Langmuir isotherm derivation

The Langmuir isotherms derived therefrom provide the basis for the formulation of rate equations. 

Langmuir-Type Relations For systems composed of solutes that individually follow Langmuir isotherms, the traditional iTuilti-component Langmuir equation, obtained via a kinetic derivation, is... 

The assumptions made to derive the Langmuir isotherm (Eq. 2.7) are well known Energetic equivalence of all adsorption sites, and no lateral (attractive or repulsive) interactions between the adsorbate molecules on the surface. This is equivalent to a constant, coverage independent, heat (-AH) of adsorption. 

The above equation reduces to the one used to derive the Langmuir isotherm (Eq. 6.26) when Xj=0 or A0=O. 

Give the essential assumptions made in deriving the BET isotherm. Ho v does the BET isotherm differ from the Langmuir isotherm ... 

Sorption. Capture of neutral organics by non-living particulates depends on the organic carbon content of the solids (9). Equilibrium sorption of such "hydrophobic" compounds can be described by a carbon-normalized partition coefficient on both a whole-sediment basis and by particle size classes. The success of the whole-sediment approach derives from the fact that most natural sediment organic matter falls in the "silt" or "fine" particle size fractions. So long as dissolved concentrations do not exceed 0.01 mM, linear isotherms (partition coefficients) can be used. At higher concentrations, the sorptive capacity of the solid can be exceeded, and a nonlinear Freundlich or Langmuir isotherm must be invoked. 

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... 

Ferrocene-based Linear Polymers. The first derivative that was studied from the electrochemical point of view was polyvinylferrocene (PVF). As illustrated in Figure 25, it displays a single oxidation process, which in some solvents is affected by problems of adsorption of the oxidation product (though not of the ideal Langmuir isotherm type discussed in Chapter 2, Section 1.6). 

This model is directly derived from the Langmuir isotherm. It assumes that the adsorbent surface consists of two different types of independent adsorption sites. Under this assumption, the adsorption energy distribution can be modeled by a bimodal discrete probability density function, where two spikes (delta-Dirac functions) are located at the average adsorption energy of the two kinds of sites, respectively. The equation of the Bilangmuir isotherm is... 

For example, although it is easy to deduce, the Langmuir isotherm (Section 6.8.8) has several drawbacks Originally it was derived for the adsorption of molecules from... 

The Langmuir isotherm can be derived from a statistical mechanical point of view. Thus, for the reaction M + Agas Aads, equilibrium is established when the chemical potential on both phases is the same, i.e., pgas = p,ads. The partition function for the adsorbed molecules as a system is given by... 

One isotherm that is both easy to understand theoretically and widely applicable to experimental data is due to Langmuir and is known as the Langmuir isotherm. In Chapter 9, we see that the same function often describes the adsorption of gases at low pressures, with pressure substituted for concentration as the independent variable. We discuss the derivation of Langmuir s equation again in Chapter 9 specifically as it applies to gas adsorption. Now, however, adsorption from solution is our concern. In this section we consider only adsorption from dilute solutions. In Section 7.9c.4 adsorption over the full range of binary solution concentrations is also mentioned. 

Since the entire derivation of the Langmuir isotherm assumes dilute solutions, the concentration c2 of the solute (denoted by c, for simplicity) rather than the activity is generally used in presenting experimental results. Figure 7.16b shows how actual experimental data might appear. 

We apply this method to the derivation of the Langmuir isotherm both to illustrate the method and to see the assumed nature of the surface energy states on which it is based. 

Until now, we have focused our attention on those adsorption isotherms that show a saturation limit, an effect usually associated with monolayer coverage. We have seen two ways of arriving at equations that describe such adsorption from the two-dimensional equation of state via the Gibbs equation or from the partition function via statistical thermodynamics. Before we turn our attention to multilayer adsorption, we introduce a third method for the derivation of isotherms, a kinetic approach, since this is the approach adopted in the derivation of the multilayer, BET adsorption isotherm discussed in Section 9.5. We introduce this approach using the Langmuir isotherm as this would be useful in appreciating the common features of (and the differences between) the Langmuir and BET isotherms. 

EXAMPLE 9.4 Kinetic-Theory-Based Description of Binary Adsorption. Assume that two gases A and B individually follow the Langmuir isotherm in their adsorption on a particular solid. Use the logic that results in Equation (46) to derive an expression for the fraction of surface sites covered by one of the species when a mixture of the two gases is allowed to come to adsorption equilibrium with that solid. 

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.)... 

Another concept is that the electronic work function changes linearly with the amount adsorbed or that the dipole moment is independent of the concentration. The (1 — 0) concept states that the rate of adsorption for a constant arrival rate is proportional to the fraction of the surface which has not yet been covered. The last two concepts permitted Langmuir to derive his famous adsorption isotherm, which has been verified by experiment in many systems (see the discussion in section IV). Langmuir s experimental work for Cs on W convinced him that the (1 — 0) law was not applicable in this system. This work also led to the concept that the energies involved in surface migration were much smaller than the energies involved in evaporation. 

Figure 9.5 Graphic representation of the Langmuir isotherm Eq. 9-4. Note that Cis max and K,h can be derived from the slope and intercept of the regression line (see also Illustrative Example 9.1).

We can compare the coverage of mobile surface species predicted by Eq. 11.99 with the coverage calculated for immobile species (Langmuir adsorption isotherm) derived in Section 11.5.3. To do this comparison, we take the low-pressure limit of Eq. 11.79, in which the denominator becomes unity. The ratio of the coverages in these two limiting cases is... 

Besides having the capital property of being able to imitate any kinetics—see Eq. 61—Astarita s uniform kinetics are often patient of a mechanistic derivation. Thus, for what Astarita and Ocone call uniform generalized Langmuir isotherm kinetic mechanism (UGLIKM), we might take 6(x) dx to be the fraction of catalytic surface occupied by the species A(x) with index in... 

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