Dimensionless concentrations second-order kinetics

The average dimensionless solid phase concentration Y can be given409 the numerical value of 0.5, and if the effect of the film mass transfer is negligible, i.e., if K f - 00, then adsorption with the nonporous HPLC sorbent in a well-packed bed is controlled by second-order kinetics.169,399 When the external film resistance Kf controls the adsorption, equilibrium is assumed to exist between the polypeptide or protein and the polypeptide- or protein-ligate complex at each point on the particle surface. 

Reactant concentrations and effectiveness factors are calculated for diffusion and second-order kinetics in isothermal catalytic pellets via the methodology described above. In each case, convergence is achieved when the dimensionless molar density of reactant A at the external surface is 1 5, where S is on the order of 10 or less. 

The second-order reaction with adsorption of the ligand (2.210) signifies the most complex cathodic stripping mechanism, which combines the voltammetric features of the reactions (2.205) and (2.208) [137]. For the electrochemically reversible case, the effect of the ligand concentration and its adsorption strength is identical as for reaction (2.205) and (2.208), respectively. A representative theoretical voltammo-gram of a quasireversible electrode reaction is shown in Fig. 2.86d. The dimensionless response is controlled by the electrode kinetic parameter m, the adsorption... 

Now suppose the tip-generated species is not stable and decomposes to an elec-troinactive species, such as in the case (Chapter 12). If R reacts appreciably before it diffuses across the tip/substrate gap, the collection efficiency will be smaller than unity, approaching zero for a very rapidly decomposing R. Thus a determination of /x//s as a function of d and concentration of O can be used to study the kinetics of decomposition of R. In a similar way, this decomposition decreases the amount of positive feedback of O to the tip, so that ij is smaller than in the absence of any kinetic complication. Accordingly, a plot of ij vs. d can also be used to determine the rate constant for R decomposition, k. For both the collection and feedback experiments, k is determined from working curves in the form of dimensionless current distance (e.g., did) for different values of the dimensionless kinetic parameter, K = kcP ID (first-order reaction) or = k a CQlD (second-order reaction). 

If we were to change the kinetics so that the first reaction was second order in A and the second reaction was first order in B, then we would see largely the same picture emerging in the graphs of dimensionless concentration versus time. There would of course be differences, but not large departures in the trends from what we have observed for this all first-order case. But what if the reactions have rate expressions that are not so readily integrable What if we have widely differing, mixed-order concentration dependencies In some cases one can develop fully analytical (closed-form) solutions like the ones we have derived for the first-order case, but in other cases this is not possible. We must instead turn to numerical methods for efficient solution. 

Subscript j is unnecessary when there is only one chemical reaction. When the kinetics are second-order, irreversible, and only a function of Ca, the dimensionless pseudo-volumetric reaction rate, based on bulk gas-phase concentrations, is... 

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