Diffusion stationary medium

Special Case Ideal Gas MixtJtes 779 Pick s Lav/ of Diffusion Stationary Medium Consisting of Tv/o Species 779... 

By considering of the appropriate element of a sphere show that the general equation for molecular diffusion in a stationary medium and in the absence of a chemical reaction is ... 

For mass transfer by molecular diffusion from a single sphere of diameter d to an infinite stationary medium, it can be shown that... 

In the hydrodynamic theory, the diffusion coefficient of a solute molecule A or single particle through a stationary medium B, DAB, is given by the Nemst-Einstein equation ... 

The validity of the generalized Langevin equation (22) is restricted to a stationary medium. In other situations, for instance when the diffusing particle evolves in an aging medium such as a glassy colloidal suspension of Laponite [8,12,55,56], another equation of motion has to be used. 

Let us consider again the particular case of a particle diffusing in a stationary medium, in order to see how the generalized Langevin equation (22) can be deduced from the more general equation (169). When the medium is stationary, the response function x (M0 reduces to a function of t — t (% (t, t ) = X (f — t j). Introducing then the causal function y(f) as defined by... 

Analysis We can consider the total molar concentration to be constant (C = Ca+ Cfl - Cfl = constant), and the container to be a stationary medium since there is no diffusion of nickel molecules (Ng - U) the concentration of the hydrogen in the container is extremely low (C D.Then the molar flow rate of hydrogen through this spherical shelf by diffusion can readily be determined from Eq. 14-28 to be. . 

Transient mass difliision in a stationary medium is analogous to transient heat transfer provided that the solution is dilute and thus the density of the medium p is constant. In Chapter 4 we presented analytical and graphical solutions for one-dimensional transient heat conduction problems in solids with constant properties, no heat generation, and uniform initial temperature. The analogous one-dimensional transient mass diffusion problems satisfy these requirements ... 

Analogy between the quantities that appear in the formulation and solution of transient heat conduction and transient mass diffusion in a stationary medium... 

The special case V - 0 corresponds to a stationary medium, which can now be defined more precisely as a medium whose mass-average velocity is zero. Therefore, mass transport in a stationary medium is by diffusion only, aud zero mass-average velocity indicates that there is no bulk fluid motion. 

Under steady conditions, the molar flow rates of species A and B can be determined directly from Eq. 14-24 developed earlier for one-diinensional steady diffusion in a stationary medium, noting that P CRJT and thus C = PIRJ" for each constituent gas and the mixture. For one-dimensional flow through a channel of uniform cross sectional area A with no homogeneous chemical reactions, they arc expressed as... 

The special case K = 0 corresponds to a stationary medium. Using Pick s law of diffusion, the total mass fluxes J = m/A in a moving medium are expressed as... 

SC Define the following tei/ns mass-average velocity, diffusion velocity, stationary medium, and moving medium. 14-76C What is diffusion velocity How does it affect the mass-average velocity Can the velocity of a species in a moving medium relative to a fuxed reference point be zero in a moving medium Explain. 

As an exercise, the reader can verify that equation (2.73) satisfies both real and imaginary parts of equation (2.70). This development represents the starting point for both the Warburg impedance associated with diffusion in a stationary medium of infinite depth and the diffusion impedance associated with a stationary medium of finite depth. 

Another limit is that of simple axial diffusion in a stationary medium (Pick s second law). In that situation the radial diffusion time a lD must be short compared with the convection time L/U or... 

Release into a stationary medium is by no means an ideal substitute for in vitro release studies. Clearly, future experiments must be conducted in stirred solutions. Diffusion coefficients, as well as the rate-limiting mechanisms for diffusion could then be more readily determined. 

Example 7.1 Consider the diffusion of speeies A through a stationary medium B around a stationary partiele. Determine Aq in Eq. (7.3.3). 

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. 

The form of the effective mobility tensor remains unchanged as in Eq. (125), which imphes that the fluid flow does not affect the mobility terms. This is reasonable for an uncharged medium, where there is no interaction between the electric field and the convective flow field. However, the hydrodynamic term, Eq. (128), is affected by the electric field, since electroconvective flux at the boundary between the two phases causes solute to transport from one phase to the other, which can change the mean effective velocity through the system. One can also note that even if no electric field is applied, the mean velocity is affected by the diffusive transport into the stationary phase. Paine et al. [285] developed expressions to show that reversible adsorption and heterogeneous reaction affected the effective dispersion terms for flow in a capillary tube the present problem shows how partitioning, driven both by electrophoresis and diffusion, into the second phase will affect the overall dispersion and mean velocity terms. 

The possibility that adsorption reactions play an important role in the reduction of telluryl ions has been discussed in several works (Chap. 3 CdTe). By using various electrochemical techniques in stationary and non-stationary diffusion regimes, such as voltammetry, chronopotentiometry, and pulsed current electrolysis, Montiel-Santillan et al. [52] have shown that the electrochemical reduction of HTeOj in acid sulfate medium (pH 2) on solid tellurium electrodes, generated in situ at 25 °C, must be considered as a four-electron process preceded by a slow adsorption step of the telluryl ions the reduction mechanism was observed to depend on the applied potential, so that at high overpotentials the adsorption step was not significant for the overall process. 

An analogous situation can be envisioned if the medium is stationary (or a fluid in laminar flow in the x direction) and the temperature difference (7) — T0) is replaced by the concentration difference (Cj — C0) of some species that is soluble in the fluid (e.g., a top plate of pure salt in contact with water). If the soluble species (e.g., the salt) is A, it will diffuse through the medium (B) from high concentration (Cj) to low concentration (Co). If the flux of A in the y direction is denoted by nAy, then the transport law is given by... 

Two other attempts, without the use of a distribution function, are worth mentioning, as these are operationally related to experiments and serve to give a rough estimate of the thermalization time. Christophorou et al. (1975) note that in the presence of a relatively weak external field E, the rate of energy input to an electron by that field is (0 = eEvd, where vd is the drift velocity in the stationary state. Under equilibrium, it must be equal to the difference between the energy loss and gain rates by an electron s interaction with the medium. The mean electron energy is now approximated as (E) = (3eD )/(2p), where fl = vd /E is the drift mobility and D is the perpendicular diffusion coefficient (this approximation is actually valid for a Maxwellian distribution). Thus, from measurements of fl and D the thermalization time is estimated to be... 

This result implicitly requires (a) that no strongly attractive/repulsive electrostatic interaction occurs as the particles approach each other, and (b) that a stationary diffusional concentration field surrounds the particle. [Note Einstein s result for spherical particle diffusivity i.e.,D = k Tlfm-qr, where is the Boltzmann constant, T is temperature in kelvin, 17 is the viscosity of the medium, and r is the radius) indicates that RuD will be approximately Ak TKyni].]... 

This equation was derived above for the movement of a liquid through a stationary solid phase. Its application here to the movement of colloidal particles under experimental conditions that render the liquid medium immobile implies that the solid particle is large compared with the dimensions of the diffuse double layer k 1. It is customary to term this movement of the solid phase electrophoresis. The phenomenon is observed with particles suspended in a liquid (Fig. 6.139). 

Integration of the stationary electro-diffusion equations in one dimension. The integration of the stationary Nernst-Planck equations (4.1.1) with the LEN condition (4.1.3), in one dimension, for a medium with N constant for an arbitrary number of charged species of arbitrary valencies was first carried out by Schlogl [5]. A detailed account of Schlogl s procedure may be found in [6]. In this section we adopt a somewhat different, simpler integration procedure. 

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