Born’s formula

From Born s formula (cf. Problem 6.4) derive an expression for the difference in the energy of solvation of a spherical ion in two solvents with different dielectric constants. 

Despite the apparent similarity of the Bohr and the Bethe stopping power formulae, the conditions of their validity are rather complimentary than the same. Bloch [23] pointed out that Born approximation requires the incident particle velocity v ze jh, the speed of a Is electron around the incident electron while the requirement of Bohr s classical theory is exactly the opposite. For heavy, slow particles, for example, fission fragments penetrating light media, Bohr s formula has an inherent advantage, although the typical transition energy has to be taken as an adjustable parameter. 

Born s idea was taken up by Kirkwood and Onsager [24,25], who extended the dielectric continuum solvation approach by taking into account electrostatic multipole moments, Mf, i.e., dipole, quadrupole, octupole, and higher moments. Kirkwood derived the general formula ... 

The lattice energy U is defined as the energy released U is therefore negative by thermodynamic convention) when a mole of the requisite free gaseous ions comes together from infinite interionic separation to make up the crystal. If N is Avogadro s number (6.0221 x 10 ), we have the Born-Lande formula ... 

A case illustrating the metaphorical approach to Alchemy is the Swiss-born master of fantasy, Paul Klee (1879-1940). Klee was early championed by the French Surrealists, and was one of the few modern painters mentioned by name in the First Manifesto of 1924. For Breton, Klee was particularly to be recommended as a pioneer of automatism, an artistic practice which we should now recognize to have arisen nearly a century earlier directly out of populist Spiritualist experiments. Klee s own writings, however, make plain the fact of a much more profound, quasi-philosophical impulse, one heavily redolent of traditional hermetic dialectics expressed by the perennial formula of the caniunctio oppositomm. Klee s transcendental and blatantly mystical aspirations were best manifested in his Schopferische Konfession (1920). As Klee claimed. 

Here, we introduce the convention of denoting the solvent by a suffix, and the solute by a superscript index. Hence, ss stands for the dielectric constant of the solvent S, and Qlon is the total charge of the ion. Surely, Born was aware that this is a crude approximation, but his formula led to a qualitative understanding of the experimentally observed values of solvation energies. Indeed, it was later used to define ionic radii Rlon, and thus it became in some way a self-fulfilling prophecy. 

Analysis and mass spectral molecular weight determination established the empirical formula, C21H26N2O3, for stemmadenine (8, 116, 117). Its UV-spectrum was characteristic of an indole (cf. ref. 55), while the IR-spectrum indicated the presence of a normal ester grouping (1718 cm-1) and the absence of any substituent in the indole aromatic ring (116). These findings were fully borne out by NMR-spectroscopy which showed the presence of an indole NH (9.3 S), four aromatic protons, and a carbo-methoxyl methyl singlet (3.79 S). A single vinyl proton quartet (5.4 S)... 

Taking into account once again the fact that the Green s tensor (r r) exhibits either a singularity or a peak at the point where = r, one can calculate the Born approximation Gg [A5 (r) E (r)] using the formula... 

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