Equal mobility

The error due to diffusion potentials is small with similar electrolyte solutions (cj = C2) and with ions of equal mobility (/ Iq) as in Eq. (3-4). This is the basis for the common use of electrolytic conductors (salt bridge) with saturated solutions of KCl or NH4NO3. The /-values in Table 2-2 are only applicable for dilute solutions. For concentrated solutions, Eq. (2-14) has to be used. 

Another issue that can be clarified with the aid of numerical simulations is that of the recombination profile. Mailiaras and Scott [145] have found that recombination takes place closer to the contact that injects the less mobile carrier, regardless of the injection characteristics. In Figure 13-12, the calculated recombination profiles arc shown for an OLED with an ohmic anode and an injection-limited cathode. When the two carriers have equal mobilities, despite the fact that the hole density is substantially larger than the electron density, electrons make it all the way to the anode and the recombination profile is uniform throughout the sample. 

Changes in the reference electrode junction potential result from differences in the composition of die sample and standard solutions (e.g., upon switching from whole blood samples to aqueous calibrants). One approach to alleviate this problem is to use an intermediate salt bridge, with a solution (in the bridge) of ions of nearly equal mobility (e.g., concentrated KC1). Standard solutions with an electrolyte composition similar to that of the sample are also desirable. These precautions, however, will not eliminate the problem completely. Other approaches to address this and other changes in the cell constant have been reviewed (13). 

The situation is more complicated when charged ions rather than uncharged molecules are transferring. In this case, a Nemst-Planck equation which includes terms for both counter-ions and mobile co-ions must be applied. The problem may be simplified by assuming that the counter-ions have equal mobility, when the relationship is ... 

The discussion clearly shows the limitations of Eq. (18) for calculating the optimal chiral selector concentration. As already mentioned, Eq. (18) was derived based on the assumption of equal mobility of both diastereomeric complexes, which indeed is not always the case. This means that Eq. 

It will be noted that liquid junction diffusion potentials can be eliminated almost completely by ensuring that the bulk of the current is carried by cations and anions possessing equal mobilities, e.g. KCl or NHiNOg. Thus by inserting a saturated solution of B.s,o. 16... 

The film thickness is a fictitious quantity and cannot be measured directly. Its magnitude is usually of the order 10 3-10 2 cm, depending on agitation. This criterion is valid only for counterions of equal mobility and infinite solution volume. However, it is a useful approximation despite its limited precision. 

As it is seen in these figures, the higher n(0), the faster the asymptotics is achieved. For the immobile reactant A and d = 1, a(t) systematically exceeds that for the equal mobilities which leads to faster concentration decay in time. The results for d = 2 and 3 are qualitatively similar. Their comparison with the one-dimensional case demonstrates that the concentration decay is now much faster since the critical exponents strive for a = 3/4 and a = 1/2 for the symmetric and asymmetric cases, respectively, which differ greatly from the classical value of a = 1. Respectively, the gap between symmetric and asymmetric decay kinetics grows much faster than in the d = 1 case. Therefore, the conclusion could be drawn that the effect of the relative particle mobility is pronounced better and thus could be observed easier in t ree-dimensional computer simulations rather than in one-dimensional ones, in contrast to what was intuitively expected in [33]. 

In calculations presented below we assume first one kind of defects to be immobile (Da = 0, k = 0) and their dimensionless initial concentration n(0) = 0.1 is not too high it is less than 10 per cent of the defect saturation level accumulated after prolonged irradiation [41]. Its increase (decrease) does not affect the results qualitatively but shorten (lengthen, respectively) the distinctive times when the effects under study are observed. To stress the effects of defect mobility, we present in parallel in Sections 6.3.1 and 6.3.2 results obtained for immobile particles A (D = 0, asymmetric case) and equal mobility of particles A and B (Da = Dq, symmetric case). In both cases only pairs of similar particles BB interact via elastic forces, (6.3.5), but not AA or AB. The initial distribution (t = 0) of all defects is assumed to be random, Y(r > 1,0) = -X (r,0) = 1 i/ = A,B. 

Unlike isotropic media, where molecules have equal mobility and conformational flexibility in all dimensions, in an organized medium their mobility and flexibility are restricted or constrained in at least one dimension. For example, the reaction cavities of a micelle and cyclodextrins are made up of a hydrophobic core and a hydrophilic exterior (Fig. 11). A highly polar boundary separates the hydrophobic core from the aqueous exterior. Such a boundary provides unique features to these media that are absent in isotropic solution. Translational motion of a guest present within the reaction cavity is hindered by the well-defined boundary. 

Minimization of the liquid junction potential is commonly carried out using a salt bridge in which the ions have almost equal mobilities. One example is potassium chloride (t+ = 0.49 and t =0.51) and another is potassium nitrate (t+ = 0.51 and / = 0.49). If a large concentration of electrolyte is used in the salt bridge this dominates the ion transport through the junctions such that the two values of have the same magnitude but opposing polarities. The result is that they annul each other. In this way values of E can be reduced to 1-2 mV. 

Let us briefly highlight the effect of equilibrium space charges on the transport in our prototype oxide (with just O and h as defects).244 The situation becomes particularly clear if we assume equal mobilities and equal bulk conductivities. Since in the case of chemical diffusion space charge splitting... 

The uptake diffusivity or diffusion times are estimated from uptake curves on the basis of the concentration in the adsorbed phase. The uptake diffusivity is multiplied by a Henry s law constant to transform it into a steady-state diffusivity based on the gas-phase concentration (105,106). If all the molecules sorbed internally are assumed to be equally mobile, the definition of the steady-state diffusivity is given by Eq. (8) ... 

The difference in ion mobilities, ion selectivity, and the effect of the diffusion potential upon co-ions as well as counter-ions make a rigorous theoretical treatment of film diffusion kinetics extremely difficult. Analytical rate equations have been proposed for monovalent ion exchange between ions of different mobility but no selectivity (cr = 1), and also for finite selectivity 1) but equal mobilities... 

Although strictly an infinite volume solution for ions of equal mobility, a useful approximation for describing ion exchange under film diffusion control with selectivity is given by ... 

The reaction half time t- j2 is the time required to reach 50% attainment of equilibrium (F = 0.5). The half times for particle and film diffusion control are equal when both mechanisms proceed at the same rate in which case the ratio of the i/2 values will equal unity. Under infinite volume boundary conditions and counter-ions of equal mobility a dimensionless rate parameter is obtained from equations 6.12 and 6.21 given by ... 

Inserting X] from Eq. [4] in Eq. [3], and subsequently from Eq. [3] for in Eq. [2], the r,/2 can be calculated. Because the resulting rate equation is rather complicated, it is not reproduced here, but will be discussed later. Equations for ion-exchange processes controlled by film diffusion were given by Helfferich (1962) for ions of different mobilities for the case 2 - 1 (no selectivity), or for aj b but then only for ions of equal mobilities. No such restrictions were assumed in the derivation of the rate equation given above. 

Each pair represents a tendency in the sample to change radius. The third of the pairs is the composition-driven interdiflfusion effect, where the two species travel in opposite directions, and the effect arises from their difference in mobility, (X — X ). If they were of equal mobility they would simply exchange places volume-for-volume in a compatible way and no stress anomalies would develop, i.e., s would be zero. As already discussed, when the mobilities are not equal, the sample tends to swell at some points and get thinner at other points, and it is the business of the stress terms to just exactly nullify this effect. 

Based upon such data, direct analysis of some fractions is possible when the components can be identified by their mobility alone. However, such an application is very limited in practice, as many subfractions have almost equal mobilities. In order to measure relative mobilities, it is practical to use pure serum albumin as reference. If the mobility of the latter is aib and the mobility of the unknown protein Ux, the quotient x/Waib is independent of the carrier medium if the migration velocity does not change during the run. In Table 3, relative mobilities... 

We minimize the liquid-junction potential by using a high concentration of a salt whose ions have nearly equal mobility, for example, KCl. 

Case (j8). Two distinguishable species A and B, characterized by equal V and occupying the same sites (equal 8) are present. No gradient in the sum of the concentrations exists, but we have opposite and numerically equal gradients in concentration for either species. There will be no net flow of particles, since (c + c ) is constant in space and time, but there can be opposite equal fluxes of both species. Under these conditions of equal mobility for both components, the rate of binary diffusion must be equal to the rate of self-diffusion for either pure component at the same total concentration. The average jumping frequency v of any particle would be... 

Current in the channel in subthreshold is a function of charge carrier concentration in the channel. The best subthreshold slope which can be observed in an FET device at room temperature is 60mV/decade, which is the slope at the edge of a Fermi distribution when it is convolved with an abrupt density of states. Because organic semiconductors exhibit a gradual rise in the density of states at the channel edge and not all carriers are equally mobile, the convolution produces a shallower rise in the carrier density and the observed subthreshold slope is worse than this value. 

---