Diffusing species

Note that the diffusion coefficient for a polymer through an environment of low molecular weight molecules is typically on the order of magnitude of 10"" m" sec". If the first subscript indicates the diffusing species, and the second the surrounding molecules, and P stands for polymer and S for small molecules, we see that the order of diffusion coefficients is Ds g > Dp g > Dp P sequence which makes sense in terms of relative frictional resistance. 

Activated diffusion of the adsorbate is of interest in many cases. As the size of the diffusing molecule approaches that of the zeohte channels, the interaction energy becomes increasingly important. If the aperture is small relative to the molecular size, then the repulsive interaction is dominant and the diffusing species needs a specific activation energy to pass through the aperture. Similar shape-selective effects are shown in both catalysis and ion exchange, two important appHcations of these materials (21). 

D o is the low pressure diffiisivity at the temperature of interest. (DizP) is a reduced diffiisivity pressure product at infinite reduced temperature and A, B, C, and E are constants. All are a function of P,. tabulated in Table 2-401. Component 1 is the diffusing species, while component 2 is the concentrated species. Critical properties are for the solvent. The pressure is given in Pa. The diffiisiv-ity is in mvsec. Errors from evaluation average near 15 percent. 

The stagnant-film model discussed previously assumes a steady state in which the local flux across each element of area is constant i.e., there is no accumulation of the diffusing species within the film. Higbie [Trans. Am. Jn.st. Chem. Eng., 31,365 (1935)] pointed out that industrial contactors often operate with repeated brief contacts between phases in which the contact times are too short for the steady state to be achieved. For example, Higbie advanced the theory that in a packed tower the liquid flows across each packing piece in laminar flow and is remixed at the points of discontinuity between the packing elements. Thus, a fresh liquid surface is formed at the top of each piece, and as it moves downward, it absorbs gas at a decreasing rate until it is mixed at the next discontinuity. This is the basis of penetration theoiy. 

There is no sharp dividing hne between pure physical absorption and absorption controlled by the rate of a chemic reaction. Most cases fall in an intermediate range in which the rate of absoration is limited both by the resistance to diffusion and by the finite velocity of the reaction. Even in these intermediate cases the equihbria between the various diffusing species involved in the reaction may affect the rate of absorption. 

Pore dijfusion in fluid-filled pores. These pores are sufficiently large that the adsorbing moleciile escapes the force field of the adsorbent surface. Thus, this process is often referred to as macropore dijfusion. The driving force for such a diffusion process can be approximated by the gradient in mole fraction or, if the molar concentration is constant, by the gradient in concentration of the diffusing species within the pores. 

Intraparticle convection can also occur in packed beds when the adsorbent particles have very large and well-connected pores. Although, in general, bulk flow through the pores of the adsorbent particles is only a small frac tion of the total flow, intraparticle convection can affec t the transport of veiy slowly diffusing species such as macromolecules. The driving force for convec tion, in this case, is the... 

The procedure in use here involves the deposition of a radioactive isotope of the diffusing species on the surface of a rod or bar, the length of which is much longer than tire length of the metal involved in the diffusion process, the so-called semi-infinite sample solution. 

Darken assumed that the accumulated vacancies were annilrilated within the diffusion couple, and that during tlris process, tire markers moved as described by Smigelskas and Kirkendall (1947). His analysis proceeds with the assumption tlrat the sum of tire two concenuations of the diffusing species (cq - - cq) remained constant at any given section of tire couple, and tlrat the markers, which indicated the position of tire true interface moved with a velocity v. 

When die conceiiuation, c, and diffusion coefficient, D of die diffusing species in die reaction product are constant, dieii die rate of growdi of die product in diickness will be given by die simple equation... 

A number of metals, such as copper, cobalt and h on, form a number of oxide layers during oxidation in air. Providing that interfacial thermodynamic equilibrium exists at the boundaries between the various oxide layers, the relative thicknesses of the oxides will depend on die relative diffusion coefficients of the mobile species as well as the oxygen potential gradients across each oxide layer. The flux of ions and electrons is given by Einstein s mobility equation for each diffusing species in each layer... 

When cations are the diffusing species (Fig. 1.81a), metal is consumed either by solution in the oxide as interstitial cations and electrons... 

Recall that Nemstian behavior of diffusing species yields a r1 /2 dependence, hi practice, the ideal behavior is approached for relatively slow scan rates, and for an adsorbed layer that shows no intermolecular interactions and fast electron transfers. 

The Chapman-Enskog method has been used to solve for steady state tracer diffusion (. ). According to the method the singlet distribution function for the diffusing species 1, present In a trace amount n nj, 1 1) In an otherwise equilibrium fluid. Is approximated by... 

Where F is the flux of diffusing species and dc/dx is its gradient of concentration c in the x direction. From the equation of continuity,... 

Such a reaction would occur if we exposed a metal surface to either oxygen or chlorine. A MX film would build up on the metal surface and growth of a film would occur by diffusion. In the initial description, we ignored vacancy and interstltleil diffusion and presented only the charged particles, M + and 0= as the diffusing species (see section 4.5.). In actuality, the metal diffuses as the interstitial, Mi2+, and the anion as Oi=. ... 

Ba2+wlll be very fast while the silicate ion will diffuse very slow (if at all). Because of the vast differences in the types of diffusing species, there is no reason to expect all of them to diffuse at the same rate, particularly when we compare electrons and vacancies. Actually, this aspect of solid state reaction has been studied in great detcdl and the Kirchendall Effect deals with this aspect. 

Calcium oxide, in proper proportion, reacts with silica to form calcium orthosilicate. In terms of the spinel case, we would expect to see the following diffusing species ... 

In this case, we have two concomitant materials, CaO and Si02, reacting together to form the compound, calcium orthosilicate, which exists as a phase boundary between the five diffusing species. We can h)T)othesize at least three cases involving diffusion. In the first case, both Ca2+ uid 0= diffuse together in the same direction. 

These tetreihedra are tied together at the corners so that a silicate "backbone" forms the structure. The metal cations form "bridges" between backbone-layers and are much more free to move. However, it is well to note that a small amount of silicate does move, but the exact nature of the diffusing specie cannot be quantitatively defined (It may depend upon the nature of the compounds being formed. Most probably, the diffusing specie is actually SiOn but the charge of each actual specie may vary). In... 

Here, we have reacted our diffusing species, Ca2+ and O-, with the initial product of our reaction, Ca2Si04 in one direction, and Si and 0= in the... 

Thus, the final possible reaction is that which forms the metasilicate stoichiometry. In the opposite direction, we also have the diffusing species, Si and 0=. TTiese react with the nearest phase to give CaSi04, as shown in the following diagram ... 

Obviously, the other side must be protected during the process to form one or the other active layer. Clearly, the silicon disc needs to be heated as well during the process to aid the diffusion process. Note that the surface will be rich in diffusing species and that the density of species declines within the mterior, forming a diffuse layer which is dense near the top and thinner in the interior of the silicon. What happens is that... 

It is important to note that the dilfnsion-layer thickness depends not only on hydro-dynamic factors but also (through the diffusion coefficient) on the nature of the diffusing species. This dependence is minor, of course, since the values of Dj differ little among the various substances, and in addition are raised to the power one-third in Eq. (4.37). 

Fig. 5. Charge separation a at an interface with restricted transport and b with freely diffusing species...

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