Ionic atmosphere radius

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. 

Within a spherical space of radius a, by definition Qy = 0, so that the value of potential of the ionic atmosphere here is constant and equal to that at point r = a ... 

AA-gpjj. Conditionally, the ionic atmosphere is regarded as a sphere with radius r. The valnes of approach the size of colloidal particles, for which Stokes s law applies (i.e., the drag coefficient 9 = where r is the liquid s viscosity) when they... 

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 ionic atmosphere can thus be replaced by the charge at a distance of Lu = k 1 from the central ion. The quantity LD is usually termed the effective radius of the ionic atmosphere or the Debye length. The parameter k is directly related to the ionic strength I... 

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. 

For very dilute solutions, the motion of the ionic atmosphere in the direction of the coordinates can be represented by the movement of a sphere with a radius equal to the Debye length Lu = k 1 (see Eq. 1.3.15) through a medium of viscosity t] under the influence of an electric force ZieExy where Ex is the electric field strength and zf is the charge of the ion that the ionic atmosphere surrounds. Under these conditions, the velocity of the ionic atmosphere can be expressed in terms of the Stokes law (2.6.2) by the equation... 

The discussion above is a description of problem that requires answers to the following (1) the determination of the distribution of ions around a reference ion, and (2) the determination of the thickness (radius) of the ionic atmosphere. Obviously this is a complex problem. To solve this problem Debye and Huckel used a rather general approach they suggested an oversimplified model in order to obtain an approximate solutions. The Debye-Huckel model has two basic assumptions. The first is continuous dielectric assumption. In this assumption water (or the solvent) is a continuous dielectric and is not considered to be composed of molecular species. The second, is a continuous charge distribution in the ionic atmosphere. Put differently, charges of the ions in the ionic surrounding atmosphere are smoothened out (continuously distributed). 

Equation (2.30) represents the potential produced by a charge Zt of ionic atmosphere at a distance 1/k. The quantity 1/k has the dimensions of length and is appropriately called the thickness (or radius) of the ionic atmosphere in a given solution. Also, k Ms called the Debye-Huckel length and is assigned symbol Erom Eq. (2.21)... 

Derive an equation for the radius of ionic atmosphere for bi-bivalent (2 2) electrolytes at 25°C with water as a solvent. Use the derived equation to calculate r-Q values for the following concentrations of salt 0.0001, 0.001, and 0.1 moI/L. Compare the results obtained with the results of Problem 2.4. 

Ks for isentropic compressibility [-(l/y)(3W3P)s]). 3. Symbol for reciprocal radius of ionic atmosphere. 

In order to consider the influence of the ionic atmosphere on the electrophoretic mobility, the theoretical electrical charge of the ion q in Equation 6.14 is replaced by the smaller effective charge <2eff and the hydrodynamic radius r by the effective radius R of the ion, which includes its ionic atmosphere ... 

Deviations from the Limiting Law. In Figure 3 results for 1 1 electrolytes with four different ion-sizes are plotted for hypothetical solutions having D = 78.54, T = 25.0°C, density = 1.0 and n = 1. The DH limiting-law is also plotted for comparison. It should be noted that the ionic atmosphere extends outwards from the surface of the central ion, and the parameter a is the mean effective ionic radius rather than the distance of closest approach of the DH theory. 

An approximate value of the radius of the ionic atmosphere r as a function of concentration, for a uni-univalent (1-1) electrolyte at 25 °C and water as solvent, considering 79 as the dielectric constant value, may be obtained from the relationship... 

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. 

What lower limit should be used for the integration In the point-charge model, one used a lower limit of zero, meaning that the ion cloud commences from zero (i.e., from the surface of a zero-radius ion) and extends to infinity. However, now the ions are taken to be of finite size, and a lower limit of zero is obviously wrong. The lower limit should be a distance corresponding to the distance from the ion center at which the ionic atmosphere starts (Fig. 3.30). 

The radius of the ionic atmosphere is l//c where is defined in the text. Work out the average distance between ions (d) in terms of the concentration in mol dm . id> Hk, then one is confronting a situation in which the radius of the atmosphere is less than the average distance between ions. Describe what this means. Derive a general expression for at which this problem (coarse grainedness) occurs for a 2 2 electrolyte. Do you think an ionic atmosphere model applies when d> l//c ... 

What is the total charge on an ionic atmosphere around an anion of valence z From the data in the text, examine logy vs. Vm, where tn is the molality of the solution, from 0 to 1 mol dm". The plots always pass through a minimum. Use the fully extended Debye-Hlickel theory, including the Bjerrum-Stokes and Robinson terms, to find the significance of the minimum at which the electrolyte concentration increases with the increase of the cation radius. 

This poses an interesting problem. The ionic atmosphere can be considered a charge sphere of radius (Fig. 4.89). The charged sphere moves under the action of... 

Fig. 4.89. The ionic atmosphere can be considered a charged sphere of radius...

Ka <1, where k is the Debye screening length and a is the radius of the micelle, diffusion coefficients at the critical micelle concentration (cmc), Dcmc. decrease with decreasing ionic strength, i.e., Ka. With further decrease in m from 0.24 to 0.18, the Dcmc value does not decrease. This fact suggests that the drag of ionic atmosphere reaches maximum when xa becomes ca. 0.2. 

For a single electrolyte a quantity k may be defined as the reciprocal of the radius of the ionic atmosphere and is proportional to the square root of the concentration. In other words, the charge in the ionic atmosphere may be visualized as being uniformly distributed over the surface of a sphere of radius 1/k. 

Calculate the effective radius of the ionic atmosphere (l/x) in a 0.1 molal aqueous solution of potassium sulfate at 25 C. The dielectric constant of water at 26 C is 78.64. 

Q.l5.2 Calculate the effective radius of the ionic atmosphere for ions in the serum. Base your answer on the work derived in Question Q.15.1... 

This result may be interpreted as showing that the ionic atmosphere has a charge —z, opposite to that of the ion on which attention is focused. Furthermore, the atmosphere has an effective radius equal to u + 1 /k. The motion of the atmosphere is in the opposite direction to that of the ion whose velocity is given by Stokes law. 

The first term, ken, is the rate of encounter between the reactants in absence of electrostatic interaction the second term represents the contribution of the electrostatic interactions at I = 0, while the last one is a correction for ionic screening at / > 0 x is the radius of the ionic atmosphere [x = 3.27 x 107 VT (cm-1)] and rtj is the radius of the encounter between the reactants. 

Mathematically the ionic atmosphere can be treated as though it were a sphere of radius, a, with all the ions of the ionic atmosphere distributed on the surface of this sphere. Electrostatic theory states that inside such a sphere the potential due to the ionic atmosphere is constant at aU points within the sphere and is equal to the value of this potential at the surface of the sphere. It follows that ... 

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