Debye-Hiickel theory quantity

It is important to realise that whilst complete dissociation occurs with strong electrolytes in aqueous solution, this does not mean that the effective concentrations of the ions are identical with their molar concentrations in any solution of the electrolyte if this were the case the variation of the osmotic properties of the solution with dilution could not be accounted for. The variation of colligative, e.g. osmotic, properties with dilution is ascribed to changes in the activity of the ions these are dependent upon the electrical forces between the ions. Expressions for the variations of the activity or of related quantities, applicable to dilute solutions, have also been deduced by the Debye-Hiickel theory. Further consideration of the concept of activity follows in Section 2.5. 

However, the quantity inside the brackets, i.e., 2c( le / (JcT, is nothing else than the familiar K2 of the Debye-Hiickel theory. The integration constant in Eq. (6.128) can be evaluated from the boundary condition that at —> 0, — /0. Therefore,... 

The above equation is known as the linearized Poisson-Boltzmann equation since the assumption of low potentials made in reaching this result from Equation (29) has allowed us make the right-hand side of the equation linear in p. This assumption is also made in the Debye-Hiickel theory and prompts us to call this model the Debye-Hiickel approximation. Equation (33) has an explicit solution. Since potential is the quantity of special interest in Equation (33), let us evaluate the potential at 25°C for a monovalent ion that satisfies the condition e p = kBT ... 

The quantity given in equation (V-58) and expressing the average distance between the nearest ions in the solution (i. e. equalling the total of radii of both ions with opposite charges and being within the range of 3—5 x 10-8 cm) determines the specific influence of the electrolytes on the activity coefficient. Because this quantity iH not directly measurable, verification of the validity of the Debye - Hiickel theory is carried out in such a manner that a value is substituted for which conforms best with the values of y+c obtained by experiments. 

To clarify the different roles played by the Helmholtz and Gibbs free energies of ionic solutions, it is relevant to reconsider the derivation of these thermodynamic quantities in the original Debye-Hiickel theory [1—4],... 

We might proceed by plotting versus m, drawing a smooth curve through the points, and constructing tangents to the curve at the desired concentrations in order to measure the slopes. However, for solutions of simple electrolytes, it has been found that many apparent molar quantities such as tp vary linearly with yfm, even up to moderate concentrations. This behavior is in agreement with the prediction of the Debye-Hiickel theory for dilute solutions. Since... 

Prior to the Debye-Hiickel theory, had been empirically introduced by Lewis as a quantity of importance in the treatment of ionic solutions. Since it quantifies the charge in an electrolytic solution, it was known as the ionic strength and given the symbol I... 

The term potential is often referred to in electrostatics. It is a quantity such that the potential difference (see below) between two points is equal to the difference between the value of the potential at one point and its value at the other. The term turns up in the Debye-Hiickel theory (see Chapter 10, especially Section 10.4.5). 

The effective ionic radius of an ion in solution is an important quantity in the discussion of the behaviour of electrolyte solutions. Mobilities are thus fundamental to both Debye-Hiickel theory and conductance theory. 

For a given concentration, the relaxation time is directly related to ( + + )/ + , to A/2+2- and to A for the given electrolyte. The relaxation time is thus afundamental quantity in any theory of conductance. In turn, the relaxation time is a property of the ionic atmosphere which is regarded as the crucial concept in the Debye-Hiickel theory of non-ideality. These statements could be summarised as ... 

The quantity 1/k, referred to as the effective ion atmosphere thickness, was introduced in the Debye-Hiickel theory of strong electrolytes, which was developed later than the Gouy-Chapman theory. 

According to Lewis, the constant A in Eq. (8.22) had to be determined empirically. On the other hand, with the aid of the Debye-Hiickel theory, it can be based on physical quantities and thus may be calculated. The theoretically derived dependence... 

Dielectric properties describe the polarization, P, of a material as its response to an applied electric field E (bold symbols indicate vectors) [1—3], In the field of solution chemistry, the discussion of dielectric behavior is often reduced to the equilibrium polarization, Pq = So(s — V) Eq (eq is the electric field constant), of the isotropic and nonconducting solvent in a static field, Eq. Characteristic quantity here is the static relative permittivity (colloquially dielectric constant ), , which is a measure for the efficiency of the solvent to screen Coulomb interactions between charges (i.e., ions) embedded in the medium. As such, enters into classical electrolyte theories, like Debye-Hiickel theory or the Bom model for solvation free energy [4, 5] and is used... 

For some reactions use can also be made of statistical-mechanical equations either (rarely) alone or in combination with some of the quantities alluded to above. These are reactions taking place in systems for which we have a model which is at once realistic enough and mathematically tractable enough to be useful. An example is the calculation of the standard equilibrium constant of a gas reaction (and thence of the yield, but only if the gas mixture is nearly enough perfect) from spectroscopically determined molecular properties. Another example is the use of the Debye-Hiickel theory or its extensions to improve the calculation of the yield of a reaction in a dilute electrolyte solution from the standard equilibrium constant of the reaction when, as is usually so, it is not accurate enough to assume that the solution is ideal-dilute. 

Debye lengths. In the Debye-Hiickel theory of monovalent salt solutions there is a characteristic length quantity x, defined by x- = 2e-n )/iaonkT), where... 

The standard electrode potential of an electrode is a very important electrochemical quantity. Conventionally, it is determined by the extrapolating the electrode potentials of extremely dilute solution along the line predicted by the Debye-Hiickel theory. Nevertheless, in the thermoelectrochemistry, it can be obtained by the measurement of the apparent enthalpy change. Based on a thermodynamic principle mentioned above (see Eq. (25)), Eq.(16) can be rewritten as... 

The last equality pertains to any solvent and any temperature. For aqueous solutions at 25°C A/mor kg = 0.510 z z. The quantity in the denominator equals k, the reciprocal of the thickness of the ionic atmosphere in the Debye-Hiickel theory [2]. This thickness is to be construed as pertaining to the distance beyond the distance of closest approach of the ions, a, to the periphery of the ionic atmosphere. Therefore, BF may be replaced by B aF, where, according to the theory ... 

The quantity 1/k determines the length scale over which the potential decays due to screening from the ion cloud in this Debye-Hiickel theory. 

Until about 1923 activity coefficients were purely empirical quantities in that when concentrations were modified by their use, correct results could be predicted for the properties of a system. We shall see that, on the basis of the Debye-Hiickel theory, to be discussed shortly, activity coefficients become rationalized and theoretically predictable quantities. 

The ionic diameters used in the Debye— Hiickel theory have been described as, ... the effective diameter of the ion in solution. Since no independent method is available for evaluating aj this quantity is an empirical parameter, but the aj s obtained are of a magnitude for ion sizes. [26]. Values for these effective ionic diameters have been experimentally evaluated and can be found in the literature [26, 27]. 

Figure 3.48 shows two ways of expressing the results of Mayer s viriai coefficient approach using the osmotic pressure of an ionic solution as the test quantity. Two versions of the Mayer theory are indicated. In the one marked DHLL + B2, the authors have taken the Debye-Hiickel limiting-law theory, redone for osmotic pressure instead of activity coefficient, and then added to it the results of Mayer s calculation of the second viriai coefficient, B. In the upper curve of Fig. 3.48, the approximation within the Mayer theory used in summing integrals (the one called hypernetted chain or HNC) is indicated. The former replicates experiment better than the latter. The two approxi-... 

In all other solutions the so called degree of dissociation, as determined from the measurement of some colligative property, merely indicates the magnitude of interionic forces, it cannot, however, be taken as a measure of the quantity of dissociated and undissociated molecules of the solute. A complete theory of strong electrolytes, at least of their diluted solutions, has been developed by Debye and Hiickel, this theory is the basis of modern electrochemistry. 

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... 

B V7 is the fundamental quantity k in the interionic attraction theory of Debye and Hiickel. 

The theory of Peter Debye and Erich Hiickel (1923) provides theoretical expressions for single-ion activity coefficients and mean ionic activity coefficients in electrolyte solutions. The expressions in one form or another are very useful for extrapolation of quantities that include mean ionic activity coefficients to low solute molality or infinite dilution. 

A theory of ionic solutions developed by Peter Debye and Erich Hiickel in 1923 (which is based on statistical mechanics and beyond the scope of this text) provides an expression for the activity. We shall only state the main result of this theory, which works well for dilute electrolytes. The activity depends on a quantity called the ionic strength /, defined by... 

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