sluggish in changing orientation during its fall so that no “stable” run could be achieved. However, many runs were performed and their time readings averaged. The experimental data for the silicon sample were as follows (19): 62.114 sec in the light liquid and 199.42, in the heavy liquid. Size, 0.42 cm; weight, 6.8 mg. The procedure, apparatus and materials used were equal to those of Ref. ( I ) . The calculated density (by Equation 2) yielded a surprisingly good result, 2.3289 as compared to 2.3290 reported in literature (7). The explanation probably lies in the fact that, as the shape factor of a compact body does not markedly differ from unity, the various fall-orientations were not significantly different in their effects, or at least, repeated themselves equally enough in the two liquids. Also, the average of many runs represented a mean orientation of even improved equality in the two liquids. This allowed Equations 4-7 to apply with adequate accuracy. Application of error analysis calculations has resulted in Api = 0.7 x 10-3, which is, too, in agreement with the experimental results.
CONCLUDING REMARKS Selecting a reference body of a density close to that of the test sample should be very useful for increasing accuracy, especially in measuring samples of high densities. In practice, this can always be easily done. When studying
fine density differences, e.g., on investigating isotopic composition or variation of lattice structure, the reference body can be made of an available modification of the same material under investigation having a well-known density. This provides a really small pt-pj value, which helps increase accuracy considerably. If, in addition, the size and shape of the reference body is made to be approximately close to those of the test sample, the liquids are chosen to have a wide density difference and high, approximately equal viscosities, and a refined timing technique is used, then an accuracy of additional several decimals may be gained. Precautions are always necessary with respect to fall runs showing a drift from the vertical axis of the cylinders. Errors from lack of laminarity and wall effect may be eliminated altogether by correction factors. The liquids should be thoroughly homogenized before use and the temperature of the liquid-filled cylinders be kept constant for a sufficiently long time prior to the fall runs. Validity tests can be included in the experimental procedure to help detect any loss of control such as temperature change in time. This can be done by including in the fall runs additional one or more reference bodies of known densities, thus checking the overall state of control of the experiment. Received for review December 8, 1971. Accepted March 19, 1973.
Computer Approach to the Continuous Variations Method for Spectrophotometric Determination of Extraction and Formation Constants Werner Likussar lnstitute of Inorganic and Analytical Chemistry, University of Graz, A-8010 Graz. Austria
A modified continuous variations method based on the use of a normalized absorbance scale is utilized to evaluate the composition of metal complexes and the corresponding formation and extraction constants, respectively. A set of absorbance readings and corresponding mole fractions is input to a computer. These data are fitted to equations describing the continuous variations plots for distinct complex models. That complex model with the best fit to the appropriate function is assigned to be the most probable one. A computer program to accomplish this calculation, Jobcon, is given in Fortran I V . The applicability and performance of this method are demonstrated by test calculations on several metal chelate systems. Some problems accompanying the proposed method are discussed and its limitations are outlined.
The form of general theoretical continuous variations curves has been discussed by different authors (1-9). A new approach to the continuous variations method has been described in a previous paper (9) in which a “normalized 1926
absorbance” term was used in a rigorous treatment to develop general equations which describe the plots obtained when the continuous variations method is applied. In the same paper ( 9 ) , a modified tangents method has been developed and it is now possible to determine the composition and formation constant of the investigated complex with more reliability than previously possible. This improved method for calculating formation constants is limitated in its application. The conditions for the application of the continuous variations method are that only one complex predominates and that the formation constants to be calculated are in a certain range. It is the author’s experience that these conditions are fulfilled very often when the continuous variations method is applied to Y . Schaeppi and W.D. Treadwell, Helv. Chirn. A c t a , 31, 577 (1948). G. Schwarzenbach, Helv. Chirn. A c t a . 32, 839 (1949). E. Asrnus, Z. Anal. Chern.. 183, 321 (1961). K. S. Klausen and F. J. Langmyhr, Anal. Chirn. A c t a , 28, 335 (1963). (5) K. S.Klausen and F. J. Langmyhr, Anal. Chirn. Acta. 40, 167 (1968). (6) G. F. Atkinson, Anal. Chern.. 44, 1098 (1972). (7) 6 . W.Budesinsky. Anal. Chirn. Acta. 62. 95 (1972). (8) A. Ringborn and L. Harju, Anal. Chirn. Acta, 5 9 , 33, 49 (1972). (9) W.Likussar and D. F. Boltz,Anal. Chern.. 43, 1265 (1971).
(1) (2) (3) (4)
ANALYTICAL CHEMISTRY, VOL. 45, NO. 11, SEPTEMBER 1973
MOLE FRACTION ( X ) Figure 1. Continuous variations plot for a weak complex system (aluminum-8-quinolinol complex) The triangular plot is a hypothetical one for a nondissociated 1 : 3 complex
studies on common spectrophotometric determinations of metal ions. If the continuous variations method is used in complex formation studies, which are more involved, a number of difficulties may arise. The following instances need to be considered. (1) Over the complete range of mole fractions, certainly only one type of complex is formed, but the complex is too weak. In the case of small formation constants, one can increase the concentration to be measured but it is still difficult to determine the correct composition of the complex in order to calculate the stability constant (see Figure 1). (2) Over the complete range of mole fractions, certainly only one type of complex is formed, but the complex is too stable. In the case of strong complex systems, it is possible to decrease the conditional constants. This decrease can be accomplished experimentally by selecting a certain pH or adding a subcomplex agent. By this approach, almost all conditional formation constants can be reduced to a value within a n appropriate range. (3) The type of complex formed is dependent on an excess of one of the components, although the experimentally chosen conditions would allow “correct” values measured in the middle range of mole fractions. At the smaller mole fractions, there should be only small amounts of metal ions in comparison with each ligand. Therefore, some complexes which contain more ligands than metal ions as m < n in M,R, may be formed there, even if each MR is to be formed as a main product for most parts of the continuous variations plot. If, for instance, MRZ‘or MR3 is formed only in such small mole fraction ranges, the curve will start with an “abnormal” curvature, “abnormal” in respect to the 1:l plot, as shown in Figure 2 . Similar anomalous plots might appear at larger mole fractions when small amounts of ligands are present compared with each metal ion. In order to obtain reliable results, one has to measure only in the middle range of the mole fractions. (4) Because the proposed method (9) for normalization uses a measured maximum absorbance value, obtained ex-
-0 MOLE FRACTION ( X I Figure 2. Continuous variations plot for a 1 : l complex (--) effectedby 1 : 2 and 1 : 3 complexes formed at small mole fractions A plot for an uninterfered: 1 : l complex (----), 1:2 complex 1 :3complex (-.-.-.-.-),
(-.e..);
perimentally by using a large excess of ligand, the same difficulty mentioned before may be experienced and incorrect values for formation constants, even in the middle range of mole fractions, could be obtained. (5) More than one species of complex is formed so it would not be possible to evaluate such a continuous variations plot. In this case, one can select the experimental conditions so that only one type of complex is formed. This can be accomplished by choosing a certain pH and wavelength or adding a masking reagent, etc. A convenient way of taking into consideration the interferences mentioned is to utilize conditional constants by including corrections by means of a coefficients, as recommended by Ringbom (IO). Meaningful results have been obtained, of course, only when the various assumptions made were valid so that the functional form of the mathematical model was also valid. Enumerating the assumptions implied or explicit up to this point, we note that we require: (a) selecting the experimental conditions so that only one species of complex predominates; (b) adding a subcomplex agent for strong complexes so that the absorbance measurements are in a convenient region; (c) independence on experimentally determined absorbance maximum value; (d) independence on a prior knowledge of the composition of the complex. The earlier (9) treatment of the problem of the determination of formation constants from continuous variations data requires a measured absorbance maximum value and the knowledge beforehand of the composition of the complex. It is a feature of this paper to fulfill the above given requirements c and d so that the method of continuous variations will be utilized more extensively. The analyst who follows the above rules strictly knows that the calculation of formation constants is trustworthy. (10) A Ringbom, ’ Complexatlon in Analytical Chemlstry, ’ Interscience, NewYork, N Y , 1963
ANALYTICAL C H E M I S T R Y , VOL. 45, NO. 11, SEPTEMBER 1973
1927
THEORETICAL Mathematical Treatment and Statement of Problem. Considering a general form for the complex formation system a t equilibrium we can write
+ nR e M,R,
mM
(1)
where M represents a metal ion, R a ligand, and MmRnis the complex formed in a molar ratio of m:n for M and R. The following equations are taken from a previous paper (9)after rearrangement of some of them.
D1,t
( Y M A - YM,)
(12)
while
D2, ( Y M A - Y M , ) / Y M A
(13)
D3,
(14)
(YMA - YM,)/YMA2
are relative test criteria for judgment. The individual Y M , values calculated by means of Equation 9 lead to the mean Y Y A value L
YMA
=
xYM,/L
(15)
1-1
c = CM + CR x = CM/C Y = A/AM X M = m / ( m n) E , = [ ( m n)X - m y ] E 2 = [ ( m n)(l - X ) - nu]
+
+
+
F(X,Y)= (m+n-li YE,-"'E2-" = 0 K' - [ ( m n)/Cl F(X,Y) = E l m E t - Ym"n"(1 - YM)'"+"'/YM = 0
+
K'
=
[(m
(2) (3) (4) (5) (6) (7)
(8)
RE
(9)
+ n)/C](m+n-l)
Y M A / [ m " n " ( l - YMA)'"+"'] (10)
where C is the constant sum of CM (total concentration of the metal) and CR (total concentration of the ligand); X is the mole fraction, expressed as the mole fraction of the metal ion; Y is the "normalized absorbance term," where A denotes the absorbance of the complex measured against the appropriate ligand blank solution and AM represents the maximum absorbance assuming nondissociation of the complex (see Figure 1); K' is the conditional constant, it can be a conditional formation constant Kr' or a conditional extraction constant KE' since the mathematical treatment is the same one; X M is the mole fraction corresponding to the maximum of the continuous variations plot (see Figure 1); YM is the maximum value of the "normalized absorbance term" (see Figure 1); and F ( X , Y) is the analytical function describing the continuous variations plot. The method of continuous variations gives primarily a set of ( X ,A ) data points and an experimentally chosen k value. In order to normalize the system, the data points are to be changed into a set of ( X , Y) values applying Equation 4.In order to be independent of the experimentally measured AM value which is required in Equation 4, attempts were undertaken to assume different AM values until the general Equations 8 or 9 are applicable for each point of the continuous variations plot. For this approach, a successive approximation of A M is necessary until certain standard criteria for formation constants (calculated for each data point using Equation 8) and YM values (calculated for each data point using Equation 9), respectively, reach a minimum. Unfortunately the procedure using Equation 8 was found to be misleading in some cases so Equation 9 was used for the further treatment. The decision whether a chosen AM value leads to a minimum is based on the parameter
where the subscript j refers to one of the three equations given below, L is the number of data points measured, and in the common least square method, D is given by 1928
The parameter PI, normally used in least square calculations, is inapplicable for this purpose and even Pz has been found to be a less reliable test datum since both parameters do not always reach the necessary minimum. The application of parameter Pa as the basis of judgment is rather unconventional but justified because the position of the minimum is not displaced. The minimum of the test parameter leads to the smallest possible error in YM. This relative error RE, as denoted in Equation 16, is assigned to be a measure for the accuracy of the method and goodnessof-fit, respectively.
loop:!
(16)
Now some remarks follow concerning random errors which might occur. The measured values for A are thought of as having an equal precision among them with P a s an average fractional error of the 2 to 5 individual absorbance readings of each data point. Only at small and at large mole fractions would the absorbance errors be sufficiently large so that a term A ( 1 - P) rather than A should be employed, provided that a better fit to Equation 9 is obtainable. The errors of the other measured values such as C and X are negligible. Because Equation 9 contains as further variables m and n, the procedure described in this paper is valid only for a distinct complex model with fixed values for m and n. Therefore, a number of models have to be tested by fixing a certain m and n combination. That model with the smallest error, ie., the best fit to the appropriate function, is the most probable one. The conditional constant K' and the X M value were calculated by means of Equations 10 and 5 , respectively, since the values for m, n, C, and YMA are known. Computer Program Jobcon. The schematic flow diagram is given in Figure 3. In the course of the complete program the following conditions were observed: K' 2 0, E1 I 0 5 Ez, and YM 5 1, to avoid absurd and merely timeconsuming computations. Since the solution of Equation 9 for YM cannot be given in an explicit form, the inner loop is for the successive approximation of each YMi value, the middle loop is for obtaining A M iteratively, and the outer loop is to repeat the calculations for the complex models selected. The approximations for YM and AM were done by subtracting consecutive increments of 1.0, 0.1, 0.01, 0.001, and 0.0001 until the test criteria were fulfilled, L e . , printout was allowed only for acceptable fits. About 5 to 50 trials were necessary to obtain YM and A M values, respectively, with an accuracy of five significant figures. Round-off error within the computer was found to be considerable in this type of least square calculations so all data were processed in the double-precision mode. This program is written to calculate values for the following 15 complex models: 1:l . . . 1:8 and 2 : l . . . 8:l. Since the program is subdivided into three subroutines, it can be easily rewritten in order to test some other complex models.
ANALYTICAL CHEMISTRY, VOL. 45, NO. 11, SEPTEMBER 1973
The computer used is Univac 494 and the language for the program is Fortran IV. Computer times were about 40 sec for each run (15 complex models) containing 10 data points. Listings are available on request. L,
WRITE
C, P, X I I I , A111
EXPERIMENTAL General Procedure. Prepare equimolar solutions of metal and ligand in a concentration so that the absorbance measurements are in a convenient region (0.1-1.0 absorbance unit). Thus, a series of solutions is prepared by mixing different volumes of equimolar solutions of the two components and diluting to a constant final volume to give solutions having identical total molar concentrations (C) but different mole fractions ( X ) .If a n extraction system is studied, maintain an equal volume of both the aqueous and the organic phase. Measure the absorbance ( A ) a t a wavelength where the complex shows a maximum absorbance against the proper blank solution. Determine the average fractional error ( P ) of the individual absorbance readings. The data points ( X , A ) and the values for L, C, and P were input to the computer program (omit values for L < 3 and 1 5 X 5 0). Test Calculations. T o test the performance and the applicability of this proposed method, several systems known to form MR, MR2, and MR3 complexes with certain metal ions and dithizone, 8-quinolinol, or ammonium 1-pyrrolidinecarbodithioate were investigated. The experimental conditions, the absorbance readings, and corresponding mole fractions of the different complex systems were taken from experiments carried out earlier (9, ZI). These data sets were input to the computer program.
RESULTS AND DISCUSSION Table I shows the computer output for the silver(1)dithizone complex system. In the top of the table the mole fraction data ( X ) ,as well as the apparent absorbances ( A ) , the number of measurements ( L ) ,the fractional error ( P ) , and the constant sum of concentrations (C) are printed. As the results of the calculation, the composition of the complex ( M : N , metal to ligand ratio), the absorbance maximum ( A M ) ,the X maximum ( X M ) value, as well as the Y maximum ( Y M A ) value, the log of formation (extraction) constant ( K ' ) , and the relative standard deviation of the method (RE) are printed next. That set of values with the minimum relative standard deviation is reprinted and is assigned to be the most probable one. The values obtained by applying the process of successive approximations to the results given in Table I are shown in Table 11. The values illustrate the effect of using the different parameters P I , Pz, and P3, according to Equations 11-14, in order to approach the best fit of a set of given data points to a theoretical continuous variations plot, i.e., the best fit to Equation 9. Table I1 shows very clearly that the parameter PI leads to a maximum a t first rather than to a minimum so this parameter is inapplicable as the basis for judgment. Even Pz tends to remain constant over a relatively wide range and in this instance a misleading maximum appears a t the third step of approximation. Only parameter P3 leads to the desired minimum (at step 21). The test parameters and extraction constants of silver, mercury, and lead dithizonate; copper, aluminum, and gallium oxinate; and copper, cobalt, cadmium, zinc, bismuth, and gallium 1-pyrrolidinecarbodithioates are summarized in Table 111. Besides the complex system, the average fractional error ( P )of the corresponding data set is given in the first row of the table. The relationship between conditional ( K E ' )and true ( K E )extraction constants was accomplished by means of Ringbom's method (10). The exact conditions have been described earlier (9, 11). The original and presently calculated K E values are compared in Table 111. The extraction constants of the mentioned dithizonate and oxinate complexes are moreover (11) W. Likussar and D. F. Boltz.AnaL Chem.. 43, 1273 (1971).
, I*I ,
L
I
THE
Ms I
FIRST
Nxl
COMPLEX
MODEL
I
b AM-IO
AM
r
I
AM
- A AM
1 VI11
I
A l l 1 / AM
I
FULFILS
VMIII
IF I Z L
YES VMA:X
VMIII / L
CALCYLATE
( E a 5.
16,
161
WRITE
WRITE
3 "?;T HE"'E S M:!h :L"EAa" 0R8E E LV A L U E ALLEST
WITH
&) Figure 3. Flow chart for computer program Jobcon
compared with values given in the literature (10, 12-16). The experimentally obtained YMA values are in excellent agreement with the calculated one using the method just proposed. The only bismuth(II1)-1-pyrrolidinecarbodithioate complex system yields two K E values which differentiates by a factor of about 3. The K E value calculated by the computer method is certainly more reliable than (12) 0. G . Koch and G . A. Koch-Dedic,"Handbuch der Spurenanalyse," Springer-Verlag,Berlin, 1964. (13) G . H. Morrison and M . Freiser. "Solvent Extraction in Analytical Chemistry,"Wiley. N e w York, N . Y.. 1957. (14) E. B. Sandell, "Colorimetric Determination of Traces of Metals," 3rd ed, Interscience. N ew York. N . Y . . 1959. (15) M . Oosting,Anal.Chim. Acta, 21, 505 (1959). (16) C. H. R: Gentry and L. G. Sherrington, Analyst (London), 71, 432 (1946).
A N A L Y T I C A L CHEMISTRY, VOL. 45, NO. 11, SEPTEMBER 1973
1929
~
~~
Table I. Computer Output Obtained by Processing a Set of Continuous Variations Data for the Silver(1)-Dithirone Complex System DATA L
X
A
0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900
0.110 0.216 0.31 7 0.41 5 0.470 0.410 0.322 0.215 0.110
9
P
c
0.005
0.00005
RESULTS M:N
AM
1:l 1:2 1:3 1: 4 1:5 1:6 1: 7 1 :8 2:1 3:l 4:l 5:l 6:l 7:l 8 :1
0.560 0.731 0.821 0.876 0.91 2 0.938 0.958 0.973 0.731 0.821 0.876 0.912 0.938 0.958 0.973
1:l
0.560
XM
0.500 0.333 0.250 0.200 0.167 0.143 0.125 0.111 0.667 0.750 0.800 0.833 0.857 0.875 0.889
YMA
Log K'
RE %
0.81 5 0.684 0.679 0.684 0.691 0.697 0.702 0.707 0.685 0.678 0.683 0.690 0.696 0.701 0.708
5.98 10.29 15.08 19.93 24.80 29.68 34.57 39.46 10.30 15.06 19.92 24.79 29.67 34.55 39.48
1.99 38.19 44.77 46.73 47.31 47.35 47.18 46.90 38.12 44.71 46.70 47.27 47.32 47.14 46.87
5.98
1.99
THE FOLLOWING VALUES ARE MOST PROBABLE
0.500
0.81 5
--
Table II. The Process of Successive Approximations (Silver( I)-Dithizone Complex System)
1930
Step
AM
YMA
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
9.0000 8.0000 7.0000 6.0000 5.0000 4.0000 3.0000 2.0000 1 .oooo 0.9000 0.8000 0.7000 0.6000 0.5900 0.5800 0.5700 0.5600 0.5599 0.5598 0.5597 0.5596 0.5595 0.5590 0.5500
0.04242 0.04773 0.05455 0.06368 0.07643 0.09558 0.12762 0.19209 0.39234 0.43936 0.50041 0.58460 0.71815 0.73774 0.75897 0.78359 0.81319 0.81355 0.81386 0.81420 0.81452 0.81489 0.81657 0.85422
P0.00687 0.00773 0.00885 0.01031 0.01236 0.01541 0.02040 0.03012 0.05385 0.05716 0.05955 0.05821 0.04015 0.03538 0.02942 0.02212 0,01622 0.01625 0.01624 0.01624 0.01621 0.01624 0.01632 0.03306
A N A L Y T I C A L CHEMISTRY, VOL. 45, NO. 11, SEPTEMBER 1973
p2
p3
log K'
0.1 6198 0.1 6195 0.16228 0.16203 0.16171 0.16121 0.15983 0.15680 0.13725 0.13010 0.11901 0.09957 0.05591 0.04798 0.03876 0.02822 0.01995 0.01998 0.01996 0.01994 0.01991
3.81 837 3.39312 2.97489 2.54438 2.1 1580 1.68667 1.25237 0.81628 0.34981 0.29610 0.23782 0.17032 0.07785 0.06507 0.05107 0.03602 0.02455 0.02453 0.02452 0.02449 0.02444
3.267 3.323 3.388 3.463 3.554 3.670 3.827 4.071 4.628 4.748 4.904 5.132 5.558 5.631 5.718 5.826 5.970 5.971 5.973 5.975 5.976
0.01992 0.01998 0.03870
0.02445 0.02447 0.04531
5.978 5.987 6.206
Table 111. Test System Parameters m:n
System Dithizonate
log KE'
log K E ~
AM
YMA
0.555 0.560
0.830 0.81 5
6.06 5.98
0.850 0.848
0.882 0.880
11.68 11.66
1.160 1.158
0.620 0.620
10.03 10.01
0.950 0.949
0.640 0.640
8.10 8.09
1.740 1.767
0.230 0.225
8.00 7.98
0.803 0.797
0.860 0.869
13.63 13.75
25.8 26.1 26.1 31 .O 30.8 30.8 39.8 39.5 39.6
0.860 0.855 0.800 0.910 0.672 1.130 1.142 0.720 0.715 0.505 0.490 0.920 0.911
0.840 0.838 0.580 0.546 0.764 0.790 0.775 0.870 0.868 0.940 0.919 0.780 0.780
10.06 10.05 9.25 9.12 14.76 10.28 10.18 7.55 7.53 16.33 15.80 11.37 11.37
23.7 23.7 11.5 11.4 18.2 13.3 13.2 12.6 12.6 33.9 33.3 22.9 22.9
lRei std dev, %
Ref
of
Ag(l)
1:l
P = 0.005
1 :2
Pb(ll)
1 :2
P = 0.004 Oxinate of Cu(ll)
1 :2
P = 0.007 AI( I I I)
1 :3
P = 0.010
Ga(lll)
1 :3
P = 0.016
16.2 16.5 16.4 43.8b 45.4 45.4 20.4 20.5 20.5
1.99
2.62
0.97
2.38
0.52
2.14
1 -Pyrrolidine-carbodithioate of Cu(ll)
P = 0.012 Co(l I ) P = 0.056 Co(II I) Cd(ll)
1 :2 1 :2
1 :3 1 :2
P = 0.078 Zn(l1)
1 :2
P = 0.020 S i ( II I)
1 :3
P = 0.031
Ga(l II) P = 0.064 a
1 :3
2.14 10.26 16.71 2.33 2.39 2.58 4.96
Extraction constants for complexes in chloroform: u = 0.1, t = 24 f 1 " C . Extraction constant for the complex in carbon tetrachloride.
that based on an experimentally determined A M value. The reason for this is outlined at point 3 and 4 a t the beginning of this paper. Table I11 illustrates the limitations of the continuous variations method. In the case of the cobalt-l-pyrrolidinecarbodithioate complex system, it can be seen that both a Co(I1) and a Co(II1) complex system are formed. The relative errors of the calculated extraction constants are between 10 and 17% and the values of the constants are rather doubtful. Such unacceptable results are very easy to recognize. There are important advantages to the use of a computer for fitting analytical functions. In the procedure proposed, all significant decisions are incorporated in the computer, the effects of human error are minimized since no additional measurement ( A M value) or decision (composition
of the complex) is necessary. Another advantage is that the computer can be easily programmed to perform computations for further complex models using one distinct set of continuous variations data. However, if a computer is not available, the previously described method (9) for calculating formation and extraction constants is applicable provided that the limitations of that approach are considered.
ACKNOWLEDGMENT The author thanks Eugen Gagliardi, Wolfgang Beyer, and Hans Raber, University of Graz, for their inspiration and useful discussions. Received for review October 20, 1972. Accepted February 12, 1973.
ANALYTICAL CHEMISTRY, VOL.
45, N O . 1 1 ,
SEPTEMBER
1973
1931
