Debye-Huckel distance

The derivation of the equations of the Debye-Huckel theory did not require differentiation between a solution of a single electrolyte and an electrolyte mixture provided that the limiting law approximation Eq. (1.3.24), was used, which does not contain any specific ionic parameter. If, however, approximation (1.3.29) is to be used, containing the effective ionic diameter ay it must be recalled that this quantity was introduced as the minimal mean distance of approach of both positive and negative ions to the central ion. Thus, this quantity a is in a certain sense an average of effects of all the ions but, at the same time, a characteristic value for the given central... 

Siveral approaches are available in the case of mixed electrolyte solutions. The Guntelberg equation can be used at very high dilutions to avoid the ambiguity in the meaning of aD, the distance of closest approach, when several electrolytes are present. This equation is empirical and has fewer terms than the Debye-Huckel extended equation. I found it to yield poor agreement with experimental results even at m = 0.01 for NaCl at 25°C (y+ caic = 0.8985 and y+ exp = 0.9024). For the Davies equation for m = 0.20 one obtains y+ calc = 0.752 andy+exo = 0.735 also for NaCl at 25°C. 

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 ... 

A modification of GB that includes the effects of dissolved electrolytes in the formalism, i.e., an extension analogous to the Poisson-Boltzmann extension of the Poisson equation, has been proposed by Srinivasan et al. (1999). In this model, the dielectric constant is a function of the interatomic distance and the Debye-Huckel parameter (Eq. (11.7)). 

One of the most important quantities to emerge from the Debye-Huckel approximation is the parameter k. This quantity appears throughout double-layer discussions and not merely at this level of approximation. Since the exponent kx in Equation (37) is dimensionless, k must have units of reciprocal length. This means that k has units of length. This last quantity is often (imprecisely) called the thickness of the double layer. All distances within the double layer are judged large or small relative to this length. Note that the exponent kx may be written x/k a form that emphasizes the notion that distances are measured relative to k in the double layer. 

The work terms wl (/ = r or p) are associated with the electrostatic work done when the reactants are brought together from infinity to a distance separated from rigid spheres. For ions of charges Zj and z2 in a medium with a dielectric constant D, w , i r or p, can be calculated on the basis of the Debye-Huckel theory (Equation 6.110). 

Taking the surface potential to be xp°, the potential at a distance x as rp, and combining the Boltzmann distribution of concentrations of ions in terms of potential, the charge density at each potential in terms of the concentration of ions, and the Poisson equation describing the variation in potential with distance, yields the Pois-son-Boltzmann equation. Given the physical boundary conditions, assuming low surface potentials, and using the Debye-Huckel approximation, yields... 

In the Debye-Huckel theory, an ion in solution is treated as a conducting sphere. The distance of closest approach of two ions is a.4 The solution beyond a... 

Measurement of the potential by means other than electro-kinetic measurement. G. S. Hartley and J. W. Roe1 point out that the potential determines the distribution of ions near a surface in the same manner as the potential just outside an ion controls the ionic atmosphere in the Debye-Huckel theory of strong electrolytes. There is a simple relation between the concentration of an ion in the layer next to a surface and in the bulk solution at a distance from the surface and the potential, so that if a means can be found of measuring the concentration of an ion in the surface and in the solution, it should be possible to estimate the potential of that surface. 

Debye length — (also - Debye-Huckel length) In the formulation of the - Debye-Huckel theory the counter ions surrounding the sample ion under consideration are substituted in an attempt of simplification by an ionic cloud. The radius of this ionic cloud or atmosphere giving the distance between the ion under consideration and the location where dq/ (1/k) is at maximum (dq is the charge enclosed in a shell of dr thickness around the ion, and k is the - Debye-Huckel parameter). The Debye length rD (LD and other symbols are also used) also is given by... 

The quantity I /at has units of length and is called the Debye length it defines the extent of the double layer, i.e., the distance in which the potential decays to I je of its initial value k is called the Debye-Huckel parameter. Hence within validity of this approximation (low surface potentials < 25 mV) the potential decreases exponentially away from the surface. 

Hence, the maximum value of the charge contained in a spherical shell (of infinitesimal thickness dr) is attained when the spherical shell is at a distance r = rc from the reference ion (Fig. 3.15). For this reason (but see also Section 3.3.9), re" is known as the thickness, or radius, of the ionic cloud that surrounds a reference ion. An elementary dimensional analysis [e.g., of Eq. (3.43)] will indeed reveal that k has the dimensions of length. Consequently, k is sometimes referred to as the Debye-Huckel length. 

Utilize the known values of the Debye-Huckel constants A and B for water to calculate the mean activity coefficients for 1 1, 1 2, and 2 2-valent electrolytes in water at the ionic strengths 0.1 and 0.01 at 25 °C. The mean distance of closest approach of the ions a may be taken as 300 pm in each case. (Constantinescu)... 

Debye-Huckel length k. Hence, if the ions diffuse to a distance k the central ion... 

The asymptotic expression for the potential of a spherical particle of radius a in a symmetrical electrolyte solution of valence z and Debye-Huckel parameter k, a large distance r from the center of the sphere may be expressed as... 

Observed molar conductivities were analyzed by assuming the ion association (ion-pair formation) between the complex ions and the counter ions in the same manner as described previously. The closest distances of approach of ions (a) in the Robinson-Stokes conductivity equation and in the Debye-HUckel equation were taken as 6.8 and 7.3 A for chlorides and perchlorates of the tris(phen) complexes 6.6 and 7.1 A for those of the tris(bpy) complex, respectively, using the effective ionic radii of the complex ions, shown in T le 1, and those of Cl (1.81 A) and C104 (2.30A). The values of ref were estimated from the ionic partial molar volumes (f i°°) by use of Glueckauf equation. > ... 

These techniques can also be developed further to deal with Stern layer problems. Note that for Z) the effective capacitance distance D is reduced by a factor (1 —0.62y/alD). If image effects are included, it can be shown that (when the zwitterions are immersed in a high dielectric medium e = 80, adjacent to a hydrocarbon surface s = 2), image effects can virtually double the electrostatic energy so that the two effects may partially cancel out. Nonetheless the observation that is reduced from the intuitive capacitance form is of some interest, as it is known that for ionic micelles use of Debye-Huckel theory (at low salt equivalent to the capacitance form) gives results for the repulsive free energy too large by a factor of 2. 

As an illustrative example taken from Russel et al. (1989), let us consider a 0.01 molar solution of sodium chloride in contact with a surface charged at a density of 5 x 10 negative charges per square meter at room temperature, 298°K. Equation (2-52) gives /c = 3 nm. The dimensionless surface potential exfJkrtT. obtained from Eq. (2-45), is —5.21, and Eqs. (2-46) and (2-49) give respectively the exact and the Debye-Huckel approximations for the potential as a function of distance from the surface. The results are plotted in Fig. 2-13. Note that since — s/ ks T > 1, the Debye-Huckel approximation is... 

Electrostatic repulsive energy, Tr, at a given interparticle distance is the work which must be performed to bring the particles to a specific point. Using the Debye-Huckel low-potential approximation and assuming equal spheres, Vr can be described by... 

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