Component partition constants

The amount by which we can separate two components as peaks in time, or bands in distance, on a chromatographic system is determined by the selectivity of the stationary phase. It is defined for specific conditions namely, temperature in GC, or mobile phase composition in LC. Selectivity, a, is related to the ratio of partition constants of components A and B, or their more easily measured adjusted retention times, as illustrated in Fig. 11.2 and calculated from Eq. (11.4). 

This equation yields an alternative relationship for the partition constant in HS in which K is found to be inversely proportional to a product of the partial pressure of component i if it were pure and its activity coefficient according to... 

Partition constants report the equilibrium distribution of components in aggregated systems but do not represent the free energy of transfer of alcohols between aggregates and the bulk aqueous or oil phases, and their values need not be independent of solution composition, especially when the alcohol concentration in the aggregates or the bulk phase is high. However, alcohol distributions expressed as mass-action binding constants in aqueous three-component microemulsions [reaction (16)] and in W/O microemulsions [reaction (17)] are independent of alcohol concentration. 

All methods of chromatography operate on the principle that the components of a mixture will distribute unequally between two immiscible phases, which is also the basis for separations by extraction (Chap. 5). The mobile phase is generally a liquid or a gas that flows continuously over the fixed stationary phase, which may be a solid or a liquid. The individual components of the mixture have different affinities for the mobile and stationary phases, so a dynamic equilibrium is established in which each component is selectively, but temporarily, removed from the mobile phase by binding to the stationary phase. When the equilibrium concentration of that substance in the mobile phase decreases, it is released from the stationary phase and the process continues. Since each component partitions between the two phases with a different equilibrium constant or partition coefficient, the components divide... 

Component partitioning between two immiscible phases is traditionally characterized by partition constants using a variety of experimental methods 18,19]. The Hansch octanol-water partition cmistant, is probably the most widely accepted physicochemical property used to characterize the polarity of compounds[20, 21] and it is also a widely used... 

The function of the column is to allow partitioning of the constituents of the sample to be separated between the stationary and mobile phases, and this is aided by having the liquid phase as a thin film with a large surface area accessible to the flow of the gas phase. As the sample (solute) passes down the column, the molecules of each component partition between the liquid and gas phases according to a distribution coefficient or constant, Kp, i.e. 

A molecule in solution can be seen as a system complying with constant F, V, and N. Let Z9) denote its partition function. Having the partition function a probabilistic interpretation, the partition function of a composite system equals the product of the components partition functions, unless they are identical and indistinguishable. Consider for instance a system composed of N identical, indistinguishable molecules in solution. In this case, the system partition function is ... 

The theory of the process can best be illustrated by considering the operation, frequently carried out in the laboratory, of extracting an orgaiuc compound from its aqueous solution with an immiscible solvent. We are concerned here with the distribution law or partition law which, states that if to a system of two liquid layers, made up of two immiscible or slightly miscible components, is added a quantity of a third substance soluble in both layers, then the substance distributes itself between the two layers so that the ratio of the concentration in one solvent to the concentration in the second solvent remains constant at constant temperature. It is assumed that the molecular state of the substance is the same in both solvents. If and Cg are the concentrations in the layers A and B, then, at constant temperature ... 

The working capacity of a sorbent depends on fluid concentrations and temperatures. Graphical depiction of soration equilibrium for single component adsorption or binary ion exchange (monovariance) is usually in the form of isotherms [n = /i,(cd or at constant T] or isosteres = pi(T) at constant /ij. Representative forms are shown in Fig. I6-I. An important dimensionless group dependent on adsorption equihbrium is the partition ratio (see Eq. 16-125), which is a measure of the relative affinities of the sorbea and fluid phases for solute. 

Lipophilicity is intuitively felt to be a key parameter in predicting and interpreting permeability and thus the number of types of lipophilicity systems under study has grown enormously over the years to increase the chances of finding good mimics of biomembrane models. However, the relationship between lipophilicity descriptors and the membrane permeation process is not clear. Membrane permeation is due to two main components the partition rate constant between the lipid leaflet and the aqueous environment and the flip-flop rate constant between the two lipid leaflets in the bilayer [13]. Since the flip-flop is supposed to be rate limiting in the permeation process, permeation is determined by the partition coefficient between the lipid and the aqueous phase (which can easily be determined by log D) and the flip-flop rate constant, which may or may not depend on lipophilicity and if it does so depend, on which lipophilicity scale should it be based ... 

The log Kow, water solubilities, and Henry s law constants of several of the components that are present in the organophosphate ester hydraulic fluids included in this profile have been measured and are presented in Tables 3-4, 3-5, 3-8, and 3-9. In general, chemicals with low Kow (log Knw <1) tend to have high water solubilities, do not sorb to sediments, and do not bioconcentrate chemicals with high Kow tend to have low water solubilities, partition to sediments and soil, and bioconcentrate in fish (Lyman et al. 1982). Most of the values presented above are for mixtures and are the average values for all of the components in the mixture. 

The first term on the right is the formula for the chemical potential of component a at density pa = na/V in an ideal gas, as would be the case if interactions between molecules were negligible, fee is Boltzmann s constant, and V is the volume of the solution. The other parameters in that ideal contribution are properties of the isolated molecule of type a, and depend on the thermodynamic state only through T. Specifically, V/A is the translational contribution to the partition function of single a molecule at temperature T in a volume V... 

The acceptance criteria for the Gibbs ensemble were originally derived from fluctuation theory [17]. An approximation was implicitly made in the derivation that resulted in a difference in the acceptance criterion for particle transfers proportional to 1/N relative to the exact expressions given subsequently [18]. A full development of the statistical mechanics of the ensemble was given by Smit et al. [19] and Smit and Frenkel [20], which we follow here. A one-component system at constant temperature T, total volume V, and total number of particles N is divided into two regions, with volumes Vj and Vu = V - V, and number of particles Aq and Nu = N - N. The partition function, Q NVt is... 

In the preceding section we have set up the canonical ensemble partition function (independent variables N, V, T). This is a necessary step whether one decides to use the canonical ensemble itself or some other ensemble such as the grand canonical ensemble (p, V, T), the constant pressure canonical ensemble (N, P, T), the generalized ensemble of Hill33 (p, P, T), or some form of constant pressure ensemble like those described by Hill34 in which either a system of the ensemble is open with respect to some but not all of the chemical components or the system is open with respect to all components but the total number of atoms is specified as constant for each system of the ensemble. We now consider briefly the selection of the most convenient formalism for the present problem. 

As in Section 5.1 for the reacting- and conserved-scalar sub-spaces, (pc can be further partitioned into two sub-spaces corresponding to N m components that vary with space/time and ( V - Y o,) components that are uniform/constant. This is illustrated in Fig. 5.5. The partitioning is most easily carried out using the SVD of d ... 

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