Debye-Hiickel theory association

Thus we have found that the screening should be more efficient than in the Debye-Hiickel theory. The Debye length l//c is shorter by the factor 1 — jl due to the hard sphere holes cut in the Coulomb integrals which reduce the repulsion associated with counterion accumulation. A comparison with Monte Carlo simulation results [20] bears out this view of the ion size effect [19]. 

The contribution of electrostatic interactions to fast association was analyzed by applying the classical Debye-Hiickel theory of electrostatic interactions between ions to mutants of bamase and barstar whose ionic side chains had been altered by protein engineering (Chapter 14).16 The association fits a two-step model that is probably general (equation 4.84). 

In 1926, Bjerrum [137] used Debye-Hiickel theory to describe ion association and took into account the interaction of ions within a short range. He introduced an ion-pair concept, gave a definition of ion pairs as neutral species formed by electrostatic attraction between oppositely charged ions in solution, and showed how ion-pair formation was dependent on the ions size (radius of ions), solvent (dielectric constant), and temperature. 

Frequently electrolytes will be associated in solution. Wu and Friedman used an approximate method of obtaining standard heats of solution in such cases. They confined all measurements to low con-centrations(<0.01m) and neglected long-range interionic effects,such as those covered by the Debye-Hiickel theory. They further assumed that there is only one (unspecified) association process occurring. The heat of solution is then given by... 

A multitude of MM-level models can be found in the literature differing from one another by the underlying assumptions of short-range interactions. The oldest one is the Debye-Hiickel theory, which does not recognize short-range interactions and association. The Debye-Hiickel theory yields the limiting laws of thermodynamic and transport properties based on the potential of mean force. 

MSA is the extension of Debye-Hiickel theory to high electrolyte concentrations using the same continuum model [397, 398). Combined with the law of mass action for ion- pair formation, MSA-MAL (mass action law approach), and finally as associative mean spherical approximation (AMSA) [399-401], it permits us to take into account any ion-complex formation. Equations for electrolyte conductivity are given by Blum et al. [402, 403]. However, the complexity of battery electrolytes hinders the application of a high concentration continuum approach. 

Solutions to the Poisson—Boltzmann equation in which the exponential charge distribution around a solute ion is not linearized [15] have shown additional terms, some of which are positive in value, not present in the linear Poisson—Boltzmann equation [28, 29]. From the form of Eq. (62) one can see that whenever the work, q yfy - yfy), of creating the electrostatic screening potential around an ion becomes positive, values in excess of unity are possible for the activity coefficient. Other methods that have been developed to extend the applicable concentration range of the Debye—Hiickel theory include mathematical modifications of the Debye—Hiickel equation [15, 26, 28, 29] and treating solution complexities such as (1) ionic association as proposed by Bjerrum [15,25], and(2) quadrupole and second-order dipole effects estimated by Onsager and Samaras [30], etc. 

Here, A is the ideal-gas contribution represents the hard-chain repulsion of the reference system. A P and A account for the Helmholtz energy contributions due to attraction (dispersion) and hydrogen bonding (association), respectively. To account for the fact that at least some of the species carry charges, this model was combined with a Debye-Hiickel theory to describe the Helmholtz energy contribution A to a system that is caused by charging the species. 

Equations [305]-[310] are strictly valid only for thermodynamic species, which are ordinarily associated with stationary points on the potential energy surface V(R), where R denotes the full set of solute coordinates. However, we also use the SMx solvation models to calculate potentials of mean force, which are called W(R,T). The gradient of W(R,T) gives the force on the solute molecule averaged over a canonical ensemble of solvent molecules and is a generalization of the one-dimensional radial potential of mean force that appears in Debye-Hiickel theory. Thus, we write... 

Another attempt to go beyond the cell model proceeds with the Debye-Hiickel-Bjerrum theory [38]. The linearized PB equation is used as a starting point, however ion association is inserted by hand to correct for the non-linear couplings. This approach incorporates rod-rod interactions and should thus account for full solution properties. For the case of added salt the theory predicts an osmotic coefficient below the Manning limiting value, which is much too low. The same is true for a simplified version of the salt free case. 

This does not mean that the Debye-Htickel theory gives the right answer when there is ion-pair formation. The extent of ion-pair formation decides the value of the concentration to be used in the ionic-cloud model. By removing a fraction 0 of the total number of ions, only a fraction 1 - 0 of the ions remain for the Debye-Hiickel treatment, which interests itself only in the free charges. Thus, the Debye-Htickel expression for the activity coefficient [Eq. (3.120)] is valid for the free ions, with two important modifications (1) Instead of there being a concentration c of ions, there is only (1 - 0)c the remainder Oc is not reckoned with owing to association. (2) The distance of closest approach of free ions is q and not a. These modifications yield... 

The major feature is a rapid decrease of A at low salt concentrations, followed by a minimum and pronounced increase. At the CP there is a substantial conductance. To interpret this behavior, we first note that the Debye- Hiickel (DH) theory itself predicts an instability regime at low T, but if compared with experiment C is far too low. Taking account for ion association considerably improves thew results. In the presence of ion association, a higher salt concentration is needed to achieve the concentration of free ions to drive phase separation, i.e. C is shifted to higher values. In particular, the Bjerrum model for ion pair association yields ... 

A concept of ion association in electrolyte solutions was introduced about eighty years ago by Bjerrum [1] in order to improve the Debye-Hiickel (DH) theory [2], In accordance with this concept an electrolyte solution is considered to be a mixture of free ions and ion clusters (usually ion pairs and some-... 

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