J. Phys. Chem. 1980, 84, 722-726
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Rules Governing Concentration Distribution in Complex Equilibrium Systems I. Nagyp6l' and M. T. Beck" Institute of Inorganic and Analytical Chemistry, and Institute of Physical Chemistry, L. Kossuth University, H-40 I0 Debrecen, Hungary (Received February 23, 1978; Revised Manuscript Received October 25, 1979)
Starting from the general equations describing homogeneous equilibrium systems, we give the derivative of the concentration distribution curves in explicit form. It has been proved that logarithmic concentration distribution curves of two-component systems are always concave. From the study of the three-component metal ion-ligand-proton systems it is concluded that the concentration minimum may occur only on the distribution of those species which are taking part in protolytic equilibria. The chemical processes responsible and the conditions necessary for the concentration minimum are outlined and illustrated.
Introduction Equilibrium systems in solution are best represented by concentration distribution curves. These are obtained by plotting the concentration of the complexes vs. the equilibrium concentration (or logarithm of equilibrium concentration) of one of the components, at the constant total concentration of the others. In case of mononuclear stepwise complex formation, the concentrations, in general, are plotted against -log [L] = pL. The rules for concentration distribution in this case are well known. The concentration of M decreases and that of the complex containing a maximum number of ligands (ML,) increases monotonously with increasing free ligand concentration. The concentration of ML, (0 < i < N) complexes are represented by maximum curves; the maxima are found at integer values of the formation curve (a = i). In case of more complicated systems (polynuclear, mixed ligand, protonated, hydroxo complexes) the rules for concentration distribution are not so clear. A number of examples prove that more than one extremum may occur in the more complicated system^."^ An interesting example was published by V6rtes et a1.l in the Sn14-CC14-DMFO (dimethylformamide) system, where the sum of the concentration of coordinatively saturated complexes showed two extrema as a function of DMFO concentration. In this case, however, a change of the solvent takes place, thus the formation constants of the complexes are also dependent on the DMFO concentration. This medium effect may result in the unusual concentration distribution, as was proved earliera6A similar example was found and interpreted by Schipperta2 Agarwall and Perrin3 published the first example where the medium effect may not be responsible for the concentration minimum, without noticing the unusual character of the distribution. This was the copper(I1)glycine-diglycine system, illustrated in Figure 1. It is seen in Figure 1that the complex CuGz has one minimum and two maxima between pH 6 and 9. It is seen moreover that extrema are only slightly visible, and the concentration of CuG, is practically constant over a wide pH range. This means that, if the rules for the concentration distribution are known, equilibrium systems could be designed in which the concentration of a given species is fairly constant, practically almost independent on the total concentration of some of the components. Institute of Inorganic and Analytical Chemistry. *Author to whom correspondence should be addressed at the Institute of Physical Chemistry. 0022-3654/80/2084-0722$01 .OO/O
The aim of the present work is to uncover the chemical background and the mathematical relations determining the shape of the distribution curves, with special regard to the concentration minimum. The mathematical considerations are given generally; the examples and their chemical interpretation, however, are restricted to threecomponent systems and to the pH dependence of the concentration distribution, because this is the most general and most informative form of the concentration distribution in the more complicated systems.
Theory The general equations of equilibrium analysis are based on expressing the mass balance with the law of mass action:
n
where k is the number of components, Tl, . .Tkare the total concentrations of the components, and cl. . .ck are the components of the system. These are, in general, the metal ion, the proton, and the deprotonated form of the ligandb), because the total concentrations defined for these may be determined analytically. n is the number of the species in the system, including the components and the different complexes formed from them. Pi = [A,]/[c1la~l.. .[ck]'jk is the formation constant of the j t h species. (The formation constant of the components are unity by definition.) Aj is any species of the system, component or complex. a,; are the stoichiometric coefficients, giving the number of ith component in the j t h species. The concentration distribution in terms of eq 1 means that the T2...Tktotal concentrations are constant and the equation related to T2.. .Tk are solved for [cz]. . .[ck] at different fixed value of [cl]. Then the [Aj] values are calculated and plotted against [cl] (or -log [q], for example, pH). In this way, for a given set of T,. . .Tk, the concentration of any species depends on [cl] only, although it is rarely possible to express the [A,] = f([cJ) function explicitly. For mathematical analysis of the concentration distribution, however, the derivative of this function is necessary. The derivatives can be expressed explicitly for any of the c2. .ck components, using the rules for the derivation of implicit functions for eq 1: 0 1980 American Chemical Society
Concentration Distribution in Equilibrium Systems
The Journal of Physical Chemistry, Vol. 84, No. 7, 1980 723
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Equation 21 is linear for d log [c,]/d log [el], thus it can be expressed by appropriate determinants or it can be calculated by well-known methods. The d log [ci]/d log [el] values may be used to extrapolate the component concentrations to the successive experimental point of the pH-potentiometric titration curves used in equilibrium analysis. This speeds up the computer programs used in A similar implicit derivation equilibrium may be used to calculate the d log [clJ/d log [PI] derivatives, from which dTl/d log PI (and dvld log @ can be calculated analytically. Thus the numerical, time-consuming derivation used in SCOGS,~LETAGROP: and similar programs may be avoided.l0 For our present purposes, however, the d log [c,]/d log [cl] derivatives are necessary, because the derivative for any of the species may be expressed1 by these:
(3)
Two-Compoinent Systems For two-component systems, the d log [czl/d log [cJ function may be expressed from eq 2 as follows: n
i.e.
6
PH
7
0
Flgure 1. Partial molar percentages of the complexes formed in the copper(I1)-glycine-diglycine system: A = glycinate, L = diglycinate, L-H = diglycinate, which has lost its proton from the peptide bond. T, mol dm-3. = T A = 2 X lom3,Tcu =
It is seen from eq 7 that the second derivative is always negative, i.e., the logarithmic concentration distribution curves for the two-component systems are always concave. Comparing eq 4 and 6 we can state, moreover, that the concentration maxima of the species follow each other in order of their increasing ajl/ap ratios, independent of the T2total concentration, of their formation constants, and of the individual value of ajl and ajz.
Three-Component Systems The most important three-component systems are those containing metal ion, ligand and proton. The complexes formed in these systems may be classified as follows: (i) complexes containing protons or having a proton deficiency (proton complexes, protonated, and hydroxo complexes as well as mixed hydroxo complexes; these are called Br~nsted species in the following); (ii) complexes with the coniposition M,L,, i.e., which do not contain protons (these are called Lewis species). The effect of the change of pH on the concentration distribution of these two types of complexes is basically different. The concentration of the Lewis species is affected by the pH only indirectly, through the deprotonation of the ligand. It is evident that if their concentration is plotted against -log [L] = pL, then the statements for the two-component systems are valid for their distribution. The free ligand concentration is a monotonous function of pH. Therefore no concentration minimum may occur in their pH-dependent distribution, and only the distribution curves are "distorted' by the pL pH coordinate transformation. The effect of pH on the distribution of the Br$nslted species is basically different. There is, on one hand, a direct reaction (protonation or proton loss) with the proton which increases or decreases their concentration. On the other hand, there is an indirect effect of pH, originating from the Lewis species formation. The change of the Lewis species concentration means that the sum of the concentration of the species taking part in the protonation processes is also changing, always in opposite direction. The formation of a Lewis species always decreases; the decomposition always increases the concentration of the Bransted species. The direct and indirect effects of the change of pH may result in unusual concentration distribution with concentration minimum. To demonstrate the meaning of the above statement and to see the conditions of the concentration minima, three different equilibrium systems will be analyzed in detail. The first example is the methylmercury-acetate-proton system studied by Rabenstein et al.4 The authors stated
-
m=l
Equation 4 expresses that the d log [cz]/d log [cl] function is monotonous. At the extremum of the concentration of any AJ species d log [A,]/d log [cl] is zero, thus
(d log; [c21/d
log
[clI)[A,]=max
= -CUJl/a]2
Differentiation of eq 5 with respect to log [el] gives
(6)
9
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The Journal of Physical Chemistry, Vol. 84, No. 7, 7980
- o l e d ~ ’ [ - 5 7 - - ~
K
i
O.’
Z M L + H ’ = M2H:, -0.01 1
2
3
2MH-,+H’:
2HL 4
5
6
1
M2H:l
PH
8
Figure 3. The d[M,H-,]/d pH function and its components in the methylmercury-acetate-proton system: (1) d[MzH..,]/d pH originating from the protolytic reactions; ( 2 ) d[MZH-,]/d pH originating from the metal-removing effect of the ML complex; (3) the sum of curves 1 and 2.
more illustrative, let us see how [M2H-,] is changed by the change of the total concentration of the metal containing Bransted species: 0 3
1
5
1
PH
Figure 2. Concentration distribution of the complexes formed in the methylmercury-acetate-proton ~ y s t e m .Upper ~ part: HL-L distribution in the absence of methylmercury. Lower part: M-MZH..l-MH-l distribution in the absence of acetate. The distribution in the three-component system is seen In the middle of the figure. T, = T, = 0.2 mol dm-3. log PnL= 4.65, log pML= 3.18, log PMH-,= -4.70, log PM~H.,
-2.33.
In this system, however d([M] + [MH-J
that M(0H) = MH-, and M2(OH) = M2H-, hydroxo complexes are formed in aqueous solution of methylmercury, and ML complex is also formed in the presence of acetate ion. Figure 2 illustrates the concentration distribution of the complexes. The upper part shows the HL-L concentration distribution in absence of methylmercury, the lower part shows the M-M2H-1-MH-1 distribution in the absence of acetate, In the middle of the figure the concentration distribution of the three-component system is illustrated. The dotted line shows the change of the sum of the concentration of Bransted species. Because of the relatively low amount of M2H-, its concentration is multiplied by a factor of 10 for illustration. It is seen that the M& concentration has three extrema as a function of pH. To separate the effects causing this distribution, we derived the d log [M,H-,]/d pH function using eq 2 and 3:
A simple algebraic rearrangement of this equation leads to the following form: [MI - [MH-lI d 1% [M,H-lI [MI + [MH-11 + 4tM2H-11 d PH ((2[MLl(2[MzH-iI([HLl - LI) + [ M I W I [Ll[MH-,I)l/Wl + [MH-,I + 4[M2H-111x (([Ll + [HLI + [MLI)([Ml + [MH-iI + 4[MzH-i1) + [MLI([Ll + [HLIN (9) This form of the equation, although more complicated, separately shows the direct and indirect effect of pH on the shape of the distribution curve. The first part of the equation is the same as that which would describe the d log [M,H-,]/d pH function in the absence of acetate, Le., this is valid for the lower part of Figure 2. The second part expresses the effect of MI, formation on the shape of the distribution curve. To make it even
-
+ 2[M,H-i])
= -d[ML]
(11)
Let us divide d log [M2H-J/d pH into two parts, which would represent it in a two-component system and which is a consequence of the formation of ML: d 1% [MZH-lI d PH
--
[MI - lMH-11 [MI -t [MH-,I + 4[M2H-,l d log [MZH-i] d[ML] (12) d[ML] dpH +
Expressing d[ML]/d pH from the general eq 2 and 3, as well as using eq 11, one can see that eq 9 and 1 2 are identical. Equation 12, however, is more illustrative, and clearly shows the secondary effect of [ML] on the distribution of [M2H-,]. Figure 3 illustrates the d[M,H-,]/d pH function. Curve 1 is calculated from the first part of eq 12 and curve 2 is calculated from the second part of eq 12. Their sum is curve 3 with zero points at the extrema of [M&] on Figure 2. It is interesting to analyze the change of the sign of the component curves 1 and 2, and to see the different chemical processes causing the sign change. In the positive range of curve 1 [MI dominates over [MH-,I, thus mainly 2M + M2H-1+ H protolytic reaction increases [M2H-,]by increasing the pH. In the negative range of curve 1 [MH-,I dominates, thus the following reaction occurs: 2MH-1+ H + M2H-1 Le., the increase of pH decreases [M,H-,]. The effect of the ML complex is just the opposite, because at that range where the ML is formed, mainly HL is present, Le., [M,H-,] is decreased by increasing pH because of the following reaction: 2ML + H + M2H-1 + 2HL
A t higher pH, where [L] dominates over [HL], the following reaction increases [M2H-,l: 2ML + MZH-l+ H + 2L
Concentration Distribution in Equilibrium Systems mole d i 3
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-1
'
Ooo3
L
h,
13.01 .
/
L
6
5
8
7
PH
Figure 5. The d[HL]/d pH function and its components in the model system: (1) d[HL]/d pH originating from protolytic reactions; (2) d[liL]/d pH originating from the ligand-removingeffect of ML,; (3) the sum of curves 1 and 2.
Figure 4 . Conclentration distribution of the species in the H,L-HL-L, M-ML, model system. Upper part: Distribution of the proton complexes in the absence of metal ion. Lower part: Distribution of the complexes in the presence of metal ion. TL = 0.025, TM = 0.01 mol log &L = 8.0, log [!H2L = 13.0, log /3MLg = 11.25.
These reactions, which are responsible for the signs of the different parts of the curves, are written in the appropriate segments of Figure 3. The second1 example shows a result of a model calculation in a HJ-HL-L, M-MLz containing system. The distribution of the H2L-HL-L species in the absence of metal ifon is seen in the upper part of Figure 4,and the distribution of the complexes formed in the three-component system is seen in the lower part of Figure 4. The dotted line shows the sum of the concentration of the Bransted species in the system. It is seen that [HL] has three extrema as a function of pH. In a manner similar to that shown above, we have split the d log [HL]/d pH function into two parts d log [IIL] [HZLI - [LI d pH [LI + [HLI + W2LI d log [HL] d[ML2] (13) d[MLzI d P H Taking into account the following +
[HLI d/HLI - -2 (14) [LI + [HLI + [H,LI d[ML,I and expressing d[ML,]/d pH from eq 2 and 3, we obtain
Figure 5 illustrates the components and the sum of d[HL]/d pH. It is clearly seen that the minimum is exhibited because the metal ion removes the ligand from the Bransted equilibrium. It is evident, moreover, that this effect may lead to a minimum only if ML2 complex formation takes place in that pH range where [HL] would be increased in the absence of metal ion. The problem which remains is to state the stoichiometric conditions for the occurrence of a minimum. Evidently, minimum may only occur if the concentration decreasing effect of the Lewis species formation may overcompensate the increasing effect of the Brdnsted equilibrium. This overcompensation m a y occur only if the formation of the Lewis species i n its main reaction is accompanied by the removal of more
ligand (for unit p H change) than the number of Brdnsted species released in the concentration increasing process. The HL concentration increasing process in this system is the following equilibrium: HZL + HL + H i.e., at [H2L]> [HL], d log [HL]/d pH would be -1. The dominating Bransted species in this pH range are HzL and HL, Le., ML2 formation takes place mainly through the following processes: M + 2HzL F+ ML2 4H
+
M
+ 2HL + MLZ + 2H
If only the first reaction took place, at [MI, [ML,] > [HZLI, d log [H,L]/d pH would be --2, but if only the second reaction took place, it would be only --1. The second reaction therefore could not overcompensate the [HL] increasing effect of the Bransted equilibria. The first reaction may overcompensate this, thus the following reaction is responsible for the [HL] minimum: M + 2H2L * MLZ 4H
+
The above condition is fulfilled in the methylmercuryacetate system too. The ML formation M + HL ML H
+
--
would result in d log [M]/d pH -1, which is transferred to [M,H-,] as d log [M,H-,]/d pH -2, while through the 1. direct formation, d log [M,H_,]/d pH The stoichiometric condition means that if only ML is formed or if ML is also formed in significant concentration then the [HL] minimum may not occur. It is known from the elementary theory of complex formation that the slope of the N = 2 formation curve agrees with the slope of the N = 1formation curve if the statistical K1/Kz= 4 relation is fulfilled. The slope of the formation curve is evidently related to the ligand-removing effect of the metal ion. Therefore it may be safely concluded that the [HL] minimum may only occur if K l / K z< 4. For sake of simplicity, no ML complex formation was assumed in the model calculation. The third example is the copper(II)-1,3-diaminopropane system'' illustrated in Figure 6. It is seen that the concentration of MLH-l exhibits two extrema. It is seen, moreover, that ML2and ML& complex formation begins only after almost complete formation of ML, thus the effects influencing the MLH-' concentration may be analyzed step by step. Parallel with ML formation, MLH-' is also formed, and its concentration increases until pH -7. Taking into account the pK values, one can see, moreover, that ML2 complex formation begins in the pH range where H2L dominates among the different forms of the ligand.
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The Journal of Physical Chemistry, Vol. 84,
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PH
Flgure 6. Concentration distribution of the complexes formed in the copper(II)-l,3diaminopropane-proton systeniVii ,T, = 4 X T, = 4 X lo-' mol dm-'. log PHL = 10.80,log PH,, = 19.91 (pKH,L = 9.11,PKHL= 10.80), log &,= 9.96,log PML = 17.23,log &H-, = 1.89,logpMLfi,_,= 5.17,log P ~ = 6.80. ~ The ~ MLH-, _ concentration ~ is multiplied by a factor of 10 for illustration, and the distribution of M2L2H-*is omitted, because this is a very minor species at this total Concentration.
Therefore the following reaction occurs, causing a quadratic decrease of ML: ML H2L ML2 2H
+
+
This overcompensates for the linear increase of MLH-l in the reaction ML F= MLH.I+ H and [MLH-,I decreases. In the pH >9 range, where already HL dominates, the reaction ML HL F= ML2 + H cannot overcompensate but only balance the MLH-l formation, Le., its concentration is constant over a wide pH range. At even higher pH, where L formation begins, these effects can neither overcompensate nor balance MLH..l formation, and its concentration increases again. It is worth noticing that not only the Brplnsted but also the Lewis species may have minimum in four-component systems (see Figure 1). Equations 2 and 3 could be used to study the effects resulting from the unusual distribution for any system, although the equations and effects are much more complicated. It is evident, however, that in multicomponent systems, the counteracting formation and removing effects are responsible for the minimum.
+
Conclusion The classification of the species formed in the threecomponent equilibrium systems, the illustrated examples,
and the considerations allow us to draw the following conclusions: In three-component metal ion-ligand-proton equilibrium systems only the Brplnsted species, taking part in protolytic equilibria, may exhibit concentration minimum on their pH-dependent distribution curves. The minimum may occur only if there is (are) Lewis species in the system. This species is formed in the pH range where the concentration of Brdnsted species would be increased because of the protolytic equilibria. The formation of the Lewis species is limited to a narrower pH range than the pH region for the increase of the Concentration of the Bransted species. The Lewis species removes more species (per unit pH) from the Br$nsted equilibria than the protolytic process would release in absence of the Lewis species. These considerations indicate that a concentration minimum could be mainly expected for metal-containing hydrolyzed species; very special conditions must be fulfilled to get concentration minimum for a simple proton complex. There are only few examples of concentration minimum in the literature. The phenomenon, however, probably occurs much more frequently than detected by the usual methods of equilibrium analysis. The metal ion and complex hydrolysis almost always unavoidably takes place as a "side reaction" beside the formation of the main, detectable species. The rules summarized can be utilized in choosing appropriate buffers to keep the concentration of a given species a t nearly constant value. The use of such equilibrium systems could be important in the design of appropriate buffers for ion-selective electrodes.12 It is well known, moreover, that many regulators are working in biological systems to keep the concentration of some important species at fairly constant value. The chemical equilibrium systems exhibiting species distribution in a maximum-minimum manner may also help to understand some features of biological regulators.
References and Notes VBrtes, A,; Gaizer, F.; Beck, M. Magy. K6m. Fob. 1973, 79, 910. Schippert, E. Inorg. Chim. Acta 1977, 2 1 , 35. Agarwall, R. P.;Perrin, D. D. "Coordination Chemistry in Solution", Hogfeldt, E., Ed.; Berlingska Boktryekeriet: Lund, 1972. Rabenstein, D. L.; Ozubko, R.;Libich, S.; Evans, G.A,; Fairhust, M. T.; Suwanprakorn, C. J. Coord. Chem. 1974, 3, 263. Rouche, M. L. D.; Williams, D. J . Chem. Soc., Dalton Trans. 1976,
1355. Nagypll, I.; Beck, M. Inorg. Chim. Acta 1975, 14, 17. ZBkBny, L.; Nagypll, I.; PLka, 1. to be submitted far publication. Sayce, I. G. Talanfa 1968, 15, 1397. SIIIBn, L. G. Acta Chem. Scand. 1962, 16, 158. Nagypll, I.; Plka, I.; ZBklny, L. %/anta 1978, 2 5 , 549. Debreczeni, F.; NagypBI, I., to be submitted for publication. Perrin, D. D.; Dempsey, B. "Buffers fqr pH and Metal Ion Control"; Chapman and Hall: London, 1974.
