Factors Controlling Equilibrium between Phases

The thermodynamics outlined above has the advantage of being universally applicable to all dilute systems and the associated disadvantage of lacking specifics. Equation 2.18, for example, relates the equilibrium concentration ratio of component i between two phases to A/xJ but it fails to specify how A/xf is to be calculated. Thermodynamics is mute concerning actual values of parameters like A/x . Indeed, on closer examination, it is found that A/x cannot be related rigorously by any means to molecular parameters. 

Statistical mechanics, the science that should yield parameters like A/x , is hampered by the multibody complexity of molecular interactions in condensed phases and by the failure of quantum mechanics to provide accurate interaction potentials between molecules. Because pure theory is impractical, progress in understanding and describing molecular equilibrium between phases requires a combination of careful experimental measurements and correlations by means of empirical equations and approximate theories. The most comprehensive approximate theory available for describing the distribution of solute between phases—including liquids, gases, supercritical fluids, surfaces, and bonded surface phases—is based on a lattice model developed by Martire and co-workers [12, 13]. 

The first step in understanding the nature of A/x is the recognition that it is composed of two unlike terms an enthalpy term and an entropy term. This follows if we recognize that /x is the gain in the Gibbs free energy per mole of added i under standard conditions, and furthermore that each G or /x, value is composed of both enthalpy (H) and entropy (TS) terms as shown in Eq. 2.4. Since A/x is the difference in two /x values, one for each of two phases, A/x can be expressed by 

For most separation systems involving a partitioning of components 

Intermolecular attractive forces fall into several categories. In each, the magnitude of the forces depends upon the properties of the two molecules (e.g., a solute and a solvent molecule) subject to attraction [14,15]. 

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Separation processes rely on various mechanisms, implemented via a unit operation, to perform the separation. The mechanism is chosen to exploit some property difference between the components. They fall into two basic categories the partitioning of the feed stream between phases and the relative motion of various chemical species within a single phase. These two categories are often referred to as equilibrium and mass transfer rate processes, respectively. Separation processes can often be analyzed with either equilibrium or mass transfer models. However, one of these two mechanisms will be the limiting, or controlling, factor in the separation and is, therefore, the design mechanism. 

The distribution constant, K, has just been discussed as the controlling factor in the partitioning equilibrium between a solute and the stationary phase. It is defined as the concentration of the solute A in the stationary phase divided by its concentration in the mobile phase. 

In this chapter, the discussion of these topics is divided into three broad categories. The first briefly considers the starting materials and how they might influence the outcome. The second topic concerns phase equilibria and the methods used to establish the situation at equilibrium. The final section discusses how the starting materials transform into the equilibrium products that is, what factors control the rates and mechanisms during the reaction and what methods and techniques are used to follow the course of the overall reaction process Clearly, there is considerable overlap between the last two topics, and the experimental techniques in common are covered in the section on equilibrium. 

Contaminant volatilization from subsurface solid and aqueous phases may lead, on the one hand, to pollution of the atmosphere and, on the other hand, to contamination (by vapor transport) of the vadose zone and groundwater. Potential volatihty of a contaminant is related to its inherent vapor pressure, but actual vaporization rates depend on the environmental conditions and other factors that control behavior of chemicals at the solid-gas-water interface. For surface deposits, the actual rate of loss, or the pro-portionahty constant relating vapor pressure to volatilization rates, depends on external conditions (such as turbulence, surface roughness, and wind speed) that affect movement away from the evaporating surface. Close to the evaporating surface, there is relatively little movement of air and the vaporized substance is transported from the surface through the stagnant air layer only by molecular diffusion. The rate of contaminant volatilization from the subsurface is a function of the equilibrium distribution between the gas, water, and solid phases, as related to vapor pressure solubility and adsorption, as well as of the rate of contaminant movement to the soil surface. 

When the amount of the sample is comparable to the adsorption capacity of the zone of the column the migrating molecules occupy, the analyte molecules compete for adsorption on the surface of the stationary phase. The molecules disturb the adsorption of other molecules, and that phenomenon is normally taken into account by nonlinear adsorption isotherms. The nonlinear adsorption isotherm arises from the fact that the equilibrium concentrations of the solute molecules in the stationary and the mobile phases are not directly proportional. The stationary phase has a finite adsorption capacity lateral interactions may arise between molecules in the adsorbed layer, and those lead to nonlinear isotherms. If we work in the concentration range where the isotherms are nonlinear, we arrive to the field of nonlinear chromatography where thermodynamics controls the peak shapes. The retention time, selectivity, plate number, peak width, and peak shape are no longer constant but depend on the sample size and several other factors. 

Because of interference from the radioactive decay of other nuclides (which are typically formed with much higher yields), extraction systems with relatively high decontamination factors from actinides, Bi, and Po must be chosen, and the transactinide activity can only be measured in the selectively extracting organic phase. For this reason, measurement of distribution coefficients is somewhat difficult. By comparing the Rf or Db detection rate under a certain set of chemical conditions to the rate observed under chemical control conditions known to give near 100% yield, distribution coefficients between about 0.2 and 5 can be determined. If the control experiments are performed nearly concurrently, many systematic errors, such as gas-jet efficiency and experimenter technique, are cancelled out. Additionally, extraction systems which come to equilibrium in the 5-10 second phase contact time must be chosen. 

The simplest case to consider is the situation that describes the contact between an n-type semiconductor and a metal. As discussed above, the factor that controls the charge transfer process is the electrochemical potential, or Fermi level, E-g, of each phase. The initial difference in the electrochemical potentials of the two phases indicates that, after contact, charge must flow between the phases in order to reach equilibrium. We will consider the case where the electrochemical potential of an isolated n-type semiconductor (Af.sc) is more negative than the electrochemical potential of an isolated metal phase... 

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