Sorption/desorption processes equations

Sorption and desorption are usually modeled as one fully reversible process, although hystersis is sometimes observed. Four types of equations are commonly used to describe sorption/desorption processes Langmuir, Freundlich, overall and ion or cation exchange. The Langmuir isotherm model was developed for single layer adsorption and is based on the assumption that maximum adsorption corresponds to a saturated monolayer of solute molecules on the adsorbent surface, that the energy of adsorption is constant, and that there is no transmigration of adsorbate on the surface phase. 

The main reasons for investigating the rates of solid phase sorption/desorption processes are to (1) determine how rapidly reactions attain equilibrium, and (2) infer information on sorption/desorption reaction mechanisms. One of the important aspects of chemical kinetics is the establishment of a rate law. By definition, a rate law is a differential equation [108] as shown in Eq. (32) ... 

While first-order models have been used widely to describe the kinetics of solid phase sorption/desorption processes, a number of other models have been employed. These include various ordered equations such as zero-order, second-order, fractional-order, Elovich, power function or fractional power, and parabolic diffusion models. A brief discussion of these models will be provided the final forms of the equations are given in Table 2. 

Figures 1 and 2 illustrate the sorption/desorption process, respectively. For the sorption of penetrants, =0. Thus, Equation (3) yields...

Equation (57) is empirical, except for the case where v = 0.5, then Eq. (57) is similar to the parabolic diffusion model. Equation (57) and various modified forms have been used by a number of researchers to describe the kinetics of solid phase sorption/desorption and chemical transformation processes [25, 121-122]. 

Metal hydride element is a complex physical item, which can be described by the continuum theory equations. There are three basic processes i) process of hydrogen filtration through a porous hydride matrix and heat-conducting inserting, ii) processes of heat application and heat abstraction from a space of hydrogenation and iii) chemical processes accompanying hydrogen sorption - desorption. 

The first order equation has also been used to describe sorption and desorption processes. Haggerty and Gorelick [43] showed that there is an equivalent distribution of first order rate constants for any distribution of diffusion coefficients. Other models that use distributions of first order rate constants are presented in the following references [44-47]. 

These equations take into consideration the concentrations of radicals bonded with the template (denoted by [A ] and [B ]) and nonbonded ([A°] and [B°]). The equations were formulated with the assumption that the process sorption-desorption of radicals onto template is reversible with equilibrium constants and K ... 

This solution is valid for the initially linear portion of the sorption (or desorption) curve when MtIM is plotted against the square root of time. These equations also demonstrate that for Fickian processes the sorption time scales with the square of the dimension. Thus, to confirm Fickian diffusion rigorously, a plot of MJM vs. Vt/T should be made for samples of different thicknesses a single master curve should be obtained. If the data for samples of different thicknesses do not overlap despite transport exponents of 0.5, the transport is designated pseudo-Fickian. ... 

Sorption relates to a compound sticking to the surface of a particle. Adsorption relates to the process of compound attachment to a particle surface, and desorption relates to the process of detachment. Example 2.2 was on a soluble, nonsorptive spiU that occurred into the ground and eventually entered the groundwater. We will now review sorption processes because there are many compounds that are sorptive and subject to spills. Then, we can examine the solutions of the diffusion equation as they apply to highly sorptive compounds. 

The mass transfer process (absorption, desorption sorption ) in gas/liquid contacting is described according to the Two-Film Theory with the general mass transfer equation ... 

All chemical reactions comprise at least two species. For models of transport processes in groundwater or in the unsaturated zone reactions are frequently simplified by a basic sorption or desorption concept. Hereby, only one species is considered and its increase or decrease is calculated using a Ks or Kd value. The Kd value allows a transformation into a retardation factor that is introduced as a correction term into the general mass transport equation (chapter 1.1.4.2.3). 

A comparison of the desorption rates at pH 7, shown in Figure 7 for the plutonium sorbed from fresh and aged solutions, indicates that the total desorption curve may be interpreted in terms of two different sorbed species. This is expressed in Equations 2, 3, and 4 as two first order processes. For both the fresh and aged systems, the relative quantities of the Ao(d or loosely-held species were almost identical, as were their desorption rate constants. It is likely that the A0<2 or tightly-held species were colloidal in size, since irreversibility is a widely known characteristic of colloid sorption. This was found to apply, for example, in the case of the sorption of colloidal americium on quartz (27). 

Here, v denotes the mean velocity of advection, and k is a rate constant of a reaction with first order kinetics. The last term in the equation R(x) is an unspecified source or sink related term which is determined by its dependence on the depth coordinate x. Instead of R(x), one might occasionally find the expression (ERj) which emphasizes that actually the sum of different rates originating from various diagenetic processes should be considered (e.g. Berner 1980). Such reactions, still rather easy to cope with in mathematics, frequently consist of adsorption and desorption, as well as radioactive decay (first-order reaction kinetics). Sometimes even solubility and precipitation reactions, albeit the illicit simplification, are concealed among these processes of sorption, and sometimes even reactions of microbial decomposition are treated as first order kinetics. 

Both adsorption from a supercritical fluid to an adsorbent and desorption from an adsorbent find applications in supercritical fluid processing.The extrapolation of classical sorption theory to supercritical conditions has merits. The supercritical conditions are believed to necessitate monolayer coverage and density dependent isotherms. Considerable success has been observed by flic authors in working with an equation of state based upon the Tofli isoterm. It is also important to note that the retrograde behavior observed for vapor-hquid phase equilibrium is experimentally observed and predicted for sorptive systems. 

Rates of adsorption and desorption in porous adsorbents are generally controlled by transport within the pore network, rather than by the intrinsic kinetics of sorption at the surface. Since there is generally little, if any, bulk flow through the pores, it is convenient to consider intraparticle transport as a diffusive process and to correlate kinetic data in terms of a diffusivity defined in accordance with Pick s first equation ... 

Equation (3.3.39) is also derived based on kinetic suggestions. In equilibrium, the rate of adsorption (in mol m s ) equals the rate of desorption, which is justified as sorption is an almost instantaneous chemical process (the potential influence of mass transfer is discussed below) ... 

The isotherms related to monomolecular sorption processes were calculated by Eq. (7.38). If < 0.7, then the calculated adsorption or desorption isotherms were equal to the experimental data. However, at

0.7 the experimental isotherms deviate from the calculated values due to the starting of multimolecular sorption processes such as multilayer adsorption and capillary condensation causing the sorption hysteresis. The diameter of mesopores of cellulose samples estimated by Kelvin s equation was in the range from 4 to 12 nm. 

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