Zero separation factor

A separation factor of one means that only one peak will be produced from our two solutes, ie the value of R is zero. 

This relation is the potential distribution theorem [73, 74], which gives a physical interpretation of the cavity function in terms of the chemical potential, and the excess interaction generated by the test particle, Y j>2 u(rv)> yia the ensemble average of its Boltzmann factor. In numerical simulation, the use of such a test-particle insertion method is of prime importance in calculating the cavity function at small distances and particularly at zero separation. Note that if the particle labeled 1 approaches the particle labeled 2, a dumbbell particle [41] is created with a bond length L = r2 n corresponding to a dimer at infinite... 

The present relations differ from the KM approximation since the factor 3 is replaced by the bridge function at zero separation. This feature does not seem to be unreasonable because, from diagrammatic expansions, B (r) = B r)/3 is supposed to be accurate only at very low densities. Eq. (112) presents two advantages at high density i) it provides a closed-form expression for Bother fluids than the HS model and ii) it allows to ensure a consistent calculation of the excess chemical potential by requiring only the use of the pressure consistency condition (the Gibbs-Duhem constraint, no longer required, is nevertheless implicitly satisfied within 1%). 

The dissociation constant of the carboxyl or amine group, the separation factor of the cation or anion of the amino acid and the distribution coefficient of zwitterion with zero net charge are considered to be approximately constant. 

As mentioned above, the neutral amino acid can exist as neutral specie HA and cation HjA in acidic solutions. The distribution coefficient of HA is replaced by HA of its separation factor. In spite of zero... 

Selectivity (or separation factor) is another important parameter when evaluating membrane separation processes. The selectivity of a membrane (aUj) is the ratio of the permeability of gas i to the permeability of gas j. Thus, for pure gases with near-zero permeate pressure, the selectivity is simply stated as... 

In general a is not equal to a due to back diffusion, caused by non-zero pressure at the permeate side, or to contributions of non-separative mechanisms to the total flow and concentration polarisation on feed or the permeate side. Also the presence of surface diffusion influences the ideal separation factor. 

Not all of these n points will cause a separate reflection some of them may have a zero structure factor, and some may be at equal distances from the reciprocal-lattice origin, i.e., correspond to planes of the same spacing. (The latter effect is taken care of by the multiplicity factor, since this gives the number of different planes in... 

Because longitudinal velocity is zero at the heated and cooled walls, the logarithm of the effective separation factor is found to be that of Eq. (14.347). For y 1 and AT/T < 1, this becomes... 

The separation factor is meaningful only with respect to a mixture of two gases. The ideal separation factor is the ratio of the permeance of the two gases measured at zero pressure, where there is no interaction or momentum exchange... 

The individual properties studied were the selectivities, separation factors, valley-to-peak ratios and overlapped fractions. The two first criteria only consider the position of the chromatographic peaks, and the latter two their position and shape. The three latter functions may vary from 0 to 1. The combined function of resolution, r, is maximized to obtain the optimum mobile phase, and its proximity to unity indicates the quality of the separation. Since the product of all observed resolutions is used, coelution will effectively cause the criterion to drop to zero. Extremely long chromatograms with a number of unnecessarily large resolution values will also be represented by low criterion values. 

The azeotropic points in homogeneous systems can be determined with the procedures introduced in Chapter 5, for example, in binary systems by looking for a composition for which the separation factor au shows a value of unity - or more general - the following objective function F shows a value of zero for binary or... 

The basic quantity of reference Ni°, which is defined as the common value that may take the two numbers of electrons and holes, is called the intrinsic number of charge carriers. Here, as well as in the simple case of linear distribution of energy states, this number is indicated with a superscript zero to avoid confusion with the number N, which can be determined in the more complex case of a material with an irregular distribution of energy states. The denominator of the argument of the exponential is the ratio of the scaling energy-per-corpuscle on the separability factor Sq and the... 

From the above discussion, it is clear that various kinetic criteria (reaction orders, Tafel slopes, and log/-AGads plots on different electrode materials, where appropriate) can be used to determine the reaction mechanism and the coverage conditions. For reactions involving protons, the separation factors (H-T or H-D) can also be used as another criterion, since the barrier heights vary characteristically with mechanism because of the relatively large zero point energy difference between the isotopes and their different quantum mechanical tunneling properties. This is evaluated in Refs. 50 and 51, and may be used to confirm other evidence. 

Since this reversal of sign is incompatible with the fundamental property that the kinetic and potential energies must be unchanged by an operation of the group, such cross terms must have zero coefficients. The vanishing of all cross terms coupling coordinates of two different species clearly puts zeros in the secular equation which break it up into separate factors, one for each species. 

---