ANALYST, JANUARY 1993, VOL. 118 97 Spectrophotometric Determination of Stability Constants of Anti-stepwise Complexes Shi-Fu Zou, Jie Zhou* and Wei-An Liangt Department of Chemistry, Shandong University, Jinan, Shandong, People‘s Republic of China There is a considerable knowledge of chemical equilibria of the stepwise formation of complexes in coordination chemistry. However, the coordination equilibria of complex formation of the type, M + R e MR s M2R e M3R e ... e M,R is less well understood. As their compositions are contrary t o the formation of stepwise complexes, they are called ’anti-stepwise complexes’. The existence of such complexes is not unusual and often causes difficulties in speciation analyses. Therefore, it is essential t o investigate further the chemical equilibria in solution.In this paper, based on Bjerrum’s function ( f i ) and the method of corresponding solutions, a method is suggested for determining the stability constants of this kind of complex. For demonstration purposes it has been used with the iron(ll1) and Eriochrome Cyanine R system and satisfactory results were obtained. This method is generally applicable and could be used for most ’anti-stepwise‘ systems. Keywords: Polynuclear complex; stability constant; iron; Eriochrome Cyanine R; spectrophotometry Stepwise coordination reactions are very common in coordi- nation chemistry and their equilibria in solution is well understood. However, for coordination equilibria of the type M + R f MR M,,R, investigations carried out so far seem insufficient. The latter, in which the complexes formed are called ‘anti-stepwise complexes’, is different from the former; as M is increased, the series from mononuclear to polynuclear complexes are formed.Although they are less widely known than the normal analytically useful complexes, they are not uncommon.1-7 The methods used in the determi- nation of the stability constants of the former can hardly be used in that of the latter, based on Bjerrum’s function (2)8 and the method of corresponding solutions,v so a method has been developed here for the latter. It has been applied to the iron(m) and Eriochrome Cyanine R system and satisfactory results were obtained. M2R f ... Theory Suppose that an anti-stepwise complexing system consists of the following reactions: M + R = MR K11 = [MR]/[M][R] (1) 2M + R = M2R K21 = [M,R]/[M]2[R] (2) (the charges are omitted for simplicity).If there are three coloured species, R, MR and M?R, in the system, EO, E~ and €2 are their respective molar absorptivities, the absorbance of the solution is expressed as A = EO[R] + E~[MR] + E ~ [ M ~ R ] CM = [MI + [MR] + 2[M?R] CR = [R] + [MR] + [M2R] (3) (4) ( 5 ) From mass balance equations, we obtain where cM and cR represent the total concentrations of the metal ion and the ligand, respectively. Substituting eqns. (4) and ( 5 ) into eqn. (3) yields A = ( E ~ - q))[MR] + (E? - EO)[M~R] + EOCR (3a) AA = A - EOCR = (11 - E,,)[MR] + (E? - E ~ ) [ M ~ R ] (3b) * Present address: Department of Basic Sciences, Shandong Agricultural University, Taian, Shandong, People’s Republic of China.t To whom correspondcncc should bc addrcsscd. We define E = AA/cR Substituting eqns. (l), (2), (3b) and ( 5 ) into eqn. (6) yields E = {(El - E*)Kll[MI + (E2 - EO)K21[Ml2V (1 + KlI[MI + &,[MI2) (6a) (7) We define 6 = (cM - [M])/cK % is different from 6, and is called the ‘average anti- coordination number’. Substituting eqns. (l), (2), (4) and ( 5 ) into eqn. (7), we obtain f i = ([MR] + 2[M,R])/([R] + [MR] + [MZR]) = (KlI[MI[RI + 2K21[M12[R1)/([Rl + Kll[M”l = (KIl[M] + 2K21[MI2)/(1 + KlI[MI + K21[M12) + K21[M12[R1> (7a) As E(), E ~ , E ~ , KI1 and K21 in eqns. (6a) and (7a) are constants under given conditions, this indicates that E and 6 are only a function of [MI. Therefore, it follows that solutions having the same E must have the same value of [MI. As % is a function of [MI only, it also follows that these solutions have the same value of f i .Hence we call these ‘anti-corresponding’ solu- tions. The experimental method to find these solutions is as follows. Several series of solutions are prepared with cR fixed and CM gradually increased. AA is measured for each of the solutions, then E is evaluated using eqn. (6). A series of graphs of E versus cM are plotted then lines parallel with the abscissa are drawn that intersect the curves at points a l , b , , ..., e l , and a2, b2, ..., e2, etc. The solutions corresponding to the points of intersec- tion are anti-corresponding solutions. They have the same When the anti-corresponding solutions have been obtained, [MI. eqn. (7) is rewritten in the form CM = 6 c R + [MI (7b) A linear plot of cM versus cR is constructed from a set of anti-corresponding solutions.The slope and the intercept of the line are % and [MI, respectively. According to the treatments mentioned above, several pairs of values ( f i , [MI) are obtained from several sets of anti-corresponding solutions. Each pair of values (%, [MI) are substituted into eqn. (7a) to give linear equations, where KI1 and Kzl are unknown values. By arbitrarily taking two of the equations to form simul- taneous equations, K l l and K21 can be evaluated.98 ANALYST, JANUARY 1993, VOL. 118 For a system that consists of more than two anti-stepwise complexes, the stability constants of these complexes can be evaluated by a similar treatment and selecting the appropriate equations for solution.Experimental Apparatus A Shimadzu Model UV-3000 recording spectrophotometer and a Model pHs-2 acidimeter were used. Reagents All chemicals were of analytical-reagent grade. A 1.00 x 10-3 rnol 1-1 stock standard solution of iron(rr1) was prepared by dissolving calculated amounts of NH4Fe- (S04)2-12H20 in 0.01 moll-' hydrochloric acid. Working standard solutions were prepared by appropriate dilution of the stock standard solution. Eriochrome Cyanine R (ECR) was purified according to the procedure of Dixon et al.,lO which uses slightly modified dextran chromatography,"ll and was used to prepare a 1.00 X 10-3 rnol 1-1 stock standard solution with a final hydrochloric acid concentration of 0.008 rnol 1-1. Working standard solu- tions were obtained by appropriate dilution of the stock standard solution.Buffer solution was prepared by mixing 0.5 rnol I-' monochloroacetic acid and 0.5 moll- 1 sodium hydroxide solutions in a suitable ratio (pH 2.75). Procedure Into each of a series of 25 ml calibrated flasks, transfer a known volume of iron(m) solution, a known volume of ECR solution, 5 ml of buffer solution and a suitable volume of 1 rnol 1-1 potassium nitrate solution (keep the ionic strength at 0.70 0.60 0.50 0.40 (0 e s n q 0.30 0.20 0.10 0 E 520 560 600 640 Wavelengthlnm Fig. 1 Absorption spectra for Fe'Il-ECR system. pH, 2.77; 2 cm cells; reagent blanks as reference. A-E, cFe = 3.20 X 10-5 rnol 1-1; cECR (xl0-6 rnol 1-1): A, 2.00; B, 4.00; C, 6.00; D, 8.00; and E, 10.00.A'-F', cECR = 4.00 x 10-5 moll-1; cFe (x10-6 mollp1): A', 2.00; B', 4.00; C', 6.00; D', 8.00; E', 10.00; and F', 12.00. CECR > C F ~ , solid lines; cECR < cFe, broken lines O . l ) , dilute to the mark with distilled water and mix. After allowing the flask to stand for 15 min, measure the absorption spectra or absorbances of the solution. Maintain the tempera- ture of the solutions at 20 ? 1 "C throughout. Results and Discussion Langmyhr and Stumpes studied the complexing reaction between iron(ir1) and ECR in detail. They found Fe(ECR) and Fe2(ECR) complexes around pH 2.7 and that the formation of the two species depended on the mole ratio between the reactants. In the presence of a large excess of iron over the ligand, the formation of 2 : 1 complexes predominated; with a large excess of the ligand, the formation of 1 : 1 complexes predominated.We obtained similar results. However, in the pH range 3.9-4.2, they reported the existence of 2 : 2 complexes, but we could not obtain the same results, because the colour of the solutions was unstable under these conditions and turbidity occurred soon after the solutions had been mixed and set aside. Therefore, in this work, the equilibria among 1 : 1 and 2 : 1 complexes and the reactants were studied at pH ==: 2.7. 0.35 m 0.30 - 0.25 - yJ 0.20 - +? s n a 0.15 - c m 0.10 - 0.05 - 0 Wavelengthlnm Fig. 2 Absorption spectra for Fe-ECR system. CECR = 5.00 X 10-6 mol 1-1; cFe (X10-5 moll-1): A, 0.40; B, 0.80; C, 1.20; D, 1.60; E, 2.40; F, 3.20; and G, 4.00. Other conditions are the same as for Fig.1 Table 1 Conditions measured and results obtained by the Asmus method (pH 2.77). Subscripts 0 and i represent the fixed and increased concentration, respectively Measuring wave- length/ No. cFe,o/mol 1-l cEcRJmol 1- nm 4MtnR;z) 1 2.00x 10-6 (0.50-2.so) x 10-5 573 1 2 3 . 2 0 ~ 10-5 (2.00-10.00) x lo-" 586 1 3 (2.00-12.00) x 10-6 4.00 x 10-5 573 1 4 (2.0W.80) X 10-5 5.00 x 10-6 586 2 cFe,l/mol 1-1 cEcR,o/mol I-' m ( M m R,,)ANALYST, JANUARY 1993, VOL. 118 99 Table 2 Values of E ( X 10-5 1 mol-1) for five series of solutions. Measuring wavelength, 586 nm; for other conditions, see Fig. 1 Curve in Fig. 3 1 0.50 1 .oo 0.109 0.201 0.280 0.345 0.400 0.456 0.498 0.538 0.574 0.589 - 2 1.50 0.083 0.161 0.230 0.294 0.343 0.401 0.445 0.486 0.514 0.545 - 3 2.00 0.071 0.136 0.194 0.248 0.293 0.338 0.378 0.415 0.446 0.474 0.496 4 2.50 0.060 0.118 0.171 0.217 0.261 0.305 0.344 0.381 0.412 0.436 0.464 5 c ~ , / ~ O - ~ mol I-' cEcR/10-5 moll-L 0.40 0.80 1.20 1.60 2.00 2.40 2.80 3.20 3.60 4.00 4.40 0.134 0.248 0.336 0.414 0.474 0.530 0.576 0.618 - - - Table 3 Values of rn and [MI for four sets of anti-corresponding solutions Corresponding By substituting [Fe] and 6 into eqn. (7a): No. of - equation solutions [Fe]/10-5 moll-' rn (2 - G)K21[M]2 + (1 - G)K1I[M] = G a1-1 1.06 0.9879 1.137 x 10-'OK21+ 0.0128 X 10-5KIl = 0.9879 7a-1 a2-e2 0.84 0.8430 0.8164 x lO-l"K21 + 0.1319 X 10-5K11 = 0.8430 7a-2 a4-e4 0.50 0.5920 0.3520 x 10-1"K21 + 0.2040 X 10-5Kll = 0.5920 7a-4 are3 0.67 0.6954 0.5856 x 10-1"KZl + 0.2041 X lO-'KIl= 0.6954 7a-3 Table 4 Values of K2, and K1, obtained by the method of anti-corresponding solutions No.of simultaneous equations 7a-1. 7a-2, 7a-1, 7a-2, 7a-1, Parameter 7a-2 7a-3 7a-4 7a-4 7a-3 Average Kll(x 10-5) 1.090 0.830 1.431 1.554 0.945 1.17 K2,(x10-l") 0.856 0.899 0.853 0.782 0.858 0.85 Absorption Spectra Three series of solutions were prepared: (1) cECR fixed and cFe gradually increased (cEcK > CFcj; (2) cFC fixed and C ~ C K gradually increased (cECR < cFe); ( 3 ) cECR fixed and cFe gradually increased to an excess. Their absorption spectra are shown in Figs. 1 and 2. The solid and broken lines in Fig. 1 represent the absorption spectra of solution series (1) and (2), respectively, and it can be seen that the wavelengths of maximum absorption differ. This shows that there may be two different complexes in the system.When cECR < cFe, complexes with a maximum at 586 nm are mainly formed; when cECK > cFe, complexes with a maximum at 573 nm are mainly formed. The regular pattern of the series of curves in Fig. 2 is identical with that in Fig. 1. The above conclusion of the existence of two complexes at pH 2.77 agrees with that reported by Langmyhr and Stumpe.5 Determination of the Compositions of the Complexes The compositions of the two complexes were determined under different experimental conditions. At pH 2.77, in the presence of a large excess of iron(iir), the composition of complexes with a maximum at 586 nm was found to be Fe2(ECR); in the presence of a large excess of ECR the composition of the complexes with a maximum at 573 nm was Fe(ECR).The calculation method used was the Asmus method" and extended by Klausen and Langmyhr.13 The conditions and the results are shown in Table 1. Determination of Stability Constants The determination of the stability constants of Fe-ECR complexes as reported5 requires the preparation of solutions under specified conditions in advance, such that only one complex actually exists and the other is neglected. It includes approximate treatments in the equilibrium calculation. It is considered that the method published previously5 is not suitable for anti-stepwise complexes where K21 approaches KI1. However, the method proposed here is not restricted by these conditions and can allow the complexes to co-exist during the determination of their stability constants.0.60 0.50 7 0.40 - E - 8 0.30 ;5 0.20 0.10 I 0.80 1.60 2.40 3.20 4.00 0 cF,/10-5 mol 1-1 Fig. 3 conditions see Table 2. Graphs of T. versus C F ~ for five series of solutions. For 3.60 1 ";p 1 3.20 2.80 7 2.40 0 - - E 2.00 LD I 0 7 1.60 1.20 W 0" 0.80 0.40 t cEcR/10-5 mol 1-1 Graphs of cFe versus cECR for four sets of anti-corresponding Fig. 4 solutions100 ANALYST, JANUARY 1993, VOL. 118 Five series of solutions were prepared as shown in Fig. 2. The AA value of each solution prepared was measured, then E was evaluated from eqn. (6). The results are given in Table 2. A series of graphs of E versus cFe are plotted in Fig. 3. As described above, several lines were drawn parallel with the abscissa, intersecting the curves at points a-e, e.g., al-el at E = 0.40 x 10s 1 mol-1 .The solutions corresponding to the points of intersection are anti-corresponding solutions, with the following compositions: a1 bl c1 dl el cECK/10-s moll-1 0.50 1.00 1.50 2.00 2.50 cF,/lO-~mol 1-1 1.53 2.00 2.44 3.05 3.49 and correspondingly for a2-e2, etc. After the anti-corresponding solutions had been obtained, graphs of cFe versus cECR were plotted according to eqn. (7b) (Fig. 4). In Fig. 4, each line represents a set of anti-corre- sponding solutions, and therefore each pair of values (6, [MI) can be obtained from its slope and intercept. The results are given in Table 3. These values were substituted into eqn. (7a) and on rearrangement yielded the linear equations given in the fourth column. Further, by taking arbitrarily two of the equations to form simultaneous equations, we obtained the values of KI1 and K2] (Table 4). The results are basically the same as those reported (at pH 2.70, K Z 1 = 2.00 x 1010 and K I 1 = 3.16 X 10s).5 The advantage of this method is that it can be applied to any situation where complexes are formed. 1 2 3 4 5 6 7 8 9 10 11 12 13 References Sato, H., Koyama, Y., and Momoki, K . , Anal. Chim. Acta, 1978, 99, 167. DoLsa, L., Saabo. A., and Beck. M. T., Acta Chim. Hung., 1971. 67, 189. Murakami, S., and Yoshino, T., Bull. Chem. Soc. Jpn., 1981, 54, 619. Semb, A . , and Langmyhr, F. J., Anal. Chim. Acta, 1966, 35, 286. Langmyhr, F. J., and Stumpe, T., Anal. Chim. Acra, 1965,32, 535. Langmyhr, F. J., and Klausen. K. S . , Anal. Chim. Acta, 1963, 29, 149. Zou, S. F., Chao, W., and Wen, S . Q., Gaodeng Xuexiao Huaxue Xuehao, 1990, 11. 240. Bjerrum, J . , Metul Ammine Formation in Aqueous Solution, Haase, Copenhagen, 1941. Orszagh, I., and Beck, M. T., Acta Chem. Scand., 1979,33,63. Dixon. E. J., Grislcy, L. M.. and Sawycr, K., Analyst, 1970,95. 945. King, H. G. C., and Pruden, G., Analyst, 1967. 92, 83. Asmus, E., FreJenius’ 2. Anal. Chem., 1960, 178, 104. Klausen, K. S., and Langmyhr, F. J., Anal. Chim. Acta, 1963, 28. 501. Paper 21037673 Received July 15, I992 Accepted August 6, 1992