Langmuir isotherm limiting behavior

Adsorption isotherms in the micropore region may start off looking like one of the high BET c-value curves of Fig. XVII-10, but will then level off much like a Langmuir isotherm (Fig. XVII-3) as the pores fill and the surface area available for further adsorption greatly diminishes. The BET-type equation for adsorption limited to n layers (Eq. XVII-65) will sometimes fit this type of behavior. Currently, however, more use is made of the Dubinin-Raduschkevich or DR equation. Tliis is Eq. XVII-75, but now put in the form... 

This means that the decomposition occurs uniformly across the surface, where the products are weakly bound and rapidly desorbed consequently, the rate-determining step is the surface decomposition step. This type of reaction shows two rate-limiting laws corresponding to the two extreme behaviors of the Langmuir isotherm. That is, at low pressure, 0A is small and proportional to the pressure, and the rate becomes first order in A(g) ... 

A (rue adsorption maximum, however, is rarely observed. Precipitation reactions cau exhibit Langmuir-type behavior. If only a limited quantity of a solute that precipitates is present, a Langmuir isotherm can result as the solute increases, that is, a sorption maximum occurs. This behavior is found at low solute concentrations, where no precipitation occurs until the solute s solubility product is reached. 

Functional Form and Limiting Behavior. The fnnctional form of the Langmuir isotherm can be rationalized by calcnlating the partition function of N indistinguishable molecules adsorbed on a solid surface, where the internal degrees of... 

Let us now examine the asymptotic behavior of the Langmuir isotherm At low values of the fluid-phase concentration Y, the term bY becomes small compared to 1, and the Langmuir isotherm approaches the limiting linear form... 

Plots of an amount of material adsorbed versus pressure at a fixed temperature are known as adsorption isotherms. They are generally classified in the five main categories described by Brunauer and his co-workers (4). In Figure 6.2 adsorbate partial pressures (P) are normalized by dividing by the saturation pressure at the temperature in question (P0). Type I is referred to as Langmuir-type adsorption and is characterized by a monotonic approach to a limiting amount of adsorption, which presumably corresponds to formation of a monolayer. This type of behavior is that expected for chemisorption. 

A simple Langmuir-Hinshelwood model explains quantitatively the steady-state behavior (4) but it fails to explain the oscillatory phenomena that were observed. The origin of the limit cycles is not clear. Rate oscillations have not been reported previously for silver catalyzed oxidations. Oxidation of ethylene, propylene and ethylene oxide on the same silver surface and under the same temperature, space velocity and air-fuel ratio conditions did not give rise to oscillations. It thus appears that the oscillations are related specifically to the nature of chemisorbed propylene oxide. This is also supported by the lack of any correlation between the limits of oscillatory behavior and the surface oxygen activity as opposed to the isothermal oscillations of the platinum catalyzed ethylene oxidation where the SEP measurements showed that periodic phenomena occur only between specific values of the surface oxygen activity (6,9). 

Figures 4.26A and 4.26B compare the results of the experimental determination of isotherms using the traditional mass balance method (MMB) and those obtained with MMC. The adsorption isotherm predicted by MMC deviates significantly from the isotherm data obtained by MMB. This may be due to the limited applicability of the Langmuir competitive model for the modeling of the adsorption behavior even of such simple systems as p-cresol and phenol in reversed-phase chromatography. Figures 4.26C and 4.26D compare the results obtained by MMB and HMMB for the same system. Over most of the concentration range, the agreement between the experimental data and the results of these two methods is...

In this equation, rt and q are the adsorbed and the mobile phase concentrations, X is a dimensionless coefficient, and is the equilibrium constant the upper limit of rt is given by N . The linear term describes the interaction of achiral carrier particles with the analyte, while the Langmuir term describes the saturation behavior of the chiral selector. This behavior can be understood by inspection of Fig. 5, which shows in an idealized form the adsorption isotherms of two enantiomers. 

Assuming a linear Henry-like absorption isotherm, for polymer membranes the permeability P. (molmm h Pa ) can be expressed as product of solubility S (mol Pa" m" ) and diffusivity D (m h ). For pore membranes, however, this simple relation is not valid because the limited pore volume the amount adsorbed does not increase linearly with the pressure (Henry-like behavior only for permanent gases at relative high temperatures) and the adsorption isotherm is usually curved (Langmuir-like). 

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