Shapes of isotherms

The classical method to study adsorption deals with establishing a correspondence between thermodynamically equilibrium amount of absorbed gas, the temperature and the partial pressure of the gas in space surrounding adsorbent. Usually, a constant temperature is maintained throughout an adsorption experiment. The shape of isotherms obtained gives information on the character of adsorption. 

The differences in the shape of isotherms have been attributed to the formation of secondary pores during framework dealumination. The hysteresis loops observed were attributed to capillary condensation in the secondary pores. 

In conclusion, the different shapes of isotherms describing equilibrium distributions of a contaminant, between geosorbents and aqueous or gaseous phases, depend on the sorption mechanism involved and the associated sorption energy. At low contaminant concentration, all models reduce to essentially linear correlation. At higher contaminant concentration, when sorption isotherms deviate from linearity, an appropriate isotherm model should be used to describe the retention process. 

Fig. 28 Zeta potential of alumina as a function of equilibrium concentration of SDS (designation based on the shape of isotherm in Fig. 31)...

Various combination of shapes of isotherms and mass transfer factors have been taken into account by solutions of the problem in the literature. One of the simpler cases was adopted by Hougen and Marshall (1947, see Figure 15.13), who took a linear isotherm and diffusion to the external surface as controlling the rate. They developed the solution in analytical form, of which several approximations that are easier to use are mentioned for instance by Vermeulen et al. (1984, p. 16.28). A graphical form of the solution appears in Figure 15.13. This shows the effluent concentration ratio,... 

Figure 11.8 Shape of isothermal, low-frequency, modulus variations against cure conversion for a stoichiometric epoxide-amine system. (1) vitrification (2) antiplasticization decreasing with conversion.

The Freundlich equation, unlike the Langmuir one, does not become linear at low concentration but remains concave to the concentration axis, nor does it show a saturation or limiting value. Roughly speaking krf gives a measure of the adsorbent capacity and slope n, of intensity of adsorption. The shape of isotherm is such that n is less than unity. 

Given the great number of Isotherms published In the literature, within the scope of the present chapter we restrict ourselves to a few arbitrary examples, merely meant to illustrate certain points. As shapes of isotherms depend sensitively on the preparation and pretreatment of the sample, especially on the outgasslng conditions, these Illustrations are not necessarily representative. 

In many physically important cases of localized adsorption, each adatom of the compact monolayer covers effectively n > 1 adsorption sites [3.87-3.89, 3.98, 3.122, 3.191, 3.214, 3.261]. Such a multisite or 1/n adsorption can be caused by a crystallographic Me-S misfit, i.e., the adatom diameter exceeds the distance between two neighboring adsorption sites, and/or by a partial charge of adatoms (A < 1 in eq. (3.2)), i.e., a partly ionic character of the Meads-S bond. The theoretical treatment of a /n adsorption differs from the description of the 1/1 adsorption by a simple Ising model. It implies the so-called hard-core lattice gas models with different approximations [3.214, 3.262-3.266]. Generally, these theoretical approaches can only be applied far away from the critical conditions for a first order phase transition. In addition, Monte Carlo simulations are a reliable tool for obtaining valuable information on both the shape of isotherms and the critical conditions of a 1/n adsorption [3.214, 3.265-3.267]. 

While high membrane water contents can be established in contact with liquid water, water vapor can be the principal mode of hydration of the PEFC membrane under some, not uncommon operation conditions. The shape of isotherms of... 

Figure 3.4. Various typical shapes of isothermal ar-time and dex/df) - time curves observed for the decompositions of solids. The alternative sets of axes A - E represent the different types of behaviour, characteristic of various reactants, that are discussed in the text.

Two factors principally determine the shape of isotherms of the electrical conductivity of binary mixtures ... 

For the determination of component concentration in equilibrium solution the columns packed by adsorbent with some stationary phase is used for the complete separation of component. The adsorption of volatile compounds on hydroxylated silica from ternary solutions was investigated by gas chromatography [3 - 5].For alt components the heats of adsorption on silica were known and it was possible to find the correlation between the heats of adsorption and the shape of isotherm of component adsorption. 

Another form is a power-series expression which can be made to conform to any of the five shapes of isotherm that have been discussed here ... 

The new capillary condensation theory, if essentially valid, claims that the shape of isotherms measured up to saturation, that is, x = PjP = 1, is determined by the pore size distribution of porous bodies, and so any theory to explain sorption isotherms by thermodynamic or kinetic mechanisms becomes meaningless except with respect to the formation of monolayer adsorption. Therefore an important problem in sorption is to investigate the pore structure of sorbent specimens, which are easily varied by varying the conditions of their preparation, and to elucidate the pore structure in relation to the material properties. 

In the above sections, we have amply shown that capillary condensation occurs in the contact zones of elementary particles and the shape of isotherms measured up to a = 1 is determined by existing pore distributions of the porous materials. In addition, it has been shown by sorption methods that porous Vycor glass and Neobead C-5 are ideal porous bodies consisting of spherical elementary particles arranged in characteristic types of packing. Based on these idealized models, we have calculated the effective diffusion coefficients in these porous bodies. 

Figure 6.22. Variation of the shape of isotherms for the crystallization of a PEO/PS block copolymer. Isotherms characterized by n = 1.180j5/(t , - Vp ) as a function of their position along the log time axis. Position defined by (t - (1)20 = 25°C. (Kovacs et...

Equations (14) and (15) are employed to calculate selectivities Sj 3 and S23 from known binary adsorption data (n, n2) and the constant C is determined from the original composition of the surfactant mixture. Some examples of the possible shapes of isotherms determined by the described procedure are given in Figures 5 and 6, and parameters used in the calculations are listed in Table 1. 

Fig 3 3 Effect of mobile adsorption and interaction of adsorbed molecules on shape of isotherm... 

So far, the results have demonstrated that nucleation is characterized decisively by non-steady-state effects. The typical shape of the isotherms illustrated in Fig. 2.9 is the result of the continuous approach of the non-steady-state nucleation rate to the equilibrium nucleation rate at any fixed temperature. The different shape of isotherms shows that the high-temperature and... 

Brunaur, Emmett, and Teller [88] and Emmett and De witt[89] developed a theory which can account for various shapes of isotherms shown in Figure 8.3. This theory is called BET theory and is based on multi-layered adsorption. It assumes that different layers of adsorptions build up at different parts of the surface by adsorption and desorption (At any time, adsorption creates monolayers on empty surface or from desorption of additional layers). This results in Langmuir isotherm (Type I in Figure 8.3) when just a single layer is used. 

The amount of a surfactant adsorbed at a constant temperature and pressure increases with the surfactant concentration to an adsorption limit or exhibits maxima and minima. Adsorption isotherms have been classified by Giles et al. [21,22] using letters and numbers to indicate subgroups (Fig. 5.1). However, two shapes of isotherms, 2L and 3L, are most common [14]. 

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