Adsorbed concentration, expression

This expression can be used to describe both pore and solid diffusion so long as the driving force is expressed in terms of the appropriate concentrations. Although the driving force should be more correctly expressed in terms of chemical potentials, Eq. (16-63) provides a qualitatively and quantitatively correct representation of adsorption systems so long as the diffusivity is allowed to be a function of the adsorbate concentration. The diffusivity will be constant only for a thermodynamically ideal system, which is only an adequate approximation for a limited number of adsorption systems. 

As for equilibrium values of as and P they are mainly dependent on relations between such parameters of the systems as initial electric conductivity of adsorbent, concentration of chemisorbed particles, reciprocal position of the energy levels of absorbate and adsorbent. Thus, during acceptor adsorption in case of small concentration of adsorption particles one can use (1.82) and (1.84) to arrive to expressions for equilibrium values of ohmic electric conductivity and the tangent of inclination angle of VAC ... 

In two limiting cases differing in the values of stationary concentration of chemisorbed radicals and initial electric conductivity of adsorbent the expression (2.94) acquires the following shape ... 

Because more of the classic adsorption studies have been done with gaseous adsorbates, concentrations are usually expressed in terms of partial pressures, hence the terms concentration and partial pressure are used somewhat interchangeably throughout this section. For gaseous species, the term saturaiian means that any increase in analyte concentration will result in spontaneous condensation of any additional analyte from the gas phase. 

Identification of the effective diffusion coefficient with the mathematical model of hatch adsorption. The model assumes that the carbon particles are spherical and porous (gp- voids fraction). Using c (kg A/m fluid inside the pores) and q (kg A/kg adsorbent) to express the concentration of the transferable species through the pores and through the particle respectively, we can write the following expression for transport flux ... 

Finite Concentration. In this concentration range, surface adsorption results in nonlinear isotherms in which partition coefficients and retention volumes are dependent upon the adsorbate concentration in the gas phase. This means that a single partition coefficient, Ks (= T/c), is insufficient to characterize the process and the differential (3r/3c)T is required. Here, T is the surface excess of adsorbate expressed in mol m z, c Is the gas phase adsorbate concentration, and T is the column temperature. Nonlinear Isotherms give asymmetric peaks, whose shapes and retention volumes depend on the concentration of the probe. 

An expression for the adsorbed concentration is obtained with the help of the Dubinin-Radushkevitch adsorption isotherm for microporous materials ... 

We have already considered in detail the reversible first order reaction. The kinetics might be quite different (as illustrated in Exercise 6.2.4) and similar expressions might be derived, but the algebraic labor would be much greater. If the adsorption and desorption are rapid in comparison with reaction we can always substitute the equilibrium adsorbed concentrations in any rate law. Thus if the reaction A B really second order in both... 

The concentration form of Eq. (8-3) can be obtained by introducing the concept of an adsorbed concentration C, expressed in moles per gram of catalyst. If represents the concentration corresponding to a complete monomolecular layer on the catalyst, then the rate of adsorption, moles/(sec) (g catalyst) is, by analogy with Eq. (8-1),... 

At steady state the rates of adsorption r, surface reaction r, and desorption are e qual. To express the rate solely in terms of fluid concentrations, the adsorbed concentrations C, Cg, Q, and C must be eliminated from Eqs. (9-15) to (9-22). In principle, this can be done for any reaction, but the resultant rate equatLOJiTnYoLves-all-the-rate-co-nstantS-/r,-a-nd-the-eq-uilibHum-constants Ki. Normally neither type of constant can be evaluated independently. Both must be determined from measurements of the rate of conversion from fluid reactants to fluid products. However, there are far too many constants, even for simple reactions, to obtain meaningful values from such overall rate data. The problem can be eased, with some confidence from experimental data, by supposing that one step in the overall reaction controls the rate. Then the other two steps occur at near-equilibrium conditions. This greatly simplifies the rate expression and reduces the number of rate and equilibrium constants tr t must be determined from experiment. To illustrate the procedure equations for the rate will be developed, for various controlling steps, for the reaction system... 

Adsorption or Desorption Controlling Still retaining the simple reaction A + B C, qI us now suppose that the adsorption of A is the slow step. Then the adsorption of B, the surface reaction, and the desorption of C will occur at equilibrium. The rate can be formulated from the adsorption equation (9-14). The adsorbed concentration in this expression is obtained from the equilibrium equations for the surface rate [Eq. (9-21)], adsorption of B [Eq. (9-23)], and desorption of C [Eq. (9-24)]. Thus... 

The adsorption isotherms for the RhMoe Anderson phase were measured at room temperature using the C, Cf and Ca values determined for M=Mo. When the concentration of Mo adsorbed (Ca) expressed as monolayer (%) was plotted against the Mo concentration in solution at equilibrium (Cf), the shape of the resulting curve was found to follow the Langmuir model [13]. Hence, by plotting the linearised form of the Langmuir equation, i. e. 

Adsorption capacity (or loading) is probably the most important characteristic of an adsorbent. More detail is provided in Section 14.3.2. The loading is the amount of adsorbate taken up by the adsorbent, per unit mass (or volume) of the adsorbent, and it depends on the fluid-phase concentration, the temperature, and other conditions (especially the initial condition of the adsorbent). Typically, adsorption capacity data are plotted as isotherms (loading of adsorbate on the adsorbent versus fluid-phase adsorbate concentration at constant temperature), isosteres, isobars, and others mentioned later. Examples are shown in Figures 14.1 through 14.3. Adsorption capacity is of paramount importance to the capital cost, because it sets the amount of adsorbent required and also the volume of the adsorber vessels the costs of both are significant if not dominant. When comparing alternate adsorbents, it is fair to express their capacity on a per unit volume basis, since that fixes... 

Surfactant Concentration. The parameter a needs some elaboration. In a very dilute system, a may equal the concentration of the adsorbate (if expressed in the same units), but that is not always true, as discussed in Section 2.2. Even if it is true, it concerns the concentration in the solution, not in the total system. This means that the concentration adsorbed, which equals F times the specific surface area of the adsorbent, has to be subtracted from the total concentration. 

It is found that the concentration on the carbon is proportional to the concentration not adsorbed usually it is proportional to a power or fractional power of the unadsorbed concentration. Expressed mathematically,... 

Before discussing the rate of adsorption as a possible controlling step, we will consider various models for reaction on the surface when the surface is at equilibrium with the gas phase. The concentrations of reactants and products on the surface are given in terms of adsorption isotherms, where the amount adsorbed is expressed as the fraction of surface (or sites) covered, rather than in concentration units such as moles/m. ... 

The second situation of surface diffusion is less well understood. It is usually represented by a Fickian-type flux expression, using the adsorbed concentration as the driving force ... 

In contrast, the stoichiometries and concentrations of different reactive complexes at the mineral surface are rarely known. The rate laws for dissolution are therefore usually expressed in terms of total adsorbate concentrations, and we know that these expressions are approximate. If hydration or hydrolysis by a water molecule of a detaching surface complex at steady state controls the rate of reaction, then rate laws such as those proposed by Furrer and Stumm (7) result. These rate laws are characterized by rates that are proportional to single adsorbate concentrations (see 8). Implicit in these rate laws is the idea that water is present in large and constant concentration and that only a single complex stoichiometry affects the reaction at the surface. 

Using the parameters obtained earlier for the system of propane on activated carbon into eq. (3.2-18g), we get the following expression for the isosteric heat of adsorption as a function of the adsorbed concentration at 283 K... 

This is the flux expression for surface diffusion written in terms of the gradient of adsorbed concentration. Eq. (9.3-6) gives the same diffusion flux of the adsorbed species, but written in terms of the gradient of the fluid species concentration. Although these two equations are mathematically equivalent, eq. (9.3-6) is more efficient from the computational point of view. 

As seen in eq.(9.5-13a) that the surface flux is driven by the gradient of the gas phase partial pressure. This is so because the two phases are in equilibrium with each other. However, if one wishes to express the surface flux in terms of the gradients of adsorbed concentration, we could make use of eq. (9.5-12), and by applying the chain rule of differentiation we get... 

Eq. (9.6-11) involves the time derivative of the average adsorbed concentration. Its form is not convenient for numerical computation, and what we will show is an alternative expression for the heat balance equation. We take the volumetric average of the mass balance equation (9.6-1) ... 

It is desirable, however, that we express the flux equation in terms of the adsorbed concentration instead of the gradient of the unknown partial pressure of a hypothetical gas phase. To do this, we simply apply the chain rule of differentiation to eq.( 10.2-5) and obtain ... 

Written in terms of this hypothetical pressure, eq. (10.5-5) does not have direct application as we do not know the hypothetical pressure directly but rather we have to solve for them from eq. (10.5-4). It is desirable, however, that we express the flux equation in terms of the adsorbed concentration as this is known from the solution of mass balance equations in a crystal. Now that the partial pressure Pi is a function of the adsorbed concentrations of all species (eq. 10.5-4), we apply the chain rule of differentiation to get ... 

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