Big Chemical Encyclopedia

Chemical substances, components, reactions, process design ...

Articles Figures Tables About

Permittivity formula

On the assumption that = 2, the theoretical values of the ion solvation energy were shown to agree well with the experimental values for univalent cations and anions in various solvents (e.g., 1,1- and 1,2-dichloroethane, tetrahydrofuran, 1,2-dimethoxyethane, ammonia, acetone, acetonitrile, nitromethane, 1-propanol, ethanol, methanol, and water). Abraham et al. [16,17] proposed an extended model in which the local solvent layer was further divided into two layers of different dielectric constants. The nonlocal electrostatic theory [9,11,12] was also presented, in which the permittivity of a medium was assumed to change continuously with the electric field around an ion. Combined with the above-mentioned Uhlig formula, it was successfully employed to elucidate the ion transfer energy at the nitrobenzene-water and 1,2-dichloroethane-water interfaces. [Pg.41]

Another factor influencing the reactivities of polar particles is their nonspecific solvation. Since both the individual particles, namely phenol and peroxyl radicals and their complex are polar, rate constants must depend on the polarity of the medium, its permittivity s, in particular. This was confirmed in experiments with mixtures of benzene and methylethyl-ketone, which showed that kq diminishes as the concentration of methylethylketone decreases provided the hydrogen bonding between the benzene and methylethylketone molecules are taken into account [10]. The dependence of ogkq on the medium permittivity s is described by the formula... [Pg.523]

In order to allow for the crystal structure itself, which will modify the interaction energy, it is possible, as a first approximation, to assume that the force of attraction is diluted in the crystal by an amount equal to its relative permittivity. The modified formula is then... [Pg.68]

Solution The discrepancy between the equation given here and Equation (27) arises from the fact that the equation above is written for cgs units whereas Equation (27) applies to SI. Remember that e = e o in Equation (27) and that the vacuum permittivity Cq usually appears with the factor 4r. When we combine Equations (6) and (26), the ratio 47r/67r reduces to 2/3. Many electrokinetic formulas differ by a factor 4ir, depending on the system of units being used, and the reader is cautioned to be aware of this difference. The presence or absence of the vacuum permittivity in the equation is the key to the system being used, although this factor is sometimes hidden, as in the case of Equation (27). [Pg.543]

Interaction Formulae for ru Typical solvent relative permittivity, e (nm) -Reff (nmf Attractive Repulsive... [Pg.238]

The above-mentioned formula applies when there is a vacuum between the plates 0 is the permittivity of the vacuum ... [Pg.240]

It is very different to calculate the energy-loss function Im[-1/ e(q, a>)] theoretically. At <7—>0 one can use the optical data. This way Lindhard90 has obtained a formula for the permittivity of the electron gas, while the authors of Ref. 100 have constructed a semiempirical formula for e(q, n>) of water. [Pg.283]

If at certain frequencies cjs the dielectric permittivity of the medium e is zero, formula (5.22) can be rewritten as211... [Pg.320]

A brief list of basic assumptions used in the ACF method precedes the detailed analysis of the results of calculations. The derivation of the formula for the spectral function is given at the end of the section. The calculations demonstrate a substantial progress as compared with the hat-flat model but also reveal two drawbacks related to disagreement with experiment of (i) the form of the FIR absorption spectrum and (ii) the complex-permittivity spectrum in the submillimeter wavelength region. We try to overcome these drawbacks in the next two sections, to which Fig. 2c refers. [Pg.79]

In this section we calculate the complex permittivity (v) and the absorption coefficient ot(v) of ordinary (H20) water and of fluoromethane CH3F over a wide range of frequencies. We shall first write down the list of the formulas useful for further calculations. [Pg.140]

In this section we have to calculate the complex permittivity s (v) and the absorption coefficient a(v) of ordinary (H2O) water over a wide range of frequencies. It is rather difficult to apply rigorous formulas because the fluctuations of the calculated characteristics occur at a small reduced collision frequency y typical for water (in this work we employ for calculations the standard MathCAD program). Such fluctuations are seen in Fig. 13b (solid curve). Therefore the calculations will be undertaken for two simplified variants of the hat model. Namely, we shall employ the planar libration-regular precession (PL-RP) approximation and the hybrid model.26... [Pg.144]

In Table V the fitted free and estimated statistical parameters are presented. For calculation of the spectral function we use rigorous formulas (130) and Eqs. (132) for the hybrid model. For calculation of the susceptibility %, complex permittivity , and absorption coefficient a we use the same formulas as those employed in Section IV.G.2 for water.29... [Pg.150]

Starting with the important example of ordinary water, we choose temperatures 22.2°C and 27°C. We compare our theory with the recorded FIR spectra [42, 56] of the complex permittivity/absorption. At low frequencies we use for this purpose an empirical formula [17] by Liebe et al. these formulas were given also in Section IV.G.2.a. The values of the employed molecular constants are presented in Table VI and the fitted parameters in Table VII. The Reader may find more information about experimental data of liquid H2O and D2O in Appendix 3. [Pg.174]

As a second example, we consider liquid fluoromethane CH3F, which is a typical strongly absorbing nonassociated liquid. For our study we choose the temperature T 133 K near the triple point, which is equal to 131 K. The relevant experimental data [43] were summarized in Table IV. As we see in Table VIII, which presents the fitted parameters of the model, the angle p is rather small. At this temperature the density p of the liquid, the maximum dielectric loss and the Debye relaxation time rD are substantially larger than they would be, for example, near the critical temperature (at 293 K). At such small (5 the theory given here for the hat-curved model holds. For calculation of the complex permittivity s (v) and absorption a(v), we use the same formulas as for water. [Pg.177]

In spite of numerous studies, the properties of liquid water are still far from been understood at a molecular level. For instance, large isotope effects are seen in some properties, such as the temperature of maximum density, which occur at 277.2 K in liquid H20 and 284.4 K in D20. The isotope shift 7.4 K will be used below with the purpose to employ the Liebe et. al. formula [17] for calculation of the low-frequency dielectric permittivity of D20 in analogous way as it used for H20. [Pg.198]

Figs. 32a-c illustrate the absorption spectra, calculated, respectively, for water H20 at 27°C, water H20 at 22.2°C, and water D20 at 22.2°C dotted lines show the contribution to the absorption coefficient due to vibrations of nonrigid dipoles. The latter contribution is found from the expression which follows from Eqs. (242) and (255). The experimental data [42, 51] are shown by squares. The dash-and-dotted line in Fig. 32b represents the result of calculations from the empirical formula by Liebe et al. [17] (given also in Section IV.G.2) for the complex permittivity of H20 at 27°C comprising double Debye-double Lorentz frequency dependences. [Pg.211]

The theory of wideband complex permittivity of water described in the review drastically differs from the empirical double Debye representation [17, 54] of the complex permittivity given for water by formula (280b). Evolution of the employed potential profiles, in which a dipole moves, explored by a dynamic linear-response method can be illustrated as follows ... [Pg.246]

A classical resonance-absorption theory [66, 67] was aimed to obtain the formulas applicable for calculation of the complex permittivity and absorption recorded in polar gases. In the latter theory a spurious similarity is used between, (i) an almost harmonic perturbed law of motion of a charge affected by a parabolic potential (ii) and the law of motion of a free rotor, this law being expressed in terms of the projection of a dipole moment onto the direction of an a.c. electric field. [Pg.269]

Now we apply the additivity approximation corresponding to Eq. (387). Namely, we sum up the contributions of three sources of dielectric loss due to (a) reorienting dipoles, (b) oscillating anions, and (c) oscillating cations. Combining Eqs. (387), (388), and (399a), we write the formula for the total complex permittivity ... [Pg.280]

In our case of 1-1 electrolytes we have the following relations for the concentrations N 1 = N = Nmn. The first term in Eq. (413a) is calculated by using the formulas presented in Section IV. However, generally speaking, the Kirkwood correlation factor now should be determined with account of ionic static permittivity as... [Pg.280]

However, the latter formula is not more applicable, if xlaa is rather long and/ or Cm is rather high, so that the zero-frequency ionic contribution Aefon(0) to permittivity is noticeable in comparison with the static permittivity es of the solution. We note that the Kirkwood correlation factor g is used for calculation of the component p in Eq. (387). Thus, even in our additivity approximation the solvent permittivity )jip is determined in this case by concentrations of both solution components. This complication leads to a new self-consistent calculation scheme. [Pg.289]

The effective medium theory consists in considering the real medium, which is quite complex, as a fictitious model medium (the effective medium) of identical properties. Bruggeman [29] had proposed a relation linking the dielectric permittivity of the medium to the volumetric proportions of each component of the medium, including the air through the porosity of the powder mixture. This formula has been rearranged under a symmetrical form by Landauer (see Eq. (8), where e, is the permittivity of powder / at a dense state, em is the permittivity of the mixture and Pi the volumetric proportion of powder / ) and cited by Guillot [30] as one of the most powerful model. [Pg.309]

For reactions in solution, the exponential term appropriate to the gas phase has to be modified to include the contribution accounting for the charges and the solvent. Calculations show that AG has to be modified by a term involving er, the relative permittivity of the solvent, and the charges on the ions. This term turns out to be the same as that appearing in the collision formula, Equation (7.3), i.e. z.KZ.ne1 / 47T o r ry, so that... [Pg.280]

For the closed description of the electron transfer in polar medium, it is necessary to express the reorganization energy in the formula (27) via the characteristics of polar media (it is assumed that the high-temperature approach can be always applied to the outer-sphere degrees of freedom). It was done in the works [12, 19, 23], and most consistently in the work of Ovchinnikov and Ovchinnikova [24] where the frequency dependence of the medium dielectric permittivity e(co) is taken into account exactly, but the spatial dispersion was neglected. [Pg.29]

In this formula, e0 = e(0) is the static permittivity, qe is the electron charge, f are the donor and acceptor coordinates. [Pg.30]

These are obtained by introducing an explicit time dependence of the permittivity. This dependence, which is specific to each solvent is of a complex nature, cannot in general be represented through an analytic function. What we can do is to derive semiempirical formulae either by applying theoretical models based on measurements of relaxation times (such as that formulated by Debye) or by determining through experiments the behaviour of the permittivity with respect to the frequency of an external applied field. [Pg.122]

When velocity gradient is absent, the above formula looks like any other formula for dielectric permittivity for a system with the only relaxation process, which is used for estimation of dielectric relaxation time. [Pg.154]

A frequency dependence of complex dielectric permittivity of polar polymer reveals two sets or two branches of relaxation processes (Adachi and Kotaka 1993), which correspond to the two branches of conformational relaxation, described in Section 4.2.4. The available empirical data on the molecular-weight dependencies are consistent with formulae (4.41) and (4.42). It was revealed for undiluted polyisoprene and poly(d, /-lactic acid) that the terminal (slow) dielectric relaxation time depends strongly on molecular weight of polymers (Adachi and Kotaka 1993 Ren et al. 2003). Two relaxation branches were discovered for i.s-polyisoprene melts in experiments by Imanishi et al. (1988) and Fodor and Hill (1994). The fast relaxation times do not depend on the length of the macromolecule, while the slow relaxation times do. For the latter, Imanishi et al. (1988) have found... [Pg.154]


See other pages where Permittivity formula is mentioned: [Pg.128]    [Pg.220]    [Pg.155]    [Pg.362]    [Pg.5]    [Pg.131]    [Pg.136]    [Pg.274]    [Pg.317]    [Pg.329]    [Pg.89]    [Pg.143]    [Pg.144]    [Pg.215]    [Pg.229]    [Pg.281]    [Pg.155]    [Pg.113]   
See also in sourсe #XX -- [ Pg.472 ]




SEARCH



Complex permittivity empirical formula

Permittance

Permittivities

Permittivity

© 2024 chempedia.info