Numerical Treatment in Determining Stability Constants by the Spectrophotometric Method of Corresponding Solutions Boiidar Grabaric, Ivan Piljac, and Ivan Filipovic Laboratory of lnorganic Chemistry, Faculty of Technology and lnstitute of lnorganic and Analytical Chemistry, University of Zagreb, Zagreb, Croatia, Yugoslavia
Numerical treatment for use with a digital computer is proposed for determining stability constants by means of an improved spectrophotometric method of corresponding solutions. This method has been employed in a study of the system of nickel(1I)-2-hydroxybutyrato complexes. Alternatively, the stability constants were refined by the Gauss method, according to the program by Tobias. Results from both methods are compared and the advantages of the method proposed in this paper are indicated.
Spectrophotometric determination of constants characterizing successive formation of mononuclear complexes in solution, using a procedure of corresponding solutions, was first proposed by Bjerrum ( I ) . However, despite the simplicity and accuracy of spectrophotometric measurements, practical use of the method of corresponding solutions has formerly been reported only with a few systems (2-7) and the stability constants given in these papers were obtained mainly by one of the known graphical methods. Because of subjective factors involved in graphical methods, these values cannot be unequivocal and should be taken merely as approximate values. This is probably the principal reason for such scarce use of the method of corresponding solutions. In two of our previous papers (8, 9), this method was applied for investigating a series of monocarboxylato complexes of cobalt(II), nickel(II), and copper(I1). Evaluation of stability constants was carried out, using the Gauss Z computer program by Tobias (IO), from the average number, il, of ligands bound to the central metal ion and free ligand concentration, [L]. In the latter procedure, however, parameters needed for calculation of il and [L] have not been determined entirely by numerical treatment. In the present paper, we propose to use a complete numerical scheme to determine il and [L], as well as for calculation of stability constants, basing the latter on least squares treatment and utilizing a digital computer. Results obtained in this way were compared with those obtained by the Tobias program. In addition to data handling, we have improved and rendered more economical our technique for preparing solutions. These are prepared by discontinuous additions of one solution directly into the other, which has been placed initially in a modified optical cell. (1) J. Bjerrurn, Kgl. Dan. Vidensk. Selsk. Mat.-Fys. Medd., 21, No. 4 (1944), (2) H. Olerup, “Jarnkloridernas Kornplexitet,” Lindstedts UniversitetsBokhandel. Lund, 1944. (3) S. Fronaeus,Acta Chem. Scand., 5, 139 (1951). (4) S.Ahrland, Acta Chem. Scand., 3, 783 (1949) (5) S.Ahrland,ActaChem. Scand., 5, 199, 1151 (1951). (6) J. Badoz-Lambling, Ann. Chim. (Paris), 8, 586 (1953). ( 7 ) B. Jezowska-Trzebiatowska, L. Pajdowski and J. Starosta, Rocz. Chem., 35,445 (1961). (8) B. Grabaric and I. Filipovic, Croat. Chem. Acta, 42, 479 (1970). (9) J. SaviC, M. Savic, and I . Fllipovic, Croat. Chem. Acta, 44, 305 (1972). (10) R. S. Tobias and M. Yasuda, J. Inorg. Chem., 2, 1307 (1963).
1932
The principle of corresponding solutions can be used to advantage, if the following conditions are met. (a) Ligand (L) absorbance must be negligible in comparison to complex (ML,) absorbance. (b) Ionic strength must be kept constant. (c) Components must neither associate, nor dissociate. (d) The Lambert-Beer law must hold for any individual complex species, i.e., Equation 1 must be obeyed, where A means absorbance; d, optical cell thickness; t n , molar absorptivity of the nth species; and [ML,] its concentration.
If the above conditions are satisfied, two or more solutions are said to be corresponding if they exhibit the same absorbance A at a fixed wavelength despite differences in metal ion ( CM) and ligand ( CL) total concentrations. Corresponding solutions possess the same percentage composition in complex species and therefore have identical values for it and [L]. The following relationships are valid for such solutions
CM’
> CM”
whence it is easy to derive the values for il and [L].
C’, - CL)’ (3) CM’ - chf” CM‘ CL” - CM” CL’ [L] = (4) CM’ - CM” The procedure required consists, first of recording absorption spectra of metal ion solutions in the presence and absence of ligands, in order to establish the wavelength at which the complex absorbs, but the free ligand does not absorb. Next, the dependence of absorbance on ligand concentration is determined a t this wavelength, keeping the metal ion concentration constant and using the same optical cell thickness d’. These measurements produce a function Ai’ = f’(CL’)i, a t total metal ion concentration CM’. At another concentration CM” and a different optical cell thickness d”, a second function is obtained Ai” = f”(CL”)i; CM“ is chosen so that CM’d’ = CM”d”. Pairs of corresponding solutions are obtained by reading CL‘ and CL” off the above curves, at the same values of absorbance, then ii and [L] are calculated using Equation; 3 and fi=
4.
Whenever this is possible, similar functions should be established at three, four, or more different metal ion concentrations. Values CL’, CL”, CL”’, etc., read off a t the same absorbance value, are then plotted against CM’, CM”, CM”’, etc. CL depends linearly on CM for each set of corresponding solutions, containing only mononuclear complexes, so that il is equal to the slope and [L] to the intercept of the line.
ANALYTICAL CHEMISTRY, VOL. 45, NO. 11, SEPTEMBER 1973
NUMERICAL TREATMENT Once A , = f(C,), values are tabulated, one can find a corresponding ligand concentration for any point on one curve by numerical interpolation on another curve. The total ligand concentrations obtained in this way and known metal ion concentrations in corresponding solutions are substituted into Equations 3 and 4 for calculation of values for IIand [L]. Parameter ft is related to the cumulative stability constants, p i , through the well-known equation
(5)
INPUT Number of points of curve 1 Number of points of curve 2 Curve 1 m A'i = f (Ci l j Cwve 2-A7 : f K ; 1; A'i
, CC;
Numerical interpolation for each polnt of curve 1 to Corresponding ligand concentration of curve 2 and inversely
where Fo([L]) stands for the Leden-Fronaeus function (11, 12),related to the stability constants via equation I
I Numerical integration and calculation I From known values for IIand [L], function Fo([L]) can be determined using Fronaeus' method (12). Since differentiating Equation 6 with respect to [L] and combining with Equation 5 gives
I of function F o ( l l I I according to eq.(7)1 1
I Weightingof F o l l t l ) by 1 I F ~ ~ l L I I m e t h oId numerical integration of II/[L] us. [L] will give the value of the integral and Fo([L]) can be easily calculated. Momoki, Sato, and Ogawa (13, 14) have derived relative statistical weights w' of function Fo([L])
w'
4
l/Fo"[L])
(8) from the law of error propagation, using results from polarographic measurements. Therefore, the weighted polynomials that are to be fitted to computed values by the least squares method are given by expressions of the form =
N:assumed rnoximal wmber of complex species
(9)
This procedure is justifiable, whenever the observed quantities differ by an order of magnitude and when the percentage error expected in each observation is presumed to be constant as was shown by Sabatini and Vacca (15). On the other hand, the function in question Fo([L]) is the same as in polarographic measurements, and the errors in spectrophotometric measurements are equal or less than those in polarographic measurements. The above weighting method was, therefore, used in the stability constants determination by the spectrophotometric method. Least squares fitting of polynomial Fo([L]) gave the statistically most probable values for the stability constants and corresponding standard errors, along with a percentage composition of complexes in solution, % ML, ( i = 1, 2, . . . ,N ) . This numerical approach to the handling of data obtained from the method of corresponding solutions has been programmed for an IBM 1130 computer and used in treatment of results with nickel(II)-2-hydroxybutyrato complexes. A block diagram for the computer program is given in Figure 1. (11) I . Leden, 2. Phys. Chem., 118A, 160 (1941). (12) S. Fronaeus, Acta Chem. Scand., 4, 72 (1950). (13) K. M o r n o k i , H. Sato, and H. Ogawa. Bull. Fac. Eng., Yokohama Nat. Univ., 16, 127 (1967). (14) K. M o r n o k i . H. Sato, and H. 0gawa.Anal. Chem., 39, 1072 (1967). (15) A . Sabatini and A. Vacca, J. Chem. Soc., Dalton Trans., 16, 1693 (1972).
M:l
least squares fitting o f F o l I L I l weighted polynomial
4
Calculation of percent composition of individual complex species lor known ligand concentration according lo Oh M l i
Blrll'
:
~100
FO (1 L IIcatc
I i = 1, 2 , . . N )
C Upi,
+
OUTPUT t i , %Hli ( i z l , . ? , . . M )
(j+ M-H.1
Figure 1. Block diagram for computer program
EXPERIMENTAL Apparatus. Absorption spectra were recorded on a Beckman Acta I11 spectrophotometer, while absorbance measurements a t fixed wavelength were carried out with a Carl Zeiss VSU 1 spectrophotometer with a reproducibility of 0.001 a t 0.4 A. Spectrophotometer cells were adapted to hold a larger volume of solution, to allow thermostating as well as mixing by bubbling air. Additions of solutions were made directly into the cell with a precision microburet (Metrohm E 451) having a reproducibility of 0.001 ml.
ANALYTICAL CHEMISTRY, VOL. 45, NO. 11, SEPTEMBER 1973
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Table I. Absorption as a Function of Total Ligand Concentration
0.1
CL‘,
CL“,
0.6
Y
z u
g
0.4
lOOO), so that curve R/[L] us. [L] is much steeper than in the case of relatively weak complexes, the numerical integration may be carried out utilizing the three-neighboring point parabolic approximation, used by Thun, Verbeek, and Vanderleen (16) in determining stability constants from potentiometric data. Polynomial Fo([L]) was then weighted and solved for the unknown stability constants by a normal least squares routine. Since the fitting is made on the basis of assuming H. Thun, F. Verbeek. and W. Vanderleen, J. Inorg. Nucl. Chem., 29, 2109 (1967).
I
1
I
I
0.2
0.1
llI,mole/I
Figure 4. Plot of fi/[L] vs. [L]
the existence of two, three, or more complex species within the studied range of ligand concentration, the calculated stability constants must be scrutinized critically. We may accept a system as a physically meaningful one, if, first, all constants associated with a given order of polynomial Fo([L]) (uiz., with a given number of complex species) are positive; second, if the standard errors are less than the constants themselves (the t-test can be used as a criterion) and finally, if graphical extrapolation of curve R/[L] us. [L] to [L] = 0 leads to the numerically ob(& = lim R/[L] us. [L] for [L] 0). In tained value for Table I1 stability constants of nickel(I1) 2-hydroxybutyrato complexes are given, assuming the existence of two, three, and four complex species. Graphical extrapolation of curve R/[L] us. [L] to [L] = 0 does not lead to a value of 32, which eliminates the assumption of two complex species. The assumption of four complex species, on the other hand, gives a negative value for 1114, i.e., -18708, and must, therefore, also be abandoned. The conditions set
-
ANALYTICAL C H E M I S T R Y , VOL. 45, NO. 1 1 , SEPTEMBER 1973
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Table 111. Stability Constants of Nickel(l1) 2-Hydroxybutyrato Complexes
P1
P2 P3
Proposed method
Graph
Gauss Za
Gauss Zb
Pot. (77)
52 f 2 821 f 15 2239 f 138
59 850 2489
49 f 1 862 f 28 1919 f 98
53 f 2 955 & 55 2092 f 162
53 f 2 781 f 10 4257 f 121
Refined stability constants obtained by the proposed method. Refined graphically obtained stability constants.
forth above for physically meaningful systems are thus met only by assuming the existence of three complex species within the studied range of ligand concentrations. In order to check the correctness of the proposed numerical treatment and, particularly, to verify the correct use of Momoki, Sato, and Ogawa’s weighting method (13, 14) of polynomial Fo( [L]), the present experimental data have been treated according to a program by Tobias (IO),based on the nonlinear Gauss-Newton least squares method, which provides for a refining of approximate stability constants directly from fi and [L] values. In one instance, approximate stability constants were obtained graphically, using the method by Fronaeus (12); otherwise the values for stability constants obtained by the method described presently were taken as the initial values for a Tobias refinement. The results are given in Table 111. Good agreement of constants obtained by either method is obvious in both instances. A comparison of the two sets of constants shows further that the present method differs from the Tobias method by being insensitive to the degree of accuracy of approximate values. Another advantage of the present treatment lies in saving computer time, since there are no reiterations involved in this method. The final accuracy is, however, practically the same with either method. So far the discussion was based on the sole existence of mononuclear complexes in solutions. But it should be kept in mind that the polynuclear complexes could be excluded with certainty only in such cases where measurements at a different wavelength, preferably even in another absorption band, yielded results agreeing with those given above. A presence of hydroxo complexes could have been detected by absorbance measurements in buffer solutions of
1936
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different acid-to-salt ratios. Differences in absorption would suggest a presence of hydroxo complexes. The system under study failed to show any dependence of absorbance on pH of buffer solutions, thus eliminating an existence of hydroxo complexes under the present experimental conditions. A still better check on the correctness of the stability constants reported in this paper would consist in attempting their determination by a different method, possibly one based on a different physicochemical principle, which can also discriminate between the existence and nonexistence of polynuclear and hydroxo complexes in solution. This has been done for the system under investigation by a potentiometric method. The results obtained in this way ( I 7) are included in Table 111. Good agreement of stability constants obtained by the two different methods and exceptionally good agreement of the constant PI, most frequently in literature considered as a criterion in comparing the stabilities of complexes, shows that this almost forgotten spectrophotometric method is simple and reliable despite its scarce use in the past. To be reliable, however, the spectrophotometric data must be subjected to statistical-numerical treatment by a digital computer, giving unequivocal results of maximum statistical probability. Such treatment also makes possible a better comparison of stabilities in different systems; it is, therefore, presently applied to a series of hydroxymonocarboxylato complexes now under study in this laboratory. Received for review December 8, 1972. Accepted March 21, 1973. (17) Unpublished results
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1973
