Linear relationships between acidity and stability in mono- andhexahydrated metal ions: a computational study

Philip George a, Jenny P. Glusker a,*, Mendel Trachtman b, Charles W. Bock a,b

a The Institute for Cancer Research, Fox Chase Cancer Center, 7701 Burholme Avenue, Philadelphia, PA 19111, USAb Department of Chemistry, Philadelphia University, Henry Avenue and School House Lane, Philadelphia, PA 19144, USA

Received 19 July 2001; in final form 9 November 2001

Abstract

Linear relationships have been established between the dissociation enthalpies of the monohydrates of Liþ, Naþ, Kþ,

Be2þ, Mg2�, Ca2þ, Mn2þ, Fe2þ, Co2þ, Ni2þ, Cu2þ, Zn2þ, Al3þ, Sc3þ, Ti3þ, V3þ, Mn3þ, Fe3þ and Ga3þ calculated at the

MP2(FULL)/6-311++G**//MP2(FULL)/6-311++G** level, the hexahydrates of Naþ, Kþ, Mg2þ, Ca2þ, Mn2þ, Zn2þ,

Al3þ, Sc3þ, Fe3þ and Ga3þ calculated at the MP2(FULL)/6-311++G**//HF/6-311++G** level, and experimentally

reported pKa values for the first ionization of the corresponding aqua-ions. In contrast to the results of Chang and

Wang [Chem. Phys. Lett. 286 (1998) 46], we find that the 3þ ion hexahydrates are more (not less) stable than the 2þ ion

hexahydrates. � 2002 Elsevier Science B.V. All rights reserved.

In a recent Letter [1] Chang and Wang reporteda linear correlation between the experimental val-ues of pKa for the ionization of the di- and triva-lent aqua-ions [2],

Mnþaq þH2O!Mnþ½OH��aq þHþ

and the binding energies, Eb, for the correspondinghexahydrated (symmetry Th) cations, M

nþ½OH2�6,calculated using density functional theory. Thebinding energy was defined as �ðEt � EaÞ where Etis the total energy of the molecule and Ea the sumof the atomic energies. An example of the rela-tionships they present is

pKa ¼ �13:0837þ 0:0216Eb ðr ¼ 0:965Þ;and they concluded that this correlation confirmsthe existence of a global linear relationship forboth main and transition group hexahydratedmetal ions.In addition to demonstrating linearity using

local (VWN) and nonlocal (BP and BLYP) densityfunctionals [3–6], they found that the hexahy-drated ions of divalent Mg, Ca, Mn, Zn and Cdwere more stable (larger value of Eb) than thehexahydrated ions of trivalent Al, Sc, Cr, Fe, Gaand In, i.e.

EbðM2þ½OH2�6Þ > EbðM3þ½OH2�6ÞThis result is surprising because their calculatedM–O distances were consistently smaller for thetrivalent ion hexahydrates, which is usually

17 January 2002

Chemical Physics Letters 351 (2002) 454–458

www.elsevier.com/locate/cplett

* Corresponding author. Fax: +1-215-728-2863.

E-mail address: jp_glusker@fccc.edu (J.P. Glusker).

0009-2614/02/$ - see front matter � 2002 Elsevier Science B.V. All rights reserved.

PII: S0009-2614 (01 )01399-9

indicative of stronger, and not weaker, binding [7].Classical electrostatic theory would also predictthe trivalent hexahydrates to be the more stable. Interms of the Born equation

Eb ¼Z2

2r

� �1

�� 1

D

�

the binding energy is proportional to Z2, where Zis the charge on the ion, r the radius, and D thedielectric constant [8–10]. Hence, assuming theionic radii to be the same, the ratio of the bindingenergies for the tri- and divalent hexahydrateswould be 9/4.To investigate further this question of the sta-

bility relationship, we decided to compare resultsfor monohydrates with those for hexahydrates.Taking enthalpies of dissociation 1, DH 0

298 (cal-

culated at the MP2(FULL)/6-311++G**//MP2(FULL)/6-311++G** level using GAUSSIAN

98 [11,12]) as a measure of the stability of the(symmetry C2v) monohydrates of Li

þ, Naþ, Kþ,Be2þ, Mg2þ, Ca2þ, Mn2þ, Fe2þ, Co2þ, Ni2þ, Cu2þ

and Zn2þ [7], augmented with additional resultsfor Al3þ, Sc3þ, Ti3þ, V3þ, Mn3þ, Fe3þ and Ga3þ, wefind the expected trend for dissociation, i.e.

DH 0298ðMþ½OH2�Þ < DH 0

298ðM2þ½OH2�Þ< DH 0

298ðM3þ½OH2�Þ

as the calculated M–O distances decrease. In ad-dition there is a good correlation between thesecalculated dissociation enthalpies and the experi-mental pKa values [2],

pKa ¼ 15:97� 0:0720DH 0298 ðr ¼ �0:975Þ;

see Fig. 1. The enthalpies of dissociation for thehexahydrates (Th) of Mg

2þ, Ca2þ, Mn2þ, Fe2þ,Zn2þ and Al3þ, Sc3þ, Fe3þ and Ga3þ were calcu-lated at the MP2(FULL)/6-311++G**//HF/6-311++G** level. Vibrational frequency analysesconfirmed that these Th structures are local minimaon their HF/6-311++G** potential energy surfaces.

1 For a series of hydrates with the same number of water

molecules the summation of their energies of atomization is a

common term in the computation of binding energies, Eb, asdefined by Chang and Wang [1]. Hence dissociation energies

(enthalpies) for the hydrates, Mnþ½OH2�m !Mnþ þ mH2O,

give the same measure of stability as Eb.

Fig. 1. Linear relationship between the calculated enthalpies, DH 0298, for the dissociation of the monohydrates and the experimental

values of pKa for the first ionization of the corresponding aqua-ions.

P. George et al. / Chemical Physics Letters 351 (2002) 454–458 455

We find the dissociation enthalpies for the triva-lent ions to be consistently larger than those forthe divalent ions, i.e.

DH 0298ðM2þ½OH2�6Þ < DH 0

298ðM3þ½OH2�6Þin complete accord with the results for the mono-hydrates, see above, but at variance with thosereported by Chang and Wang [1]. This findingraises the possibility that there is a mistake of somekind in their computation of Eb for the divalentand trivalent hexahydrates. We suspect that anincorrect sign may have been used in a summationof thermal data.We have also carried out similar calculations of

the dissociation enthalpies of the monovalenthexahydrates Naþ½OH2�6 and Kþ½OH2�6. The Thform of Naþ½OH2�6 is a transition state on the HF/6-311++G** potential energy surface so thissymmetry constraint was relaxed to obtain anappropriate local minimum (S6) [13]. As shown inFig. 2 the values of DH 0

298 plotted versus pKa are inaccord with a good linear relationship no matterwhether the regression analysis is based only onthe data for the 2þ and 3þ ions, i.e.

pKa ¼ 16:86� 0:02082DH 0298 ðr ¼ �0:943Þ;

or whether the data for the 1þ ions are also in-cluded, i.e.

pKa ¼ 16:43� 0:02001DH 0298 ðr ¼ �0:964Þ:

The ionization reaction. A consideration ofthe corresponding metal ion hydroxides (as op-posed to hydrates) relative to the overall ion-ization reaction shows that the stability of thehydroxide plays a dominant role in determiningthe acid strength of the metal ion [14]. Forexample, in the case of the ionization of themonohydrate

Mnþ½OH2� !Mnþ½OH�� þHþ

ðmonohydrate ionizationÞ

the thermochemical summation

Mnþ½OH2� !Mnþ þH2O

DH 0298 ðmonohydrate dissociationÞ

H2O! OH� þHþ DH 0298 ðwater ionizationÞ

Fig. 2. Linear relationship between the calculated enthalpies, DH 0298 for the dissociation of the hexahydrates and the experimental

values of pKa for the first ionization of the corresponding aqua-ions. The line is based on the data for all the ions, see text.

456 P. George et al. / Chemical Physics Letters 351 (2002) 454–458

Mnþ þOH� !Mnþ½OH��DH 0

298 ðmonohydroxide formationÞ

gives

DH 0298 ðmonohydrate ionizationÞ¼ þDH 0

298 ðmonohydrate dissociationÞþ DH 0

298 ðwater ionizationÞþ DH 0

298 ðmonohydroxide formationÞ

Since dissociation reactions are bond breakingprocesses, enthalpies of dissociation are positiveentities. The value of DH 0

298 for monohydrate dis-sociation thus makes an adverse (positive) contri-bution to DH 0

298 for monohydrate ionization,whereas the value of DH 0

298 for monohydroxideformation makes a favorable (negative) contribu-tion. Calculated values for the monohydrates [7]and monohydroxides [14] of Liþ, Naþ, Kþ, andBe2þ; Mg2þ; Ca2þ; Mn2þ; Fe2þ; Co2þ; Ni2þ;Cu2þ,and Zn2þ are listed in columns two and threeof Table 1. The value of DH 0

298 for monohydroxidedissociation is greater than the value of DH 0

298

formonohydrate dissociation, making the enthalpydifference fDH 0

298 ðmonohydrate dissociationÞ �DH 0

298 ðmonohydroxide dissociationÞ ¼ DH 0298 ðmono-

hydrate dissociationÞ þ DH 0298 ðmonohydroxide

formationÞg negative in all cases, see column 4.Since DH 0

298 for water ionization is the same for all

the ions (see above) this enthalpy difference (incolumn 4 of Table 1) determines the relative valuesof DH 0

298 for monohydrate ionization. For the 1þions the differences are appreciably smaller thanthose for the 2þ ions with mean values of )135and )318 kcal/mol, respectively, thereby indicatingthat the 2þ monohydrates are significantlystronger acids than the 1þ monohydrates. It maybe noted that although the span of pKa values forthe corresponding aqua-ions is far smaller, seecolumn 5, the 2þ aqua-ions are also the strongeracids.In conclusion these results substantiate the ex-

pectation that there is a global relationship be-tween acidity and the stability of either themonohydrated or hexahydrated metal ions. How-ever, we find that the relevant linear equation has anegative slope, and not the positive slope proposedby Chang and Wang [1].

Acknowledgements

We thank the Advanced Scientific ComputingLaboratory, NCI-FCRF, for providing super-computer time. This work was supported by grantsCA10925 and CA06927 from the National Insti-tutes of Health, and by an appropriation from theCommonwealth of Pennsylvania. The contents ofthis Letter are solely the responsibility of the

Table 1

Values of DH 0298 for the dissociation of the monohydrates [7] and the formation of the monohydroxides [14], and the sum

fDH 0298 ðhydrate dissociationÞ þ DH 0

298 ðhydroxide formationÞg, in kcal/mol, calculated at the MP2(FULL)/6-311++G**//

MP2(FULL)/6-311++G** level, together with experimental pKa values for the corresponding aqua-cations [2]

Cation DH 0298

(hydrate dissociation)

DH 0298

(hydroxide formation)

DH 0298 ðhydrate dissociationÞ

þDH 0298 (hydroxide formation)

pKa

Liþ 34.0 )187.0 )153.0 13.64

Naþ 23.6 )154.2 )130.6 14.18

Kþ 16.7 )139.2 )122.5 14.46

Be2þ 140.5 )505.5 )365.0 5.4

Mg2þ 77.5 )374.5 )297.0 11.44

Ca2þ 51.9 )330.5 )278.6 12.85

Mn2þ 75.6 )383.0 )307.4 10.59

Fe2þ 85.5 )403.9 )318.4 9.5

Co2þ 96.6 )416.4 )319.8 9.65

Ni2þ 92.9 )413.8 )320.9 9.86

Cu2þ 102.3 )433.2 )330.9 8.0

Zn2þ 94.7 )421.1 )326.4 8.96

P. George et al. / Chemical Physics Letters 351 (2002) 454–458 457

authors and do not necessarily represent the offi-cial views of the National Cancer Institute.

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