Bulk reactions rate constants

In this section the foundations of the theory underlying chemical kinetics are presented. Based on the diffusion equation to describe Brownian motion together with Smoluchowski s theory [ 1, 2], a thorough derivation of the bulk reaction rate constant for neutral species for both diffusion and partially diffusion controlled reactions is presented. This theory is then extended for charged species in subsequent sections. 

Figure 13. Numerically calculated PMC potential curves from transport equations (14)—(17) without simplifications for different interfacial reaction rate constants for minority carriers (holes in n-type semiconductor) (a) PMC peak in depletion region. Bulk lifetime 10" s, combined interfacial rate constants (sr = sr + kr) inserted in drawing. Dark points, calculation from analytical formula (18). (b) PMC peak in accumulation region. Bulk lifetime 10 5s. The combined interfacial charge-transfer and recombination rate ranges from 10 (1), 100 (2), 103 (3), 3 x 103 (4), 104 (5), 3 x 104 (6) to 106 (7) cm s"1. The flatband potential is indicated.

As already has been mentioned mass transfer of ozone from the gas phase to the liquid phase may be enhanced by the chemical reactions of ozone with components A and B and by the decay of ozone. The effect of this enhancement in mass transfer on the selectivity will be discussed now semi-quantitatively13. To that aim we consider a gas phase in contact with a liquid phase. The liquid phase consists of a thin stagnant film at the interface with the gas phase, and a liquid bulk phase. We assume that the ozone is completely converted in the stagnant liquid film. This is for example the case if we have to deal with a high reaction rate constant and a relatively high concentration of one of the pollutants in the liquid film. Figure 5 gives a schematically presentation of this situation. 

The liquid bulk is assumed to be at chemical equilibrium. Contrary to gas-liquid systems, for vapour-liquid systems it is not possible to derive explicit analytical expressions for the mass fluxes which is due to the fact that two or more physical equilibrium constants m, have to be dealt with. This will lead to coupling of all the mass fluxes at the vapour - liquid interface since eqs (15c) and (19) have to be satisfied. For the system described above several simulations have been performed in which the chemical equilibrium constant K = koiAo2 and the reaction rate constant koi have been varied. Parameter values used in the simulations are given in Table 5. The results are presented in Figs 9 and 10. 

Due to chemical conversion in the liquid-phase mass transfer film the mass flux of A at the vapour-liquid interface and the mass flux of A at the boundary between this film and the liquid bulk will differ. Figures 9(a) and (b) show the values of these fluxes as a function of the reaction rate constant ko for equilibrium constants K = 1 and X = 100. The... 

Fig. 9. The molar flux of component A at the vapour-liquid interface (°) and at the boundary between mass transfer film and liquid bulk (S) as function of reaction rate constant in case (a) the mass transfer coefficients are equal and (b) the mass transfer coefficients are different.

In Equation (4.31) the rate constant is either the reaction rate constant or the transport rate constant, depending on which rate controls the dissolution process. If the reaction rate controls the dissolution process, then k. t becomes the rate of the reaction while if the dissolution process is controlled by the diffusion rate, then k j becomes the diffusion coefficient (diffusivity) divided by the thickness of the diffusion layer. It is interesting to note that both dissolution processes result in the same form of expression. From this equation the dependence on the solubility can be seen. The closer the bulk concentration is to the saturation solubility the slower the dissolution rate will become. Therefore, if the compound has a low solubility in the dissolution medium, the rate of dissolution will be measurably slower than if the compound has a high solubility in the same medium. 

In a chemical combustion reaction the amount of the original materials entering into the reaction, the amounts of the combustion products that form and the quantity of heat generated are in definite, strictly constant relations to one another. Denoting the bulk reaction rate by F, we express all the other quantities in terms of it ... 

On metal surfaces, well-ordered structures are often formed [7, 79-82], At sufficiently high temperatures (> 300 K) and pressures (> 10 6 Torr), oxygen diffuses into the catalyst bulk [8, 61, 73 76, 83-91] in Pd, as many as 300 monolayers are dissolved [83] whereas in Ir and Pt the number of dissolved monolayers is slightly lower [78, 85 87]. The oxygen dissolved in the subsurface layer changes the reaction rate constants considerably (see Table 3 [86]). Finally, under certain conditions, oxygen adsorption can lead to surface reconstruction [7, 92]. Various types of oxygen adsorption over Pt metals have been studied in detail by Savchenko [7]. 

The reaction rate constant is k2 = 0.016 m6/mol2/s [9, 10, 11, 14]. Considering similar mass transfer and diffusion coefficients of NO and oxygen, Hao2 = 1-5 is obtained. This small value shows that the reaction is slow and takes place mainly in the bulk of the liquid. We recall that reaction (12.3) is undesired, because it consumes the iron-EDTA complex. 

Let us now consider a catalytic packed bed reactor , i.e. a tubular reactor filled with a grained catalyst through which the gas mixture flows. With the particle diameter of the catalyst, dp, an additional dimensionless number dp/d is added to the pi-space the Reynolds number is now expediently formed with dp. The reaction rate is related to the unit of the bulk volume and characterized by an effective reaction rate constant ko,eff = k . The thermal conductivity (k) also has to be valid for the gas/bulk solids system and diffusion can be considered as being negligible (Sc is irrelevant). The complete pi-space is therefore ... 

A schematic of the flow-through nanohole array concept is shown in Fig. 13a. Figure 13b shows computationally predicted biomarker transport within the nanoholes for in-hole average fluid velocities of 1 pm/s and 1 cm/s (as indicated). Reaction rate constants characteristic of surface-based antibody-antigen reactions (with reaction rate constant k - 10 /M/s) [69] were applied at the nanohole walls. For the low average velocity, diffusion of the biomarker (with diffusivity D - 4x10 m s ) to the nanohole surface is effectively complete in one diameter. This result reflects the rapid diffusion characteristic of nanoconfinement. For the higher flow rate case, the absorption of the analyte stream is delayed however, over 90% bulk adsorption of analyte is attained with the flow rates and nanohole... 

In Table 2, the relative steady-state surface concentration (with respect to the maximum steady-state concentration defined by the lowest surface reaction rate constant used in the study) as a function of surface reaction rate constant and pseudo-first-order rate constant for the bulk reaction is given. 

Such an approach could explain, though in a semiquantitative way, the behavior of the systems studied earlier, when a minimum was observed on the kf versus solvent composition dependence. Analysis of the change in the rate of reaction, expressed as a product of the electrode reaction rate constant and the reactant concentration in the surface phase, c, in mixtures of water with acetone reveals a deep minimum, which corresponds to the greatest difference in composition of the surface and bulk phases. 

The simulations were performed in using the reaction rate constants obtained from the autoclave experiment. The bulk diffusion coefficients for reactant and intermediates were estimated... 

In Eq. (7-1) is the usual mass-transfer coefficient based on a unit of transfer surface, i.e., a unit of external area of the catalyst particle. In order to express the rate per unit mass of catalyst, we multiply k by the external area per unit mass, a. In Eq. (7-2) k is the reaction-rate constant per unit surface. Since a positive concentration difference between bulk gas and solid surface is necessary to transport A to the catalyst, the surface concentration Cj will be less than the bulk-gas concentration Q. Hence Eq. (7-2) shows that the rate is less than it would be for = Q. Here the effect of the mass-transfer resistance is to reduce the rate. Figure 7-1 shows schematically how the concentration varies between bulk gas and catalyst surface. 

The same reasoning suggests that there will be a temperature difference between bulk fluid and catalyst surface. Its magnitude will depend on the heat-transfer coefficient between fluid and catalyst surface, the reaction-rate constant, and the heat of reaction. If the reaction is endothermic,... 

In reality, a typical catalyst pellet will be a porous solid that may be quite complicated or even irregular in shape with a large number of catalytic reaction sites distributed throughout. However, to simplify the problem for present purposes, the catalyst pellet will be approximated as being spherical in shape. Furthermore, we will assume that the catalyst pellet is uniform in constitution. Thus we assume that it can be characterized by an effective reaction-rate constant kef that has the same value at every point inside the pellet. In addition, we assume that the transport of reactant within the pellet can be modeled as pure diffusion with a spatially uniform effective diffusivity To Author simplify the problem, we assume that the transport of product out of the pellet is decoupled from the transport of reactant into the pellet. Finally, the concentration of reactant in the bulk-phase fluid (usually... 

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