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In convective diffusion to a rotating disk, the characteristic velocity Vo is given by the product of the disk radius r, as a characteristic dimension of the system, and the radial velocity a>, so that the Reynolds number is given by the equation Re = (2.7.32)
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It follows from Eq. (2.7.29) for / = r, and Eqs (2.7.13), (2.7.27) and (2.7.32) that Sh = 0.62 Re1/2Sc1/3 (2.7.33) When the relationship between the material flux and the parameters of the system can be calculated directly by solution of the appropriate differential equations, the criterion equation (2.7.30) has little significance. However, this is not possible in the great majority of practical systems, and thus the empirically determined criterial equation is of general validity for physically similar systems. It can form a basis for designing larger equipment on the basis of experiments with model systems. References
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Agar, J. N., Diffusion and convection at electrodes, Faraday Soc. Discussions, 1, 26 (1947). Cussler, E. L., see page 20. Heyrovsky, J., and J. Kuta, Principles of Polarography, Academic Press, New York, 1966. Ibl, N., and O. Dossenbach, Convective mass transport, CTE, 6, 133 (1983). Levich, V. G., see page 20. Rousar, I., K. Micka, and A. Kimla, Electrochemical Engineering, Elsevier, Amsterdam, Vol. 1, 1985, Vol. 2, 1986.
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3 Equilibria of Charge Transfer in Heterogeneous Electrochemical Systems
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Again, the consideration of the electrical potential in the electrolyte, and especially the consideration of the difference of potential in electrolyte and electrode, involve the consideration of quantities of which we have no apparent means of physical measurement, while the difference of potential in pieces of metal of the same kind attached to the electrodes is exactly one of the things which we can and do measure. J. W. Gibbs, 1899 3.1 Structure and Electrical Properties of Interfacial Regions
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In contrast to the previous chapters, dealing with the electrochemical properties of a single phase, this and the next three chapters will be concerned with electrochemical systems consisting of two or more phases in contact, at least one of which is an electronic or electrolytic conductor. The second phase may be either another electrolytic conductor (this case will be considered the most extensively), or another electronic conductor, a dielectric or a vacuum. Two phases in contact are separated by a surface called a phase boundary or interface (Fig. 3.1). The interface is unambiguously defined, for example, in the case of a metal-electrolyte solution interface. Surprisingly enough, even in the case of a liquid-liquid interface where the liquids are partially miscible the transition from one phase to the other is rather sharp. The term 'interface' will be distinguished from a related term, the interfacial region or interphase. This term denotes the region between the two phases where the properties vary markedly in contrast to those in the bulk of the phases. In the case of electrically conducting phases, charge distribution occurs in this region. The main criterion of the electrochemical properties of the interface is whether electric charge can pass sufficiently rapidly across it in both directions, or whether it is impermeable to charge transfer and thus has the 144
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Fig. 3.1 Structure of the interphase
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properties of a capacitor without leakage. For the first type of interface, the electrical potential difference between the two phases is a result of charge transfer across the interface; these interfaces are termed non-polarizable. In this case, charge transfer involves the transfer of electrons from a metal conductor to an electron acceptor in the second phase, the emission of electrons into a liquid phase with formation of solvated electrons, the transfer of ions from one phase into the other and similar processes occurring in the opposite direction. In the second situation, charge distribution occurs through charging of the two phases with opposite charges from an external source, unequal adsorption of ions with opposite charges on the two sides of the interface (including purely electrostatic attraction and repulsion of ions at the boundary between oppositely charged phases), adsorption and orientation of molecular dipoles and polarization of species in the inhomogeneous force field in the interfacial region. Such an interface is termed polarizable. The term electrical (electrochemical) double layer is often used for the corresponding interface. An electrical double layer is, of course, present even at non-polarizable interfaces. In this case, however, it cannot be charged (i.e. charge separation cannot be attained) by introduction of charge from an external source, but only through charge transfer across the interface. The electrical double layer will be dealt with in 4. This chapter will include equilibria at non-polarizable interfaces for a metal or semiconductor phase-electrolyte system (a galvanic cell in the broadest sense) and for two electrolytes (the solid electrolyte-electrolyte solution interface, or that between two immiscible electrolyte solutions).
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