Transport of matter

A membrane is defined as an intervening phase separating two phases forming an active or passive barrier to the transport of matter. Membrane processes can be operated as (1) Dead-end filtration and (2) Cross-flow filtration. Dead-end filtration refers to filtration at one end. A problem with these systems is frequent membrane clogging. Cross-flow filtration overcomes the problem of membrane clogging and is widely used in water and wastewater treatment. 

If a liquid system containing at least two components is not in thermodynamic equilibrium due to concentration inhomogenities, transport of matter occurs. This process is called mutual diffusion. Other synonyms are chemical diffusion, interdiffusion, transport diffusion, and, in the case of systems with two components, binary diffusion. 

From the applications point of view, mutual diffusion is far more important than self-diffusion, because the transport of matter plays a major role in many physical and chemical processes, such as crystallization, distillation or extraction. Knowledge of mutual diffusion coefficients is hence valuable for modeling and scaling-up of these processes. 

Introduction.—Statistical physics deals with the relation between the macroscopic laws that describe the internal state of a system and the dynamics of the interactions of its microscopic constituents. The derivation of the nonequilibrium macroscopic laws, such as those of hydrodynamics, from the microscopic laws has not been developed as generally as in the equilibrium case (the derivation of thermodynamic relations by equilibrium statistical mechanics). The microscopic analysis of nonequilibrium phenomena, however, has achieved a considerable degree of success for the particular case of dilute gases. In this case, the kinetic theory, or transport theory, allows one to relate the transport of matter or of energy, for example (as in diffusion, or heat flow, respectively), to the mechanics of the molecules that make up the system. 

The mechanisms by which this interaction occurs may be divided into two distinct groups (S4) first, the hydrodynamic behavior of a multiphase system can be changed by the addition of surface-active agents, and, as a result, the rate of mass transfer is altered secondly, surface contaminants can interfere directly with the transport of matter across a phase boundary by some mechanism of molecular blocking. 

Diffusion—transport of matter as a result of differing values of the chemical potential of a given component at various sites within the system, or in the system and its surroundings. Obviously, the particles... 

In Chapter 11, we indicated that deviations from plug flow behavior could be quantified in terms of a dispersion parameter that lumped together the effects of molecular diffusion and eddy dif-fusivity. A similar dispersion parameter is usefl to characterize transport in the radial direction, and these two parameters can be used to describe radial and axial transport of matter in packed bed reactors. In packed beds, the dispersion results not only from ordinary molecular diffusion and the turbulence that exists in the absence of packing, but also from lateral deflections and mixing arising from the presence of the catalyst pellets. These effects are the dominant contributors to radial transport at the Reynolds numbers normally employed in commercial reactors. 

The equation describing the steady-state material balance for tubular packed bed reactors can be obtained from the more general relation (12.7.28) by omitting the terms corresponding toTadial transport of matter. Hence the material balance relation becomes... 

Experimental determination of Ay for a reaction requires the rate constant k to be determined at different pressures, k is obtained as a fit parameter by the reproduction of the experimental kinetic data with a suitable model. The data are the concentration of the reactants or of the products, or any other coordinate representing their concentration, as a function of time. The choice of a kinetic model for a solid-state chemical reaction is not trivial because many steps, having comparable rates, may be involved in making the kinetic law the superposition of the kinetics of all the different, and often unknown, processes. The evolution of the reaction should be analyzed considering all the fundamental aspects of condensed phase reactions and, in particular, beside the strictly chemical transformations, also the diffusion (transport of matter to and from the reaction center) and the nucleation processes. 

We have already discussed ion association in Section 6.2. In that section we referred to evidence for the existence of ion clusters from static techniques such as IR, Raman, EXAFS and X-ray diffraction. In this section we examine ion association from the point of view of dynamics, concentrating in particular on electrochemical measurements which reveal the presence of ion clusters. Because ion association is so intimately connected to the transport of matter and charge through polymer electrolytes, it seems appropriate to consider these two topics in the same section. 

The fundamental processes involved in a solid state reaction are twofold. First, there is the reaction itself - the breaking and forming of bonds. Second, there is the transport of matter to the reaction zone. A number of models aiming to describe solid state reactions exist. They are generally based on sigmoidal kinetic curves. The general form of the kinetic equation is as follows ... 

Advection The large-scale mass transport of matter. 

Flux The transport of matter or energy through a given surface area or volume in a given unit of time. 

The lipid bilayer forms a barrier to transport of matter into and out of the cell. This barrier function is essential since cells need to be able to control their internal milieu, regardless of the external environment. (Some antibiotics work by disrnpting the barrier function of bacterial membranes see Chapter 23). At the same time, some communication of signals and materials across the bilayer must occur. Special mechanisms to do this are a key property of biological membranes. More specifically, these mechanisms are the province of proteins that one finds in these membranes. 

These waves correspond to a transport of matter in the system. The flux of X at a given point in space on the average over one period is of the form... 

Dr. Sanfeld refers to phenomena that concern membranes and surface behaviors rather than bulk solutions whether such behaviors play a role in the motion of fluids in cells remains an open question. On the other hand, the coupling of chemical and hydrodynamic effects is not a necessary condition to the existence of rotations and macroscopic transport of matter. We have seen in the lecture of Professor Prigogine that such effects can result more simply from the coupling of chemical reactions with transport phenomena like diffusion. 

Anisothermal Homogeneous Diffusion. Using the reasonable simplifications that the flow of heat is much faster than the transport of matter and that thermal kinetic effects can be neglected, we can dispense with the effect of changing temperature on diffusion within a phase simply by using a reduced time r = tD/R2. Use of a reduced radius p = r/R... 

Such transformations have been extensively studied in quenched steels, but they can also be found in nonferrous alloys, ceramics, minerals, and polymers. They have been studied mainly for technical reasons, since the transformed material often has useful mechanical properties (hard, stiff, high damping (internal friction), shape memory). Martensitic transformations can occur at rather low temperature ( 100 K) where diffusional jumps of atoms are definitely frozen, but also at much higher temperature. Since they occur without transport of matter, they are not of central interest to solid state kinetics. However, in view of the crystallographic as well as the elastic and even plastic implications, diffusionless transformations may inform us about the principles involved in the structural part of heterogeneous solid state reactions, and for this reason we will discuss them. 

When regions of dissimilar chemical potential are created, solute molecules will move between them until a homogeneous condition is restored. This relaxation process involves a temporary net mass transport across some imaginary plane. The transport of matter from a region of higher chemical potential to one of lower chemical potential is the process of diffusion. The motive force behind this movement is maximization of entropy. The fully relaxed system is in its most random configuration. 

In a batch reactor the reactants and the catalyst are placed in the reactor which is then closed to transport of matter and the reaction is allowed to proceed for a given time whereupon the mixture of unreacted material together with the products is withdrawn. Provision for mixing may be required. 

Thermodynamic systems are parts of the real world isolated for thermodynamic study. The parts of the real world which are to be isolated here are either natural water systems or certain regions within these systems, depending upon the physical and chemical complexity of the actual situation. The primary objects of classical thermodynamics are two particular kinds of isolated systems adiabatic systems, which cannot exchange either matter or thermal energy with their environment, and closed systems, which cannot exchange matter with their environment. (The closed system may, of course, consist of internal phases which are each open with respect to the transport of matter inside the closed system.) Of these, the closed system, under isothermal and iso-baric conditions, is the one particularly applicable for constructing equilibrium models of actual natural water systems. 

Two objections to this discussion may be expected. First, it is well-known that there is no true equilibrium in the ocean. Why then bother to talk about this equilibrium model I agree that there is no complete equilibrium in the oceans and that everything interesting we observe is caused by lack of equilibrium. Nevertheless, it may be worthwhile to try to determine what the equilibrium model would look like—i.e., what the solid phases and composition of the solution would be. Perhaps the equilibrium model will be a useful first approximation the next step would then be to discuss how it is disturbed by various processes radiation, life, transport of matter, etc. Finally, the model may be proved useless, but this has not yet been done, and perhaps one would learn many interesting things in refuting it. 

What problems face the theory of combustion The theory of combustion must be transformed into a chapter of physical chemistry. Basic questions must be answered will a compound of a given composition be combustible, what will be the rate of combustion of an explosive mixture, what peculiarities and shapes of flames should we expect We shall not be satisfied with an answer based on analogy with other known cases of combustion. The phenomena must be reduced to their original causes. Such original causes for combustion are chemical reaction, heat transfer, transport of matter by diffusion, and gas motion. A direct calculation of flame velocity using data on elementary chemical reaction events and thermal constants was first carried out for the reaction of hydrogen with bromine in 1942. The problem of the possibility of combustion (the concentration limit) was reduced for the first time to thermal calculations for mixtures of carbon monoxide with air. Peculiar forms of propagation near boundaries which arise when normal combustion is precluded or unstable were explained in terms of the physical characteristics of mixtures. 

It remains to be shown that the reaction as measured here is not limited by diffusion. For this purpose Wheeler s test (37) is again useful. The result obviously depends somewhat on the amount of catalyst applied, but Wheeler s X j value was found to be between 4 X 10-4 sec.-1 (when 1 mg. Ni was used) and 4 X 10-3 sec.-1 (for 10 mg. Ni), while the rates measured were in the range of 10-5-10-4 sec.-1. Although in some experiments we have hence been approaching the conditions for diffusion limitation, the rates measured were in general true reaction rates. Obviously, by avoiding difficulties with transport of heat, difficulties connected with transport of matter were automatically taken care of. 

Diffusion, the basis of the solution-diffusion model, is the process by which matter is transported from one part of a system to another by a concentration gradient. The individual molecules in the membrane medium are in constant random molecular motion, but in an isotropic medium, individual molecules have no preferred direction of motion. Although the average displacement of an individual molecule from its starting point can be calculated, after a period of time nothing can be said about the direction in which any individual molecule will move. However, if a concentration gradient of permeate molecules is formed in the medium, simple statistics show that a net transport of matter will occur... 

Alan Allnatt s research interests at Western Ontario have been concerned with the statistical mechanics of the transport of matter through crystals. His earliest work centered on obtaining methods for calculating the equilibrium distributions and thermodynamic properties of the point defects (vacancies, interstitials, solutes) that make transport possible. He first studied dilute systems, so the methods could be largely analytical. The methods for ionic crystals,... 

A thermodynamic system is an arbitrary volume of matter without any transportation of matter across its surface. 

Because the flow of electric current always involves the transport of matter in solution and chemical transformations at the solution-electrode interface, local behavior can only be approached. It can be approximated, however, by a reference electrode whose potential is controlled by a well-defined electron-transfer process in which the essential solid phases are present in an adequate amount and the solution constituents are present at sufficiently high concentrations. The electron transfer is a dynamic process, occurring even when no net current flows and the larger the anodic and cathodic components of this exchange current, the more nearly reversible and nonpolarizable the reference electrode will be. A large exchange current increases the slope of the current-potential curve so that the potential of the electrode is more nearly independent of the current. The current-potential curves (polarization curves) are frequently used to characterize the reversibility of reference electrodes. 

Wall flux density (area-related collision rate). A measure of the rate of flow of matter from one part of a system to another is given by the flux (flow rate per unit area). For the transportation of matter, this is given by ... 

Knowledge of the concentration of defects and molar disturbance enthalpies would permit calculation of the actual free energy of the solid, and also the chemical potential. These can be measured by using either solution calorimetry or differential scanning calorimetry. An example of the excess energy was given as 20-30 kj mol-i in mechanically activated quartz. Different types of reactions demand different defect types. For example, Boldyrev et al. [25] state a classification and provide examples for solid reactions with different mechanisms and necessary solid alterations. Often, reaction rates in solids depend strongly on the mass transport of matter. Lidi-ard [26] and Schmalzried [27] each provide reviews on transport properties in mechanically treated solids. The increased amount of defects allows a faster transport of ions and atoms in the solid structure. 

An open system is one which can undergo all the changes allowed for a closed system and in addition it can lose and gain matter across its boundaries. An open system might be one phase in an extraction system, or it might be a small-volume element in an electrophoretic channel. Such systems, which allow for the transport of matter both in and out, are key elements in the description of separation processes. 

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