Electrolytes dilute solution theory

The water distribution within a polymer electrolyte fuel cell (PEFC) has been modeled at various levels of sophistication by several groups. Verbrugge and coworkers [83-85] have carried out extensive modeling of transport properties in immersed perfluorosulfonate ionomers based on dilute-solution theory. Fales et al. [109] reported an isothermal water map based on hydraulic permeability and electro-osmotic drag data. Though the model was relatively simple, some broad conclusions concerning membrane humidification conditions were reached. Fuller and Newman [104] applied concentrated-solution theory and employed limited earlier literature data on transport properties to produce a general description of water transport in fuel cell membranes. The last contribution emphasizes water distribution within the membrane. Boundary values were set rather arbitrarily. 

Garcia etal. [41] developed a two-dimensional porous electrode model and accounted for potential and charge distributions in the electrolyte. They employed transport equations derived from dilute solution theory, which is generally not adequate for LIB systems. The stress generation effect is built into the 2D DNS modeling framework with a simplified, sphere-packed electrode microstmcture description. 

Dilute solution theory is not often used in the treatment of lithium batteries, because most electrolytic solutions used in lithium batteries exhibit concentrated behavior. However, dilute solution theory becomes useful for cases such as the examination of side reactions such as redox shuttles for overcharge protection, because concentrated solution theory becomes more complicated when there are more than three species (anion, cation, and solvent) in solution. 

Concentrated solution theory includes interactions among aU species present in solution whereas dilute solution theory assumes that ions interact only with the solvent and not with other ions. In addition, dilute solution theory assumes that aU activity coefficients are unity. There is substantial evidence that both liquid and especially polymer electrolytes used in lithium batteries exhibit concentrated behavior [12,13,14,15]. 

For impurity species present in dilute concentrations, some may find it more convenient to treat the species using dilute solution theory, which accounts only for interactions of the dilute species with the solvent Rigorously, Equation 12 was derived for a binary electrolyte with no impurity species in the solution. While it is not completely rigorous to treat one species with dilute solution theory while treating the main electrolyte with equations derived from concentrated solution theory in the absence of the impurity species, the error may be small. The flux of the dilute species is given by Equation 5. The mass balance for the main electrolyte remains unchanged. If 2 is defined by Equation 3, then Us must be defined as a function of the concentration of the impurity species in order to include the concentration overpotential of the impurity species in the kinetic expression. Equation 53. The Nemst equation. Us = Uf + /JTln( Ci cf). is often used to account for the concentration overpotential of dilute species i. If Og is defined by Equation 6, then Us should not be defined as a function of solution composition. 

Nearly all pit models have been based on transport equations which strictly apply in solutions much more dilute than those usually found in pits, which exceed 1 M and often approach saturation in the metal chloride salt. The fundamental shortcoming of dilute solution transport theory is that it accounts only for interactions between ions and solvent molecules, and not between pairs of ions. Ion-ion interactions are manifested, for example, by deviations of the solution conductivity from values predicted by dilute solution theory, which become appreciable at concentrations as low as 0.01 M." This section will examine specific inaccuracies resulting from the dilute solution approximation, and point out cases where the use of concentrated solution transport models is tractable. Dilute and concentrated solution approaches will be compared in the context of a simple example of a one-dimensional pit with passive sidewalls. The metal and electrolyte solution were taken to be aluminum in 0.1 M NaCl. There are no cathodic reactions or homogeneous reactions in the pit, and the solution composition at the pit mouth is that of the bulk solution. This example was described in more detail in an earlier publication. This example is chosen because of its simplicity and since the behavior of the dilute solution model may be familiar to readers. 

The solutions of (6) and (11) require three concentration-dependent transport properties, D, t+, and k, along with the electrolyte activity coefficient in Eq. (11). These properties may be measured using well-known techniques. The three independent transport coefficients D, t+, and k may be related to the diffiisiv-ities Do-, Do+, and D+ from (5). In the dilute solution model there would be only two diffusivities, D+ and D, equivalent to Do- and Do+. There is no analog of D+ in dilute solution theory because ion-ion interactions are not considered. When D+ is significant, the predictions of the potential in dilute solution models deviate strongly from those in concentrated solution models. These deviations caimot be compensated by adjusting D+ and D from their values at infinite dilution. However, since the properties D and t+ in (6) and (7) do not depend on D+, the predictions of the concentration are not affected by ion-ion interactions. 

Debye-Hiickel theory The activity coefficient of an electrolyte depends markedly upon concentration. Jn dilute solutions, due to the Coulombic forces of attraction and repulsion, the ions tend to surround themselves with an atmosphere of oppositely charged ions. Debye and Hiickel showed that it was possible to explain the abnormal activity coefficients at least for very dilute solutions of electrolytes. 

It seems appropriate to assume the applicability of equation (A2.1.63) to sufficiently dilute solutions of nonvolatile solutes and, indeed, to electrolyte species. This assumption can be validated by other experimental methods (e.g. by electrochemical measurements) and by statistical mechanical theory. 

Battery electrolytes are concentrated solutions of strong electrolytes and the Debye-Huckel theory of dilute solutions is only an approximation. Typical values for the resistivity of battery electrolytes range from about 1 ohmcm for sulfuric acid [7664-93-9] H2SO4, in lead—acid batteries and for potassium hydroxide [1310-58-3] KOH, in alkaline cells to about 100 ohmcm for organic electrolytes in lithium [7439-93-2] Li, batteries. 

It is important to realise that whilst complete dissociation occurs with strong electrolytes in aqueous solution, this does not mean that the effective concentrations of the ions are identical with their molar concentrations in any solution of the electrolyte if this were the case the variation of the osmotic properties of the solution with dilution could not be accounted for. The variation of colligative, e.g. osmotic, properties with dilution is ascribed to changes in the activity of the ions these are dependent upon the electrical forces between the ions. Expressions for the variations of the activity or of related quantities, applicable to dilute solutions, have also been deduced by the Debye-Hiickel theory. Further consideration of the concept of activity follows in Section 2.5. 

The precipitate should be washed with the appropriate dilute solution of an electrolyte. Pure water may tend to cause peptisation. (For theory of washing, see Section 11.8 below.)... 

These large increases in rate might be attributed to the operation of a neutral salt effect, and, in fact, a plot of log k versus the square root of the ionic strength, fi, is linear. However, the reactants, in this case, are neutral molecules, not ions in the low dielectric constant solvent, chloroform, ionic species would be largely associated, and the Bronsted-Bjerrum theory of salt effects51 52, which is valid only for dilute-solution reactions between ions at small n (below 0.01 M for 1 1 electrolytes), does not properly apply. 

Further simphfication of the SPM and RPM is to assume the ions are point charges with no hard-core correlations, i.e., du = 0. This is called the Debye-Huckel (DH) level of treatment, and an early Nobel prize was awarded to the theory of electrolytes in the infinite-dilution limit [31]. This model can capture the long-range electrostatic interactions and is expected to be valid only for dilute solutions. An analytical solution is available by solving the Pois-son-Boltzmann (PB) equation for the distribution of ions (charges). The PB equation is... 

As a result of these electrostatic effects aqueous solutions of electrolytes behave in a way that is non-ideal. This non-ideality has been accounted for successfully in dilute solutions by application of the Debye-Huckel theory, which introduces the concept of ionic activity. The Debye-Huckel Umiting law states that the mean ionic activity coefficient y+ can be related to the charges on the ions, and z, by the equation... 

According to modem views, the basic points of the theory of electrolytic dissociation are correct and were of exceptional importance for the development of solution theory. However, there are a number of defects. The quantitative relations of the theory are applicable only to dilute solutions of weak electrolytes (up to 10 to 10 M). Deviations are observed at higher concentrations the values of a calculated with Eqs. (7.5) and (7.6) do not coincide the dissociation constant calculated with Eq. (7.9) varies with concentration and so on. For strong electrolytes the quantitative relations of the theory are altogether inapplicable, even in extremely dilute solutions. 

Van t Hoff introduced the correction factor i for electrolyte solutions the measured quantity (e.g. the osmotic pressure, Jt) must be divided by this factor to obtain agreement with the theory of dilute solutions of nonelectrolytes (jt/i = RTc). For the dilute solutions of some electrolytes (now called strong), this factor approaches small integers. Thus, for a dilute sodium chloride solution with concentration c, an osmotic pressure of 2RTc was always measured, which could readily be explained by the fact that the solution, in fact, actually contains twice the number of species corresponding to concentration c calculated in the usual manner from the weighed amount of substance dissolved in the solution. Small deviations from integral numbers were attributed to experimental errors (they are now attributed to the effect of the activity coefficient). 

This theory of the diffuse layer is satisfactory up to a symmetrical electrolyte concentration of 0.1 mol dm-3, as the Poisson-Boltzmann equation is valid only for dilute solutions. Similarly to the theory of strong electrolytes, the Gouy-Chapman theory of the diffuse layer is more readily applicable to symmetrical rather than unsymmetrical electrolytes. 

If solutions of two electrolytes are brought into contact there is, generally speaking, a potential difference between them, just as there is one at the interface mercury-electrolyte in the capillary electrometer. This potential difference has been shown by Nemst to depend on the differences in the concentrations and the migration velocities of the ions. Smith uses dilute solutions containing equivalent amounts of KI and KC1 the kation is thus the same in both solutions, and the migration velocities of the I and Cl ions are nearly equal, so that, according to Nemst s theory, there should be no potential difference or double layer at the interface. These... 

Electrolytic chemistry and solution theory continued to be a principal source of speculation about chemical bonding. As John Servos has noted, it was Noyes s attempt to visualize the difference between strong and weak electrolytes, to explain anomalies in the dilution law, that led him to make a distinction between "electrical molecules" and "chemical molecules" in the early 1900s. 

Because the theory of the liquid state is not nearly so well developed as the kinetic theory of gases, estimation methods for liquid diffusion coefficients are not as reliable as those used for gases. For dilute solutions of non-electrolytes, one widely used correlation is that due to Wilke and Chang[48]... 

According to modem theory, many strong electrolytes are completely dissociated in dilute solutions. The freezing-point lowering, however, does not indicate complete dissociation. For NaCl, the depression is not quite twice the amount calculated on the basis of the number of moles of NaCl added. In the solution, the ions attract one another to some extent therefore they do not behave as completely independent particles, as they would if they were nonelectrolytes. From the colligative properties, therefore, we can compute only the "apparent degree of dissociation" of a strong electrolyte in solution. 

The two mass action equilibria previously indicated have been used in conjunction with a modified form of the Shedlovsky conductance function to analyze the data in each of the cases listed in Table I. Where the data were precise enough, both K2 and K were calculated. As mentioned previously, the K s so evaluated are practically the same as those obtained for ion pairing in solutions of electrolytes in ammonia and amines. This is encouraging since it implies a fairly normal behavior (in the electrolyte sense) for dilute solutions of metals. Further support of the proposed mass action equilibria can be found in the conductance measurements of sodium in NH8 solutions with added salt. Bems, Lepoutre, Bockelman, and Patterson (4) assumed an additional equilibrium between sodium and chloride ions, associated to form NaCl, to compute the concentration of ionic species, monomers, and dimers when the common ion electrolyte is added. Calculated concentrations of conducting species are employed in the Onsager-Kim extension of the conductance theory for low-field conductance of a mixture of ions. Values of [Na]totai ranging from 5 X 10 4 to 6 X 10 2 and of the ratio of NaCl to [Na]totai ranging from zero to 28.5 are included in the calculations. 

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