Analytica Chimica Acta, 226 (1989) 323-329Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

Short Communication

323

DETERMINATION OF STABILITY CONSTANTS OF COMPLEXSYSTEMS BY A LINEAR PLOT METHOD

HENG LI* andJIXIANG WANG

Department of Food Science and Technology, Shanghai Fisheries University, Jun Gong Road334, Shanghai (China)

(Received 28th November 1988)

Summary. A linear plot method for determining the stability constants of complex systems ispresented. It gives high accuracy because side-reactions of both the metal ion and ligand and thedifference between the total ligand and free ligand concentration are incorporated in the generalequations. This method is also suitable for determining the constant of complexes of very lowstability. The effectiveness is demonstrated on the barium-glycine (GLY) system, the zinc-thi­ocyanate (SCN) system and the lead-fluoride system by spectrophotometry and potentiometry.The values of the complex stability constants are logPl= -0.374 (Ba-GLY system), logPI=0.72and logP2= 1.86 (Zn-SCN system) and logPl= 1.39, logP2=2.56 (Pb-F system).

There are two main methods for determining the stability constants of metalcomplexes [1], one based on the function AM(L) (M = metal, L=ligand) andthe other based on the complex formation function ii. In the former method,the free ligand concentration is usually assumed to be equal to the total ligandconcentration, CL ~ [L], in cases where CL » CM throughout the experiments.The values of the overall complex stability constants Pi (i = 1, 2, 3, ...) areobtained by a graphical method, plotting (AM(L) -1) j [L],(AM (L)- I - Pd L ]) j [L ]2 or other parameters against [L]. In the lattermethod, the complex formation function nmay include the side-reaction coef­ficient of the ligand. The complex stability constants can be determined by agraphical method from the half-value points or from the linearized form [1].Although these methods have been improved [2-6], there is no general equa­tion that incorporates the side-reactions from both the metal ion and the li­gand and the difference between the total ligand and free ligand concentration.This communication presents a modification of the above methods. The methodis also suitable for the study of complexes which have very low stability con­stants that are difficult to determine by conventional methods. The effective­ness of this method is demonstrated on barium-glycine, zinc-thiocyanate andlead-fluoride systems by both spectrophotometry and potentiometry; satisfac­tory results were obtained.

0003-2670/89/$03.50 © 1989 Elsevier Science Publishers B.V.

324

TheoreticalThe following scheme shows the reactions between a metal ion M and ligand

L in a complex system:

M+L=ML Pl= [ML]f[M] [L]

M+2L=M~ P2= [M~l/[M] [L]2

M+iL=MLi Pi= [MLd/[M] [L]i

(1)

(2)

(3)

(i = 1, 2, 3, ..., n), where Pi is the overall complex stability constant of the ithcomplex. In addition to these main reactions, there may also be side-reactionsof MandL:

M+aA=MAa , M+gG=MGg ••,

L+bB=LBb , L+hH=LHh •••

(4)

(5)

(a, g, b,h = 1, 2, 3, ...). The side-reaction coefficient [1] is written as follows:

aM=aM(A) +aM(G )+... - (m-l)

aL=aL(B) +L(H)+ ... - (l-l)

(6)

(7)

where m and l are the number of side-reactions.The total concentrations of M species (CM ) and L species (CL) in the so­

lution, whether they are complexed or not, are given by

n

eM = [M] (I Pi[L]i+ a M)i=1

n

CL- [L]aL= [M] I i[L] iPii = 1

By combining Eqns. 8 and 9, the following equations can be obtained:

y naM P + nIl P ._(l+i-n) [LP1= (l-n) [L] 1 j =1 1+) I-n

(8)

(9)

(10)

i-1-nYi= (Yi-1-Pi-1) (i-n)[L]

(i=2, 3, ..., n) and

Xi=i~1-=n[L]l-n

325

(11)

(12)

(j=1, i=1, 2, ..., n), where n= (CL- [L]ad/CMand [L] is the concentrationof the free ligand in the solution.

The values of overall complex stability constants Pi (i = 1, 2, 3, ..., n) can beobtained by the graphical method, plotting Yi versus Xi; the intercept is Pi' Onthe other hand, the function AM(L), which includes side-reactions, can be writ­ten as follows:

AM(L)=CM/[M] =aM + PdLL+ P2j[L]j + P3j[LlJ +... (13)

where [M] is the concentration of metal ion in the solution, which can bemeasured, e.g., by polarography or with metal ion-selective electrodes. When

n

j=1, then [L]l=CdaL' when j=2, [Lb= (CL- L iPidM][LH it«: andi=l

n

when j=j, [LL= (CL- L iPiU-1) [M] [L]}_l )/aL until IPij- PiU-1) I/Pij< fi=l

(e.g., f = 5%). Equation 13 takes into account the side-reactions and the dif­ference between the total ligand and free ligand concentrations, which is nor­mally not included in the function AM(L)'

The values of complex stability constants can be obtained by a graphicalmethod and by iterative calculations. It can be seen that the larger the differ­ence between CLand [L], the larger is the number of iterations required andthe more accurate are the results.

DiscussionExample 1. Determinations on the barium-glycine system by spectrophoto­

metry. Harju [7] used a spectrophotometric method to determine the stabilityconstants ofcomplexeswith very lowstability, but only the formation 1:1metal­glycine (M-GLY) complexes was studied. The values of the stability constants({J;., in Table 1) were calculated directly. The values of Yi and Xi obtainedfrom Eqns. 10 and 12 are also given in Table 1. The values oflog[H], volumeof 0 H - added and A are taken from Harju's paper [7]. In the evaluation, theconcentration of free ligand in the solution is calculated as (A - 0.068)/18.07and the total concentration of ligand or metal ion as 0.02X50/ (50+ V) or0.333X50/ (50+ V), where Vis the volume of OH-. The side-reaction coeffi­cients of ligand and metal ion with hydrogen and chloride ions are obtainedfrom the expressions aL(H) = 1+ 109

.567+1og[Hl and aM(OH, en = 10°.051+

1O-log[Hl-13.3 (aM(OH) = 1+ 10-1og[Hl-13.3), respectively [1,7,8].

326

TABLE 1

Spectrophotometric determination of fJB..-GLY (CGLy =0.0200 M, CBa = 0.333 M, 1= 1.0 M, A. = 225nm, EL = 18.07 1 moI- 1 cm -1, Abackground = 0.068, Vo= 50.00 ml, aBa(Cl) = 10°.051 _ 1)

-Iog[H]V,OH- (mI)A[L] X10-2 (M)nXlO-3

XiX 10-2

v,LogPlLogP;[7]

10.3490.850.3461.5385.3133.0850.3908

-0.408-0.427

10.6260.920.3581.6056.6943.2210.4732-0.325-0.342

10.9591.020.3731.6886.2423.3860.4202

-0.377-0.397

11.2591.160.3781.7166.2763.4420.4173

-0.380-0.402

11.6281.500.3801.7276.1903.4640.4134

-0.384-0.415

From the values of the function nand the relationship between YI and X,shown in Table 1, it can be seen that the complex reaction of the Ba-GLYsystem under the experimental conditions is a one-step process, i.e., i= 1, andthe average value of 10gPIis - 0.374, which is in agreement with Harju's study[7] (log P~ shown in Table 1) . When the concentration of glycine in the so­lution increases, Ba(GLYb Ba(GLYh or other complexes may be formed.The other values of the overall stability constants Pi (i=2, 3, ..., n) of com­plexes can be obtained by the graphical method, plotting Y i against Xi' It isdifficult to determine their stability constants accurately by the conventionalmethods [1-9]. The experimental data can also be dealt with by the regressionmethod.

Example 2. Determinations on the Zn-SCN system with a thiocyanate ion­selective electrode. Small volumes of 0.942 M NaSCN solution were added to asolution containing 49.98 ml of 0.0261 M ZnS04' Potentiometric analysis wascarried out by using a Model 217 calomel reference electrode and a thiocyanateion-selective electrode [10], which respond to the SCN ion concentration inthe solution. The potential was measured by using a Model 750 Ionic Meter(Fisher Scientific) with stirring at 25±0.5°C.

The values of Y i and Xi can be calculated from the experimental data bymeans of Eqns. 10 and 12. The values of both aM and aL are 1 in this systembecause there are no side-reactions. The relationship between Y i and Xi inTable 2 is shown in Fig. 1. The value of the overall complex stability constantPI is obtained from YI~Xlt andp2 from Y2~X2; the results are given in Table3.

Example 3. Determinations on the Pb-F system with ion-selective electrodes.In Table 4, experimental data taken from Hefter's studies [4], in which theside-reactions were neglected are incorporated. In the calculation of X, and Y1

by means of Eqns. 10 and 12, the following values are used: for the total con­centration of metal or ligand, 3.12X10-2X25/(25+V) or 4XIO-4x

327

TABLE 2

Calculation of Yi and Xi for the Zn-SCN system (LogPI =0.72)

V (ml) [SCN] (M) YI Xl Y2 X2

0.50 0.0080682 6.365 0.01655 64.341 0.012200.75 0.0118372 7.472 0.02472 87.848 0.018011.00 0.0156593 7.905 0.03326 78.327 0.023951.25 0.0192605 8.894 0.04182 85.945 0.029651.50 0.0229745 9.329 0.05087 79.197 0.035571.75 0.0266790 9.710 0.06027 73.169 0.041552.00 0.0302765 10.306 0.07000 71.511 0.047462.50 0.0373822 11.542 0.09089 68.671 0.059393.00 0.0445905 12.364 0.11376 62.094 0.071703.50 0.0525806 11.257 0.13628 43.710 0.084874.00 0.0583141 15.543 0.16948 60.436 0.096564.50 0.0646727 18.792 0.20794 64.882 0.109235.00 0.0714502 20.925 0.24973 62.570 0.122465.50 0.0774396 27.211 0.31806 68.889 0.136026.00 0.0836098 34.899 0.41118 71.985 0.150226.50 0.0892396 54.713 0.61420 80.450 0.165517.00 0.0952485 88.933 0.99733 83.858 0.181407.50 0.1016619 114.74 1.69924 82.060 0.19724

1d" m1m 0Y2

o Io 0·1 0·2

-x

Fig. 1. Relationship between Yi and Xi'

400

200

Fig. 2. Plot of A ij for the lead (II) -fluoride system for two successive steps in the iteration procedure.

v/ (25+ V), and for the side-reaction coefficients, aM= 1.002 and aF= 1.014at pH 5.0 [1]. The value of Y' was taken from Hefter's paper [4], which meanswhen aM =aF =1; these values are higher than the result from this method. Itcan be seen that Y I increases when aM increases and YI decreases when aLincreases, from Eqn. 10. The larger aM and aL, the more the effect becomesmanifest in the experimental results.

From the values of the function n and the relationship between Y I and K,shown in Table 4, it is obvious that under the experimental conditions the

328

TABLE 3

Stability constants of the Zn-SCN system

Method

SCN - electrodeSCN- electrodeSCN- electrodeSpectrophotometricPolarographic

·1= ionic strength.

TABLE 4

2525202030

1 MNaClO4

1 MNaCl04

0.1 MKNOs1MNOi2MKNOs

0.72,1.860.54,1.881.070.440.5, 1.32, 1.32, 2.62

Ref.

This work10111213

Calculation of values of Yi and Xi (T=25°C; 1= 1.00 mol l" ': ([Pb2+ ]lot)initiol = 3.12 X 10-2 M;(volume)initioi=25.00 ml; titrant, 1= 1.00 mol l" ': [NaF] =4.00X 10-4 M)

Volume added [F-] (10-6M) nXlO- 4 X IX lO- 6 YI Y'(ml)

0.30 2.64 0.67 5.28 25.4 25.80.50 4.32 1.13 8.64 26.3 26.70.70 5.98 1.60 12.0 26.8 27.11.05 9.48 2.17 19.0 23.0 23.41.40 12.2 2.99 24.4 24.6 25.01.80 15.8 3.72 31.6 23.6 24.12.30 20.0 4.70 40.0 23.5 24.03.00 26.0 5.92 52.0 22.8 23.34.00 34.1 7.66 68.2 22.5 23.05.00 41.2 9.57 82.4 23.3 23.8

Result: PI = 24.0 or logPl = 1.38

complex reaction of the lead (II) - F system here is a one-step reaction, i.e., i= 1.When the concentration of fluoride in the solution increases, PbF2, PbFi orother species may be formed. The other values of the overall stability constantsPi (i = 2, 3, ...) of the complexes can be obtained by the graphical method, plot­ting Yi against Xi'

Table 5 and Fig. 2 show the results for the lead(II)-fluoride system mea­sured by using a lead ion-selective electrode. The reference electrode is a dou­ble-junction calomel reference electrode containing a salt bridge of 1 M potas­sium nitrate. The initial concentration of lead (II) nitrate was 4X10-4 M, theinitial concentration of sodium fluoride was as shown in Table 5, the pH of thesolutions was 5.5 and the ionic strength was maintained throughout the ex­periments at 1.0with sodium perchlorate. All measurements were carried witha Model 811 digital pH-millivolt meter (Orion Research) with stirring at25:!:0.5 0 C.

329

TABLE 5

Analysis of A ii function for the lead (II)-fluoride system

CF - (M) [Pb2 + 1(10- 4 M) [F-ll (M) An A21 [F-jz (M) A 12 A22

0.0800 0.7692 0.0796 52.70 356.8 0.0791 53.03 359.40.0700 0.9027 0.0697 49.19 357.4 0.0692 49.52 360.20.0600 1.072 0.0597 45.67 357.9 0.0593 46.00 361.10.0500 1.289 0.0498 42.13 358.0 0.0494 42.47 361.80.0400 1.573 0.0398 38.64 360.3 0.0395 38.93 362.90.0300 1.950 0.0299 34.99 357.5 0.0296 35.35 363.10.0200 2.450 0.0199 31.54 363.8 0.0197 31.84 367.30.0100 3.121 0.00995 27.80 351.8 0.00985 28.09 353.8

Results: Pn = 24.3; P12 = 24.6; P21 = 358.5; P22 =361.7

The values of AM(L)ij and [F- Lwere calculated by Eqn. 13, then the overallstability constants were obtained by the graphical method, plotting Aij against[F- L. In the calculation the coefficients aF(H) = 1+103

.15 X10-5

.5 = 1.0045 and

aM(OH)=1 +106.2 X 105

.5

-14= 1.0050 were used [1].

The results in Table 5 and Fig. 2 for two iterative steps were almost identicalbecause there was only a small difference between the values of [F-] 1 and[F-] 2' It can be seen that the values of A ij and Pij increase when aL increasesor [L L decreases. The relationships between A2j and [F- L indicates thatPbF; does not form.

In conclusion, it can be stated that the method presented here can be suc­cessfully applied to determine complex stability constants with higher accu­racy by taking into account the side-reactions of both M and L, and also thedifference between the total ligand and free ligand concentrations. Comparedwith other methods, the method is simpler and more accurate.

REFERENCES

1 J. Inczedy, Analytical Applications of Complex Equilibria, Wiley, New York, 1976.2 E.R. Still, Anal. Chim. Acta, 116 (1980) 77.3 S.-H. Shi and H.-L. Zhu, Fenxi Huaxue, 14 (1986) 817.4 G. Hefter, J. Electroanal. Chem., 39 (1972) 345.5 D. Midgley, Analyst, 112 (1987) 557.6 S.-H. Shi, Y.-W. Bai, L.-N. Wang and W.-F. Zhang, Fenxi Huaxue, 13 (1985) 646.7 L. Harju, Talanta, 34 (1987) 817.8 S.K. Patil and H.D. Sharma, Can. J. Chem., 47 (1969) 3851.9 A.M. Bond and G. Hefter, J. Electroanal. Chem., 42 (1973) 1.

10 X.-O. Wang and J.-Y. Yin, Chem. Sensors, 7 (1987) 33.11 M. Polasek and M. Bartusek, Scr. Fac. Sci. Nat. Univ. Purkynianae Brun., 2 (1970) 109.12 K.B. Yatsimirskii and V.D. Tetyushkina, Zh. Inorg. Chim., 2 (1975) 320.13 R.E. Frank and D.N. Hume,J. Am. Chem. Soc., 75 (1953) 1736.