Linear adsorption isotherm, assumption

The system of differential equations is too complex to be solved analytically. Assumptions of a linear adsorption isotherm can be used to obtain analytical solutions, but this approach is generally not applicable to describe affinity chromatography experiments. Several numerical techniques arc used to solve the system of partial differential equations. The other method is to use an analytical solution with simplifying approaches [32] that describe the adsorption process with a single step and a lumped mass transfer coefficient [27],... 

The assumption of a linear adsorption isotherm, = K in Equation 4b, may be valid under low solution concentrations (21). However, Rao and Davidson (22) showed that this assumption could produce errors within a factor of 2 or 3. Additionally, K can be estimated from soil 0M content (22). Consequently, the solution... 

This equation has been theoretically derived by Liaw (Liaw et al 1979) by the assumption of a parabolic profile of the loading within a pellet. Note that the Glueckauf equation is only valid for a linear adsorption isotherm (dX/dc i = const). 

Thus Eq. (3.3.74) is only valid for the point where we reach a mean adsorption loading of approximately 50% of the maximum equilibrium value. In addition, Eq. (3.3.74) was deduced based on the assumption of a linear adsorption isotherm, and is therefore only a very rough estimation for a Langmuir isotherm and so on, as shown below. 

In the simplest case of a fixed bed of adsorbent particles, the following mass transport processes are considered axial dispersion in the interparticle fluid phase, fluid-to-particle mass transfer, intrapaitide diffusion, and a first-order, reversible adsorption in the interior of the particle. The last step corresponds to a linear adsorption isotherm with a finite adsorption rate. This assumption includes the case of inflnitdy fast adsorption rate. 

In reality, none of the above assumptions apply to activated carbon, even approximately, so it comes as a surprise that the Langmuir equation can be used to linearize adsorption isotherms. In fact, assumptions 2 and 3 probably compensate for each other, as adsorbate-adsorbate interactions increase with increasing coverage at the same time as enthalpies of adsorption decrease with increasing coverage. 

Show that the results may be interpreted on the assumptions that the solids are completely mixed, that the gas leaves in equilibrium with the solids and that the adsorption isotherm is linear over the range considered. If the flowrate of gas is 0.679 x 10-6 kmol/s and the mass of solids in the bed is 4.66 g, calculate the slope of the adsorption isotherm. What evidence do the results provide concerning the flow pattern of the gas ... 

The speed of the adsorption wave can be readily derived by introducing the linear isotherm assumption and the chain mle derivative of q with respect to t. The wave speed results because the assumptions turn Eq. (9.10) into a kinematic wave equation and the wave speed W is instantly recognized as ... 

Pankow (1987) showed that this assumption of a linear Langmuir isotherm is consistent with the pioneering work of Junge (1977) on the adsorption of SOCs on particles. The fraction of the total SOC in the atmosphere adsorbed on aerosol particles, denoted , was hypothesized by Junge to be related to the surface area (ST, cm2 per cm3 of air) of the TSP and the saturation vapor pressure of the SOC by... 

Geochemical models of sorption and desorption must be developed from this work and incorporated into transport models that predict radionuclide migration. A frequently used, simple sorption (or desorption) model is the empirical distribution coefficient, Kj. This quantity is simply the equilibrium concentration of sorbed radionuclide divided by the equilibrium concentration of radionuclide in solution. Values of Kd can be used to calculate a retardation factor, R, which is used in solute transport equations to predict radionuclide migration in groundwater. The calculations assume instantaneous sorption, a linear sorption isotherm, and single-valued adsorption-desorption isotherms. These assumptions have been shown to be erroneous for solute sorption in several groundwater-soil systems (1-2). A more accurate description of radionuclide sorption is an isothermal equation such as the Freundlich equation ... 

The evidence is indirect, but fairly convincing it rests mainly on the fact that if the linear part of the adsorption isotherms, corresponding to the building up of layers beyond the first, is extrapolated back to zero pressure, and the area of the surface calculated on the assumption that the extrapolated amount of gas adsorbed is in a closely packed monolayer on the surface, consistent results are obtained with all the gases for the area of the surface. 

Because of the assumption made above, the solution is limited to the linear region of analyte partitioning and adsorption isotherms. The analyte distribution between two liquid phases (eluent and adsorbed phase) at equilibrium could be described as follows ... 

The relationships of Equations 5 and 2 are unquestionably valid for unlimited surface coverage on ideal external open (flat, planar, accessible) surfaces ranging from nil at E to infinity at E=0. All of the inherent assumptions (tabulated above) are equally valid as models for physical adsorption in internal constricted regions. These are classically denoted as ultramicropores ( 2 nm), micropores(<2 nm), mesopores (2<1000nm) and macropores (very large and difficult to define with adsorption isotherm). In these instances there are finite concentration limits corresponding to the volume (space, void) size domain(s). Although caution is needed to deduce models from thermodynamic data, we can expect to observe linear relationships over the respective domains. The results will be consistent with, albeit not absolute proof of the models. 

This analytical solution revealed well the qualitative critical current behavior seen experimentally (Fig. 32), but did not fully predict the quantitative dependence of the critical current at a Pt anode catalyst on Pco in the anode feed [66]. The latter required assuming a Temkin adsorption isotherm for CO at the anode catalyst, as originally suggested in Ref. 67. By using a Temkin isotherm, Eq. (36) allows the free energy of CO adsorption to decrease linearly with co This assumption is in agreement with literature data for CO adsorption on Pt group metals ... 

In this chapter a lumped dynamic model of a porous catalyst pellet is developed on the basis of the active site theory and assuming equilibrium adsorption-desorption according to a linear Langmuir isotherm. This model is compared with a previous pseudo-homogeneous model due to Liu and Amundson (1962). Next, the assumption of equilibrium adsorption-desorption is relaxed and the effect of both activated as well as non-activated adsorption is presented. The rate of adsorption is treated in very simple terms under the Langmuir postulates as discussed earlier. 

From the above equations, models of different degrees of sophistication can be obtained. The simplest physically realistic one is that based on the assumption of equilibrium adsorption-desorption with linear Langmuir adsorption isotherm, which will be presented first, followed by the general non-equilibrium adsorption-desorption model. 

The assumption of a linear isotherm is, of course, valid only over a limited concentration range. The use of the full adsorption isotherm may require numerical solution of the problem the results of such treatments are in qualitative agreement with that for the linearized isotherm (48, 49) (see Figure 13.5.4). The rate of attainment of equilibrium is clearly seen to depend on the bulk concentration C, however. 

Up to this point the treatment is strictly thermodynamic. However, the next steps necessarily demand the use of extra-thermodynamic assumptions. The first of them is the choice of an adsorption isotherm, which in general may be written as Pa = f( , 0). The most commonly adopted isotherm is the Frumkin isotherm, Eq. (1), assuming arbitrarily a linear or a quadratic dependence of upon E. The next necessary assumption concerns the dependence of p upon E, for which the following expression is usually adopted P = Pn,axexp -5( - moU, where b is a constant. Thus if a certain adsorption isotherm px = f( , 0) is selected, the partial derivative of In x with respect to E at constant 0 can be calculated and therefore Eq. (4) after integration yields the relationship = g( ). From this relationship the dependence of y upon E is obtained by integration and the differential capacity C is calculated fromC = dCT /d . 

In eq. (9.2-lb), we have assumed that the pore diffusivity as well as the surface diffusivity are independent of position, time as well as concentration. The surface diffusivity is known to exhibit a very strong concentration dependence over the range where adsorption isotherm is nonlinear. For the linear isotherm dealt with here, the assumption of constant surface diffusivity is usually valid. 

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