Debye-Hiickel theory equation

The diffusional rate constant kD is calculated on the basis of the Debye-Hiickel theory (Equation 6.107), where the distance tr is the sum of A and B radii in the hard-sphere approximation. 

The expression obtained for in Debye-Hiickel theory (equation (3.8.26)) is... 

This distribution function has already appeared in the Debye-Hiickel theory (Equation 10.22) when the Maxwell-Boltzmann distribution was used to describe the distribution of the ions of the ionic atmosphere around the central j ion (see Figure 10.16). Because the distribution function is given in terms of the distance between an ion of the ionic atmosphere and the central reference j ion it is termed a radial distribution function. This reflects the spherical symmetry assumed in the Debye-Hiickel theory. 

The ion cloud in the solution modifies the result of Equation 3.77. Exact solution of the Poisson-Boltzmann equation (Equation 3.70) is known for the salt-free solutions containing only the counterions (Alfrey et al. 1951). As expected, the electric potential falls off smoothly with the radial distance, and there exists a counterion cloud near the cylinder. In order to get insight into the basic nature of the electrostatics in salty electrolyte solutions around a charged thin cylinder, we linearize Equation 3.70 to get the Debye-Hiickel theory (Equation 3.71). Solving this equation with the boundary conditions that the electric field vanishes far away from the cylinder and that it is given by Equation 3.76 at the surface of the cylinder, the result is... 

Although it is not possible to measure an individual ionic activity coefficient,, it may be estimated from the following equation of the Debye-Hiickel theory ... 

For more concentrated solutions (/° 5 >0.3) an additional term BI is added to the equation B is an empirical constant. For a more detailed treatment of the Debye-Hiickel theory a textbook of physical chemistry should be consulted.1... 

The first assumption of the Debye-Hiickel theory is that is spherically symmetric. With the elimination of any angular dependence, the Poisson equation (expressed in spherical-polar coordinates) reduces to... 

Experience shows that solutions of other electrolytes behave in a manner similar to the examples we have used. The conclusion we reach is that the Debye-Hiickel equation, even in the extended form, can be applied only at very low concentrations, especially for multivalent electrolytes. However, the behavior of the Debye-Hiickel equation as we approach the limit of zero ionic strength appears to give the correct limiting law behavior. As we have said earlier, one of the most useful applications of Debye-Hiickel theory is to... 

When the Debye-Hiickel theory applies, equation (9.104) becomes... 

Figure 9.5 Extrapolation of the cell emf data of G. A. Linhart [/. Am. Chem. Soc., 41, 1175-1180 (1919)] to obtain E° for the Ag/AgCl half-cell. The dashed line gives the limiting slope as predicted by Debye-Hiickel theory. LHS = Left-hand side of equation see text.

Chapters 7 to 9 apply the thermodynamic relationships to mixtures, to phase equilibria, and to chemical equilibrium. In Chapter 7, both nonelectrolyte and electrolyte solutions are described, including the properties of ideal mixtures. The Debye-Hiickel theory is developed and applied to the electrolyte solutions. Thermal properties and osmotic pressure are also described. In Chapter 8, the principles of phase equilibria of pure substances and of mixtures are presented. The phase rule, Clapeyron equation, and phase diagrams are used extensively in the description of representative systems. Chapter 9 uses thermodynamics to describe chemical equilibrium. The equilibrium constant and its relationship to pressure, temperature, and activity is developed, as are the basic equations that apply to electrochemical cells. Examples are given that demonstrate the use of thermodynamics in predicting equilibrium conditions and cell voltages. 

Equation (7.44) is known as the third approximation of the Debye-Hiickel theory. Numerous attempts have been made to interpret it theoretically, hi these attempts, either individual simplifying assumptions that had been made in deriving the equations are dropped or additional factors are included. The inclusion of ionic solvation proved to be the most important point. In concentrated solutions, solvation leads to binding of a significant fraction of the solvent molecules. Hence, certain parameters may change when solvation is taken into account since solvation diminishes the number of free solvent molecules (not bonded to the ions). The influence of these and some other factors was analyzed in 1948 by Robert A. Robinson and Robert H. Stokes. 

APPENDIX A Derivation of the Main Equation of Debye-Hiickel Theory... 

This is Eq. (7.33), the main equation of the first version of Debye-Hiickel theory. 

Equations (87)-(89) apply in aqueous solutions of two electrolytes in which the interaction potentials are conformal. For example, the assumptions utilized in the extensions of the Debye-Hiickel theory (e.g. water is considered as a continuous dielectric medium of dielectric constant D, that the cation-anion repulsive potential is that of hard spheres, and that all the... 

Edwards et al. (6) made the assumption that was equal to 4>pure a at the same pressure and temperature. Further theyused the virial equation, truncated after the second term to estimate pUre a These assumptions are satisfactory when the total pressure is low or when the mole fraction of the solute in the vapor phase is near unity. For the water, the assumption was made that <(>w, , aw and the exponential term were unity. These assumptions are valid when the solution consists mostly of water and the total pressure is low. The activity coefficient of the electrolyte was calculated using the extended Debye-Hiickel theory ... 

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 ... 

Remember that Table 11.3 contains some useful numerical values of k at different concentrations of various electrolytes. Equation (10) is the basic relationship of the Debye-Hiickel theory and may be integrated as follows. The variable x is introduced with the following definition ... 

The part of the activity coefficients depending on k (Equation 33) can be simplified further if suitable average values are introduced. If the restriction d a is sufficiently well fulfilled, the term in y is small compared with the term in <5. The latter therefore represents the main influence caused by the presence of dipoles since the terms in are from ionic charges. They are identical with terms of corresponding order in the Debye-Hiickel theory. 

The diffuse layer is described by the Gouy—Chapman theory of 1913 [21, 22], which is based on the same equations as the Debye—Hiickel theory of 1923 for electrolytes, which describes the electrostatic potential around an ion in a given ionic atmosphere [23]. 

The contribution of electrostatic interactions to fast association was analyzed by applying the classical Debye-Hiickel theory of electrostatic interactions between ions to mutants of bamase and barstar whose ionic side chains had been altered by protein engineering (Chapter 14).16 The association fits a two-step model that is probably general (equation 4.84). 

Debye-Hiickel theory also predicts other thermodynamic properties. The equation for the osmotic coefficient66 equivalent to equation (11.69) for 7 ) is... 

Chapter 18 describes electrolyte solutions that are too concentrated for the Debye-Hiickel theory to apply. Gugenheim s equations are presented and the Pitzer and Brewer tabulations, as a method for obtaining the thermodynamic properties of electrolyte solutions, are described. Next, the complete set of Pitzer s equations from which all the thermodynamic properties can be calculated, are presented. This discussion ends with an example of the extension of Pitzer s equations to high temperatures and high pressures. Three-dimensional figures show the change in the thermo-... 

The activity a2 of an electrolyte can be derived from the difference in behavior of real solutions and ideal solutions. For this purpose measurements are made of electromotive forces of cells, depression of freezing points, elevation of boiling points, solubility of electrolytes in mixed solutions and other characteristic properties of solutions. From the value of a2 thus determined the mean activity a+ is calculated using the equation (V-38) whereupon by application of the analytical concentration the activity coefficient is finally determined. The activity coefficients for sufficiently diluted solutions can also be calculated directly on the basis of the Debye-Hiickel theory, which will bo explained later on. 

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. 

Debye-Hiickel limiting law — The equation based on the - Debye-Hiickel theory providing mean activity coefficients / for ions of charge z+ and z- at - ionic strength I in dilute solutions lg/ = ().S09 z+ z /l, when mol and dm3 units are used. 

Onsager equation — (a) - Debye-Hiickel-Onsager equation, see also - Debye-Hiickel-Onsager theory. 

The solubility product is equal to Ca x AI(OH)4 x [OH , where curly brackets denote species activities and [OH ] may be replaced by 2[Ca ] - [A1(0H)4"]. As the concentrations are low, activity coefficients may be calculated from simplified Debye-Hiickel theory. Solubility products may thus be obtained from experimental data (NI8,B118,BI 19.C48) (Table 10.2). The variations in solubility products with temperature may be represented by empirical equations of the form... 

Equation 2 was proposed by Lewis and Randall (6) even before the advent of the Debye-Hiickel theory for the evaluation of E° by the extrapolation of LHS of Equation 2 against f(m) because by definition y —> 1 as m —> 0. However, such extrapolations failed to yield precise values of E° because the plots did not produce the desired straight lines. The most common method of extrapolation is to substitute an expression for ln y from the Debye-Hiickel theory (7), and the two most frequently used substitutions are... 

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