Computer calculation of equilibrium concentrations in mixtures of metal ions and complexing species

Talanta,1967.

Vol.

14, pp. 833 to 842.

Pergamon

Press

Ltd. Printedin NorthernIreland

COMPUTER CALCULATION OF EQUILIBRIUM CONCENTRATIONS IN MIXTURES OF METAL IONS AND COMPLEXING SPECIES D. D. PERRIN and I. G. SAYCE Department of Medical Chemistry, Australian National University, Canberra, Australia (Received 2 December 1966. Accepted 6 March 1967)

Summary-A method is described for calculating equilibrium concentrations of all species in multi-metal-multi-l&and mixtures from the pH of the solution, the total concentration of each metal and each complexing agent, and the relevant equilibrium constants (p& values and stability constants). No restriction is imposed on types of possible complexes, which can include mixed, hydrolysed, protonated and polynuclear species. Two examples are given. One of these comprises a lo-metal-104igand system (195 equilibrium constants). In the second system, mixed complex formation is important. The computer program is given. KNOWLEDGE of equilibrium concentrations in solutions of metal ions and complexforming species is important to the analyst in many ways. Thus, he may wish to

establish the most favourable conditions for a determination, to assess the extent to which other species interfere, to develop new methods of analysis, or to compare the possible accuracies of existing procedures. The law of mass action provides the necessary basis for such calculations, given the relevant stability constants. However, when more than one type of metal ion and several different ligands are present at the same time, an iterative approach must be used. Such an approach, although possible by conventional means, is extremely laborious unless a digital computer is used. Recently,l a computer program* was described for calculating the equilibrium concentrations of all species in a multi-metal-multi-ligand mixture, provided that the complexes formed were of the types ML, ML,, Mb, . . . ML,. The only information needed was the pH of the solution, the total concentration of each metal and complexing agent, and the relevant equilibrium constants (pK, values and stability constants). Such systems are also important as models for biological fluids. A comparable (but not general) procedure has been published for the computation of metal binding in the 2-metal-2-chelate system K+, Mg2+, ATP, EDTA.2 We have now developed an entirely general treatment for calculating concentrations in the presence of all types of metal complex equilibria, including the formation of mixed species (such as ML’L” or M’LM”), hydrolysed species [such as M(OH), M,(OH), or M(OH)L], protonated species [such as LM(HL) or M(HL),], and polynuclear species [such as M,Ls or M2(0H),L2]. METHOD If M”, Mb, MC, . . . represent the different kinds of metal ions, and Lr, LB, Lt, . . . are different kinds of ligands, any complex that can be formed from them can be represented by (M”),(Mb)s(Mc), . . . (L’),(Ls),(Lt), . . . (OH),, where CC,j?, y . . . p, u, T . . .

may be positive integers or zero, and w may be a positive integer (for hydrolysed species), zero, or a negative integer (for protonated species). The concentration of * The spelling program is used instead of programme out of deference to workers in the U.S. who first applied the term to computers. 10

833

834

D. D. PERRINand I. G. SAYC~

any one of these complexes (including protonated l&and species and hydrolysed metal ions) is then given by cj = f+[M”]“[Mb]B[MC]Y.. . [L’]“[L”l”[Lt]’ . . . [OH]“, where PI is the overall formation constant, so that the total concentration of metal i is:

wherepi is the number of metal ions of M’ in the species& A similar set of equations can be written for total l&and concentrations. Using the program described in the Appendix, the computer reads in from cards the relevant set of values a, /?, y . . . w, for each species, and the corresponding equilibrium constant. It then calculates free metal and ligand concentrations by the following iterative method. Beginning with the crude approximation that complex formation is negligible (so that [M’] = [Mi]r, and [Lx] can be calculated from [LxJr by using the appropriate pZG values), the machine computes the quantity on the right-hand side of equation (1) for each kind of metal ion, and similarly for each complexing species, to give quantities that can be designated as [Mi]Flc and [Lx]ylc, respectively. The initial estimates of [M’] and [Lx] are then replaced by [M’]([M’],/[Mil”,B1c)l’aand [Lx]([Lx],/ [Ls]$lc)l’a. With these new estimates the calculations are repeated, to obtain better values of [M’] and [Lx]. The process is continued until all values of [M’EiC and of [LX]F1cdiffer from the corresponding values of [Mi]r and of [Lx]r by less than a specified quantity. (For the results given in Tables I and II this was chosen to be 0401% of [MilT and of [LXIT.) Within this accuracy, the final values of [M’] and [Lx] satisfy all equations for metal and ligand concentrations. They are then used to compute the equilibrium concentrations of all species, and the results are finally printed out in tabular form. DISCUSSION

The program has been tested for a system containing ten different kinds of metal ions and ten substituted iminodiacetic acids of the types R-N(CH&OO-),, and having a total of 195 species for which equilibrium constants are available.3 These include pK, values for the ligands and for hydrolysed metal ions, and constants for protonated and hydrolysed complexes. An IBM 360/50 electronic computer took 7-Omin to calculate all of the equilibrium concentrations at pH 10 for a solution in which the total concentration of each complexing species was 0402M and the total concentration of each metal ion was OXtOlM. These results are given in Table I. They are of some interest in providing a quantitative measure of the relative metal-binding abilities of these ligands under the particular experimental conditions chosen, and also in demonstrating preferences among the different metal ions for individual ligands. Thus, magnesium, calcium and manganese form mainly their 1: 1 complexes with the ligand where R is -OOC-CH,-, whereas the most stable 1: 1 complexes of cobalt, nickel, copper and zinc are with the ligand where R is NH,CH,-CH,--. On the other hand, complex formation is only slight when R is NC-CHaor (CH&&--CH,CH,-. Strontium and barium show little discrimination between the four ligands for which R is HO-CH,--CH,--, -OOC-CH,and NH,---CO-CH,. (CH,),C-CH,-CH,--, As a further test of the program, it has been applied to the published data* for mixed complex formation by nickel and zinc with pyruvate and glyoxalate ions and

Computer cakulation of equilibrium concentrations TABLE

835

I.-DISTRIBUTION OF METALS IONS AMONG SUBSTITUTED IMINODIACETIC ACIDS

Species L HL H& H,L MgL CaL SrL BaL MnL MnL, COL Co(HL) COL, Co(OH)L NiL Ni(HL) NL, Ni(OH)L CUL CULS Cu(OH)L Cu(OHi),L ZnL ZnLn Zn(OH)L Zn(OH)BL Zn(OH)Ls CdL CdLp

Ll

L2

L3

L4

L5

L6

L7

L8

L9

LlO

126 56 -

324 17 -

223 20 -

1922 -

269 22 -

2: -

17 9 -

1272 1 -

1879 -

796 30 -

2

loxs 173 237 164 76 141 5

X

9 11 9 21 38 1

5x5

X6

2 4 8 1

9.5 192 156 50 136 4

”

3xq 7 17 22 22 24 17

8 : 1 62 1 214

5;

4x6

533 197 211 300 x 77 X

2; X -

2xg 8 1

7x7

G

x

2:

lo:

7

4

14:,

X9

2

222

x

X

lib

1

X3

”

92 1

5 39 X

1 x x

2:

X I

9

13 29 2L 8 44 17

3 x

5:

X

X

X

X

X

X

X

:; 4

X

X

16

3 186

-

1;

80

” -

X

6 416

17x4

x

1 X

”

2; X

-

-x

33 X

X

X

X

x

3 1 x

X

X

X

X

X

X

X

91

X

1;

X

X

2

X

13

;; 15 1

-

”

-

5611 19:

8

2 231

6

X

1

X

X

5

4x6 IO 14

X

X

246

X

x

X

X 1" X

21: x

145 170 186 71 18 2

:7 13 33 35 4 1 -

X

1 X

9: X

3x4 z

x

2:

-

12

At pH 10. Ah concentrations in @f. The total wncentrations of each metal ion and wmpkxing species were 1000 @W and 2000 pIif, respectively. Con~ntmtions of free metal ions (rounded off to 1 @f); Mg, 121; Ca, 12; Sr, 5; Ba, 193, Mn, 1; Co, 0; Ni, 0; Cu, 0; Zn, 0; Cd, 4. An x indicates that no stability constant was available for a particular species, or that the species was not known, whereas a dash implies that the concentration was kss than 05pM. Constants were also included for MnOH*, CoOHf, NiOH+, Cu,(OH)$+, CuaW% s+,ZnOH+ and CdOH+. The ligands were all of the type R-N(CH&OO-),, where R is CHa--, HO--CH~--CHz-, (CH&C-CH~-CHB, (CH&&-CH~-CH~--, CH,S-CH~--CH,--, NHp--CH~-CH~-, -OOCXH~--, NHp-COXHa---, NC-CHs--, and C~H&&CD-NH-CH*--CH,-, for ligands l-10 respectively. Equilibrium concentrations of 28 different species were calculated in 16 set for a solution at pH 9 and @OOlM in each metal ion and complexing species. These results are summarized in Table II, rounded off to 1 ,&f. An alternative program has also been developed to solve simul~eously the (m + n) non-linear equations that can be written to express total ~on~ntrations in systems conning m kinds of metal ions and n kinds of complex@ species. The method chosen makes use of a Newton-~aphson procedure,5 which wehavegenera~zedtodea~ iteratively with (m + n) unknowns. This technique had already proved useful for the simultaneous solution of two equations in the refinement program GAUSS6 (which was developed from a similar program written by Tobias’) for dealing with equilibria in systems of one metal and one ligand. When the method was employed with the lo-metal-IO-ligand system, convergence was not satisfactory, possibly because the available precision of the computer was insufficient for the calculations. However, for systems containing only small numbers of metals and ligands (up to about three

glycine.

836

D. D. PERRINand I. G. SAYCE TABLE II.-MIXED

COMPLEXFORMATIONBY NICKEL AND ZINC

IONS

Species

Glycine

Pyruvate

Glyoxalate

L HL HIL NiL NiL, NiLs ZnL ZnL, ZnL,

2.48 12 382 67 143 5 -

818 -

879 -

X

X

Wgly)(pyrW Ni(glyMpyruv) WgWdpyruv)~ Wgly)(glyOx)

Ni(gly),(glyox) Ni(gly),(glyox),

4 1: X

81

15 3 90 15 1

Zn(gly)(pyruv)

3 X

% 3 X X

6:

Zn(glyh(pyruv) Zn(glyh(pyruv)2

1

Zn(gly)(glyOx)

9

Zn(gly),(glyox) Zn(gly),(glyox),

-

At pH 9. All concentrations in ,uM. Total concentrations of each metal ion and complexing species, 1000 @J. Free nickel ion, 341 pM; free zinc ion, 764 ,uM.

of each) calculations were appreciably faster than by the other program. The two types of program are thus complementary. The simpler program deals reliably with the more complicated systems, while the Newton-Raphson approach calculates more rapidly the relevant concentrations in simpler systems. A continuing handicap to the application of either of these programs to equilibria in metal-ligand mixtures is the small number of recorded stability constants for mixed complexes and also for protonated and hydrolysed species. Existing computer programs such as GAUSS6 and LETAGROPs can be used to calculate stability constants for polynuclear, protonated or hydrolysed complexes but hitherto no general program has been available for similar computations for mixed species, notwithstanding their great importance for a quantitative understanding of metallo-enzymesubstrate interactions. For this reason we have also developed a general program for calculating the stability constant of any type of mixed complex from titration data, provided the constants for the other complexes are known. This program uses the Newton-Raphson method. Thus, in principle, it should now be possible to obtain constants for all of the types of complexes that need to be considered in multi-metalmulti-ligand systems. These programs have further applications, particularly in the quantitative investigation of metal ion equilibria in biological systems, in metal ion toxicity studies, and in dealing with simultaneous buffering with respect to more than one metal ion. Zusammenfaasung-Eine Methode zur Berechnung der Gleichgewichtskonzentrationen aller Spezies in Mischungen mehrerer Metalb und mehrerer Liganden aus dem pH der Losung, den Gesamtkonzentrationen jedes Metalls und jedes Komplexbildners und den einschlagigen Gleichgewichtskonstanten (p&-Werten und Stabilit&tskonstanten) wird beschrieben. Die Typen moglicher Komplexe unterliegen keiner Beschrankung: es kiinnen gemischte, hydrolysierte, protonierte und mehrkernige Spezies vorkommen. Zwei Beispiele werden angegeben. Eines davon enthiilt ein System aus 10 Metallen und 10 Liganden Im zweiten System ist die Bildung (195 Gleichgewichtskonstanten). gemischter Komplexe von Bedeutung. Das Rechenprogramm wird angegeben.

Computer calculation of equilibrium concentrations

837

Rt&n11&0n d&it une m&bode pour calculer les concentrations d’bquilibre de toutes les esp&ces dans des melanges multi-metauxmulti-ligands a partir du pH de la solution, de la concentration totale de chaque metal et de chaque agent de ~mplexation et des constantes d%quilibre correspondantes (valeurs de pK, et constantes de stabilite). Aucune restriction n’est impo& quant aux types des complexes possibles, qui peuvent inclure des esp&es mixtes, hydroly&es, proton&es ou polynucleaires. On donne deux exemples. L’un d’eux comprend un systeme lo-m&am+10-ligands (195 constantes d’equilibre). Dam le second systeme, la formation de complexe mixte est importante. On donne le programme de computation. REFERENCES D. D. Perrin, Nature, 1965,206, 170. J. Botts, A. Chashin and L. Schmidt, Biochem., 1966,5, 1360. G. Schwarzenbach, G. Anderegg, W. Schneider and H. Serm, Helo. Chim. Acta, 1955,38,1147. D. L. Leussing and E. M. Harma, J. Am. Chem. Sot., 1966,88,693. D. D. McCracken and W. S. Dom, Numerical Methods and Fortran Programming, p. 144., Wiley New York, 1964. 6. D. D. Perrin and I. G. Sayce, .r. Chem. Sot., A, 1967,82. ‘7. R. S. Tobias and M. Yasuda, Znorg. C&m., 1963,2, 1307. 8. N. Ingri and L, G. Sillen, Arkiu Kemi, 1964,23,97, and references therein. 1. 2. 3. 4. 5.

APPENDIX The computer program described above is in two parts. The main program, which we have called COMICS (concentrations of metal ions and complexing species), is used to read the data, to set up initial approximations for the free metal and free ligand concentrations, and later to print out the results. The iterative procedure for adjusting the free metal and free @and con~ntrations is carried out in a subroutine called COGS (con~trations ofgeneralised .rpecies), which returns the computed concentrations of free metal ions, free ligands, and of all complex species to the main program for tabulation, after a satisfactory degree of convergence has been achieved. The program, as listed below, is intended to calculate concentrations at a series of pH values. In order to speed the calculation for the second and subsequent pH values, the initial approximations for the concentrations of free metals and free ligands for the nth point are taken to be the final values of these quantities for the (n - I)th point. The listing of the program includes a description of the input data, and is followed by a list of the variables used, and by a simplified fIow sheet showing the calculation method. Computer program COMICS PROGRAM

COMICS

INPUT DATA NUMBER OF SETS OF EXPERIMENTAL

CONDITIONS

TO BE RUN

TITLE OF FIRST (SECOND, THIRD ETC.) E~E~~E~ ANY CHARACTERS IN COLS. l-80. NUMBER OF LIGANDS, OF METALS, AND OF COMPLEX SPECIES FORMED (INCLUDING PROTONATED FORMS OF LIGAND, AND HYDROLYSED METAL IONS), (212,13) A CARD FOR EACH COMPLEX SPECIES LISTING THE NUMBER OF MOLECULES OF LIGAND (1), (2), (3) ETC. UP TO (IO) THE NUMBER OF IONS OF METAL (l), (2) (3) ETC. UP TO (10) THE NUMBER OF IIYDROXYL IONS (A POSITIVE INTEGER). OR OF PROTONS (A NEGATIVE INTEGER) THE LOGARITHM GF THE CUMULATIVE ASSOCIATION CONSTANT OF THE SPECIES ~, (2112,8X,F8.4) THE TOTAL CONCENTRATION OF EACH LIGAND (lOF8.4) THE TOTAL CONCENTRATION OF EACH METAL (lOF8.4) A SERIES OF CARDS BEARING PH AND INDEX INDEX = 0 FOR ALL BUT LAST CARD OF EXPERIMENT WHEN INDEX = 1 (F10.4,Il)

D. D. PERRINand I. G. SAYCE

838 8

s.ooo1

THEN RETURN TO ITEM 2 UNTIL TOTAL NUMBER MENTS (GIVEN BY ITEM 1) HAS BEEN REACHED.

OF EXPERI-

DIMENSION C(200), Yl(lO), Y2(10), Y3(10), Y4(10), BTOT(lO), 1 CLTOT(lO), ITX(lO), VX(lO), ML(10,200), MM(10,200), MN(200), 2 AL(10,200), 2AM(10,200), AN(200), B(200), E(200), DM(lO), DMY(lO), 3 TITLE(20) COMMON

s.ooo2

C,Yl,Y2,Y3,Y4,BTOT,CLTOT,TX,VX,lML,MM,MN,AL,

1 AM,AN,NL,NM,N,B,UX,IPT s.ooo3 s.ooo4 s.ooo5 S.0006 S.O0@7 S.0008 s.ooo9

f 10 11

s.0010 s.0011 s.0012 s.0013

12 17 30 33

s.0014 s.0015 S.0016 s.0017 S.0018 s.0019 s.0020 s.0021

34 35 36

S.0022 S.0023 S.0024 S.0025 S.0026 S.0027 S.0028 S.0029 s.0030 s.0031 S.0032 s.0033 s.0034 s.0035 S.0036 s.0037 S.0038 s.0039 :z! s:oo42 s.0043 S.0044 s.0045 S.0046 s.0047 S.0048 s.0049 s.0050 s.0051 S.0052

: 6

iI 130 131 999

106

3 7

4 13 14 15 16

FORMAT (12) FORMAT (2112,8X,F8.4) FORMAT (1X, 13,2X,21(2X,I2),3X,F8.4) FORMAT (lOF8.4) FORMAT (24H TOTAL CONCN. OF METAL (,12,4H) = ,E10.3) FORMAT (25I-I TOTAL CONCN. OF LIGAND (,12,4H) = ,E10*3) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ I 2HC7,9X 1,2HC8,9X,2HC9,8X,3HClO) FORMAT (2I2,13) FORMAT (F10.4,11) FORMAT (111’ PH = ‘,F6*3) FORMAT (8X,93HLl M L2 L3 L4 L5 L6 L7 L8 L9 LlO Ml 1 M2 M3 lM4 M5 M6 M7 M8 M9 Ml0 OH LOG.BETA I) FORMAT (‘FREE METALS) FORMAT (8X,lO(lX,lPE1@3)) FORMAT (‘FREE LIGANDS’) FORMAT (‘COMPLEX SPECIES) FORMAT (1X,13,1H-,13,10(1X,1PE10~3)) FORMAT (2OA4) FORMAT (lX,2OA4//) FORMAT (lH1) WRITE (3,999) READ (1,l) NJ NJD=O READ (1,130) (TITLE(I),I=1,20) WRITE (3,131) (TITLE(I),I=1,20) WRITE (3,33) READ (1,12) NL,NM,N DO 7 J=l,N READ (1,2) (ML(I,J),I=l,lO),(MM(I,J),I=l,lO),MN(J),E(J) WRITE (3,6) J,(ML(I,J),I=l,lO),(MM(I,J),I=I ,lO),MN(J),E(J) READ (1,8) (CLTOT(I),I=I,lO) READ (1,8) (BTOT(I),I=l,lO) WRITE (3,lO) (I,CLTOT(I),I=l ,NL) WRITE (3,9) (I,BTOT(F),I=l ,NM) HX=ALOG(lO.O) IPT=l DO 4 J=l,N AN(J)=MNQ DO 4 I=l,lO AL (1,J) =ML(I,J) AM(J,J) =MM(I,J) DO 13 I=l,N B(I)=EXP(HX*E(I)) DO 14 I=l,NM Y1(I)=BTOT(I)*0@OOO1 DO 15 I=l,NL Y3(I)=CLTOT(I)*O~OOOO1 READ (1,17) PH,INDEX WRITE (3,30) PH UX=EXP(HX*PH) IF (IPT-1) 18,18,27

839

Computer calculation of equilibrium concentrations

5.0036

Sets

for use

2.0053

If consldenng

calculation

did

pH (unlikely calculates

first

not converge

if no errors

initial

up constants later.

5.0049

Reads

PH and

INOEX.

pH, or if for last

present)

approximations;

VX(I)=BT0T(I) and TX(I) which

is calculated

selects

protonated

by a routine

all species

forms

which

of ligand(I)

If calculation

converged

satisfactorily

for last

TX(I)

are

pH are used

and uses

estimates

TX(I)-CLT0T(I)/(l+(H+lB~+(H+1*8~+...) where

81.B2

association

etc.

are cumulative

constants

acid

for liga"d(1).

and free

lfgand

Returns next

FIG. I.-Flow

and VX(I)

Co"Ce"tratiO"S.

to calculate

set of data.

sheet for computer program COMICS.

from

as initial

at present

pH,

last

pH.

840

D.D.

PERRIN and1.G.

SAYCE

(Continued) 18 19

SO053 s.0054

23

DO 19 I=l,NM VX(I)=BTOT(I) DO‘20 I=l,Ni, DMY(I)= 1.0 DO 22 J=l,N IF (ML(I,J)) 22,22,200 DO 21 K==l,NM IF (MM(K,J)) 22,21,22 CONTINUE DM~I)=(EXP(HX*B(J)))*UX**MN(J) DMY(~=DMY(I)+DM(I) CON~NU~ CONTINUE DO 23 I==l,NL TX(I>=CLTOT(I)/DMY(I)

27

CALL COGS

5.0055 S.0056 s.0057 S.0058 s.0059 S.0060 S.0061 S.0062 s.0063 S.0064 S.0065 S.0066 S.0067

200 21

22 20

C S.0068 C 5.0069 5.0070 SO071 S.0072 s.0073 s.0074 s.0075 S.0076 s.0077 SO078 SO079 S.0080 S.0081 S.0082 S.0083 S.0084 5.0085 S.0086 SO087 So088 s.0001

40

41 42 31

32

WRITE (3,ll) WRITE (3,34) WRITE (3,35) (VX(I),I= 1,NM) WRITE (3,36) WRITE (3,35) (TX(I),I=l,NL WRITE (3,37) KP=O KP=KP+l KN=lO*KP KM=KN-9 IF (KN-N) 41,42,42 ;I$;: ($43) KM.KN,(C(I),I=KM,KN) WRITE (3,43) KM,N(C(I),I=KM,N) IF (INDEX) 31,16,31 NJD =NJD + 1 WRITE (3,999) IF (NJD-NJ) 106,32,32 STOP END SUBROUTINE COGS

C DlMENSION TERM(ZOO), TERN(200),C(200),Y1(10),Y2(10),Y3(10), 1 Y4(10),1BTOT(1O),CLTOT(1O),TX(1O),VX(1O),ALO(1O),BO(1O),TY(lO), 2 W(10),2ML(10,200),~(10,200),MN(200)AL(10,2~),AM(10,2~), 3 AN(2~),B(2~)

s.0002

C COMMON C,Yl,Y2,Y3,Y4,BTOT,CLTOT,~,~,lML,MM,MN,AL, 1 AM,AN,NL,NM,N,B,UX,IPT

s.0003

s.0004 S.0005

C

99 998

FORMAT FORMAT

(‘NUMBER OF ITERATIONS = ‘,14) (‘ITERATION DID NOT CONVERGE’)

C 5.0006 s.0007 S.0008 s.0009 s.0010 s.0011 s.0012 s.0013 s.0014 So015

1 2 3

4

NZT=O DO 1 K=l,N TERM(K)=~(K)*~**MN(K) DO 3 K=I ,N TERN(K)=TERM(K) DO 4 K=l,N DO 4 J=l,NM TERN(K)=TERN(K)*VX(J)**MM(J,K) DO 5 K=l,N DO 15 J=l,NL

Computer calculation of equilibrium concentrations

5.0007 using

Calculates

Current

ligand

concentration

estimates

of species

of free metal

841

(J)

and free

concentrations

C.=~.IMalulMblB~Mcly...[L'lPILSla[Ltl~..[OH]W --J J

9 5.0021

Calculates

improved

value

total

concentration

for concentration

of metal(I),

of free

metal(I),

and VY(I)

t 5.0025

calculates

l60(1)-9~0T(I)/

= IM'I~~'~

- IM~I~

t 5.0028

Calculates

improved

5.0032

value

total

concentration

for concentration

calculates

of ligand(I).

of free

and

ligand(I),TY(I)

IAL~(I)-cLTDT(I)I=[L~~S~'~

_ILXIT

I 5.0033

I

exits

Checks

that calculation

if no convergence

5.0034

Tests

convergence

after

whether has been

is not in loop; 999 cycles

I

satisfactory achieved

Achieved

Not achieved

FIG. 2.-Flow

sheet for sub-routine

COGS.

List of variables used in program COMICS NJ NJD TITLE(I) NL NM

Number of separate jobs to be run with program. Number of jobs done. Array used to print headings. Number of ligands in system under study. Number of metals in system under study. Number of complex species in system under study (including protonated forms of ligand, and hydrolysed metal species). Number of molecules of ligand(I) in species(J), in fixed and floating point. Number of atoms of metal(I) in species(J), in flxed and floating point. Number of protons in species(J), in tixed and floating point. Log of cumulative association constant for species(J). Total concentration of ligand(1).

842 S.0016 s.0017 S.0018 s.0019 s.0020 s.0021

D. D. PERRINand I. G. SAYCB

15 TERN(K)=TERN(K)“TX(J)**ML(J,K) 5 C(K) =TERN(K) NIT==NIT+ 1 DO 7 I=l,NM BO(I) =VX(I) DO 8 K=l,N 8 Bq~=BO(~+AM(I,K)*C~) ::ZZ ~TIO=~~~BTOT(~ VY(1) =VX(I)/SQRT(RATIO) 5.0024 So025 7 Y2(I)= ABS(BO(I)-BTOT(1)) DO 9 I=l,NL 5.0026 S.0027 ALO(I)=TX(I) S.0028 DO IO K=l,N s.0029 10 ALO(I)=ALO(I)+AL(I,K)*C(K) RATIO=ALO(I)/CLTOT(I) s.0030 s.0031 TY(I)=TX~)/SQRT(~TIO) S.0032 9 Y4(1) = ABS(AL0~) -CLTOT(I)) IF (NIT-999) 11,11,999 s.0033 s.0034 11 DO 12 I=l,NM IF (Yl(I)-Y2(1)) 14,12,12 5.0035 S.0036 12 CONTINUE DO 13 I=l,NL S.0037 S.0038 IF (Y3(1)-Y4(1)) 14,13,13 13 CONTINUE s.0039 s.0040 IPT=IPT+l So041 WRITE (3,99) NIT RBTURN s.0042 14 DO 16 I=l.NL s.0043 16 TX(I)=TY(I) S.0044 DO 17 I=l.NM s.0045 17 VX(1) =VY(I) S.0046 5.0047 GOT02 999 WRITE (3,998) S.0048 IPT=l s.0049 s.0050 RETURN S.0051 END Total con~en~tion of metal(I). BTOT(1) Loge l@O. HX Exponential form of constant E(1). B(I) Indicator used in obtaining initial approximations for free metal and free IPT l&and concentrations. IPT = 1 for first point (or if last point was nonconvergent). IPT > 1 otherwise. Convergence parameter for total metal(I). Yl(I) (Calculated - experimental) concentration of total metal(I). Y2(I) Convergence parameter for total ligand(I). Y3(I) (Cakulated - experimental) concentration of total l&and(I). Convergence ~40) is assumed to be satisfactory when Y2(1) < Yl(I) and Y4(I) < Y3(I) for al1 vaIues of I. PH Index = 0 for all cards bearing pH vaIues, up to last card when index = 1. Exponential form of pH. Current estimate of concentration of free metal(I). Current estimate of concentration of free ligand(I). Dummv variables used in calculation of initial estimate of TX(I). DM(I), DMY(1) Intege; used in tabulation routine. KM, KN, KP Concentration of complex species(I). Dummy variables used for calculation of species concentrations C(J). Variable used in station of cakulated total metal(I). Variable used in summation of calculated total l&and(I). Calculated total metal(I)/experimental total metal(I), or calculated total ligand(I)/experimental total l&and(I). Variable used to store improved value of VXfl). VY(I) Variable used to store improved value of TX(I). TY(I)