Debye limit

Debye s theory, considered in Chapter 2, applies only to dense media, whereas spectroscopic investigations of orientational relaxation are possible for both gas and liquid. These data provide a clear presentation of the transformation of spectra during condensation of the medium (see Fig. 0.1 and Fig. 0.2). In order to describe this phenomenon, at least qualitatively, one should employ impact theory. The first reason for this is that it is able to describe correctly the shape of static spectra, corresponding to free rotation, and their impact broadening at low pressures. The second (and main) reason is that impact theory can reproduce spectral collapse and subsequent pressure narrowing while proceeding to the Debye limit. 

Analogous investigations of the HF+BFs+NaBF system showed that in this case B has to be taken as equal to zero, so that the mean activity coefficient is approximately given by the Debye limiting law (21)... 

The values for g obtained from Equation 4.22 do not seem to be very different from those obtained from Equation 4.20, as shown in Table 4.1 and Figure 4.3a. It is easy to see that g must be equal to /2 as Os tends to zero by expanding the exponentials in the linear approximation (Debye limit). Naturally, Equation 4.15 gives us i = 1 in this limit, as an uncharged layer does not expel co-ions and salt is equally distributed between regions I and II. However, as shown in Table 4.2 and Figure 4.3b, the predicted salt-fractionation effect obtained by substituting Equation 4.22 into Equation 4.15 is markedly different from the Donnan equilibrium. 

Much attention has been given to the relaxation of various properties toward their equilibrium values, especially in the glasses literature. Kohlrausch first proposed a stretched exponential form as a description of viscoelasticity, while Williams and Watts suggested the same form for dielectric relaxation exp[-(t/r) ], where 0 < 9 < 1 and 0 = 1 corresponds to the Debye limit. The master equation solution, Eq. (1.41), has a decaying multiexponential form that could lead to a wide variety of behavior depending upon the system. 

It is evident in Fig. 25b that the ratio has a value close to 3 at high temperatures (T > 1.0) and declines steadily below T 1.0 until it reaches a value nearly equal to unity at low temperatures. While the Debye model of rotational diffusion, which invokes small steps in orientational motion, predicts the ratio ti/t2 to be equal to 3, a value for this ratio close to 1 is taken to suggest the involvement of long angular jumps [146, 147]. The ratio was observed to deviate from the Debye limit at lower temperatures in a recent molecular dynamics simulation study as well [148]. The onset temperature was thus found to mark the breakdown of the Debye model of rotational diffusion [145]. Recently, the Debye model of rotational diffusion was also demonstrated to break down for calamitic liquid crystals near the I-N phase boundary due to the growth of the orientational correlation [149]. 

When p is small the rotors execute many free rotation cycles between collisions (tc > T/) and the theory then gives for Ci(t), the free-rotor function Q, (f) for times shorter than tc. On the other hand when p is large (tc < t/), the rotors can execute only a small fraction of a full cycle before they are interrupted by a collision and should perform a kind of random walk in angle space. The theory then gives a rotational diffusion limit albeit not necessarily the Debye limit. 

Equation (1.69) is also known as the Debye-Hiickel limiting law, and the coefficient A is called its marginal coefficient or the coefficient of Debye limiting law. Value of the constant is determined from equation... 

For spherical particles, the excess Rayleigh ratio AR(K) over that of the continuum solvent is given in the Rayleigh-Debye limit by ... 

Consider a solution of a single strong electrolyte whose molality is m. From equations (10 53) and (10 54) it is evident that the Debye limiting law can be written in the form... 

In the extremely dilute solution where the Debye limiting law may be applied, the term on the lefb-hcmd side of this relation is given by equation (10 86). Mctking this substitution we obtain... 

This approximation enables us to replace the exponential in relation [4.43] with a limited expansion, whieh Debye limits to the second term -i.e. ... 

Low-temperature behaviour. In the Debye model, when T < 0q, the upper limit, can be approximately... 

The situation for electrolyte solutions is more complex theory confimis the limiting expressions (originally from Debye-Htickel theory), but, because of the long-range interactions, the resulting equations are non-analytic rather than simple power series.) It is evident that electrolyte solutions are ideally dilute only at extremely low concentrations. Further details about these activity coefficients will be found in other articles. 

The upper limit of the dimensionless variable Vp, is typically written in tenns of the Debye temperature 9, as... 

Low-temperature behaviour. In the Debye model, when T upper limit, can be approximately replaced by co, die integral over v then has a value 7t /15 and the total phonon energy reduces to... 

The solution detennines c(r) inside the hard core from which g(r) outside this core is obtained via the Omstein-Zemike relation. For hard spheres, the approximation is identical to tire PY approximation. Analytic solutions have been obtained for hard spheres, charged hard spheres, dipolar hard spheres and for particles interacting witli the Yukawa potential. The MS approximation for point charges (charged hard spheres in the limit of zero size) yields the Debye-Fluckel limiting law distribution fiinction. 

The theory of strong electrolytes due to Debye and Htickel derives the exact limiting laws for low valence electrolytes and introduces the idea that the Coulomb interactions between ions are screened at finite ion concentrations. 

Wlien KC) < i (i.e. at very low concentrations), we have the Debye-Htickel limiting law distribution fiinction ... 

The Debye-Htickel limiting law predicts a square-root dependence on the ionic strength/= MTLcz of the logarithm of the mean activity coefficient (log y ), tire heat of dilution (E /VI) and the excess volume it is considered to be an exact expression for the behaviour of an electrolyte at infinite dilution. Some experimental results for the activity coefficients and heats of dilution are shown in figure A2.3.11 for aqueous solutions of NaCl and ZnSO at 25°C the results are typical of the observations for 1-1 (e.g.NaCl) and 2-2 (e.g. ZnSO ) aqueous electrolyte solutions at this temperature. 

Figure A2.3.11 The mean aetivity eoeffieients and heats of dilution of NaCl and ZnSO in aqueous solution at 25°C as a fiinotion of z zjV I, where / is the ionie strength. DHLL = Debye-Htiekel limiting law.

In the limit of zero ion size, i.e. as o —> 0, the distribution functions and themiodynamic fiinctions in the MS approximation become identical to the Debye-Htickel limiting law. 

The osmotic coefficients from the HNC approximation were calculated from the virial and compressibility equations the discrepancy between ([ly and ((ij is a measure of the accuracy of the approximation. The osmotic coefficients calculated via the energy equation in the MS approximation are comparable in accuracy to the HNC approximation for low valence electrolytes. Figure A2.3.15 shows deviations from the Debye-Htickel limiting law for the energy and osmotic coefficient of a 2-2 RPM electrolyte according to several theories. 

Figure A2.3.15 Deviations (A) of the heat of dilution /7 and the osmotie eoeflfieient ( ) from the Debye-Htiekel limiting law for 1-1 and 2-2 RPM eleetrolytes aeeordmg to the DHLL + B2, HNC and MS approximations.

In Debye solvents, x is tire longitudinal relaxation time. The prediction tliat solvent polarization dynamics would limit intramolecular electron transfer rates was stated tlieoretically [40] and observed experimentally [41]. 

If the coefficients dy vanish, dy = 28y, we recover the exact Debye-Huckel limiting law and its dependence on the power 3/2 of the ionic densities. This non-analytic behavior is the result of the functional integration which introduces a sophisticated coupling between the ideal entropy and the coulomb interaction. In this case the conditions (33) and (34) are verified and the... 

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