Adsorption rate equations

For a system with n components (including nonad-sorbable inert species) there are n — 1 differential mass balance equations of type (17) and n — 1 rate equations [Eq. (18)]. The solution to this set of equations is a set of n — 1 concentration fronts or mass transfer zones separated by plateau regions and with each mass transfer zone propagating through the column at its characteristic velocity as determined by the equilibrium relationship. In addition, if the system is nonisothermal, there will be the differential column heat balance and the particle heat balance equations, which are coupled to the adsorption rate equation through the temperature dependence of the rate and equilibrium constants. The solution for a nonisothermal system will therefore contain an additional mass transfer zone traveling with the characteristic velocity of the temperature front, which is determined by the heat capacities of adsorbent and fluid and the heat of adsorption. A nonisothermal or adiabatic system with n components will therefore have n transitions or mass transfer zones and as such can be considered formally similar to an (n + 1)-component isothermal system. 

The rate of formation of the affinity complex is often described by the second-order Langmuir adsorption rate equation... 

Chase [32] used the adsorption rate-limited model [Eqs. (7 —(11)[ to analyze the experimental breakthrough curves in affinity chromatography. This empirical approach assumes that all the rate-limiting processes can be represented by an apparent single second-order Langmuir adsorption rate equation in which k is considered a lumped" parameter. 

For smface-reaction-limited mechanisms, we use the adsorption rate Equation (10-72) to obtain... 

To obtain the solution for the breakthrough curve it is necessary to solve Eq. (8.32), subject to these boundary conditions, together with the appropriate adsorption rate equation, which must be consistent with the equilibrium isotherm. The various mathematical models differ only in the form of the rate expression and the choice between Eqs. (8.7) and (8.32) to represent the fluid phase mass balance. Some of the solutions which have been obtained are summarized in Table 8,1. 

The particle mass balance yields the adsorption rate equation for each component, which may be written in the generalized form... 

To satisfy all of the four conditions at the same time, the exchange probability kernel must be a logical sum of the sum and product kernels. For species exchange problems, A generalized adsorption rate equation is therefore expressed as follows ... 

Multi-site adsorption model.Upon the assumption that there are two sites with different natures, in the course of TPD, and they only take place adsorption and desorption, it can be imagined that following adsorption-desorption process can occur on the surface (Fig. 7.17). The adsorption rate equation for each adsorption site is ... 

Combining Eqs (5) and (9) we arrive at a simple, yet fairly general adsorption rate equation ... 

For the case where k - is strictly constant, the adsorption rate equation, Eq. (11), can be easily integrated ... 

In practice the kinetics are usually more complex than might be expected on this basis, siace the activation energy generally varies with surface coverage as a result of energetic heterogeneity and/or sorbate-sorbate iateraction. As a result, the adsorption rate is commonly given by the Elovich equation (15) ... 

Chemical Equihbrium When A is not in adsorptive equilibrium, it is assumed to be in chemical equilibrium, with.p =p, JK py. This expression is substituted for p wherever it appears in the rate equation. Then... 

Reaction Special condition Basic rate equation Driving force Adsorption term... 

Commonly used forms of this rate equation are given in Table 16-12. For adsorption bed calculations with constant separation factor systems, somewhat improved predictions are obtained using correction factors f, and fp defined in Table 16-12 is the partition ratio... 

Various Langmiiir-Hinshelwood mechanisms were assumed. GO and GO2 were assumed to adsorb on one kind of active site, si, and H2 and H2O on another kind, s2. The H2 adsorbed with dissociation and all participants were assumed to be in adsorptive equilibrium. Some 48 possible controlling mechanisms were examined, each with 7 empirical constants. Variance analysis of the experimental data reduced the number to three possibilities. The rate equations of the three reactions are stated for the mechanisms finally adopted, with the constants correlated by the Arrhenius equation. 

The latter kind of formulation is described at length in Sec. 7. The assumed mechanism is comprised of adsorption and desorption rates of the several participants and of the reaction rates of adsorbed species. In order to minimize the complexity of the resulting rate equation, one of the several rates in series may be assumed controlling. With several controlling steps the rate equation usually is not exphcit but can be used with some extra effort. 

The adsorption of carbon monoxide retards the reduction reaction with the rate constant k, followed by the desorption reaction with a rate constant k in the overall rate equation... 

Kinetic theories of adsorption, desorption, surface diffusion, and surface reactions can be grouped into three categories. (/) At the macroscopic level one proceeds to write down kinetic equations for macroscopic variables, in particular rate equations for the (local) coverage or for partial coverages. This can be done in a heuristic manner, much akin to procedures in gas-phase kinetics or, in a rigorous approach, using the framework of nonequihbrium thermodynamics. Such an approach can be used as long as... 

To get the equilibrium sticking coefficient we assume that at an ambient pressure Pq the adsorbate is in equilibrium at a temperature T with partial coverages Hq, m, and Iq. We then increase the pressure slightly to p = Pq- - AP and linearize the rate equations in the increase in the precursor coverages Am = (m) —m and Al = (/) — Iq. If adsorption into and desorption from the precursors is much faster than transitions from the precursors into the adsorbed state, we can ignore terms proportional to An = n) -6 on the right-hand side of Eqs. (70-72) and also assume that the precursors will be in a steady state. It has been shown that the sticking... 

In this equation, Mp is the monomer concentration within forming particles, pa is the adsorption rate of oligomeric radicals by the forming particles, Vp is the volume fraction of forming particles within the system, and kp and k, are the rate constants of propagation and termination, respectively. 

In contrast to consecutive reactions, with parallel competitive reactions it is possible to measure not only the initial rate of isolated reactions, but also the initial rate of reactions in a coupled system. This makes it possible to obtain not only the form of the rate equations and the values of the adsorption coefficients, but also the values of the rate constants in two independent ways. For this reason, the study of mutual influencing of the reactions of this type is centered on the analysis of initial rate data of the single and coupled reactions, rather than on the confrontation of data on single reactions with intergal curves, as is usual with consecutive reactions. 

The results obtained showed, again, that the form of the rate equations and the values of their constants, obtained by the study of isolated reactions, are valid also in the coupled system. This was also confirmed by the observed agreement between the calculated and the experimental integral data (94)- Kinetic results and the analysis of the effect of reaction products revealed that adsorption of the reaction components was competitive and that all the compounds involved in the three reactions were adsorbed on the same sites of the catalytic surface. 

In particular, reactions in heterogeneous catalysis are always a series of steps, including adsorption on the surface, reaction, and desorption back into the gas phase. In the course of this chapter we will see how the rate equations of overall reactions can be constructed from those of the elementary steps. 

Writing the rate equations for the adsorption desorption equilibrium for each layer, we obtain... 

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