Calculation of all equilibrium concentrations in a system of competing complexation

149

SHORTCOMMUNICATIONS (- 058 V) and that of the tetrathionate-thiosulphate couple is 0.08V.8 Although sulphite is produced by oxidation of hydrosulphite with ferricyanide, it does not interfere in the determination because the rate of oxidation of hydrosulphite is much greater than that of sulphite under the conditions specified. However, it is imperative that the solution of hydrosulphite be stirred efficiently while titrant is being added. Inefficient stirring leads to high results, owing to the slow oxidizing action of ferricyanide on sulphite. If the solution is continuously or fairly regularly stirred. the error is less than 0.3%. but if stirring is virtually omitted, errors of up to almost + 5% occur. Tris(4,7-dihydroxy-1-lo-phenanthroline)iron(II) failed to function as an indicator in titrations of cobalt(H) sulphate, ascorbic acid, and sodium hypophosphite with potassium ferricyanide, and functioned very poorly as an indicator in the titration of sodium hydrosulphite with potassium octacyanomolybdate(V1). Addition of octacyanomolybdate(VI) to a solution containing the indicator gave an amber colour, and the end-point, from amber to yellow, was so gradual that it was almost impossible to detect. It is sus-

petted that formation of a mixed complex, possibly dicyano-bis(4,7-dihydroxy-1,lO-phenanthroline)iron(II), is involved. REFERENCES 1. D. P. Poe and H. Diehl, Talanta, 1974, 21. 1065. 2. Idem. ibid, 1976, 23. 141. 3. R. G. Bates, Determination of pH, p. 71. Wiley-Interscience. New York. 1973. 4. P. George, G. I. H. Hanania and W. A. Eaton, in Hemes and Hemoproteins, B. Chance, R. W. Estrabrook and T. Yonetani, p. 269. Academic Press, New York, 1966. 5. M. N. Hale and M. G. Mellon, J. Am. Gem. Sot., 1950, 72. 3217.

6. G. Charlot, Memoires Present& a la Socie’tP Chimique, 1939. 6. 977. 7. K. Jellinek, Das Hydrosulfit, p. 71. Enke, Stuttgart,

1912. 8. W. M. Latimer, Oxidation Potenti&, 2nd Ed., PrenticeHall. New York, 1952.

Summary-The formal reduction potential of the tris(4,7-dihydroxy-1,lO-phenanthroline)iron(III,II) couple is - 0.06 V in the pH range l&l 3, not - 0.11 V as reported earlier. The couple forms an excellent visual oxidation-reduction indicator for the titration of sodium hydrosulphite with potassium ferricyanide in alkaline solution.

T&ma.

Vol. 23. pp

149-152. Pergamon

Press, 1976. Prmted !n Great Britam

CALCULATION OF ALL EQUILIBRIUM CONCENTRATIONS SYSTEM OF COMPETING COMPLEXATION

IN A

G. GINZBURG Chemistry Department, Ben Gurion University of the Negev, Beer-Sheva, Israel (Received 22 February 1975. Accepted 8 April 1975)

It is often necessary to calculate the equilibrium concentration of each species in a multicomponent system of metal ions and ligands. As a rule, the starting data are the total (analytical) concentration of each reactant, the stability constant of each complex and the pH of the final solution. Some programs have been described for solving this problem.‘*’ The following iterative method is used in the computer program COMICS:’ the equilibrium concentration of each species is found at the kth approximation ([At]); from these, the total concentration of each of the reactants is calculated (A?“); then the (k + l)th approximate equilibrium concentrations are determined by the equation: [A!“]

= [A:]

c,1 $

I’*

(1)

The concentration of this complex is given by: cj = p, fi [Ai]“>~ i=*

(j= 1,...,N)

(3)

where Ai are the M different reactants, which may be metal ions, complexing agents, H+ and OH- ions; mij are the stoichiometric coefficients of the ith reactants in the jth complex; pj is the cumulative stability constant and cj the equilibrium concentration ofthe jth complex. For the system described by equations (2) and (3) it is possible to set up M equations for the material balance:

ATOT = CA,1+

5mijcj

(i = I,.

, M)

j= 1

(4)

where A:“’ is the total concentration of the ith reactant. With this program it turns out that the iterations do not always converge. Therefore it is of interest to examine the criteria for convergence in a more general case. The present work will consider a somewhat modified program which gives improved intermediate results. Let us write the jth complex formation reaction in the following way:

where ATOTi is the total concentration of the ith reactant. Following COMICS,’ the approximation to ATOT, given by the kth iterative step will be denoted by ACALC;. Using equations (3) and (4) it will be shown that

ii, m;jAi = I(A,),,(A,),~...(A~M)mMI (i.= l,...,N)

The (k + I)th iteration is obtained by modifying the method employed in COMICS,’ from which the following general-

(2)

ACALC! = [A”] + i mijPj. i j=1

i= 1

[A”]“”

(5)

(i= l,...M)

150

SHORT COMMUNICATIONS

Table 1. Calculation of the equilibrium concentrations in a multicomponent system. The total concentration of each reactant is equal to 5 x 10m3M.pH = 8.0. The stability constants were taken from refs. 4 and 5. Equilibrium concentrations below JO-‘M were not included. NTA is the tertiary anion of nitrilotriacetic acid Equilibrium concentration, 10-6M

2.65 4.75 6.19 7-12 6% 5.14 4.15 7.65 IO.54 1267 2.37 4.XI 7.31 9.46 16.12 6.47 2.3 4.4 14.37 15.44 9.80 15.3 12.68 IO.45 14.6 9.73 5.0

919.6 65.1 1.0 _393.8 700 305 23.1 416 64.4 11.5 0.91 146 0.73 79,4 0.74 329

1

2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

Logarithm of stability constant

3380 1270 20.8 3660 49.6 3160 1.42 x 1O-5 562

H -

-. -

I

1 1 1 I 1

2 3 4 5 6

-

-

-

1 1

-

1

1

-

1

1 -

3 4

-

-

-

1

-

_-_

-

1

-. -

2 3 4

-

2 3 4 --

1

I 1 1 I

4

1

-

-

1 1

-

1

-

-

-

I 1

1

1 .-

1

--

--

1

1

-

1

1

-

-

1 -

-

I 2 1 1

-

-

-

-

-

I

-

1 1 I -_

I -

-

-

-

--

1

1

In conformity with condition (1l), equation (9) for the first iterative cycle may be rewritten:

l/P = [A”], A$As 1

1

-

-. 1

NH,

-

--

-. -

NTA

-

-

ization of equation (1) is obtained: [A”“]

Composition of species cu Zn Cd

OH

(6)

c; = (1 - h)([Ai] + E;) - [Ai] = E: - h([AJ $ E:) (12)

where [A”] and [A:“‘] are consecutive approximations to the equilibrium concentration of the ith reactant. Let us write [A:] as [A;] = Ai + 6: (7)

Owing to condition (1 I), a value of p may be selected such that: ]h([AJ + $‘)I < k$‘/ or k!j < ]eyi (13)

,1

It follows from equations (5) (6) and (7) that

L

1/P

ATOT,


IcyI> Ie!I >

[Ai] + E: + f llzij.pj. fi ([As] + c:)mr1 i i=, j=1

(8)

Dividing the numerator and the denominator of the first bracketed expression in equation (8) by ATOT,, one obtains for the first iteration: 1 l/P (9) (iAil + 6) - CM ci= 2 where

Cl

V M [AJ + ej’ + c mij. a,. n ([A{] + e:)‘“,~ j= L L=1 Lp = --__> 0 ATOT, It is known3 that for L > 0: (L)-*=

1 --ii

where

limh=O

e-7

> /Ef(

[Ai] + E: + f pnijj?, fi ([AJ + c:)mrJ

1

x (CM + 4 - [M

If [Afl and e: were already chosen, these values in the (k + 1)th iteration would be determined by equation (8). Furthermore:

(IO)

(11)

IL;/ = *

j= 1

I[Aill + letI + s--

111

ATOT,

L 1=1

i ~ijfij i ([A,] + E~,PJ j= ATOTi

ICAJI + It-O1 + t piijbj fi ([Ail +
151

SHORT COMMUNICATIONS

Table 2. Dependence of pX on the quantity of CL?* added to a “buffer” system of ZnX-/Zn’+ (Cznx*= 0.3M; CNTA= 0.2M)

pH = 6.00 pH = 1OQO

px= px =

0

2 x lo-‘+

10.10 7.79

10.10 7.19

An iterativecomputer programwas developed to calculate the equilibrium concentrations of all species in a system of many competing complexation reactions in a single phase; this program is based on equations (5) and (6). The total analytical concentration of each reactant was used as a first approximation of its equilibrium concentration. The use of generalized forms for the equations of complexation reactions [equation (2)] and a series of other transformations made it possible to reduce the number of commands to 55 from the 140 used in COMICS. It decreases the number of iterations and the computer time. The usefulness of this program has been proved on a series ofmulticomponent systems. As an example the results of one of these calculations is given in Table 1. For the parameter p(i) it is enough, as a rule, to employ the largest stoichiometric coefficient of the ith reactant in equation (2). The rate of convergence slows up slightly as p(i) increases. For the system given in Table 1, for example, the parameters p(i) were chosen in accordance with the above; then the parameter for each reactant of the same system was arbitrarily assigned the value p = 5. The numberof iterations increased in the latter case From 151 to 476. It is Frequently necessary to fix the concentration of one reactant (either metal or ligand) of a system in such a way that it does not change during the course of a reaction. Traditionally. a large excess is introduce& so that its concentration is not altered substantially by the reaction. A new procedure6 using an auxiliary “buffering” system eliminates this necessity. For example, let [X3-J be fixed throughout the course of the reaction cu2+ + x3-

$

cux-

where X3- is the tertiary nitrilotriacetate N(CH,COO):-. The auxiliary reaction Zn’+ +X3-

*

(15) anion,

ZnX-

(16)

for which the stability constant B is expressed by

Pm =

CZnXl [Zn2+][X3-]

may be used to fix the concentration From (17) is obtained

(17)

of X- in the system.

CznXl 1% hx = PX + log CZn2+]

(18)

Equation (18) shows that it is possible to fix various low concentrations of X3- by use of relatively large concentrations of ZnX- and Zn’+. Copper(I1) ions may be added to this “buffer” solution as long as Ccuz+ < [Zn”],[ZnX-] and reaction (15) will proceed with [X3-] constant. The program COMPLEX was used to calculate the equilibrium concentrations in the system described by equations (15) and (16), taking into account the possible hydrolysis of all reactants. The results, as presented in Table 2, demonstrate that a wide range of total Cu2+ concentrations cause no change in [X3-]. Acknowledgement-The author thanks A. Lichtman Ch. Tobias for their valuable assistance.

and

Total [Cu”], M 5 x 10-4 1 x 10-3 10.10 1.19

10.11 7.80

2 x 1o-3 10.11 7.80

REFERENCE’S I.

2. 3. 4. 5. 6.

D. D. Perrin and I. G. Sayce, Talanta 1967,14. 883. G. A. Cumme, ibid., 1973, 20. 1009 and references therein. G. M. Fichtengoltz, A Course in Di@rential and Integral Cahhs, Vol. 1, p. 22. Nauka, Moscow, 1966. K. B. Yatzimirskii and V. P. Vasilyev, Instability Constants of Complex Compounds, (in Russian), Moscow, 1959. L. G. Sillkn and A. E. Martell, Stability Constants, 2nd Ed., Chemical Society, London, 1964. G. Ginzburg, in the press. APPENDIX

The computer program COMPLEX is used to read data, to make consecutive approximations for adjusting the equilibrium concentrations of reactants and complexes and to print the results after a satisfactory degree of convergence. The listing of the program includes a description of the input data. In this program the Following variables were used: Number of separate experiments. NE Number of reactants (up to 18). M Array used to print headings. TITLE(I) Analytical concentration of each reactant. ATOT Equilibrium concentration of I-th reactant, AK(I), A(I) as used by iteration, and as calculated. Calculated total concentration of I-th ACALC(1) reactant. Number of complex species in system N under study (including protonated and hydrolysed forms). Number of units of reactant (I) in comMNLJ) plex (J). MOH(J), MH(J) Number of hydroxyls and protons in complex species (J). Equilibrium concentration of J-th comC(J) plex. Logofcumulative stability constant of J-th E(J) complex. SUM(J) Conditional stability constant of species(J) at a given pH in the log form. The same, but in the exponential Form. TERM(J) Degree of convergence of the I-th reactant. Y1(I) Absolute difference between real and Y2(I) calculated total concentration of I-th reactant. Parameter p (see text) For I-th reactant. BK(I) INDEX Index = 0 for all cards bearing pH values, up to last card when index = 1. Computer program COMPLEX C INPUT DATA Cl NUMBER OF SETS OF EXPERIMENTS TO BE RUN (12) C2 NUMBER OF REACTANTS AND OF COMPLEX SPECIES FORMED (INCLUDING PROTONATED AND HYDROLYSED FORMS), (212) C 3 A SERIES OF CARDS WITH THE TITLE OF REACTANT, ITS TOTAL CONCENTRATION AND PARAMETER P (A5,E15.3,F10.5)

SHORT COMMUNICATIONS

152

C4

A

CARD FOR EACH COMPLEX SPECIES LISTING THE LOGARITHM OF THE CUMU-

LATIVE ASSOCIATION CONSTANT OF THE SPECIES, THE NUMBER OF MOLECULES OF REACTANT (I), (2) ETC. UP TO (Is), THE

PROGRAM DIYENSION

NUMBER OF HYDROXYL IONS, THE NUMC5

BER OF PROTONS (F10.3,2012) A SERIES OF CARDS BEARING PH AND INDEX. INDEX = 0 FOR ALL BUT LAST CARD

OF EXPERIMENT WHEN INDEX = I (FlO.4.11)

COMPLEX (INPUT,OUTPUT1 TITLE(20~rATOT(18) rY1 (181 rHAf18r501

lA(18) rAK(l8) sMOHf50) rMH(50) rTERM(50) rC(SQ) 100 FORHAT(4012) 101 FORHAT~A5rE15.3,F10.5) 103 FORMATtF10.3rSX~2212) 104 FORMAT(F10.4,Il) READ 190,NE DO 1 IK=l*NE READ loO.MvN DO 2 I=lrM READ lOlrTITLE(I) *ATOT tBKf1) YltI)=ATOTII)*,0001 2 A(I)=ATOT(I) DO 3 J=lrN 3 READ 103,EfJ) s (MA{ IIJ) rI=lrM) ,NOH(J)rMHfJ) 13 READ 104,PHtINDEX NIT=0 DO 4 J=lsN SUM=E(JI-MH(JfOPH*MOH(J)a(PH-I4,) 4 TERM{ J) =IO,**SUN 93 CONTINUE DO 25 J=lrN = TERM(J) 25 C(J) DO 5 J = 1rN 00 5 I = 1rM = C(J)*AlI)**MA(ItJ) 5 C(J) NIT=NIT+l DO 6 I=lrM ACALCfI) = A(I) DO 36 J = 1rN 36 ACALC (I) = ACALCfI) * MA(I,J)*C(J) AK(I)=A(I)~(ATOT(II/ACALC(I)~~~~l./BK(I)) 6 Y2fI)=AEStACALCfI)-ATOT( IFtNIT-1999!16r16s7 7 PRtNl 205qPH GO TO 11 16 DO 9 I=lrM IFfYl (I)-Y2(1)) 149999 14 DO 17 IN=lrM 17 A(IN)=AKfIN) GO TO 93 9 CONTINUE PRINT 209rPHgNIT 11 PRINT ~o~~(I~I~K~I)~TITL&~I)~ATOT(I)~A~I)~I=~~M) PRINT 207, (TITLE(I) rI=lrM) DO 12 J=lrN 12 PRINT 208* C(J)~ELJIIMH(J)~MOH(J)*(MA~~~=~~H)

IF(INDEX.NE.l)GO TO 13 1 CONTINUE 205 FORMAT{//* Ptl=*,F7.3r*OX~” 206 FORMAT~Il0,F10.5.A10~2El5~3~ 207 FORHAT(/~XI*C*~SXI * LOG 208 209

FORMAT(2E11.3r1215) FORMAT{//* PH=**F7.3r40X**NUM5ER STOP END

ITERATION EETA*r3Xs*

DID

rY2(18) rACALC(ltl)

NOT

H*,4X,*OH OF

,E(SOl, sBK(lS)

CONVERGE*/) “I

180A5/)

ITERATION=“rI4/)

Summary-An iterative method and a computer program are presented for calculating equilibrium concentrations of all species in a multicomponent system of many competing complexation reactions. The initial data required are: pH, total concentration of each reactant, stability constant of each complex, and pK, values. Convergence of the iterations is proved. As an example a system of 7 reactants and 27 complexes is given. A second example for the use of this program, including so-called “ligand buffering”, is also shown.