where qm (mgg"1) is the maximum amount of adsorbate in the adsorbent and b (l mg" ) is the equilibrium constant related to the enthalpy of the process. It can be easily derived considering that at equilibrium the amounts (per unit time) of molecules adsorbed and desorbed are equal, that is, the rates of both reactions are the same.

Four hypotheses are behind this formulation. First adsorbate cannot attach indifferently on the surface, but only on particular sites that satisfy appropriate conditions, that is, they are able to establish a link with the adsorbate (by way of van der Waals forces or chemical bonds). Second, adsorption occurs only in a monolayer, that is, once a site is occupied by a molecule, it cannot bind to another one: this leads to the saturation shape of the isotherm. Third, sites are energetically equivalent, that is, the energy involved is the same in all sites. Fourth, the molecules adsorbed are not able to influence adsorption (and desorption) because they do not interact with the other molecules of adsorbate. The last two assumptions can be summarized stating that adsorption occurs homogeneously (even if discreetly) on the surface.

Langmuir isotherm is shown in Figure 2. The shape of the curve clearly depends on parameters; however, usually the saturation is reached quite fast.

The equation for the Freundlich isotherm is q = aCl!n where a (mgg—*) is the mass adsorbed with a unitary concentration and n is an empirical constant usually greater than 1. As for Langmuir, Freundlich model is not empirical but it is theoretically based. Among all the assumptions behind this model, it is important to mention two of them.

First, adsorption is no more monolayer, but several layers of adsorbate can be attached on the adsorbent. The mathematical formulation does not even provide for an asymptote: adsorbent would never saturate and would continuously bind to adsorbate. This is obviously not true in nature, and in fact Freundlich isotherm is not able to fit experimental data when concentration of adsorbate is high.

The second important assumption is that the energy required for adsorption is not constant, but it varies and it is exponentially distributed. The strength of bonds is not homogeneous due either to the physicochemical characteristics of sites or to the number of molecules already adsorbed. In particular, more the number of molecules bound to a site, less probable is that another molecule binds to the same site because an (exponentially) higher energy is required.

Figure 2 Isotherms of Langmuir (solid line), Freundlich (dashed), and BET (dotted). They describe the amount of adsorbate expressed in term of mass fraction q (or fraction of sites occupied 6 for BET) depending on its concentration C (or on pressure of gaseous adsorbate relative to saturation pressure).

Figure 2 Isotherms of Langmuir (solid line), Freundlich (dashed), and BET (dotted). They describe the amount of adsorbate expressed in term of mass fraction q (or fraction of sites occupied 6 for BET) depending on its concentration C (or on pressure of gaseous adsorbate relative to saturation pressure).

This is well clear also by looking at the Freundlich curve (Figure 2): the slope decreases when concentration of adsorbate increases.

The parameter values for both isotherms are usually estimated by interpolation of empirical observation. Parameters change in response to a variation of temperature, leading to a decrease of the fraction adsorbed when temperature rises, and vice versa. In fact, higher temperature means stronger vibrational motion, that is, a higher probability for adsorbate to break weak van der Waals bonds. The slope of both curves decreases, and Langmuir curve reaches the asymptote for higher concentration of adsorbate.

Always from empirical observation, it appears that Freundlich's model is usually more appropriate to describe adsorption from liquid solutions, whereas the Langmuir's one tends to fit better the data on adsorption of gases.

Another isotherm that fits well the data of adsorption of gases on solids is the BET isotherm (after Brunauer, Emmet, and Teller). The main assumption of this model is that the first layer of adsorbate can adsorb on its another layer, and this adsorption is also regulated by a Langmuir equation:

where 0 is the fraction of sites on the adsorbent that have adsorbed a molecule, z is the ratio between the pressure of adsorbate and its saturation pressure at the same temperature, and c is a constant that depends on enthalpy of desorption and vaporization.

As in the Freundlich's model, BET isotherm implies an endless adsorption, even if in this case the slope of the curve increases with the pressure of adsorbate (Figure 2).

There are several other isotherms, sometimes specific for some adsorbate-adsorbent couples, for instance, the Redlich-Peterson, the Sips, and the Temkin isotherms.

A simpler approach is to assume that adsorption is linearly dependent on the concentration of adsorbate:

where k is the partition coefficient. This simple relationship has clearly a strong limitation, given that q increases very fast with concentration. However, for diluted adsorbate, this model can easily be used with small or no error: indeed, it is an approximation of the Langmuir model in conditions far from saturation. The value of the partition coefficient can be estimated via empirical observation, or, for nonionic compounds like organic chemicals, can be estimated through k — f k k fockoc cz where foc is the fraction of carbon in the adsorbent and koc is the partition coefficient of adsorbate in the organic carbon. The latter can be estimated directly from the well-known kow (partition coefficient of a substance in an octanol-water mixture).

Due to the particular affinity of organic chemicals and organic carbon, it is assumed that adsorption occurs only on the carbonic fraction (i.e., sites are located only on this fraction).

Even if adsorption is usually a process at the equilibrium, sometimes it is also important to study its dynamics, for instance, in those systems where the contact between adsorbate and adsorbent is too small (e.g., rains percolating through an arid soil). The assumption and methodology to derive the dynamic law for adsorption are similar to the theory of the two films used for absorption: the single film considered to separate the bulk of solution where adsorbate is uniformly diluted due to eddy diffusion, and the film near the surface of adsorbent, where only molecular diffusion occurs (Figure 3).

In this case the second film is not necessary because adsorption is a surface process; molecules stick to the surface and do not enter the solid structure. Nonetheless, there are two resistances that contrast the motion of adsorbate: the first is the resistance due to stagnant film (i.e., molecular diffusion, proportional to the ratio between the molecular diffusivity DAB and the thickness 6 of the film); the other is the resistance due to motion into the pores of solid. In order to complete

Liquid bulk

Liquid film

Liquid film

Eddy diffusion

Figure 3 Conceptual model used to derive the equation of the dynamics of adsorption: a stagnant liquid (or gaseous) film, where only molecular diffusion occurs, surrounds a particle of porous adsorbent immersed in a liquid (or gaseous) bulk where concentration of solute is constant due to eddy diffusion.

adsorption till the equilibrium, molecules need to move through tortuous pores to reach all the sites available.

Mass transfer in the fluid film is described by qc , ,

where kf is the mass-transfer coefficient in the fluid film, a is the ratio between area and volume of adsorbent, and c. the concentration at the interface. This is the same equation as used in absorption case, except for a that comes into play because adsorption is only a surface process and as consequence of mass conservation processes.

Adsorbate flows into the porous adsorbent to reach the sites that are hidden more deeply in the particle; this leads to the mass-transfer equation in solid. Assuming spherical shape, the equation is sq dt'~

Ds r2 0r

Sr where Ds is the diffusion coefficient of adsorbate in the adsorbent and r is the radius of the particle.

To simplify the solution of the system of the two masstransfer equations, the latter can be simplified assuming that the solid particle also behaves as if a solid film was present, that is, sq st

where ks is the mass-transfer coefficient in the solid, q is the concentration of adsorbate in the solid volume (and not mass as previously), and qi the concentration at the interface.

At the interface, adsorbate in fluid and solid are at equilibrium, which, for simplicity and with the constraint already mentioned, is assumed to be linear, that is, qi = mCi

Equalizing mass-transfer fluxes (no accumulation is permitted at the interface), after several calculations it is possible to describe the flux of adsorbate that sticks on the surface with the following equation:

Given the assumed linear equilibrium, the equation can be rewritten as qq sT

where q and C* are the concentrations, respectively, on solid and fluid at equilibrium.

Once again, like in absorption, the driving force of global mass transfer is the distance from the equilibrium, and the total resistance is equal to the inverse of the sum of the inverse of single resistances.

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