Sorbents Langmuir isotherms

Most solid sorbents rely on vapors being sorbed by a physical adsorption mechanism the substance enters the internal pores of the sorbent and is held there by attractive forces considerably weaker and less specific than those of chemical bonds. These weakly attractive forces facilitate desorption for subsequent analysis. The mechanisms for physical adsorption have been studied extensively and are described mathematically by equations such as the Langmuir isotherm. 

Figure 11.15 Observed sorption of dodecylpyridinium on a soil (EPA-12) exhibiting an overall cation exchange capacity of 0.135 mol-kg"1. Two Langmuir isotherms (defined with particular values of C,s max and K/l, recall Eq. 9-5) are placed on the data to illustrate how different portions of the observed isotherm may reflect the influence of different materials in the complex soil sorbent or possibly different mechanisms (data from Brownawell et al., 1990).

This model is not only incomplete (e.g., it makes no prediction of a or t), it is clearly oversimplified, and it is well known that the Langmuir isotherm is not a very good description of most sorbent-sorbate interactions, particularly at high pressures (charcoal is an important exception). It is, nevertheless, quite valuable as a starting point for more sophisticated generalizations and as a framework by which to form a qualitative evaluation of numerical data. 

Interpretation of the adsorption behavior of polypeptides and protein with nonporous HPLC sorbents can thus be based on Eqs. (140)—(143) in which the film mass transfer and surface interaction rates are both considered finite. Simplified cases can be derived from these two relationships for fixed-bed performance409,410,412 where the equilibrium relationship can be expressed by the Langmuir isotherm. Under these isothermal conditions, attainable adsorption capacity of the adsorbent q, which is the amount of the protein retained by the adsorbent when the column reaches saturation, can be expressed by... 

The Langmuir isotherm was developed to describe sorbents with a limited number of sorption sites on their surface... 

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. 

Figure 11.18 The influencing factors of pH (a), initial concentration (b), sorbent dosage (c), contact time (d), the Freundlich isotherm (e), and the Langmuir isotherm (f) of Ag(I) adsorption by sawdust ( ) and poly(3-methyl thiophene)/sawdust composite (). Reprinted from Ref. [83] with permission from WUey InterScience (2009).

The Langmuir isotherm is represented by a hyperbola of the type schematically shown in fig. 15. (Quantity of sorbate per g (or per unit surface) of the sorbent plotted against the pressure). The curvature is concave to the pressure axis in contrast to Fig. 13. This is in conformity with the positive heat effect assumed. 

There is a formal analogy between this result and the vapour pressure isotherms of solutions (section a). This becomes clear if we take into consideration that in the relevant systems the molecular weight of the volatile component will always be very small as compared to that of the sorbent. The molecular fraction n/(rio h will then remain small and nearly proportional to n/n. For small values of n/n the hyperbola of the athermal ideal mixture will also appear as a straight line. The difference which remains between the Langmuir isotherm for heats of absorption approaching zero on the one hand and the vapour pressure of the ideal mixture on the other hand is not due to a difference in energetic conditions but to the finite number JV of available places where the vapour can be bound, in contrast to the unlimited number of molecules which can be stored at mixing. 

For the process of pressure-swing adsorption of a nontrace binary gas mixture, Rege and Yang (2001) have defined a dimensionless sorbent selection parameter S based on the Langmuir isotherm parameters ( 7Aff)ma>i At,(< B,.)max Bt for species A and B, respectively (see equation (3.3.112 b)) ... 

Assume linear isotherm behavior corresponding to the lower-pressure region of the Langmuir isotherm. Show that this sorbent selection parameter S, based on the product of the selectivity ratio (( Air)max At / (<7B.j)max Bt) and the ratio of the working capacity of the two species A and B (working capacity is the difference between the adsorbed amounts at adsorption (high) pressure Ph and at desorption (low) pressure Pi, preferably with the gas mbcture) (maybe described as follows ... 

Adsorption isotherms are used to quantitatively describe adsorption at the solid/ liquid interface (Hinz, 2001). They represent the distribution of the solute species between the liquid solvent phase and solid sorbent phase at a constant temperature under equilibrium conditions. While adsorbed amounts as a function of equilibrium solute concentration quantify the process, the shape of the isotherm can provide qualitative information on the nature of solute-surface interactions. Giles et al. (1974) distinguished four types of isotherms high affinity (H), Langmuir (L), constant partition (C), and sigmoidal-shaped (S) they are represented schematically in Figure 3.3. 

Adsorption isotherms play a key role in either the design of the adsorption-based process for the disposal of wastes containing VOCs or modeling the catalytic oxidation process. The equilibrium data for mesoporous sorbents are fitted to combined model of Langmuir and Sips equations. This hybrid isotherm model with four isotherm parameters... 

Competition between adsorbing species for the same site on a sorbent has been modeled with the so-called competitive Langmuir model. In this model an individual isotherm equation with its own constants is written for each species sorbed by a given sorbent (see Table 10.7). 

Experiments to determine constants for the isotherms are usually performed in laboratory batch tests, that is in vessels that contain a known total amount of sorbent solid mixed with a known total amount of a potentially sorbing aqueous species. The approach is often to systematically vary sorbent and/or sorbate concentrations and pH, for example, in a series of centrifuge tubes. After centrifugation the amount sorbed is then deterimined by its difference from the total sorbate added (cf. Catts and Langmuir 1986). 

The Langmuir and Freundlich isotherms are the most common isotherms it is likely that the constants in these isotherms for many sorbate-sorbent pairs have already been measured (see Tables 3.1 and 3.2, respectively). 

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