Debye-Hiickel atmosphere

Na ions bound to the polyion per charged group. One could even get carried away and interpret the dependence of Equation (25) on + as a decrease in the number of bound Na" due to the competition of the Mg " ions with the Na" for sites on the polyion. Such an interpretation, however, is completely wrong the effects are due to the diffuse Debye-Hiickel atmosphere. Actually, in this case, the effect persists even if ( = 0, that is, even when all activity coefficients are unity the electroneutrality constraint given by Equation (12) is sufficient to cause an asymmetric distribution of small ions. 

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. 

At this point an interesting simplification can be made if it is assumed that r, as representing the depth in which the ion discrimination occurs, is taken to be just equal to 1/x, the ion atmosphere thickness given by Debye-Hiickel theory (see Section V-2). In the present case of a 1 1 electrolyte, k = (8ire V/1000eitr) / c /, and on making the substitution into Eq. XV-7 and inserting numbers (for the case of water at 20°C), one obtains, for t/ o in millivolts ... 

The Debye-Hiickel formula for the activity coefficient of an ion was developed by a consideration of ion atmosphere effects.10 It starts with an electrostatic expression for the free energy of interaction for one ion with one mole of others ... 

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... 

The Debye-Hiickel theory suggests that the probability of finding ions of the opposite charge within the ionic atmosphere increases with increasing attractive force. 

Another arena for the application of stochastic frictional approaches is the influence of ionic atmosphere relaxation on the rates of reactions in electrolyte solutions [19], To gain perspective on this, we first recall the early and often quoted triumph of TST for the prediction of salt effects, in connection with Debye-Hiickel theory, for reaction rates In kTST varies linearly with the square root of the solution ionic strength I, with a sign depending on whether the charge distribution of the transition state is stabilized or destabilized by the ionic atmosphere compared to the reactants. 

A new theory of electrolyte solutions is described. This theory is based on a Debye-Hiickel model and modified to allow for the mutual polarization of ions. From a general solution of the linearized Poisson-Boltzmann equation, an expression is derived for the activity coefficient of a central polarized ion in an ionic atmosphere of non-spherical symmetry that reduces to the Debye-Hiickel limiting laws at infinite dilution. A method for the simultaneous charging of an ion and its ionic cloud is developed to allow for ionic polarization. Comparison of the calculated activity coefficients with experimental values shows that the characteristic shapes of the log y vs. concentration curves are well represented by the theory up to moderately high concentrations. Some consequences in relation to the structure of electrolyte solutions are discussed. 

Normally, the validity of the Debye—Hiickel theory extends little further than kR <1. At room temperature, this requires ionic concentrations < 0.1 mol dm-3 for univalent ions in water, 0.03moldm-3 for univalent ions in ethanol or <0.01 mol dm-3 for univalent ion in ethers. In these cases, ions may be regarded as point particles and the strong repulsive core potential ignored. Furthermore, the time taken for non-reactive ions to diffuse far enough to establish an ionic-atmosphere around an ion, which was suddenly formed in solution containing only univalent ions, is... 

Bearing in mind these limitations on the Debye—Hiickel model of electrolytes, the influence of ionic concentration on the rate coefficient for reaction of ions was solved numerically by Logan [54, 93] who evaluated the integral of eqn. (56) with the potential of eqn. (55). He compared these numerical values with the predictions of the Bronsted— Bjerrum correction to the rate of a reaction occurring between ions surrounded by equilibrated ionic atmospheres, where the reaction of encounter pairs is rate-limiting... 

The ionic atmosphere model leads to the extended Debye-Hiickel equation, relating activity coefficients to ionic strength ... 

The diffuse layer is described by the Gouy—Chapman theory of 1913 [21, 22], which is based on the same equations as the Debye—Hiickel theory of 1923 for electrolytes, which describes the electrostatic potential around an ion in a given ionic atmosphere [23]. 

In the limit of an infinite micellar radius, i.e. a charged planar surface, the salt dependence of Ge is solely due to the entropy factor. A difficult question when applying Eq. (6.13) to the salt dependence of the CMC is if Debye-Hiickel correction factors should be included in the monomer activity. When Ge is obtained from a solution of the Poisson-Boltzmann equation in which the correlations between the mobile ions are neglected, it might be that the use of Debye-Hiickel activity factors give an unbalanced treatment. If the correlations between the mobile ions are not considered in the ionic atmosphere of the micelle they should not be included for the free ions in solution. 

Debye-Hiickel developed a theory for the activity coefficients of an ionic solution at a molecular level. A selected ion in the ideally diluted solution is statistically well distributed and there are no interactions between ions present in the solution. In contrast, the ion in the concentrated solution is surrounded by the excess of counter ions in the vicinity of the ion, as the counter ions are attracted by Coulombic forces, while ions of the same charge are repelled. Thus, ion atmosphere is created. As a result, there is a difference in reversible work between the concentrated wrev and dilute solutions wrev ideal ... 

However, for higher concentrations the model is no longer valid, and further, the approximations, i ZlUkT < 1, cannot be valid close to the ion i [Eq. (XV.7.2)]. Bjerrum made the proposal that any pair of ions whose interaction is of the order of 2kT or more should be considered as an ion pair, not as independent ions, and that the Debye-Hiickel treatment should be reserved only for the free ions separated by distances sufficiently large that their interaction is less than this. If we call this distance tb and neglect the ion atmosphere around such an ion pair, then for the ion pair and... 

One could use the Debye-Hiickel ionic-atmosphere model to study how ions of opposite charges attract each other, (a) Derive the radial distribution of cation ( +) and anion (nj concentration, respectively, around a central positive ion in a dilute aqueous solution of 1 1 electrolyte, (b) Plot these distributions and compare this model with Bjerrum s model ofion association. Comment on the applicability of this model in the study of ion association behavior, (c) Using the data in Table 3.2, compute the cation/anion concentrations at Debye-HUckel reciprocal lengths for NaCl concentrations of lO and 10 mol dm", respectively. Explain the applicability of the expressions derived. (Xu)... 

This problem is best approached by considering the electrostatic free energy, F, oi protein molecule due to its net charge and its interaction with an ion atmosphere. For a spherical ion of uniform charge density, according to the Debye-Hiickel theory (Cohn and Edsall, 1943, p. 473)... 

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