1544
J. Phys. Chem. 1982, 86,1544-1551
Xoc,- = 76.3; Doc = DONa+ = 1.33 X
Do* = Docr = 2.03 and v = 1900. The ternary diffusion coefficients estimated from binary transport data by this procedure are listed in Table 11. The agreement with experimental values is reasonable, even encouraging, considering the gross simplificationsz5made in the analysis. Equations can be developed in a similar fashion to estimate from binary data the diffusion coefficients of polyelectrolyte solutions with several added salt comp~nents.~JJ~~~~ Finally, how large is the electric potential gradient &/dx associated with a polyelectrolyte concentration gradient dc/dx? For a dilute salt-free solution of polyelectrolytes X
lob5;Dp = RTXp/(Ffvlzcl)= 1.2 X
(25) Activity coefficients, electrophoretic and time of relaxation effects, changes in f with added salt, viscoeity chan ea of the solutions, and
possible confiiational changes of the polyionF7 are all neglected. (26)D. G. Leaist and P. A. Lyons, J. Phys. Chem., 85, 1756 (1981).
Since fvXoc
>> X P / f u , we obtain
(At 25 "C, RTIF = 0.026 V.) For an aqueous solution of our NaPSSA sample with the concentration decreasing by 50% over a distance of 1cm, a4ldx is about 10 V/cm. For aqueous NaC1, and the same concentration gradient, &$/ax is only -0.004 V/cm. Acknowledgment. Acknowledgement is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for support of this research. We are grateful to Dr. J. Tan of Eastman Kodak Co. for providing purified NaPSSA.
New Electrostatic Model for Calculation of the Energies for Hydration of the Univalent Gaseous Ions B. Thimme Gowda and Sidney W. Benson' Department of Chemistry, W o c a r b o n Research Institute, University of Southern California, Los Angeles, California 90007 (Received: October 27, 198 1)
Potential energies of alkali metal cation-water molecule interactions have been calculated by using a new model for the dipole moment of the water molecule and for a variety of hydrate structures involving from one to six water molecules. The total hydration energy was represented as a sum of electrostatic, polarization, dispersion, and repulsion energy terms. The calculation uses the repulsive parameters and the van der Waals constant computed independently by us to estimate the repulsive and dispersion energy terms. The results are in good agreement with the experimentalgas-phasedata and with other results in the field. Most significantly, a potential function obtained solely from crystal structure data, omitting the dispersion terms, gave just as good fit to the hydration energies. From the log-log plots of hydration energies vs. ionic radii, we have formulated empirical equations for both Pauling's and Ladd's radii for univalent ions with which hydration energies of other simple ions can be predicted if we know their radii. The values so predicted for the negative ions are in good agreement with the experimental values, indicating the consistency of the model.
Introduction One of the key problems in the physical chemistry of ionic solutions has been that of providing a detailed molecular model for the ion-solvent interaction. While the interaction seems predominantly of the ion-dipole type, it has been evident for a long time that polarization and repulsive interactions must play a major role. In the absence of accurate ab initio calculations ( f l kcal), which seem beyond present capabilities, any proposed empirical potential will have a large number of undetermined parameters. With the availability of reasonably accurate gas-phase thermochemical data on ion-solvent interactions of the type M+.(S), with n = 1-6,'J it becomes possible to eliminate some of the arbitrariness in such potential functions. In the present paper we shall outline an approach that examines an empirical potential function chosen to fit a sufficient number of available physical and thermochemical data such that the arbitrariness of the parameters is eliminated. (1)I. Dzidic and P. Kebarle, J. Phys. Chem., 74,1466 (1970). (2)M. Ar~hadi,R.Yamdagni, and P. Kebarle, J. Phys. Chem., 74,1475 (1970). 0022-3654/02/2006-1544$0 1.2510
Ion-Solvent Potential Function Since the pioneering work of Bernal and Fowler3 on molecular models for aqueous solutions there have been many appro ache^^-'^ mostly concerned with predicting heats of solvation of ions. These have had mixed success in correlating AH of solvation with molecular parameters, and all have had difficulties in assigning parameters to the repulsive energy functions. Generally Lennard-Jones potentials or experimentally determined potentials for the inert-gas pair interactions have been used to estimate the repulsive energies. But these are poor approximations subject to large errors. This turned out to be one of the (3)J. D. Bernal and R. H. Fowler, J. Chem. Phys., 1, 515 (1933). (4)D. D. Eley and M. G. Evans, Trans. Faraday Soc., 34,1093(1938). (5)A. D. Buckingham, Disc. Faraday Soc., 24, 151 (1957). (6)J. S.Muirhead-Gouldand K. J. Laidler, Trans. Faraday Soc., 63, 944 (1967);63,953 (1967). (7)I. Eliezer and P. Krindel, J. Chem. Phys., 57, 1884 (1972). (8) F.J. Garrick, Philos. Mag., 9, 131 (1930);10,76 (1930). (9)G.J. Doyne and R. G. Caldwell, Report AD 635,522,Stanford Research Institute, 1966. (10)(a) K. G.Spears and S. H. Kim, J.Phys. Chem., 80,673 (1976); (b) K.G.Spears, J.Phys. Chem., 81, 186 (1977).
0 1982 American Chemical Society
The Journal of Physical Chemistry, Vol. 86, No. 9, 1982 1545
Hydration of Univalent Gaseous Ions
main reasons for the failure of Dzidic and Kebarle's calculations' in estimating hydration energies. Their values for AH(hydration) for one or two water molecules are generally about 70% larger than their own experimental values obtained from mass spectrometric measurements in the gas phase.' They felt it unlikely, and we agree, that the experimental determinations could be in error by such a large margin. It is more likely that some assumptions made in their calculations are inappropriate. (Very recently Sunner and Kebarle" have corrected the values of Dzidic and Kebarle for K+(H20),.) Later Spears12employed the Rittner potential13 to calculate the repulsive potentials of the alkali halide diatomics. Combining these results with independent data on ion-rare-gas scattering, he has estimated the repulsive potentials of the ions. He then evaluated the repulsive potential of water by using K+-H20 experimental potential energy from Dzidic and Kebarle.' The main drawback of his method of selection of the repulsive potentials is that he has used the repulsive potential of K+-K+, obtained by using a geometric mean of the potentials with the rare-gas repulsive potentials. Because these potentials obtained by different groups from the scattering experiments vary widely, it is difficult to decide which repulsive energy is correct. Kim and Spears'O used one set of parameters so obtained to calculate the repulsive energy between the ions and the water molecules and then the hydration energy. Further, some of their assumptions seem questionable. For example, they take the distance between the ion and the water molecule as equal to the distance from the ion nucleus to the center of mass of water. The center of mass of water is, however, not the center of the dipole of water. In spite of these, many of the values1°J2 are in good agreement with the experimental values because they make use of one of the experimental values of AH(so1vation) to adjust their parameters for the calculation. We felt it worthwhile to carry out detailed calculations on ion hydration energies by evaluating separately the repulsive parameters and van der Waals constant. We have also used a new model to represent the total permanent dipole moment of the water molecule. Since the bond dissociation energies and spectroscopic data for the alkali halide molecules, ionization potentials for the alkali atoms, and electron affinities for the halogen atoms are well-known to good accuracy ( f l kcal), we used the Rittner potential V(R)and ita fmt and second derivatives to obtain the repulsive and dispersion parameters A , C, and p. The V(R),includes the electrostatic interRittner p~tential,'~ actions between the ions (charge-charge interaction, charge-dipole interaction, dipole-dipole interaction, and polarization energy stored in the induced dipoles), a repulsion term of the exponential form, a van der Waals attraction term. When the experimental data are used, corrections must be made for kinetic energy terms representing the difference in translational, rotational, and vibrational energies between the molecule and the free ions from which it is composed, including zero-point energy:
V ( R )=
+ a2)e2 --2ala2e2+ A e.p( ~ 4 R7
- e2 - - (a1
R
2
-;)
-
c
R6
V ( R )- 1/,R,T + ( N / 2 ) h v + R,TL(a In Q,/dT) (11)J. Sunner and P. Kebarle, J.Phys. Chem., 85,327 (1981). (12)K.G. Spears, J. Chem. Phys., 57, 1842 (1972). (13)E. S.Rittner, J. Chem. Phys., 19, 1030 (1951).
+
where V(R,) = -(De I - E ) , Q, = (1 - e-hv/kT)-l(the vibrational partition function), (t3V/t3R)Re= 0, and (t32V/t3R2)Re = K , the vibrational force constant for the diatomic molecule with frequency v and reduced mass p , where v = ( 1 / 2 ? r ) ( ~ / p ) ' e/ ~is, the electronic charge, R is the interionic distance, R, is the gas constant, a1and a2 are the polarizabilities of the positive and negative ions, respectively, A is the repulsive potential coefficient, p is the repulsive potential parameter, C is the van der Waals constant, h and FE are Planck's and Boltzmann's constants, respectively, De is the bond dissociation energy to atoms, I is the ionization potential of the alkali atom, E is the electron affinity of the halogen atom, and N is the Avogadro constant. Bond dissociation energies, ionization potentials, and electron affinities were taken from ref 14 and Rosenstock et al.15 Recently, very precise spectral data have become available for the alkali halides. Vibrational frequencies to evaluate the force constanta and the interionic distances were taken from Gordy et al.16 Since there is no close agreement among the polarizabilities computed by different group^,^^-^^ we have used the experimentally deof the alkali halides to termined dipole momenta (pe1pt1)27 calculate the polarizabilities of alkali cations and halide anions from the expre~sion'~ pexptl
= eRe -
R,4e(al + a2)+ 4R,ea1a2 R,6 - 4al~u2
where e, Re, a', and a2have the same meaning as before. While doing so we have fixed the polarizability of the Na+ ion as 0.312 A3 (Pirenne and K a r t h e ~ s e rto ~ ~get ) the polarizabilities of other positive and negative ions. These values were used in the present calculation of A , p, and C. The effect of polarizabilities on the values of the parameters A , p, and C has been discussed in a separate paper.% We used the values of A , p, and C of alkali fluoride to evaluate the repulsion and dispersion energies between the corresponding alkali ion and the oxygen atom of water. While this seems arbitrary, it is not unreasonable because the repulsions and dispersions between the alkali (+) ions and the oxygen atom of water are similar to those between the alkali (+) ions and fluoride ion as the latter and water have the same number of valence electrons. This was unavoidable due to the complexity of the situation. The details of our electrostatic model to predict the total hydration energies will be discussed later. The radii of Fand H 2 0 are also very close. (14)"Handbook of Chemistry and Physics",61st edition, CRC Press, Inc., 1980-1981, Boca Raton, FL. (15)H. M. Roaenstock, K. Draxl, B. W. Steiner, and J. T. Herron, J . Phys. Chem. Ref. Data, Suppl., No. 1, 6 (1977). (16)J. R. Rusk and W. Gordy, Phys. Rev., 127,817(1962);P.Clouser and W. Gocdy, ibid., 134,A863 (1964);S.E.Veazy and W.Gordy, ibid., 138,A1303 (1965);E.Pearson and W. Gordy, ibid., 177,52 (1969). (17)K.Fajans and G. Jws, Z.Phys., 23, 1 (1924). (18)M. Born and W. Heisenberg, Z.Phys., 23,388 (1924). (19)L. Pauling, Roc. R. SOC.(London), Ser. A 114,191 (1927). (20)J. E.Mayer and M. G. Mayer, Phys. Reu., 43,605 (1933). (21)J. R. Teasman, A. H. Kahn,and W. Shockley, Phys. Reu., 92,890 (1953). (22)A. R. Ruffa, Phys. Rev., 130,1412 (1963). (23)J. Pirenne and E. Kartheuser, Physica (Amsterdam), 31, 284 (1965). (24)A. J. Michael, J. Chem. Phys., 51,5730 (1969). (25)J. N.Wilson and R. M. Curtis, J. Phys. Chem., 74,187 (1970). (26)V. G. Solomon&, J. Stmct. Chem., (Engl. Transl.), 19,860(1978). (27)F. J. Lovas and E. Tiemann, J. Phys. Chem. Ref. Data, 3,609 (1974),and references therein. (28)B. T. Gowda and S. W. Benson, J. Phys. Chem., 86,847 (1982).
1546
The Journal of Physical Chemistry, Vol. 88, No. 9, 1982
Separately, we have also estimatedz8the repulsive parameters B and u for alkali halide crystals for the NaCl structure, by employing recently reported values of the lattice energies, which were taken from ref 14 and from
u=--AMe2 + 6 B ex.(-:) R
(
Gowda and Benson TABLE I: Optimized Ion-Oxygen Distances and the Corresponding Induced Dipole Moments crystal parameters gas-phase parameters
=0
where U is the lattice energy for the alkali halide crystals, e is the electronic charge, AM is the Madelung constant,'* and R is the interionic distance in the alkali halide crystals, taken from Smith and Cain29and Pa~ling.~OA factor of 6 appears because each ion has six nearest neighbors. The dispersion energy we felt to be small enough in crystals to omit in our calculations of B and u from the above equations. Since these parameters were obtained from crystal data that neglected the dispersion-energy term, we have also neglected the dispersion energy while calculating hydration energies with these parameters. B and u values of the alkali fluoride crystals have been used in calculating the repulsive energy of the corresponding ion and oxygen atom of water as described earlier. Further, we have also computed the repulsive parameters for alkali halide crystals with the sodium chloride structure by taking into account the dispersion i n t e r a ~ t i o n . Hydration ~~ energies computed from these parameters, as above, are almost the same as the values obtained from repulsive parameters computed by neglecting dispersion interaction. The difference in dispersion and repulsion energy terms between the ions and the water molecules estimated by using the dispersion and repulsive parameters from the crystal data turns out to be approximately the same as the repulsion energy between the ions and water molecules estimated from repulsive parameters obtained by neglecting dispersion energy. So the neglect of dispersion energy while the crystal parameters are used will have little or no effect on the overall values. As will be seen later, the ion hydration energies obtained with our electrostatic model by using these two sets of parameters from alkali fluoride crystals and gas molecules agree with each other and also with the experimental values. Our ability to duplicate the experimental data with the simplest model indicates the value of our electrostatic method and the relative unimportance of the dispersion energies. The main reason for the success of the simple electrostatic treatment appears to be the use of repulsive and dispersive parameters derived from well-established, independent, experimental data and treated in a self-consistent fashion. Most molecular theories of hydration consider the water molecule as a sphere containing a certain number of point charges or as suggested by Buckingham? characterized by its radius, dielectric constant, polarizability, dipole moment, etc. Either a fixed point charge or a point dipole model was used to represent the permanent dipole of the water molecule. In the point charge model, the true behavior of the water molecule was approximated by a set of three point charges and a polarizable sphere. The negative charge was located at the oxygen atom and the positive charges were at the two hydrogen atoms. The magnitudes of the charges were calculated32from the observed dipole moment of water, HOH bond angle, and O-H bond distance. Thus the hydrogen atoms were as(29)C. s. Smith and L. s. Cain, J. Phys. Chem. Solids, 36,205(1975). (30)L. Pauling, 'The Nature of the Chemical Bond", 3rd Edition, Cornell University Press, Ithaca, NY, 1960. (31) (a) F. Hajj, J . Chem. Phys., 44,4618 (1966);(b) R. Narayan, J . Phys. Chem. Solids, 38, 1097 (1977). (32) F. Sheehan, 'Physical Chemistry",Allyn and Bacon, Inc., Boston, 1961.
without dispersion energy
with dispersion energy
Li+.(OH,) Li+,(OH,), Li+,(OH,), Li+.(OH,), Li+.(OH,), Li+.(OH,),
1.79 1.83 1.87 1.90 1.93 1.94
Na+,(OH,) Na+.(OH,), Na+.(OH,), Na+.(OH,), Na+.(OH,), Na+.(OH,),
2.25 2.27 2.30 2.33 2.35 2.37
1.18 1.09 0.96 0.84 0.72 0.59
2.08 2.11 2.15 2.18 2.22 2.23
1.36 1.23 1.07 0.91 0.76 0.62
2.08 2.11 2.15 2.19 2.22 2.24
1.36 1.23 1.07 0.91 0.76 0.62
K+.(OH,) K+.(OH,), K+.(OH,), K+.(OH,), K+.(OH,), K+.(OH,), Rb+.(OH,) Rb+.(OH,), Rb+.(OH,), Rb+.(OH,), Rb+.(OH,), Rb+.(OH,),
2.63 2.64 2.67 2.69 2.71 2.73
0.89 0.85 0.76 0.68 0.60 0.51
2.57 2.59 2.61 2.64 2.66 2.68
0.93 0.87 0.79 0.70 0.61 0.52
2.59 2.61 2.64 2.66 2.69 2.71
0.92 0.86 0.78 0.70 0.61 0.52
2.78 2.80 2.82 2.84 2.87 2.89
0.81 0.76 0.70 0.63 0.56 0.48
2.76 2.77 2.79 2.81 2.84 2.86
0.82 0.78 0.71 0.64 0.56 0.49
2.81 2.82 2.85 2.87 2.90 2.92
0.79 0.75 0.69 0.62 0.55 0.48
Cs+.(OH,) Cs+.(OH,), Cs+.(OH,), Cs'.(OH,), Cs+.(OH,), Cs'.(OH,),
2.94 2.96 2.98 3.00 3.02 3.04
0.73 0.69 0.64 0.58 0.52 0.45
2.97 2.98 3.00 3.02 3.04 3.06
0.72 0.68 0.63 0.57 0.51 0.45
3.10 3.12 3.14 3.17 3.20 3.23
0.66 0.63 0.58 0.53 0.48 0.42
1.76 1.55 1.30 1.08
0.88 0.68
sumed to each carry +0.33 electronic charges while the oxygen atom carried -0.66 charge. In the point dipole model, the total dipole was taken to be a point dipole located on the HOH angle bisector and midway between the oxygen atom and the bisector of a line drawn between the two hydrogen atoms. Method I. In the present calculation we have used a new model for the permanent dipole moment of the water molecule. The HOH angle was taken as 104.5', O-H bond distance as 0.96 A, and the permanent dipole moment of water as 1.85 D. Atomic polarizabilities3 of hydrogen and oxygen were used to represent the polarizability of water. In the present model we split the total dipole moment of water, 1.85 D, into two parts, 1 D for the lone pair of the oxygen atom and the rest (0.85 D)for calculation of the bond charges (31). x =
2
X
0.85 X 10-l8 esu cm X cos 52.25 X
0.96
cm
-
esu The oxygen dipole was taken to be located at the center of the oxygen atom. Each hydrogen atom was assumed to carry +0.723 X esu located at the position of the hydrogen nuclei, while the oxygen atom carried -1.446 X esu located at its center. In all cases the water molecules were considered to be oriented with respect to the ion in such a way that the ion was on the extension of the bisector of the HOH angle and the oxygen atom was 0.723 12 X
(33) J. 0. Hirschfelder, C. F. Curtiss, and R. B. Bird, 'Molecular Theory of Gases and Liquids, Wiley, NY, 1954,p 950.
The Journal of Physical Chemistry, Vol. 86, No. 9, 1982 1547
Hydration of Univalent Gaseous Ions
TABLE 11: Potential Parameters Used To Compute
I
I
Repulsion and Dispersion Energy Contributions between Water and Ions from gas-phase data from crystal data
lOV,
ion
A,eV
p,A
eVcm6
B,eV
o,A
Li+
1401.8 2404.3 2732.7 2875.9 3096.1
0.2504 0.2738 0.2996 0.3071 0.3175
11.49 47.13 87.54 89.88 117.71
951.3 5140.8 13822.2 17092.5
0.2761 0.2561 0.2438 0.2528
Na+
K' Rb+ Cs+
pointing toward the ion. The distance between the center of the ion to the center of oxygen (RIo)was computed by optimizing our potential function (Table I). The distance between the center of the ion to the center of hydrogen (RH) was computed by using the ion-oxygen distance, the HOH bond angle, and the 0-H bond distance. The distance between the ion and the water molecule was taken as being equal to the distance from the center of the ion to the center oxygen plus 0.294 A (RIw). The distances between the water molecules themselves were measured from center to center by assuming the center of the dipole lay on the bisector of the HOH angle and 0.294 A from the center of oxygen atom. We computed the energies for M+(HzO)o,,and M+(HzO),+, where M+ = Li+, Na+, K+, Rb+, and Cs+ and n = 1-6. For clusters with n > 1 a number of structures can be assumed. We used a symmetrical dimer, a planar symmetrical trimer, a tetrahedron, a trigonal bipyramid, and an octahedron for n = 2, 3, 4, 5, and 6, respectively. The ion was located in the center while the water molecules were at the comers of the regular polyhedra mentioned above. The potential energy of a cluster consisting of a central ion and n water molecules relative to the energy of the ion and the water molecules at infinity was computed by = EIODP + EIOB+ EIHB + EIOP+ EIHP+ EDIS+ EREP + EDDP where EIoDp = - poe/RIoz and stands for the energy due to the ion-oxygen permanent dipole attractions ( p o is the dipole moment of oxygen atom due to the lone pair of electrons and e is the electronic charge); EIoB = -2xe/RIo for the ion-oxygen bond charge attractions (-2x is the bond charge on the oxygen atom of water); E m = +2xe/RIH for the ion-hydrogen bond charge repulsions (+x is the bond charge on each hydrogen atom of water); EIoP= -aoe2/ 2R104 and EIHP= -aHez/RIH4for the ion-induced dipole attractions (polarization energy) (a0and aHare the atom polarizabilitiess of oxygen and hydrogen respectively);EDLs = C/RIo6 for the dispersion energy between the ion and the oxygen atom, ignoring the negligible dispersion between the ion and the hydrogen atoms and water-water dispersion forces (the constant C was used from the corresponding alkali fluoride as described earlier); and EREp = A exp(-RIo/p), the energy due to the ion-oxygen electronic repulsions (the negligible repulsions between the ion and hydrogen atoms and water molecules themselves were ignored). The repulsive and dispersive parameters used are listed in Table 11. EDDp is the energy term representing dipole-dipole repulsion forces. The dipole-dipole repulsion forces are those due to permanent dipole and induced dipole repulsions. We have computed the induced dipole moment by taking into account the total electric field at one water molecule in terms of the ionic charge and the permanent and induced dipole moment of all other water molecules (Table I). To simplify the calculation, point dipole and
Figure 1. Plot of A€,,, as a function of ion-oxygen distances (RIo).
point polarizability of water was used to calculate the dipole-dipole interactions. However, dipole-dipole repulsions for Li+ clusters were computed by taking contributions due to permanent and induced dipole moments of oxygen, induced dipole moments of hydrogen atoms, and bond charges on oxygen and hydrogen. The value so obtained for EDDp is close to the value obtained by using the complete dipole on water as a whole. So we did not continue the latter calculations with the other clusters. The ion-oxygen distances computed by minimizing our potential functions increased as we go from clusters with n = 1to n = 6 (Table I) and become almost equal to the sum of the Pauling30 or Ladd34ionic radii of the ion and oxygen with n = 5 or 6. This is not surprising as the ionic radii have been computed for the coordination number 6. This also supports our choice of potential function. The ion-oxygen distances increase as we increase n because as the water molecules become crowded they are pushed apart due to dipole-dipole repulsions. I t is also interesting to note that the potential energies around the minimum energies are not so sensitive to the ion-oxygen distances (Figure l),changing the latter by 0.1 A either way changes the energy by little (
