Equilibria partitioning

Partitioning coefficients describe the phase equilibrium of one solute between the two phases. At equilibrium, there is no net transfer. When there is a net transfer, it is proportional to the difference from equilibrium. Equilibrium is therefore always important, because it is the result of transport it is where the chemical concentrations are going. Example 8.1 will show us that even a common term, such as relative humidity, can be related to Henry s law. 

On a humid day in August, the radio announcer says that the relative humidity is 70%. Relate this to Henry s law constant for water. At the same temperature, why does 70% relative humidity feel so much warmer than 30% relative humidity  

Because the Henry s law constant for water is on the order of 10 Ab.HjO -Hh20-Ag,h20, and equation (E8.1.1) becomes 

Relative humidity RH) is related to Henry s law constant through the relationship implied in equation (E8.1.3), or 

A relative humidity of 70% means that we are at 70% of air-water equilibrium or 

Diffusive transport between phases can be described mathematically as the product of the departure from equilibrium and a kinetic term  

When a small amount of a chemical is added to two immiscible phases and then shaken, the phases will eventually separate and the chemical will partition between the two phases according to its solubility in each phase. The concentration ratio at equilibrium is the partition coefficient  

In the laboratory, we usually determine Kl2 from the slope of C versus C2 over a range of concentrations. Partition coefficients can be measured for essentially any two-phase system air-water, octanol-water, lipid-water, particle-water, and so on. In situ partition coefficients also can be measured where site-specific environmental conditions might influence the equilibrium phase distribution. 

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Of particular interest has been the study of the polymer configurations at the solid-liquid interface. Beginning with lattice theories, early models of polymer adsorption captured most of the features of adsorption such as the loop, train, and tail structures and the influence of the surface interaction parameter (see Refs. 57, 58, 62 for reviews of older theories). These lattice models have been expanded on in recent years using modem computational methods [63,64] and have allowed the calculation of equilibrium partitioning between a poly-... 

Among all possible partitions in the above expression, the equilibrium partition eorresponds to the most probable partition, for whieh dP = 0. Evaluating tiiis differential yields the following relation ... 

The separation of components by liquid-liquid extraction depends primarily on the thermodynamic equilibrium partition of those components between the two liquid phases. Knowledge of these partition relationships is essential for selecting the ratio or extraction solvent to feed that enters an extraction process and for evaluating the mass-transfer rates or theoretical stage efficiencies achieved in process equipment. Since two liquid phases that are immiscible are used, the thermodynamic equilibrium involves considerable evaluation of nonideal solutions. In the simplest case a feed solvent F contains a solute that is to be transferred into an extraction solvent S. 

Gmehhng and Onken (Vapor-Liquid Equilibrium Data Collection, DECHEMA, Frankfurt, Germany, 1979) have reported a large collection of vapor-liqnid equilibrium data along with correlations of the resulting activity coefficients. This can be used to predict liqnid-hqnid equilibrium partition ratios as shown in Example 1. 

Condition (273) is the requirement that at the center of the bubble the concentrations and the temperature must be finite, and condition (274) follows from the condition that the net average flux is zero on the surface r = b which encloses each bubble. Condition (275) refers to the interfacial concentrations and the temperature on both phases, which are related through known equilibrium partition coefficients mf. Hence... 

A production process has evolved from this original work, and is presently used for extracting americium from kilogram amounts of plutonium metal. This process is based upon equilibrium partitioning (by oxidation-reduction reactions) of americium and plutonium between the molten chloride salt and the molten plutonium phase. The chemistry of this process is indicated by the following reactions ... 

After phase separation, two sets of equations such as those in Table A-1 describe the polymerization but now the interphase transport terms I, must be included which couples the two sets of equations. We assume that an equilibrium partitioning of the monomers is always maintained. Under these conditions, it is possible, following some work of Kilkson (17) on a simpler interfacial nylon polymerization, to express the transfer rates I in terms of the monomer partition coefficients, and the iJolume fraction X. We assume that no interphase transport of any polymer occurs. Thus, from this coupled set of eighteen equations, we can compute the overall conversions in each phase vs. time. We can then go back to the statistical derived equations in Table 1 and predict the average values of the distribution. The overall average values are the sums of those in each phase. 

Concentrations in Water and Particles. In order to obtain the rates of reaction, the concentrations of the two monomers and the chain transfer agent in the water and polymer phases were calculated using equilibrium partition coefficients (H). ... 

CASE II - An Easily VectorIzable Problem Equilibrium Partitioning of Freely Jointed Chains A Monte Carlo Simulation... 

Johansson and coworkers [182-184] have analyzed polyacrylamide gel structure via several different approaches. They developed an analytical model of the gel structure using a single cylindrical unit cell coupled with a distribution of unit cells. They considered the distribution of unit cells to be of several types, including (1) Ogston distribution, (2) Gaussian distribution of chains, and (3) a fractal network of pores [182-184]. They [183] used the equilibrium partition coefficient... 

Using the formalism of statistical mechanics, Giddings et al. [135] investigated the effects of molecular shape and pore shape on the equilibrium distribution of solutes in pores. The equilibrium partition coefficient is defined as the ratio of the partition function in the pore... 

When a two- or higher-phase system is used with two or more phases permeable to the solute of interest and when interactions between the phases is possible, it would be necessary to apply the principle of local mass equilibrium [427] in order to derive a single effective diffusion coefficient that will be used in a one-equation model for the transport. Extensive justification of the principle of local thermdl equilibrium has been presented by Whitaker [425,432]. If the transport is in series rather than in parallel, assuming local equilibrium with equilibrium partition coefficients equal to unity, the effective diffusion coefficient is... 

According to Eq. (32.10), the distribution potential corresponding to the equilibrium partition of the electrolyte RX is independent of the electrolyte concentration. On the other hand, when more than two ions are involved in the partition equilibrium, there always exists a thermodynamic relationship between the potential difference and the concentrations of ions present. More specifically, let us consider an ITIES with a different electrolyte in each phase. 

Solute Flux Solute partitioning between the upstream polarization layer and the solvent-filled membrane pores can be modeled by considering a spherical solute and a cylindrical pore. The equilibrium partition coefficient 0 (pore/bulk concentration ratio) for steric exclusion (no long-range ionic or other interactions) can be written as... 

The equilibrium partition of ions present in the system gives rise to the equilibrium Galvani potential difference A (p = y (w) — (o) between the phases w and o (Nernst potential) [7,8]... 

During equilibrium crystal growth from a melt a U-series parent and daughter will be incorporated according to their equilibrium partition coefficients, Dp and D respectively ... 

One possibility for increasing the minimum porosity needed to generate disequilibria involves control of element extraction by solid-state diffusion (diffusion control models). If solid diffusion slows the rate that an incompatible element is transported to the melt-mineral interface, then the element will behave as if it has a higher partition coefficient than its equilibrium partition coefficient. This in turn would allow higher melt porosities to achieve the same amount of disequilibria as in pure equilibrium models. Iwamori (1992, 1993) presented a model of this process applicable to all elements that suggested that diffusion control would be important for all elements having diffusivities less than... 

Quigley MS, Honeyman BD, Santschi PH (1996) Thorium sorption in the marine enviromnent equilibrium partitioning at the Hematite/water interface, sorption/desorption kinetics and particle tracing. Aquat Geochem 1 277-301... 

Here Jta(x) denotes the a-th component of the stationary vector x of the Markov chain with transition matrix Q whose elements depend on the monomer mixture composition in microreactor x according to formula (8). To have the set of Eq. (24) closed it is necessary to determine the dependence of x on X in the thermodynamic equilibrium, i.e. to solve the problem of equilibrium partitioning of monomers between microreactors and their environment. This thermodynamic problem has been solved within the framework of the mean-field Flory approximation [48] for copolymerization of any number of monomers and solvents. The dependencies xa=Fa(X)(a=l,...,m) found there in combination with Eqs. (24) constitute a closed set of dynamic equations whose solution permits the determination of the evolution of the composition of macroradical X(Z) with the growth of its length Z, as well as the corresponding change in the monomer mixture composition in the microreactor. 

In Figure 10.10a, it can be seen that for porous membranes, the partial pressure and concentration profiles vary continuously from the bulk feed to the bulk permeate. This is not the case with nonporous dense membranes, as illustrated in Figure 10.10b. Partial pressure or concentration of the feed liquid just adjacent to the upstream membrane interface is higher than the partial pressure or concentration at the upstream interface. Also, the partial pressure or concentration is higher just downstream of the membrane interface than in the permeate at the interface. The concentrations at the membrane interface and just adjacent to the membrane interface can be related according to an equilibrium partition coefficient KM i. This can be defined as (see Figure 10.10b) ... 

Unlike the previous kinetics imposed by the sink condition, steady-state transport kinetics under non-sink conditions will lead to equilibrium partitioning between the aqueous phase of the donor and receiver compartments and the cell mono-layer. In contrast to the sink condition wherein CR 0 at any time, under nonsink conditions CR increases throughout time until equilibrium is attained. As previously stated in Eqs. (1) and (3), the rate of mass disappearing from the donor solution is... 

Two classes of mathematical models have been developed those which are specific and attempt to describe the transport and degradation of a chemical in a particular situation and those which are general or "evaluative" and attempt to generally classify the behavior of chemicals in a hypothetical environment. The type of modeling discussed here, equilibrium partitioning models, fall into the latter category. Such models attempt, with a minimum of information, to predict expected environmental distribution patterns of a compound and thereby identify which environmental compartments will be of primary concern. 

Soil Diffusion. Water-soluble material In the soil Includes material dissolved In the soil water, material dissolved In the soil air, and material adsorbed to the soil solids. The soil water-soil air equilibrium partitioning Is described by Henry s law ... 

The equilibrium partitioning between soli water and soil solids is described by ... 

None (purely steric) Equilibrium partitioning in pores Size exclusion or gel permeation chromatography... 

Lincoff, A. H., Gossett, J. M. (1984) The determination of Henry s law constants for volatile organics by equilibrium partitioning in closed systems. In Gas Transfer at Water Surfaces. Brutsaert, W., Jirka, G. H., Eds., pp. 17-26, D. Reidel Publishing Co., Dordrecht, The Netherlands. 

Cheng, W.-H., Chu, F.-S., Liou, J.-J. (2003) Air-water interface equilibrium partitioning coefficients of aromatic hydrocarbons. Atmos. Environ. 37, 4807 -815. 

Dewulf, J., van Langenhove, H., Grare, S. (1999) Sediment/water and octanol/water equilibrium partitioning of volatile organic compounds temperature dependence in the 2-25°C range. Water Res. 33, 2424—2436. 

De Seze, G., Valsaraj, K.T., Reible, D.D., Thibodeaux, L.J. (2000) Sediment-air equilibrium partitioning of semi-volatile hydrophobic organic compounds. Part 2. Saturated vapor pressures, and the effects of sediment moisture content and temperature on the partitioning of polyaromatic hydrocarbons. Sci. Total Environ. 253, 27-44. 

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