Laws Langmuir equation

Over the usual hmited range of conditions, a power law rate equation often appears to be as satisfactory a fit of the data as a more complex Langmuir-Hinshelwood equation. The example of the hydrogenation of oc tenes is shown in Fig. l-2d and l-2e, and another case follows. 

This is the important Hill-Langmuir equation. A. V. Hill was the first (in 1909) to apply the law of mass action to the relationship between ligand concentration and receptor occupancy at equilibrium and to the rate at which this equilibrium is approached. The physical chemist I. Langmuir showed a few years later that a similar equation (the Langmuir adsorption isotherm) applies to the adsorption of gases at a surface (e g., of a metal or of charcoal). 

If we apply the law of mass action to this form of antagonism, the proportion of inhibitory sites occupied by the antagonist will be given by the Hill-Langmuir equation ... 

Hill-Langmuir equation and the application of the law of mass action to the kinetics of drug-receptor interaction ... 

The Langmuir equation is derived here from application of the mass law, in a similar way as the surface complex formation equilibria were derived in Chapter 2. In principle at a constant pH there is no difference between a Langmuir constant and a surface complex formation constant. 

For a formal kinetic description of vapour phase esterifications on inorganic catalysts (Table 21), Langmuir—Hinshelwood-type rate equations were applied in the majority of cases [405—408,410—412,414,415]. In some work, purely empirical equations [413] or second-order power law-type equations [401,409] were used. In the latter cases, the authors found that transport phenomena were important either pore diffusion [401] or diffusion of reactants through the gaseous film, as well as through the condensed liquid on the surface [409], were rate-controlling. 

FIG. 18.8 Typical isotherm plots of sorbed concentration versus ambient vapour pressure. (I) Henry s law, S is a constant (II) Langmuir equation (III) Flory-Huggins equation (IV) BET equation, site saturation at point B (from C.F. Rogers, 1985). 

Vj, is the specific pore volume of the material typical values are 200-400 cm /kg for zeolites and up to 1000 cm /kg for activated carbon, n is the actual number of molecules contained in the micropores the excess adsorption subtracts from n the number of molecules which would have been present in the micropores at the bulk density in the absence of adsorption. The (oversimplified) case when absolute adsorption is described by the Langmuir equation and the gas obeys the perfect gas law p - PjRT) has been worked out in detail for the isotherms and thermodynmnic functions (enthalpy, entropy, etc.) [2j. 

When A"Ce 1, the Langmuir isotherm reduces to a linear form analogous to Henry s Law. Sorbate-sorbent pairs that display a Type 1 isotherm can use the Langmuir equation. 

The Hougen-Watson kinetic model that is consistent with the Langmuir-Rideal mechanism can be obtained from the rate law in equations (14-63) and (14-64) via the following modification of each generic term ... 

It should be emphasized that permeation through a polymer involves three steps, as shown in Fig. 14.1 (1) the permeant dissolves at the polymer interface, (2) the permeant diffuses within the polymer film from the side of high concentration toward the low concentration side, and (3) the permeant diffuses out from the opposite polymer interface. These steps are always present in any system regardless of whether D and S follow Pick s and Henry s laws, respectively, or not. For instance, the solubility of CO2 in PET follows the Langmuir-Henry s law model, Equation 14.7, and P is given by,... 

The next paper we will comment on in this section is a letter by Nakatami et al. [147] in which they describe a microscale technique to study the dynamics of adsorption. This technique, the single-microparticle injection, is basically an optical method that uses Lambert-Beer law to follow the concentration of methylene blue on the surface of a silica gel microparticle. Their main conclusions are that equilibrium is attained within 20 min, Langmuir equation describes the experimental adsorption isotherm, methylene blue molecules penetrate into the pores and the whole process is controlled by adsorbate diffusion in water. 

It is the Langmuir equation of a single adsorption molecule. Note that the fraction of surface sites can be determined experimentally by measuring adsorbed volumes in a system of constant pressure and temperature, according to the ideal gas law, since... 

The second and fourth assumptions are seldom valid, as solvent molecules do adsorb to the monolayer of solvent attached to the aaivated adsorbent — forming layers. But, in the same way that the ideal gas law is useful, so also is the Langmuir Equation (isotherm). It is often the first choice for a model of the adsorption process because it has been shown to be valid for, and produces information about, the number of activated sites on (not within) a monolayer of an adsorbent. 

Applications of Hobson s Method.—Zeldowich s kernel approximation [equation (6)] discussed in Section 2 has been generalized by Hobson to account for adsorbate lateral interactions. The major problem associated with the Langmuir equation is its inability to predict phase transitions. Rather than use the step approximation to the local isotherm, Hobson chooses a combination of a Henry s law isotherm and a step approximation , as shown in Figure 1(c) ... 

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