a fixed internal reference and accepts circular cuvets of about 12-ml capacity. Matched cuvets were etched at 7.0 ml and the test performed directly in the cuvettes. This avoids the transfer of the solutions from the test tubes to capped Beckman cells. This modification also has the advantage that the internal reference eliminates any error caused by the darkening of the reference solution during the time that the other solutions are measured. One further modification of the test involved the column used to prepare the reagent. Instead of using a column filled with Aquasorb to clear the reagent, the same column was tightly packed with ovendried glass wool. The sensitivity of the test is at its greatest
with this column to the extent that 2 pg of water is easily determined, but 35 kg is the maximum detectable amount because of the high absorbance (0.85) of solutions at this concentration. It is though that the sensitivity of the reagent treated with the Aquasorb column is lessened by small amounts of P20~from the Aquasorb in the reagent solution which compete with the lead tetraacetate for water. RECEIVEDfor review May 22, 1970. Accepted August 6, 1970. Work supported by the U S . Army Mobility Equipment Research and Development Center, Fort Belvoir, Va., under Contract No. D A 44-009-AMC-1386 (T).
Least Squares Curve-Fitting Method for End-Point Detection of Chelatometric Titration with Metal Indicator Hisakuni Sat0 and Kozo Momoki Laboratory for Industrial Analytical Chemistry, Faculty of Engineering, Yokohama National Unicersity, Ooka-machi, Minami-ku, Yokohama-shi, Japan
For obtaining an accurate end point for a chelatometric titration with indicator, where 1:l chelates (MI and MY) are formed, a numerical calculation method by the least squares method using a digital computer i s presented. Because the conditional formation constants (KJiI and KAIY)for the chelates formed in a titration are also obtained as the results, the present method is more useful for the titration systems in which these values are not known beforehand. To ascertain its utility, the method is applied to the titration of MgZi in the absence and in the presence of 1M KCI or NaCI. In the latter case, KMI and KIIY are considerably reduced and the end point detection has been difficult. Precise and accurate results are obtained rapidly. Some problems accompanying the present method are also discussed.
PHOTOMETRIC STEP-INDICATION ( I ) employed widely in chelatometric titration has been discussed theoretically or practically on the optimal conditions for a sharp indication (2-6) or on the methods to determine the end point in the curved titration plot once obtained (7-10). The end point is often located by linear extrapolation. Musha, Munemori, and Ogawa ( 7 ) extended the linear extrapolation method in acid-base titration by Higuchi, Rehm, and Barnstein ( 1 1 ) to chelatometry. This method seems to be (1) H. Flaschka and P. Sawyer, Tulanta, 9, 249 (1962). (2) G. Schwarzenbach, “Die komplexometrische Titration,” Ferdinand Enke Verlag, Stuttgart, 1955. (3) J. M. H. Fortuin, P. Karsten, and H. L. Kies, Anal. Chim. Acta, 10, 356 (1954). (4) H. Flaschka and S. Khalafallah,Z . Anal. Chem., 156,401 (1957). (5) C. N. Reilley and R. W. Schmid, ANAL.CHEM., 31, 887 (1959). (6) M. Tanaka and G. Nakagawa, Anal. Cliim. Acta, 32, 123 (1965). (7) S. Musha, M. Munemori, and K. Ogawa, Bull. Chem. SOC. Jup., 32, 132 (1959). (8) A. Ringbom, “Complexation in Analytical Chemistry,” Interscience Publishers, New York, N. Y., 1963. (9) E. Still and A. Ringbom, Anal. Chim.Acta, 33, 50 (1965). (IO) E. Still, Suom. Kemistilehti E , 41,33 (1968). (11) T. Higuchi, C. Rehrn, and C. Barnstein, ANAL.CHEM.,28, 1506 (1956).
very ingenious giving the correct end point easily without knowing about the conditional formation constants (KaIIand KJIY). However, the linearity of this plot was often found to be questionable ascribed to the dilution effect during the titration as well as to the incomplete treatment of the concentration of indicator. In other graphical methods, by Ringbom (8) and by Still (IO) for example, KJIIand K3ly must be known beforehand for the accompanying calculation. The present paper deals with a rigorous least squares method (curve fitting method) for obtaining an accurate end point rapidly using a digital computer. The conditional constants are also obtained as the results of this method. Therefore, these values need not be known beforehand. To make the calculation program as general as possible, the theoretical equation (3, 4) to fit a set of titration data is modified to take the dilution effect into account. The present method is applied to the typical Mg-Calmagite-EDTA system and to the more difficult Mg-Calmagite-EDTA-NaC1 (or KCl) system. Although computer calculation of similar titration procedures has been described in the recent book by Dyrssen, Jagner, and Wengelin (IZ), their treatments are essentially based on the tactics by Ringbom (8) as quite different from ours, as will be seen. THEORETICAL
Fundamental Equations. Fortuin, Karsten, and Kies ( 3 ) derived the theoretical equation for the photometric titration curve, in which only 1 :1 complex formations were assumed between metal ion (M) and indicator (I) as well as metal ion and titrant (Y). Although they neglected the dilution effect during a titration, many authors have treated the photometric titration based upon the same equation. If the dilution ~~
~
~~
(12) D. Dyrssen, D. Jagner, and F. Wengelin, “Computer Calcula-
tion of Ionic Equilibria and Titration Procedures with Specific Reference to Analytical Chemistry,” Almqvist & Wiksell, Stockholm, 1968.
ANALYTICAL CHEMISTRY, VOL. 42, NO. 13, NOVEMBER 1970
1477
et€ect is taken newly into account, a more general expression for the titration curve is obtained as follows: CY - = (1
C ni
CI --(l-cY)-Cni
CniKniI
a
X
where a=
g
. A - Ani1 AI - AMI CY =
( A = -log T )
.u
fY ~
va
m
C M = va
be obtained. However, this does not mean that the unknown parameters should necessarily be taken as mar, KXI, and KIIY in the course of the least squares calculation. Another way of taking unknowns as mar, l/Knrr, and KnrIIKIrY is also possible, and more advantageous than the former way in view of the convergent calculation. Now we have to consider the precisions of u and a for solving observation equations by the least squares method. The measured values for L’ are thought as having an equal precision among them, while the precision of a can be evaluated by the law of propagation of errors (13) as,
(3)
or (4)
U
g=l+va
and millimoles of metal ion to be determined concentration of titrant solution, mole/liter concentration of indicator in the solution to be titrated, mole/liter v, = initial volume of the solution to be titrated, ml L ‘ = added volume of titrant solution, ml a = ratio of the free indicator concentration to CI AMI = absorbance by the complex MI when the concentration of the latter is CI AI = absorbance by the free indicator when the concentration of the latter is CI I n the usual photometric titration procedures, the absorbance A (or the transmittance T ) of a solution is directly measured with the added volume of titrant solution, u. However, it is also usual that the theoretical equation is conveniently represented as Equation 1 t o simplify the expression where the variables included are made as relative. It should be noted that a in Equation 1 must be calculated with the absorbance values corrected for the dilution effect as in Equation 2. In this point, the reference of the absorbance values should be defined for Equation 2, where solvent water was here taken as the reference for uncolored titrant solution used in the experiment. In Equation 2, the absorbance by metal ion is neglected as usual. Least Squares Curve-Fitting Method. When a set of titration data is given, the unknown parameters, mnf, KMI, and K M Yin Equation 1, could b e evaluated by the least squares adjustment of the theoretical equation (curve-fitting method). Although the direct observables in the titration are u and A (or T), it is more convenient to introduce a relation between u and a , F(L‘,a ) ,for the curve fitting method. Substituting Equations 3 and 4 into Equation 1, we obtain F ( V , a ) = fy VO
‘
u
-
(y -
Cr(1
0
{ln
AMI
- AI))
(8)
where the contribution of u O 2 to ua2 is neglected as small. Under these data conditions, the least squares calculation can be suitably made bq the Deming’s method (13) which employs a first order Taylor expansion in order to obtain a linear function of the parameters and takes the weight of each observation equation into account. This weight for the equation, pi1/2,can be represented as,
The errors of the other measured values such as V0,fy, and C1 are neglected as small in Equation 9. Here, let the unknown parameters be denoted simply with a, h, and c. With suitable first approximations (u,, bo,and c,) for the parameters, the normal equations to obtain the corrections (6a, 66, and 6c), can be written as:
- a) -
This equation is regarded as the observation equation for the present least squares method. When a data set, { o r , a i ) , (i = 1 , 2 , . . , ,m), is given, rn numbers of observation equations are written for the three unknowns. The unknown parameters in Equation 6 are mi, KMI, KbfY,and the most probable values for these parameters must 1478
where ux2means the variance of X. When the spectrophotometric measurements are made with a usual transmittancelinear instrument, the errors in the transmittance are considered to remain as constant and independent of the magnitude of the measured values. So substituting Equation 2 into Equation 7-2 after making the appropriate differentiations, we obtain
(13) W. E. Deming, “Statistical Adjustment of Data,” John Wiley and Sons Inc., New York, N. Y . , 1943.
ANALYTICAL CHEMISTRY, VOL. 42, NO. 13, NOVEMBER 1970
where
Solving the normal equations, the new approximations for the parameters can be obtained as (a, - 6a), etc. Then the calculation must be iterated until convergence is obtained. The Determination of An11 and AI. I n the above treatment, we behave as if the absorbances in Equation 2 are easily measured in the experiment. Strictly speaking, however, the actual determinations of A X I and A I present another difficulty because these cannot usually be measured directly in the experiment as is evident from their definitions. Of these, A I can rather easily be obtained by averaging the absorbances which are measured after the equivalence point and corrected for the dilution effects to become almost constant. On the other hand, the value of A M Ican not be estimated easily, although an experimental procedure for the estimation has been described by Ringbom (8). For this a device, by which AarIcan be estimated iteratively, is taken into the computer program. The absorbance at u = 0 (A,) is usually close to An11 in the case of CJ1> CI. Therefore, measured A, can be used as the first approximation for AXI at the initial stage of the least squares calculation. Substituting the calculation results, which are more or less approximate ones, into Equation 6, the value of CY at u = 0 (CY,) can be calculated. This CY, is then applied to Equation 2, where g = 1 , as
I
I
4 read : initial values for the parameters
I
I
[ 5 transmittance + absorbance I
16 I dilution correction
I
I
I
calculation of sum of
‘--a?$==converge ?
12 calculation of a,
13 amendment of AMI
no
Since the value of A M I ,thus obtained, is also the approximate one, the least squares calculation must be repeated until the resulted values for Ah11 converge. Now, the precisions of TMIand TI, which are transformed inversely from so obtained AJIIand A I , are no longer the same as that of T . The CY values near 0 and 1 are relatively more erroneous and the estimation of the variance of CY with using Equation 8 must not be correct statistically. In the present paper, however, Equation 8 was used as it is to avoid further complication of the treatments. For this, the CY value of the curve-fitted data points was restricted arbitrarily within the range from 0.05 to 0.95 in the computer calculation. Computer Program. Figure 1 is the flow diagram for the calculation program, where the inner loop is for the successive approximation in the Deming’s method and the outer loop is for obtaining A M Iiteratively as mentioned above. The computer used is HITAC 5020E, and the language for the program is Fortran IV. The program will be available to persons desiring it.
*@ Figure 1. Flow diagram for computer programs method (17) with magnesium ion. Buffer solution (pH 10) was the solution of “821 and NH3 (2). At each titration, 2 nil of this buffer solution was used. Apparatus. Photometric titrations were performed mainly with Hirama’s spectrophotometer (type 6) with some modifications for the titration. A cubic type cell of 3.1-cm width was used. The output signal of the photometer (transmittance linear) was measured with a pen recorder, LER-12A made by Yokogawa Electric Works. Photometric titrations in the ultraviolet region were carried out with Shimazu’s QV-50 spectrophotometer and its accessory for titration. Metrohm piston buret (manual type, full capacity of 5 ml) were used for the titration and for taking an aliquot of metal ion solution. The pH of the titrated solution was measured with Toa Denpa’s HM-SA pH meter. RESULTS AND DISCUSSION
EXPERIMENTAL
Reagents. Disodium salt of EDTA was recrystallized once from water-alcohol solution (14). Solutions of 0.01M EDTA were standardized against the standard zinc(1I) solution, and 0.01M solutions of magnesium chloride against the EDTA solution, at pH 10 by the photometric titration in the ultraviolet region without using metal indicator (15). Calmagite was recrystallized twice according to the procedure by Lindstrom and Isaac (16). The effective concentration of Calmagite solution was determined by the molar ratio (14) W. J. Blaedel and H. T. Knight, ANAL.CHEM., 26, 741 (1954). (15) P. B. Sweetser and C. E. Bricker, ibid., p 195. (16) F. Lindstrom and R. Isaac, Tulunta, 13, 1003 (1966).
An example of the computer output is shown in Table I. In the top of the table, the titration system and the titration conditions are printed. Titration data as well as the apparent absorbances (AS), the absorbances corrected for the dilution effect (ASC), and CY values are printed next. As the results of the calculation, the most probable values and their standard deviations for the three parameters are printed. The iteration numbers for the Deming’s method and for the estimation of AarI are shown at ITERATION NUMBER 1 and 2, respectively. Other additional information including the weights (17) K. Momoki, J. Sekino, H. Sato, and N. Yamaguchi, ANAL. CHEM., 41, 1286 (1969).
ANALYTICAL CHEMISTRY, VOL. 42, NO. 13, NOVEMBER 1970
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Table 1. Computer Output Obtained by Processing a Set of Titration Data for Mg2+in 1M Solution of KCI SYSTEM MAGNESIUM CALMAGITE EDTA EXPERIMENTAL NUMBER 69.7.15. NO 1 CONDITION OF TITRATION CONCENTRATION OF TITER SOLUTION 0.10258E-01 CONCENTRATION OF INDICATOR 0.1208E-04 VOLUME OF TITRATED SOLUTlON 50.0 PH 10.3 WAVE LENGTH 520 DATA 27 NUMBER OF OBSERVED POINTS NUMBER OF POINTS USED FOR FITTING 19 I V(ML.) T AS, ASC ALPHA 1 0.8378 0.300 0.261 0.8428 0.0714 2 0.8263 0.400 0.268 0.8329 0.0908 3 0.500 0.275 0.8143 0.8224 0.1112 0.600 0.7980 4 0.286 0.8076 0.1402 5 0.700 0.7658 0.308 0.7766 0.2009 6 0.750 0.326 0.7412 0.7523 0.2483 7 0.780 0.7216 0.341 0.7329 0.2862 8 0.7042 0.800 0.355 0.7154 0.3204 0.820 0.6839 9 0.372 0.6951 0.3602 10 0.6578 0.840 0.395 0.6689 0.4114 11 0.860 0.6270 0.6378 0.4721 0.424 12 0.880 0.465 0.5869 0.5973 0.5513 13 0.518 0.900 0.5405 0.5502 0.6433 14 0.910 0.551 0.5132 0.5226 0.6974 15 0.586 0.4865 0.920 0.4955 0.7504 16 0,930 0.623 0.4599 0.4685 0.8032 17 0.940 0.659 0.4355 0.4437 0.8516 18 0.950 0.690 0.4156 0.4234 0.8912 19 0.960 0.712 0.4019 0.4096 0.9182 20 0.980 0.741 0.3846 0.3921 0.9524 21 1.ooo 0.754 0.3770 0.3846 0.9672 22 1,100 0.775 0.3651 0.3731 0.9895 23 0.3617 0.781 1.200 0.3704 0.9948 24 0.788 1.400 0.3581 0.3682 0.9992 25 0.790 1.600 0.3568 0.3682 0.9992 26 1.800 0.3551 0.793 0.3679 0.9997 27 0.796 2.000 0.3535 0.3676 1.0003 RESULTS VALUE ST. DEV. METAL ION PRESENT (MG.) 0.23556 0 00008 CONDITIONAL CONSTANT KI 0.1000E+06 0.4685E+03 CONDITIONAL CONSTANT KY 0.4225€+08 0.8037Ef06 AUXILIARY VALUES TITER CONSUMED (ML.) 0.9445 0.0003 METAL ION CONCENTRATION (M) 0.1938E-03 ASCMI 0.8793 ASCI 0.3678 DASCI 0.0006 ITERATION NUMBER 1 7 ITERATION NUMBER 2 3 0.22951E-05 0.22951E-05 SUM OF SQUARES OF RESIDUALS 1.000 WEIGHT-RATIO OF MEASURABLE VALUES WEIGHTS AND RESIDUALS FqI) I PY(1) PWI) VX(I) VY(U 1 0.45952E-01 0.23661E-04 -0.49190E-02 0.97681E-05 0.11659€+05 2 0.48082E-01 0.12000E-05 -0.14882E-03 0.18411E-06 0.31465E+05 3 0 . 50447E-01 0.73189E+05 0.20469E -06 0.30972E-05 -0.24614E-03 0.17231E-05 4 0.53994E-01 0.19241E+06 0.68425E-04 -0.32394E-02 5 0.62329E-01 0.86206€+06 -0.46374E-04 0.95346E-03 -0.26120E-06 -0.22719E -06 6 0.69919E-01 0.20342E+07 -0.95093E-04 0.11715E-02 0.37912E-07 7 0.76795E-01 0.35058E+07 0.27337E-04 -0.23642E-03 -0.75505E-07 8 0.83696E-01 0.52314E+07 -0 . 81200E- 04 0.52680E-03 0.74999E- 08 9 0.92719E-01 0.76009€+07 0.11724E-04 -0.5599lE-04 -0.79035E-07 10 0.10613E+00 0.10859€+08 -0.17631E-03 0.58834E-03 0.32044E- 07 11 0.12511E+00 0.14405E+08 0.9482OE-04 -0.21542E-03 12 0.15614€+00 0.17893€+08 - 0.24662E-03 0.35617E-03 -0.67133E -07 0.26905E-07 13 0.20292€+00 0.20369E+08 0.11253E-03 -0.10151E-03 -0.26484E-07 14 0.23623E+OO 0.21261E+08 -0.11554E-03 0 . 81142E-04 0.32358E-07 15 0.27264€+00 0.21866E+08 0.14524E-03 -0 . 81491E-04 16 0.31087Et00 0.22276€+08 0.29600E-03 -0.13641E-03 0.64746E -07 17 0.34517E+00 0.22524€+08 0.24245E-03 -0.96263E-04 0.52451E-07 18 0.37026E+00 0.22647E+08 -0.13311E-03 0.48342E- 04 -0.28650E-07 19 0.38463€+00 0.22679€+08 -0.13226E-03 0.46430E- 04 -0.28428E- 07 ALPHA MIN. 0.050 ALPHA MAX. 0.950 1480
ANALYTICAL CHEMISTRY, VOL. 42, NO. 13, NOVEMBER 1970
Table 11. The Process of the Iterative Calculations to the Results in Table I SYSTEM MAGNESIUM CALMAGITE EDTA NO 1 EXPERIMENTAL NUMBER 69.7.15. VALUES ON EACH APPROXIMATION STEP KI/KY l/KI SUM1 ASCMI METAL ION 0.10000E-01 0.10000E-04 0 . lOOOOE -04 0.8530 0.27308E-2 0.47178E-04 0.80142E-05 0.96711E-05 0.25141E -02 0.65434E-04 0.79379E-05 0.96674E-05 0.25156E-02 0.66348E -04 0.96672E-05 0.59343E-05 0.25157E-02 0.79342B-05 0.66390E- 04 0.96672E-05 0.25157E -02 0.79342E-05 0.66392E -04 0.96672E-05 0,25157E-02 0.66392E -04 O.79342E-05 0.96672E-05 0.25157E-02 0.79342E-05 0.96672E-05 0.8742 0.25551E-02 0.13707E-04 0.94454E-05 0.96750E-05 0.25286E-02 0.10275E-04 0.94735E-05 0.96770E -05 0.25282E-02 0.10225E-04 0.94740E -05 0.96771E-05 0.25282E-02 0.10225E-04 0.94740E-05 0.96771E - 05 0.25282E-02 0.10225E-04 0.94740E -05 0.96771E-05 0.25282E-02 0.8782 0.94740E-05 0.96771E-05 0.24039E -02 0.35458E-05 0.98849E -05 0.96864E-05 0.24013E-02 0.33055E-05 0.98879E-05 0.96866E-05 0.24013E-02 0.33038E-05 0.98879E-05 0.96866E -05 0.24013E-02 0.33041E-05 0.9888OE-05 0.96866E-05 0.24013E-02 0.33039E-05 0.98879E-05 0.96866E-05 0.33038E-05 0.98879E-05 0.24013E-02 0.96866E- 05 0.24013E-02 0.33039E-05 0.98879E-05 0.96866E-05 0.33038E-05 0.98879E-05 0.2401 3E-02 0.96866E-05 0.33038E-05 0.24013E-02 0.98879E-05 0.96866E- 05 0.24013E-02 0.98879E-05 0.8793 0.96866E-05 0.23675E-02 0.99998E-05 0.23382E- 05 0.96891E- 05 0.23671E-02 0.10000E- 04 0.22953E-05 0.96892E-05 0.23671E-02 0.10000E-04 0.22950E-05 0.96892E-05 0.23671E- 02 0.lOOOOE-04 0.96892E-05 0.22949E-05 0.23671E- 02 0.22951E-05 0.96892E- 05 0. lOOOOE-04 0.1OOOOE -04 0.23671E-02 0.22951E-05 0.96892E-05 0.23671E-02 0 . lOOOOE-04 0.22951E-05 0.96892E-05 Table 111. Calculation Results for the Titration Data of Mg-Calmagite-EDTA System No. Mg taken, mg Mg found, mg Relative dev, Kiri* X IO-' 1 0.04644 0.04672 f 0.00009 +0.60 3.05 i 0.04 2 0.04644 0.04687 i 0.00012 +O. 92 3.07 i 0.05 3 0.11501 0.11539 f 0.00015 $0.33 2.98 f 0.06 4 0.11501 0.11510 f 0.00027 $0.07 3.01 f 0.10 5 0.22928 0.22950 f 0.00015 +0.09 2.80 i 0.05 6 0.22928 0.22916 f 0.00021 -0.05 2.80 i 0.07 7 0.45783 0.457524~0.00021 -0.06 2.66 i 0.06 8 0.45783 0.45752 f 0.00024 -0.06 2.58 i 0.07 9 1.1435 1.14198 f 0.00033 -0.12 2.46 f 0.09 -0.09 2.53 i 0.07 10 1.1435 1.14242 f 0.00024 Over-all standard deviation:
KUY* X 4.04 f 2.04 4.75 f 3.91
3.61 f 0.81 3.22 + 1.50 3.67 i 0.45 3.48 f 0.51 3.69 =!c 0.30 3.65 & 0.36 3.43 f 0.16 3.40 f 0.12
Experimental conditions: V , = 50 ml, CI = 1.29 X 10d6M,fy = 1.0078 X 10-2M, pH 10.0
for c y f [PY (I) ] and each observation equation [P (I) 1, and residuals for ut [VX (I) 1, at [VU (I) ] and each observation equation [FO (I)] are also printed. The process of the successive calculations to the results given in Table I is shown in Table 11. The values in the first line of numerical part indicate A , and the first approximations for the unknown parameters. Values in the last row, sum of squares of residuals, show clearly the convergence of the calculation. In Table 111, the calculated results for the titration of different amounts of MgZ+in the absence of neutral salt are given. The values listed as the confidence interval for the
Mg found, which is defined as 3u, are very small. This means the fitness of the curve fitting is excellent. The differences between Mg taken and Mg found given in Table I11 are also very small. Moreover, such small deviations are thought to be rather inevitable, taking the buret performance into account. In this connection, the piston buret used for the titration was examined by weighing each 0.5 ml of draining water. Assuming the functional relationship between the volume of the draining water, V , (ml), and the buret scale, V , (ml) as V , = a . v, (13) the coefficient u was found to be 1.0010 with the standard deviation of 0.0008.
ANALYTICAL CHEMISTRY, VOL. 42, NO. 13, NOVEMBER 1970
1481
Table IV. Calculation Results for Titration Data of Mg-Calmagite-EDTA-KCI Deviation, No. Mg taken, mg Mg found, mg mg 1 0.23063 0.23556 i 0.00024 0,00493 2 0.45918 0.46420 i 0.00033 0.00502 3 0.68773 0.69269 i 0.00048 0.00496 4 0.22375 0.22702 f 0.00117 0.00327 5 0.22375 0.22605 i 0.00123 0.00230 6 0.44676 0.44955 f 0.00252 0.00279 7 0.66977 0.67256 i 0.00279 0.00279 Experimental conditions: NO. 1-3: KC1 l.OM, CI = 1.21 X lO-'M,fy = 1.0258 X 10-'M, pH 10.3. No. 4-7: NaCl l.OM, CI = 1.15 x 10-6M,fY = 1.0078 x lO-'M, pH 10.5. Table V. Some Examples of Calculation Results Changing the Range of CY Value No.
1
2
3
Range of (Y value 0.01-0,99 0.05-0.95 0 . 1 -0.9 0 . 2 -0.8 0 . 01-0.99 0.05-0.95 0 . 1 -0.9 0 . 2 -0.8 0.01-0.99 0.05-0.95 0 . 2 -0.8
Mg found, mg KI* x lo-' 0.11516 3.02 0.11514 3.00 0.11514 3.00 0.11517 2.98 0.45761 2.64 0.45752 2.66 0.45751 2.67 0.45757 2.68 1,14259 2.49 1,14242 2.53 1.14245 2.53
Ky* x 10-8 3.85 3.22 3.42 3.53 3.80 3.69 3.77 4.57 3.49 3.40 3.58
The values for KMIand &IY in Table I11 are considered to be not so correct, mainly because the p H and the temperature of the solution being titrated could not be kept constant during a titration. The p H of the solution decreased about 0.2 unit in a titration because of vaporization of NH, and the absorption of COZin the air. The temperature of the titrated solution increased about 2 degrees in a titration. In addition, since the ratio, KnrI/K>ry,is fairly small in the titration system, the information for determining Kary is not included sufficiently in the titration data, the a value of which has been restricted in the range from 0.05 to 0.95. However, even considering these situations, the values for KnrI and KJrY in Table I11 are found to have a tendency of decreasing with the increase of the concentration of Mg2+. In Table IV, the calculation results for the more difficult problems, i.e., the titration of Mg2+ in 1M solution of potassium chloride o r sodium chloride. From the titration curves, KJII and KJIY were found t o be reduced considerably by the presence of these salts. The calculated results in Table IV show this situation definitely. Nevertheless, the end points are obtained precisely as shown in the values for confidence interval. In the case of the presence of sodium chloride, these values are somewhat larger than others. However, such results are considered to be rather excellent, because Knry is very small.
1482
and Mg-Calmagite-EDTA-NaCI Systems Karl X 1.00 i 0.01 0.939 i 0.006 0.911 =k 0.009 0.712 f 0.501 0.754 f 0.017 0.669 i 0.030 0.606 i 0.029
K~~ x 10-7 4.23 i 0.08 4.64 i 0.04 5.37 i 0.07 0.501 i 0.008 0.480 Jr 0.008 0.449 i 0.018 0.442 =k 0.018
Although the deviations between Mg taken and Mg found are considerably large, these are almost constant in No. 1-3 (KCl) and in No. 4-7 (NaCl). These deviations are due to the alkaline earth impurities which are carried into the titrated solution by the neutral salts. Traces of heavy metal ions, which are also carried into the titrated solution and make the curve fitting poor, were masked by using potassium cyanide and triethanolamine. Now, it will be obvious that the present curve fitting method is still more useful when KIII and KJrY can not be estimated from literature, o r when we have not thought of the actual changes under certain titration conditions. Certainly, the appropriate initial values for the unknown parameters must be given as input information before starting the calculation by Deming's method. The successive approximation in Deming's method will converge when the initial values are suitable. However, the calculation converged in most cases to the same solution with using any set of the initial values which were in the range of i.10 for the true mnr, f 1 in log unit for the true ~ / K Jand ~ I K>rI/Km. Although such values for the ranges of the initial values are arbitrarily chosen, the initial values can be presumed empirically within the range from the titration curve. As mentioned previously, the a value of the titration data used for the curve fitting was arbitrarily restricted in the range from 0.05 to 0.95. The same data as No. 4,7, and 10 in Table I11 were processed changing the range of a value, and the results are shown in Table v. The influence of changing the range is so small that the approximate treatment of using Equation 8 may practically be permissible. It was found by thin-layer chromatography that Calmagite used as indicator included small quantities of colored impurities. These can be considered to have almost no influence o n the present results. Finally, a modern digital computer does not take more than 10 seconds in most cases to process a set of titration data. Therefore, the present curve-fitting method will be more useful in the time-sharing age. RECEIVED for review March 4, 1970. Accepted July 1, 1970.
ANALYTICAL CHEMISTRY, VOL. 42, NO. 13, NOVEMBER 1970
