Diffusivities under equilibrium conditions

Since there is no net diffusion under equilibrium conditions, then- p hole current is equal to the p —> n hole current. These equilibrium currents are analogous to the equilibrium exchange currents at an electrode/solution interface. They represent the exchange of holes across the junction between the n- and p-types of material and will be designated by the symbol i0fl. This i0 will now be examined more carefully. 

Single-component diffusion under equilibrium conditions can be monitored either by labeling some of the molecules or by following their trajectories. Considering the diffusion flux of the labeled molecules, again a proportionality relation of the type of eq 2 may be established. The factor of proportionality is called the coefficient of self-diffusion (or tracer diffusion). In a completely equivalent way [2], the self-diffusion coefficient may be determined on the basis of Einstein s relation... 

Self-diffusion under equilibrium conditions may also be monitored in multicomponent systems. Again, with both eqs 2 and 3 a self-diffusion coefficient (of a particular component) may be defined. This coefficient depends on the nature and the concentration of all molecular species involved as well as on the nature of the catalyst particle. 

Analogous considerations apply to spatially distributed reacting media where diffusion is tire only mechanism for mixing chemical species. Under equilibrium conditions any inhomogeneity in tire system will be removed by diffusion and tire system will relax to a state where chemical concentrations are unifonn tliroughout tire medium. However, under non-equilibrium conditions chemical patterns can fonn. These patterns may be regular, stationary variations of high and low chemical concentrations in space or may take tire fonn of time-dependent stmctures where chemical concentrations vary in botli space and time witli complex or chaotic fonns. 

If a sedimentation experiment is carried out long enough, a state of equilibrium is eventually reached between sedimentation and diffusion. Under these conditions material will pass through a cross section perpendicular to the radius in both directions at equal rates downward owing to the centrifugal field, and upward owing to the concentration gradient. It is easy to write expressions for the two fluxes which describe this situation ... 

Let us consider another situation where a force (or forces) is not compensated on a time average. Then the particles upon which the force is exerted become transported in the medium. This translocation phenomenon changes with time. Particle transport, of course, also occurs under equilibrium conditions in homogeneous media. Self-diffusion is a process that can be observed and its velocity can be measured, provided that a gradient of isotopically labelled species is formed in the system at constant composition. 

The performance of the BioCD under assay conditions has been tested using several gold standard systems. These are assays of anti-rabbit and anti-mouse IgG systems, prostate specific antigen (PSA), and haptoglobin. Incubations have been performed under equilibrium conditions without transport limitation, and also under transient conditions as ambient assays that are diffusion limited. Ambient assays are performed in practice, while equilibrium assays provide more information about the performance of the antibodies and provides a quantitative estimate for equilibrium dissociation constants. 

Under equilibrium conditions, the chemical potential of diffusible hydrogen equals the chemical potential of trapped hydrogen, i.e.. 

Atoms taking part in diffusive transport perform more or less random thermal motions superposed on a drift resulting from field forces (V//,-, Vrj VT, etc.). Since these forces are small on the atomic length scale, kinetic parameters established under equilibrium conditions (i.e., vanishing forces) can be used to describe the atomic drift and transport, The movements of atomic particles under equilibrium conditions are Brownian motions. We can measure them by mean square displacements of tagged atoms (often radioactive isotopes) which are chemically identical but different in mass. If this difference is relatively small, the kinetic behavior is... 

During vaporization of non-stoichiometric refractory carbides each element vaporizes at a different rate which is dependent on surface composition or relevant activities at the surface. When the initial bulk composition is near C/M = 1, the vaporization of C is much greater than that of M. As a result, the surface C content decreases and eventually approaches a constant value, which we will call the steady-state CVC (ssCVC). At the ssCVC, the vapor composition is nearly equal to the initial bulk composition. As C diffusion to the vaporizing surface reduces the C content of the bulk material, the surface composition asymptotically approaches the equilibrium CVC (eCVC). The rate at which eCVC is approached depends on the relative magnitudes of C vaporization and diffusion. When the eCVC has been reached, the surface and bulk C/M ratios are equal to the vapor composition. The intersection of the solid eCVC map with the solidus boundary of the monocarbide phase determines where melting occurs under equilibrium conditions for a particular atmosphere. 

The use of these diffusion models to progress the evaluation process of a food packaging plastic will be discussed shortly. In those cases where assessment by mass balance considerations under equilibrium conditions, including partitioning effects, does not provide a clear picture of the plastics conformity status, then the different diffusivities of polymer types and the influence of the migrant molecule size or its molecular weight on its mobility within a plastic can be taken into account to achieve more distinguished views on QM/SML ratios. 

The relationship between the transport diffusivity (D), as measured under non-equilibrium conditions in an uptake experiment and the tracer self diffusivity (Ds), measured under equilibrium conditions in an NMR experiment, has been discussed by Ash and Barrer(30) and Karger(31,32)t who show that... 

The comparison between Equations 5.42 and 5.43 shows that the Gibbs adsorption equation can be expressed either in terms of a, and a or in terms of o, E , and a. Note that Equations 5.42 and 5.44 are valid under equilibrium conditions, whereas Equation 5.43 can be used also for the description of dynamic surface tension (Section 5.2.2) in the case of surfactant adsorption under diffusion control, assuming local equilibrium between adsorptions E and subsurface concentrations of the respective species. 

In addition to a vacancy-driven process, diffusion under irradiation may also be enhanced by the formation and diffusion of other point defects, such as selfinterstitials, divacancies, and other defect aggregates, which are not present under equilibrium conditions. A general statement for the atomic diffusion coefficient can be written in terms of the various point defects as... 

NMR PFG measurements determine the tracer or self-diffusivity (D ) under equilibrium conditions with no concentration gradient. n any sorption rate measurement it is the transport diffusivity under the influence of a concentration gradient which is measured. In general these two quantities are not the same but the relationship between them can be established from irreversible thermodynamics. (17,18) In the low concentration limit the thermodynamic correction factor vanishes and the transport and self diffusivities should approach the same limit. Since ZLC measurements are made at low concentrations within the Henry s Law region the diffusivity values should be directly comparable with the NMR self-dif fusivities. ... 

Depending on the column configurations (packed, capillary, etc.) several formulations of Eq. (30) have been suggested. In the present case column parameters must be designed so as to magnify the effects of slow diffusion in the stationary phase. This is quite easily achieved with polymer stationary phases since their diffusion coefficients are usually smaller by two orders of magnitude than those of low molecular weight liquids. It should also be noted that measurements must be performed under equilibrium conditions, i.e., at temperatures in excess of Tg + 50°. 

Bulk sorbate concentration under equilibrium conditions, mmolg Concentration of diffusate in the gas phase, mmolg ... 

For most engineering alloys, the ambient temperature only corresponds to a small fraction of the melting temperature, Tm-As outlined above, this implies a very low solid-state diffusivity under these conditions that impedes the establishment of complete equilibrium of the alloy electrode according to Eq. (3). At anodic... 

The composition of the diffusion zone will depend on the binary equilibrium diagram. Under equilibrium conditions, all such systems conform to the phase rule... 

---