Gronwall

A great deal of attention has been paid to the question of the necessity of having logarithmic terms in the expansion (Gronwall 1937, Bartlett 1952, 1955, Fock 1954, Hylleraas 1955), but Kino-shita has pointed out that although such terms may be convenient from the. numerical point of view (Hylleraas and Midtdal 1956), they are not necessarily required by the form of the Schrodinger equation itself. 

Gronwall, A. Dextran and Its Use in Colloidal Solutions Academic Press, New York, 1957. 

An alternative would be the hyperspherical coordinate system, introduced into the study of the ground-state helium by Gronwall [86], developed for nuclear reactions by Delves [87, 88], adopted in molecular reactive collisions by Smith [89], and initiated applications to two-electron excited QBSs by Macek [90]. The hyperradius p and one of the hyperangles, the radial hyperangle a, are defined by... 

For glyco- and mucoproteins the usual reagent is periodic acid-Schiff reagent (abbreviated PAS). This method was initiated by Koiw and Gronwall (K14) in 1952. Quantities of protein about four times as large as usual are applied to the paper (G19, K15, K16, S15). 

K14. Koiw, E., and Gronwall, A., Staining of protein-bound carbohydrates after electrophoresis of serum on filter paper. Scand. J. Clin, iz Lab. Invest. 2, 244... 

K15. Koiw, E., Wallenius, G., and Gronwall, A., L utflisation clinique de l electro-phorese sur papier filtre. Comparaison avec l electrophorese tube en U selon la methode de Tiselius. Bull. soc. chim. biol. 33, 1940 (1951). 

A2. Aronsson, T., and Gronwall, A., Improved separation of serum proteins in paper electrophoresis. A new Electrophoresis buffer. 2nd Intern. Congr. Clin. Chem., Stockholm (1957). 

Gronwall A, Ingelman B, Mosimann H (1945) Uppsala Lakarforening Forh 51 397... 

This dilemma led us to investigate the feasibility of a nonlinear extrapolation for the evaluation of E°. For the sake of brevity, we will not reproduce here the detailed mathematical derivations of Gronwall, LaMer, and Sandved s extended terms of the Debye-Hlickel theory. One can find these derivations in their original paper (4) or in Hamed and Owens classic monograph. For a description of the basic assumptions and the physical model of the interionic attraction theory one should consult Hamed and Owens monograph as well as the work of Gurney (8,9). 

AG°, AGt°, AGt° = standard free energy change asterisk indicates that the solvent contains an organic component or it may be nonaqueous GLS = Gronwall, LaMer, and Sandved k = Boltzmann constant m = molality N = Avogadro s number... 

Electromotive force measurements of the cell Pt, H2 HBr(m), X% alcohol, Y% water AgBr-Ag were made at 25°, 35°, and 45°C in the following solvent systems (1) water, (2) water-ethanol (30%, 60%, 90%, 99% ethanol), (3) anhydrous ethanol, (4) water-tert-butanol (30%, 60%, 91% and 99% tert-butanol), and (5) anhydrous tert-butanol. Calculations of standard cell potential were made using the Debye-Huckel theory as extended by Gronwall, LaMer, and Sandved. Gibbs free energy, enthalpy, entropy changes, and mean ionic activity coefficients were calculated for each solvent mixture and temperature. Relationships of the stand-ard potentials and thermodynamic functons with respect to solvent compositions in the two mixed-solvent systems and the pure solvents were discussed. 

In the application of equation (69) an arbitrary value of a is chosen so as to give calculated activity coeflScients which agree with those derived by direct experiment the proper choice of a is made by a process of trial and error until a value is found that is satisfactory over a range of concentrations. There is no doubt that the Gronwall-LaMer-Sandved extension represents an important advance over the simple Debye-Hilckel treatment, for it frequently leads to more reasonable values of the mean ionic diameter. The validity of equation (69) has been tested by a variety of activity measurements and the results have been found satisfactory were it not for the tedious nature of the calculations it would probably be more widely used. 

Ion-Association.—A device, proposed by Bjerrum, for avoiding the difliculty of integrating the Poisson equation when it is not justifiable to assume that Ziep/kT is much smaller than unity, involves the concept of the association of ions to form ion-pairs (cf. p. 96). It may be remarked that, in a sense, a solution, such as that of Gronwall, Sandved and LaMer, of the differential equation resulting from the use of the complete expression for the electrical density, makes the Bjerrum treatment unnecessary. The results obtained are, nevertheless, of interest, especially in connection with their application to media of low dielectric constant. 

---