Mass related distribution coefficients

The selectivity of a gel, defined by the incremental increase in distribution coefficient for an incremental decrease in solute size, is related to the width of the pore size distribution of the gel. A narrow pore size distribution will typically have a separation range of one decade in solute size, which corresponds to roughly three decades in protein molecular mass (Hagel, 1988). However, the largest selectivity obtainable is the one where the solute of interest is either totally excluded (which is achieved when the solute size is of the same order as the pore size) or totally included (as for a very small solute) and the impurities differ more than a decade in size from the target solute. In this case, a gel of suitable pore size may be found and the separation carried out as a desalting step. This is very favorable from an operational point of view (see later). 

Baskaran and Santschi (1993) examined " Th from six shallow Texas estuaries. They found dissolved residence times ranged from 0.08 to 4.9 days and the total residence time ranged from 0.9 and 7.8 days. They found the Th dissolved and total water column residence times were much shorter in the summer. This was attributed to the more energetic particle resuspension rates during the summer sampling. They also observed an inverse relation between distribution coefficients and particle concentrations, implying that kinetic factors control Th distribution. Baskaran et al. (1993) and Baskaran and Santschi (2002) showed that the residence time of colloidal and particulate " Th residence time in the coastal waters are considerably lower (1.4 days) than those in the surface waters in the shelf and open ocean (9.1 days) of the Western Arctic Ocean (Baskaran et al. 2003). Based on the mass concentrations of colloidal and particulate matter, it was concluded that only a small portion of the colloidal " Th actively participates in Arctic Th cycling (Baskaran et al. 2003). 

The second complicating factor is interfacial turbulence (1, 12), very similar to the surface turbulence discussed above. It is readily seen when a solution of 4% acetone dissolved in toluene is quietly placed in contact with water talc particles sprinkled on to the plane oil surface fall to the interface, where they undergo rapid, jerky movements. This effect is related to changes in interfacial tension during mass transfer, and depends quantitatively on the distribution coefficient of the solute (here acetone) between the oil and the water, on the concentration of the solute, and on the variation of the interfacial tension with this concentration. Such spontaneous interfacial turbulence can increase the mass-transfer rate by 10 times 38). 

The Distribution Coefficient. As a measure of the sorption and retarding capacity of the rock and clay the mass related distribution coefficient Kj [ms/kg] was used, defined as... 

Measured mass related distribution coefficients for granite... 

Mass related distribution coefficients for granite. (Particle size 0.063-0.105 mm, contact time ... 

Mass related distribution coefficients for Tc and U under non-oxidizing conditions (25 C, Aquog)... 

In order to permit sizing a tower, data must be available of the height of a transfer unit (HTU). This term often is used interchangeably with the height equivalent to a theoretical stage (HETS), but strictly they are equal only for dilute solutions when the ratio of the extract and raffinate flow rates, E/R, equals the distribution coefficient, K = xE/xR (Treybal, 1963, p. 350). Extractor performance also is expressible in terms of mass transfer coefficients, for instance, KEa, which is related to the number and height of transfer units by... 

Complexity in multiphase processes arises predominantly from the coupling of chemical reaction rates to mass transfer rates. Only in special circumstances does the overall reaction rate bear a simple relationship to the limiting chemical reaction rate. Thus, for studies of the chemical reaction mechanism, for which true chemical rates are required allied to known reactant concentrations at the reaction site, the study technique must properly differentiate the mass transfer and chemical reaction components of the overall rate. The coupling can be influenced by several physical factors, and may differently affect the desired process and undesired competing processes. Process selectivities, which are determined by relative chemical reaction rates (see Chapter 2), can thenbe modulated by the physical characteristics of the reaction system. These physical characteristics can be equilibrium related, in particular to reactant and product solubilities or distribution coefficients, or maybe related to the mass transfer properties imposed on the reaction by the flow properties of the system. 

Dj are distribution coefficients at equilibrium at feed-membrane and membrane-strip interfaces Relation for the overall mass transfer resistance can be derived [11,25,91] ... 

The modeling procedure can be sketched as follows. First an approximate description of the velocity distribution in the turbulent boundary layer is required. The universal velocity profile called the Law of the wall is normally used. The local shear stress in the boundary layer is expressed in terms of the shear stress at the wall. From this relation a dimensionless velocity profile is derived. Secondly, a similar strategy can be used for heat and species mass relating the local boundary layer fluxes to the corresponding wall fluxes. From these relations dimensionless profiles for temperature and species concentration are derived. At this point the concentration and temperature distributions are not known. Therefore, based on the similarity hypothesis we assume that the functional form of the dimensionless fluxes are similar, so the heat and species concentration fluxes can be expressed in terms of the momentum transport coefficients and velocity scales. Finally, a comparison of the resulting boundary layer fluxes with the definitions of the heat and mass transfer coefficients, indiates that parameterizations for the engineering transfer coefficients can be put up in terms of the appropriate dimensionless groups. 

This equation is simply a rearrangement of the distribution constant equation that relates the mass sorbed to the stationary phase divided by the mass in the solution phase. The authors note that the amount of analyte sorbed by the coating is proportional to the initial analyte concentration in both Eqs. (12.1) and (12.2). However, the additional term of A"V, is now present in the denominator of Eq. (12.2). This term decreases the amount of solute sorbed (nJ when this term is comparable in size to When it is much smaller than then only the volume of sample is important. As KV becomes much greater than V2, then the terms KV in numerator and denominator cancel, and one is left with the conclusion that the majority of the original analyte, C20, is sorbed. Thus, the extraction is quantitative at this point. In practice, the authors have found that for 90% of the sample to be sorbed into the coating, the distribution coefficient must be about an order of magnitude greater than the phase ratio, V2/V,. For this to occur, the K must be approximately 1000, which is equivalent to compounds with an octanol-water partition coefficient ( ow) of approximately the same value, or log of 3. 

The theory of adsorption at porous adsorbents predicts the existence of a finite critical energy of adsorption e, where the macromolecule starts to adsorb at the stationary phase. Thus, at > the macromolecule is adsorbed, whereas at e < e the macromolecule remains unadsorbed. At e = Ec the transition from the unadsorbed to the adsorbed state takes place, corresponding to a transition from one to another separation mechanism. This transition is termed critical point of adsorption and relates to a situation, where the adsorption forces are exactly compensated by the entropy losses TAS = AH [2, 7]. Accordingly, at the critical point of adsorption the Gibbs free energy is constant (AG = 0) and the distribution coefficient is Kj = 1, irrespective of the molar mass of the macromolecules. The critical point of adsorption relates to a very narrow range between the size exclusion and adsorption modes of liquid chromatography. It is, therefore, very sensitive towards temperature and mobile phase composition. 

We use the hydrolysis of A into P and Q as an illustration. Examples are the hydrolysis of benzylpenicillin (pen G) or the enantioselective hydrolysis of L-acetyl amino acids in a DL-mixture, which yields an enantiomerically pure L-amino acid as well as the unhydrolysed D-acetyl amino acid. In concentrated solutions these hydrolysis reactions are incomplete due to the reaction equilibrium. It is evident that for an accurate analysis of weak electrolyte systems, the association-dissociation reactions and the related phase behaviour of the reacting species must be accounted for precisely in the model [42,43]. We have simplified this example to neutral species A, P and Q. The distribution coefficients are Kq = 0.5 and Kp = K = 2. The equilibrium constant for the reaction K =XpXQ/Xj = 0.01, where X is a measure for concentration (mass or mole fractions) compatible with the partition coefficients. The mole fraction of A in the feed (z ) was 0.1, which corresponds to a very high aqueous feed concentration of approximately 5 M. We have simulated the hydrolysis conversion in the fractionating reactor with 50-100 equilibrium stages. A further increase in the number of stages did not improve the conversion or selectivity to a significant extent. Depending on the initial estimate, the calculation requires typically less than five iterations. 

For both SFC and SEC, the solute distribution coefficient between stationary and mobile phases is K, which is related to the standard free energy difference for a solute in the two phases. SFC is an enthalpy-controlled process, so that the retention parameter k is positive, and retention volumes, Fr, for the members of a homologous series in SFC are related to molecular mass, M ... 

Liquid-liquid extraction is an equilibrium process between two immiscible phases, described by a equilibrium constant, usually called the distribution constant or partition coefficient. The distribution constant, K, is defined as the ratio of the concentration of the substance in the two phases at equilibrium K = Cu/Cl, where Cu is the concentration of analyte in the less dense (upper) phase and Cl its concentration in the more dense (lower) phase). The distribution or partition ratio, G, defined as the ratio of the solute masses in the two phases (G = mi]/mi, where ntu is the mass of analyte in the less dense phase and Wl its mass in the more dense phase), is often more useful. The distribution constant and distribution ratio are related through the phase ratio, X by G = KV. The phase ratio is the volume ratio of the liquid phases (V = volume of the less dense phase/volume of the more dense phase). 

Equilibrium relations. Even when mass transfer is occurring equilibrium relations are important to determine concentration profiles for predicting rates of mass transfer. In Section 10.2 the equilibrium relation in a gas-liquid system and Henry s law were discussed. In Section 7.1C a discussion covered equilibrium distribution coefficients between two phases. These equilibrium relations will be used in discussion of mass transfer between phases in this section. 

To account for deviation from thermodynamics, under real conditions an effective distribution coefficient ke F is defined (Equation 7.3). The equation is similar to Equation 7.2, but the effective distribution coefficient results from parameters measured under real crystallization conditions. That is, the parameters Xir,ip s as the impurity content in the solid phase and ximp.i, as the impurity content in the liquid phase are values obtained from the separation process performed. In contrast, the parameters in Equation 7.2 are directly related to the phase diagram. Thus, the effective distribution coefficient also comprises the influence of the crystallization kinetics, in particular the crystal growth rate and mass transfer limitations. 

Chemicals in water can sorb to sediment or soil in a reversible process that reflects the attraction and adhesion of molecules to solids. (Less commonly considered in environmental mass balances, some air pollutants can sorb to particulates in the atmosphere.) The n-octanol/water partition coefficient (K ) of a substance provides a crude indication of the tendency to partition to solids from water a high value indicates that a substance is hydro-phobic/lipophilic and would tend to sorb to solids. More sophisticated tests determine a distribution coefficient (KJ or adsorption isotherm to relate the concentration in solution to the concentration sorbed to solids. The sorption coefficient is the ratio between the concentration of a chemical in soil to the concentration in water which is in contact with the soil. Normalized to the organic carbon content of the soil, this coefficient becomes (K = I i/ fraction organic carbon in soil) [4]. 

Solid-phase mass tnmsfer The various mass-transfer processes that may occur within the solid can frequently be jointly described in terms of a mass-transfer coefficient kg (based on external surface) or an effective diffusivity Dg. For a constant or nearly constant uilibrium distribution coefficient and spherical particles, the former can be related to the diffusivity through the approximation [30]... 

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