Diffusivities concentrated solutions

Diffusion and Mass Transfer During Leaching. Rates of extraction from individual particles are difficult to assess because it is impossible to define the shapes of the pores or channels through which mass transfer (qv) has to take place. However, the nature of the diffusional process in a porous soHd could be illustrated by considering the diffusion of solute through a pore. This is described mathematically by the diffusion equation, the solutions of which indicate that the concentration in the pore would be expected to decrease according to an exponential decay function. 

At very low concentrations of water, or in foods held below the free2ing point of water, physical conditions may be such that the available water may not be free to react. Under these conditions, the water may be physically immobi1i2ed as a glassy or plastic material or it may be bound to proteins (qv) and carbohydrates (qv). The water may diffuse with difficulty and thus may inhibit the diffusion of solutes. Changes in the stmcture of carbohydrates and proteins from amorphous to crystalline forms, or the reverse, that result from water migration or diffusion, may take place only very slowly. 

R is rate of reaction per unit area, a is interfacial area per unit volume, S is solubiHty of solute in continuous phase, D is diffusivity of solute, k is rate constant, kj is mass-transfer coefficient, is concentration of reactive species, and Z is stoichiometric coefficient. When Dk is considerably greater (10 times) than Ra = aS Dk. 

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. 

In many process design applications like polymerization and plasticization, specific knowledge of the thermodynamics of polymer systems can be very useful. For example, non-ideal solution behavior strongly governs the diffusion phenomena observed for polymer melts and concentrated solutions. Hence, accurate modeling of... 

Mechanistically, in approximately neutral solutions, solid state diffusion is dominant. At higher or lower pH values, iron becomes increasingly soluble and the corrosion rate increases with the kinetics approaching linearity, ultimately being limited by the rate of diffusion of iron species through the pores in the oxide layer. In more concentrated solutions, e.g. pH values of less than 3 or greater than 12 (relative to 25°C) the oxide becomes detached from the metal and therefore unprotective . It may be noted that similar Arrhenius factors have been found at 75 C to those given by extrapolation of Potter and Mann s data from 300°C. 

Equation 10.96 does not apply to either electrolytes or to concentrated solutions. Reid, PRAUSNITZ and Sherwood"7 discuss diffusion in electrolytes. Little information is available on diffusivides in concentrated solutions although it appears that, for ideal mixtures, the product /xD is a linear function of the molar concentration. 

The beauty of the reptation model is that it is able to make predictions about molecular flow both in solution and at fracture by assuming that the molecules undergo the same kind of motions in each case. For both self-diffusion in concentrated solutions and at fracture, the force to overcome in pulling the polymer molecule through the tube is assumed to be frictional. 

The mobility ratio equal to the diffusion ratio in this equation would naturally follow from application of the Nemst-Einstein equation, Eq. (88), to transport gels. Since the Nemst-Einstein equation is valid for low-concentration solutes in unbounded solution, one would expect that this equation may hold for dilute gels however, it is necessary to establish the validity of this equation using a more fundamental approach [215,219]. (See a later discussion.) Morris used a linear expression to fit the experimental data for mobility [251]... 

The various physical methods in use at present involve measurements, respectively, of osmotic pressure, light scattering, sedimentation equilibrium, sedimentation velocity in conjunction with diffusion, or solution viscosity. All except the last mentioned are absolute methods. Each requires extrapolation to infinite dilution for rigorous fulfillment of the requirements of theory. These various physical methods depend basically on evaluation of the thermodynamic properties of the solution (i.e., the change in free energy due to the presence of polymer molecules) or of the kinetic behavior (i.e., frictional coefficient or viscosity increment), or of a combination of the two. Polymer solutions usually exhibit deviations from their limiting infinite dilution behavior at remarkably low concentrations. Hence one is obliged not only to conduct the experiments at low concentrations but also to extrapolate to infinite dilution from measurements made at the lowest experimentally feasible concentrations. 

It also follows from what was said that a zeta potential will be displayed only in dilute electrolyte solutions. This potential is very small in concentrated solutions where the diffuse edl part has collapsed against the metal surface. This is the explanation why electrokinetic processes develop only in dilute electrolyte solutions. 

The GC-MS data of fraction 1 revealed a strong peak of verticilla-4(20),7,ll-triene (compound 1) accompanied by small amounts of cembrane A and cembrane C. To purify the violet spot and isolate compound 1, it was necessary to reduce the solvent strength. In the mobile phase dichloromethane-hexane (9 + 1 v/v), the development time decreases, which leads to minor diffusion of the zone. The zone of (compound 1) was marked by X = 254 nm UV light. To exclude the impurities, the separation process had to be repeated several times. The zone was removed from the glass plate and eluted from die adsorbent with dichloromethane. The concentrated solution achieved was applied onto a TLC plate as well as injected onto a GC column the... 

To describe the diffusion of solutes in the rhizosphere, where concentration gradients change with time, /, as well as space, mass conservation is invoked with the spatial geometry appropriate for the cylindrical root (8) ... 

Diffusion in solution is the process whereby ionic or molecular constituents move under the influence of their kinetic activity in the direction of their concentration gradient. The process of diffusion is often known as self-diffusion, molecular diffusion, or ionic diffusion. The mass of diffusing substance passing through a given cross section per unit time is proportional to the concentration gradient (Fick s first law). 

Introduction of a water-soluble ionic substance into the vascular system results in an increase in the number of particles in the bloodstream as the contrast substance dissolves. The body possesses several internal regulation systems and, when perturbed by an injection, attempts to restore the concentrations of substances in the blood to their normal or preinjection levels. To re-equilibrate the system, water from the cells of surrounding body tissue moves into the blood plasma through capillary membranes. This transfer of water is an example of osmosis, the diffusion of a solvent (water) through a semipermeable membrane (the blood vessels) into a more concentrated solution (the blood) to equalize the concentrations on both sides of the membrane. To accommodate the increase in... 

It can be seen from the table that, in dilute solutions, the diffuse layer may extend some hundreds of angstroms out from the electrode. In contrast, in more concentrated solutions, i.e. 0.1 M, the diffuse layer thickness decreases to < 10 A not much more than the thickness of the Helmholtz layer. As CH has no concentration dependence it remains constant on changing the concentration however, from equations (2.22) and (2,23), CGC decreases as the concentration of the electrolyte increases. Thus, at low concentration ... 

The diffusion layer theory, illustrated in Fig. 15B, is the most useful and best-known model for transport-controlled dissolution. The dissolution rate here is controlled by the rate of diffusion of solute molecules across a diffusion layer of thickness h, so that kT kR in Eq. (40), which simplifies to kx = kT. With increasing distance, x, from the surface of the solid, the concentration, c, decreases from cs at x = 0 to cb at x = h. In general, c is a nonlinear function of x, and the concentration gradient dddx becomes less steep as x increases. The hyrodynamics of the dissolution process has been fully discussed by Levich [104]. In a stirred solution, the flow velocity of the liquid dissolution medium increases from zero at x = 0 to the bulk value at x = h. 

B/Si is the molecular diffusion term and relates to diffusion of solute molecules within the mobile phase caused by local concentration gradients. Diffusion within the stationary phase also contributes to this term, which is significant only at low flow rates and increases with column length. As B is proportional to the diffusion coefficient in the mobile phase, the order of efficiency at low flow rates is liquids > heavy gases > light gases. 

We have already met the concept of error propagation a few times when dealing with the change of variable formulas for probability distribution, but let us try to illustrate it with a simple example. We want to measure the diffusion coefficient Q) of uranium in a glass by maintaining at a specific temperature and for a specific time t the surface of one long glass rod in contact with a concentrated solution of uranium. We admit without further justification (see Section 8.5) that the depth x of uranium... 

---