Estimation of the Solubility Dependence of Aluminate Salts of Alkali

solubility of lithium aluminate with its structure, as reflected by the ion radius. For this purpose, local density functional. (LDF) molecular orbita...
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J. Phys. Chem. 1996, 100, 6531-6542

6531

Estimation of the Solubility Dependence of Aluminate Salts of Alkali Metals on Ion Radii of Alkali Metals by LDF Molecular Orbital Calculations Toshiaki Matsuo,*,† Kinya Kobayashi,‡ and Kazutami Tago‡ Power & Industrial Systems R&D DiVision, Hitachi Ltd. 7-2-1 Omika-cho, Hitachi, Japan, and Hitachi Laboratory, Hitachi Ltd. 7-1-1 Omika-cho, Hitachi, Japan ReceiVed: October 26, 1995; In Final Form: January 29, 1996X

The addition of a lithium salt forms an insoluble lithium aluminate salt film on an aluminum surface, effectively preventing aluminum corrosion. Other aluminate salts of alkali metals lack this behavior, being easily dissolved into water. In this study, the reason for the difference in behavior is investigated by applying molecular orbital calculation to clusters of ions and molecules. The difference in solubility among aluminate salts of alkali metals is attributed to the difference in binding energy between the anion cluster and the alkali metal ion cluster with the first layer of water molecules surrounding the alkali metal ion. Calculations show that this difference of binding energy arises from the difference in the sum of the cohesive energy, the variation of the coordinate bond energy between an alkali metal ion and water molecules surrounding it, and the interaction energy between those water molecules and an anion, in a salt molecule cluster. These three factors, in turn, derive from the difference in ion radius of the alkali metals. The binding energy in lithium aluminate is about 25 kJ/mol larger than that in sodium aluminate and about 45 kJ/mol larger than in potassium aluminate. The binding energy of antimonate salts of alkali metals is also calculated as a second example in which the value of the ion radii causes the insolubility, to verify the explanation. Then, the binding energy in the sodium antimonate is about 5 kJ/mol larger than in the lithium antimonate, and about 15 kJ/mol larger than that in the potassium antimonate. The ratio of the lithium ion radius to the aluminum ion radius is 1.4, nearly as much as that of the sodium ion radius to the antimony ion radius. This value of ion radii ratio made the binding energy largest of the aluminate or antimonate salts.

1. Introduction Cement solidification is a conventional method to treat many kinds of radioactive wastes generated from nuclear power plants and to produce waste packages.1 When these wastes include aluminum materials, the latter corrode in the highly alkaline conditions of the cement paste because of dissolution of the thin Al2O3 film on the aluminum surfaces as follows

Al2O3 + 3H2O + 2OH- f 2Al(OH)4-

(1)

The dissolution leads to the oxidation of metallic aluminum and generation of hydrogen gas. When the wastes include many aluminum materials, this gas generation causes voids and cracks in the waste packages and reduces the compression strength of the waste products. It has been reported that an addition of LiNO3 to the cement prevents alkaline corrosion of aluminum, because lithium ions of LiNO3 form an insoluble lithium aluminate salt film on the aluminum surface which protects the Al2O3 layer from dissolution.2 But it is not known why only lithium, among the alkali metals, can produce the insoluble aluminate salt. If lithium addition is most appropriate for the formation of the insoluble salt, the reason for it should be clarified. The purpose of this study is to relate theoretically the low solubility of lithium aluminate with its structure, as reflected by the ion radius. For this purpose, local density functional (LDF) molecular orbital calculations3-6 were applied to the aluminate salts of alkali metals to estimate the difference in their binding energy as reflected by the difference in the alkali metal ion radius. In section 2.2, methods used to estimate the binding energy by molecular orbital calculation are explained. X

Abstract published in AdVance ACS Abstracts, March 15, 1996.

0022-3654/96/20100-6531$12.00/0

Then, how this difference in the binding energy leads to the difference in their solubility was discussed by using some classical theories of solvation, entropy, and electric interaction, which are introduced in section 2.3. The calculation results are described in section 3. The results of the solubility estimation are discussed in section 4.1. Then, the most appropriate corrosion inhibitor for aluminum is discussed on the basis of these results in section 4.2. 2. Computational Methods and Models Generally, when the solid phase and the aqueous phase are in equilibrium the chemical potentials in these two states are equal. The aqueous phase of ionic crystals in a solution is considered to be that of cations and anions surrounded by solvent molecules, and these ions are dispersed uniformly and move freely in the solvent. The dissolution of the solute goes on until the difference in the chemical potentials of the two state reaches 0. When the ionic crystal consists of one kind of cation and anion, the difference between the free energy per mole of the solid phase and that of aqueous phase, with the concentration of the solute in the solution is described as follows:7

∆µ ) ∂ Gsolid/∂ n - ∂ Gaqueous/∂ n - 2RT ln C ) 0 (2) The third term in this equation is attributed to the entropy of mixing, and the concentration of the cations or the anions in the solution is denoted as C, which corresponds to the solubility of the solid. The solubility of the solute can be obtained by the modification of eq 2:

C ) exp{-(∂ Gsolid/∂ n - ∂ Gaqueous/∂ n)/2RT} (mol/mol solvent) (3) © 1996 American Chemical Society

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Matsuo et al.

Figure 1. Model for theoretical estimation of solubility.

2.1. Contributors to the Solubility of Salts. In this study, the content of the free energy is considered on the assumption that the terms in the numerator in the exponent of eq 3, ∂Gsolid/ ∂n and ∂Gaqueous/∂n, can be described as follows:

∂Gsolid/∂n ) (enthalpy in the solid phase) T·(entropy in the solid phase) Z (enthalpy of one salt molecule) + (electric interaction energy between salt molecules) - T· (entropy of one salt molecule) - T· (entropy of salt molecule alignment in the solid phase) (4a) ∂Gaqueous/∂n Z (enthalpy of ions) - T· (entropy of ions) - T· (entropy of ion alignment in the aqueous phase) + (electric interaction free energy between ions) + (free energy of Born-type ion hydration) (4b) In eq 4a, the free energy in the solid phase is considered as the sum of the free energy of one salt molecule and the electric interaction energy between the salt molecules because the crystal structure of the lithium aluminate preservation film on the surface layer facing the solvent is not known exactly. The above interpretation of the free energy in the solid phase allows molecular orbital calculation to be applied to the salt molecule clusters and to estimate the electric interaction energy in the ionic crystal as the electric interaction energy between the salt molecules. From eqs (4a and 4b), the difference between ∂Gsolid/ ∂n and ∂Gaqueous/∂n is obtained:

∂Gsolid/∂n - ∂Gaqueous/∂n ) ∆H - T(∆Scom + ∆S*) + (∆GBorn - Φ) - ∆GDebye-Huckel (5) Notations in eq 5 are (Figure 1) as follows: ∆H, difference of the enthalpy between one salt molecule and ions (one cation and one anion); GDebye-Huckel, free energy of electric interaction among ions in the aqueous phase; ∆GBorn, free energy of Borntype ion hydration in the aqueous phase; Φ, energy of electric interaction of one salt molecule with the others in the solid phase; T∆Scom and T∆S*, energy accompanied by the entropy

variation between one salt molecule and ions (∆Scom, entropy variation derived from the difference of the distribution function about alignment between solid phase and aqueous phase, communal entropy; and ∆S*, entropy variation derived from the difference of the degree of freedom between solid phase and aqueous phase, activation entropy). From eq 3, the large difference in free energy between the solid phase and the aqueous phase leads to the insolubility of the solute. In this study, it was considered that the insolubility of lithium aluminate in water compared with other aluminate salts of alkali metals (i.e. the large difference of ∂Gsolid/∂n ∂Gaqueous/∂n between lithium aluminate and other aluminate salts of alkali metals) contributes to the large difference in ∆H among them, and that the large difference in ∆H is caused by the ratio of the ion radii between the alkali metal and aluminum leading to the strength of ionic bond between them, because one of the biggest differences between lithium ions and other alkali metal ions seems to be their ion radii (the ratio of rLi+/rAl3+ ) 1.4, rNa+/rAl3+ ) 2.1, rK+/rAl3+ ) 2.9). So, it was expected that lithium aluminate has a unique structure and a strong ionic bond between lithium ions and aluminate ions compared with the other aluminate salts of alkali metals. To estimate the difference in the binding energy among the aluminate salts caused by the difference in the alkali metal ion radius, LDF molecular orbital calculations3-6 were applied to multiatomic clusters of aluminate salt molecules (LiAl(OH)4, NaAl(OH)4, and KAl(OH)4), hydrated alkali metal ions, and aluminate ions. The discussion then focused on how this difference in the binding energy leads to the difference in their solubility, by using some classical theories of solvation, entropy, and electric interaction. In addition, the binding energy of antimonate salts of alkali metals was also calculated as another example in which it is considered that the ion radii determine the solubility of the solute (rLi+/rSb5+ ) 1.0, rNa+/rSb5+ ) 1.5, rK+/rSb5+ ) 2.1), to confirm the generalization of the phenomenon, where sodium salt has the least solubility in water among antimonate salts of alkali metals. 2.2. Methods To Estimate the Binding Energy by Molecular Orbital Calculations. The LDF molecular orbital calculation method applied in our investigation was ESPACDF developed by Kobayashi et al.,3-6 on the basis of the KohnSham self-consistent-field method. In this calculation method, the local density-functional method is applied to the determination of cluster structures, and the generalized gradient approximation is applied to the energy calculations. The molecular orbital in a molecule, ψm(R), is expressed by the linear combinations of the Slater-type atomic orbitals, {χq(r),q)1,K},

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like ψm(R) ) ∑qCqmχq(R), and each orbital energy m and the set of coefficients Cqm are obtained by calculation of the eigenvalue equation FC ) SC. The elements of these matrices are given as follows.

Fpq ) ∫ db r χp(b) r H(b) r χq(b) r ) N

∑ ∑ ωa(i) Ωa(i) χp(i) H(i) χq(i)

(6)

a)1 i)1 N

Spq ) ∑ ∑ ωa(i) Ωa(i) χp(i) χq(i)

(7)

a)1 i)1

H(b) r )-

∆r 2

N

Za

a)1

|b r -B Ra|

-∑

b + ∫ dx

F(x b) |x b - b| r

r (8) ηxc[F(b)]

where H(i) is a Hamiltonian, ωa, Ωa are weighting coefficients, Fpq is the resonance integral, Spq is the overlap integral, F is electron density, and η is the exchange potential. In our calculations, triple atomic orbitals (neutral atomic orbital, one 2s, and one 2p orbitals) were chosen for hydrogen atoms, double atomic orbitals (neutral atomic orbitals and ionized atomic orbital) with one 3d orbitals for oxygen atoms, and double atomic orbitals for other elements. Also, the pseudopotential approximation with relativistic effects taken into account was applied to the orbitals of the antimony atom except the 5s and 5p orbitals. A total of 6496 of fine mesh points per atom was used for integration. Although it is often said that the use of LDA for geometry optimization is likely to lead to shortened distance and over-binding, our calculations evaluate the energy of hydrogen bond within the accuracy of the experimental errors (2.94 kJ/mol Z 1 × 10-3 hartree),5,6 and the distance of hydrogen bond, between two oxygen atom (OsH‚‚‚O), within the accuracy of the experimental errors, too.8 That is because the gradient of potential between H‚‚‚O is much milder in hydrogen bonds than in the other kinds of bonds so that the very small error of the distance between them does not matter. Enthalpy values of clusters (salt molecules, cations, and anions) were obtained by molecular orbital calculations to evaluate the binding energy between a cation and an anion in a salt molecule, as the difference of enthalpy between the solid phase (the salt molecule) and the aqueous phase (the cation and the anion). An aluminate ion and an antimonate ion were calculated as Al(OH)4- and Sb(OH)6- which are the forms in which they exist in the aqueous solution.9,10 The initial geometries of Al(OH)4- and Sb(OH)6- for calculations were a tetrahedron made of four oxygen atoms placing one aluminum atom in its center, and a octahedron made of six oxygen atoms placing one antimony atom in its center, as known well. In these geometries, each hydrogen atom binding to these oxygen atoms faced in the opposite direction against the centered atom with leaning to one of the neighboring oxygen atoms, or to the center of a pair of the neighboring two oxygen atoms. The enthalpies of cations were calculated in the hydrated forms, i.e. with one layer of water molecules surrounding the alkali metal ions, Li(OH2)4+, Na(OH2)4+, K(OH2)6+,11,12 which included the energy of the coordinate bonds between the alkali metal ions and the water molecules. The initial geometries of Li(OH2)4+, Na(OH2)4+, and K(OH2)6+ for calculations were tetrahedrons made of four oxygen atoms placing a lithium or sodium atom in their center, and that of K(OH2)6+ was a octahedron made of six oxygen atoms placing a potassium atom in its center. In these geometries, two hydrogen atoms in each water molecule

were directed against the centered atom with the face of the H2O triangle to its side, and with leaning of the H2O triangle to one of the neighboring oxygen atoms, or to the center of a pair of the neighboring two oxygen atoms. The enthalpies of salt molecules were calculated with these coordinate bonds between the alkali metal ions and the water molecules, too. The initial geometries of the salt molecules were pairs of aluminate (or antimonate) ion and one of these three hydrated alkali metal ions with their structure obtained with the above calculations. In these geometries, the hydrated alkali metal ion faced every possible side of the aluminate tetrahedron (or antimonate octahedron), or faced every possible face. Among many structures of each cluster we tried, we decided the structures with the lowest enthalpies as consequences. We expected that some hydrogen bonds of the hydrated alkali metal ion and an anion (an aluminate or an antimonate ion) with water molecules around them would be broken by the formation of a salt molecule, so that the energy of these broken hydrogen bonds was subtracted from the calculated binding energy between the cation and the anion in the salt molecule. After that, we explained the binding energy between them from the results of molecular orbital calculations in relation to the difference in the salt molecule structures, especially the distance between the cation and the anion in the salt molecule, caused by the different ionic radii. 2.3. Methods To Estimate the Other Contributors. All of the terms except for ∆H in eq 5 were theoretically estimated (Table 1) so that we could discuss whether the difference of ∆H really determines the difference of solubility of both aluminate salts and antimonate salts. Evaluation of each term in eq 5 is explained as follows. 2.3.1. Electric Interaction of One Ion with Ions Surrounding It in Solution. The electric interaction of one ion with ions surrounding it in a dilute solution can be estimated from DebyeHuckel theory,13 where the free energy for the distribution of ions around i-ion in the solution governed by a MaxwellBoltzmann distribution, is obtained per 1 mol of i-ions as follows:

∆GDebye-Huckel ) -

zi2e2NA κ 2 1 + κai

κ2 )

4πe2N ∑cizi2 1000 kB T (9)

where NA is Avogadro’s number, e is electron charge, ci is the concentration of solute (mol/L), zi is the charge number of nuclei, ai is ion radius, and kB is Boltzmann constant. 2.3.2. Free Energy of Born-type Ion Hydration in the Aqueous Phase. For a solvent like water with a high dielectric constant which reflects the electric dipole moment of the solvent molecule, a polarized charge is generated on the boundary surface between ions and the solvent molecules. Because of the electric interaction between the ions and that polarized charge, the free energy of the aqueous phase is lowered. This variation of the free energy can be evaluated approximately by Born’s theory,14 in which the free energy variation is taken for the electric energy variation between 1 mol of ions in a vacuum and those in a medium with the relative dielectric constant . The variation of the free energy can be obtained from the following equation:

NAzi2e2 1 -1 ∆GBorn ) 2ri 

(

)

(10)

In eq 10, ri corresponds to the radius of the cavity formed by the presence of i-ion.

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TABLE 1: Interactions to Which the Energy Variation Is Attributed to Both Solid and Aqueous Phases interactions

energies to be estimated

methods

a.Solid State between cation and anion between salt molecules entropy variation between ions between ions and solvent entropy variation

binding energy electric potential energy

molecular orbital calculation linear combination of electric potential

b.Liquid State electric potential energy between ions solvation energy entropy of commonality activation entropy entropy of mixing

Debye-Huckel’s theory Born’s theory Kirkwood’s theory theory of absolute reaction rate in eq 2

Usually, the relative dielectric constant of the first layer of water molecules surrounding an alkali metal ion is about 5, significantly different from that of pure water ()79.5, which can be considered the relative dielectric constant in the second layer of water molecules and in all layers there after) because the water molecules lose a degree of freedom in a strong electric field in the first layer.15 It has been reported that, by taking this double-layer model of the water molecules around alkali metal ions into account, theoretical estimation succeeded in the free energy of Born-type ion hydration with an accuracy of 2%.16 This double-layer model does not have to be taken into account to estimate the free energy of Born-type ion hydration around anions because there are no coordinate bonds.16 In this study, we calculated all the salt molecules and alkali metal ions with the first layer of water molecules surrounding the alkali ions, so that eq 10 could be applied to the solvent from the second layer. The positions of water molecules in the second layer around the alkali metal ions were decided by molecular orbital calculations, and we considered that the region of an isotopic solvent with the relative dielectric constant  starts from the oxygen atom’s position in a water molecule in the second layer. The values of ri for the alkali metal ions were defined as the distance between the position of the alkali metal ion and the position of that oxygen atom in the second layer of water molecules (Figure 2a). Outside the alkali metal ion with one layer of water molecules, the position of the oxygen atom in the second layer of the water molecules is determined on the extrapolated line from the oxygen atom through a hydrogen atom in one water molecule of the first layer. The distance between the oxygen atom in the second layer of the water molecules and the hydrogen atom in the first layer of the water molecules is equal to the distance of hydrogen bond. The effect of the hydrogen atoms is weakened by an oxygen atom in the third water molecule layer and all subsequent layers. Consequently, the effect of the polarized charge on the surface of the dielectric substance (i.e. the solution) comes from the oxygen atoms in the second layer of water molecules. To estimate the free energy of Born-type ion hydration around the aluminate and the antimonate ions, the positions of water molecules in the first layer around them, which have hydrogen bonds with the aluminate and antimonate ions, were obtained from molecular orbital calculations. And we considered that the region of an isotopic solvent with the relative dielectric constant  starts from the hydrogen atom’s position of a water molecule in the first layer. The values of ri for the aluminate and the antimonate ions were defined as the distance between an aluminum or an antimony atom and a hydrogen atom (Figure 2b,c). 2.3.3. Electric Interaction Energy between Salt Molecules in the Solid Phase. In the solid phase, the electric interaction energy between salt molecules has to be taken into consideration. This electric interaction energy Φ of the salt molecule on the surface of a solid was estimated using the assumed alignment

Figure 2. Methods to determine the radius of cavity: (a) hydrated alkali metal ion; (b) aluminate ion; and (c) antimonate ion. The distance parameters are given in angstroms (1 Å ) 0.1 nm).

of the salt molecules shown in Figure 3. The symbol d corresponds to the distance between the alkali metal ion and

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J. Phys. Chem., Vol. 100, No. 16, 1996 6535

Figure 3. Model for estimation of electric interaction energy between salt molecules in the solid phase.

the aluminum or antimony atom in the salt molecule, and the symbol d0 corresponds to the distance between the salt molecules, which is defined as the maximum sum of the distance between the alkali metal atom and the hydrogen atom of a water molecule in the first layer surrounding the alkali metal ion, the distance between the aluminum or antimony atom and an oxygen atom in the aluminate or antimonate ion, and the length of the hydrogen bond. δi is defined as 1 when i is even and -1 when i is odd. The energy of electric interaction is obtained from the following equation:

Φ)

∑ i,j,k

δi+j

[x

2

-

(i + j )d + k (d + d0)

kg0 i2+j2+k2*0

2

2

2

2

2

1

x(i

2

-

+ j )d + {k(d + d0) + d) 1 2

2

x(i

2

2

+ j )d + {k(d + d0) - d) 2

2

2

]

(11)

As seen in Figure 3, the assumed surface layer structure essentially consists of the ordered salt molecules, and the reason why this structure was chosen is explained below. The aluminate and salt molecules of alkali metals are known to exist in the shape of dimers in their supersaturated solution. The dimer is composed of two molecules associated with each other by a hydrogen bond.17 It is expected that the ionic bond between the hydrated alkali metal ion and the aluminate ion in the dimer is the most effective of all interactions, and that the ionic interaction between them occurs first and then the dimers are crystallized by the interaction and the hydrogen bond among the dimers in the surface layer. The same story is also expected to occur for the antimonate salts, because the dimer of antimonate salt molecules is formed in their supersaturated solution, too.18 We then considered the model which consists of the alignment of the salt molecules associated with each other by hydrogen bonds, with the effects described below. Our calculations gave the positions of the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) in each cluster. The HOMO lies on an oxygen

atom which coordinates an aluminum (or antimony) atom from the opposite side to the facing hydrated alkali metal ion in each salt molecule. The LUMO lies on a hydrogen atom which coordinates the alkali metal atom from the opposite side to the facing aluminate (or the antimonate) ion in each salt molecule. Then, we considered that one salt molecule is joined to the next salt molecule in succession to make a straight chain. On the other hand, two electric dipoles, in general, tend to orient toward antiparallel directions each other to lower the electric potential. So a bundle of the chains is assumed where one chain is oriented against the direction of its neighboring chains. Next, it was taken into account how many neighboring chains exist around one chain. We assumed two conditions. The first one was that each hydrogen atom in the chains has its hydrogen bond with oxygen atoms of aluminate (or antimonate) ions in the chain directly or indirectly (that means “through the other water molecules outside the chains”) as the characteristics of the surface layer. The second one was that when the chains get together to make a bundle, chains with the same direction do not share the region of hydrogen bond around them (otherwise the hydrogen atoms of the chains are left alone in the shared region). The region of hydrogen bond around them was defined as the region between the hydrogen atom in the hydrated alkali metal ion and the oxygen atom in the second layer of water molecules around the hydrated alkali metal ion, as seen in Figure 2. In Figure 3, r1 and r2 are the distance between the alkali metal atom and the oxygen atom in the second layer of water molecules, and the distance between the aluminum (or the antimony) atom and the oxygen atom in the aluminate (or the antimonate ion, respectively. If r2/r1 > 1, six neighboring chains can exist around one chain. If 1 > r2/r1 > (21/2 - 1) Z 0.41, four neighboring chains can exist around one chain. And if r2/r1 < 0.41, less than four neighboring chains can exist around one chain. For example, r1 of Li(H2O)4+ and r2 of Al(OH)4are 0.41 and 0.18 nm, respectively, and r2/r1 ) 0.44. So, for the case of (H2O)4LiAl(OH)4, four neighboring chains can exist around one chain. As a result, it was found that four neighboring chains exist around one chain for all of kinds of salt molecules we calculated, and we obtained the model in Figure 3. 2.3.4. Energy Accompanying the Difference in Entropy between the Solid and Aqueous Phases. To consider the entropy variation between the solid and aqueous phases, a path through a intermediate state was assumed between them. This intermediate state is the state of the salt molecules moving freely, like melted salt. We considered that the entropy variation between the solid and intermediate states could be attributed to the difference of the spatial alignment order (communal entropy), and the entropy variation between the intermediate and aqueous phases could be attributed to the variation in the degrees of freedom during the formation process of the salt molecule from the cation and the anion (activation entropy). So, in this study, the entropy variation between the solid and aqueous phases was estimated as the sum of these two entropy factors. The communal entropy per mole can be estimated by eq 12 because the ratio of the distribution function in the aqueous phase and that in the solid phase is expressed as (Vn/n!)/(V/n)n, where the distribution function of the molecules and their number are V and n.19

∆Scom ) kB ln(Vn/n!)/(V/n)n Z R (when n ) NA) (12) Next, the activation entropy was estimated through the absolute reaction rate theory.20,21 In this theory, when E* is

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notated as the activation energy, with zero point energy subtracted from the experimentally measured activation energy Ea, the reaction velocity k is described as k ) A exp(-E*/RT), in which A is expressed as follows, on the assumption that the transmission coefficient is 1:

A ) (kBT/h)Qmole/(QcationQanion)NA (dm3/mol s) (13) In this study, the unit volume was defined as 10-3 m3, considering that 1 mol of salt molecules occupies the space of 10-3 m3. In eq 13, Qcation, Qanion, Qmole are the distribution functions of a cation made up of m atoms, an anion made up of n atoms, and a salt molecule made up of (m + n) atoms. They are described approximately with the distribution functions of transition, rotation, and vibration per degree of freedom: qt, qr, qv.

Qcation ) qt3qr3qv3m-3

(14a)

Qanion ) qt3qr3qv3n-3

(14b)

Qmole ) qt3qr3qv3(m+n)-6

(14c)

The distribution functions qt, qr, qv are expressed as follows:

qt3 ) (2π m kB T)3/2/h3 (unit volume)

(15a)

qr3 ) π1/2 (8π2kB T)3/2 (IAIBIC)1/2/h3

(15b)

qv ) {1 - exp(hν/kB T)}-1 Z 1

(15c)

As mentioned above, the unit volume in eq 15a was 10-3 m3 per mole. The moments of inertia IA, IB, IC of each cluster were estimated from the structures of the clusters obtained by molecular orbital calculations. On the other hand, the reaction velocity can be expressed with the activation free energy ∆G*, as k ) (kBT/h) exp(-∆G*/RT), where ∆G* ) ∆H* - T∆S* ) Ea - RT - T∆S*, and ∆H* is the activation enthalpy () Ea - RT). Then, using the Arrhenius equation22 d(ln k)/dT ) Ea/ RT2 and k ) A exp(-E*/RT), we get the relation Ea ) E* 2RT, because A is proportional to 1/T2 from eqs 13-15. Finally, the relation ∆G* ) E* - 3RT - T∆S* is obtained. This gives the following equation of reaction velocity.

k ) (kBT/h) exp(3 + ∆S*/R) exp(-E*/RT)

(16)

With eqs 14 and 16, the activation entropy is obtained as follows:

∆S* ) R[ln [(Qmole/QcationQanion)‚NA] - 3]

(17)

3. Results of Calculations The binding energies of the alkali metal ions with aluminate or antimonate ions were calculated on the basis of the molecular orbital calculation results, and they were explained in relation to the salt molecules produced in eq 18. In our calculations, we paid much attention to the initial geometries of the salt molecules, especially where the hydrated alkali metal ion was associated to the aluminate (or the antimonate) ion. Their faces and sides are the possibilities for the places they meet. In the case of potassium antimonate, its enthalpy had the lowest value when the potassium ion met the antimonate ion on one of the faces the antimonate ion’s octahedron. When the potassium ion met the antimonate ion

Figure 4. Dependence of binding energy on ion radius of alkali metals: (a) aluminate salts and (b) antimonate salts.

on one of its sides, the hydrated potassium ion moved to one of its faces of the antimonate ion’s octahedron during calculation. In the case of lithium aluminate, its enthalpy had the lowest value when the lithium ion met the aluminate ion on one of the sides of the aluminate ion’s tetrahedron. But when the lithium ion met the aluminate ion on one of the faces of the aluminate ion’s tetrahedron, it had a local minimum geometry with the enthalpy about 30 kJ/mol higher than the lowest one. In this case, Li+ and Al3+ repelled each other to distort the tetrahedron of Al(OH)4- because the distance became close between them by that initial geometry. This distortion led to the higher enthalpy at the local minimum geometry. Appendix A gives the results of molecular orbital calculations about enthalpy of the ions or salt molecules, and about the characteristics of their structures which appear in eq 18

Li(OH2)4+ + Al(OH)4 f (H2O)4LiAl(OH)4

(18a)

Na(OH2)4+ + Al(OH)4- f (H2O)4NaAl(OH)4 (18b) K(OH2)6+ + Al(OH)4- f (H2O)6KAl(OH)4

(18c)

Li(OH2)4+ + Sb(OH)6- f (H2O)4LiSb(OH)6 (18d) Na(OH2)4+ + Sb(OH)6- f (H2O)4NaSb(OH)6 (18e) K(OH2)6+ + Sb(OH)6- + H2O f (H2O)7KSb(OH)6 (18f) The estimation process of binding energy is detailed in appendix B. 3.1. Aluminate Salts. Figure 4a relates the binding energy between the alkali metal ion and the aluminate ion to the alkali metal ion radius, for the aluminate salts. The lithium salt, with the smallest ion radius among them, has the largest binding

Estimation of the Solubility Dependence energy and the binding energy decreases as the ion radius gets larger, (rLi+ < rNa+ < rK+). The binding energy in the lithium salt is about 25 kJ/mol larger than that in the sodium salt, and about 45 kJ/mol larger than that in the potassium salt. To find out the reason why, the binding energy was divided into three terms and analyzed: the electric interaction energy of a waterstripped alkali metal ion with an aluminate ion (the cohesive energy), the variation of coordinate bond energy through the formation process of the salt molecule, and the electric interaction energy of an aluminate ion with water molecules surrounding the alkali metal ion. These three terms were estimated by individual molecular calculations. To estimate the cohesive energy, the enthalpy of the salt molecule cluster, which is stripped of the water molecules surrounding the alkali metal ion and keeps the same relative positions of atoms as those in the salt molecule, was calculated first. Then, the difference of its enthalpy from the sum of enthalpy of the aluminate ion and the water-stripped alkali metal ion was obtained. To estimate the variation of the coordinate bond energy, the calculated enthalpy of the hydrated alkali metal ion cluster, which has the same relative positions of atoms as those in the salt molecule, was compared with the enthalpy of the hydrated alkali metal ion in the aqueous phase. And to estimate the interaction energy between the aluminate ion and the water molecules surrounding the alkali metal ion, the enthalpy values of three clusters were calculated: clusters of the salt molecule from which the alkali metal ion had been taken away, of the aluminate ion, and of the water molecules. In Figure 5, cohesive energies, variations of the coordinate bond energy, and the interaction energies between the aluminate ion and the water surrounding the alkali metal ions, are plotted as a function of the alkali metal ion radius. The three components were obtained from molecular orbital calculations. In Figure 5a, the distance between the alkali metal atom and the aluminum atom in the salt molecule increases so that the cohesive energy drops, as the radius of the alkali metal ion increases. In Figure 5b, the energy reduction of the coordinate bond is largest for the lithium salt, because lithium has the smallest ion radius among the alkali metals, which leads to the shortest distance between the alkali metal atom and aluminum atom, giving the water molecules the largest change in their alignment. In Figure 5c, the interaction between the aluminate ion and two further water molecules (in the cases of lithium aluminate and sodium aluminate) or four further water molecules (in the case of potassium aluminate) from the aluminate ion than the other two water molecules, is much larger in the lithium salt. This result is caused by the shortest distance between the lithium and aluminum atoms of the lithium salt, which leads to the shortest distance between the hydrogen atoms in these water molecules and the oxygen atoms in the aluminate ion among the aluminate salts of alkali metals. Then, with the interaction between the aluminate ion and the other two water molecules added to it, the total interaction energy, between the aluminate ion and water molecules surrounding the alkali metal ion, is the largest for the lithium salt molecule. As a consequence, the sum of the energy variation of the coordinate bond and the interaction between the aluminate ion and the water molecules surrounding the alkali metal ion is independent of the ion radius of the alkali metal so that the binding energy between the aluminate ion and the hydrated alkali metal ion decreases in a manner similar to the cohesive energy, in proportion to the increase in the alkali metal ion radius. 3.2. Antimonate Salts. Figure 4b relates the binding energy between the alkali metal ion and the antimonate ion to the alkali metal ion radius for the antimonate salts. Among the antimonate

J. Phys. Chem., Vol. 100, No. 16, 1996 6537

Figure 5. Contributors to the binding energy in aluminate salts of alkali metals; (a) cohesive energy; (b) energy variation of coordinate bonds between alkali metal ion and water molecules, and (c) interaction energy between the water molecules surrounding the alkali metal ion and the aluminate ion.

salts, the sodium salt has a slightly larger binding energy than the lithium and potassium salts, differing from the tendency in the aluminate salts, although the ratio of ion radii rNa+/rSb5+ ) 1.5 Z rLi+/rAl3+. The binding energy in the sodium salt is about 5 kJ/mol larger than that in the lithium salt, and about 15 kJ/ mol larger than that in the potassium salt. To find out the reason, the binding energy was divided into three terms, as well: the cohesive energy, the variation of coordinate bond energy, and the electric interaction energy of an antimonate ion with the water molecules surrounding the alkali metal ion. These three terms were estimated by molecular orbital calculations individually, the same as for the aluminate salts. In Figure 6, the cohesive energies, the variations of the coordinate bond energy, and the interaction energies between

6538 J. Phys. Chem., Vol. 100, No. 16, 1996

Matsuo et al.

Figure 7. Solvation free energy of ions.

Figure 6. Contributors to the binding energy in antimonate salts of alkali metals: (a) cohesive energy; (b) energy variation of coordinate bonds between alkali metal ion and water molecules; (c) interaction energy between the water molecules surrounding the alkali metal ion and the antimonate ion.

between two oxygen atoms than Al(OH)4- because of the difference in the coordinate number of OH around them. This leads a little longer distance between the lithium atom and the antimony atom in the lithium antimonate salt molecule, than that between the lithium atom and the aluminum atom in the lithium aluminate salt molecule. Then, this difference causes a longer distance from the hydrogen atoms of the two further water molecules to the oxygen atoms of the antimonate ion in lithium antimonate, than that of the aluminate ion in lithium aluminate. That is the reason for the small interaction between the antimonate ion and two further water molecules in lithium aluminate, although the distance in lithium antimonate is shortest among the antimonate salts. Then, the sum of the cohesive energy, the energy variation of the coordinate bond, and the interaction between the antimonate ion and the water molecules surrounding the alkali metal ion is flattened among the three antimonate salts because of the larger size of the antimonate ion than that of aluminate ion. That explains why no larger binding energy between the antimonate ion and the hydrated alkali metal ion is seen in the antimonate salts, which differs from the tendency of the aluminate salts. In consequence, the binding energy becomes slightly larger than the energies of the others when the ratio of the alkali metal ion radius and the antimony ion radius is 1.5 for sodium antimonate, approximately as much as that for lithium aluminate. To summarize the above results, the binding energies becomes largest for lithium aluminate among the three aluminate salts, and for sodium antimonate among the three antimonate salts, because when the ratio of the alkali metal ion radius to the aluminum or the antimony ion radius becomes approximately 1.5, the sum of the cohesive energy, the energy variation of the coordinate bond, and the interaction between these anions and the water molecules is largest among them. 4. Rough Estimation of Solubility and Discussion

the antimonate ion and the water molecules surrounding the alkali metal ions are plotted as a function of the alkali metal ion radius. In Figure 6a, the distance between the alkali metal atom and antimonate atom in the salt molecule becomes longer so that the cohesive energy drops as the radius of the alkali metal ion gets larger. This is the same tendency as seen for the aluminate salts (Figure 5a). In Figure 6b, the energy reduction of the coordinate bond becomes largest for the lithium salt, just as for the aluminate salts. But Figure 6c does not show any greater interaction between the antimonate ion and two further water molecules from the antimonate in the lithium salt, which is seen for lithium aluminate. This result is explained as follows. The ion radius of Sb5+ (0.074 nm) is larger than that of Al3+ (0.053 nm), and Sb(OH)6- has a narrower space

4.1. Solubility Estimation. The values of the cavity radius ri for the hydrated alkali metal ion clusters, the aluminate, and antimonate ions were obtained from molecular orbital calculations. In Figure 7, the free energy of Born-type ion hydration of the hydrated alkali metal ions, the aluminate ion, and the antimonate ion are plotted against the alkali metal ion radius. In the hydrated alkali metal ions, the free energy of Born-type ion hydration becomes larger as ion radius gets smaller (that means a salt with smaller alkali metal ions is more soluble). In Figure 8, the estimated electric interaction energy between the salt molecules in the solid phase is plotted. Figure 9 shows the results of the activation entropy estimation. The difference of the binding energy and the energy of the above effects gives

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Figure 8. Energy of electric interaction between salt molecules in the solid.

Figure 9. Variation of activation entropy through the salt formation process.

the value of chemical potential to determine the solubilities of the salts. In Figure 10, the sum of the above energy is plotted against the alkali metal ion radius. The binding energy is also plotted for comparison. It is expected that the trend of the chemical potential of the salts follows that of the binding energy because the variation of binding energy is larger than the sum of the three energies. This can be confirmed in Figure 11, which gives the trend of the chemical potential, the sum of the binding energy, the solvation energy, the electric energy between the salt molecules in the solid phase, and the energy from entropy variations. These results and eqs 3 and 9 give the solubility of each salt. In Figure 11, the numerically estimated solubilities of the salts are also plotted as a function of the alkali metal ion radius. In proportion to the binding energy which is explained in relation to the ratio of ion radii, the solubility is the least for lithium aluminate among the three aluminate salts and for sodium antimonate among the three antimonate salts. From these results, we confirm that the difference of ∆H determines the solubility of the aluminate and antimonate salts. These calculated solubilities differ from the experimentally measured values (for lithium aluminate 8.0 × 10-4 mol/L at pH 10,2 for sodium antimonate 5.0 × 10-3 mol/L at pH 14,10 for potassium antimonate 1.6 × 10-2 mol/L at pH 14,21 see section 4.2). It is considered that the effect of pH should be added to compare the calculated solubilities with the experimental ones. Also, a more detailed estimation may be required for the electric interaction between the salt molecules in the solid phase because it is expected that the hydrogen bonds around the salt molecule in the solid phase are weaker with less mobility than those around ions in the aqueous phase, which can lead to the underestimation of the solubilities. More

Figure 10. Comparison of the binding energy with the sum of solvation energy, energy accompanied by entropy variation, and interaction energy between salt molecules in the solid: (a) aluminate salts and (b) antimonate salts.

Figure 11. Dependence of chemical potential difference between solid phase and aqueous phase calculated solubility of salts on ion radius of alkali metal.

consideration about limitations in the computational approach, approximations and uncertainties taken in this study is required to make the estimated results quantitatively more accurate. 4.2. Solubility Measurements of Antimonate Salts. It has been known that among the antimonate salts of alkali metals, sodium antimonate has the smallest solubility. But unfortunately, not all of their experimental values could not be looked over. So they were measured through the experiments as follows. At first, 100 mL of 0.1 mol/L LiOH solution, 0.1 mol/L NaOH solution, and 0.1 mol/L KOH solution were prepared, and 0.2 g of Sb2O5 was soaked in each alkaline solution for one month at room temperature. All of the substances were made by Wako Pure Chemical Industry, Ltd. These solutions

6540 J. Phys. Chem., Vol. 100, No. 16, 1996 were then filtered and the concentrations of antimonate ions in them were measured by ICP analysis (Hitachi P-5200). From the experiments, we estimated the solubility products of lithium antimonate, sodium antimonate, and potassium antimonate. They were [Li+][Sb(OH)6-] ) 2.5 × 10-4, [Na+][Sb(OH)6-] ) 3.0 × 10-5, and [K+][Sb(OH)6-] ) 3.0 × 10-4. For sodium antimonate and potassium antimonate, these measured solubility products are nearly as much as the referred ones ([Na+][Sb(OH)6-] ) 2.48 × 10-5 5, [K+][Sb(OH)6-] ) 2.5 × 10-4 23). In this paper, the solubilities of the salts were considered as the roots of the solubilities. For sodium aluminate and potassium aluminate, about 3.5 × 10-2 mol/L of sodium aluminate is dissolved into water, and potassium aluminate is more soluble than sodium aluminate.24 4.3. Consideration of the Corrosion Inhibitor Characteristics for Aluminum. We can choose the characteristics which a cationic corrosion inhibitor for aluminum should have in an alkaline solution as described below. (1) It should form an insoluble salt by the reaction with aluminate ions in the alkaline solution; i.e. preservation film formed on aluminum surface must be insoluble. (2) It should dissolve readily and greatly in alkaline solutions (the concentration should be 0.1 M,2 because the pH of cement paste is about 13, i.e. the concentration of OH-, is 0.1 M, and it is easily anticipated that the formation reaction of the preservation film will not succeed with a lower concentration of the corrosion inhibitor than that of OH-). Some cations other than lithium ion can fulfill the first condition, like Ca2+, and Mg2+. But they do not satisfy the second condition; only alkali metal cations satisfy the second condition. From the above discussion, lithium ions satisfy both conditions. A soluble lithium salt like LiNO3 then is most suitable for the inorganic corrosion inhibitor of aluminum in an alkaline solution. 5. Conclusions The addition of a lithium salt formed an insoluble lithium aluminate salt film on aluminum surfaces in alkaline solutions, preventing aluminum corrosion. All others were easily dissolved into the alkaline solutions. Reasons for the behavior were investigated by applying molecular orbital calculations to clusters of ions and molecules in relation to the ion radius of the alkali metal and the electric binding energy. The difference in solubility among aluminate salts of alkali metals was attributed to the difference of binding energy between the anion cluster and the alkali metal ion cluster, with the first layer of water molecules surrounding the alkali metal ion. Calculations showed that this difference in binding energy contributed to the difference in the sum of the cohesive energy, the variation of the coordinate bond energy between alkali metal ions and water molecules surrounding them, and the interaction energy between those water molecules and anions among them. These three factors, in turn, derive from the difference in ion radius of the alkali metals. The binding energy between the lithium ion and the aluminate ion in lithium aluminate was about 25 kJ/mol larger than that between the sodium ion and the aluminate ion in sodium aluminate, and about 45 kJ/mol larger than that between the potassium ion and the aluminate ion in potassium aluminate. The reason why the binding energy in the lithium salt was larger than those in the other aluminate salts of the alkali metals was due to the ratio of the lithium ion radius to the aluminum ion radius () 1.4) making the sum of the cohesive energy, the variation of the coordinate bond energy between alkali metal ions and water molecules surrounding them, and the interaction energy between those water molecules and aluminate ions,

Matsuo et al. largest among the aluminate salts of alkali metals. This leads to the difference of the solubility among the salts in water. The binding energy of antimonate salts of alkali metals was also calculated. The sodium salt had the least solubility among the antimonate salts. The ratio of the sodium ion radius to the antimony ion radius () 1.5) was as large as that of the lithium ion radius to the aluminum ion radius, which made the binding energy in the sodium salt about 5 kJ/mol larger than that in the lithium salt, and about 15 kJ/mol larger than that in the potassium salt. From these results, the relationship between the size of alkali metal ions and anions resulted in the insolubility of the salt. Because the alkali metal ions can only exist in an alkaline solution in amounts equal to that of OH- in it, and among the alkali metal ions, only lithium can produce the insoluble salts with aluminate ions, soluble lithium salts are most appropriate for the inorganic corrosion inhibitor of aluminum in cement paste. Appendix A. Structure and Enthalpy of Clusters The enthalpy of each cluster and some geometric parameters which are characteristic to each cluster are described in this section and Chart 1. The distance parameters and the enthalpy are given in angstroms and hartrees (1 Å ) 0.1 nm, 1 hartree ) 2.6255 × 106 J/mol). cluster (H2O)4LiAl(OH)4 (H2O)4NaAl(OH)4 (H2O)4KAl(OH)4 + 2H2O (H2O)4LiSb(OH)6 (H2O)4NaSb(OH)6 (H2O)7KSb(OH)6 Al(OH)4Sb(OH)6Li(OH2)4+ Na(OH2)4+ K(OH2)6+ H2O (H2O)2

enthalpy -851.913 423 49 to 8.409 628 44 -1 005.883 661 8 to 9.294 161 65 -1 594.136 220 07 to 11.860 317 7 -766.140 176 19 to 8.936 683 9 -920.130 193 91 to 9.812 623 88 -1 584.086 638 28 to 13.248 345 3 -541.857 961 49 to 4.687 693 08 -456.091 479 34 to 5.214 266 39 -309.841 031 96 to 3.730 934 8 -463.825 520 07 to 4.610 131 5 -1 052.089 171 05 to 7.172 666 54 -75.649 490 79 to 0.884 259 14 -151.312 728 57 to 1.764 293 72

Energy of a hydrogen bond: EH2O-H2O ) -0.009523 hartree () -5.97568 kcal/mol). Appendix B. Calculation of the Binding Energy The loss of hydrogen bond energy by the combination of cation and anion is taken into account to calculate the binding energy. The values of binding energy are all described in hartrees. salts ∆E(H2O)4LiAl(OH)4 ∆E(H2O)4NaAl(OH)4 ∆E(H2O)6KAl(OH)4 ∆E(H2O)4LiSb(OH)6 ∆E(H2O)4NaSb(OH)6 ∆E(H2O)KSb(OH)6

calculations of binding energy H(H2O)4LiAl(OH)4 - HLi(OH2)4+ - HAl(OH)4- - 3EH2O-H2O H(H2O)4NaAl(OH)4 - HNa(OH2)4+ - HAl(OH)4- - 3EH2O-H2O H(H2O)6KAl(OH)4 - HK(OH2)6+ - HAl(OH)4- - 3EH2O-H2O H(H2O)4LiSb(OH)4 - HLi(OH2)4+ - HSb(OH)6- - 3EH2O-H2O H(H2O)4NaSb(OH)4 - HNa(OH2)4+ - HSb(OH)6- - 3EH2O-H2O H(H2O)7KSb(OH)6 - HK(OH2)6+ -HSb(OH)6- - HH2O - 7EH2O-H2O

values -0.1769 -0.1680 -0.1605 -0.1706 -0.1729 -0.1670

Supporting Information Available: Tables of the positions of atoms in each cluster described in orthogonal coordinates (11 pages). Ordering information is given on any current masthead page.

CHART 1

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6542 J. Phys. Chem., Vol. 100, No. 16, 1996 References and Notes (1) Kikuchi, M.; Izumida, T.; Tsuchiya, H.; Chino, K.; Nishi, T. Proceedings of The 1st JSME/ASME Joint International Conference on Nuclear Engineering, Nov 4-7, 1991, Tokyo, Vol. 2, p 437. (2) Matsuo, T.; Nishi, T.; Matsuda, M.; Izumida, T. J. Nucl. Sci. Technol. 1995, 32, 912. (3) Tago, K.; Kumahora, H.; Kobayashi, K. Int. J. Supercomput. Appl. 1988, 2, 58. (4) Kobayashi, K.; Kurita, N.; Kumahora, H.; Tago, K. Phys. ReV. 1991, A43, 5810. (5) Kobayashi, K.; Kurita, N.; Kumahora, H.; Tago, K. Phys. ReV. 1992, B45, 13690. (6) Kobayashi, K.; Kurita, N.; Tago, K. Phys. ReV. 1996, in press. (7) Murell, J. N.; Boucher, E. A. Properties of liquids and solutions; Wiley Japan, Inc.: New York, 1984; p 142. (8) Marcus, Y. Introduction to liquid state chemistry; John Wiley & Sons Ltd.: New York, 1977; p 36. (9) Nagayama, M. Boshoku Gijutsu (Corrosion PreVention Technology) 1970, 12, 491 (in Japanese). (10) Gorshtein, G. I.; Lisichkin, I. N. Russ. J. Phys. Chem. 1976, 50, 641.

Matsuo et al. (11) Shannon, R. D.; Prewitt, C. T. Acta Crystallgr. 1969, B25, 925. (12) Shannon, R. D. Acta Crystallgr. 1976, A32, 751. (13) Debye, P.; Huckel, W. Phys. Z. 1923, 24, 185. (14) Das, B. Bull. Chem. Soc. Jpn. 1994, 67, 1217. (15) Hasted, J. B.; Riston, D. M.; Collie, C. H. J. Chem. Phys. 1948, 16, 1. (16) Stokes, R. H. J. Am. Chem. Soc. 1964, 86, 979. (17) Sibata, Y.; Kimura, K. X-1-1. Aluminum Muki-Kagaku Zensho (Encyclopedia of Inorganic Chemistry) 1975, 280, (written in Japanese). (18) Sibata, Y.; Kimura, K. IV-4 Sb. Muki-Kagaku Zensho (Encyclopedia of Inorganic Chemistry) 1954, 176, (written in Japanese). (19) Marcus, Y. Introduction to liquid state chemistry; John Wiley & Sons Ltd.: New Yor, 1977; p 7. (20) Eyring, H. J. Chem. Phys. 1935, 3, 107. (21) Pelzer, H.; Wigner, E. Z. Phys. Chem. 1932, B15, 445. (22) Arrhenius, S. Z. Phys. Chem. 1989, 4, 226. (23) Abe, M.; Ito, T. Bull. Chem. Soc. Jpn. 1968, 41, 333. (24) Sibata, Y.; Kimura, K. X-1-1 Aluminum. Muki-Kagaku Zensho (Encyclopedia of Inorganic Chemistry) 1975, 285, (written in Japanese).

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