Debyes treatment

The basic ingredients of Debye treatment are the following Newton s law, 

Combining eq.(5.62) with the hydrodynamic continuity equation for each ion  

The potential given by eq.(5.69) is concentration dependent at low concentration, it increases with the concentration and above some critical concentration (above 10 Af) it reaches a plateau value. 

Consequently, at large c (low ijJTu) 1 0 is independent of c, which yields the plateau value. 

Debye s treatment leads to a plateau for the potential o, at high c. However, it is observed experimentally that some salts do not exhibit any plateau, but rather a continuously decreasing pattern. As the frequency is typically on the order of 100 kHz for a wave lenght of 1 cm and a pressure amplitude around 0.1 Atm in water, absorption effect or the ionic association could not explain this decreasing. The consideration of non ideal terms gives an explanation of this behaviour. 

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The Debye treatment is not easily extended to higher concentrations and special methods are required to... 

From fig. 6.2(a) it is immediately apparent that the specific heat of ice differs considerably from that predicted by a simple Debye treatment. Fig. 6.2(6) shows the deviation from the T behaviour expected at very low temperatures and there is no sign of a flattening of the specific heat at temperatures up to the melting point. The limiting value of Cp/ T as T approaches 0 °K does. 

The atoms in a Debye solid are treated as a system of weakly coupled harmonic oscillators. Normal modes with wavelengths that are large compared to the atomic spacing do not depend on the discrete nature of the crystal lattice, and consequently these normal modes can be obtained by treating the crystal as an isotropic elastic continuum. In the Debye treatment of a solid all of the normal modes are treated as elastic waves. The partition function for a Debye solid cannot be obtained In closed form, but the thermodynamic functions for a Debsre solid have been tabulated as a function of 9p/T- For the pair of Isotopic metals Li(s)... 

From Eq. (3) the frequency distribution can be calculated following the Debye treatment by making use of the fact that an actual atomic system must have a limited number of frequencies, limited by the number of degrees of freedom N. The distribution p(v) is thus simply given by Eq. (4). This frequency distribution is drawn in the sketch on the right-hand side in Fig. 2.36. The heat capacity is calculated by using a properly scaled Einstein term for each frequency. The heat capacity function for one mole of vibrators depends only on Vj, the maximum frequency of the distribution, which can be converted again into a theta-temperature, j. Equation (5) shows that at temperature T is equal to R multiplied by the one-dimensional Debye... 

This expression shows that the low-temperature heat capacity varies with the cube of the absolute temperature. This is what is seen experimentally (remember that a major failing of the Einstein treatment was that it didn t predict the proper low-temperature behavior of Cy), so the Debye treatment of the heat capacity of crystals is considered more successful. Once again, because absolute temperature and dy, always appear together as a ratio, Debye s model of crystals implies a law of corresponding states. A plot of the heat capacity versus TIdo should (and does) look virtually identical for all materials. 

The concentration of salt in physiological systems is on the order of 150 mM, which corresponds to approximately 350 water molecules for each cation-anion pair. Eor this reason, investigations of salt effects in biological systems using detailed atomic models and molecular dynamic simulations become rapidly prohibitive, and mean-field treatments based on continuum electrostatics are advantageous. Such approximations, which were pioneered by Debye and Huckel [11], are valid at moderately low ionic concentration when core-core interactions between the mobile ions can be neglected. Briefly, the spatial density throughout the solvent is assumed to depend only on the local electrostatic poten-... 

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... 

Further simphfication of the SPM and RPM is to assume the ions are point charges with no hard-core correlations, i.e., du = 0. This is called the Debye-Huckel (DH) level of treatment, and an early Nobel prize was awarded to the theory of electrolytes in the infinite-dilution limit [31]. This model can capture the long-range electrostatic interactions and is expected to be valid only for dilute solutions. An analytical solution is available by solving the Pois-son-Boltzmann (PB) equation for the distribution of ions (charges). The PB equation is... 

Finally, it must be recalled that the transport properties of any material are strongly dependent on the molecular or ionic interactions, and that the dynamics of each entity are narrowly correlated with the neighboring particles. This is the main reason why the theoretical treatment of these processes often shows similarities with models used for thermodynamic properties. The most classical example is the treatment of dilute electrolyte solutions by the Debye-Hiickel equation for thermodynamics and by the Debye-Onsager equation for conductivity. 

The discussion here follows the more quantitative treatment given by Debye and Bueche, although we seek to avoid their model consisting of a sphere throughout which the segment density is uniform, and beyond which it is zero. 

FIG. 4 The calculated internal energy of a 1-1 salt (line) is compared with the corresponding simulation results (open circles) obtained by Van Megen and Snook (Ref. 20). The Debye-Hiickel (DH, dashed line) and corrected Debye-Hiickel (CDH, full line) theory were used together with a GvdW(I) treatment of the uncharged hard-sphere mixture. The ion diameter was 4.25 A, the temperature was 298 K and the dielectric constant e was 78.36. 

Thus we can see that a combination of van der Waals treatment of hard sphere excluded volume and Debye-Huckel treatment of screening with a minor generalization to account for hole correction of electrostatic interactions yields quite accurate bulk thermodynamic data for symmetrical salt solutions. 

The Wd estimated from the RASCI is compared with other theoretical calculations [46, 54] in Table V. The present DF estimate of Wd deviates by 21%(29%) from the SCF (RASSCF) estimate of Kozlov et al. [46] and by 17% from the semiempirical result of Kozlov and Labzovsky [54], while our RASCI result departs by 6% (3%) from the SCF-EO (RASSCF-EO) treatment of Kozlov et al. [46] and is in good agreement with the semi-empirical result of Kozlov and Labzovsky. At this juncture, we emphasize that our computed ground state dipole moment of BaF (p(. = 3.203 debyes) is also reasonably close to experiment p(. = 3.2 debyes (see Table 5 of Ref. 38). 

More rigorous Debye-Hiickel treatment of the activity coefficient... 

In dilute solutions it is possible to relate the activity coefficients of ionic species to the composition of the solution, its dielectric properties, the temperature, and certain fundamental constants. Theoretical approaches to the development of such relations trace their origins to the classic papers by Debye and Hiickel (6-8). For detailpd treatments of this subject, refer to standard physical chemistry texts or to treatises on electrolyte solutions [e.g., that by Harned... 

Among other applications of electrolyte solution theory to defect problems should be mentioned the application of the Debye-Hiickel activity coefficients by Harvey32 to impurity ionization problems in elemental semiconductors. Recent reviews by Anderson7 and by Lawson45 emphasizing the importance of Debye-Hiickel effects in oxide semiconductors and in doped silver halides, respectively, and the book by Kroger41 contain accounts of other applications to defect problems. However, additional quantum-mechanical problems arise in the treatment of semiconductor systems and we shall not mention them further, although the studies described below are relevant to them in certain aspects. 

As soon as the concentration of the solute becomes finite, the coulombic forces between the ions begin to play a role and we obtain both the well-known relaxation effect and an electrophoretic effect in the expression for the conductivity. In Section V, we first briefly recall the semi-phenomenological theory of Debye-Onsager-Falkenhagen, and we then show how a combination of the ideas developed in the previous sections, namely the treatment of long-range forces as given in Section III and the Brownian model of Section IV, allows us to study various microscopic... 

The ion-ion electrostatic interaction contribution is kept as proposed by PITZER. BEUTIER estimates the ion - undissociated molecules interactions from BORN - DEBYE - MAC. AULAY electric work contribution, he correlates 8 and 8 parameters in PITZER S treatment with ionic standard entropies following BROMLEY S (9) approach and finally he fits a very limited (one or two) number of ternary parameters on ternary vapor-liquid equilibrium data. 

While Debye and HUckel recognized the short-range repulsive forces between ions by assuming a hard-core model, the statistical mechanical methods then available did not allow a full treatment of the effects of this hard core. Only the effect on the electrostatic energy was included—not the direct effect of the hard core on thermodynamic properties. 

An important series of papers by Professor Pitzer and colleagues (26, 27, 28, 29), beginning in 1912, has laid the ground work for what appears to be the "most comprehensive and theoretically founded treatment to date. This treatment is based on the ion interaction model using the Debye-Huckel ion distribution and establishes the concept that the effect of short range forces, that is the second virial coefficient, should also depend on the ionic strength. Interaction parameters for a large number of electrolytes have been determined. 

These equations do not reduce to the Debye-Huckel model for dilute solutions and are thus only justified for the treatment of very concentrated solutions. 

In the conventional Debye-Huckel treatment the equilibrium radial distribution function for a pair of reactants g(r) is simply equal to exp(-w/RT) with w given by (15)... 

Salt effects in kinetics are usually classified as primary or secondary, but there is much more to the subject than these special effects. The theoretical treatment of the primary salt effect leans heavily upon the transition state theory and the Debye-Hii ckel limiting law for activity coefficients. For a thermodynamic equilibrium constant one should strictly use activities a instead of concentrations (indicated by brackets). 

A polyelectrolyte solution contains the salt of a polyion, a polymer comprised of repeating ionized units. In dilute solutions, a substantial fraction of sodium ions are bound to polyacrylate at concentrations where sodium acetate exhibits only dissoci-atedions. Thus counterion binding plays a central role in polyelectrolyte solutions [1], Close approach of counterions to polyions results in mutual perturbation of the hydration layers and the description of the electrical potential around polyions is different to both the Debye-Huckel treatment for soluble ions and the Gouy-Chapman model for a surface charge distribution, with Manning condensation of ions around the polyelectrolyte. 

This equation contains the activity coefficients 71, 72, and 7. Recall from the Debye-Hiickel treatment of ionic interactions in dilute solutions that the magnitude of these coefficients shows the following dependence on ionic strength fi for a solution of electrolytes ... 

An equation for estimating how the activity coefficient of an ion in dilute solutions is influenced by ionic strength. See Debye-Huckel treatment... 

To understand why depends on solution conditions, we must recognize that the behavior of solutes in solution depends upon the presence of other similar and/or dissimilar solutes, and electrolytes (/. e., charged solutes) are especially affected by the presence of all ionic species in solution. Unless we can account for these effects on the value of a for each substance, we cannot know the effective concentration of a substance at any particular analytical concentration, and we cannot comprehend the thermodynamic properties of these substances in solution. The Debye-Hilckel treatment offers us a means for estimating a. ... 

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