Adsorption isotherms, equation

The preceding derivation, being based on a definite mechanical picture, is easy to follow intuitively kinetic derivations of an equilibrium relationship suffer from a common disadvantage, namely, that they usually assume more than is necessary. It is quite possible to obtain the Langmuir equation (as well as other adsorption isotherm equations) from examination of the statistical thermodynamics of the two states involved. 

The equation of state for a solid film is often ic= b - aa (note Section IV-4D). Derive the corresponding adsorption isotherm equation. Plot the data of Problem 11 according to your isotherm equation. 

In Table 10.2, this correlation is shown, comparing solid phase concentration calculated from the retention times of the fronts, and using the adsorption isotherm equation. 

Bi-layer adsorption is not uncommon and the development of the bi-layer adsorption isotherm equation is a simple extension of that used for the mono-layer equation. The Langmuir equation for bi-layer adsorption is as follows ... 

The linear relationship between H NMR transverse relaxation rate and (1 av) is shown in Figure 30 for pregelled potato starch (Hills et al., 1999). The change in slope at about 0.90 c/w corresponds to the bulk water break (i.e., the removal of bulk water) in a corresponding adsorption isotherm. Equation... 

For sufficiently low coverages (such as those expected for low bulk concentrations, e.g. trace pollutants), the adsorption isotherms, equation (2), revert to linear ones ... 

Ym = l/6m n = 1). In general, the larger is the number of adjustable parameters in an adsorption isotherm equation, the better its fit to experimental data is likely to be. 

The adherence of experimental sorption data to an adsorption isotherm equation provides no evidence as to the actual mechanism of a sorption process. 

Adsorption from liquids is less well understood than adsorption from gases. In principle the equations derived for gases ought to be applicable to liquid systems, except when capillary condensation is occurring. In practice, some offer an empirical fit of the equilibrium data. One of the most popular adsorption isotherm equations used for liquids was proposed by Freundlich 21-1 in 1926. Arising from a study of the adsorption of organic compounds from aqueous solutions on to charcoal, it was shown that the data could be correlated by an equation of the form ... 

The relationship between the activity of adsorbed ions fiQO in Eqn. 5-21 and the adsorption coverage 6i is known as an adsorption isotherm. Equations 5-23 and 5-24 show simple adsorption isotherms ... 

The adsorption data is often fitted to an adsorption isotherm equation. Two of the most widely used are the Langmuir and the Freundlich equations. These are useful for summarizing adsorption data and for comparison purposes. They may enable limited predictions of adsorption behaviour under conditions other than those of the actual experiment to be made, but they provide no information about the mechanism of adsorption nor the speciation of the surface complexes. More information is available from the various surface complexation models that have been developed in recent years. These models represent adsorption in terms of interaction of the adsorbate with the surface OH groups of the adsorbent oxide (see Chap. 10) and can describe the location of the adsorbed species in the electrical double layer. 

The adsorption isotherm —Equation (8) —associated with this surface equation of state is called the Henry law limit, in analogy with the equation that describes the vapor pressure of dilute solutions. The constant m, then, is the adsorption equivalent of the Henry law constant. When adsorption is described by the Henry law limit, the adsorbed state behaves like a two-dimensional ideal gas. 

Equation 3 was verified experimentally (3) over wide ranges of temperature and equilibrium pressure for the adsorption of various vapors on active carbons with different parameters for the microporous structure. For adsorption on zeolites, this equation fitted the experimental results well only in the range of high values of 0 (4, 5, 6, 7). Among other equations proposed for the characteristic curve (4, 5, 8, 9, 10) we chose to use the Cohen (4) and Kisarov s (10) equation, which starts from the following adsorption isotherm equation ... 

Langmuir Equation. It is probable that adsorbed layers have a thickness of a single molecule because of the rapid decrease in intermolecular forces with distance. The Langmuir adsorption isotherm equation is... 

An important question is how much of a material is adsorbed to an interface. This is described by the adsorption function T = /(/, T), which is determined experimentally. It indicates the number of adsorbed moles per unit area. In general, it depends on the temperature. A graph of T versus P at constant temperature is called an adsorption isotherm. For a better understanding of adsorption and to predict the amount adsorbed, adsorption isotherm equations are derived. They depend on the specific theoretical model used. For some complicated models the equation might not even be an analytical expression. 

The most simple type, A, is that of a linear increase. It is described by the Henry adsorption isotherm equation ... 

Type B is very common. It is concave with respect to the abscissa. Most surfaces are heterogeneous. There are adsorption sites, which have a high affinity, and regions, which have a low affinity. The high affinity sites are occupied first, which accounts for the steep increase at low pressure. Another reason is sometimes a lateral repulsion between adsorbed molecules. This type of adsorption isotherm is described by the Freundlich1 adsorption isotherm equation [366] ... 

Type E is common for the adsorption of gases. Usually the first concave part is attributed to the adsorption of a monolayer. For higher pressures more layers adsorb on top of the first one. Eventually, if the pressure reaches the saturation vapor pressure, condensation leads to macroscopically thick layers. It can be described by the BET adsorption isotherm equation Eq. (9.37) (see below). 

Alternatively, the Langmuir adsorption isotherm equation can be expressed by the number of adsorbed moles per gram or surface area... 

Which theory is suitable for a certain application The adsorption theory of Henry is applicable at low pressure. This, however, is natural since it can be viewed as the first term in a series of the adsorption function. A widely used adsorption isotherm equation is the BET equation. It usually fits experimental results for 0.05 < P/P0 < 0.35. For very small pressures the fit is not perfect due to the heterogeneity. For higher pressures the potential theory is more suitable at least for flat, homogeneous adsorbents. It often applies to P/Po values from 0.1 to 0.8. Practically for P/Po > 0.35 adsorption is often dominated by the porosity of the material. A more detailed description of adsorption is obtained by computer simulations [382],... 

An example of the adsorption to one such material is shown in Fig. 9.16. The siliceous material, called MCM-41, contains cylindrical pores [397], With increasing pressure first a layer is adsorbed to the surface. Up until a pressure of P/Po 0.45 is reached, this could be described by a BET adsorption isotherm equation. Then capillary condensation sets in. At a pressure of P/Po 0.75, all pores are filled. This leads to a very much reduced accessible surface and practically to saturation. When reducing the pressure the pores remain filled until the pressure is reduced to P/Pq rs 0.6. The hysteresis between adsorption and desorption is obvious. At P/Po 0.45 all pores are empty and are only coated with roughly a monolayer. Adsorption and desorption isotherms are indistinguishable again below P/Po 0.45. 

A most important feature of the models upon which adsorption isotherm equations, such as those above, are based is a characteristic assumption relating to heat of adsorption and surface coverage. Several factors merit consideration in this respect. 

Show that the data fit a Langmuir adsorption isotherm equation and calculate the area occupied by each adsorbed dodecanol molecule at limiting adsorption. 

Experimental studies of the partitioning of sodium into aragonite, calcite, and dolomite have been carried out by White (1977, 1978), who concluded that the mode of substitution was as Na2CC>3 rather than as Na bicarbonate or hydroxide. The behavior of Na+ does not follow that normally described by a simple partition coefficient, but rather coprecipitation is better described by a Freundlich-type adsorption isotherm equation, implying that adsorption is important. 

Determination of the Micropore Volume 6.7.3.1 Dubinin Adsorption Isotherm Equation... 

The Dubinin adsorption isotherm equation is a good tool for the measurement of the micropore volume. This isotherm can be deduced with the help of Dubinin s theory of volume filling, and Polanyi s adsorption potential [11,26], The Dubinin adsorption isotherm equation has the following form [11]... 

It is possible, as well, to express the Dubinin adsorption isotherm equation in linear form... 

Langmuir-Type and Fowler-Guggenheim-Type Adsorption Isotherm Equations... 

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