370
YOTES
0
+
c m c lg/mll
I 01
02
0 3
0 4
05
Fig. 1.
1.01. 65
second virial at the lower concentrations (c < 0.1 g./ml.). Thus the orientation effect is much larger than the 12% calculated on the basis of noninteracting rigid rod-like molecules. This is in line with the strong attractive forces operative between sucrose molecules as calculated by Stigter. Interpretation of depolarization data in conjunction with the total scattered intensity as measured by Rgotherefore can furnish additional, useful information regarding the structure of solutions. Above a concentration of about 0.2 g./ml. (Y starts to fall and even seems to become negative, which would correspond to a tendency for perpendicular arrangement of the molecules (diatropic case). It is very doubtful, however, whether the calculation of cy is meaningful in this range. Maron and Lou’s data have not been corrected for secondary scattering.8 Moreover, it is possible that at these high concentrations the internal field uncertainty invalidates eq. 2 . In the above the optical activity of the sucrose molecules has not been taken into account. In natural light, however, any effect due to the asymmetry of the polarizability tensor should be negligible. Although at present no data exist, measurable differences in the light scattering envelope can perhaps be found if right and left circularly polarized incident light is used. Such studies are planned by Professor Heller at Wayne State University. (8) J. Kraut and R-. Dandliker, J . Polymer A c t , 18, 571 (1955); E. P. Geiduschek, zbzd., IS, 408 (1951).
TWO URANYL PEROXIDES
Fig. 2 .
BY LOUISSILVERMAN .4ND ROBERT h.SaLLACH
In this equal,ion, the “compressibility ratio” p is defined by
Atomics Internatzonal A Dzvzszon of North Amerncan Auzatzon, Inc. P.O. Box 509, Canoga P a r k , Calzfornza Recezued J u l y 16, 1960
In a recent investigation of the precipitation of uranium peroxide from concentrated uranyl sulfate and can be related to the osmotic pressure by solutions (such as used in the Atomics International “water boiler reactors”), and in certain analytical separations of uranium as peroxide, it was found that two different crystalline forms of uranium where n is the number of moles/ml., cy is an inte- peroxide could be precipitated readily. gral over the orientation correlation function The first is obtained by precipitation from hot h(r,$) and the radial distribution function g(r) solutions (1 to 2 JI U02S04,pH 2, 80’). Chemical of the solute is analysis showed that it is a di-hydrate, U04.2Ha0 cy == f 2ar2dr :in $d+ y ( r ) h(r,+) (theoretical analysis 70.47, uranium, 9.467* per€2 is the molecular anisotropy (a - b ) 2 / ( a 2b)2, oxide oxygen; found 70.2% uranium, 9.6y0 pera and b being the principal polarizabilities of the oxide oxygen). The second is obtained when precipitation is sucrose molecule. 62 can be obtained by extrappH 2 ) . olating 5 p / ( 6 - ’ i p ) to infinite dilution where cy = made at room temperature (1to 2 ill uozso4, By chemical analysis it is a tetra-hydrate, uo4. OandP = 1. In Fig. 1 Maron and Lou’s depolarization data 4Hz0 (theoretical analysis : 63.6% uranium, 8.55% are plotted. ‘Thestraight line obtained then is used peroxide oxygen; found 62.3% uranium, 8.7% to calculate CY from eq. 1, 2 and 3. The results peroxide oxygen). The tetra-hydrate can be converted easily to the are given in Fig. 2 . Restricting ourselves to condi-hydrate by digestion in acidified water (pH 2) a t centrations below 0.20 g./ml. LY is found to be positive and increasing, which means that there is an 80’ for 4 hours. (SOY0 conversion in 45 minutes.) The procedure for preparing these compounds is as follows: increasing tendency for the sucrose molecules to orient themselves parallel to each other (paratropic To a solution of U02SOc (1 to 2 M with sufficient sulfuric acid to lower the pH to 2 ) at the selected temperature, an case). The d u e of a is seen to csnstitute at least excess of 30%.H?Oz is added, causing essentially all the ura50yc of ( I ’ p - 1). The latter quantity is the nium to precipitate. The thick precipitate is immediately p - 1 = Jg(7)rG
+
filtered and generously washed with water, followed by several washings of methanol. The compound now is air dried at room temperature.
X-Ray Daraction Pattern Discussion.-Based on the analytical chemical results (analysis for U, analysis for HzOzand conversion to U308),there are two hydrates, of which the tetrahydrate easily can be converted to the dihydrate by boiling in p H 2 water a t 80'. The comparisons of the X-ray diffraction patterns for these materials show: (1) that Watt's1 dihydrate X-ray patterns are essentially correct, ( 2 ) that Zachariasen'sz early "tetrahydrate" material is really the dihydrate, and (3) that Dunn's3 "dihydrate" material is really the tetrahydrate. (1) G. W. Watt, S. L. ichorn and J. L. hIarley, J . A m . Chem. Soc., 72, 3341 (1950). ( 2 ) W. H. Zachariasen. General Physlcs Research Report, Part I, CP-1249, January 28, 1942, p. 14. (3) H. W.Dunn, "X-Ray Diffraction Data for Sonie Uranium Compounds," ORNL-2092, August 16, 1966, p. 41.
THE APPLICATION OF HIGH-SPEED COMPUTERS TO THE LEAST SQUARES DETERMIXATION OF THE FORMhTlON CONSTAXTS OF THE CHLORO-COMPLEXES OF TIN(I1)' BY8. W. RABIDEAU AND R. H. MOORE University of Cali/ornia, Los Alamos Scientific Laboratory, Los Alamos, New Mezico Received July 18, 1960
Bjerrum2 has emphasized the need not only to investigate new systems in the study of the stability of complex ions, but also has appealed for increased accuracy of measurement and for the determination of the temperature coefficients of the complexing reactions. Equally important to progress in this field, perhaps, is the selection of the best method wit,h which to carry out the analysis of the data. We would like to suggest that the significance of measurements of even the highest quality often may be improved by the application of the least squares crit,erion t,o the data. Rydberg, e t C L Z . , ~ , ~ have shown how digital comput'ers may be used in the least squares analysis of data for the case in which the coefficients of the function defining the formation constants of the successive complex ions appear in a linear form. However, wit'h developments in programming and the availability of high-speed computers, it is not necessary to restrict the applicat,ion of the method to the linear cases and, in fact, there are often advantages in the use of the non-linear form of the defining function. If t,he function in question is non-linear in t'he coefficients, the usual methods of solving the normal equations do not apply. In t,his case, as Moore and Zeigler5 have demonstrated, the iterative method (1) (a) This work was done under the auspices of the U. S. Atomic Energy Commission. (b) Presented in part a t the Northwest Regional Meeting of the American Chemical Society, Richland, Waahington, June 17, 1960. (2) J. Bjerrum, Chem. Revs., 4 6 , 381 (1950). (3) J. Rydberg. J. C. Sullivan and W. F. Miller, d c t a Chem. Scand., 13, No. 10, 2023 (1969). (4) J. Rydberg and .J. C. Sullivan, ibid., 13, No. 10, 2067 (1959).
due to Gauss and Seidel can be applied to obt,ain the least squares solutions. 111 an ideal method for the calculation of the successive formation const,ant,s (a) full use is made of all the daba, (b) the subjectivity of the computational method is removed, (e) the maximum amount of informat,ion is obtained from t,he data, including the determination of uncertainty estimates, and (d) appropriate weights can be conveniently applied if necessary. The usual graphical methods, such as the Leden6 method, have difficulty in meeting any of these requiremenbs; however, it is believed that all these specifications are admirably met by the suitably programmed high-speed computer. As a first example of' the use of the non-linear least squares solution of equations pertaining to the chloro-complexes of Sn(II), t,he work of Duke and Court,ney7 was considered. With tin amalgam electrodes, these authors det,ermined t,he effect of added chloride ion on the potential of a concentrat'ion cell. Perchlorates were added t.0 one compartment. and chlorides to the other under the conditions of constant acidity (2.00 M) and ionic strength (2.03 31). The expression which t'hey obt,ained and used for the graphical and determinant method of evaluating the format,ion constants can be written as exp(-nFE/RT) = 1
+ pl[C1-] + . . + p,,[Cl-]" ,
(1)
The amount of chloride ion in the t,in chloro-coinplexes was neglect.ed since the total Sn(I1) concentration was 5 X M and became smaller as a result of dilution. If tshenatural logarit'hm of equation l is taken it, follows tha,t - E = ( R T / n F ) ln(1
+ pl[C1-] +
,
..
+ ,%[Cl-ln)
(2)
By t,his transformation we have obtained an expression which is non-linear in t,he coefficients: however, we have also obt,ained the equation in a form which permits us t,o minimize the sums of the squares of t.he differences between the directly measured quantit.y, E , and the function on the right side of equation 2. That is, Q is minimized and takes the form
c 21
.Wi[Ei
i=l Q
=
(21
- F(CI-)]
- n)
0)
where w i is t'he weight factor, F(C1-) represents the quantity on the right side of equation 2 and the number 21 refers to the fact that t,here were 21 data points in Duke and Courtney's data.7 Inasmuch as the electromotive force measurements were made with equal precision, it can be shown that whatever constant weight wi is applied, it will cancel in the subsequent treatment of the equations; accordingly it is convenient to assign a unit weight to all the data points. However, this is not the (5) R. H. Moore and R. K. Zeigler, Los illamos Scientific Laboratory report, LA-2367, October 15, 1959. Available from the Office of Technical Services, U. S. Dept. of Commerce, Washington 25, D. C., $2.25. This reference contains a useful bibliography to the literature of least squares methods and discusses functions in which the parameters appear both linearly and non-linearly. (6) I. Leden, 2. physih. Chem., 8188, 160 (1941). (7) F. R. Duke and W. G. Courtney, I0u.a State College J . Sci., 24, 397 (1950).
