Rate Langmuir isotherm

For noncoustaut diffusivity, a numerical solution of the conseiwa-tion equations is generally required. In molecular sieve zeohtes, when equilibrium is described by the Langmuir isotherm, the concentration dependence of the intracrystalline diffusivity can often be approximated by Eq. (16-72). The relevant rate equation is ... 

For nonlinear systems, however, the evaluation of the flow rates is not straightforward. Morbidelli and co-workers developed a complete design of the binary separation by SMB chromatography in the frame of Equilibrium Theory for various adsorption equilibrium isotherms the constant selectivity stoichiometric model [21, 22], the constant selectivity Langmuir adsorption isotherm [23], the variable selectivity modified Langmuir isotherm [24], and the bi-Langmuir isotherm [25]. The region for complete separation was defined in terms of the flow rate ratios in the four sections of the equivalent TMB unit ... 

In essence, we have used the Langmuir isotherms for the adsorbing and desorbing species. By substituting the coverages into the rate expression for the ratedetermining step we obtain... 

Often, mnltistep reactions are enconntered where a reactant j first becomes adsorbed on the electrode, then is converted electrochemically (or chemically) to a desorbing prodnct. We shall consider the case where the electrochemical step involving adsorbed particles is rate determining. With a homogeneons electrode surface and without interaction forces between the adsorbed particles [i.e., in conditions when the Langmuir isotherm (10.14) can be apphed], the assumption can be made that the rate of this step is proportional not to the bulk concentration Cy j but to the surface concentration Aj or to the degree of surface coverage 0 hence. 

Electrochemical reaction rates are also influenced by substances which, although not involved in the reaction, are readily adsorbed on the electrode surface (reaction products, accidental contaminants, or special additives). Most often this influence comes about when the foreign species I by adsorbing on the electrode partly block the surface, depress the adsorption of reactant species j, and thus lower the reaction rate. On a homogeneous surface and with adsorption following the Langmuir isotherm, a factor 10, will appear in the kinetic equation which is the surface fraction free of foreign species 1 ... 

Derivation of the Langmuir Equation— Adsorption of a Single Species. The kinetic approach to deriving a mathematical expression for the Langmuir isotherm assumes that the rate of adsorption on the surface is proportional to the product of the partial pressure of the adsorbate in the gas phase and the fraction of the surface that is bare. (Adsorption may occur only when a gas phase molecule strikes an uncovered site.) If the fraction of the surface covered by an adsorbed gas A is denoted by 0Ay the fraction that is bare will be 1 — 0A if no other species are adsorbed. If the partial pressure of A in the gas phase is PA, the rate of adsorption is given by... 

Figure 4.2 Current-potential curve for a Langmuir isotherm whose potential dependence is given by Eq. (4.7). Such curves are obtained by a slow potential sweep. The absolute value of the current depends on the sweep rate (see text).

Show that these data are consistent with the view that the catalyst is saturated with hydrogen and that the reaction rate is given in terms of the Langmuir isotherm. 

At steady-state conditions, the two rates must be equal and the well-known Langmuir isotherm is obtained ... 

When this happens the adsorption phenomenon follows the so-called Langmuir isotherm and the current intensity at the maximum of the forward peak is correlated with the parameter To (that represents the quantity of Ox, in mol m-2, initially adsorbed on the electrode surface) and with the scan rate according to the relationship ... 

Equilibrium is reached when the absorption and the desorption rate are equal. The maximum occupation is then given by the Langmuir isotherm ... 

The superficial characteristics of atmospheric corrosion products of steel depend on the type of atmosphere where the sample has been exposed. The way of adsorption of the corrosion products obtained in the coastal atmosphere is polymolecular due to a higher content of salts. This makes easier the presence of water in the metal-corrosion products interface and determines a high corrosion rate. The adsorption of water of a corrosion product formed in a rural zone obeys a Langmuir isotherm, i.e. a monomolecular adsorption takes place. It causes a lower corrosion rate. 

It was assumed that the equilibrium follows a Langmuir isotherm with competitive adsorption of ions (the coverages of which are potential dependent), then the rate of reduction was formulated as... 

The common shape observed for Tq as a function of the substrate or catalyst concentration depending on the primary events depicted in Eqs. (la)-(lf) is normally reported of saturative type as if the rate would be determined by adsorption properties of the substrate on the catalyst surface according to a Langmuir isotherm ... 

Probably the reason why the (1 — 0) concept has received such widespread credence is that Langmuir was able to derive his famous adsorption isotherm on the basis of this concept. Since the Langmuir isotherm equation has been experimentally verified in many cases, it was felt that the (1 — 0) concept must be essentially correct. This again is fallacious reasoning, since in the derivation two other assumptions are necessarily made which are not in accord with recent experiments. These are ( ) the rate of evaporation is proportional to 0 and (2) one can treat the experimental data as if the surface were homogeneous. Because of this situation, it is desirable that someone derive the Langmuir isotherm equation on more realistic assumptions. (See ref. 10a.)... 

Schuldiner (1959) studied the effect of Hi pressure on the hydrogen evolution reaction at bright (polished) Pt in sulphuric acid. The mechanism of the reaction was assumed to be as in equations (3.3) and (3.4). The step represented by equation (3.3) was assumed to be at equilibrium at all potentials and equation (3.4) represented the rate-determining step. The potentials were measured as overpotentials with respect to the hydrogen potential, i.e. the potential of the H +/H2 couple in the solution (0 V vs. RHE). The experiments were performed at 25CC, where the adsorption of both forms of hydride was assumed to follow the Langmuir isotherm. Thus, the current for the forward (cathodic) and reverse (anodic) reactions in equation (3.3) can be written as ... 

In heterogeneous systems, the rate expressions have to be developed on the basis of (a) a relation between the rate and concentrations of the adsorbed species involved in the rate-determining step and (b) a relation between the latter and the directly observable concentrations or partial pressures in the gas phase. In consequence, to obtain adequate kinetic rate expressions it is necessary to have a knowledge of the reaction mechanism, and an accurate means of relating gas phase and surface concentrations through appropriate adsorption isotherms. The nature and types of adsorption isotherm appropriate to chemisorption processes have been discussed in detail elsewhere [16,17] and will not be discussed further except to note that, in spite of its severe theoretical limitations, the Langmuir isotherm is almost invariably used for kinetic interpretations of surface hydrogenation reactions. The appropriate equations are... 

Prediction of the breakthrough performance of molecular sieve adsorption columns requires solution of the appropriate mass-transfer rate equation with boundary conditions imposed by the differential fluid phase mass balance. For systems which obey a Langmuir isotherm and for which the controlling resistance to mass transfer is macropore or zeolitic diffusion, the set of nonlinear equations must be solved numerically. Solutions have been obtained for saturation and regeneration of molecular sieve adsorption columns. Predicted breakthrough curves are compared with experimental data for sorption of ethane and ethylene on type A zeolite, and the model satisfactorily describes column performance. Under comparable conditions, column regeneration is slower than saturation. This is a consequence of non-linearities of the system and does not imply any difference in intrinsic rate constants. 

Isotherms. When a fiber is immersed in a dyebath, dye moves from the external phase into Lhe fiber. Initially the rate is quick but with time this slows and eventually an equilibrium is reached between the concentration of dye in the fiber and the concentration of dye in the dyebath. For a given initial dyebath concentration of a dye under given dyebath conditions, e.g.. temperature, pH, and conductivity, there is an equilibrium concentration of dye in fiber, D, and dye in the dyebath external solution, D,. Three models describe this relationship simple partition isotherm. Freundlich isotherm, and Langmuir isotherm. 

It is interesting, however, to compare the different dissolution constants. K, obtained with the simple Langmuir isotherm model with experimental data of dissolution rates over a 24-hour period. Figure 15 gives concentrations obtained consecutively with fresh solvent every 24... 

An unusual dependence for adsorption on a uniform surface arises when it is assumed that the rate of change of the surface is considerably slower than the rate of adsorption (see figure). If adsorption and desorption occur rapidly, the state of the surface remains practically unaltered and we then get an adsorption isotherm corresponding to a non-uniform surface with a distribution p(E, 0) of the heat of adsorption (curve 1) [3]. However, when the time interval is considerable, slow adsorption accompanies changes in the properties of the surface, and the amount of gas adsorbed approaches that given by the Langmuir isotherm (curve 2, point B), which describes a state of complete equilibrium (see above). 

The diffusion of the electroactive ions is both physical and due to electron transfer reactions.45 The occurrence of either or both mechanisms is a function of the electroactive species present. It has been observed that the detailed electrochemical behaviour of the electroactive species often deviates from the ideal thin film behaviour. For example, for an ideal nemstian reaction under Langmuir isotherm conditions there should be no splitting between the anodic and cathodic peaks in the cyclic voltammogram further, for a one-electron charge at 25 °C the width at half peak height should be 90.6 mV.4 In practice a difference between anodic and cathodic potentials may be finite even at slow scan rates. This arises from kinetic effects of phase formation and of interconversion between different forms of the polymer-confined electroactive molecules with different standard potentials.46... 

Pshezhetskii et al. (17) have recently expressed the view that in the equation for the rate of dehydrogenation, containing the function p in the form of the Langmuir isotherm, z is the ratio of the rate constants of some partial reactions. 

There is an interesting parallel between substrate binding and adsorption. Since each enzyme molecule has one active site, and since these active sites all have the same structure, we can think of enzyme molecules in solution as a surface with many equivalent adive sites. In this case, k2 in the Michaelis-Menten kinetics (Eq. 5.1 see Chapter 2 for a detailed discussion) represents the rate of adsorption, k x the rate of desorption, and k2 the rate of the surface readion followed by fast product desorption. Moreover, this system fits the assumptions of the Langmuir isotherm (all sites identical, one molecule per site, no lateral interadions) even better than the adive sites on some real solid catalysts ... 

Langmuir (1918) laid down a systematic and theoretically grounded adsorption isotherm, the well-known Langmuir isotherm, which has served as a basis for a number of future isotherms, such as the BET isotherm, described in 1.4.1.3. The basic assumptions of the Langmuir adsorption model include monolayer coverage, localized adsorption (i.e., the specific adsorption sites exist and the interactions are between a specific molecule and the site), and homogeneous materials. In addition, the heat of adsorption is independent of coverage. He proposed that the rate of adsorption is proportional to the dimensionless pressure of the adsorbate, p/po (= p ), where po is the saturated vapor pressure, and the... 

The assumption of monolayer adsorption in the Langmuir isotherm model is unrealistic in most cases, and a modification to multilayer adsorption should be considered. In 1938, Brunauer, Emmett, and Teller modified the Langmuir approach of balancing the rates of adsorption and desorption for the various molecular layers [Brunauer et al., 1938], This approach is known as the BET method. The BET isotherm assumes that the adsorption of the first layer has a characteristic heat of adsorption A Ha and the adsorption and desorption on subsequent layers are controlled by the heat of condensation of the vapor, A Hc. The derivation of the BET equation is beyond the scope of this book however, a common form of the BET equation is given as... 

---