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).
VOl. 65
372 case if equation 1, the linear form, were used in setting up the normal equations. In the application of the non-linear program as prepared for the IBM-704 to the data, there are two fixed parameters: namely, unity, the first term in the polynomial, and the constant, RTInF. In addition, there are n parameters corresponding to the successive formation constants for the Sn(I1) chloro-complexes to be evaluated. With minor programmatic alterations, it is possible to allow n to assume increasing integral values. It has been considered that a given parameter was not significant if twice the standard deviation of the parameter encompassed zero. It has been found that the number of significant parameters as judged by this criterion with Duke and Courtney's data is equal to three. In Table I are given the results of the compu'ations. I LEAST S Q ~ A R EVSA L L E OF ~ F O R M A T COMTAYTS IO~ OF Sn(II) CHLORO-COiJIPLEXES AT 25" AT AU IONIC STRENGTH OF 2 03
If we represent by v t*he average number of chloride ions per tin atom in the complexes, then the uncomplexed chloride ion [Cl-] can be shown to be related to the total chloride ion [Cl-1, by [Cl-] = [Cl-lt
Ua
41 = 11 4 4 2 = 52 3 43 = 31 -1
0 26 1.8 2 3
Y
=
d In (fo
-
+
8n++ f HZ(3
SnOH+
+ H+
(ti)
Following the notation of Vanderzee and Rhodes, the equilibrium quotient for this reaction is defined as h. There is also the possibility of the formation of mixed complexes of the form SnOH
+
nC1-
SnOI-I(C1-)n+l-n
(6)
the equilibriuim quotient expression for which is given the notation 6,. It can be shown that the total tin(I1) concentration, [SnII], is related to the uiicomplexed bivalent tin, [Sn++],by the relation [SnII] = [Sn++] Z,(p,
+ S,?b/[H+])[Cl-]n
(7)
Since the total stannous ion concentration is the same in the two cell compartments, the relationship bet\$een the cell e.m.f. and the uncomplexed chloride ion concentration can be written ( 8 ) C E Vander ree a d (1962).
L).
E. Rhodeb, J . Am. Chem. Soc , 74, 3552
(11)
l ) / d In [Cl-] = AI[Cl-] . . . kAk[C1-Ik 1 Ai[Cl-] . . . Ak[C1-lk (12)
+
++ ++
With the substitution of the expression for v in equation 12 into equation 11 and simplifying, a IC l t h polynomial is obtained Ak[C1-lk+l - Ak([Cl-]t - Ic[SnII]1 [C1-Ik (AL~ - A ~ l ( [ C l - ] t- ( k - I)[SnII])1 [CI-]k-l . . .
+
+
( I - Al([Cl-]t
A comparison nith the values obtained by the authors7 is not posrible since they assumed a total of four significant formation constants to be defined by their data. It is to be recognized that the standard deviations given in Table I are a measure of the precision m t h which the data are represented by equation 2, it is only by repeated experiments that it is posqible to demonstrate whether these uncertainty estimates are realistic. Vanderzee mid Rhodes8 also have made electromotive force measurements with a concentration cell method to evaluate the formation constants of the Sn(I1) chloro-complexes. In addition to the equilibria Sn-' + nC1SnCIn+2-n (4) the equilihriu,m constants for which we have assigned the drsignntion p,, there is also the hydrolysis reaction of Sn++ which may be important in solutions of sufficiently low acidity, i.e.
p[SnrI](f0 - l ) / f o
Also, the average number of chlorine atoms per tin atom is
TABLE
Standard deviation
-
-
+
[SnII]))[Cl-] - [Cl-It = 0
+
+
(13)
Preliminary values of the Ak coefficients were obtained with the least squares solution of equation 9 with the assumption that [Cl-] = [Cl-1,. These quantities are not equal in the work of Vanderzee and Rhodes (whereas they were considered to be equal in the experiments of Duke and Courtney) because SnII concentrations of about 0.01 M were used and were maintained constant a t the various chloride ion concentrations, hence, a small but significant quantity of chloride ion is present in the tin complexes. The Ak-coefficients then were inserted in equation 13. The roots of equation 13 were obtained with an iterative numerical program developed at Los Alamos Scientific Laboratory by Mr. J. K. Everton. It is called the "floating point polynomial solver." KO ambiguity was involved in trying to decide which root to select since in each instance the number of positive real roots turned out to be one. After obtaining the first approximation to the uncomplexed chloride ion concentration by solving equation 13, these concentrations were resubstituted into equation 9 to get a second approximation to the A-coefficients. This process was repeated by the computer until convergence within 1 part in lo6was obtained. I n Table I1 are given the results of the least squares calculations of the A-coefficients with Vanderzee and Rhodes' data. It was found that three A-coefficients were sufficient to describe the system, even a t 45'. However, at 45', it appears that from the similarity of the &coefficients a t 0.500 M acid at 35 and 45', additional data at concentrations of chloride ion above 0.6 M would have been desirable. In a comparison of the least squares values of the A-coefficients with those determined by Vanderzee and Rhodes, it is noted that there is fairly good greement in the AI results, less satisfactory agreement in the A , values and poor agreement in the As terms.
Feb., 1961
373
XOTES
TABLE I1 thetical restriction on the reactivity of molecules just after the instant of energization) the rate conLEASTSQUARES VALUESOF A-COEFFICIENTS AS FUNCTIONS stant may pass through a maximum, and then OF TEMPERATURE AT p = 3.0 Tyw.,
C.
: ;
Ai
O
I
~
Ai
OP
At
UP
0.500 9 . 0 3 0 . 1 32.1 1 . 0 21.8 1 . 6 .lo0 Y,27 . 2 25.6 1 . 9 26.8 3.2 25 .500 13.70 .08 49.7 0.8 45.6 1 . 4 ,100 12.97 .09 44.7 0 . 9 32.4 1 . 6 35 ,500 15.92 .09 64.4 1 . 0 57.5 1 . 8 .lo0 13.74 .07 55.9 0 . 8 37.9 1 . 4 45 .500 18.3 .4 64.1 4 . 1 109.9 8 .lo0 15.8 .2 70.7 1.6 52.5 3 a The authors regard these numbers only as approximate standard deviations because the procedure used in this particular problem involved more than the usual least squares methods. Because of the modification of the chloride ion concentration together with the A-coefficients, straight-forward least squares error analysis does not apply. 0’
With the values of A 2 and i13as a function of acidity, Vanderzee and Rhodes calculated the hydrolysis constant, h, as a function of temperature. However, with the results shown in Table 11, inconsistent values of the hydrolysis constant were obtained. Accordingly, it was decided to calculate the heats for the A-coefficients since an accurate value of the hydrolysis const,ant of Sn(I1) does not appear to be available. Values of 2800, 3100 and 4770 cal./mole were obtained from the temperature dependence of A1, Az and A s at the 0.500 M acidity level. These values are in essential agreement with those report.ed7 far the corresponding Pcoefficients. T H E HIGH PRESSURE LIMIT OF UNIMOLECULAR REACTIONS BYM.C. FLOWERS AND H. M. FREY Chemistry DepaTtment, The Univcraity, Southampton, England Received J u l y 66,1960
Of the large number of homogeneous decompositions that have been studied kinetically in the gas phase, few are simple, most involve complex steps. In addition there is often a heterogeneous component of the decomposition as well as the simultaneous occurrence of side reactions. As a result few experimental data are available to test the predictions of the various detailed theories of unimolecular reactions. The data that, are available for such cases as X2O6,lcyclopropane2 and cyclobutane3 support the prediction of the theories that at sufficiently low pressures a decrease in the apparent first-order rate constants of unimolecular processes will occur. Until now no unimolecular reaction has been studied over a pressure range extending from the region where the “fall off” of the rat)e constant begins to pressures very many times greater than this. This region is of added interest since it has been suggested4 recently that (subject to a hypo(1) R. L. Mills and H. S. Johnston, J . A m . Chem. Soe., 7 3 , 938 (1951). (2) H. 0. Pritoliard, R. G. Sowden and A. F. Trotman-Diekenson, Proc. Roy. S O C .(London), g l S A , 416 (1953). (3) C. T. Qenaux, F. Kern and W. D. Walters, J . A m . Chem. Soc., 75, 6196 (1953).
diminish, with increasing pressure. We wish to report a study in this pressure range on the thermal isonierization of 1,l-dimethylcyclopropane. This has been shown to be a unimolecular reaction5 whose apparent first-order rate constant begins to decrease below 10 mm. The apparatus and analytical technique have been described elsewhere.6 Runs were carried out at 460.4’ in the pressure range 16 to nearly 1600 mm. The rate constants obtained are shown below. Pressure, mm. 10‘ k, uec.-l Pressure, mm. 104 k, sec. -1 Pressure, mm. lo4 k, see. -1
16 2 44 253 2.46 1008 2.41
39 2 43 283 2 42 1057 2.43
55 2.47 567 2 40 1447 2 44
54 2.43 294 2 44 1240 2.41
104 2.44 774 2 41 1596 2 44
217 2 43
Increasing the pressure one hundred and fiftyfold from the point where the high pressure limit is first reached has no effect on the rate constant. It must be concluded that the high pressure rate constant of this unimolecular reaction does not go through a maximum. The authors thank the Esso Petroleum Company for the award of a research studentship to A1.C.F. (4) D. J. Wilson, J . Phys. Chem., 64, 323 (1960). (5) M. C. Flowers and H. M. Frey, J . Chem. Soc., 3953 (1959). (6) M C. Flowers and H. M. Frey, Pioc Roy. Soc (London),2S78, 122 (1960).
-
A STUDY OF EQUILIBRIA I N T H E SYSTEM IODINE CYANIDE-POTASSIUM IODIDEWATER-HEPTL4KE BY G. LAP ID US^ AND G. M. HARRIS Department of Chemistry, University of Bufalo, B U ~ U Z O 14, N . Y. Received July 26, 1960
In some earlier work in this Laboratory,2 the almost instantaneous exchange of iodine between ICN and K I in aqueous solution was explained in terms of a rapid establishment of the equilibrium ICN H + + II* + HCN (1)
+
Jr
along with the well-known iodide/iodine equilibration I2
+ I-
13-
(2)
.li search of the literature showed there to be considerable disagreement in regard to the magnitude of K1. I
