The Migration of Ions

If Rs is the resistance of the cell containing a solution of known conductivity then 

In precision work great care must be taken to eliminate effects due to electrolysis and those due to variation in temperature. Controlling the temperature is a particularly difficult problem because of the heating effect of the current. Water of extreme purity (conductivity water) must be used, since stray impurities in the water can produce sensible variations in the value of the conductivity of the solution. The contribution of water itself to the conductivity must be subtracted from the measured value for the solution. 

Kohlrausch established that electrolytic solutions obeyed Ohm s law accurately once the effect of the electrolysis products was eliminated by using high-frequency alternating current. Kjohlrausch also showed from the experimental data that the conductivity of a solution couId Be mposed of separate contributions from each ion this is known as Kohlrausch s law of the independent migration of ions. 

Consider an electrolyte with the formula, which is completely dissociated 

Let iV+ and iV be the number of positive and negative ions per cubic metre, respectively. Let their velocities be and v-, and their charges be z + and z e. Then by the fundamental law of transport, Eq. (30.11), the current density is 

Transference Numbers.—The quantity of electricity qi carried through a certain volume of an electrolytic solution by ions of the tth kind is proportional to the number in unit volume, i.e., to the concentration d in gram-ions or moles per liter, to the charge Zi carried by each ion, and to the mobility w , i.e., the velocity under unit potential gradient (cf. p. 58) thus 

This fraction is called the transference number, or transport number, 

The quantities and c-j5, which represent the equivalent concentrations of the ions, are equal, and hence for this type of electrolyte, which has been most frequently studied. 

The speed of an ion in a solution at any concentration is proportional to the conductance of the ion at that concentration (p. 80), and so the transference number may be alternatively expressed in the form 

Three methods have been generally employed for the experimental determination of transference numbers the first, based on the procedure originally proposed by Hittorf (1853), involves measurement of changes of concentration in the vicinity of the electrodes in the second, known as the moving boundary method, the rate of motion of the boundary between two solutions under the influence of current is studied (cf. p. 116) the third method, which will be considered in Chap. VI, is based on electromotive force measurements of suitable cells. 

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The term electrolyte is also used to denote a medium, such as a liquid solution or a porous solid, that can conduct electricity by the migration of ions. 

By definition the partial current density ij is the number of charges that in unit time cross the unit cross-sectional area due to the migration of ions j that is,... 

The task of a cell separator is to impede a direct mixing of anolyte and catholyte and to decrease diffusion, but at the same time the migration of ions has to be possible without a too high voltage drop. Naturally, a compromise of these requirements has to be found. 

With the study of the migration of hydrogenium ions (H ) in a phenolphthalein gel by Lodge in 1886 and the description of the migration of ions in saline solutions by Kohlraush in 1897, a basis was set for the development of a new separation technique that we know today as electrophoresis. Indeed, several authors applied the concepts introduced by Lodge and Kohlraush in their methods and when Arne Tiselius reported the separation of different serum proteins in 1937, the approach called electrophoresis was recognized as a potential analytical technique. Tiselius received the Nobel Prize in Chemistry for the introduction of the method called moving boundary electrophoresis. ... 

Electrophoresis, which is discussed in Chapter 26, is the migration of ions in solution under the influence of an electric field. Ion mobility spectrometry is gas-phase electrophoresis, which separates ions according to their size-to-charge ratio. Unlike mass spectrometry, ion mobility spectrometry is capable of separating isomers. 

Normally, ionic solids have very low conductivities. An ordinary crystal like sodium chloride must conduct by ion conduction since it does not have partially filled bands (metals) or accessible bands (semiconductors) for electronic conduction. The conductivities that do obtain usually relate to the detects discussed in the previous section. The migration of ions may be classified into three types. 

In contrast to the pseudo 3-D models, tmly multi-dimensional models use, in general, finite element or finite volume CFD (Computational Fluid Dynamics) techniques to solve full 3-D Navier-Stokes equations with appropriate modifications to account for electrochemistry and current distribution. The details of electrochemistry may vary from code to code, but the current density is calculated almost exclusively from Laplace equation for the electric potential (see Equation (5.24)). Inside the electrolyte, the same equation represents the migration of ions (e g. 0= in SOFC), elsewhere it represents the electron/charge transfer. In what follows, we briefly summarize a commonly used multi-dimensional model for PEM fuel cells because of its completeness and of the fact that it also addresses most essential features of SOFC modeling. 

Electrodialysis In this process, dialysis is carried out under the influence of electric field (figure 3) Potential is applied between the metal screens supporting the membranes. This speeds up the migration of ions to the opposite electrodes. So, dialysis is greatly accelerated. 

The measurement of the ohmic resistance for an electrolyte is more difficult than it is for a metal wire. A variety of chemical and physical processes occur at the electrode-solution interface that must be separated from the voltage drop associated with the migration of ions through the bulk electrolyte. 

In Fig. 9 (a-c), TSD traces for some epoxy-aromatic amine networks in the glassy state are shown. It is clearly seen that in the glassy state there exist several different types, at least four, of TSD peaks (Fig. 9c). A special measurement has shown that all TSD peaks are a result of dipole relaxation but not of the migration of ions or heterocurrents. From Fig. 9 a and b it becomes clear that the yl peak belongs to unreacted epoxy groups the P peaks, undoubtedly, are related to some motions of aliphatic chains. The relaxation time as a function of temperature has been measured for two peaks — yt and p2 (dotted lines in Fig. 9c). Both peaks have been refined by... 

The second microwave heating mechanism arises from the migration of ions in the electric field. The resulting current from the oscillating ions gives rise to heat in the familiar way, following the i2r law, where / is the current and r reflects the resistance or impedence to ionic movement through collisions with other ions and molecules present in the medium. Ionic conduction is important in situations where the ions are free to move to some extent. 

In these cells a - diffusion potential develops between a and ji. If the migration of ions is allowed but the diffusion potential is minimized by adding another electrolyte in high concentration (it also causes the decrease of fcation to approximately zero)... 

Only the two end chambers of the stack are fitted with electrodes. The separation results from the migration of ions under the influence of the electric field established by the cell voltage between the end electrodes. Cations and anions of the electrolyte can migrate in opposite directions toward the anode and cathode, respectively. Due to the alternating arrangement of the membranes,... 

In a solution with a cation 1 and an anion 2, one can considered two cases for which the thermal diffusion potential is defined. One case is at the initial state, when the concentration is uniform and a temperature gradient is established and a second case when the steady state is reached, in other words, when the migration of ions by the temperature gradient has ceased. 

In studies where a knowledge of the diffusion of metallic ions in polymers is important, one often wishes to measure a profile of the concentration as a function of depth. Neutron activation cannot be used to measure these profiles directly, but if the sample can be cut into thin slices with a microtome, these can be analysed individually to construct the profile. In our laboratory this technique is used extensively to study the migration of ions into the polyethylene insulation of high-voltage cables (10). These impurities contribute to the degradation with use of the electrical properties of the cable. 

Fig. 4.2. The migration of ions resulting from a gradient of electrostatic potential (i.e., an electric field) in an electrolyte. The electric field is produced by the application of a potential difference between two electrodes immersed in the electrolyte. The directions of increasing electrostatic potentials and of ionic migration are shown below the diagram.

This then is an elementary picture of diffusion. The next task is to consider the phenomenon of conduction, i.e., the migration of ions in an electric field. 

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