Chemical potential mean ionic

Like its chemical potential, the activity of an individnal ion cannot be determined from experimental data. For this reason the parameters of electrolyte activity % and mean ionic activity are nsed, which are defined as follows ... 

All quantities in Eq. (12.6) are measurable The concentrations can be determined by titration, and the combination of chemical potentials in the exponent is the standard Gibbs energy of transfer of the salt, which is measurable, just like the mean ionic activity coefficients, because they refer to an uncharged species. In contrast, the difference in the inner potential is not measurable, and neither are the individual ionic chemical potentials and activity coefficients that appear on the right-hand side of Eq. (12.3). 

The electrostatic potential y(r) is a physical observable, which can be determined experimentally by diffraction methods as well as computationally. It directly reflects the distribution in space of the positive (nuclear) and the negative (electronic) charge in a system. V (r) can also be related rigorously to its energy and its chemical potential, and further provides a means for defining covalent and ionic radii" ... 

The chemical potential of an individual ion in solutions can never be measured because it depends on the other ions in the solution with which it interacts. We can, however, discuss p — p,/v, the mean ionic chemical potential, or the average chemical potential of an ion produced from the electrolyte. Writing p, j i, p, and p,- each in the form p = p° + RT In a gives... 

The mean chemical potential of a pair of cations and anions can be estimated from the ionic dissociation equilibrium shown as an example in Eqs. 8.42 and 8.45 ... 

In the case of an - electrolyte dissociating in solution as Aj,+ Bj, < is+Az+ + z/ Bz where v+z+ = v z to ensure electroneutrality, and the total number of particles formed by each molecule is v = v+ + z/, then the only activity that can be measured is that of the complete species, and the individual ions cannot be assigned meaningful chemical potentials. Under these circumstances, a mean activity coefficient is defined through the equation yv = y++ yvs. Since individual ionic chemical potentials are not measurable, it has become conventional to assign to the chemical potential of the hydrogen ion under standard conditions the value of zero, allowing relative chemical potentials for all other ions to be formulated. 

What is the significance of these quantities fi, x, and It is obvious they are all average quantities—the mean chemical potential the mean standard chemical potential the mean ionic mole fraction x, and the mean ionic-activity coefficient f. In the case of and fi% the arithmetic mean (half the sum) is taken because free energies are additive, but in the case of x and f , the geometric mean (the square root of the product) is taken because the effects of mole fraction and activity coefficient on free energy are multiplicative. 

Second, there is no possibility of separately determining either a+ or d. For, by definition, /r+ = (9G/9/i+)r,p, that is, the determination of requires addition of only positive ions to the solution, while holding the concentration of anions fixed. However, this step, which violates the Law of Electroneutrality, cannot be carried out operationally. Consequently, one cannot deal with individual ionic activities rather, as Eq. (4.1.3c) shows, the ionic activities occur in such a manner that only their geometric mean or appropriate logarithmic sum is involved. Third, since thermodynamic descriptions must be confined to measurable properties we may regard the quantity, /x -F i Tlna on the left of Eq. (4.1.3a) as an effective chemical potential that is to be used to represent the behavior of the electrolytes. On the other hand, one does not wish to ignore the ionic nature of the solution hence it is customary to write... 

Mean Chemical Potentials and Mean Ionic Activity Coefficients. 

In general, corresponding to the mean chemical potential (27.19) there corresponds a mean ionic activity coefficient given by... 

For most insertion compounds, the interaction of intercalated ions with each other in the host lattice is not negligible. In order to simply consider the contribution of ionic interaction in Equation (5.8), it is often assumed that each ion experiences a mean interaction or energy field from its neighboring ions, based on a mean-field theory [10]. According to this approximation, the contribution to the chemical potential is proportional to the fraction of sites occupied by the ions 5, and hence the interaction term is introduced into Equation (5.8) as... 

Then the mean ionic chemical potential fi is defined by... 

The chemical potential of the electrolyte, /u-e, is directly related to its activity and to its mean ionic activity coefficient, ye (on the molal scale) or yE (on the molar... 

An electron-conducting material brought into contact with an ionically conducting phase establishes an electrode. In case of semiconductors in addition to electrons, holes may act as means of charge propagation. The ionically conducting phase may be an electrolyte solution composed of a dissociated electrolyte and a solvent, an ionic liquid, a molten salt, or a solid electrolyte. At the established interphase an equilibrium is established. Assuming at the instance of contact a non-equilibrium between both phases and the chemical potentials of the involved species, two ]u, possibilities may be considered one is depicted below (Fig. 1) ... 

To conclude this section on the DH theory, we would like to point out that these last two criticisms (neglecting short range repulsive interactions and linearizing the PBE) are the only valid criticisms. In fact the McMillan-Mayer theory (MMM) showed that, provided a correct definition of the "effective interaction potential" is given, the molecular structure of the solvent needs not to be considered explicitly(1) in calculating the thermodynamic properties of ionic solutions. This conclusion has very important consequences the first one is that, as the number density of ion in a typical electrolyte solutions is of the order of 10"3 ions/A, then the solution can be considered as a dilute ionic gas as a consequence the theories available for gases can be used for ionic fluids, provided the "effective potential" (more often called potential of the mean force at infinite dilution) takes the place ot the gas-gas interaction potential. Strictly this is true only in the limit of infinite dilution, but will hold also at finite concentrations, provided the chemical potential of the solvent in the given solution is the same as in the infinitely dilute solutions. This actually... 

In ionic systems, since the positive and negative ions always come in pairs, physically it is not possible to measure the chemical potentials j,Ag+ nnd pci-separately only the sum can be measured. A similar problem arises for the definition of enthalpy and Gibbs free energy of formation. For ions, these two quantities are defined with respect to a new reference state based on the H" " ions, as described in Box 8.1. For the chemical potential, a mean chemical potential is defined by... 

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