Radius of the ionic atmosphere

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. 

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)... 

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... 

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 ... 

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... 

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. 

The quantity, has units m, i.e. has the units of a length and as such it is often stated that /c represents the extent of the ionic atmosphere around the central j cation. Such a description can only be given if the value of k is measured from r = a, since the ionic atmosphere must always lie at distances, r > a. Alternatively it has been stated that k represents the reciprocal of the effective radius of the ionic atmosphere, again with the proviso that this is measured from r = a. However, these could be misleading statements, as is shown by the diagrams below. 

The potential (p corresponding to the charge Q is given by equation (6.34) as — QK/4Txs e, where l/ic is the radius of the ionic atmosphere. The work is thust... 

Thus at a distance 1 Jk the potential has dropped by a factor of (1/e). This distance may be used as a measure of the extension of the double layer and is often loosely called the thickness of the double layer. According to the theoretical equations it has the value /k = [ekT/e l.CizlY and is identical with the parameter introduced in the Debye-Hiickel theory of electrolytes in which /k is identified with the radius of the ionic atmosphere. Of particular importance in colloid science is the fact that the thickness of the... 

To summarize, the proton transfer reaction can be broken into three distinct parts Diffusion of the reactants to within the radius of the ionic atmosphere accelerated diffusion to within the encounter distance and subsequent conversion of the encoimter complex to products. For reactions in which the equilibrium is rapidly established within the encounter complex, the rate equations are dominated by the diffusion process. This results in the loss of information about the dynamics of the encounter complex. For such a reaction some information can be obtained about the ionic radius by varying the ionic strength and using an electrostatic theory (such as is done for Deby-Hiickel activity coefficients) to calculate the effect of shielding by the ions. ... 

A key quantity in the Debye-Huckel theory, leading to the values of the constants A and B, is the screening length, k, the average reciprocal of the radius of the ionic atmosphere surrounding an ion in the solution, made up essentially by ions of the opposite charge. The square of this quantity is proportional to the ionic strength of the solution and also to the reciprocal of the product w T ... 

K has the dimension of reciprocal length. This is the reason for referring to k as the Debye radius. As we shall soon see, this may be interpreted as the effective radius of the ionic atmosphere. 

This justifies the interpretation of k as the effective radius of the ionic atmosphere. 

In this presentation, the total electrical potential at R is the sum of two contributions— the direct potential due to a at the center and an effective screening potential, having the opposite sign to that of the first term—which arise from the ionic atmosphere. The latter may be viewed as the constant potential within a sphere of radius k due to a net charge of - Za6 on its surface. This is another interpretation of the effective radius of the ionic atmosphere. 

This radius r is known as the radius of the ionic atmosphere around an ion k. It is the distance at which we find the maximum of the electrical... 

Table 4.1 gives the value of the radius of the ionic atmosphere for these two values of the ionic strength. 

If, for the radius of the ion, we choose the value of 30 nm, this means that for an ionic strength of 1 mole/l, in water at ambient temperature, the radius of the ionic atmosphere is of the order of magnitude of the radius of the ion,... 

Nevertheless, let us remember that Debye and Hiickel s law supposes the ions are spherical, which can be accepted for sufficiently dilute solutions in which the radius of the ionic atmosphere is larger than a and therefore in which the ions are relatively far removed from one another. If the ions come closer together, the hypothesis of sphericity becomes trickier to accept for a large number of t5q)es of ions - particularly for polyatomic ions. 

T = Xc,zf being the ionic concentration and s, the viscosity and the dielectric constant of the solvent, respectively , the reciprocal average radius of the ionic atmosphere a, the mean distance of closest approach of ions

[Pg.8]

K is the inverse radius of the ionic atmosphere d is the minimum distance between the anion and cation 8, is the solvent relative permittivity (dielectric constant)... 

The radius of the ionic atmosphere shown as large cycles in Figure 1.3 and k (m"0 is defined as follows ... 

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