Langmuir expression

This has the advantage that the expressions for the adsotbed-phase concentration ate simple and expHcit, and, as in the Langmuir expression, the effect of competition between sorbates is accounted for. However, the expression does not reduce to Henry s law in the low concentration limit and therefore violates the requirements of thermodynamic consistency. Whereas it may be useful as a basis for the correlation of experimental data, it should be treated with caution and should not be used as a basis for extrapolation beyond the experimental range. 

In the simplest case, a linear relationship may exist between solutes in soil solution and those on the solid pha.se, such that C, = hCr, where C, is the. solute concentration on the solid phase and h defines the buffer power. In perhaps more typical situations, the relationship might be defined by a more complicated, concentration-dependent relationship such as a Freundlich or Langmuir expression here the instantaneous buffer power may be defined as the tangent to the exchange isotherm i.e., h = dCJdCi. Some care is needed when defining this buffer power (7). 

By combining surface-reaction rate laws with the Langmuir expressions for surface coverages, we can obtain Langmuir-Hinshelwood (LH) rate laws for surface-catalyzed reactions. Although we focus on the intrinsic kinetics of the surface-catalyzed reaction, the LH model should be set in the context of a broader kinetics scheme to appreciate the significance of this. 

This equation, containing the Langmuir expression for the adsorption (surface complex formation) of phenol on the Mn(III)(hydr)oxide, is similar to the principles discussed for reductive dissolution of Fe(IH)(hydr)oxide. 

The first term on the right side of Eq. (19) is the Langmuir expression for the number of moles of species 1 which adsorb without competition on the surface area proportional to (Q, - Q2). The second term represents the number of moles of species 1 adsorbed on the surface area proportional to Q2 under competition with species 2 and is based on the Langmuir model for competitive adsorption. The number of moles of species 2 adsorbed on the surface area proportional to Q2 and under competition with species 1 can be calculated from Eq. (20). 

For many photoelectrochemical events, Langmuirian kinetics govern substrate adsorption. That is, the observed photochemically induced reaction rate is found to be proportional to the concentration (or pressure) of the reactant, which in turn can be related to the surface coverage 9 by the Langmuir expression, Eq. (10),... 

By assuming the Langmuir expression for the evaporation of a droplet with the Rosin-Rammler size distribution law, Sacks (74) found that the theoretical evaporation rate of a kerosine spray was about 100 times the experimentally observed values. He concluded that the Langmuir expression is based on the single drop and neglects the vapor pressure of the surrounding air, which would tend to inhibit vaporization in a spray. Consideration of the effects of dissociation of combustion products plus the effects of thermal conductivity for the vapors enabled Graves (33) to derive a theoretical curve for combustion rate which compared favorably with experimental data. However, the use of Probert s analysis to determine combustion efficiency, yielded efficiencies which were much lower than experimentally observed results. 

Equations (20) and (21) are Langmuir expressions for adsorption and desorption rates. If the equilibrium is reached with respect to the process (19), then, in accordance with mass action law,... 

Whereas the charging approach could be applied to any geometry but only at constant surface charge or potential, the Langmuir expression could be employed for any surface conditions but only for parallel planar plates. The addition of electrostatic, entropic, and chemical contributions would allow the calculation of the free energy of interaction for systems of any shape and any surface conditions, if one could derive a general expression for the chemical free energy contribution. 

Using the electrical potential in the middle between the two plates, the force per unit area can be calculated using the Langmuir expression [20]... 

If only the small ions are present in the system, the above equation reduces to the Langmuir expression... 

Since is much more strongly adsorbed than is CO and covers nearly the entire surface at pressures greater than 10 atm (Hayward and Trapnell, 1964), the Langmuir expression for the rate R simplifies to ... 

Thermal ionization from hot surfaces has been used extensively to produce positive ions for isotopic analysis The efficiency of positive ion formation for selected alkali and other elements is controlled by the Saha- Langmuir expression ... 

In contrast to the appendix, the area A is now written explicitly. Equation [3.4.33] needs no further comment, but the Langmuir expression, [3.4.34] does. In [I.A1.2b] it was written as =-fcTln(l-fl), where is the area of one site. Recall that the Langmuir adsorption Isotherm and equation of state were derived... 

It is instructive to consider some limiting cases of [3.4.56]. For monomers (N = 1) in an athermal solvent [x =0), only the logarithmic term remains the sum reduces to the term for z = 1 (trains). For dilute solutions (0 1), we thus obtain =fcrin(l-0j), which is identical to the Langmuir expression [3.4.34] with 0 = 6(1) = 0. For monomers in a poorer solvent (where multilayers may form due to attraction between the monomers) the layers z > 1 also give a contribution, according to the last term on the r.h.s. of [3.4.56[. 

Using independently measured parameters for Cr, Selim and Amacher (1988) showed fair agreement between the second-order, two-site model and experimental Cr BTC (Fig. 10-6). Adsorption isotherms obtained from batch studies were modeled with the two-site Langmuir expression to provide model parameters,/(fraction of type-1 sites) and Sj (maximum number of sites). 

The parameters F, and K are the same as in Tables 5.2 through 5.4. Setting Q = 0 (assuming equilibrium surface-subsurface), from each expression in Table 5.5 we deduce the respective equilibrium adsorption isotherm in Table 5.2. In addition, for P = 0 the expressions for Q related to the Frumkin and van der Waals model reduce, respectively, to the expressions for Q in the Langmuir and Volmer models. For Fj F both the Frumkin and Langmuir expressions in Table 5.5 reduce to the Henry expression. 

The effective osmolality of cell lysate was fit to a Langmuir expression, where OSM is the maximum stabilizing effect and K is the equilibrium constant for interaction of the stabilizers with the protoplasts. The resulting equation,... 

A sequential procedure was adopted to develop rate models which would overcome shortcomings of this type, employing modified Langmuir expressions empirically rather than power law models. 

In this mechanism, the pressure differential between the surface of the particle and the neck area results in a net matter transport, via the gas phase from the surface to the neck. The evaporation rate (in molecules of MX per square meter per second), is given by the Langmuir expression... 

On A1203, Menon (90) found a retarding effect of water at 200°C this could be described by the Langmuir expression... 

Expressed in terms of the sticking probability, the Langmuir expression is simply... 

For intermediate partial pressures of hydrogen, the full Langmuir expression would be needed to determine the surface concentration and the rate of diffusion as a function of hydrogen partial pressure. The exponent of the partial pressure of hydrogen would lie between 0 and 0.5. 

The first term defines the simple linear partitioning process while the second term is a Langmuir expression defining distribution into the irreversible compartment as a function of compartment capacity, the fraction of this compartment ifaf) filled at the time of exposure and the distribution factor Aioc(irr), which appears to be relatively constant. The capacity of the irreversible compartment, expressed per unit of SOC, shows an inverse relation to Aiow. however, the utility of this relation will depend on further evaluation of how widely it can be applied. 

At higher loadings (beyond the Henry s law region) the equilibrium isotherms for microporous adsorbents are generally of Type I form in Brunauer s classification [2]. Several different models have been suggested to represent such isotherms, the simplest being the ideal Langmuir expression [3] ... 

Although the simple Langmuir expression provides a useful qualitative representation of the equilibrium behavior of many systems it is generally not quantitatively reliable, especially at higher loadings. There have therefore been numerous attempts to develop more accurate models, a few of which are noted here. 

Such an expression contains four independent constants, so it will obviously provide a better fit to experimental data than the simple two-constant Langmuir expression. However, such a model makes physical sense for systems such as the adsorption of polar (or quadrupolar) molecules on a cationic zeolite, where the most favorable sites are those associated with the exchangeable cations and the less favorable sites correspond to adsorption elsewhere on the framework or simply within the micropores. For example, it has been shown that the analysis of equilibrium isotherms for CO2 on various different forms of zeolite A yields site densities that are consistent with structural information [4]. 

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