Jt = - A grad c, - - ^ Ci in .NET

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Jt = - A grad c, - - ^ Ci
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For a single electrolyte, Eqs (2.5.37) and (2.3.14) for y = 0 and Eqs (2.4.4) and (2.4.5) yield the equation
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where D+ and D_ are the diffusion coefficients of the cation and anion. The overall material flux of the electrolyte J = = (2.5.39) v + v_ where v+ and v_ are the stoichiometric coefficients of the anion and of the cation in the salt molecule. From Eqs (2.5.37) to (2.5.39) we then obtain Z)+D_(v+ + v_) v_D+ + v + D_ 5
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Diffusion of a single electrolyte ('salt') is thus characterized by an effective diffusion coefficient
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Replacement of the diffusion coefficients by the electrolytic mobilities according to Eq. (2.3.22) yields the Nernst-Hartley equation:
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It could be expected for less dilute solutions that an expression for the effective diffusion coefficient could be derived theoretically from Eq. (2.5.42) by using correction terms for U+ and U_ taken from the Debye-Huckel-Onsager theory of electrolyte conductivities. However, it has been demonstrated experimentally that this approach does not lead to a correct description of the dependence of the diffusion coefficient on the concentration. The mobility of ions in the diffusion process varies far less with changes in the concentration than when charge is transported in an external electric field, and the effect of the concentration on the mobility can be either retarding, zero or accelerating, depending on the type of salt (increasing concentration always reduces ion mobility in an external electric field). This difference between the two transport phenomena is a result of the fact that diffusion is connected with the movement of cations and anions in the same direction, so that faster species are retarded by slower species, and vice versa, whereas during electric current flow, oppositely charged ions move in opposite directions and the two types of ions retard one another. The electrophoretic effect caused by the retardation of the movement of the central ion by the moving ionic atmosphere is thus of a different magnitude in diffusion. The time-of-relaxation effect is completely absent, as the symmetry of the ionic atmosphere is not disturbed during diffusion. As mentioned, the gradient of the diffusion electric potential is suppressed in the case of diffusion of ions present in a low concentration in an excess of indifferent electrolyte ('base electrolyte'). Under these conditions, the simple form of Fick's law (2.3.18) holds for the diffusion of the given ion. The
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Table 2.4 Diffusion coefficients D x 106 (cm2 s l) determined by means of polarography or chronopotentiometry at various indifferent electrolyte concentrations c (mol dm"3) at 25 C. The composition of the indifferent electrolyte is indicated for each ion. (According to J. Heyrovsky and J. Kuta) Ag + c 0.01 0.1 1.0 3.0 KNO3 15.85 15.32 15.46 KNO3 18.2 16.5 Tl + KC1 17.4 15.7 13.5 NaCl 17.7 15.0 9.2 IO 3 NaOH 6.54 5.13 4.18 KC1 10.15 9.89 9.36 NaCl 10.01 8.92 7.24 Pb 2+ KNO3 8.76 8.28 8.02 KC1 8.99 8.67 9.20 8.14 Cd2+ KNO3 6.90 6.81 KC1 8.15 7.15 7.90 7.90 Fe(CN) 6 3 KC1 7.84 7.62 7.63 7.36
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Zn 2+ c 0.01 0.1 1.0 3.0 KNO3 6.60 6.38 6.20 KC1 6.76 6.73 7.23 7.69
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Fe(CN)6 KC1 6.50 6.50 6.20
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Table 2.5 Diffusion coefficients (cm2 s l x 10 5) of electrolytes in aqueous solutions at various concentrations c (mol dm~ 3 ). (According to H. A. Robinson and R. H. Stokes) Temperature Electrolyte LiCl NaCl KC1 KC1 KC1 RbCl CsCl LiNO, NaNO 3 KC1O4 KNO, AgNO 3 MgCl2 CaCl2 SrCl2 BaCl2 Li2SO4 Na 2 SO 4 Cs2SO4 MgSO4 ZnSO 4 LaCl3 K 4 Fe(CN) 6 ( C) 25 25 20 25 30 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 0 1.366 1.610 1.763 1.993 2.230 2.051 2.004 1.336 1.568 1.871 1.928 1.765 1.249 1.335 1.334 1.385 1.041 1.230 1.569 0.849 0.846 1.293 1.468 0.001 1.345 1.585 1.739 1.964 2.013 1.845 1.899 1.187 1.263 1.269 1.320 0.990 1.175 1.489 0.768 0.748 1.175 0.002 1.337 1.576 1.729 1.954 2.011 2.000 1.535 1.841 1.884 1.169 1.243 1.248 1.298 0.974 1.160 1.454 0.740 0.733 1.145 c 0.003 1.331 1.570 1.722 1.945 2.174 2.007 1.992 1.296 1.835 1.879 1.719 1.158 1.230 1.236 1.283 0.965 1.147 1.437 0.727 0.724 1.126 1.213 0.005 1.323 1.560 1.708 1.934 2.161 1.995 1.978 1.289 1.516 1.829 1.866 1.708 1.213 1.219 1.265 0.950 1.123 1.420 0.710 0.705 1.105 1.184 0.007 1.318 1.555 1.925 2.152 1.984 1.969 1.283 1.513 1.821 1.857 1.698 1.201 1.209 1.084 0.010 1.312 1.545 1.692 1.917 2.144 1.973 1.958 1.276 1.503 1.790 1.846 1.188
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diffusion coefficient, of course, depends on the concentration of the indifferent electrolyte. A similar situation occurs in tracer diffusion. This type of diffusion occurs for different abundances of an isotope in a component of the electrolyte at various sites in the solution, although the overall concentration of the electrolyte is identical at all points. Since the labelled and the original ions have the same diffusion coefficient, diffusion of the individual isotopes proceeds without formation of the diffusion potential gradient, so that the diffusion can again be described by the simple form of Fick's law. Experimental methods for determining diffusion coefficients are described in the following section. The diffusion coefficients of the individual ions at infinite dilution can be calculated from the ionic conductivities by using Eqs (2.3.22), (2.4.2) and (2.4.3). The individual diffusion coefficients of the ions in the presence of an excess of indifferent electrolyte are usually found by electrochemical methods such as polarography or chronopotentiometry (see Section 5.4). Examples of diffusion coefficients determined in this way are listed in Table 2.4. Table 2.5 gives examples of the diffusion coefficients of various salts in aqueous solutions in dependence on the concentration.
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118 2.5.5 Methods of measurement of diffusion coefficients Both steady-state and non-steady-state methods are used in the determination of diffusion coefficients. In the steady-state methods (dc/dt = 0) the flux J and the concentration gradient dc/Sx are measured directly and the diffusion coefficient D is calculated according to Fick's first law, often without assuming that D is constant. In non-steady-state methods, all the parameters of diffusion flux change with position and time and thus equations resulting from integration of Fick's second law must be employed with various boundary conditions, depending on the experimental arrangement. In integration, when necessary, a suitable form of the function D(x, t) must be assumed. Usually the diffusion coefficient is assumed to be constant and thus the final integrated equations can be used only for small concentration ranges in which the change in the diffusion coefficient can be neglected. A steady-state method. In the diaphragm method the diffusion process is restricted to a porous diaphragm (a glass frit with a pore size of about lOjUm, separating compartments 1 and 2 of volumes V1 and V2 (cf. Eq. (2.5.18). The diaphragm is the actual diffusion space with an effective cross-section of pores A and an effective length /. The compartments are filled with solutions of concentrations c and c2 at the start of the experiment, and are thoroughly stirred during the experiment. After a short time the solutions penetrate into the pores of the diaphragm and the solution diffuses from 1 to 2 if c\>c\. In the steady state there is no accumulation of the solute in the pores of the diaphragm, i.e. the same amount of solute which leaves the compartment 1 in a certain time interval passes into the compartment 2. The flux of the substance is constant through the entire thickness of the diaphragm but it decreases with time because the concentration difference between compartments 1 and 2 decreases. After the experiment has been running for a certain time, the contents of both compartments are analysed and the concentrations c[ and c2 are determined. From these data we may compute the average value of the diffusion coefficients between the concentrations (c + c[)/2 and (c2 + c2)/2. The time dependence of the concentration in 1 and 2 is obtained by solving the differential equations
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