ASSOCIATION OF BISPYRIDINIUM CATIONS WITH POLYCARBOXYLIC ACIDS
Gas Imperfection Energy --
The quantity @Tidea1 - ETreal)/Tc for the gas is a function that depends only on reduced properties of the gas and is a measure of the net attractive energy between the molecules. It was calculated for HgCl2 by the same method that was used for BiC4 (see eq 4 of ref 4). The values obtained for @Tidea’ ETreal)/Tc for HgC12 gas are shown as a function of reduced temperature compared to several other substancesz1in Figure 5. The curve for mercuric chloride is quite close to that for carbon dioxide. This is not surprising because of the similarity in their molecular structuresboth are linear symmetrical molecules of point group Dmh. For both molecules the intermolecular interaction energy must arise primarily from quadrupolequadrupole attractions. On the reduced basis of Figure 5, differences between interaction energies for molecules of the same type are suppressed, and only
2995
qualitative differences between types of intermolecuIar potentials influence the relative shapes of the curves. Water and ammonia are seen, on this basis, to have similar shapes that are different from the form of the carbon dioxide and mercuric chloride curves. The curve for BiC4 has approximately the same shape as the HzO and NH3 curves but is displaced downward. Their similarity probably originates in the fact that these three molecules are dipolar. The displacement probably arises because HzO and NH3 are hydrogen bonded, while BiC1, is not. More detailed consideration of the relation between the nature of intermolecular potentials and plots of the type of Figure 5 should probably be made from statistical-thermodynamic considerations.
Acknowledgment. The X-ray search for a change in structure of solid HgCl2 in the neighborhood of 428°K was performed by Dr. Malcolm Barlow. Mr. W. J. Silva performed some of the experimental work.
The Association of Bispyridinium Cations with Polycarboxylic Acids1
by H. Morawetz and A. Y. Kandanian Polylner Research Imtitute, Polytechnic Institute of Brooklyn, Brooklyn, New York (Received Bpril 21, 1966)
The binding of a,w-polymethylenebispyridinium bolaform cations to partially neutralized poly(acry1ic acid), as well as isotactic and syndiotactic poly(methacry1ic acid) was studied by a dialysis equilibrium technique. The variables explored included the concentration of polyions and bolions, the spacing of the charges on the bolions, the degree of ionization of the polymeric acid, and the concentration of added uni-univalent salt. The binding of Mg2+ by the polyions differs most strikingly from the binding of the bolions in its much lower sensitivity to the concentration of added uni-univalent electrolyte.
I t is commonly observed that counterions in sohtions of polyelectrolytes have unusually low activity coefficients. Several factors contribut,e to this effect. Part of the inactivation may be ascribed Do long-range electrostatic forces which are, of course, particularly powerful in systems containing polyions with high charge densities, Another part is a consequence of ion-pair formation of counterions with ionized groups
carried by the polyion or the formation of complex ions, frequently considered together under the term of “site binding.” These phenomena have been fully (1) Abstracted from a doctoral dissertation submitted by A. Y . Kandanian to the Graduate School, Polytechnic Institute of Brooklyn, June 1964. We are grateful for a fellowship received by A. Y . Kandanian from the American Machine and Foundary Co. and for financial assistance of this work by Grant GM-05811 of the National Institutes of Health.
Volume 70,Number 9 September 1966
2996
H. MORAWETZ AND A. Y.
discu~sed,~ and . ~ Lyons and Kotin4 have recently considered the problem of how long-range electrostatic interactions and site binding may be differentiated from experimental data. Frequently, such a distinction cannot be made unambiguously, but a quantitative measure of site binding can be obtained for the interaction of transition metal cations with a polycarboxylic acid, where complex formation is evidenced by characteristic changes in the absorption spectrum6 and in the titration curve."p6 It can also be demonstrated by dialysis equilibrium measurements under conditions where a simple electrolyte, which does not participate in complex formation, is in large excess over the polyelectrolyte.6b In the present study we have used the dialysis equilibrium technique to determine the association of the doubly charged species P2, P3, P4, P10, and PX with partially ionized poly(acry1ic acid) and with poly(methacrylic acid) samples of different stereoregularity.
n=5 3,4,10 (P2, P3,P4,P10)
Q Such species, in which two ionic charges are separated by several uncharged atoms have been designated as "bolaform ions"' (or bolions), and it seems that their interaction with polyions has been investigated previously only by Ehrenpreis and his collaborators,s who reported on the formation of insoluble complexes when certain bolaform cations were added to solutions of acidic polysaccharides. This work suggested that complex formation may depend critically on a correspondence of the spacing of the charges in the polyion and the bolaform counterion but changes in solubility are known to be a rather unreliable criterion of counterion binding by p o l y i ~ n s . ~The ~ ~ present study employed, therefore, the dialysis equilibrium method to give an unambiguous measure Of the extent Of sitebinding of bolaform counterions.
Experimental Section Preparation of Bispyridinium ,"&.
The synthesis Of the bispyridinium may be by the procedure used for l,l'-ethylenebispyridinium bromide' Two grams of dry ethylene bromide was heated a t 50" for 10 min with an excess of freshly distilled pyridine. The bispyridinium salt precipitated and was collected on a Bfichner funnel. It was recrystallized twice from 95y0 ethanol and dried a t 95" at a pressure of 2 mm. The J o u m l of Ph&cal Chemistry
KANDANIAN
The other salts were prepared in the same manner using the appropriate a,w-dibromides. The melting points of the dibromides were: P2, 290" (lit.10 295'); P3, 240" (lit." 242.5"); P4, 237' 237-239'); P10, 196' (lit." 196.5-198") ; PX, 280' (lit.13281-2820). I n addition, 8-phenylethylpyridinium (Q)bromide was prepared to serve as a singly charged analog of P2. It had a melting point of 125" (lit.14 125-126'). Bromine analyses for the six salts were (theoretical values in parentheses) : 46.25% (46.18), 44.45% (44.38), 42.56% (42.72), 34.82% (34.91), 37.77% (37.88), and 30.04% (30.28). Polymeric Acids. Freshly distilled acrylic acid was polymerized to a 30% conversion in butanone solution (15% by wt of monomer) at 60' using azobisisobutyronitrile initiator. The polymer was purified by extensive dialysis in bags of regenerated cellulose (Fisher Scientific Co.), until the dialyzate was neutral. It was then freeze dried for storage. Syndiotactic poly(methacrylic acid) (s-PMA) was prepared by irradiating a mixture of 15 ml of glacial methacrylic acid and 30 ml of dry methanol with 10 Mrads of y rays from a Co@J source at -78". Isotactic poly(methacry1ic acid) (i-PMA) was derived from poly(methy1 methacrylate) prepared by n-butyllithium-catalyzed polymerization of methyl methacrylate in toluene solution at -78". The poly(methy1 methacrylate) was dissolved in cold 96% sulfuric acid and warmed for 30 min to 60'. After cooling, the solution was poured slowly into a 100-fold excess of water, precipitating the polymer, which was purified by washing and prolonged dialysis
(2) H. Morawetz, FOTtSChT. Hochpolymer. Forsch., 1 , 1 (1958). (3) S. A. Rice and M. Nagasawa, "Polyelectrolyte Solutions," Academic Press Inc., New York, N. Y., 1961,Chapters 5 and 9. (4) J. W.Lyons and L. Kotin, J . Am. Chem. SOC., 87, 1670 (1965). (5) (a) F. T.Wall and S. J. Gill, J . Phys. Chem., 58, 1128 (1954); (b) A. M. Kotliar and H. Morawetz, J . Am. Chem. Soc., 77, 3692 (1955); (c) H.Morawetz, J . Polymer Sci., 17, 442 (1955); (d) H. Morawetz and E. Sammak, J . Phys. Chem., 61, 1357 (1957). (6) (a) H. P. Gregor, L. B. Luttinger, and E. M. Loebl, ibid., 59, 34 (1955); (b) M. Mandel and J. C. Leyte, J . Polymer Sci., A2,2883, 3771 (1964). (7) 0.V. Brody and R. M. Fuoss, J . Phys. Chem., 60, 156 (1956). (8) (a) 9. Ehrenpreis and M. M. Fishman, Biochem. Biophys. Acta' 44, 577 (1960); (b) 5. Ehrenpreis and M. G. Kellock, ibid., 45, 525 (1960). (9) U. P. S t r a w D. Woodside, and P. Wineman, J . Phys. Chem., 61, 1353 (1957). (10) J. A. Gautier and J. Renault, Compt. Rend., 225, 882 (1947). (11) J. L. Hartwell and M. A. Poporelskin, J , Am. Chem, Sot., 7 2 , 2040 (1950). (12) R. E.Lyle and J. J. Gardikes, ibid., 77, 1291 (1955). (13) F. Kohrnke, Ber., 71B, 2583 (1938). (14) s. Sugasawa and N. Sugimoto, ibid.,72,977 (1939);F. Kohrnke, ibid.,72,2000 (1939);B.Riegel and H. Wittcoff, J . Am. Chem. Soc., 68, 1805 (1946).
2997
ASSOCIATION OF BISPYRIDINIUM CATIONS WITH POLYCARBOXYLIC ACIDS
(the un-ionized isotactic polymeric acid is insoluble in water). The completion of the hydrolytic process was checked by infrared spectroscopy of the neutralized polymeric acid, which showed no absorption a t 1730 cm-I. The stereoregularity of the isotactic poly(methy1 methacrylate) was characterized before hydrolysis by the high-resolution nmr method of Bovey and Tiers;I6 it was found to contain 90% of isotactic triads. The syndiotactic poly(methacry1ic acid) was methylated with diazomethanele to obtain a sample suitable for nmr analysis. It was found to contain 85% of syndiotactic triads. Determination of Dialysis Equilibria. Solutions were prepared containing partially neutralized polyacid with the desired concentrations of sodium bromide and the salt of a cation whose interaction with the polyanion was to be studied. This solution (10 ml) was placed inside a dialysis tube (Fisher Scientific Co. regenerated cellulose); the tube was looped and placed in a bottle with both ends of the tube hanging over the rim. The bottle contained 50 ml of a sodium bromide solution of the same concentration as the inside of the dialysis tube and the levels of the two solutions were essentially identical. The stoppered bottles were thermostated a t 24 f 1" and shaken once a day. About 4 days was required to attain equilibrium. The concentrations of the bispyridinium ions inside and outside the dialysis bag mere determined by optical density measurements a t 259 mp, where the molar extinction coefficient was found to be 4700 per pyridinium residue. The concentration of the bolion in the dialysate was designated (Bel?+), equal to the concentration of the free bolions. The bound bolion concentration (Bok,'+) was obtained as the difference in the concentrations inside and outside the dialysis tube. The chemical stability of the bispyridinium salts was checked by keeping a solution of P2 for 192 hr a t pH 10.2 and 25'. Since Hoff mann degradation with liberation of free pyridine would lead to a 60% reduction in the optical density, the absence of any change in the absorbance of the solution was taken as proof that degradation was negligible even in solutions much more basic than those employed in the dialysis experiments. Adsorption of the bispyridiniiim salts on the dialysis membrane was negligible and satisfactory material balances were obtained from concentrations determined inside and outside the dialysis tube. In experiments in which the dialysis equilibrium of llIg2+ was investigated, the magnesium concentration was determined spectroscopically as the complex with eriochrome blark T."
Results and Discussion Over the range of the experimental conditions studied, the ratio r = (Bolbz+)/(Boltz+) of bound free bolions was found to be independent of the stoichiometric bolion concentration. This is illustrated in Figure 1 for solutions containing P2 and half-neutralized poly(acry1ic acid) (PAA, CY = 0.5) and would be expected as long as the concentration of the anionic sites attached to the polymer chains is in large excess over the concentration of the bolions. Similar series in which the PAA concentration was varied a t a constant degree of neutralization of the polymeric acid showed that r is proportional to the concentration of the polyanions (see Table I), indicating that the chain molecules act as adsorption sites which are independent of each other. The dependence of r on the spacing of the cationic groups in the bolions is shown in Figure 2 for the binding to half-neutralized 0.02 N polyacids in the presence of 0.02 N NaBr. It is striking how the three polymeric acids differ from one another, with the isotactic PMA less efficient and the syndiotactic PMA more efficient than PAA in its binding of the bolaform counterions. Only with PAA does the binding clearly decrease with increasing separation of the charges of the bolions; with i-PhlA the dependence on charge separation is very small and with s-PMA the affinity for the doubly charged cations actually increases slightly with increasing separation of their two charges. The behavior of PAA is the one to be expected on electrostatic grounds; with PMA it may be assumed that hydrophobic bonding is an additional factor and this would become more important as the length of the hydrocarbon chain between the two pyridinium residues
lo4 ( BOL): Figure 1. Equilibrium between bound and free P2 bolions in 0.01 N PAA at a degree of ionization LY = 0.5 and in the presence of 0.02 N NaBr.
(15) F.A. Bovey and G. V. D. Tiers, J. Polymer Sci., 44,173 (1960). (16) 8. Katchalsky and H. Eisenberg, ibid., 6, 145 (1951). (17) A. Young and T . R. Sweet, Anal. Chem., 27,418 (1955).
Volume 70,Number 9
September 1966
H. MORAWETZ AND A. Y.
2998
8-
R
II II I\
6-
r 4-
i -PMA
2-
I
I
I
2
4
No. of
I
I
6
8
IO
BOlr2+ $- (Nabf)z
is being extended. The dependence of the site-binding efficiency on the stereoisomerism of the chain is difficult to interpret. It should be noted, in any case, that the bolion PX, in which the spacing of the pyridinium residues is similar to that to be expected from 1,6-hexamethylenebispyridinium, is bound more tightly to all of the polyanions than expected on the basis of its charge separation. This may be accounted for by the rigidity of the PX structure, which reduces the loss of entropy when the two cationic centers are restrained to lie close to two ionized groups of the polyanions. The extent to which hydrophobic bonding contributes to the binding of PX may be assessed from results obtained with its singly charged analog Q. Under conditions comparable to those illustrated in Figure 2, the binding ratio T for Q is 0.38 with s-PR’IA and 0.26 with PAA. It would be expected that an increase in the concentration of added sodium bromide would reduce the ex-
Table I: The Dependence of P2 Binding to Half-Neutralized PAA on the Normality of Carboxylate Groups Carried by the Polymer Acid ( C p a ) CNaBr
CPU
r
rlCpa
0.02 0.02 0.02 0.04 0.04 0.04 0.06 0.06 0.06
0.02 0.01 0.005 0.02 0.01 0.005 0.02 0.01 0.005
8.03 4.58 2.27 3.29 1.55 0.80 1.02 0.80 0.40
401 458 454 164 155 160 51 80 80
The Journal
of
tent of binding of the bispyridinium ions. The dependence of the binding ratio r on the total Na+ concentration (from the neutralization of the polyacid and from the added salt) was found to be of a very simple form, with r(Na+)2 having a nearly constant value for any given polyacid, degree of neutralization, and bolaform counterion. This generalization is illustrated in Table I1 on the binding of P2 and PX by the halfionized polymeric acids. It suggests that the number of countercharges site-bound near the ionized groups of the polyion is independent of whether these charges belong to univalent ions or to bolions. The equilibrium of this process may, therefore, be represented by
carbon atoms between cationic charges
Figure 2. Dependence of the binding ratio r on the spacing of the cationic charges of bolions P2, P3, P4, PX, and P10. Polymeric acid concentration 0.02 N ; degree of ionization a = 0.5; NaBr Concentration 0.02 N ) .
Physicat Chemistry
KANDANIAN
BOlb2+
+ 2Naf+
(1)
We are representing here the bound sodium ion as (Nab+)z to emphasize that on binding of a bolion two Na+ ions are released from site binding to the same macromolecule. We should note that in the process represented by eq 1 the charge of the polyion with the site-bound counterions remains unchanged. This conclusion is supported experimentally by the observation Table 11: Dependence of the Binding of P2 and PX by Half-Ionized 0.02 N Polymeric Acids on the Concentration of Na + Counterions Bolion
Polymer
P2
PAA
P2
i-PMA
P2
S-PMA
PX
PAA
PX
i-PMA
PX
S-PMA
(Ns, +), N
0.03 0.05 0.07 0.03 0.05 0.07 0.03 0.05 0.07 0.03 0.05 0.07 0.03 0.05 0.07 0.03 0.05 0.07
I
4.58 1.55 0.80 3.02 0.88 0.40 4.15 1.52 0.73 3.85 1.32 0.73 3.34 1.04 0.53 7.45 3.03 1.46
104t(Na +)* 104r(Nay+)*
41.2 38.8 39.2 27.2 22.0 19.6 37.4 38.0 35.8 34.7 33.0 35.8 30.1 26.0 26.0 67.0 75.7 71.5
28.6 31.4 33.8 18.9 17.8 16.9 25.9 30.8 30.8 24.1 26.7 30.8 20.9 21.1 22.4 46.6 61.2 61.7
that addition of small amounts of bolion salts to partially ionized polycarboxylic acids does not lead to an appreciable drop of pH, as would be expected if bolion binding reduced the polyion charge and increased, in consequence, the acidity of the carboxyl groups. Since the number of Na+ counterions bound to a polyanion appears to depend little on the concentration of
ASSOCIATION OF BISPYRIDINIUM CATIONSWITH POLYCARBOXYLIC ACIDS
added uni-univalent electrolyte, r should be inversely proportional to (Nat+)2,rather than the square of the total Na+ concentration, and the last column of Table I1 lists values of r(Naf+)2based on the assumption that half of the charges carried by the half-ionized polymeric acids are associated with site-bound counterions. l9 It may be noted that for a series involving P2 binding, r(Nar+)2 varies within less than 20% for variations of r by factors between 5.7 and 7.5. For the binding of PX, a more nearly constant value is obtained for r(Na+)2. The question now arises as to how the binding ratio depends on the density of anionic groups attached to the polyanion. We may divide the process represented in eq 1 into two stages as shown in Figure 3. I n stage a, one end of the bolion is bound to a fixed charge of the polyion and a bound sodium ion is simultaneously released (not necessarily from the same site). The equilibrium of this stage should depend on the concentration of ionized groups attached to all of the polyions in the system, i.e. (B0l2+b)’( Na+t)/ (Bo12+f) Cpa = &
(2)
where (Bo12+b)’represents the concentration of bolions site-bound to a single polyion charge. However, once a bolion is attached a t one end to a given polyion, the probability of step b depends only on conditions within the domain of that one polyion. The simplest assumption is to set the probability of step b proportional to the fraction a of ionized monomer residues. Since the second bolion charge has to compete with free sodium ions for the binding site, the equilibrium of this step should be of the form
(Bo12+b)”(Na+t)/(Bo12+b)’a = Kz
(3)
where (Bo12+b)I ’ represents the concentration of doubly bound bolions. Assuming that substantially all bound bolions interact with two binding sites, the bolion binding ratio r = (Bol2+b)/(Bol2+r)would be expected to depend on polyion concentration and ionization and on the concentration of free monovalent ions as predicted from
r
=
KCpa2/(Na+r)2
(4)
The dependence of the binding ratio r on the degree of ionization a of the various polymeric acids was measured in a series of experiments in which the polymer concentration was varied inversely with a so as to keep the concentration of ionized carboxyl C p a at a constant value. The results obtained in duplicate determinations of the dialysis equilibrium are listed in Table 111. We may see that the behavior of the various systems investigated does not conform to the prediction
2999
(a)
(b)
Figure 3. Schematic representation of the two stages in the binding of a bolion by a polyion.
of eq 4 but that the dependence of bolion binding on the degree of ionization of the polyion exhibits characteristic differences for the three polymeric acids. I n the case of PAA, r / C p a rises to a maximum at about a =
Table 111: Binding of P2 to Polymeric Acids as a Function of the Degree of Ionization of the Polyanion Polymeric acid
PAA
i-PMA
S-PMA
r/Cw l0Vp
a
(Na+) = 0.03 N
10.00 4.00 2.00 1.66 1.33 4.00 2.00 1.33 4.00 2.00 1.33
0.10 0.25 0.50 0.60 0.78 0.25 0.50 0.75 0.25 0.50 0.75
18212 235 i.13 458h9 523 i t 3 7 462&9 125 i 1 30316 360139 536i.9 415 i. 16 5581:7
(Na+)=
10612 92 =t 1 1 5 4 1 12 17918 143i.5 9i.l SSfO 143 1 5 22312 151 i . 9 1 7 2 i 12
0.05 N
(Na+) = 0.07 N
50i: 1 7912 67&5 3 1 3 40f 1 59f 7 117f4 74f 1 7515
0.6 and decreases somewhat a t higher degrees of ionization. With i-PMA, the bolion is bound extremely weakly a t a = 0.25, particularly a t the higher concentrations of simple electrolyte. This may be related to the low solvation of the polymer which is water insoluble in the un-ionized state. Finally, with s-PMA, r/Cpa changes little as the degree of ionization is changed, with possibly a slight tendency for ~ / C p a :t o (18) I . Alexandrowica, J. Polymer Sci., 43, 337 (1960); 56, 115 (1962). (19) J. R. Huizenga, P. F. Grieger, and F. T. Wall, J . Am. C h m . SOC.,72, 4228 (1950).
Volume YO, Number 9 September 1968
H. MORAWETZ AND A. Y. KANDANIAN
3000
pass through a shallow minimum around a = 0.5. We see then that (with the exception of i-PMA a t low ionization) the bolion binding increases generally more slowly than in proportion to a*. The complicated behavior observed may reflect not only conformational transitions of the polymeric acid@ but also the dependence of the binding of univalent counterions on the degree of ionization. It is of interest to compare the binding of the bolaform cations discussed above with the chelation of a divalent cation such as Mg2+. The characteristics of Mg2+ binding were found to have a surprisingly low sensitivity to the concentration of added sodium bromide. For instance, with 0.02 N half-ionized PAA, the binding ratio r was 13.6, 11.1,and 7.3 for concentrations of Na+ counterions of 0.03, 0.05, and 0.07 N , respectively. The characteristic ratios r/Cpa for the binding of Mg2+ to isotactic and syndiotactic PMA a t various degrees of ionization are listed in Table IV. It was found that r / C P a remains constant for i-PMA as a is increased from 0.25 to 0.50, but drops off for a = 0.75. For the syndiotactic polymer, r/Cpa is doubled as CY rises from 0.25 to 0.50, but decreases again for a = 0.75. At the higher degrees of ionization, the magnesium ion is bound much more strongly to the syndiotactic polyacid. It was pointed out previously that the affinity of i-PMA and s-PJIA for Mg2+is thus in an order opposite to that found for the affinity of these two polymers for Cu2+.21 This specificity seems to indicate that neighboring carboxylate groups along the polymer chain,
Table IV : Binding of Mgz+by Isotactic and Syndiotactic PMA in the Presence of 0.04 N NaBr r/Cpa a
l0Wp
i-PMA
s-PMA
0.25 0.50 0.75
4.00 2.00 1.33
1370 =!= 100 1360 f 60 910 f 70
1510 f 110 3200 =k 180 2350 zk 350
The Journal of Physical Chmistry
with a well-defined mutual spatial relationship, are involved in chelate formation. The relatively small dependence of the binding of Mg*+ by PAA on the concentration of simple electrolyte deserves some comment. The pH of half-neutralized 0.02 N PAA was found to be 5.9 and 5.5 in the presence of 0.02 and 0.06 N NaBr, respectively. Thus, the increase in the ionic strength leads to an increase of the apparent ionization constant of the polymeric acid by a factor of 2.5. It is well understood that this effect is a consequence of the reduction of the excess electrostatic free energy associated with the removal of a hydrogen ion away from the field of the polyanion.22 Since the Mg2+ ion bears a double charge, the association constant of this ion with half-ionized PAA might be expected to be reduced by a factor of (2.fQ2 = 6.25 as the NaBr concentration rises from 0.02 to 0.06 N . This prediction is not a t all in accord with the experimental results, which gave a reduction of r by a factor of less than 2. We should note that the argument based on considerations of the electrostatic free energy assumes implicitly that the binding of an R4g2+ ion by PAA reduced the negative charge of the polyanion by 2 units. It neglects the possibility that binding of doubly charged cations may be accompanied by the release of site-bound univalent cations.23 It is apparent that further progress in the understanding of chelation equilibria involving polyelectrolytes will require experimental evidence bearing on this point.
Acknowledgment. We are grateful to Professor E. M. Loebl for helpful discussions of this work. (20) M. Nagasawa, T. Murase, and K. Kondo, J . Phys. Chem., 69, 4005 (1965); G. Barone, V. Crescenzi, and F. Quadrifoglio, Ric. Sci. 35, (11-A), 1069 (1965). (21) J. J. O'Neill, E. M. Loebl, A. Y. Kandanian, and H. Morawetz, J. Polymer Sci., A3, 4201 (1965). (22) A. Katchalsky and S. Lifson, ibid., 16, 409 (1953). (23) Under the experimental conditions used in this study, the number of hydrogen ions released as a consequence of Mg2 + binding is negligible compared to the number of Mg*+ ions bound.
