Adsorption Langmuir isotherm for

When the functional form of the correlation is suggested by theory, there is a great deal more confidence that the correlation can be extrapolated into regions of P that have no experimental data, and can be used for other families of compounds other than the training set S. Examples of theory-suggested functional forms include the van der Waals equation of state for gases, the Langmuir isotherm for adsorption and catalysis, and the Clausius-Clapeyron equation for the vapor pressure of liquids. 

Figure 7. Langmuir isotherms for adsorption of pyridine from cyclohexane onto ferric oxide. Conditions heat of adsorption, —8.8 kcal/mol and area per pyridine, 87 A2. Key A, 45°C and O, 25°C. (Reproduced, with permission, from Ref. 33. Copyright 1981, NACE.)...

Langmuir isotherm for adsorption of molecular carbon monoxide... 

Remark.- The homographic law with the root of the pressure comes from equilibrium of the Langmuir isotherm for adsorption. The choice of another isotherm wonld lead to another law of pressure, for example, the Temkin isotherm would give a logarithm law of pressure for a mode with an external interface reaction as the rate determining step. 

Example 4 Application of Isotherms Thomas [Ann. N.Y. Acad. Sci., 49, 161 (1948)] provides the following Langmuir isotherm for the adsorption of anthracene from cyclohexane onto alumina ... 

The exemplary peak profiles, simulated with use of Equation 2.20 for the linear, Langmuir, and the anti-Langmuir isotherms of adsorption are presented in Figure 2.21. 

A typical N2 adsorption measurement versus relative pressure over a solid that has both micropores and mesopores first involves essentially a mono-layer coverage of the surface up to point B shown in isotherm IV (lUPAC classification) in Figure 13.1. Up to and near point B the isotherm is similar to a Langmuir isotherm for which equilibrium is established between molecules adsorbing from the gas phase onto the bare surface and molecules desorbing from the adsorbed layer. The volume of adsorbed N2 that covers a monolayer volume, hence the surface area of N2 can then be determined from the slope of the linearized Langmuir plot when P/V is plotted against P ... 

It should be apparent — since an adsorption isotherm can be derived from a two-dimensional equation of state —that an isotherm can also be derived from the partition function since the equation of state is implicitly contained in the partition function. The use of partition functions is very general, but it is also rather abstract, and the mathematical difficulties are often formidable (note the cautious in principle in the preceding paragraph). We shall not attempt any comprehensive discussion of the adsorption isotherms that have been derived by the methods of statistical thermodynamics instead, we derive only the Langmuir equation for adsorption from the gas phase by this method. The interested reader will find other examples of this approach discussed by Broeckhoff and van Dongen (1970). 

The Exchange Isotherm. For adsorption at constant pH the amount of selenite on the surface is related to the solution concentration by an equation of the Langmuir type (Equation 1). 

The calculated adsorption isotherm based on the Langmuir equation for adsorption of gases is... 

Aluminum oxides As(V) and As(lll) adsorption on activated alumina pH dependence, kinetics, and column breakthrough. Regeneration by desorbing with NaOH. Modeling with pH-dependent Langmuir isotherm (for As) and surface complexation model (for protons) Ghosh and Yuan (1987)... 

Show that, following the derivation of the Langmuir isotherm, for the case of two different species competing for monolayer adsorption sites,... 

Let us first focus on a nonreactive system with constant separation factors. Typical examples are distillation processes with constant relative volatilities or adsorption processes described by competitive Langmuir isotherms. For nonreactive systems with constant separation factors, the constant pattern waves and spreading waves are... 

There are two special cases of the Langmuir isotherm. For very low concentrations (i.e., K2[A] 1), the specific adsorption is directly proportional to the remaining concentration of the solute in solution ... 

Figure 8. Langmuir plot for 2,4,5-T In accord with the Langmuir model for adsorption equilibrium, a linear trace is obtained for the isotherm for 2,4,5-T when (X) 1 is vlotted vs. (Ceq) 1. Intercept of plot is (Xm) 1, and the slope is (bXm) K...

For type a curves the Intercept n° may be interpreted as n°, that is the real value of n° In a complete layer from which all 1 Is expelled (n = 0). This is the plateau value attained by the individual Isotherm of 2, Indicated by the dashed curve In fig. 2.23. When an assumption Is made about the molecular cross-section from n° the specific surface area can be obtained. In principle, this method Is not different from finding from, say the plateau In a Langmuir Isotherm, the only difference being that horizontal Langmuir plateaus (for adsorption from dilute solution) are replaced by linear upper parts, approaching zero at 1 (for excess adsorption from binary mixtures). 

Fig. 2 shows equilibrium isotherm for adsorption of glutathione on GhF. The experimental equilibrium isotherm for adsorption of glutathione is independent of the initial concentration of glutathione. Glutathione may be adsorbed by chemisorption and the equilibrium isotherm may be expressed by the Langmuir equation ... 

Before examining the NMR, it is useful to consider some adsorption and desorption studies of this standard catalyst. Hydrogen chemisorption data have been modeled by a Langmuir isotherm for pressures between 1 and 80 Torr (Fig. 26a) (80) but also by a Temkin isotherm at pressures between 0.1 and 100 Torr (Fig. 26b) (48). The Langmuir isotherms in Fig. 26a give H/Pt ratios for monolayer adsorption of 1.17 (circles) and 1.06 (triangles). The break point in the Temkin isotherm of Fig. 26b corresponds to H/Pt = 0.82 (48). (The part of the isotherm below 0.1 Torr has not often... 

In Eq. (3). is the flux when intracrystalline transport is rate controlling, Af, is the real flux. 8 AE is the difference between the activation energy for escape from within the crystal to the externally adsorbed layer and the activation energy for diffusion. It is generally a positive quantity [35]. K and K represent the Langmuir parameters for adsorption on the external and internal surfaces, respectively. When internal and external adsorption isotherms are taken to be identical or are within Henry s law range, Eq. (3) becomes... 

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