Fig. 1.5 Arrhenius plot of diffusion coefBcients versus reciprocal temperatures for various minerals. Data from phases reacted under wet conditions are given as solid lines, whereas dry conditions are represented by dashed lines. Note that the rates for dry systems are generally lower and have higher activation energies (steeper slopes). (Modified after Cole and Chakraborty 2001)...

In a heterogeneous process, a sequence of steps involving diffusion and chemical reaction is involved. For example, a reactant must diffuse to the surface of a solid catalyst before adsorption and chemical reaction can occur. Either diffusion or the chemical reaction may be rate-controlling and, over a sufficiently large temperature range, the Arrhenius plot of In k vs. 1/T is no longer straight, but curved. 

Fig. 3. The Arrhenius plot for a heterogeneous reaction showing regions in which the rate is diffusion-controlled and reaction-controlled.

It is known that more than 30 reactions are needed to reproduce the radiation-induced reactions occurring in pure water. Intensive measurements with a pulse radiolysis method have been done at elevated temperature up to 300°C [25 2], and the temperature dependence of some reactions does not exhibit a straight line but a curved one in Arrhenius plot. These examples are the reactions of the hydrated electron with N2O, NOJ, NO2, phenol, Se04, 8203 , and Mn [33,35], and two examples, egq + NOJ and ejq -i- NOJ, are shown in Fig. 2. The rate constant for the reaction of hydrated electron with NOJ is near diffusion-controlled reaction at room temperature and is increasing with increasing temperature. Above 100°C, the rate does not increase and reaches the maximum at 150°C, and then decreases. Therefore the curve is concave upward in Arrhenius plot. 

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