Diffusion and Phase Equilibrium

Thus far in this chapter, we have described diffusion as a means of controlling or retarding release of solutes like drugs or flavors. These solutes may initially be present as a solution or a suspension. If the suspension dissolves quickly, diffusion is key. We now turn to other cases of diffusion in two-phase systems. Like controlled release, these cases can show dramatic and unexpected behavior. Like controlled release, this behavior is the consequence of rapid chemical reactions. 

The effects we want to discuss are exemplified by the dissolution of slaked lime, Ca(OH)2, in aqueous solutions of a strong acid like HCl  

How this dissolution proceeds depends on the relative speed of diffusion and reaction. When the bulk of the solution next to the solid is rapidly stirred, the acid can diffuse to the solid s surface very quickly. It then reacts with the solid s surface. If the solid is essentially impermeable, containing a very few pores, then any ions produced by the dissolution are quickly swept back into the bulk solution. Because diffusion and chemical reaction occur sequentially, the overall dissolution rate is like that of a heterogeneous reaction, depending on the sum of the resistances of diffusion and of reaction (see Chapter 16). Such a process represents an important limit of corrosion and is that usually studied. 

Alternatively, the solution next to the solid may not be well stirred, and the solid may be highly porous, as shown schematically by Fig. 19.4-1. In this case, the acid 

To calculate the dissolution rate of the porous solid r, we write continuity equations for calcium ions (species 1) and protons (species 2)  

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Auerbach, S.M. Theory and simulation of jump dynamics, diffusion and phase equilibrium in nanopores. Int. Rev. Phys. Chem. 2000,19,155-198. 

MIK Mikhailov, Yu.M., Ganina, L.V., Chalykh, A.E., and Knlichikhin, S.G., Mutual diffusion and phase equilibrium in polysulfone-solvents systems (Russ.), Vysokomol. 

In addition to the enhanced diffusivity effect, another issue needs to be taken into account when considering stationary-phase mass transfer in CEC with porous particles. The velocity difference between the pore and interstitial space may be small in CEC. Under such conditions the rate of mass transfer between the interstitial and pore space cannot be very important for the total separation efficiency, as the driving mechanism for peak broadening, i.e., the difference in mobile-phase velocity within and outside the particles, is absent. This effect on the plate height contribution II, s has been termed the equilibrium effect [35], How to account for this effect in the plate height equation is still open to debate. Using a modified mass balance equation and Laplace transformation, we first arrived at the following expression for Hc,s, which accounts for both the effective diffusivity and the equilibrium effect [18] ... 

Generally speaking, two principal mechanisms operate in the biology of membrane processes, such as membrane transport and permeation. One involves a network of active sites and operates by metabolic energy, and it is referred to as active another is directed by passive diffusion, and it is called passive. This passive mechanism is determined by various aspects of lipid dynamics and lipid-protein interactions, and it can be described in quantitative terms of chemical and phase equilibrium and molecular physics. However, in the highly anisotropic... 

When a surfactant-water or surfactant-brine mixture is carefully contacted with oil in the absence of flow, bulk diffusion and, in some cases, adsorption-desorption or phase transformation kinetics dictate the way in which the equilibrium state is approached and the time required to reach it. Nonequilibrium behavior in such systems is of interest in connection with certain enhanced oil recovery processes where surfactant-brine mixtures are injected into underground formations to diplace globules of oil trapped in the porous rock structure. Indications exist that recovery efficiency can be affected by the extent of equilibration between phases and by the type of nonequilibrium phenomena which occur (J ). In detergency also, the rate and manner of oily soil removal by solubilization and "complexing" or "emulsification" mechanisms are controlled by diffusion and phase transformation kinetics (2-2). 

But the application of the LC technique for the measurement of liquid diffusion in molecular sieves was rather limited (18, 19). The recently developed LC technique using a commercial HPLC system (13), with many advantages over the conventional batch techniques, enables us to determine liquid phase diffusion and adsorption equilibrium in molecular sieve crystals in a simpler, more accurate and rapid way. 

Electrodialysis—electric charge and ionic mobility Liquid membranes—diffusivity and reaction equilibrium Electrophoresis—electric charge and ionic mobility Chromatographic separations—depends on type of stationary phase Gel filtration—molecular size and shape... 

The experimental methods used for the study of thermodynamic parameters such as Xi and 0 of a polymer in dilute solutions are numerous. Among them are intrinsic viscosity, light scattering, diffusion, sedimentation, vapor pressure, and phase equilibrium. Here we discuss vapor pressure and phase equilibrium, leaving the other methods for later chapters. 

Influence of Chemical Reactions on Uq and When a chemical reaction occurs, the transfer rate may be influenced by the chemical reac tion as well as by the purely physical processes of diffusion and convection within the two phases. Since this situation is common in gas absorption, gas absorption will be the focus of this discussion. One must consider the impacts of chemical equilibrium and reac tion kinetics on the absorption rate in addition to accounting for the effec ts of gas solubility, diffusivity, and system hydrodynamics. 

Combined Pore and Solid Diffusion In porous adsorbents and ion-exchange resins, intraparticle transport can occur with pore and solid diffusion in parallel. The dominant transport process is the faster one, and this depends on the relative diffusivities and concentrations in the pore fluid and in the adsorbed phase. Often, equilibrium between the pore fluid and the solid phase can be assumed to exist locally at each point within a particle. In this case, the mass-transfer flux is expressed by ... 

High mass-transfer rates in both vapor and hquid phases. Close approach to eqiiilihriiim. Adiabatic contact assures phase eqiiilihriiim, Only moderate mass-transfer rate in liquid phase, zero in sohd. Slow approach to equilibrium achieved in brief contact time. Included impurities cannot diffuse out of solid. Sohd phase must be remelted and refrozen to allow phase equilibrium. 

The general theoretical treatment of ion-selective membranes assumes a homogeneous membrane phase and thermodynamic equilibrium at the phase boundaries. Obvious deviations from a Nemstian behavior are explained by an additional diffusion potential inside the membrane. However, allowing stationary state conditions in which the thermodynamic equilibrium is not established some hitherto difficult to explain facts (e.g., super-Nemstian slope, dependence of the selectivity of ion-transport upon the availability of co-ions, etc.) can be understood more easily. 

In this approach, it is assumed that turbulence dies out at the interface and that a laminar layer exists in each of the two fluids. Outside the laminar layer, turbulent eddies supplement the action caused by the random movement of the molecules, and the resistance to transfer becomes progressively smaller. For equimolecular counterdiffusion the concentration gradient is therefore linear close to the interface, and gradually becomes less at greater distances as shown in Figure 10.5 by the full lines ABC and DEF. The basis of the theory is the assumption that the zones in which the resistance to transfer lies can be replaced by two hypothetical layers, one on each side of the interface, in which the transfer is entirely by molecular diffusion. The concentration gradient is therefore linear in each of these layers and zero outside. The broken lines AGC and DHF indicate the hypothetical concentration distributions, and the thicknesses of the two films arc L and L2. Equilibrium is assumed to exist at the interface and therefore the relative positions of the points C and D are determined by the equilibrium relation between the phases. In Figure 10.5, the scales are not necessarily the same on the two sides of the interface. 

Kishinev ski/23 has developed a model for mass transfer across an interface in which molecular diffusion is assumed to play no part. In this, fresh material is continuously brought to the interface as a result of turbulence within the fluid and, after exposure to the second phase, the fluid element attains equilibrium with it and then becomes mixed again with the bulk of the phase. The model thus presupposes surface renewal without penetration by diffusion and therefore the effect of diffusivity should not be important. No reliable experimental results are available to test the theory adequately. 

Phase transition occurs at a state of thermodynamic equilibrium, inducing a change in the microstructure of atoms. However, corrosion is a typical nonequilibrium phenomenon accompanied by diffusion and reaction processes. We can also observe that this phenomenon is characterized by much larger scales of length than an atomic order (i.e., masses of a lot of atoms), which is obvious if we can see the morphological change in the pitted surface. 

Figure 3.6. Spatial variation of the electrochemical potential, jl02-, of O2 in YSZ and on a metal electrode surface under conditions of spillover (broken lines A and B) and when equilibrium has been established. In case (A) surface diffusion on the metal surface is rate limiting while in case (B) the backspillover process is controlled by the rate, I/nF, of generation of the backspillover species at the three-phase-boundaries. This is the case most frequently encountered in electrochemical promotion (NEMCA) experiments as shown in Chapter 4.

A detailed description of AA, BB, CC step-growth copolymerization with phase separation is an involved task. Generally, the system we are attempting to model is a polymerization which proceeds homogeneously until some critical point when phase separation occurs into what we will call hard and soft domains. Each chemical species present is assumed to distribute itself between the two phases at the instant of phase separation as dictated by equilibrium thermodynamics. The polymerization proceeds now in the separate domains, perhaps at differen-rates. The monomers continue to distribute themselves between the phases, according to thermodynamic dictates, insofar as the time scales of diffusion and reaction will allow. Newly-formed polymer goes to one or the other phase, also dictated by the thermodynamic preference of its built-in chain micro — architecture. 

Similar convection-diffusion equations to the Navier-Stokes equation can be formulated for enthalpy or species concentration. In all of these formulations there is always a superposition of diffusive and convective transport of a field quantity, supplemented by source terms describing creation or destruction of the transported quantity. There are two fundamental assumptions on which the Navier-Stokes and other convection-diffusion equations are based. The first and most fundamental is the continuum hypothesis it is assumed that the fluid can be described by a scalar or vector field, such as density or velocity. In fact, the field quantities have to be regarded as local averages over a large number of particles contained in a volume element embracing the point of interest. The second hypothesis relates to the local statistical distribution of the particles in phase space the standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell-Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference co-moving with the fluid. Especially the second assumption may break dovm when gas flow at high temperature or low pressure in micro channels is considered, as will be discussed below. 

Reactive intermediates in solution and in the gas phase tend to be indiscriminant and ineffective for synthetic applications, which require highly selective processes. As reaction rates are often limited by bimolecular diffusion and conformational motion, it is not surprising that most strategies to control and exploit their reactivity are based on structural modihcations that influence their conformational equilibrium, or by taking advantage of the microenvironment where their formation and reactions take place, including molecular crystals. ... 

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