Rate laws adsorption

Where E is appreciable, adsorption rates may be followed by ordinary means. In a rather old but still informative study, Scholten and co-workers [130] were able to follow the adsorption of N2 on an iron catalyst gravimetrically, and reported the rate law... 

Ref. 205). The two mechanisms may sometimes be distinguished on the basis of the expected rate law (see Section XVni-8) one or the other may be ruled out if unreasonable adsorption entropies are implied (see Ref. 206). Molecular beam studies, which can determine the residence time of an adsorbed species, have permitted an experimental decision as to which type of mechanism applies (Langmuir-Hinshelwood in the case of CO + O2 on Pt(lll)—note Problem XVIII-26) [207,208]. 

The Langmuir-Hinshelwood picture is essentially that of Fig. XVIII-14. If the process is unimolecular, the species meanders around on the surface until it receives the activation energy to go over to product(s), which then desorb. If the process is bimolecular, two species diffuse around until a reactive encounter occurs. The reaction will be diffusion controlled if it occurs on every encounter (see Ref. 211) the theory of surface diffusional encounters has been treated (see Ref. 212) the subject may also be approached by means of Monte Carlo/molecular dynamics techniques [213]. In the case of activated bimolecular reactions, however, there will in general be many encounters before the reactive one, and the rate law for the surface reaction is generally written by analogy to the mass action law for solutions. That is, for a bimolecular process, the rate is taken to be proportional to the product of the two surface concentrations. It is interesting, however, that essentially the same rate law is obtained if the adsorption is strictly localized and species react only if they happen to adsorb on adjacent sites (note Ref. 214). (The apparent rate law, that is, the rate law in terms of gas pressures, depends on the form of the adsorption isotherm, as discussed in the next section.)... 

The course of a surface reaction can in principle be followed directly with the use of various surface spectroscopic techniques plus equipment allowing the rapid transfer of the surface from reaction to high-vacuum conditions see Campbell [232]. More often, however, the experimental observables are the changes with time of the concentrations of reactants and products in the gas phase. The rate law in terms of surface concentrations might be called the true rate law and the one analogous to that for a homogeneous system. What is observed, however, is an apparent rate law giving the dependence of the rate on the various gas pressures. The true and the apparent rate laws can be related if one assumes that adsorption equilibrium is rapid compared to the surface reaction. 

The apparent activation energy is then less than the actual one for the surface reaction per se by the heat of adsorption. Most of the algebraic forms cited are complicated by having a composite denominator, itself temperature dependent, which must be allowed for in obtaining k from the experimental data. However, Eq. XVIII-47 would apply directly to the low-pressure limiting form of Eq. XVIII-38. Another limiting form of interest results if one product dominates the adsorption so that the rate law becomes... 

Here we illustrate how to use kinetic data to establish a power rate law, and how to derive rate constants, equilibrium constants of adsorption and even heats of adsorption when a kinetic model is available. We use the catalytic hydrodesulfurization of thiophene over a sulfidic nickel-promoted M0S2 catalyst as an example ... 

The numerator of Eq (1) is that of a homogeneous mass action rate law. The denominator seems to have been added to account for adsorption of some of the participants. 

For simplicity, up to now, first-order kinetics have been assumed, but obviously other rate laws may apply. Further complications can be generated by the presence of multiple paths for M on a variety of sites exhibiting different kinetics [5,11] or sequential enzymatic processes [100], Some complexes, labelled as lipophilics , have been shown to cross the membrane without the need for specific pre-adsorption sites [5,11,18,19,50] see also Chapters 5, 6 and 10 in this volume. Fortin and Campbell [76] have recently reported the accidental uptake of Ag+ induced by thiosulfate ligand. 

Chemical adsorption or surface complexation as given in Eq. (6.21) attempts to relate to the "collision" factor A in Eq. (6.12) to the surface concentrations of adsorbed ions. By analogy to the treatment of activated processes the following "general" rate law for the rate of nucleation of the mineral (AB) on a foreign surface could then be proposed... 

Interface and colloid science has a very wide scope and depends on many branches of the physical sciences, including thermodynamics, kinetics, electrolyte and electrochemistry, and solid state chemistry. Throughout, this book explores one fundamental mechanism, the interaction of solutes with solid surfaces (adsorption and desorption). This interaction is characterized in terms of the chemical and physical properties of water, the solute, and the sorbent. Two basic processes in the reaction of solutes with natural surfaces are 1) the formation of coordinative bonds (surface complexation), and 2) hydrophobic adsorption, driven by the incompatibility of the nonpolar compounds with water (and not by the attraction of the compounds to the particulate surface). Both processes need to be understood to explain many processes in natural systems and to derive rate laws for geochemical processes. 

Reaction kinetics. The time-development of sorption processes often has been studied in connection with models of adsorption despite the well-known injunction that kinetics data, like thermodynamic data, cannot be used to infer molecular mechanisms (19). Experience with both cationic and anionic adsorptives has shown that sorption reactions typically are rapid initially, operating on time scales of minutes or hours, then diminish in rate gradually, on time scales of days or weeks (16,20-25). This decline in rate usually is not interpreted to be homogeneous The rapid stage of sorption kinetics is described by one rate law (e.g., the Elovich equation), whereas the slow stage is described by another (e.g., an expression of first order in the adsorptive concentration). There is, however, no profound significance to be attached to this observation, since a consensus does not exist as to which rate laws should be used to model either fast or slow sorption processes (16,21,22,24). If a sorption process is initiated from a state of supersaturation with respect to one or more possible solid phases involving an adsorptive, or if the... 

We have reviewed today s knowledge of the mechanisms for growth of electrolyte crystals from aqueous solution Convection, diffusion, and adsorption ( ) mechanisms leading to linear rate laws, as well as the surface spiral mechanism (parabolic rate law) and surface nucleation (exponential rate law). All of these mechanisms may be of geochemical importance in different situations. 

The transport of disulfoton from water to air can occur due to volatilization. Compounds with a Henry s law constant (H) of <10 atm-m /mol volatilize slowly from water (Thomas 1990). Therefore, disulfoton, with an H value of 2.17x10" atm-m /mol (Domine et al. 1992), will volatilize slowly from water. The rate of volatilization increases as the water temperature and ambient air flow rate increases and decreases as the rate of adsorption on sediment and suspended solids increases (Dragan and Carpov 1987). The estimated gas- exchange half-life for disulfoton volatilization from the Rhine River at an average depth of 5 meters at 11 °C was 900 days (Wanner et al. ] 989). The estimated volatilization half-life of an aqueous suspension of microcapsules containing disulfoton at 20 °C with still air was >90 days (Dragan and Carpov 1987). 

Kinetic data presented in Fig. 22.2 for the adsorption of metobromuron were fitted to the pseudo-first-order and the pseudo-second-order rate laws. Linear form of pseudo-first-order model can be formulated as... 

As discussed in Section 4.1, most redox reactions reach equilibrium only slowly if they are not catalysed. Oxidation of Fe + is catalysed by adsorption of Fe + onto Fe(OH)3 formed in the reaction, so Equation (4.36) only holds for the initial rates of reaction. Tamura et al. (1976) studied the oxidation of a solution of Fe + at different controlled pHs near neutral and with varying additions of Fe(OH)3. The reaction obeyed the rate law... 

Understanding the kinetics of contaminant adsorption on the subsurface solid phase requires knowledge of both the differential rate law, explaining the reaction system, and the apparent rate law, which includes both chemical kinetics and transport-controlled processes. By studying the rates of chemical processes in the subsurface, we can predict the time necessary to reach equilibrium or quasi-state equilibrium and understand the reaction mechanism. The interested reader can find detailed explanations of subsurface kinetic processes in Sparks (1989) and Pignatello (1989). 

Based on these rate laws, various equations have been developed to describe kinetics of soil chemical processes. As a function of the adsorbent and adsorbate properties, the equations describe mainly first-order, second-order, or zero-order reactions. For example. Sparks and Jardine (1984) studied the kinetics of potassium adsorption on kaolinite, montmorillonite (a smectite mineral), and vermiculite (Fig. 5.3), finding that a single-order reaction describes the data for kaolinite and smectite, while two first-order reactions describe adsorption on vermiculite. 

At sufficiently high concentrations of the transported solute particles, the surface coverage becomes important and non-linear laws for the rate of adsorption should be used. 

The adsorption of reactants onto catalytic active sites is the first step of the pure catalytic process. The form of the rate law is closely connected to the mechanism adopted. Specifically, if the reactant AB is considered to be adsorbed as a molecule, it can be represented by... 

Transient deposition of hydrocarbons on zeolites during a cold start operation of the DOC can be modeled by the rate laws based on Langmuir or Temkin mechanism for physical adsorption/desorption (cf., e.g. Goralski et al., 2000 Koltsakis and Stamatelos, 2000 Kruglov and Aris, 1995 Kryl et al., 2005 Twigg, 2006). The rate laws for the adsorption and desorption of hydrocarbons are then... 

The MR rate law relies on the assumption that the SCR reaction is governed by a redox mechanism and therefore predicts a kinetic dependence on oxygen. It has been derived assuming that (i) two types of sites for NH3 adsorption (acidic non-reducible sites) and for NO + NH3 activation/reaction (redox sites, associated with vanadium), respectively, prevail on the catalyst surface (ii) NH3 blocks the redox sites (iii) reoxidation of the redox sites is rate controlling. 

It is clearly recognized that on oxide semiconductors various types of chemisorption can and do occur as a result of various types of electron exchange between adsorbent and adsorbate. Slow rates of adsorption may be due to the conditions of this exchange. The logarithmic rate law, however, seems to represent a number of different processes (bulk or surface diffusion, activation or deactivation of catalytic surfaces, chemisorption). It appears futile to explain this empirical relation in terms of a unique mechanism. 

Many theories of adsorption, following Langmuir, have assumed that the rate of adsorption is proportional to (1 — 0), i.e., to the fraction of the surface which is bare or not yet covered. Langmuir first proved the (1 — 0) law by measuring experimentally how the thermionic work function

changed with time as thorium reached the surface of a tungsten filament at a constant rate (10). He then assumed that tp decreased linearly with 0 and thus deduced that dd/di was proportional to (1 — 0). But this assumption has been shown to be incorrect for such cases as Cs on W, Ba on W, SrO on W, and other systems. Hence it follows that the (1 — 0) law is not valid. The experiments described above for N2 on W not only show that dQ/dt is not proportional to (1 — 0), but they show by a direct experiment that dd/dt for a constant arrival rate is independent of 0 between 0 = 0 and 1.0. 

Another concept is that the electronic work function changes linearly with the amount adsorbed or that the dipole moment is independent of the concentration. The (1 — 0) concept states that the rate of adsorption for a constant arrival rate is proportional to the fraction of the surface which has not yet been covered. The last two concepts permitted Langmuir to derive his famous adsorption isotherm, which has been verified by experiment in many systems (see the discussion in section IV). Langmuir s experimental work for Cs on W convinced him that the (1 — 0) law was not applicable in this system. This work also led to the concept that the energies involved in surface migration were much smaller than the energies involved in evaporation. 

The particular form for the reaction rate law invoked above has been justified in a number of ways. One interpretation is that a second adsorbed reactant may also be involved in the final step, but that the adsorption and desorption of this species occurs on a much faster timescale than that of P or of the reaction step. Thus, if this second reactant is denoted R, which may be polyatomic and adsorb onto n surface sites, the kinetics become... 

The simple model just discussed shows multistability even when the system is clean but requires the involvement of a poison for oscillations. One reason for this is that the latter is needed to provide a second independent surface concentration, so we theij. have a two-variable system. It was mentioned in 12.3.1 that implicit in the rate law used above may be the adsorption of a second reactant which participates in the reaction step. The latter did not provide a second concentration variable there since its adsorption and desorption processes were assumed to be on a very much faster (instantaneous) timescale. 

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