Potential from ionic atmosphere

The most characteristic properties of ions are their abilities to move in solution in the direction of an electrical field gradient imposed externally. The conductivity of an electrolyte solution is readily measured accurately with a 1 kHz alternating potential in a virtually open circuit, in order to avoid electrolysis. The molar conductance of a completely dissociated electrolyte is A2 = A2°° - 2 + EC2 In C2 + J iR ) C2 — J" R")c2, where S, E, f, and f are explicit expressions, containing contributions from ionic atmosphere relaxation and electrophoretic effects, the latter two depending also on ion-distance parameters R. The infinite dilution can be split into the limiting molar ionic conductivities by using experimentally measured transport numbers extrapolated to infinite dilution, t+° and i °° = 1 - <+°°. For a binary electrolyte, Aa = 2+°° -I- and = i+ A2. Values of the limiting ionic molar conductivities in water at 298.15 K [1] are accurate to 0.01 S cm mol (S = Q ). 

At the shear plane, fluid motion relative to the particle surface is 2ero. For particles with no adsorbed surfactant or ionic atmosphere, this plane is at the particle surface. Adsorbed surfactant or ions that are strongly attracted to the particle, with their accompanying solvent, prevent Hquid motion close to the particle, thus moving the shear plane away from the particle surface. The effective potential at the shear plane is called the 2eta potential, It is smaller than the potential at the surface, but because it is difficult to determine 01 To usual assumption is that /q is effectively equal to which can be... 

The most important parameters of the ionic atmosphere are the charge density Qv r) and the electrostatic potential /(r) at the various points. Each of these parameters is understood as the time-average value. These values depend only on distance r from the central ion, not on a direction in space. For such a system it is convenient to use a polar (spherical) coordinate system having its origin at the point where the central ion is located then each point can be described by a single and unique coordinate, r. 

We can see from this equation that the potential / at the point r = 0 has the value that would exist if there were at distance 1/k a point charge -zj or, if we take into account the spherical symmetry of the system, if the entire ionic atmosphere having this charge were concentrated on a spherical surface with radius 1/k around the central ion. Therefore, the parameter = 1/k with the dimensions of length is called the ejfective thickness of the ionic atmosphere or Debye radius (Debye length). This is one of the most important parameters describing the ionic atmosphere under given conditions. 

We can find the potential of the ionic atmosphere by subtracting from the overall value of potential /(r) in accordance with Eq. (7.32) the value of potential of the central ion ... 

The theory of Debye and Hiickel started from the assumption that strong electrolytes are completely dissociated into ions, which results, however, in electrical interactions between the ions in such a manner that a given ion is surrounded by a spherically symmetrical distribution of other ions mainly of opposite charges, the ionic atmosphere. The nearer to the central ions the higher will be the potential U and the charge density the limit of approach to the central ion is its radius r = a. 

The Debye-Hiickel limiting law is the least accurate approximation to the actual situation, analogous to the ideal gas law. It is based on the assumption that the ions are material points and that the potential of the ionic atmosphere is distributed from r = 0 to r->oo. Within these limits the last equation is integrated by parts yielding, for constant k, the value ezk/Aite. Potential pk is given by the expression... 

In view of this equation the effect of the ionic atmosphere on the potential of the central ion is equivalent to the effect of a charge of the same magnitude (that is — zke) distributed over the surface of a sphere with a radius of a + LD around the central ion. In very dilute solutions, LD a in more concentrated solutions, the Debye length LD is comparable to or even smaller than a. The radius of the ionic atmosphere calculated from the centre of the central ion is then LD + a. 

Although the surface potential, /, the electrical potential due to the charge on the monolayers, will clearly affect the actual pressure required to thin the lamella to any given dimension, we shall assume, for the purpose of a simple illustration, that 1 Ik, the mean Debye-Huckel thickness of the ionic double layer, will influence the ultimate thickness when the liquid film is under relatively low pressure. Let us also assume that each ionic atmosphere extends only to a distance of 3/k into the liquid when the film is under a relatively low excess pressure from the gas in the bubbles this value corresponds to a repulsion potential of only a few millivolts. Thus, at about 1 atm pressure ... 

The conventional viewpoint, which assumes that the ionic atmosphere is spherically symmetric, does not take account of the inevitable effects of ionic polarization. From an analysis of the general solution (19), however, it is evident that the ionic atmosphere must be spherically symmetric for nonpolarizable ions, and the DH model is therefore adequate. (Moreover, in very dilute solution polarization effects are negligibly small, and it does not matter whether we choose a polarizable or unpolarizable sphere for our model.) But once we have made the realistic step of conferring a real size on an ion, the ion becomes to some extent polarizable, and the ionic cloud is expected to be nonspherical in any solution of appreciable concentration. Accordingly, we base our treatment on this central hypothesis, that the time-average picture of the ionic solution is best represented with a polarizable ion surrounded by a nonspherical atmosphere. In order to obtain a value for the potential from the general solution of the LPBE we must first consider the boundary conditions at the surface of the central ion. 

The constant Doo vanishes as expected because the potential VL(r,0,Laplace potential, VL, results from all the multipoles induced in the surface of the central ion by the structured ionic atmosphere, and vanishes at infinite dilution as required. 

The absorption of species from the atmosphere is common to all electrolyte solutions and clearly the absorption of water is the biggest issue. This is not solely confined to ionic liquids, however, as all electroplaters who deal with aqueous solutions of acids know, if the solution is not heated then the tank will overflow from absorption of atmospheric moisture after some time. In the aqueous acid the inclusion of water is not a major issue as it does not significantly affect the current efficiency or potential window of the solution. Water absorption is also not such a serious issue with eutectic-based ionic liquids and the strong Lewis acids and bases strongly coordinate the water molecules in solution. The presence of up to 1 wt.% water can be tolerated by most eutectic-based systems. Far from having a deleterious effect, water is often beneficial to eutectic-based liquids as it decreases the viscosity, increases the conductivity and can improve the rate of the anodic reaction allowing better surface finishes. Water can even be tolerated in the chloroaluminate liquids to a certain extent [139] and it was recently shown that the presence of trace HQ, that results from hydrolysis of the liquid, is beneficial for the removal of oxide from the aluminum anode [140]. 

Measurement of the potential by means other than electro-kinetic measurement. G. S. Hartley and J. W. Roe1 point out that the potential determines the distribution of ions near a surface in the same manner as the potential just outside an ion controls the ionic atmosphere in the Debye-Huckel theory of strong electrolytes. There is a simple relation between the concentration of an ion in the layer next to a surface and in the bulk solution at a distance from the surface and the potential, so that if a means can be found of measuring the concentration of an ion in the surface and in the solution, it should be possible to estimate the potential of that surface. 

Once again (see Section 3.3.9), one can use the law of superposition of potentials to obtain the ionic-atmosphere contribution V doud potential at a distance r from the central ion. From Eq. (3.46), i.e.,... 

The first term on the right of equation (16) is the potential at a distance r due to a given point ion when there are no surrounding ions the second term must, therefore, represent the potential arising from the ionic atmosphere. It is seen, therefore, that the potential due to the ionic atmosphere, is given by... 

To proceed further, one must separate the contribution to cp due to the ionic atmosphere from the contribution that the ion makes itself in the absence of other ions, that is, the so-called self-atmosphere potential. The latter quantity is given by... 

Considerable effort has been made to develop a model for the parameter on the basis of statistical theories using simple electrostatic concepts. The first of these was proposed by Bjerrum [25]. It contains important ideas which are worth reviewing. He assumed that all oppositely charge ions within a certain distance of a central ion are paired. The major concept in this model is that there is a critical distance from the central ion over which ion association occurs. Obviously, it must be sufficiently small that the attractive Coulombic forces are stronger than thermal randomizing effects. Bjerrum assumed that at such short distances there is no ionic atmosphere between the central ion and a counter ion so that the electrostatic potential due to the central ion may be calculated directly from Coulomb s law. The value of this potential at a distance r is... 

The potential due to the ionic atmosphere at the surface of the ion, i.e. at a distance a/2 from the centre of the central reference ion. Non-ideality in electrolyte solutions is a result of electrostatic interactions obeying Coulomb s Law. The potential energy of such interactions is given in terms of ... 

The potential due to the ionic atmosphere at a distance, a, from the central reference ion. This dehnes the distance around the reference ion into which no ion can penetrate, and is thus the distance from the centre of the reference ion at which the ionic atmosphere starts. 

The aim is to calculate the mean ionic activity coefficient from the non-ideal part of the free energy. This is done in terms of the electrostatic potential energy of the coulomhic interactions between the ion and its ionic atmosphere. These interactions give rise to non-ideality. 

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