Adsorption equilibrium expression

Many organic compounds and noxious gases, such as H2S, follow Langmuir adsorption behavior so that Eq. (37) can be converted into the familiar Langmuir-Hinshelwood form by substituting the adsorption equilibrium expression for... 

Chemical Equihbrium When A is not in adsorptive equilibrium, it is assumed to be in chemical equilibrium, with.p =p, JK py. This expression is substituted for p wherever it appears in the rate equation. Then... 

Adsorption is a dynamic process in which some adsorbate molecules are transferring from the fluid phase onto the solid surface, while others are releasing from the surface back into the fluid. When the rate of these two processes becomes equal, adsorption equilibrium has been established. The equilibrium relationship between a speeific adsorbate and adsorbent is usually defined in terms of an adsorption isotherm, which expresses the amount of adsorbate adsorbed as a fimetion of the gas phase coneentration, at a eonstant temperature. 

Another problem which can appear in the search for the minimum is intercorrelation of some model parameters. For example, such a correlation usually exists between the frequency factor (pre-exponential factor) and the activation energy (argument in the exponent) in the Arrhenius equation or between rate constant (appears in the numerator) and adsorption equilibrium constants (appear in the denominator) in Langmuir-Hinshelwood kinetic expressions. 

For different acceptor particle adsorption isotherms expressions (1.85) - (1.89) provide various dependencies of equilibrium values of <7s for a partial pressure P (ranging from power indexes up to exponential). Thus, in case when the logarithmic isotherm Nt InP is valid the expression (1.85 ) leads to dependence <75 P" often observed in experiments [20, 83, 155]. In case of the Freundlich isotherm we arrive to the same type of dependence of - P" observed in the limit case described by expression (1.87). 

Category I. Adsorption equilibrium is maintained at all times, and the overall rate of reaction is governed by the rate of chemical reaction on the surface. The expressions developed for 0 in Section 6.2 can be used in this case. 

Hougen- Watson Models for Cases where Adsorption and Desorption Processes are the Rate Limiting Steps. When surface reaction processes are very rapid, the overall conversion rate may be limited by the rate at which adsorption of reactants or desorption of products takes place. Usually only one of the many species in a reaction mixture will not be in adsorptive equilibrium. This generalization will be taken as a basis for developing the expressions for overall conversion rates that apply when adsorption or desorption processes are rate limiting. In this treatment we will assume that chemical reaction equilibrium exists between various adsorbed species on the catalyst surface, even though reaction equilibrium will not prevail in the fluid phase. 

We should also point out that the adsorption equilibrium constants appearing in the Hougen-Watson models cannot be determined from adsorption equilibrium constants obtained from nonreacting systems if one expects the mathematical expression to yield accurate predictions of the reaction rate. One explanation of this fact is that probably only a small fraction of the catalyst sites are effective in promoting the reaction. 

The applicability of Eqs. (4.27) - (4.30) is somewhat restricted because the bulk concentration is assumed to be constant because either its depletion is negligible (adsorbed quantity quantity present in the system) or because it is kept constant by a steady state mechanism. Analytical expressions of F as a function of time for situations where the approach to adsorption equilibrium is accompanied by a corresponding adjustment of c are available only for a few relatively simple cases. 

Because the inverse Debye length is calculated from the ionic surfactant concentration of the continuous phase, the only unknown parameter is the surface potential i/io this can be obtained from a fit of these expressions to the experimental data. The theoretical values of FeQx) are shown by the continuous curves in Eig. 2.5, for the three surfactant concentrations. The agreement between theory and experiment is spectacular, and as expected, the surface potential increases with the bulk surfactant concentration as a result of the adsorption equilibrium. Consequently, a higher surfactant concentration induces a larger repulsion, but is also characterized by a shorter range due to the decrease of the Debye screening length. 

Adsorption data are frequently presented as a plot of the amount of adsorbate taken up per unit weight or area of the adsorbent vs the equilibrium concentration remaining in the gaseous or solution phase (adsorption isotherm) pH, temperature and electrolyte concentration are held constant. Depending upon the purpose of the investigation, the extent of adsorption is expressed either as amount of adsorbate vs. surface area of adsorbent, as fraction adsorbed, or, in some cases, as a distribution coefficient, K. ... 

Based on the Langmuir-Hinshelwood expression derived for a unimolecular reaction system (6) Rate =k Ks (substrate) /[I + Ks (substrate)], Table 3 shows boththe apparent kinetic rate and the substrate concentration were used to fit against the model. Results show that the initial rate is zero-order in substrate and first order in hydrogen concentration. In the case of the Schiff s base hydrogenation, limited aldehyde adsorption on the surface was assumed in this analysis. Table 3 shows a comparison of the adsorption equilibrium and the rate constant used for evaluating the catalytic surface. 

The absolute rate, i.e. the amount adsorbed per unit time, is a function of the diffusion coefficient, i.e. the relative rate expressed by U(t), as well as of equilibrium expressed by the available or active sites for adsorption or ion exchange. Since for t2 > /,... 

EXAMPLE 9.4 Kinetic-Theory-Based Description of Binary Adsorption. Assume that two gases A and B individually follow the Langmuir isotherm in their adsorption on a particular solid. Use the logic that results in Equation (46) to derive an expression for the fraction of surface sites covered by one of the species when a mixture of the two gases is allowed to come to adsorption equilibrium with that solid. 

Ihe simplest expression for adsorption equilibrium, for an adsorbate A, is given... 

An attempt has been made to summarize the available literature for comparison of adsorption constants and forms of the equations used. Table XV presents a number of parameters reported by different authors for several model compounds on CoMo/A1203 in the temperature range 235-350°C (5,33,104,122,123,125-127). The data presented include the adsorption equilibrium constants at the temperatures employed in the studies and the exponential term (n) of the denominator function of the 0 parameter that was used in the calculation. The numbers shown in parentheses, relating to the value of n, indicate that the hydrogen adsorption term (Xh[H2]) is expressed as the square root of this product in the denominator. Data are presented for both the direct sulfur extraction site (cr) and the hydrogenation site (t). 

Equation 11.81 is thus an expression for the Langmuir adsorption equilibrium constant in terms of the surface and gas molecular partition functions, qs and qg, respectively. 

The establishment of the adsorption equilibrium is rapid in comparison with any disturbance of it due to the removal of the molecules in chemical change. Equating therefore, the rate of evaporation and the rate of condensation we find an expression for a. 

If an adsorbed chemical group (anchor) is more strongly bound to the surface than a solvent molecule would be at that site, an equilibrium expression may be written for the displacement of solvent by adsorbate. Adsorption is particulady strong if the chemical nature of the adsorbed group is similar to that of the particle surface for example, in aqueous systems perfluoroalkane groups adsorb well on polytetrafluoroethene particles and aromatic... 

To illustrate the analogy more clearly, it is necessary to consider the derivation of the Langmuir adsorption isotherm. We can incorporate the above assumptions into an equilibrium expression which equates the rate of adsorption racis to that of desorption rdes of gas molecules of type J. The desorption rate is directly proportional to the fraction of monolayer sites occupied /, and is expressed as... 

The condition of the general adsorption equilibrium, which does not assume the existence of high affinity binding sites, is again described by the equality of the chemical potential of the species in the sample phase jix and at the surface jisx-The general adsorption equilibrium has the form of (1.1) and the equilibrium constant (K) can be expressed in a similar manner as above ... 

The Langmuir adsorption isotherm is based on the characteristic assumptions that (a) only monomolecular adsorption takes place, (b) adsorption is localised and (c) the heat of adsorption is independent of surface coverage. A kinetic derivation follows in which the velocities of adsorption and desorption are equated with each other to give an expression representing adsorption equilibrium. 

At equilibrium the net rate is zero, and we can define an adsorption equilibrium constant (Ka) to produce the following expressions that define what is typically called Langmuir isotherm behavior ... 

Figure 1.3 shows a plot of 0 versus partial pressure for various values of the adsorption equilibrium constant. These show that as the equilibrium constant increases for a given pressure, we increase the surface fraction covered, up to a value of 1. As the pressure increases, we increase the fraction of the surface covered with A. But we have only a finite amount of catalyst surface area, which means that we will eventually reach a point where increasing the partial pressure of A will have little effect on the amount that can be adsorbed and hence on the rate of any reaction taking place. This is a kind of behavior fundamentally different from that of simple power-law kinetics, where increasing the reactant concentration always leads to an increase in reaction rate proportional to the order in the kinetic expression. 

It should be understood that this rate expression may in fact represent a set of diffusion and mass transfer equations with their associated boundary conditions, rather than a simple explicit expression. In addition one may write a differential heat balance for a column element, which has the same general form as Eq. (17), and a heat balance for heat transfer between particle and fluid. In a nonisothermal system the heat and mass balance equations are therefore coupled through the temperature dependence of the rate of adsorption and the adsorption equilibrium, as expressed in Eq. (18). 

The adsorption equilibrium constant Kou was calculated in term of Ku using the expression... 

Changes with time in the isotopic exchange reaction rate between H2 and HDO(v) over a hydrophobic Pt-catalyst induced by the addition of HN03 were studied experimentally The HN03 poisoning was found to be reversible and was well explained in terms of the competitive adsorption of HNO, with H2 or HDO onto the catalytic active sites. The adsorption equilibrium for HN03 could be expressed by the Frumkin-Temkin equation and the time evolution of the activity was well expressed by the Zeldovich rate equation. 

When a substance moves away from a solution and accumulates at the surface of a solid, the concentration of the solute remaining in solution is in dynamic equilibrium with the accumulated concentration at the surface. This distribution ratio of the solute in solution and at the surface is a measure of the adsorption equilibrium. A mathematical expression that relates the quantity of the solute adsorbed in a solid surface per unit weight of solid adsorbent to a function of the solute concentration remaining in solution at a fixed temperature is known as the adsorption isotherm. 

For practical heterogeneous catalyst kinetics this principle has the following consequence. Usually, the assumption of a homogeneous surface is not valid. It would be more realistic to assume the existence of a certain distribution in the activity of the sites. From the above, certain sites will, however, contribute most to the reaction, since these sites activate the reactants most optimally. This might result in an apparently uniform reaction behaviour, and can explain why Langmuir adsorption often provides a good basis for the reaction rate description. This also implies that adsorption equilibrium constants determined from adsorption experiments can only be used in kinetic expressions when coverage dependence is explicitly included otherwise they have to be extracted from the rate data. 

If adsorption equilibrium is established rapidly and the adsorbed and bulk species remain in equilibrium throughout the reaction, cAads can be expressed in terms of a suitable isotherm. This allows the differential kinetic equation to be integrated. For example, if Henry s law adsorption is presumed to apply [43]... 

Note Since the model is linear for the special case considered, the same equation is also satisfied by the other three variables.) The following observations may be made from Eq. (98) that expresses the dimensionless dispersion coefficient A (i) The first term describes dispersion effects due to velocity gradients when adsorption equilibrium exists at the interface. We note that this expression was first derived by Golay (1958) for capillary chromatography with a retentive layer, (ii) The second term corresponds to dispersion effects due to finite rate of adsorption (since this term vanishes if we assume that adsorption and desorption are very fast so that equilibrium exists at the interface), (iii) The effective dispersion coefficient reduces to the Taylor limit when the adsorption rate constant or the adsorption capacity is zero, (iv) As is well known (Rhee et al., 1986), the effective solute velocity is reduced by a factor (1 + y). (v) For the case of irreversible adsorption (y — oo and Da —> oo), the dispersion coefficient is equal to 11 times the Taylor value. It is also equal to the reciprocal of the asymptotic Sherwood number for mass transfer in a circular... 

Adsorption equilibrium which we have assumed can be described in two ways u = f(v) and v = g(u). The advantages and limitations of these two expressions were discussed earlier (J2). In this work we shall assume that the function f(v) is monotonic and use primarily the function v = g(u). 

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