Debye length, reciprocal

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. 

Calculate the Debye-Htickel reciprocal lengths for the following solutions (a) lO MNaCl, (b)O.l MNaCl,(c) 10 MCaClj,and(d) 10 A/CaS04.(Kim)... 

Compute the ionic activity of0.0001 M KCl. What change will be caused to the activity of KCl if 0.01 mole of ZnClj is added to 1000 ml of the above solution With the added salt, will the Debye-Htickel reciprocal length increase or decrease Why (Xu)... 

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)... 

An intelligent inspection of expression (4.316) shows that the mobility u of ions is not a constant independent of concentration. It depends on the Debye-Huckel reciprocal length k. But this parameter a is a function of concentration (see Eq. 3.84). Hence, Eq. (4.316) shows that the mobility of ions is a function of concentration, as was suspected (Section 4.6.1) on the basis of the empirical law of Kohlrausch. 

K has the dimension of a reciprocal length and k is called the Debye length . For aqueous solutions of symmetrical electrolytes Eq. (8) reduces to ... 

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. 

The definition of the limit for dilute solutions is obviously important for the proper use of available chemical data. Moreover, as this property is also of particular importance for the sol-gel process discussed later it deserves further analysis. In the Debye-Hiickel ionic-cloud model, the influence of dissolved ions on a particular central ion is evaluated by comparing the mean distance between ions to the dimension of the ion cloud surrounding it. The latter is evaluated by the so-called Debye (or Debye-Hiickel reciprocal) length (1/k), which may be approximatively written for aqueous solutions at 25°C as ... 

We deduce for an expression which coincides (in the limit case of negligible interaction between nearest neighbors) with the reciprocal of the characteristic Debye-Hiickel length. Following current methods we establish the relation between x and the given experimental results. We study the... 

The renormalized reciprocal screening length for the nth harmonic is the combination of Debye screening length and structural screening length due to the charge periodicity ... 

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. 

For originally non-charged particles dispersed in polar media, the surface preferential adsorption of ions, surface molecular group dissociation, isomorphic substitution, adsorption of polyelectrolytes, and even surface polarization will make them behave similarly to charged particles. The Stem layer and diffuse layer comprise what is commonly known as the electrical double layer, the thickness of which depends on the type and concentration of the ions in the suspension as well as on the particle surface. A parameter, called the Debye-Huckel parameter k, is used to characterize electrical double layer thickness. K has the dimension of reciprocal length. For smooth surfaces in simple electrolytes. 

One key quantity of the contribution A is the ion-specific inverse Debye screening length k which compares to the MSA F [Eq. (4)] k describes the reciprocal distance within which the long-range forces among ions are shielded by water, expressed as ... 

The most important parameter in equations (7.37) to (7.39) is k, which has the dimension of the reciprocal of length. In water at 25 C, = 0.329 /7A". Distance is the Debye-Huckel length and represents the thickness of the diffuse layer. This happens to be a misnomer because, over distance /c , the potential decreases only by j/exp(l) = tpd/2.1, but, in the weak potential approximation, the diffuse layer [equation (7.38)] can be treated as a parallel-plate capacitor Q = 6K with plates separated by distance /c . The variation in the potential in the solution, as a function of the distance from the surface, depends on the concentration and the charge of the ions present in the electrolyte (Figure 7.6). 

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