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A correlation between the overall stability constants of metal complexes—I
J. inorg, nucl. C h e m . , 1974, Vol. 36, pp. 107-113. Pergamon Press. Printed in Great Britain.
A CORRELATION BETWEEN THE OVERALL STABILITY CONSTANTS OF METAL COMPLEXES--I CALCULATION OF THE STABILITY CONSTANTS USING THE FORMATION FUNCTION h M. H. MIHAILOV Bulgarian Academy of Sciences, Institute of Nuclear Physics, Sofia 13, Bulgaria (Received 18 June 1973)
Abstract--A mathematical model concerned with the formation of a stepwise series of mononuclear complexes of the metal M "+ and the ligand L- has been formulated. The model leads to the following expression connecting the overall stability constants of the M L , complexes formed in the solution : an
~, = A " .n- ~.
(1)
where ://. is the overall stability constant of the M L , complex; n is the number of ligands in a particular complex, (I ~< n ~< N); N is the coordination number of the complex forming metal M; A and a are constants. The applicability of Eqn (1) has been demonstrated for nine metal-ligand systems. The results obtained indicate that for these systems the formation function ~ calculated using a set of stability constants related according to Eqn (1) coincides satisfactorily with the fi calculated using the/~, values reported for the given metal-ligand system.
INTRODUCTION
SINCE the basic equations related to stepwise complex equilibria were derived[l], numerous experimental studies of the equilibria in solution have been made. Special attention has been paid to the determination of the stability constants which quantitatively characterize the complexing reactions. A description of the mathematical methods used for this purpose can be found in various review articles[2, 3] and monographs[4-6-]. A large body of data on the stability constants of metal complexes are presented in the compilation of Sillrn and Martell[7]. It was to be expected that some relationship between the stability constants of complexes in a given metalligand system would be found. The pioneer work was due to Bjerrum[1]. Later attempts have been made by Van Panthaleon van Eck[8] and Babko[9]. Yatsimirskii has compared[ 10] the accuracy with which the relations derived by these three authors could describe the experimental data for the aluminium fluoride system
and has found that the best results are obtained when using the Van Panthaleon van Eck's empirical equation. He also demonstrated that this relation is a good approximation for about seventeen metal-ligand systems. The stability of metal complexes depends on many factors such as electrostatic, structural, nature of bonding and many others. So it is hard to believe that some relationship of general validity for the whole variety of the metal-ligand system could possibly be derived with our poor knowledge of these factors at present. So a derivation of a relationship valid for some more restricted class of metal-ligand systems is still of great interest. Such a relationship might lead to some systematic classification of the metal-ligand systems and increase our understanding of certain physical and chemical properties. So our interest was focussed on the construction of a mathematical model for the complex formation process in aqueous solution, which could allow the problem stated above to be solved for as many metal-ligand systems as possible. In this paper a simplified mathematical model of the
107
108
M. H. MIHAILOV
equilibrium in aqueous solution is described. This model leads to the possibility of deriving a relationship connecting the overall stability constants of the metal complexes in a given metal-ligand system. The validity of this relation has been demonstrated on more than forty metal-ligand systems. Some of them are presented in the present paper.
THEORY Let us consider the complex formation process of the metal M m+ (m is the charge of the cation M) and the univalent ligand L - , arising from the dissociation of the strong electrolyte CL in an aqueous solution of a constant ionic strength, maintained by addition of the inert electrolyte CL'. Let the system under consideration satisfy the following requirements: (i) the concentration of the metal M m÷ is sufficiently low so that the formation of polynuclear complexes could be neglected. (ii) the metal M m+ does not participate in side reactions with the ligand L ' - and O H - ions. (iii) the ionic strength of the solution is sufficiently low so the formation of outer sphere complexes (ion-pairs) between the anionic species and the C+-ions could be neglected. (iv) the activity coefficients of all species are constants so concentration terms could be used instead of activities in all equations. Then, the formation of stepwise series of metal complexes could be expressed by the following generalized equation:
M + nL ~ ML,,.
(1)
The stoichiometric stability constant of the ML. complex is defined as: /~. =
[ML,] -
-
[M][L]"
(2)
where: [ ] are concentrations; n is the number of ligands. (Here and throughout the paper the charges will be dropped for simplicity.) But the cation M and the ligand L are hydrated in aqueous solution, so the complex formation process may be regarded as a substitution of water molecules from the first coordination sphere of the metal M with negatively charged ligands[11-13], rather than a simple addition reaction as (1). Thus, we could write: -
MSN + nL backwardML.Sn_. + nS
(3)
where: S denotes water molecule; N is the coordination number of M. Here the reaction is restricted to the first coordination sphere of M. For simplicity the water molecules in the second and third hydration spheres[14] are omitted and the hydration sphere of L is omitted. The interaction between a metal ion and a negatively charged ligand in solution depends on their solvation [15, 16], the structure of the solvent in the solvation spheres of the ions[17, 18], the relative ease with which ions of opposite charge displace the water molecule
from the coordination sphere[19, 20]. All these effects as well as others may be important. To make direct calculations for such a complicated system taking into account all these factors is an extremely difficult task. So, for our purpose we felt it better to consider the total effects, making some crude simplifying approximations when deriving the main equations. The assumptions necessary for the validity of the relations which will be derived are as follows: (i) The complex formation process (forward reaction) involves substitution of water molecules coordinated to the metal ion M in its first coordination sphere by negatively charged ligands L. (ii) The dissociation of the complex ML,S N_. (backward reaction) is a substitution of the ligands L coordinated to the metal M in its first coordination sphere with water molecules. (iii) The coordination sites of the metal ion M are equivalent. (iv) The metal ions M, ligands L, all metal complexes and electrolytes ions are randomly distributed in the solution and undergo random motion. (v) The forward reaction proceeds because of short range forces defined as follows: the force directed from the metal ion M towards the ligand L or from the central group M of the complex ML.SN_. towards L, by which M attracts L is denoted bYfM~z and the force directed from L towards M or towards the central group M of the complex ML.SN_., by which L attracts M is denoted byfz~ M. We shall assume that these forces remain constant irrespective of the number of ligands coordinated to the metal ion M in its first coordination sphere. In the most general case:f~t~L :~ fL~M because the hydration shells of M and L contain different numbers of water molecules and have a different structure. (vi) The backward reaction proceeds under the action of the short range forces defined as follows: the force directed from the central group M of the complex ML.S N_. towards S by which M attracts the water molecule is denoted as fM~S and the force directed from a water molecule towards the central group M of the complex ML,SN_. by which the water molecule attracts M is denoted by fS~M' We shall assume that these forces are also independent on the number of ligands in the complex species. (vii) The progress of a reaction (forward or backward) is a result of an effective collision between the interacting particles. We shall define as an effective collision one in which the colliding species first have a suitable energy to approach within some distance for which the attractive forces are effective and second have the proper orientation. (viii) The number of M ions and complex species ML.SN_. which have a suitable energy for the forward reaction will be proportional to their concentration in the solution and to the temperature. Thus, for a given temperature the number of metal ions M, which have a suitable energy for the forward reaction will be equal to KM[M], for the complex ML this number will be KML[ML], for the complex ML. it will be KMz.[ML.] and for the ligand L it will be given by: KzEL ] respectively. By analogy, the number of the complex species which have a suitable energy for the backward reaction at a given temperature will be propor-
109
Stability constants of metal complexes--I tional to their concentrations in the solution with proportionality factors K'ML, K'ML. . . . . . K'ML" respectively. The number for the water molecules will be Ks[S]. We shall assume that: KML = KML~ = . . . . KML. = Kx, which corresponds to the assumption that the heat of displacement of a water molecule in the complex MLnS N_. by ligand L is independent on the number of ligands in the given complex• If the same can be assumed for the backward reaction we can write: K~L = K~L. = . . . . K~L ~ = K~:. The metal ion M cannot be regarded as a member of the complex series, its first coordination sphere being symmetrical and containing only water molecules. So considering the proportionality factor KM we supposed that the following relation holds: K M = A . Kx, where A is a constant. (ix) In each complex the domains available to the attractive force of the central group M and to those of each ligand are equivalent. So, a collision between complex species ML,SN_, and a ligand L will be effective only in the case that L approaches the complex in the domain where the attractive force of the central group f u ~ L is dominant• (x) The hydrated metal ion M and the hydrated complex ML,SN_, have approximately equal volumes. Now, we shall define the probability of the complex M L being found in the solution as a difference between the forward and backward reaction probabilities. If the forward reaction probability is expressed as a product of the numbers of M ions and L ligands having a suitable energy for reaction and the corresponding attractive forces we obtain:
where: ao = K~fM~LKLfL~M = const. d = Kj, fM.sKsfs.M[S] = const.
By analogy, the probability of ML2 species being found in the solution will be given by: a2
1 A ~ (1 - d)2[M] [L] 2.
PML2 = ~ "
Z!
(7)
The factor ½ arises because, according to assumptions (vii) and (ix) the probability of an effective collision between the complex MLSN_ 1 and the ligand L, leading to ML2SN_2 is only half the probability of an effective collision between the metal ion M S N and the ligand L. For the next complex in the series, M L a S N_ 3, this factor must be multiplied by ½ and so in general the factor 1 becomes - - ' n!
In an analogous way the following relation for the probability of the M L . S ~ _ . complex (n > 2) being found in the solution can be derived: 1
an
P~L. = C--MM"A ..~ (1 - d)"[M] [L]"
(8)
or:
1
an
PML. = -~M" A ~. [M] [L]"
(8')
where: a = ao(I - d). If we define the total probability as:
1 pfo~. = ~ .
AKxfM~L[M]KLfL~M[L]
n=N P = PM + E PML. = 1
(4)
(9)
n=l
where: CM is the total (analytical) concentration of the complex forming metal M. The backward reaction probability will be given by the product of the number of M L species which have a suitable energy for the reaction and the number of water molecules having a suitable energy for the reaction and the corresponding attractive forces. If the concentration of the M L complexes in the solution is expressed by Eqn (4) we could write: 1 Pback. ~ ~MM "
. K ~ f M ~ s A K x f M ~ L [ M ] K L f L ~ M [ L I K s f s~M[S].
(5)
Then, the probability PML of finding the M L complex in the solution will be given by: 1
1
{ 1 - K'&.~Kff~.,[S]}
= - - . A ' ao(1 - d)[M] [L]
c,,,
[M] PM-
(10)
Cu
will be the probability of the M ion to be found in the solution. Expressing PM and PM~ through Eqns (10) and (8') respectively we get the following expression for Eqn (9):
P
=
nl.n " } =1. 1 [ M I ~ I + A n f =rLl" C~ l .=1
(11)
Multiplying both sides of Eqn (11) by CM, the following expression for the distribution of M (central group) in the solution is obtained:
CM [M]f1~
.=N a"
l
Obviously, if Eqn (12) is equated to the conventional expression for the distribution of M (central group) in the solution[4] we get the following equation for the stability constant ft, of the complex M L , :
PML ~- Prof. - Pback. = ~-'~" • AI(.f,~.~[M]K~f~,[L]
then:
an
(6)
fl,, = A - - - . n!
(13)
110
M. H. MIHAILOV
Then, the complexity function as defined by Froneus [21] will be transformed to:
Table 1. Comparison of calculated formation functions for the cupric-chloride system
n=N a n
F = 1 + A ~. ~ . [L]" n=l
(14)
•
and the degree of formation or the formation function of the system h will be given by: /~=A
~ (nn=l
1)'[L]" 1 + A "
~
"(15)
n=l
CALCULATION OF THE OVERALL STABILITY CONSTANTS USING THE FORMATION FUNCTION OF THE S Y S T E M - - S
It was interesting to see how efficiently the relations derived in the previous part of this paper will describe the complex formation process in aqueous solution. F o r this purpose we have turned our attention to the metal halogenide systems as the most exhaustively studied m e t a l - l i g a n d systems. F o r example, in the compilation of Sill6n a n d Martell[7] they represent a third of all listings for inorganic ligands. Thus, in this group it was not difficult to find systems which satisfy the requirements encountered in the first part of this paper. We have selected only systems for which more than three stability constants are reported. We have not included systems for which only the first two stability constants fll and f 1 2 respectively are reported because for such systems two constants could always be found to satisfy Eqn (13). But this fact would not serve as support for the validity of the relation previously derived. The calculations have been performed as follows: from the values of ft, reported in the literature[7] for a given m e t a l - l i g a n d system the formation function fi was calculated using the conventional expression[l]. Then, from two values of fi taken at two ligand concentrations, the constants A and a were determined using Eqn (15). F r o m the constants A and a determined the corresponding ft, were calculated using Eqn (13). To check the correctness of the values of a a n d A constants determined, the formation function ~ was now recalculated using the set of stability constants derived according to Eqn (13). The formation functions h for the systems: Cu 2+ - C1-[22], Z n 2+ - C1-[23], Cd 2+ C1-[24], In 3+ - C1-[25, 26], G a 3+ - C1-[27], P b 2+ C1-[28], Z n z+ - Br-[23] and Cd 2+ - Br-[29] were analyzed in the way described above. In Tables 1-9 the formation functions hi" which were calculated using the ~, values obtained by the present m e t h o d and correlated according to Eqn (13) are compared with the/~*, calculated using the ft, values reported in [7] for the corresponding systems. The discrepancy at each ligand concentration was estimated by the quantity An = I/fi*(fi* - ~f)100. For the most part of the systems the representative values of the formation functions were calculated for ligand concentration up to a value close to that of the ionic strength of the solution. Only for the Cu 2 + - C I - system, which had been studied at constant ionic strength equal to 0-69, the formation functions were calculated up to EL] - 4 M. Irrespective
(L) M
h*
h~
Ah (%)
0.050 0.075 0.100 0.200 0.300 0.400 0.500 0.600 0.690 1-000 1.500 2.000 2.500 3.000 3.500 4.000
0.337 0.444 0.512 0.771 0-922 1.054 1-173 1.281 1.384 1.700 2-133 2-467 2.718 2.907 3.037 3.163
0.319 0.427 0.515 0.764 0.936 1.075 1.198 1.312 1.400 1.712 2.124 2.449 2.700 2.893 3.043 3-167
5.34 3.83 -0.59 0.91 - 1.52 - 1.99 -2.13 -2.42 - 1.16 -0.71 0.42 0.73 0.66 0-48 - 0.20 -0-13
* Calculated by using the//, values reported in [22]: fll = 9.55;//2 = 4"90;//3 = 3"55;//4 = 1.00. t Calculated using the //, values obtained by the present method: fll = 8-60; l/2 = 6.10;//3 = 2'89;//4 = 1'03; A = 6'056 and a = 1-42. of the fact that the results in Table 1 show satisfactory coincidence of b o t h formation functions over the whole ligand region, we consider such extrapolation as not quite justifiable because at higher ionic strengths side Table 2. Comparison of calculated formation functions for the zinc-chloride system (L) M
fi*
~
A~ (%)
0-05 0.10 0.15 0.20 0-25 0-30 0-35 0.40 0-45 0.50 0.55 0-60 0.65 0.69
0-219 0.379 0-503 0.606 0.695 0.774 0-848 0.917 0-983 1.048 1.112 1.175 1.239 1.290
0-219 0-381 0.509 0.616 0.709 0.792 0.867 0.937 1.002 1-064 1-124 1.181 1-236 1.279
0-00 -(~53 - 1.25 - 1.65 - 2-01 - 2.33 - 2.24 -2.18 - 1.93 - 1.53 - 1.08 -0-51 0-24 0.85
* Calculated using the ft, values reported in [23]: //1 = 5.25;/12 = 3.09;//3 = 0.65;//4 = 1.51. "["Calculated using the //, values obtained by the present method://1 = 5.20;//2 = 3.38;//3 = 1.46; f14 = 0'48; A = 4'0 and a = 1"3.
Stability constants of metal complexes--I Table 3. Comparison of calculated formation functions for the cadmium-chloride system
11'.
Table 5. Comparison of calculated formation functions for the indium-chloride system
(L) M
h*
~t
Ah (%)
(L) M
h*
fit
Ah (%)
0.0010 0-0050 0.0075 0-010 0-050 0.075 0-10 0.50 0-75 1.00 1.50 2.00 2.50 3.00
0.091 @344 0.449 0.530 1.020 1.161 1.276 2.494 2-931 3-201 3-496 3-643 3.727 3.781
0.096 0.353 0.456 0.535 0.993 1.125 1.237 2.518 2.944 3-202 3.478 3.618 3.700 3.753
-4.35 -2.62 - 1.56 -0.94 2.65 3-10 3.06 -0.96 -0.44 -0.03 0.51 0-68 0-72 0.74
0.001 0.003 0.005 0.007 0.01 0.03 0.05 0-07 0.10 0.20 0-30 0.40 0-50 0.60 0.70
0.193 0.443 0.607 0.721 0.851 1.278 1.491 1.676 1.788 2-081 2.245 2.354 2.434 2.496 2.545
0-193 0.430 0-575 0-676 0-785 1.151 1.375 1-556 1.773 2.217 2.441 2.570 2.653 2.710 2.751
0.00 2-93 5.27 6-24 7.76 9.94 7.78 7.16 0.84 -6.54 -8.73 -9-18 -9.00 -8-57 - 8.09
* Calculated using the //, values reported in [24]: //~ = 100; //2 = 4 0 0 ; //3 = 504;//4 = 1008. t Calculated using the ft, values obtained by the present method: fix = 105; [/2 = 312; fla = 619; //4 = 920; A = 17.68 and a = 5-94.
reactions could take place, w h i c h are n o t taken into a c c o u n t in the p r e s e n t model. G a l l i u m chloride a n d c a d m i u m chloride are the o t h e r exceptions, which h a d n o t been studied at c o n s t a n t ionic s t r e n g t h of the solution a n d so for these systems the c o n d i t i o n s for a constancy of the activity coefficients are n o t fulfilled.
Table 4. Comparison of calculated formation functions for the indium-chloride system (L) M
h*
ht
Ah (%)
0.005 0.007 @01 0.03 0.05 0.07 0.10 0.20 0-30 0.40 0.50 0-60 0-69
0.571 0-700 0-849 1-354 1.596 1.753 1.915 2.212 2.370 2.473 2.545 2.600 2.639
0.601 0.705 0.820 1.223 1.477 1.676 1.907 2.339 2.452 2.649 2.718 2.765 2.796
-5.25 -0.71 3.42 8-51 7.46 4.39 0-42 -5.74 -3.46 -7.12 -6.80 -6.35 -5.95
* Calculated using the ft, values reported in [25]: fix = 186; f2 = 4677;//3 = 13,800 t Calculated using the //, values obtained by the present method: //1 = 250; f2 = 2750; f3 = 20.166; A = 11.36 and a = 22.0.
* Calculated using the ft, values reported in [26]: f l = 229; f2 = 4266; f3 = 8913. t Calculated using the f , values obtained by the present method: f l = 234; f 2 = 2083; f3 = 12,356; A = 13.15 and a = 17.80. Table 6. Comparison of calculated formation functions for the gallium-chloride system (L) M
h*
fit
Ah ("~)
0.5 1.0 1-5 2.0 2.5 3-0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8-0 8.5 9-0 9-5 10.0
0.114 0.208 0.287 0.356 0.416 0.469 0.517 0.561 0.601 0.637 0.671 0.703 0.733 0-761 0-787 0-812 0.836 0-860 0-882 0.903
0.106 0.196 0.275 0-346 0-409 0.466 0.519 0.568 0.613 0-656 0-696 0.734 0.771 0.806 0-839 0.872 0.904 0.933 0.963 0.992
7.02 5.77 4.18 2.81 1.68 0.64 -0.39 1.25 -2.00 -- 2.98 - 3.73 - 4-41 -5.18 - 10.25 -6.61 -7.39 - 8.01 - 8.49 -9-18 - 9-86
* Calculated using the ft, values reported in [27]: fll = 0'251;fl2 = 5.01 × 10-3;fl3 = 3-16 x 10-5;fl4 = 1.58 x 10 -6. t Calculated using the ft, values obtained by the present method: fll = 0.229; f12 = 6.59 x 10-3; f13 = 12.65 x 10-5; fl, = 1.80 × 10-6; A = 3.97 and a = 0-0576.
112
M . H . MIHAILOV
Table 7. Comparison of calculated formation functions for the lead-chloride system (L) M
h*
h~
Ah (%)
(L) M
fi*
h?
Ah (%)
0'1 0.2 0.3 0.4 0.5 0.6 0.7 0-8 0.9 1.0
0-563 0.892 1.169 1.412 1-623 1-804 1.956 2.083 2-190 2.279
0.531 ~922 1.228 1.470 1.664 1-821 1.950 2.055 2.143 2.218
5.68 -3.36 -5.05 -4-11 -2.53 -0.94 0.31 1.34 2.15 2.68
0.050 0.075 0.10 0.20 0-30 0-40 0.50 0.60 0-70 0.80
0.796 0.970 1.106 1.532 1.883 2.231 2.448 2.665 2.744 2.990
0.766 0'941 1.081 1.541 1.915 2.271 2.479 2.683 2.734 2.982
3.77 2.96 1.90 -0.61 - 1.72 - 1-80 - 1-27 - 0.67 0.36 0.26
* Calculated using t h e / 3 , values reported in [28]: /31 = 9.3;/32 = 7.1;/33 = 22.4. t Calculated using the /3. values obtained by the present m e t h o d : fll = 6.50; /32 = 12.67; f13 = 16.50; A = 1-67 and a = 3.9.
N e v e r t h e l e s s , t h e stability c o n s t a n t s w h i c h were f o u n d by t h e p r e s e n t m e t h o d o b e y E q n (13). I n c o n c l u s i o n it m u s t be said t h a t in all t h e s y s t e m s s t u d i e d a satisfactory fit b e t w e e n t h e f o r m a t i o n funct i o n s c a l c u l a t e d u s i n g t h e ft, v a l u e s r e p o r t e d in t h e c o m pilation of Sill6n a n d Martell[7] a n d t h o s e o b t a i n e d w h e n u s i n g t h e ft, v a l u e s o f the p r e s e n t m e t h o d w a s achieved. So, t h e r e s u l t s p r e s e n t e d in T a b l e s 1 - 9 s u p p o r t the validity of t h e r e l a t i o n s derived in this p a p e r for all t h e s y s t e m s a n a l y z e d .
Table 8. Comparison of calculated formation functions for the zinc-bromide system
r
Table 9. Comparison of calculated formation functions for the c a d m i u m - b r o m i d e system
(L) M
fi*
fit
Ah (%)
0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.69
0.155 0.226 0.293 0.357 0.418 0.477 0.534 0.588 0.642 0-694 0.744 0.794 0.833
0.155 0-226 0.293 0.358 0-419 0.479 0.537 0.592 0-647 0-700 0.751 0.801 0.841
0.00 0.00 0.00 -0.28 -0.24 - 0.42 - 0.56 -- 0-68 - 0-78 - 0-86 - 0-94 - 0-88 - 0-96
* Calculated using the ft, values reported in [23]: /31 = 1-66;/32 = 0-794;/3a = 0.182;/34 = 0.100. t Calculated using the /3, values obtained by the present method: /31 = 1.65; /32 = 0.792; /33 = 0 - 2 5 3 ; /34 = 0.061 ; A = 1.72 and a = 0.96.
* Calculated using the /3, values reported in [29]: /31 = 36.31;/32 = 126; f13 = 145;/34 = 339. t Calculated using the /3, values obtained by the present method: /31 = 34; /32 = 100; /3a = 195; /34 = 283; A = 6.0 a n d a = 5.6.
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26. A. I. Busev and N. A. Kanaev, Vest. mosk. gos. Univ. Set. Mat., No. 1, 135 (1959). 27. K. A. Kraus, P. Nelson and G. W. Smith, J. phys. Chem. 58, 11 (1954). 28. P. Kivalo and R. Luoto, Acta chem. fenn. 30B, 163 (1957). 29. P. Kivalo and P. Ekari, Acta chem.fenn. 30B, 116 (1957).
A CORRELATION BETWEEN THE OVERALL STABILITY CONSTANTS OF METAL COMPLEXES--I CALCULATION OF THE STABILITY CONSTANTS USING THE FORMATION FUNCTION h M. H. MIHAILOV Bulgarian Academy of Sciences, Institute of Nuclear Physics, Sofia 13, Bulgaria (Received 18 June 1973)
Abstract--A mathematical model concerned with the formation of a stepwise series of mononuclear complexes of the metal M "+ and the ligand L- has been formulated. The model leads to the following expression connecting the overall stability constants of the M L , complexes formed in the solution : an
~, = A " .n- ~.
(1)
where ://. is the overall stability constant of the M L , complex; n is the number of ligands in a particular complex, (I ~< n ~< N); N is the coordination number of the complex forming metal M; A and a are constants. The applicability of Eqn (1) has been demonstrated for nine metal-ligand systems. The results obtained indicate that for these systems the formation function ~ calculated using a set of stability constants related according to Eqn (1) coincides satisfactorily with the fi calculated using the/~, values reported for the given metal-ligand system.
INTRODUCTION
SINCE the basic equations related to stepwise complex equilibria were derived[l], numerous experimental studies of the equilibria in solution have been made. Special attention has been paid to the determination of the stability constants which quantitatively characterize the complexing reactions. A description of the mathematical methods used for this purpose can be found in various review articles[2, 3] and monographs[4-6-]. A large body of data on the stability constants of metal complexes are presented in the compilation of Sillrn and Martell[7]. It was to be expected that some relationship between the stability constants of complexes in a given metalligand system would be found. The pioneer work was due to Bjerrum[1]. Later attempts have been made by Van Panthaleon van Eck[8] and Babko[9]. Yatsimirskii has compared[ 10] the accuracy with which the relations derived by these three authors could describe the experimental data for the aluminium fluoride system
and has found that the best results are obtained when using the Van Panthaleon van Eck's empirical equation. He also demonstrated that this relation is a good approximation for about seventeen metal-ligand systems. The stability of metal complexes depends on many factors such as electrostatic, structural, nature of bonding and many others. So it is hard to believe that some relationship of general validity for the whole variety of the metal-ligand system could possibly be derived with our poor knowledge of these factors at present. So a derivation of a relationship valid for some more restricted class of metal-ligand systems is still of great interest. Such a relationship might lead to some systematic classification of the metal-ligand systems and increase our understanding of certain physical and chemical properties. So our interest was focussed on the construction of a mathematical model for the complex formation process in aqueous solution, which could allow the problem stated above to be solved for as many metal-ligand systems as possible. In this paper a simplified mathematical model of the
107
108
M. H. MIHAILOV
equilibrium in aqueous solution is described. This model leads to the possibility of deriving a relationship connecting the overall stability constants of the metal complexes in a given metal-ligand system. The validity of this relation has been demonstrated on more than forty metal-ligand systems. Some of them are presented in the present paper.
THEORY Let us consider the complex formation process of the metal M m+ (m is the charge of the cation M) and the univalent ligand L - , arising from the dissociation of the strong electrolyte CL in an aqueous solution of a constant ionic strength, maintained by addition of the inert electrolyte CL'. Let the system under consideration satisfy the following requirements: (i) the concentration of the metal M m÷ is sufficiently low so that the formation of polynuclear complexes could be neglected. (ii) the metal M m+ does not participate in side reactions with the ligand L ' - and O H - ions. (iii) the ionic strength of the solution is sufficiently low so the formation of outer sphere complexes (ion-pairs) between the anionic species and the C+-ions could be neglected. (iv) the activity coefficients of all species are constants so concentration terms could be used instead of activities in all equations. Then, the formation of stepwise series of metal complexes could be expressed by the following generalized equation:
M + nL ~ ML,,.
(1)
The stoichiometric stability constant of the ML. complex is defined as: /~. =
[ML,] -
-
[M][L]"
(2)
where: [ ] are concentrations; n is the number of ligands. (Here and throughout the paper the charges will be dropped for simplicity.) But the cation M and the ligand L are hydrated in aqueous solution, so the complex formation process may be regarded as a substitution of water molecules from the first coordination sphere of the metal M with negatively charged ligands[11-13], rather than a simple addition reaction as (1). Thus, we could write: -
MSN + nL backwardML.Sn_. + nS
(3)
where: S denotes water molecule; N is the coordination number of M. Here the reaction is restricted to the first coordination sphere of M. For simplicity the water molecules in the second and third hydration spheres[14] are omitted and the hydration sphere of L is omitted. The interaction between a metal ion and a negatively charged ligand in solution depends on their solvation [15, 16], the structure of the solvent in the solvation spheres of the ions[17, 18], the relative ease with which ions of opposite charge displace the water molecule
from the coordination sphere[19, 20]. All these effects as well as others may be important. To make direct calculations for such a complicated system taking into account all these factors is an extremely difficult task. So, for our purpose we felt it better to consider the total effects, making some crude simplifying approximations when deriving the main equations. The assumptions necessary for the validity of the relations which will be derived are as follows: (i) The complex formation process (forward reaction) involves substitution of water molecules coordinated to the metal ion M in its first coordination sphere by negatively charged ligands L. (ii) The dissociation of the complex ML,S N_. (backward reaction) is a substitution of the ligands L coordinated to the metal M in its first coordination sphere with water molecules. (iii) The coordination sites of the metal ion M are equivalent. (iv) The metal ions M, ligands L, all metal complexes and electrolytes ions are randomly distributed in the solution and undergo random motion. (v) The forward reaction proceeds because of short range forces defined as follows: the force directed from the metal ion M towards the ligand L or from the central group M of the complex ML.SN_. towards L, by which M attracts L is denoted bYfM~z and the force directed from L towards M or towards the central group M of the complex ML.SN_., by which L attracts M is denoted byfz~ M. We shall assume that these forces remain constant irrespective of the number of ligands coordinated to the metal ion M in its first coordination sphere. In the most general case:f~t~L :~ fL~M because the hydration shells of M and L contain different numbers of water molecules and have a different structure. (vi) The backward reaction proceeds under the action of the short range forces defined as follows: the force directed from the central group M of the complex ML.S N_. towards S by which M attracts the water molecule is denoted as fM~S and the force directed from a water molecule towards the central group M of the complex ML,SN_. by which the water molecule attracts M is denoted by fS~M' We shall assume that these forces are also independent on the number of ligands in the complex species. (vii) The progress of a reaction (forward or backward) is a result of an effective collision between the interacting particles. We shall define as an effective collision one in which the colliding species first have a suitable energy to approach within some distance for which the attractive forces are effective and second have the proper orientation. (viii) The number of M ions and complex species ML.SN_. which have a suitable energy for the forward reaction will be proportional to their concentration in the solution and to the temperature. Thus, for a given temperature the number of metal ions M, which have a suitable energy for the forward reaction will be equal to KM[M], for the complex ML this number will be KML[ML], for the complex ML. it will be KMz.[ML.] and for the ligand L it will be given by: KzEL ] respectively. By analogy, the number of the complex species which have a suitable energy for the backward reaction at a given temperature will be propor-
109
Stability constants of metal complexes--I tional to their concentrations in the solution with proportionality factors K'ML, K'ML. . . . . . K'ML" respectively. The number for the water molecules will be Ks[S]. We shall assume that: KML = KML~ = . . . . KML. = Kx, which corresponds to the assumption that the heat of displacement of a water molecule in the complex MLnS N_. by ligand L is independent on the number of ligands in the given complex• If the same can be assumed for the backward reaction we can write: K~L = K~L. = . . . . K~L ~ = K~:. The metal ion M cannot be regarded as a member of the complex series, its first coordination sphere being symmetrical and containing only water molecules. So considering the proportionality factor KM we supposed that the following relation holds: K M = A . Kx, where A is a constant. (ix) In each complex the domains available to the attractive force of the central group M and to those of each ligand are equivalent. So, a collision between complex species ML,SN_, and a ligand L will be effective only in the case that L approaches the complex in the domain where the attractive force of the central group f u ~ L is dominant• (x) The hydrated metal ion M and the hydrated complex ML,SN_, have approximately equal volumes. Now, we shall define the probability of the complex M L being found in the solution as a difference between the forward and backward reaction probabilities. If the forward reaction probability is expressed as a product of the numbers of M ions and L ligands having a suitable energy for reaction and the corresponding attractive forces we obtain:
where: ao = K~fM~LKLfL~M = const. d = Kj, fM.sKsfs.M[S] = const.
By analogy, the probability of ML2 species being found in the solution will be given by: a2
1 A ~ (1 - d)2[M] [L] 2.
PML2 = ~ "
Z!
(7)
The factor ½ arises because, according to assumptions (vii) and (ix) the probability of an effective collision between the complex MLSN_ 1 and the ligand L, leading to ML2SN_2 is only half the probability of an effective collision between the metal ion M S N and the ligand L. For the next complex in the series, M L a S N_ 3, this factor must be multiplied by ½ and so in general the factor 1 becomes - - ' n!
In an analogous way the following relation for the probability of the M L . S ~ _ . complex (n > 2) being found in the solution can be derived: 1
an
P~L. = C--MM"A ..~ (1 - d)"[M] [L]"
(8)
or:
1
an
PML. = -~M" A ~. [M] [L]"
(8')
where: a = ao(I - d). If we define the total probability as:
1 pfo~. = ~ .
AKxfM~L[M]KLfL~M[L]
n=N P = PM + E PML. = 1
(4)
(9)
n=l
where: CM is the total (analytical) concentration of the complex forming metal M. The backward reaction probability will be given by the product of the number of M L species which have a suitable energy for the reaction and the number of water molecules having a suitable energy for the reaction and the corresponding attractive forces. If the concentration of the M L complexes in the solution is expressed by Eqn (4) we could write: 1 Pback. ~ ~MM "
. K ~ f M ~ s A K x f M ~ L [ M ] K L f L ~ M [ L I K s f s~M[S].
(5)
Then, the probability PML of finding the M L complex in the solution will be given by: 1
1
{ 1 - K'&.~Kff~.,[S]}
= - - . A ' ao(1 - d)[M] [L]
c,,,
[M] PM-
(10)
Cu
will be the probability of the M ion to be found in the solution. Expressing PM and PM~ through Eqns (10) and (8') respectively we get the following expression for Eqn (9):
P
=
nl.n " } =1. 1 [ M I ~ I + A n f =rLl" C~ l .=1
(11)
Multiplying both sides of Eqn (11) by CM, the following expression for the distribution of M (central group) in the solution is obtained:
CM [M]f1~
.=N a"
l
Obviously, if Eqn (12) is equated to the conventional expression for the distribution of M (central group) in the solution[4] we get the following equation for the stability constant ft, of the complex M L , :
PML ~- Prof. - Pback. = ~-'~" • AI(.f,~.~[M]K~f~,[L]
then:
an
(6)
fl,, = A - - - . n!
(13)
110
M. H. MIHAILOV
Then, the complexity function as defined by Froneus [21] will be transformed to:
Table 1. Comparison of calculated formation functions for the cupric-chloride system
n=N a n
F = 1 + A ~. ~ . [L]" n=l
(14)
•
and the degree of formation or the formation function of the system h will be given by: /~=A
~ (nn=l
1)'[L]" 1 + A "
~
"(15)
n=l
CALCULATION OF THE OVERALL STABILITY CONSTANTS USING THE FORMATION FUNCTION OF THE S Y S T E M - - S
It was interesting to see how efficiently the relations derived in the previous part of this paper will describe the complex formation process in aqueous solution. F o r this purpose we have turned our attention to the metal halogenide systems as the most exhaustively studied m e t a l - l i g a n d systems. F o r example, in the compilation of Sill6n a n d Martell[7] they represent a third of all listings for inorganic ligands. Thus, in this group it was not difficult to find systems which satisfy the requirements encountered in the first part of this paper. We have selected only systems for which more than three stability constants are reported. We have not included systems for which only the first two stability constants fll and f 1 2 respectively are reported because for such systems two constants could always be found to satisfy Eqn (13). But this fact would not serve as support for the validity of the relation previously derived. The calculations have been performed as follows: from the values of ft, reported in the literature[7] for a given m e t a l - l i g a n d system the formation function fi was calculated using the conventional expression[l]. Then, from two values of fi taken at two ligand concentrations, the constants A and a were determined using Eqn (15). F r o m the constants A and a determined the corresponding ft, were calculated using Eqn (13). To check the correctness of the values of a a n d A constants determined, the formation function ~ was now recalculated using the set of stability constants derived according to Eqn (13). The formation functions h for the systems: Cu 2+ - C1-[22], Z n 2+ - C1-[23], Cd 2+ C1-[24], In 3+ - C1-[25, 26], G a 3+ - C1-[27], P b 2+ C1-[28], Z n z+ - Br-[23] and Cd 2+ - Br-[29] were analyzed in the way described above. In Tables 1-9 the formation functions hi" which were calculated using the ~, values obtained by the present m e t h o d and correlated according to Eqn (13) are compared with the/~*, calculated using the ft, values reported in [7] for the corresponding systems. The discrepancy at each ligand concentration was estimated by the quantity An = I/fi*(fi* - ~f)100. For the most part of the systems the representative values of the formation functions were calculated for ligand concentration up to a value close to that of the ionic strength of the solution. Only for the Cu 2 + - C I - system, which had been studied at constant ionic strength equal to 0-69, the formation functions were calculated up to EL] - 4 M. Irrespective
(L) M
h*
h~
Ah (%)
0.050 0.075 0.100 0.200 0.300 0.400 0.500 0.600 0.690 1-000 1.500 2.000 2.500 3.000 3.500 4.000
0.337 0.444 0.512 0.771 0-922 1.054 1-173 1.281 1.384 1.700 2-133 2-467 2.718 2.907 3.037 3.163
0.319 0.427 0.515 0.764 0.936 1.075 1.198 1.312 1.400 1.712 2.124 2.449 2.700 2.893 3.043 3-167
5.34 3.83 -0.59 0.91 - 1.52 - 1.99 -2.13 -2.42 - 1.16 -0.71 0.42 0.73 0.66 0-48 - 0.20 -0-13
* Calculated by using the//, values reported in [22]: fll = 9.55;//2 = 4"90;//3 = 3"55;//4 = 1.00. t Calculated using the //, values obtained by the present method: fll = 8-60; l/2 = 6.10;//3 = 2'89;//4 = 1'03; A = 6'056 and a = 1-42. of the fact that the results in Table 1 show satisfactory coincidence of b o t h formation functions over the whole ligand region, we consider such extrapolation as not quite justifiable because at higher ionic strengths side Table 2. Comparison of calculated formation functions for the zinc-chloride system (L) M
fi*
~
A~ (%)
0-05 0.10 0.15 0.20 0-25 0-30 0-35 0.40 0-45 0.50 0.55 0-60 0.65 0.69
0-219 0.379 0-503 0.606 0.695 0.774 0-848 0.917 0-983 1.048 1.112 1.175 1.239 1.290
0-219 0-381 0.509 0.616 0.709 0.792 0.867 0.937 1.002 1-064 1-124 1.181 1-236 1.279
0-00 -(~53 - 1.25 - 1.65 - 2-01 - 2.33 - 2.24 -2.18 - 1.93 - 1.53 - 1.08 -0-51 0-24 0.85
* Calculated using the ft, values reported in [23]: //1 = 5.25;/12 = 3.09;//3 = 0.65;//4 = 1.51. "["Calculated using the //, values obtained by the present method://1 = 5.20;//2 = 3.38;//3 = 1.46; f14 = 0'48; A = 4'0 and a = 1"3.
Stability constants of metal complexes--I Table 3. Comparison of calculated formation functions for the cadmium-chloride system
11'.
Table 5. Comparison of calculated formation functions for the indium-chloride system
(L) M
h*
~t
Ah (%)
(L) M
h*
fit
Ah (%)
0.0010 0-0050 0.0075 0-010 0-050 0.075 0-10 0.50 0-75 1.00 1.50 2.00 2.50 3.00
0.091 @344 0.449 0.530 1.020 1.161 1.276 2.494 2-931 3-201 3-496 3-643 3.727 3.781
0.096 0.353 0.456 0.535 0.993 1.125 1.237 2.518 2.944 3-202 3.478 3.618 3.700 3.753
-4.35 -2.62 - 1.56 -0.94 2.65 3-10 3.06 -0.96 -0.44 -0.03 0.51 0-68 0-72 0.74
0.001 0.003 0.005 0.007 0.01 0.03 0.05 0-07 0.10 0.20 0-30 0.40 0-50 0.60 0.70
0.193 0.443 0.607 0.721 0.851 1.278 1.491 1.676 1.788 2-081 2.245 2.354 2.434 2.496 2.545
0-193 0.430 0-575 0-676 0-785 1.151 1.375 1-556 1.773 2.217 2.441 2.570 2.653 2.710 2.751
0.00 2-93 5.27 6-24 7.76 9.94 7.78 7.16 0.84 -6.54 -8.73 -9-18 -9.00 -8-57 - 8.09
* Calculated using the //, values reported in [24]: //~ = 100; //2 = 4 0 0 ; //3 = 504;//4 = 1008. t Calculated using the ft, values obtained by the present method: fix = 105; [/2 = 312; fla = 619; //4 = 920; A = 17.68 and a = 5-94.
reactions could take place, w h i c h are n o t taken into a c c o u n t in the p r e s e n t model. G a l l i u m chloride a n d c a d m i u m chloride are the o t h e r exceptions, which h a d n o t been studied at c o n s t a n t ionic s t r e n g t h of the solution a n d so for these systems the c o n d i t i o n s for a constancy of the activity coefficients are n o t fulfilled.
Table 4. Comparison of calculated formation functions for the indium-chloride system (L) M
h*
ht
Ah (%)
0.005 0.007 @01 0.03 0.05 0.07 0.10 0.20 0-30 0.40 0.50 0-60 0-69
0.571 0-700 0-849 1-354 1.596 1.753 1.915 2.212 2.370 2.473 2.545 2.600 2.639
0.601 0.705 0.820 1.223 1.477 1.676 1.907 2.339 2.452 2.649 2.718 2.765 2.796
-5.25 -0.71 3.42 8-51 7.46 4.39 0-42 -5.74 -3.46 -7.12 -6.80 -6.35 -5.95
* Calculated using the ft, values reported in [25]: fix = 186; f2 = 4677;//3 = 13,800 t Calculated using the //, values obtained by the present method: //1 = 250; f2 = 2750; f3 = 20.166; A = 11.36 and a = 22.0.
* Calculated using the ft, values reported in [26]: f l = 229; f2 = 4266; f3 = 8913. t Calculated using the f , values obtained by the present method: f l = 234; f 2 = 2083; f3 = 12,356; A = 13.15 and a = 17.80. Table 6. Comparison of calculated formation functions for the gallium-chloride system (L) M
h*
fit
Ah ("~)
0.5 1.0 1-5 2.0 2.5 3-0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8-0 8.5 9-0 9-5 10.0
0.114 0.208 0.287 0.356 0.416 0.469 0.517 0.561 0.601 0.637 0.671 0.703 0.733 0-761 0-787 0-812 0.836 0-860 0-882 0.903
0.106 0.196 0.275 0-346 0-409 0.466 0.519 0.568 0.613 0-656 0-696 0.734 0.771 0.806 0-839 0.872 0.904 0.933 0.963 0.992
7.02 5.77 4.18 2.81 1.68 0.64 -0.39 1.25 -2.00 -- 2.98 - 3.73 - 4-41 -5.18 - 10.25 -6.61 -7.39 - 8.01 - 8.49 -9-18 - 9-86
* Calculated using the ft, values reported in [27]: fll = 0'251;fl2 = 5.01 × 10-3;fl3 = 3-16 x 10-5;fl4 = 1.58 x 10 -6. t Calculated using the ft, values obtained by the present method: fll = 0.229; f12 = 6.59 x 10-3; f13 = 12.65 x 10-5; fl, = 1.80 × 10-6; A = 3.97 and a = 0-0576.
112
M . H . MIHAILOV
Table 7. Comparison of calculated formation functions for the lead-chloride system (L) M
h*
h~
Ah (%)
(L) M
fi*
h?
Ah (%)
0'1 0.2 0.3 0.4 0.5 0.6 0.7 0-8 0.9 1.0
0-563 0.892 1.169 1.412 1-623 1-804 1.956 2.083 2-190 2.279
0.531 ~922 1.228 1.470 1.664 1-821 1.950 2.055 2.143 2.218
5.68 -3.36 -5.05 -4-11 -2.53 -0.94 0.31 1.34 2.15 2.68
0.050 0.075 0.10 0.20 0-30 0-40 0.50 0.60 0-70 0.80
0.796 0.970 1.106 1.532 1.883 2.231 2.448 2.665 2.744 2.990
0.766 0'941 1.081 1.541 1.915 2.271 2.479 2.683 2.734 2.982
3.77 2.96 1.90 -0.61 - 1.72 - 1-80 - 1-27 - 0.67 0.36 0.26
* Calculated using t h e / 3 , values reported in [28]: /31 = 9.3;/32 = 7.1;/33 = 22.4. t Calculated using the /3. values obtained by the present m e t h o d : fll = 6.50; /32 = 12.67; f13 = 16.50; A = 1-67 and a = 3.9.
N e v e r t h e l e s s , t h e stability c o n s t a n t s w h i c h were f o u n d by t h e p r e s e n t m e t h o d o b e y E q n (13). I n c o n c l u s i o n it m u s t be said t h a t in all t h e s y s t e m s s t u d i e d a satisfactory fit b e t w e e n t h e f o r m a t i o n funct i o n s c a l c u l a t e d u s i n g t h e ft, v a l u e s r e p o r t e d in t h e c o m pilation of Sill6n a n d Martell[7] a n d t h o s e o b t a i n e d w h e n u s i n g t h e ft, v a l u e s o f the p r e s e n t m e t h o d w a s achieved. So, t h e r e s u l t s p r e s e n t e d in T a b l e s 1 - 9 s u p p o r t the validity of t h e r e l a t i o n s derived in this p a p e r for all t h e s y s t e m s a n a l y z e d .
Table 8. Comparison of calculated formation functions for the zinc-bromide system
r
Table 9. Comparison of calculated formation functions for the c a d m i u m - b r o m i d e system
(L) M
fi*
fit
Ah (%)
0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.69
0.155 0.226 0.293 0.357 0.418 0.477 0.534 0.588 0.642 0-694 0.744 0.794 0.833
0.155 0-226 0.293 0.358 0-419 0.479 0.537 0.592 0-647 0-700 0.751 0.801 0.841
0.00 0.00 0.00 -0.28 -0.24 - 0.42 - 0.56 -- 0-68 - 0-78 - 0-86 - 0-94 - 0-88 - 0-96
* Calculated using the ft, values reported in [23]: /31 = 1-66;/32 = 0-794;/3a = 0.182;/34 = 0.100. t Calculated using the /3, values obtained by the present method: /31 = 1.65; /32 = 0.792; /33 = 0 - 2 5 3 ; /34 = 0.061 ; A = 1.72 and a = 0.96.
* Calculated using the /3, values reported in [29]: /31 = 36.31;/32 = 126; f13 = 145;/34 = 339. t Calculated using the /3, values obtained by the present method: /31 = 34; /32 = 100; /3a = 195; /34 = 283; A = 6.0 a n d a = 5.6.
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