Equilibrium multicomponent system surfaces

Fowle and Fein (1999) measured the sorption of Cd, Cu, and Pb by B. subtilis and B. licheniformis using the batch technique with single or mixed metals and one or both bacterial species. The sorption parameters estimated from the model were in excellent agreement with those measured experimentally, indicating that chemical equilibrium modeling of aqueous metal sorption by bacterial surfaces could accurately predict the distribution of metals in complex multicomponent systems. Fein and Delea (1999) also tested the applicability of a chemical equilibrium approach to describing aqueous and surface complexation reactions in a Cd-EDTA-Z . subtilis system. The experimental values were consistent with those derived from chemical modeling. 

When all the SE s of a solid with non-hydrostatic (deviatoric) stresses are immobile, no chemical potential of the solid exists, although transport between differently stressed surfaces takes place provided external transport paths are available. Attention should be given to crystals with immobile SE s which contain an (equilibrium) network of mobile dislocations. In these crystals, no bulk diffusion takes place although there may be gradients of the chemical free energy density and, in multicomponent systems, composition gradients (e.g., Cottrell atmospheres [A.H. Cottrell (1953)]). 

In addition to equilibrium between the liquid-phase water in the sample and the vapor phase, the internal moisture equilibrium of the sample is important. If a system is not at internal moisture equilibrium, one might measure a steady vapor pressure (over the period of measurement) that is not the true water activity of the system. An example of this might be a baked good or a multicomponent food. Initially out of the oven, a baked good is not at internal equilibrium the outer surface is at a lower water activity than the center of the baked good. One must wait a period of time in order for the water to migrate and the system to come to internal equilibrium. It is therefore important to remember the restriction of the definition of water activity to equilibrium. 

For multicomponent systems, the expression for y here employed may be shown equivalent to that involved in the cluster diagram technique (6), which is currently being employed in a variety of problems. The present derivation shows that the starting expressions satisfy the thermodynamic consistency relation embodied by the adsorption isotherm. It is, however, important to observe that any direct application of these alternative rigorous approaches, which is of necessity of an approximate nature, leads to some violation of the complete internal equilibrium conditions. Similarly, calculations of surface tension which employ the adsorption equation as a starting point invariably violate mechanical equilibrium in some order of approximation. 

Using the conventional approach, it is difficult to develop a conceptual feeling for the reactions occurring in a metal-multi-ligand system. The conventional approach yields the equilibrium concentration of all species, and in a multicomponent system it is virtually impossible to simultaneously comprehend all of the equilibrium reactions which are occurring. The approach developed here combines all of the reactions of a given stoichiometry into a single stability function which can be simply described in terms of a three-dimensional surface. This provides a simplified and unified visualization of the net equilibrium in a multicomponent mixture. The constant values which these functions adopt under... 

For non-porous catalyst pellets the reactants are chemisorbed on their external surface. However, for porous pellets the main surface area is distributed inside the pores of the catalyst pellets and the reactant molecules diffuse through these pores in order to reach the internal surface of these pellets. This process is usually called intraparticle diffusion of reactant molecules. The molecules are then chemisorbed on the internal surface of the catalyst pellets. The diffusion through the pores is usually described by Fickian diffusion models together with effective diffusivities that include porosity and tortuosity. Tortuosity accounts for the complex porous structure of the pellet. A more rigorous formulation for multicomponent systems is through the use of Stefan-Maxwell equations for multicomponent diffusion. Chemisorption is described through the net rate of adsorption (reaction with active sites) and desorption. Equilibrium adsorption isotherms are usually used to relate the gas phase concentrations to the solid surface concentrations. 

Rates of Diffusion. When the solid and liquid of a multicomponent system are in thermodynamic equilibrium, the composition of the solid wul usually differ from that of the liquid. When the system is submitted to further melting or crystallization, the composition of at least one of the phases will change in the vicinity of the contact surface. Diffusion tends to equalize the concentration differences occurring both in the solid and in the liquid phases and should, therefore, be promoted. 

To calculate thermodynamic equilibrium in multicomponent systems, the so-called optimization method and the non-linear equation method are used, both discussed in [69]. In practice, however, kinetic problems have also to be considered. A heterogeneous process consists of various occurrences such as diffusion of the starting materials to the surface, adsorption of these materials there, chemical reactions at the surface, desorption of the by-products from the surface and their diffusion away. These single occurrences are sequential and the slowest one determines the rate of the whole process. Temperature has to be considered. At lower substrate temperatures surface processes are often rate controlling. According to the Arrhenius equation, the rate is exponentially dependent on temperature ... 

A curious example is that of the distribution of benzene in water benzene will initially spread on water, then as the water becomes saturated with benzene, it will round up into lenses. Virtually all of the thermodynamics of a system will be affected by the presence of the surface. A system containing a surface may be considered as being made up of three parts two bulk phases and the interface separating them. Any extensive thermodynamic property will be apportioned among these parts. For example, in a two-phase multicomponent system, the extra amount of an i component that can be accom-mondated in the system due to the presence of the interface ( ) may be expressed as Qi Qii where is the total number of molecules of i in the whole system, Vj and Vjj are the volumes of phases I and II, respectively, and Q and Qn are the concentrations of i in phases I and II, respectively. The surface (excess) concentration of i is defined as Fj = A, where A is the surface area. At equilibrium, the chemical potential of any component is the same in each bulk phase and at the surface. The Gibbs adsorption equation, which is one of the most widely used expression in surface and colloid science is shown in Eq. (2) ... 

Now, in a multicomponent system, the variation of the chemical potential with space can be expressed in terms of the molar fractions, or concentrations as function of space. Further the velocity of the particles can be expressed in terms of a material flux across an imaginary perpendicular surface to the respective axis. In this way, the equation of diffusion can be derived from thermodynamic arguments. We emphasize that we have now silently crossed over from equilibrium thermodynamics to irreversible thermodynamics. 

Multicomponent systems may also involve the selective adsorption of one component at the SL interface. Since the component that lowers the interfacial tension will be preferentially adsorbed, the rate of the adsorption process can affect the local tension and the contact angle. In many systems, the rate of adsorption at the solid surface is found to be quite slow compared to the rate of movement of the SLV contact line. As a result, the system does not have time for the various interfacial tensions to achieve their equilibrium values. Most surfactants, for example, require several seconds to attain adsorption equihbrium at a LV interface, and longer times at the SL interface. Therefore, if the hquid is flowing across fresh solid surface, or over any surface at a rate faster than the SL adsorption rate, the effective values of olv and osl (and therefore 6) will not be the equilibrium values one might obtain from more static measurements. More will be said about dynamic contact angles in later chapters. 

In the general case of size-composition-dependent surface energy contribution for equilibrium two-phase state, one must solve the above given system of equations with complementary parameters and terms da/dr and da/dC. Furthermore, the rule may be applied to a multicomponent system as well when a new phase is not determined by strong stoichiometric composition that is, there exists the solubility interval on the diagram Gibbs free energy density-concentration (Ag(Q — C). As was mentioned before in the presented case. Equation 13.A.4 is applied to nucleation and separation of nanoparticles in which the composition of the new phase is a function of size. 

The surface energy can also decrease because of the segregation of various components into the interface layer, whether by concentration of admixtures relocated from the main phase, or by adsorption from the surroundings. In accordance with the formulation of the general potentiafO, the following relation can be written for the interface in a multicomponent system at equilibrium S, a dnt - SdT + A dy= < = 0, or, dy = - Fi d/Ui + dT, where T, = and = S/A are important experimentally measurable... 

In this section we have assumed that the adsorption isotherm of an adsorbate is unaffected by the presence of constituents other than the adsorbate in the fluid mixture. If such ideaiir> is assumed for the Langmuir isotherm developed in the previous example, you could use the derived expression for any gaseous system containing carbon tetrachloride and the same activated carbon. In reality, however, the presence of other solutes that have an affinity for the carbon surface alters the CCI4 equilibrium behavior. An accurate system representation would require data or models for the complete multicomponent mixture. 

At multicomponent surface solutions the existence of large solvent clusters Sc affects the mathematical formulation of the equilibrium equations as follows. Suppose that the adsorbed layer is composed of polar solvent molecules, which form clusters Sc bigger than the adsorbate molecules, and N monomeric adsorbate species, Aj, A2,. .., A, which may be N distinct states of the same adsorbate or N different adsorbate molecules. According to the arguments presented in [17] and above, the adsorption process may be described by the following system of equilibrium equations ... 

Several formalisms have been developed leading to what may be called practical thermodynamics. These treatments include the analog of solution thermodynamics, where the adsorbent and the adsorbate are considered as components in a two-phase equilibrium [6]. Another way to study the system is to use the surface excess approach, whereby the properties of the adsorbed phase are determined in terms of the properties of the real two-phase multicomponent... 

In the Gibbs adsorption equation for multicomponent surface layers (2.22) the value of p for soluble components can refer both to the bulk and to the surface layer (as equilibrium exists), and for the insoluble components to the surface layer only. For systems with one insoluble and one soluble component, denoted by subscripts 1 and 2, respectively, and the assumption that the area per mole of the insoluble component 1 is Aj = l/F, Eq. (2.22) can be rewritten as... 

The following is a summary, based on Ref. 1, of the types of molecular interactions that are important in understanding the structure and phase behavior of surfaces and interfaces. Because they are multicomponent, the interactions in systems with surfaces and interfaces are often related to the interactions between molecules in a particular type of medium. This is particularly important for self-assembling systems composed of surfactants or polymers, where the interactions and the subsequent equilibrium structures are strongly influenced by the type of solvent. 

The processes discussed in this chapter demonstrate the great variety of phase equilibrium that can arise beyond the basic vapor-liquid problems discussed in most of the previous chapters. Many other systems could be included The adsorption of gases onto solids (used in the removal of pollutants from air), the distribution of detergents in water/oil systems, the wetting of solid surface by a liquid, the formation of an electrochemical cell when two metals make contact are all examples of multiphase/multicomponent equilibrium. They all share one important common element their equilibrium state is determined by the requirement that the chemical potential of any species must be the same in any phase where the species can be found. These problems are beyond the scope of this book. The important point is this The mathematical development of equilibrium (Chapter 10) is extremely powerful and encompasses any system whose behavior is dominated by equilibrium. 

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