Equilibrium conditions, adsorption

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. 

Picture the transfer, under equilibrium conditions, of dn mole of adsorptive from the bulk liquid where its chemical potential is /i , to a... 

The standard methods of drying can be classified as deposition of the moisture as either water or ice decomposition of the water chemical precipitation absorption adsorption mechanical separation and vaporization. The completeness with which dryness can be accomplished by any process depends upon the factors controlling the equilibrium conditions achieved in the operation. A brief discussion of each method is first given. 

Adsorption for gas purification comes under the category of dynamic adsorption. Where a high separation efficiency is required, the adsorption would be stopped when the breakthrough point is reached. The relationship between adsorbate concentration in the gas stream and the solid may be determined experimentally and plotted in the form of isotherms. These are usually determined under static equilibrium conditions but dynamic adsorption conditions operating in gas purification bear little relationship to these results. Isotherms indicate the affinity of the adsorbent for the adsorbate but do not relate the contact time or the amount of adsorbent required to reduce the adsorbate from one concentration to another. Factors which influence the service time of an adsorbent bed include the grain size of the adsorbent depth of adsorbent bed gas velocity temperature of gas and adsorbent pressure of the gas stream concentration of the adsorbates concentration of other gas constituents which may be adsorbed at the same time moisture content of the gas and adsorbent concentration of substances which may polymerize or react with the adsorbent adsorptive capacity of the adsorbent for the adsorbate over the concentration range applicable over the filter or carbon bed efficiency of adsorbate removal required. 

To solve for ka and kd explicitly, we need one more equation. We can get this from the consideration of an equilibrium condition. When the concentration on the surface of the adsorbent is no longer changing, then rates of adsorption and desorption are equal. From thiswefind ... 

Eqs. (1,4,5) show that to determine the equilibrium properties of an adsorbate and also the adsorption-desorption and dissociation kinetics under quasi-equilibrium conditions we need to calculate the chemical potential as a function of coverage and temperature. We illustrate this by considering a single-component adsorbate. The case of dissociative equilibrium with both atoms and molecules present on the surface has recently been given elsewhere [11]. 

The present Section, which provides an outline of selected relevant topics in electrochemistry, is intended primarily as an introduction to aqueous corrosion for those readers whose basic training has not involved a study of electrochemistry. The scope of electrochemistry is enormous and cannot be treated adequately here, but there are now a number of excellent books on the subject, and it is hoped that this outline will serve to stimulate further study. The topics selected are as follows a) the nature of the electrified interface between the metal and the solution, (b) adsorption, (c) transfer of charge across the interface under equilibrium and non-equilibrium conditions, d) overpotential and the rate of an electrode reaction and (e) the hydrogen evolution reaction and hydrogen absorption by ferrous alloys. For reasons of space a number of important topics, such as the electrochemistry of electrolyte solutions, have been omitted. 

In actual operation, the adsorption bed is not at equilibrium conditions. Also, there is loss of bed capacity due to ... 

Suspension Model of Interaction of Asphaltene and Oil This model is based upon the concept that asphaltenes exist as particles suspended in oil. Their suspension is assisted by resins (heavy and mostly aromatic molecules) adsorbed to the surface of asphaltenes and keeping them afloat because of the repulsive forces between resin molecules in the solution and the adsorbed resins on the asphaltene surface (see Figure 4). Stability of such a suspension is considered to be a function of the concentration of resins in solution, the fraction of asphaltene surface sites occupied by resin molecules, and the equilibrium conditions between the resins in solution and on the asphaltene surface. Utilization of this model requires the following (12) 1. Resin chemical potential calculation based on the statistical mechanical theory of polymer solutions. 2. Studies regarding resin adsorption on asphaltene particle surface and... 

For h < 26, the situation is much more complex. One not only needs to know 4>(z) for each layer, but how 4>(z) changes as the two particles approach, i.e. 4>(z,h) this may well depend on the time-scale of the approach, i.e. the equilibrium path may not be followed. Scheutjens and Fleer (25) in an extension of their model for polymer adsorption have analysed the situation for two interacting uncharged parallel, flat plates carrying adsorbed, neutral homopolymer, interacting under equilibrium conditions. Only a semi-quantitative picture will be presented here. 

Equilibrium conditions for the adsorption of second, third, fourth, etc., segments from molecules already attached to the surface can be written down in similar fashion and lead to the following expression for the l step ... 

The partial pressure of H2S on a volumetric basis in the atmosphere in equilibrium with a water phase of sulfide (H2S + HS ) is at a pH of 7, approximately equal to 100 ppm (gS m-3)-1 (Figure 4.2). It is clear that under equilibrium conditions, much lower concentrations than those corresponding to the values shown in Table 4.6 may result in odor and human health problems. This is also seen from the fact that Henry s constant for H2S is rather high, //H2S =563 atm (mole fraction)-1 at 25°C (Table 4.1). However, under real conditions in sewer networks, conditions close to equilibrium rarely exist because of, for example, ventilation and adsorption followed by oxidation on the sewer walls. Typically, the gas concentration found in the sewer atmosphere ranges from 2-20% and is normally found to be less than 10% of the theoretical equilibrium value (Melbourne and Metropolitan Board of Works, 1989). 

According to this model, the total rate of adsorbed molecules (dn/dt) is given by the algebric sum of the two processes occurring at the interaction sites adsorption and desorption. The equilibrium condition is reached once one equalizes the other. 

At equilibrium the rate of all elementary reaction steps in the forward and reverse directions are equal therefore, this condition provides a check point for studying reaction dynamics. Any postulated mechanism must both satisfy rate data and the overall equilibrium condition. Additionally, for the case of reactions occurring at charged interfaces, the appropriate model of the interface must be selected. A variety of surface complexation models have been used to successfully predict adsorption characteristics when certain assumptions are made and model input parameters selected to give the best model fit (12). One impetus for this work was to establish a self-consistent set of equilibrium and kinetic data in support of a given modeling approach. 

It may prove possible to apply titration calorimetry data in one further direction. If AG can be estimated for SAL-goethite complexation and reaction enthalpies can be obtained under equilibrium conditions, then an entropy change for this reaction can also be derived. This can only be done, however, if the adsorption reaction can be shown to be reversible. Since this has not been proven as yet in our systems, such thermodynamic extensions of titration calorimetry can only be speculative at this time. 

In 1938, Brunauer, Emmett and Teller(12) and Emmett and de Witt(13) developed what is now known as the BET theory. As in the case in Langmuir s isotherm, the theory is based on the concept of an adsorbed molecule which is not free to move over the surface, and which exerts no lateral forces on adjacent molecules of adsorbate. The BET theory does, however, allow different numbers of adsorbed layers to build up on different parts of the surface, although it assumes that the net amount of surface which is empty or which is associated with a monolayer, bilayer and so on is constant for any particular equilibrium condition. Monolayers are created by adsorption on to empty surface and by desorption from bilayers. Monolayers are lost both through desorption and through the adsorption of additional layers. The rate of adsorption is proportional to the frequency with which molecules strike the surface and the area of that surface. From the kinetic theory of gases, the frequency is proportional to the pressure of the molecules and hence ... 

In analogy with the previous discussion, the adsorption isotherms for both, the pairing ion and the counter ion are obtained from the equilibrium condition for the corresponding electrochemical potentials according to the scheme ... 

However, it is also the case that slopes different from mr 1 can occur even under equilibrium conditions, due to the nature of the molecular interactions involved in adsorption on a surface. For example, Goss and Schwarzenbach (1998) propose a modified formulation of Eq. (VV) that more explicitly takes into account the interactions between the SOC and the surface on a molecular level. Thus, based on work by Goss (1997), they express the relationship between a gas-particle partition coefficient K fds, defined as the adsorbed con-... 

McBain accounted for hysteresis by assuming that the pores contained a narrow opening and a wide body, the so-called bottle-neck shape. His model asserts that during adsorption the wide inner portion of the pore is filled at high relative pressures but cannot empty until the narrow neck of the pore first empties at lower relative pressures during desorption. Therefore, for bottle-neck pores the adsorption isotherm corresponds to the equilibrium condition. However, the model proposed by McBain ignores the question of how condensation into the wider inner portion of the pore can occur once the narrow neck has been filled at low relative pressures. 

The activation energy for desorption comprises the heat of adsorption and the activation energy of adsorption, (see Fig. 1), but, as the adsorption of alkali metals and most gases on clean metal surfaces is non-activated, the activation energy of desorption is, in fact, equal to that of adsorption. Two classes of measurements have been made (1) those in which desorption occurred without subsequent readsorption, and (2) those where equilibrium conditions were approached during the desorption process. A true desorption velocity is observed in the first case only. 

Some components in a gas or liquid interact with sites, termed adsorption sites, on a solid surface by virtue of van der Waals forces, electrostatic interactions, or chemical binding forces. The interaction may be selective to specific components in the fluids, depending on the characteristics of both the solid and the components, and thus the specific components are concentrated on the solid surface. It is assumed that adsorbates are reversibly adsorbed at adsorption sites with homogeneous adsorption energy, and that adsorption is under equilibrium at the fluid- adsorbent interface. Let (m" ) be the number of adsorption sites and (m 2) the number of molecules of A adsorbed at equilibrium, both per unit surface area of the adsorbent. Then, the rate of adsorption r (kmol m s ) should be proportional to the concentration of adsorbate A in the fluid phase and the number of unoccupied adsorption sites. Moreover, the rate of desorption should be proportional to the number of occupied sites per unit surface area. Here, we need not consider the effects of mass transfer, as we are discussing equilibrium conditions at the interface. At equilibrium, these two rates should balance. Thus,... 

Carbon dioxide isotherms at —78°C. are also shown in Figure 3. The adsorption of CO2 was relatively slow however, adsorption and desorption points taken at 24-hour intervals seem to represent equilibrium conditions. The pressure scale for CO2 isotherms is given as observed pressure divided by an extrapolated liquid vapor pressure at —78°C. (A plot of the logarithm of vapor pressure of liquid CO2 as a function of reciprocal absolute temperature is linear, and this straight line was the basis of extrapolation. At —78°C. the... 

Reversibility. Apparent irreversibility phenomena of ion exchange in NaX were studied with zinc and cobalt ions using a temperature-variation method described in the experimental section. In view of the high selectivity of NaX for bivalent cations at low zeolite loading, the concentration of bivalent ions in the equilibrium solution is quite sensitive to small changes in the surface composition. In fact, the adsorption removal of bivalent cations at low loading, below 0.2, is quantitative or nearly so (99.5% or better). Consequently the value of the equilibrium concentration is an ideal criterion for assessing either reversibility or equilibrium conditions. 

In these cases, the standard free energy of adsorption can be obtained from the equilibrium condition and is a simple exponential function of the potential which does not depend significantly on the charge distribution at the interface for an uncharged adsorbate. The chemisorption thus corresponds to a vertical shift in the free energy curves as depicted in Fig. 12 and affects the energy of activation [76]. 

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