Arrhenius rate

Intrinsic Kinetics. Chemisorption may be regarded as a chemical reaction between the sorbate and the soHd surface, and, as such, it is an activated process for which the rate constant (/ ) follows the familiar Arrhenius rate law ... 

The Arrhenius rate theory, an empirical derivation, holds for the sterilization process ... 

The symmetry coefficient = —P d nk/dAE is usually close to j, in agreement with the Marcus formula. Turning to the quantum limit, one observes that the barrier transparency increases with increasing AE as a result of barrier lowering, as well as of a decrease of its width. Therefore, k grows faster than the Arrhenius rate constant. At 7 = 0... 

Deposition of TiN by the thermal decomposition of tetrakis(dimethylamido)titanium (TDMAT) in a nitrogen atmosphere (as opposed to ammonia) was characterized by a simple Arrhenius rate expression. Adequate deposition rates and good step coverage were achieved for 3/1 aspect ratio holes, 0.40 micron in size. A reactor model was designed,... 

In these equations the independent variable x is the distance normal to the disk surface. The dependent variables are the velocities, the temperature T, and the species mass fractions Tit. The axial velocity is u, and the radial and circumferential velocities are scaled by the radius as F = vjr and W = wjr. The viscosity and thermal conductivity are given by /x and A. The chemical production rate cOjt is presumed to result from a system of elementary chemical reactions that proceed according to the law of mass action, and Kg is the number of gas-phase species. Equation (10) is not solved for the carrier gas mass fraction, which is determined by ensuring that the mass fractions sum to one. An Arrhenius rate expression is presumed for each of the elementary reaction steps. 

Pure PHEMA gel is sufficiently physically cross-linked by entanglements that it swells in water without dissolving, even without covalent cross-links. Its water sorption kinetics are Fickian over a broad temperature range. As the temperature increases, the diffusion coefficient of the sorption process rises from a value of 3.2 X 10 8 cm2/s at 4°C to 5.6 x 10 7 cm2/s at 88°C according to an Arrhenius rate law with an activation energy of 6.1 kcal/mol. At 5°C, the sample becomes completely rubbery at 60% of the equilibrium solvent uptake (q = 1.67). This transition drops steadily as Tg is approached ( 90°C), so that at 88°C the sample becomes entirely rubbery with less than 30% of the equilibrium uptake (q = 1.51) (data cited here are from Ref. 138). 

Table 4.2 Typical fuel production for a zeroth-order arrhenius rate...

Raising a mixture of fuel and oxidizer to a given temperature might result in a combustion reaction according to the Arrhenius rate equation, Equation (4.1). This will depend on the ability to sustain a critical temperature and on the concentration of fuel and oxidizer. As the reaction proceeds, we use up both fuel and oxidizer, so the rate will slow down according to Arrhenius. Consequently, at some point, combustion will cease. Let us ignore the effect of concentration, i.e. we will take a zeroth-order reaction, and examine the concept of a critical temperature for combustion. We follow an approach due to Semenov [3],... 

K = Kq exp(—E /T), an Arrhenius rate expression Ao feed concentration, gmol cm-3 7o feed temperature, K A tank concentration, gmol cm-3 T tank temperature, K... 

With the importance of the devolatilization process to solid particle combustion and the complexity of the chemical and physical processes involved in devolatilization, a wide variety of models have been developed to describe this process. The simplest models use a single or multiple Arrhenius rates to describe the rate of evolution of volatiles from coal. The single Arrhenius rate model assumes that the devolatilization rate is first-order with respect to the volatile matter remaining in the char [40] ... 

In Fig. 10, experimental data for these reactions are presented. Two-parameter (Arrhenius) rate coefficient expressions for these three reactions were determined to be 2.1 x 10 exp(483/ T)cmVniol-s (Fontijn and Zellner, 1983), 3.97 x 10 exp(250/ 7 cm mole-s (Slagle et al, 1984), and 1.2 X 10 exp(830/ 7 cm /mol-s (Russell et al, 1988b), respectively. These fits, indicated by solid lines, are also shown in Fig. 10. The rates for the latter two reactions were also determined to be pressure-independent, suggesting that under the experimental conditions studied, the lifetime of the adduct must be short relative to collision times. 

Fig. 10. Arrhenius plots for the reactions HOj + NO OH + NOj (Howard, 1980), C2H3 + O2 CHjO + CHO (Slagle et ai, 1984), and C Clj + 02 =iC0Cl2 + COCl (Russell et al, 1988). Solid lines represent two-parameter (Arrhenius) rate coefficients fits to the experimental data (see text).

Figure 3.3. Schematic of direct and precursor-mediated dissociation processes on a typical adiabatic PES (given by the solid line). Solid arrow labeled S represents direct dissociation and that labeled a represents trapping into a molecular adsorption well. Dashed arrows represent competing thermal (Arrhenius) rates for desorption (kd) and dissociation (kc) from the molecular well.

The temperature dependence of the rate constant for the step A -> B leads to the term /(0) in the dimensionless mass- and heat-balance eqns (4.24) and (4.25). The exact representation of an Arrhenius rate law is f(9) — exp[0/(l + y0)], where y is a dimensionless measure of the activation energy RTa/E. As mentioned before, y will typically be a small quantity, perhaps about 0.02. Provided the dimensionless temperature rise 9 remains of order unity (9 < 10, say) then the term y9 may be neglected in the denominator of the exponent as a first simplification. 

FONI model (a) unique (b) single hysteresis loop or breaking wave (c) isola (d) isola + hysteresis loop (e) mushroom. With the full Arrhenius rate law and the provision pre-heating or cooling, two additional patterns are found (f) reversed hysteresis loop and (g) reversed hysteresis loop + isola. Also shown are various degenerate loci corresponding to parameter values on the boundaries or special points in the parameter plane (see Fig. 7.5). 

Figure 7.5 is quantitatively correct only for the special case of the exponential approximation to the Arrhenius rate law. However, the figure is also qualitatively correct for the exact Arrhenius form with non-zero y, provided y < 4. No new stationary-state patterns are introduced. 

With the full Arrhenius rate law, an extra unfolding parameter y is introduced. Even then, however, the appropriate stationary-state condition and its derivatives for the winged cusp cannot be satisfied simultaneously (at least not for positive values of the various parameters). Thus we do not expect to find all seven patterns. 

We may also note, for the special case / = 1, that the locus described by eqns (10.58) and (10.59) is exactly that corresponding to the boundary between unstable focus and unstable node for the well-stirred system. This seems to be a general equivalence between the existence of unstable nodal solutions in the well-stirred system and the possibility of diffusion-driven pattern formation in the absence of stirring. We have seen in chapter 5 that unstable nodes are not found in the present model if the full Arrhenius rate law is used and the activation energy is low, i.e. iff <4 RTa. In that case we would also not expect spatial instability. 

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