Equilibria, adsorption

When molecules are distributed between the free state and the adsorbed state, the equilibrium situation generally obeys Boltzmann s law 

Ni is the number of molecules of component i Ui is its partial molar energy 

Qj is the degeneracy of the adsorbed or free state (= bulk phase) indicated by the subscripts a and b, respectively 

The degeneracy Qj is defined as the number of states accessible to the molecules of i in the system, at given M . Equation 14.11 can be written as 

At constant temperature and volume fi = p, where p is the chemical potential of /, so that 

Notably, adsorption and desorption steps are usually very fast. In the limit of very large rate constants, V reac,i large the pore concentrations and the solid loadings are in equilibrium, and connected through an isotherm equation  

Equation 6.34 can be derived from Equation 6.31 by setting t/ reac.i equal to zero, which corresponds to equally fast adsorption and desorption steps (a dynamic chemical equilibrium). 

To formulate the accumulation term of the balance equation, an isotherm equation must be used in Equation 6.7 avoiding the need for distinctly different terms as used in Equations 6.5 and 6.6. 

All considered terms of the mass balance of a quite comprehensive model for liquid chromatographic column have now been specified in detail. Table 6.1 summarizes them. 

In the following the relevant models for liquid chromatography are derived in a bottom-up procedure related to Eigure 6.2. To illustrate the difference between these models their specific assumptions are discussed and the level of accuracy and their field of application are pointed out. In all cases the mass balances must be complemented by initial and boundary conditions (Section 6.2.7). Eor the so-called transport-dispersive model a dimensionless representation will also be presented below. 

In batch adsorption a component gets distributed in two phases where one of the phases is solid. The differential affinity of various soluble molecules for specific types of solids is the basis of adsorption. In this process, equilibrium is approached between a solid phase, often called the resin or stationary phase, and the soluble molecules in liquid or gas phase. The liquid or gas phase is then called the mobile phase[6]. 

Since in many cases, electrodeposition in the presence of an additive does not use up the additive (no incorporation of the additive in the deposit), one can conclude that the adsorption equilibrium is a dynamic one. In a dynamic adsorption equilibrium state, the adsorbed molecules continually desorb at a rate equal to the rate at which dissolved molecules from the solution become adsorbed. If the rates of the adsorption and desorption processes are high and of the same order of magnitude as that of the cathodic deposition process, then no incorporation, entrapment, of additives in the deposit will occur. However, if they are much smaller, additive molecules will be entrapped in the deposit via propagating steps 

The description of the partial pressure exerted by a sorbate, or a mixture of sorbates, when they reside on the sorbent surface, at some given temperature is what we speak of as adsorption equihbrium. For a single adsorbate (adsorbing molecular species) we require three state variables to completely describe the equilibrium the temperature, the sorbed phase concentration or loading and the partial pressure exerted by the sorbed phase are very convenient variables to use. As more adsorbable compounds are added to the problem we require additional information to adequately describe the problem. That information is the specification of the mole fractions of the adsorbable compounds in both the gas and sorbed states. 

The reader will find adsorption equilibrium relationships presented in any of three typical forms. The form of equilibrium most frequently presented is the isotherm, the partial pressure as a function of the loading at a given constant sorbent temperature. An isobar implies a chart of the loading as a function of the temperature while the partial pressure of the sorbate is held constant. 

An isostere is a line describing the equilibrium partial pressure, sometimes the dew point, as a function of the temperature of the sorbent. The most common form of isosteric plot is to display the log of the partial pressure as a function of the reciprocal absolute temperature with loading or sorbed phase concentration held fixed. 

Equilibrium relationships are typically mathematical functions that serve to specify the pressure at a given loading and temperature or conversely they may serve to describe the loading state when given a temperature and partial pressure. 

In addition to describing the partial pressure or loading states adsorption equilibrium must also include a description of the heat of adsorption. One may argue that the description of the partial pressure does, through the Clausius-Clapeyron 

In practical operations, maximum capacity of adsorbent cannot be fully utilized because of mass transfer effiects involved in actual fluid-solid contacting processes. In order to estimate practical or dynamic adsorption capacity, however, it is essential, first of all. to have information on adsorption equilibrium. Then kinetic analyses are conducted based on rate processes depending on types of contacting processes. The most typical of the rate steps in solid adsorbents is the intraparticle diffusion which is treated in the next chapter. 

Since adsorption equilibrium is the most fundamental property, a number of studies have been conducted to determine I) the amount of species adsorbed under a given set of conditions (concentration and temperature) or 2) how selective adsorption takes place when two or more adsorbable components coexist. There are many empirical and theoretical approaches. Only several simple relations, however, can be applied in later treatments on kinetic description of adsorption. These relations are sometimes insufficient for predicting adsorption isotherms under a new set of operating conditions. Thus more sophisticated trials on sound thermodynamics or on substantial models have been proposed by many authors. 

A basic review is given here. For more detailed discussions refer to Ross and Olivier (1964) and Ruthven (1984). A recent publication by Myers (1988) also gives adsorption equilibrium data available in the literature. 

When a solid surface is exposed to a gas, the molecules of the gas strike the surface of the solid. Some of the striking molecules stick to the solid surface and become adsorbed while the others rebound. Initially the rate of adsorption is large as the whole surface is bare but as more and more of the surface becomes covered by the molecules of the gas, the available bare surface decreases and so does the rate of adsorption. However, the rate of desorption, which is the rate at which adsorbed molecules rebound from the surface, increases because desorption takes place from the covered surface. As time passes, the rate of adsorption continues to decrease while the rate of desorption increases until an equilibrium is reached between the rate of adsorption and the rate of desorption. At this stage the solid is in adsorption equilibrium with the gas, and the rate of adsorption is equal to the rate of desorption. It is a dynamic equilibrium because the number of molecules sticking to the surface is equal to the number of molecules rebounding from the surface. 

For a given adsorbate-adsorbent system, the equilibrium amount adsorbed x/m is a function of pressure and temperature i.e.. 

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Systems involving an interface are often metastable, that is, essentially in equilibrium in some aspects although in principle evolving slowly to a final state of global equilibrium. The solid-vapor interface is a good example of this. We can have adsorption equilibrium and calculate various thermodynamic quantities for the adsorption process yet the particles of a solid are unstable toward a drift to the final equilibrium condition of a single, perfect crystal. Much of Chapters IX and XVII are thus thermodynamic in content. 

As also noted in the preceding chapter, it is customary to divide adsorption into two broad classes, namely, physical adsorption and chemisorption. Physical adsorption equilibrium is very rapid in attainment (except when limited by mass transport rates in the gas phase or within a porous adsorbent) and is reversible, the adsorbate being removable without change by lowering the pressure (there may be hysteresis in the case of a porous solid). It is supposed that this type of adsorption occurs as a result of the same type of relatively nonspecific intermolecular forces that are responsible for the condensation of a vapor to a liquid, and in physical adsorption the heat of adsorption should be in the range of heats of condensation. Physical adsorption is usually important only for gases below their critical temperature, that is, for vapors. 

If we consider the case of a gas in adsorption equilibrium with a surface, there must be no net free energy change on transporting a small amount from one region to the other. Therefore, since the potential represents the work done by the adsorption forces when adsorbate is brought up to a distance x from the surface, there must be a compensating compressional increase in the free energy of the adsorbate. Thus... 

In conclusion, any observation of slowness in attainment of physical adsorption equilibrium should be analyzed with caution and in detail. When this has been done, the phenomenon has either been found to be due to trivial causes or else some unsuspected and interesting other effects were operative. 

The course of a surface reaction can in principle be followed directly with the use of various surface spectroscopic techniques plus equipment allowing the rapid transfer of the surface from reaction to high-vacuum conditions see Campbell [232]. More often, however, the experimental observables are the changes with time of the concentrations of reactants and products in the gas phase. The rate law in terms of surface concentrations might be called the true rate law and the one analogous to that for a homogeneous system. What is observed, however, is an apparent rate law giving the dependence of the rate on the various gas pressures. The true and the apparent rate laws can be related if one assumes that adsorption equilibrium is rapid compared to the surface reaction. 

Adsorptive Equilibrium The fraction of the surface covered by A at equilibrium is... 

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... 

Table 7-2 summarizes the cases when all substances are in adsorptive equilibrium and the surface reac tion controls. In Table 7-3, substance A is not in adsorptive equilibrium, so its adsorption rate is controUing. 

TABLE 7-2 Surface-reaction Controlling (Adsorptive Equilibrium Maintained of All Participants)... 

Adsorbed-Solution Theoiy The common thennodynamic approach to multicomponent adsorption treats adsorption equilibrium in a way analogous to fluid-fluid equilibrium. The theory has as its basis the Gibbs adsorption isotherm [Young and Crowell, gen. refs.], which is... 

Thus, a plot of the apparent diffusivity versus the linear adsorption equilibrium constant should be linear so long as Dp and D,i remain constant. 

The local equilibrium curve is in approximate agreement with the numerically calculated profiles except at very low concentrations when the isotherm becomes linear and near the peak apex. This occurs because band-spreading, in this case, is dominated by adsorption equilibrium, even if the number of transfer units is not very high. A similar treatment based on local eqnihbrinm for a two-component mixture is given by Golshau-Shirazi and Gniochou [J. Phys. Chem., 93, 4143(1989)]. 

Displacement Development A complete prediction of displacement chromatography accounting for rate factors requires a numerical solution since the adsorption equilibrium is nonlinear and intrinsically competitive. When the column efficiency is high, however, useful predictious can be obtained with the local equilibrium theoiy (see Fixed Bed Transitions ). 

Various Langmiiir-Hinshelwood mechanisms were assumed. GO and GO2 were assumed to adsorb on one kind of active site, si, and H2 and H2O on another kind, s2. The H2 adsorbed with dissociation and all participants were assumed to be in adsorptive equilibrium. Some 48 possible controlling mechanisms were examined, each with 7 empirical constants. Variance analysis of the experimental data reduced the number to three possibilities. The rate equations of the three reactions are stated for the mechanisms finally adopted, with the constants correlated by the Arrhenius equation. 

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. 

Valenzuela, D. and Myers, A.L., "Adsorption Equilibrium Data Handbook", Prentice Hall, (New Jersey) 1989 ISBN 0- 13-003815-3... 

The simplest mode of IGC is the infinite dilution mode , effected when the adsorbing species is present at very low concentration in a non-adsorbing carrier gas. Under such conditions, the adsorption may be assumed to be sub-monolayer, and if one assumes in addition that the surface is energetically homogeneous with respect to the adsorption (often an acceptable assumption for dispersion-force-only adsorbates), the isotherm will be linear (Henry s Law), i.e. the amount adsorbed will be linearly dependent on the partial saturation of the gas. The proportionality factor is the adsorption equilibrium constant, which is the ratio of the volume of gas adsorbed per unit area of solid to its relative saturation in the carrier. The quantity measured experimentally is the relative retention volume, Vn, for a gas sample injected into the column. It is the volume of carrier gas required to completely elute the sample, relative to the amount required to elute a non-adsorbing probe, i.e. 

Valenzuela, D. P. and Myers, A. (1989). Adsorption Equilibrium Data Handbook. Prentice Hall, Englewood Cliffs, NJ. 

The model parameters, in addition to the adsorption equilibrium parameters, are ... 

The chromatographic resolution of bi-naphthol enantiomers was considered for simulation purposes [18]. The chiral stationary phase is 3,5-dinitrobenzoyl phenyl-glycine bonded to silica gel and a mixture of 72 28 (v/v) heptane/isopropanol was used as eluent. The adsorption equilibrium isotherms, measured at 25 °C, are of bi-Langmuir type and were proposed by the Separex group ... 

For nonlinear systems, however, the evaluation of the flow rates is not straightforward. Morbidelli and co-workers developed a complete design of the binary separation by SMB chromatography in the frame of Equilibrium Theory for various adsorption equilibrium isotherms the constant selectivity stoichiometric model [21, 22], the constant selectivity Langmuir adsorption isotherm [23], the variable selectivity modified Langmuir isotherm [24], and the bi-Langmuir isotherm [25]. The region for complete separation was defined in terms of the flow rate ratios in the four sections of the equivalent TMB unit ... 

The separation of bi-naphthol enantiomers can be performed using a Pirkle-type stationary phase, the 3,5-dinitrobenzoyl phenylglycine covalently bonded to silica gel. Eight columns (105 mm length) were packed with particle diameter of 25 0 fiva. The solvent is a 72 28 (v/v) heptane isopropanol mixture. The feed concentration is 2.9 g for each enantiomer. The adsorption equilibrium isotherms were determined by the Separex group and already reported in Equation (28) [33]. 

It was found that [5-7] the rate of flocculation of particles produced by the bridging action of polymer is the slower process and, consequently, the rate-determining step. The primary adsorption of polymer is fairly rapid, but the slow attainment of the adsorption equilibrium under agitation arises at least in part from the breakdown of floes offering new surfaces for adsorption. Thus, the bridging step is slow because a polymer adsorbed on one particle must find another particle having a free surface available to complete the bridge. 

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