Ionic atmosphere thickness

The action of external electric field on the free disperse system results in particle motion (electrophoresis). The electrophoretic velocity, vE, is not a function of ( -potential only, but also depends on the particle radius, r, and the type of electrolyte present in the system. However, it turns out (see fine print further down) that all of these factors can be simultaneously accounted for by the numerical coefficient, kt, introduced into the Helmholtz-Smoluchowski equation (V.26). If the particles are spherical, k, changes from 2/3 for particles smaller compared to the ionic atmosphere thickness (kt 1) to 1 for large particles ( kt 1). Consequently, the particle flux due to the applied electric... 

It was shown by D. Henry that for particles of arbitrary shape at any ratio of particle size to the ionic atmosphere thickness, 8 = 1/k, the Helmholtz-Smoluchowski equation can be written in a generalized form as ... 

We need to know the ionic atmosphere thickness not only in estimating the ionic component, but also in estimating the diffusion component of the disjoining pressure. 

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. 

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)... 

Although the surface potential, /, the electrical potential due to the charge on the monolayers, will clearly affect the actual pressure required to thin the lamella to any given dimension, we shall assume, for the purpose of a simple illustration, that 1 Ik, the mean Debye-Huckel thickness of the ionic double layer, will influence the ultimate thickness when the liquid film is under relatively low pressure. Let us also assume that each ionic atmosphere extends only to a distance of 3/k into the liquid when the film is under a relatively low excess pressure from the gas in the bubbles this value corresponds to a repulsion potential of only a few millivolts. Thus, at about 1 atm pressure ... 

The quantity k, also known as the Debye length, has the dimensions of distance and is an approximate measure of the thickness of the ionic atmosphere over which the electrostatic field of the ion extends with an appreciable strength. The k term can be calculated from the following relationship... 

Up to now, the charge density at a given distance has been discussed. The total excess charge contained in the ionic atmosphere that surrounds the central ion can, however, easily be computed. Consider a spherical shell of thickness dr at a distance r from the origin, i.e., from the center of the reference ion (Fig. 3.13). The charge dq in this thin shell is equal to the charge density times the volume AnP dr of the shell, i.e.. 

Thickness of Ionic Atmosphere (nm) at Various Concentrations and for Various Types of Salts... 

The excess charge on an ionic atmosphere varies with distance out from the central ion. Show that the net change in the charge of a spherical shell of thickness dr is... 

What distance jc is to be used hi other words, when can the ionic cloud be declared to have dispersed or relaxed These questions may be answered by recalling the description of the ionic atmosphere where it was stated that the charge density in a dr-thick spherical shell in the cloud declines rapidly at distances greater than the... 

Utilize the calculated values of the thickness of the ionic atmosphere R in 0.1 N solutions of a univalent electrolyte in (a) nitrobenzene, (b) ethyl alcohol, and (c) ethylene dichloride to calculate the relaxation times of the ionic atmospheres. (Constantinescu)... 

The net charge of the atmosphere is, of course, equal in magnitude but opposite in sign to that of the central ion the charge density will obviously be greater in the immediate vicinity of the latter and will fall off with increasing distance. It is possible, nevertheless, to define an effective thickness of the ionic atmosphere, as will be explained shortly. 

According to the definition of k, i.e., equation (12), the thickness of the ionic atmosphere will depend on the number of ions of each kind present in unit volume and on their valence. If c is the concentration of the ions of the ith kind expressed in moles (gram-ions) per liter, then... 

The thickness of the ionic atmosphere is thus seen to be of the order of iO " cm. it decreases with increasing concentration and increasing valence of the ions present in the electrolyte, and increases with increasing dielectric constant of the solvent and with increasing temperature. The value of 1/k in Angstrom units for solutions of various types of electrolytes at concentrations of 0.1, 0.01 and 0.001 moles per liter in water at 25 are given in Table XXII. 

TABLE XXII. THICKNESS OF THE IONIC ATMOSPHERE IN WATER AT 25 ... 

Conductance with High Potential Gradients.—When the applied potential is of the order of 20,000 volts per cm., an ion will move at a speed of about 1 meter per sec., and so it will travel several times the thickness of the effective ionic atmosphere in the time of relaxation. 

Although the importance of the ionic strength was first realized from empirical considerations, it is now known to play an important part in the theory of electrolytes. It will be observed that equation (12) on page 83, which gives the reciprocal of the thickness of the ionic atmosphere according to the theory of Debye and Hiickel, contains the quantity where n is the number of ions of the zth kind in unit volume... 

In order to avoid confusion with the use of the symbol k for the i eciprocal of the thickness of the double layer or ionic atmosphere, the symbol is employed here for the specific conductance. 

Before these can be solved, it is necessary to know the concentration of anions, A, in the surface phase. Herein lies the great difficulty in applying this treatment. An arbitrary thickness must be chosen for the depth of the surface phase within which all the surface-active material is concentrated and which also includes all the ionic atmosphere. The former consideration alone would lead to possibly quite a small figure for the depth, S, but the latter requires us to extend the value somewhat to include the whole cloud of counter-ions. If the ionic strength of the under-... 

Dimensional analysis shows that k has units of reciprocal length, and it is called the Debye-Huckel reciprocal distance. It depends on the ionic strength of the solution, the dielectric properties of the solvent, and temperature. For an aqueous solution containing a 1-1 electrolyte at a concentration of 1 M (1000 mol m ) at 25°C, K is equal to 3.288 nm h As will be seen below, 1/k corresponds to the effective thickness of the ionic atmosphere, which would be 304 pm for a 1 M solution. 

An important parameter obtained in the solution of this problem is F, which is related to the thickness of the ionic atmosphere ... 

One can expect (see fine print further) that the greater diffusivity of the counter-ion layer as compared to that established in Helmholtz model, would only affect the velocity distribution profile of the displacement of individual fluid layers in the direct vicinity of the solid surface. The experimentally observed velocity of the mutual motion of the phases with respect to each other, v0, determined, as in Helmholtz model, by the potential change significantly (curve 2 approaches the same limiting value as curve 7 ). This is also confirmed by the fact that the distance between the capacitor plates, 8, which is the only parameter defining the geometry of the system in the Helmholtz model, is not present in the final expression.4 The thickness of the ionic atmosphere, k 1, may be used as the parameter closest to the distance 8, i.e. 8=1/k. 

There are, therefore, theories that relate the (pd and analogous theories exist for the absolute values of which can not be experimentally measured, the -potential is a directly measurable quantity, which, along with the thickness of the ionic atmosphere, is an important characteristic of the diffuse part of the electrical double layer. 

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