Kirkwood equation

Another way to obtain a relative permitivity is using some simple equations that relate relative permitivity to the molecular dipole moment. These are derived from statistical mechanics. Two of the more well-known equations are the Clausius-Mossotti equation and the Kirkwood equation. These and others are discussed in the review articles referenced at the end of this chapter. The com-... 

Table II shows clearly the large differences between various theories for many-electron systems. The Kirkwood-Muller equation always yields somewhat too large coefficients for the atoms which are the only spherical systems but the London equation deviates by a greater amount on the low side. The Slater-Kirkwood equation gives a high value for He but yields coefficients smaller than the empirical ones for all other cases.

The discussion in the previous section indicated that Nett values would be expected to exceed substantially the actual number of outer shell electrons. Table IV amply confirms this conclusion. Since N appears to the power in the Slater-Kirkwood equation, the deviations are exaggerated. Thus, in making a very similar treatment a few years ago in which slightly different empirical potentials were used, the writer31 found substantially smaller effective N values. For the same reason a relatively crude effective... 

In view of the complications of the intermolecular potential (as compared to the interatomic potential of the rare gas atoms) the comparisons for molecules in Tables II, III, and IV should be judged with caution. The apparent discrepancies from the theories for single atoms can be misleading. An example is the calculation for CH4 on the Slater-Kirkwood theory where Table IV shows the absurd value of 24 for the effective number of electrons. Pitzer and Catalano32 have applied the Slater-Kirkwood equation to the intermolecular potential of CH4 by addition of all the individual atom interactions and, with N = 4 for carbon and 1 for hydrogen, obtained agreement within 5 per cent for the London energy at the potential minimum. 

The Slater-Kirkwood equation (Eq. 39) was selected with N = 4 for carbon and N = 1 for hydrogen. The success of the equivalent calculation for the intermolecular interaction of CH4 molecules was mentioned in the previous section. Atoms, rather than bonds, were chosen as the basis for the calculation because the location of the atom centers is unambiguous and the approximation of isotropic polarizability is better for an atom than for a bond. Possible deviations from isotropic polarizability are discussed in Section V. Ketelaar19 gives for the atomic polarizabilities of hydrogen and carbon a = 0.42 and 0.93x 10-24 cm3, respectively. The resulting equation for the London energy is... 

In addition to the calculations in Table VI which are based upon Eq. 29, one may use the Slater-Kirkwood equation (Eq. 39). If one takes effective N values less by one than those given in Table IV for the rare gas atom adjacent to each halogen, the resulting London energies are very nearly the same as those in Table VI. 

The dependence of the rate constants (kp and kt) on the dielectric constant s is described by the Kirkwood equation [87] ... 

The effect of the medium on the rates and routes of liquid-phase oxidation reactions was investigated. The rate constants for chain propagation and termination upon dilution of methyl ethyl ketone with a nonpolar solvent—benzene— were shown to be consistent with the Kirkwood equation relating the constants for bimolecular reactions with the dielectric constant of the medium. The effect of solvents capable of forming hydrogen bonds with peroxy radicals appears to be more complicated. The rate constants for chain propagation and termination in aqueous methyl ethyl ketone solutions appear to be lower because of the lower reactivity of solvated R02. .. HOH radicals than of free RO radicals. The routes of oxidation reactions are a function of the competition between two R02 reaction routes. In the presence of water the reaction selectivity markedly increases, and acetic acid becomes the only oxidation product. 

A good description of variations in k2 and k6 values is given by the Kirkwood equation (3) ... 

When methyl ethyl ketone is diluted by benzene, the ratio io2/u i becomes lower. The amount of products formed by ROo decomposition increases. The ratio of constants for elementary steps k2/k1 = (u /t i)-[Rff] changes according to the Kirkwood equation (Figure 7). The /x2 ... 

The effect of relative permitivity D on the cyclo-trimerization of phenyl isocyanate was studied in the solvent system acetonitrile (AN) - ethylacetate (EA) using cyclic sulfonium zwitterion VI as catalyst. It was found that the rate constant increased with the increase of relative permitivity D of the solvent system, the experimental data correlate well with the Kirkwood equation ... 

The value of B in equation 11.1 for the interaction of two atoms i and j may be calculated from the Slater-Kirkwood equation,... 

The value of B, the attractive potential, in the 6-12 equation (11.1), as calculated from the Slater-Kirkwood equation for two of the indicated atoms or groups interacting. 

The method of calculating of complex mixtures using the above equation would theoretically be correct only if the mixture behaves like an ideal solution. Since, most solvent mixtures may exhibit a high degree of intermolecular association af such system would lead to a deviation from the experimental data. The simpliLed Onsager-Kirkwood equation provides only a good approximate dielectric constant for mixed solvent systems. 

One approach (Melander, 1977) treats the protein as a dipole and applies the Kirkwood equation (Kirkwood, 1943), which attributes the influence of ionic strength to both electrostatic and hydrophobic forces [Eqs. (8.63) or (8.64)]. 

Table 2.1 Parameters of Huoss-Kirkwood equation for 8 and y relaxations of PCHEM. (From ref. [33])...

Hendrickson (1961) and Scott and Scheraga (1965, 1966a, b, c) have developed procedures for obtaining the constants of eqs. 5 and 6. The coefficients ci3- or ei3- of the attractive terms are obtained by using the Slater-Kirkwood equation... 

The nonelectrostatic terms include van der Waals, polarization, repulsive, and zero-point energies. The van der Waals energy has been calculated in several ways with results ranging from 5 to 15 kcal./mole. Examples of these results are indicated in Table I. In the first two entries polarizabilities were taken from Bottcher (3), and the nitrate group was treated as a single entity with an ionization potential of 99 kcal./mole for the London equation (23) and an effective electron number of 24 for the Slater-Kirkwood equation (23). A simple CsCl type of lattice was assumed. 

For the van der Waals energy in N205 we have again adopted the Slater-Kirkwood equation to individual atom pairs. The nitrate group was used with the electron numbers and polarizabilities noted previously. For N02+, estimated polarizabilities of 0.7 A.3 for N and 0.5 A.3 for O were based on the charges (20) N = +.58 and O = +.21. Since N02+ contains a total of only 16 outer electrons we have arbitrarily chosen electron numbers of 6 and 5, respectively, for N and O. The resulting van der Waals energy is 13.2 kcal./mole. 

For purely electrostatic solute/solvent interactions, the Kirkwood equation, Eq. (4-27) [56], is applicable, which relates the standard molar Gibbs free energy of transfer of spherical dipolar molecules of radius r and dipole moment // from the gas phase (fir = 1) to a continuous medium of relative permittivity r-... 

However, the most recent discussions favour these high values of g although values of the order of 20% lower had ori nally been favoured. This is because the Frohlich equation [equation (1)] differs from the earlier version of Kirkwood, and treats the inner field in a more nearly correct manner. It is no longer necessary to make a calculation of the HjO dipole moment in its surroundings in the liquid, as had been necessary in the application of the Kirkwood equation. The dipole moment of the free molecule, /i = 1.84D, is used in equation (1), together with = 1.80 at 293 K. This leads to a value of = 2.82, which is sufficiently close to that calculated from the computer dynamics model to warrant optimism for future calculations. The exact choice of will continue to present difficulties until the far-i.r. data are complete over a wide range of temperature. 

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