Diffusion in electrolyte systems

Despite the low value of the separation factor, ultracentrifugation is a viable commercially used technique for separation of the isotopes of uranium. In view of the small separation factor and low capacity per unit, a commercial plant will have a few million centrifuges (Von Halle, 1980 Voight, 1982)  

There are many applications in chemical engineering where diffusion of charged species is involved. Examples include ion exchange, metals extraction, electrochemical reactors, and membrane separations. There is an excellent textbook in this area (Newman, 1991). Here we will be content to show that the treatment of electrolyte diffusion follows naturally from the generalized treatment of diffusion given in Section 2.3. 

In mixtures of electrolytes the generalized Maxwell-Stefan equations 

The in these equations denote ionic mole fractions. In general, the ionic mole fractions will differ from the undissociated electrolyte mole fractions. To illustrate this fact consider an aqueous solution of sulfuric acid. Let us take 1 m of solution with kmol of H2SO4 and kmol of H2O. The mole fraction of the undissociated species are 

Note that the mole fraction of H2O has decreased when considered in terms of ionic species. This is an important point to bear in mind and the reader is advised to study Newman (1991) for further discussion. For mixed ion systems there will be contributions to Ci from various ionic species. For example, in the system with mixed salts HCl and BaCl2 the concentration of chloride ion, Cq-= Chci + Baci2 Example 2.4.2). 

For electrostatic potentials and electric current of charged ionic species, we start with the fundamental Gibbs equation 

Equation (6.119) indicates that the chemical work in electrolytes contains a chemical term fidNj and an electrical term ZjFipdNj and the sum is called the electrochemical potential jlI of the ionic species i 

If we have a phase in which the composition is identical at points 1 and 2 but t/q P //2, then we have 

We may also use the change in the Gibbs free energy in terms of the chemical potential 

It is often useful to express the electrochemical potential as a sum of explicit terms of activity and electrostatic potential as follows 

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As in the treatment of diffusion in nonionic systems it is usual to define diffusion fluxes Ji with respect to a specified reference velocity. For diffusion in electrolyte systems the most commonly used reference velocity is the solvent velocity... 

Figure 2.9. Schematic diagram of a two compartment diffusion cell. The experiments by Vinograd and McBain (1941) on diffusion in electrolyte systems were carried out in apparatus of this kind.

Chapter 1 serves to remind readers of the basic continuity relations for mass, momentum, and energy. Mass transfer fluxes and reference velocity frames are discussed here. Chapter 2 introduces the Maxwell-Stefan relations and, in many ways, is the cornerstone of the theoretical developments in this book. Chapter 2 includes (in Section 2.4) an introductory treatment of diffusion in electrolyte systems. The reader is referred to a dedicated text (e.g., Newman, 1991) for further reading. Chapter 3 introduces the familiar Fick s law for binary mixtures and generalizes it for multicomponent systems. The short section on transformations between fluxes in Section 1.2.1 is needed only to accompany the material in Section 3.2.2. Chapter 2 (The Maxwell-Stefan relations) and Chapter 3 (Fick s laws) can be presented in reverse order if this suits the tastes of the instructor. The material on irreversible thermodynamics in Section 2.3 could be omitted from a short introductory course or postponed until it is required for the treatment of diffusion in electrolyte systems (Section 2.4) and for the development of constitutive relations for simultaneous heat and mass transfer (Section 11.2). The section on irreversible thermodynamics in Chapter 3 should be studied in conjunction with the application of multicomponent diffusion theory in Section 5.6. 

Diffusion in electrolyte systems 323 6.3.1 Phenomenological approach in electrolyte systems... 

Several topics in diffusion have arbitrarily been excluded from the following discussion diffusion in electrolytic solutions (F3, G9, HI, 02, R3, Yl), diffusion in ionized gases (K4), diffusion in macromolecular systems (W2), diffusion through membranes (F14), use of diffusional techniques in isotopic separations (S18), diffusion in metals (S8), and neutron diffusion (F2, G5, H15, W15). 

Churchill, London (1946), Chapter 1 8) H.S, Harned, ChemRevs 40, 461-522 (1947) (Quantitative aspect of diffusion in electrolytic solutions) 9) R.B. Dean, ChemRevs 41, 503-23(1947) (Effects produced by diffusion in aqueous systems containing membranes) 10) D.A. Hougen K.M. Watson, "Chemical Process principles , Part 3, "Kinetics Catalysts , Wiley, NY (1947), Chap 20 11) Perry (1950), pp 522-59 (by... 

In contrast to the situation for gases, there are no satisfactory theoretical methods for predicting diffusivities in liquid systems. Different approaches are needed, depending on whether the solutions are electrolytic or nonelectrolytic. Most studies have been devoted to the estimation of diffusion coefficients in very dilute solutions. However, some papers report substantial variations with increasing concentrations of the diffusing solute. The theories and the experimental methods available for estimating diffusivities in liquids are well reviewed by Kamal and Caqjar (Kl), Nienow (N8), Bretsznajder (B24), Tyn (T12), Dullien et al. (E4, G6), and Simons and Ponter (S29). 

Equations 2.3.18 together with Eqs. 2.3.10 defining the generalized driving force are the starting point for the analysis of diffusion in systems where external force fields influence the process the ultracentrifuge, for example, in electrolyte systems and in porous media where pressure gradients become important. We examine the first two of these topics in the Sections 2.3.3 and 2.4. 

Consider diffusion in the system KCl - H2O at 25°C. Potassium chloride is a strong electrolyte and complete dissociation into K and CD ions will take place ... 

In electrolyte systems the components diffiise as ionic species and the effictive ionic diffiisivities are related to the nature and concentration of the other ions present. Taylor and Krishna [1993] give as an exanple the diffiision of an aqueous solution of HCl and BaCl2. The infinite dilution diffusion coefficients for the individual ions in water are ... 

The study of diffusion processes of electrolytes and non-electrolytes in aqueous solutions is important for fundamental reasons, helping to imderstand the nature of aqueous electrolyte stmcture, for practical applications in fields such as corrosion, and provide transport data necessary to model diffusion in pharmaceutical applications. Although no theory on diffusion in electrolyte or non-electrolyte solutions is capable of giving generally reliable data onO, there are, however, estimating pmposes, whose data, when compared with the experimental values, will allow us to take off conclusions on the nature of the system. 

Although being one of the key determinants for the power output, very few diffusion data in liquid electrolytes are available. This has not changed since 1947, when Harned wrote There are few domains of physical science in which so much experimental effort over nearly a century has yielded so little accurate data as the field of diffusion in liquid systems [480]. 

The molten salt electrolyte also contributes to the safety behavior of ZEBRA cells. The large amount of energy stored in a 700 g cell, which means about 30 kWh in a 300 kg battery, is not released suddenly as heat as be expected in a system with liquid electrodes such as the sodium sulfur cell. In the case of accidental destruction of ZEBRA cells, the sodium will react mainly with the molten salt, forming A1 sponge and NaCl. -The diffusion of the NaAICI ... 

Here, / is the electric field, k is the electrical conductivity or electrolytic conductivity in the Systeme International (SI) unit, X the thermal conductivity, and D the diffusion coefficient. is the electric current per unit area, J, is the heat flow per unit area per unit time, and Ji is the flow of component i in units of mass, or mole, per unit area per unit time. 

It was shown in Section 1.8 that in addition to ion migration, diffusion and convection fluxes are a substantial part of mass transport during current flow through electrolyte solutions, securing a mass balance in the system. In the present chapter these processes are discussed in more detail. 

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