Application of the quasi-random lattice model to rare-earth halide solutions for the computation of their osmotic and mean activity coefficients

JOURNAL OF RARE EARTHS, Vol. 32, No. 10, Oct. 2014, P. 979

Application of the quasi-random lattice model to rare-earth halide solutions for the computation of their osmotic and mean activity coefficients Elsa Moggia* (Department of Naval, Electric, Electronic and Telecommunications Engineering, Faculty of Engineering, University of Genoa Via Opera Pia 11A, 16145 Genoa, Italy) Received 12 December 2013; revised 4 April 2014

Abstract: This work dealt with the computation of the mean activity coefficients of rare-earth halide aqueous solutions at 25°C, by means of the Quasi Random Lattice (QRL) model. The osmotic coefficients were then calculated consistently, through the integration of the Gibbs-Duhem equation. Using of QRL was mainly motivated by its dependence on one parameter, given in the form of an electrolyte-dependent concentration, which was also the highest concentration at which the model could be applied. For all the electrolyte solutions here considered, this parameter was experimentally known and ranged from 1.5 to 2.2 mol/kg, at 25 °C. Accordingly, rareearth halide concentrations from strong dilution up to 2 mol/kg about could be considered without need for best-fit treatment in order to compute their osmotic and mean activity coefficients. The experimental knowledge about the parameter was an advantageous feature of QRL compared to existing literature models. Following a trend already observed with low charge electrolytes, a satisfactory agreement was obtained with the experimental values for all the investigated rare-earth chlorides and bromides. For the sake of compactness, in this work the considered rare-earth halides were all belonging to the P63/m space group in their crystalline (anhydrous) form. Keywords: mean activity coefficient; osmotic coefficient; pseudo lattice model; rare earth halide solution

This work deals with the application of the Quasi Random Lattice model (QRL)[1–3] to aqueous electrolyte solutions, where solutes are rare-earth chlorides or bromides. QRL is a general approach to calculate the mean molal activity coefficient γ±, at the experimental level of description[4]. The model embodies one adjustable parameter, indicated with clim (molar scale, mol/L), defined as the (upper) concentration at which the electrolyte should exhibit γ±=1, at the experimental temperature and pressure. In general, for a given electrolyte QRL considers concentrations up to clim. If saturation occurs at a lower concentration, then clim behaves as the adjustable parameter for the whole available concentration range. The structure and thermodynamic properties of dense ionic solutions represent an extremely interesting up-todate topic, currently under very active scrutiny due to the strong interest in highly concentrated solutions, ionic liquids and molten salts in different fields of science and technology. In particular, aqueous solutions of the electrolytes considered in the following are important in the field of rare earths[5–7]. Despite their importance, however, there is some lack of knowledge about their thermodynamic properties, as outlined by recent works concerning solubility[8–10] and transport properties[11,12]. Moreover, their peculiarities do not allow for any foregone conclusion when applying a model, though general or successful with lower charge electrolytes[13,14]. With 3:1 systems,

often models work less well, for example, results from the Primitive Model (PM) plus the Hypernetted Chain (HNC) are much less satisfactory for 3:1 electrolytes than for 2:1 electrolytes[13]. Fortunately, the results obtained by the application of QRL to 3:1 electrolytes show the same quality and agreement with experimental data as previous results for symmetric and 2:1 electrolytes. This is an important achievement of this work, which consolidates the research interest in QRL. From the experimental point of view, the work relies on published literatures[13–17], where issues were discussed concerning, in particular, the (indirect) evaluation of the mean activity coefficient from isopiestic measurements of the osmotic coefficients[13–15,18], since there is some lack of data for 3:1 solutions. The osmotic and activity coefficients reported in literatures by different authors are unfortunately affected by a larger spreading for 3:1 electrolytes than usually observed with 1:1 or 2:1 electrolytes. These experimental issues are not yet completely overcome, and for this reason some of the rare-earth halide solutions were not included in this work. From the theoretical point of view, some formal and general refinements of QRL are proposed in the following. In particular, the main formula for calculating ln(γ±) is given in a closed form. Integral formulae were proposed[1–3] involving elliptic functions, which provide a powerful tool, however more suitable for numerical pur-

* Corresponding author: Elsa Moggia (E-mail: [email protected]; Tel.: + 39 3402536867) DOI: 10.1016/S1002-0721(14)60172-1

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poses. A further refinement concerns the analytical behaviour and the continuity requirement that must be satisfied by the statistical parameters defined in QRL, an important aspect that may cause theoretical issues with more accentuated effects on the modelling of high charge-asymmetry electrolytes. Dealing with 3:1 electrolytes, some computational cost is due to their Madelung constants, which enter into QRL for describing some aspects of the pseudo-lattice approach, and which are scarcely reported in literature in comparison with lower-charge systems. As previously said, the QRL results for rare-earth halides are confirming the satisfactory trend already observed with other electrolytes. This may be motivated by the fact that the model is corroborated by experimental information available about the parameter, indeed clim is known for various electrolytes, and in particular for the rare-earth halide solutions here considered, whose clim values range from 1.5 to 2 mol/L (at 25 °C); that is, concentrations up to 40%–50% of the saturation values[15,17] can be used without need for best-fit treatment. The availability of experimental information about the parameter in a relevant number of cases is an advantageous feature of QRL compared to existing one-parameter theories, mainly based on PM[4]. In reason of their simplifying assumptions, one-parameter theories are not fully exhaustive, however they offer a compromise solution to the difficult problem of preserving the thermodynamic consistency of results while limiting the number of fitting (arbitrary) parameters to use. Indeed, some theoretical inconsistency is compensated by the adoption of one parameter, typically in the form of a suitable ion-size diameter. Unfortunately, even small changes in the adjustable diameter can significantly affect the fitting quality, with more evidence dealing with 3:1 electrolytes. High-quality results are obtainable by most theories when dilute solutions are concerned[4], but there is no simple rule for extending their use to higher concentrations, where a best-fit value from a model is usually too inaccurate for another one even when the models are so intimately related such as DH (Debye-Hückel) and DHX (Exponential Debye-Hückel Theory)[19]. This problem is generally afforded by adding more parameters[20–25], often with extended Pitzer equations[20,21,23]. However, parameters are usually to interpret numerically more than physically. And the advantages are not obvious by using theories that adopt effective dielectric constants, because of their experimental inaccessibility or disagreement with measured values, which is often the case with 3:1 electrolytes[22].

and solvent molecules, assembled by means of their short-range interactions, each grouping having a fixed overall charge whereas its size depends on concentration (at given pressure and temperature). A carrier is delimited by free solvent, in some analogy with a solvated ion whose spherical shape (in the PM frame) is delimited by the surrounding continuous medium. Within a carrier, cluster-like behaviour and ion association cannot be distinguished from each other, still in analogy with PM since such a distinction is not available from the sole concept of ion-size diameter. However, the simplifying modellization of a carrier, introduced at mesoscopic level, allows for considering explicitly only carrier-carrier interactions. The overall charge of a carrier is that nominal of one solute ion, and N co-solvent molecules in a volume V correspond to ν+N carriers with charge Q+, and ν–N carriers with charge Q(the overall electro-neutrality holds true). The carrier population is described in a fixed space-frame where the solution volume is conceptually divided into (ν++ν–)N ~ cells, that is, each cell has volume R 3 , where R~ is the mean inter-ionic distance. 1 R = ρ = N/V (1) 3 ( ν+ + ν− ) ρ

1 QRL model QRL[1–3] is based on the concept of “effective carriers” of charge. Carriers are suitable groupings of solute ions

It must be outlined that, as well as the DH Theory, QRL assumes full dissociation of ions at strong dilution, so the model starts from considering N solute molecules fully dissociate (and a carrier can be depicted as a solvated ion, in a highly dilute solution). Full dissociation at strong dilution is verified by most electrolytes but few exceptions (for example some metal sulphates) that may show a significant ion association in such a concentration range. These situations are still tractable with QRL by means of the additional use of association constants at strong dilution and according to the procedure discussed in[2]. However, these cases are not considered further in this work since, for the rare earth halides investigated, there is no experimental evidence of important ion association in highly dilute solutions[13,14]. It is to remind that, when speaking about a carrier, some local formation (solute+solvent) is considered, which is moving within the solution. So, the average behaviour of all carriers interesting a given cell is evaluated, such a behaviour being mathematically described by a charge density that accounts for both the physical extent of carriers and for their movements inside the solution. An “effective carrier” is what is described by such a charge density, so the “size” of an effective carrier is to intend as a statistical property, which is measured from a reference space point, and not from a reference ion position as usually done with PM theories[4]. Let RA be the reference space point individuating the cell A. Along each principal axis of the space frame, the size of the effective carrier A is estimated by means of the (linear)

Elsa Moggia, Application of the quasi-random lattice model to rare-earth halide solutions for the computation of …

standard deviation provided by the charge density, indicated with U and measured from RA.

⎛ r − RA 2 ⎞ exp ⎜ − ⎟ 3 2 3 ⎝ 2U ⎠ 2π U QA

GA (r) =

⎛ r − RA ⎞ ⎟ ⎝ 2U ⎠

QAerf ⎜ VA (r ) =

(2)

4πε r − RA

QA=Q+ or Q– In Eq. (2), r is a generic space position, GA (r) is the charge density (Gaussian distribution) describing the effective carrier referring to the cell A, and with overall charge QA. (Note, the assumption of Gaussian distribution is valid in the studied ρPT region, although not completely general since in near critical region other Lévy stable distributions should be rigorously to consider). VA(r) is the potential due to GA (r) . In Eq. (2), the Error function (erf) appears, and ε is the dielectric constant of the solvent. The procedure to calculate γ± is then based on the calculation of the electrostatic contribution to the chemical potential[4]. The mean electrostatic energy of interaction E (with respect to the whole effective-carrier population) is obtained summing up the contributions from each effective carrier due to its interactions with the others (calculations, not reported here for brevity, are of the same kind as those in Ref. [1]). Finally, ln(γ±) follows from the relationship[1] ln(γ±)=E/kBT, where kB is the Boltzmann Constant, and T the absolute temperature [K]. 1 1 1 QB QA ⎛R ⎞ ln ( γ ± ) = erf ⎜ AB ⎟ ∑ RAB k BT 2 ( ν+ + ν− ) N 4 πε B,A ⎝ 2U ⎠ A≠B

(3) In Eq. (3), RAB= RA–RB, and RAB=|RA–RB|. Concerning the set of RA points, QRL assumes an ionic lattice configuration in order to arrange cells and their effective carriers consistently with the electrolyte charge and composition, so each RA corresponds to the spatial coordinates of a lattice vertex A. Concerning U, formula below comes straightforwardly from the QRL theory[1,3]. 2

U = UD U =

πα

, c / clim < exp( − k ); UD

lg k ( clim / c )( c / clim )

k e ln( k )

1/ k

, c / clim ≥ exp( − k );

(4)

πα

=

2

In Eq. (4), c is the salt concentration (molar scale), UD =

(ν

εk BT

+

+ ν− ) ρ Q+ Q−

is the usual DH Screening

Length[4], α is the Madelung Constant of the ionic lattice, and the constant k is found by solving the last expression from the knowledge of α. (Note, Madelung constants are

981

usually given with respect to the shortest distance between two ions of opposite sign[26], whereas, with QRL, they are given with respect to R~ ). As known, the conventional Madelung constant of an ionic crystal can be ν α + ν− α − expressed as α = + + . Strictly speaking, in 2 Eq. (4) the single-ion Madelung constant should be used, that is, α+ or α−. With asymmetric electrolytes, in general one has α+≠α− , so U should vary between positive and negative effective carriers, that is, U+ and U− should be considered separately. However, to avoid formal complications and also numerical issues due to additional asymmetry effects, the procedure[1–3] assumed a common U for all the effective carriers, found by means of considerations about the analytical continuity of U+ and U− with respect to the concentration range. In this work, an alternative approach is proposed, that is, U is calculated using α into Eq. (4), α being immediately interpreted as an average value between α+ and α−. This approach is much more satisfactory from the analytical point of view, in particular concerning the partial derivatives of U (they are not considered here, but they will be discussed in future works since they are important in the theoretical investigation of the volumetric and thermal properties of electrolyte solutions). It must be stressed out that U is related to the spatial spread of the charge distribution, and derives from a combined effect of physical size and movement of carriers, but in Eq. (4) there is no need for a complete knowledge of these two contributions separately in order to calculate γ±. Moreover, U is generally large, so Gaussian densities (describing the effective carriers) can spread out widely from their reference cells, and this attributes an ancillary role to any reference lattice. At low-dilution (c→clim), large U values are due to large-size carriers, thus not localizable just at RA points. Conversely, at strong dilution, large U values arise from small-size carriers (like solvated ions), which are, however, widely moving within the solvent. These considerations justify the use of a quasi-lattice approach even in the strong-dilution case, although pseudo-lattice models[27–32] are more often applied to concentrated solutions and ionic liquids, where recent experimental evidence is confirming the theoretical approach[32]. However, the energy sum in Eq. (3) implies that the number of ions is still high enough, and this brings up to a criticism arising from using a quasi-lattice approach even in the exact infinite-dilution limit[33], that is, ρ→0 and UD→∞. Actually, at the infinite dilution no reticular scheme is adopted by QRL, and it can be shown that QRL collapses into the DH Limiting Law[33], as expected, although a detailed discussion of this point is beyond the aims of this work. Accordingly, though Eqs. (3, 4) should not be applied straightforwardly when ρ→0, they are valid at any prac-

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tical strong dilution. As said above, carriers are local structures embodying short-range interactions (though not explicitly modelled), in principle either ion-ion or ion-solvent forces, although inter-ionic dispersive forces are expected to exert a limited influence in the considered concentration regime for the rare-earth halides. Moreover, for high concentrations (effective) carriers are allowed to overlap, which is also a way to consider ion association within the QRL frame. An explicit treatment of short-range forces in the pseudo-lattice formalism was proposed in Refs. [28,31], where contributions were included from ion-ion forces and from dielectric-gradient interactions between ions and solvent dipoles, which yielded quadratic and linear terms with respect to c within the main equation for ln(f±) (f± is the mean rational activity coefficient). Such contributions were proposed on thermodynamic basis, although a best-fit treatment was needed due to the presence of three adjustable parameters (the electrostatic term was set proportional to c1/3, as in most pseudo-lattice models[27–32]). Eq. (3) can be simplified. Indeed, from geometrical considerations, only few positive (B+) and negative (B−) points are needed in practice, and Eq. (3) can be rewritten as follows, after indicating with Z+ and Z– the minimum numbers of B+ and B− points to use.

rare-earth chlorides or bromides belonging to the P63/m space group (hexagonal, No. 176) in their crystalline (anhydrous) state. Concerned cations are La3+, Nd3+, Eu3+, Gd3+, and Pr3+. To proceed with Eq. (3), a reference lattice must be chosen, a straightforward choice being the P63/m. With P63/m, it is also convenient to put:

1

ln ( γ ± ) =

⎡ ⎢ ν+ ⎢Z+ ⎣⎢

1

R=

log 24 ( clim / c )( c / clim )

1/ 24

Eq. (8) comes straightforwardly from the use of α into Eq. (4) to deal with asymmetric electrolytes. The RA positions (not reported here for brevity) are then expressed in terms of R, according to the Wyckoff positions for the P63/m space group. Cations are located at “2c” Wyckoff sites (1/3, 2/3, 3/4), and (2/3, 1/3, 1/4); anions are located at “6h” Wyckoff sites (u, v, 1/4), (1-v, u-v, 1/4), (1-u+v, 1-u, 1/4), (u-v, u, 3/4), (1-u, 1-v, 3/4), and (v, 1-u+v, 3/4). With P63/m, one has Z+=2 and Z–=6. The corresponding B+ and B– points are listed below. ⎛ 1 3C ⎞ ⎛ 2 C⎞ + , , ⎟ B 2 = R ⎜ 1, B1+ = R ⎜ 0, ⎟ 3 4 ⎠ 3 4⎠ ⎝ ⎝

1

k BT 2 ( ν+ + ν− ) 4 πε QB QA

+

+

i

⎛ RAB ⎞ ν− ⎟+ − ⎝ 2U ⎠ Z

+

+

i

i=Z

+

erf ⎜

QB QA

−

−

i

i

i

i

1

1

1

2

1

kBT 2( ν+ + ν− ) 2π εU 3

ν− -

Z

i=Z

−

i

−

−

i

ln( γ± ) =

erf ⎜

−

∑∑Q Q exp −

Bi

i=1 A

A

(

∫ 0

+

ν+

i=Z

+

Z

∑∑Q Q exp +

Bi

i =1 A

+

)

A≠Bi

R2 −u2 / 2U2 du

−

AB i

−

A≠Bi

A

(

(7)

R

C 3

The corresponding Madelung constants for the electrolytes are available in literatures[36–39] and depend on the (a, c, u, v) parameters of the ionic crystals, summarized in Table 1. In what follows, C=2c/a for notational brevity. The P63/m parameters change very little in going from one electrolyte to another, so one finds α≈4.9 for all, and k≈24 from Eq. (4), which, in practice reduces to (noting that, since exp(–24)≈4×10–11, it is c/clim>exp(–k) with almost any c): UD U≈ (8 )

⎤ ⎛ RAB ⎞ ⎥ ⎟⎥ ∑ ∑ R ∑ ∑ R A A i =1 i =1 ⎝ 2U ⎠ ⎥ AB AB A≠B A≠B ⎦ (5) For numerical purposes, Eq. (5) is rewritten conveniently as follows. i=Z

1 3

−

(

)

Bi = R Xi ,Yi 3, Zi C , i from 1 to 6

X1=2u–v; X2=1–u+2v; X3=2–u–v; X4=u–2v; X5=1–2u+v; X6=u+v–1 2 2 2 −R u / 2U + + Y1=v; Y2=1–u; Y3=u–v; Y4=u; Y5=1–v; Y6=1–u+v; AB i Z1=Z2=Z3=1/4; Z4=Z5=Z6=3/4 (9) Eq. (6) can be rewritten after changing the integration variable from u to q= Ru / 2U , and after introducing the following auxiliary functions, for notational brevity.

(6) In Eq. (6), sums can be rearranged so as to obtain fast convergence. From a strictly mathematical point of view, this methodology is similar to that for computing Madelung constants basing on elliptic functions[34,35], and in this sense the computational complexity of QRL is comparable to that of the Madelung energy of ionic crystals.

)

F (ξ , q ) ≡

2

q

2

n

)

G ( ξ , λ , q ) ≡ F (ξ , q ) F (λ , 3q ) + F (ξ − 1, q ) F (λ − 1, 3q )

(10) Table 1 Lattice parameters

2 Rare-earth halide solutions In the following, aqueous solutions at 25 °C are considered, with ν+=1, ν–=3, Q+=3Q, Q–=–Q (Q=absolute electron charge). In all the investigated cases, solutes are

∑ exp ( − ( 2n + ξ )

a

Electrolyte

a/nm

c/nm

u

v

LaCl3 a NdCl3 b EuCl3 b GdCl3 b PrCl3 c LaBr3 d PrBr3 c

0.7483 0.7399 0.7375 0.7366 0.7416 0.7951 0.7937

0.4375 0.4242 0.4132 0.4106 0.4276 0.4501 0.4396

0.382 0.388 0.389 0.389 0.388 0.390 0.386

0.287 0.301 0.301 0.301 0.301 0.300 0.299

From Ref. [36]. b From Ref. [37]. c From Ref. [38]. d From Ref. [39]

Elsa Moggia, Application of the quasi-random lattice model to rare-earth halide solutions for the computation of …

The final formula for ln(γ±) is given in Eq. (11). ln ( γ ± ) =

1

3Q

2

3

8 k BT

π εR

⎧ ⎫ ⎛ 2G (0, 0, q) − i =3 G ⎛ X , Y − 2 , q ⎞ + ⎞ ⎪ ⎪ ∑ ⎜ ⎟ i i ⎜ 3 ⎠ ⎟ ⎝ i =1 ⎪ ⎜ ⎟−2 ⎪ ⎪ ⎪ θ3 ( Cq ) ⎜ i,k =3 G ( X i − X k , Yi − Yk , q ) ⎟ ⎪ ⎪ ∑ ⎜ ⎟ 3 ⎜ i,k =1 ⎟ ⎪ ⎪ R ⎝ i≠k ⎠ 2U ⎪ ⎪⎪ ⎪ dq + ⎨ ⎬ ∫ 0 ⎪ ⎪ i =6 ⎛ 3G ⎛ 0, 1 , q ⎞ − 2 G ⎛ X , Y − 2 , q ⎞ + ⎞ ⎪ ⎪ ⎜ ⎟ ∑ ⎜ i i ⎟ ⎜ 3 ⎠ ⎟⎪ ⎝ 3 ⎠ i=4 ⎝ ⎪ ⎟⎪ ⎪ F ⎛⎜ 1 , Cq ⎞⎟ ⎜⎜ i = 6 ⎟⎪ k =3 , , G X − X Y − Y q ( ) ⎪ ⎝2 ⎠⎜ i k i k ⎟⎪ ∑ ⎪ 3 ⎜ ⎟⎪ k =1 ⎪⎩ ⎝ i=4 ⎠⎭ (11) In Eq. (11) the Third Jacobi Theta Function[34] θ3(z,ξ)=

∑e

i 2 nz

n

2

ξ is also evidenced. In the present context,

n

one has z=0, ξ=Cq. θ3(0,Cq)≡θ3 (Cq)=F(0, Cq)+F(–1, Cq)

(12) Eq. (12) outlines the role of the elliptic functions in the QRL formalism. Note, Eq. (11) derives from quite long manipulations of sums (not reported here for brevity), which can be afforded using the fact that: ∑ exp

( R2Uu ) ( F(ξ, q)F(λ, −

2 2 AB 2

=

3q) + F (ξ − 1, q) F (λ − 1, 3q)

)

A,A ≠ B

( F (ω, Cq ) + F (ω −1, Cq ) ) (13) In Eq. (13) the coefficients {ξ, λ, ω}, where ω is either 0 or ½, depend on the coordinates of each B point (Eq. (9)), while the term F(ω,Cq)+F(ω–1,Cq) can be either θ3(Cq) or 2F(1/2,Cq) (noting that F(1/2,q)=F(–1/2,q)). In principle, 8×8=64 terms of the form of Eq. (13) should be calculated, however the P63/m lattice geometry allows reducing the number of terms to calculate, yielding Eq. (11). In particular, the following rules are found useful for simplification purposes. X1–1=–X5; X2–1=–X4; X3–1=–X6; Y1–1=–Y5; Y2–1=–Y4; Y3–1=–Y6 (14) As suggested by Eq. (11), formulae for ln(γ±) are more or less handling depending on the reference-lattice geometry although, as a first-order lattice effect due to large U, γ± is expected not to change significantly with the reference lattice, analogously to 2:1 electrolytes[3]. So, in principle, simpler lattice geometries, e.g. those from the cubic system, may be adopted also for the rare-earth halide solutions here considered. However, the presented results from the P63/m offer a basis for future comparisons and more general investigations concerning the QRL dependence on the reference lattice geometry, with application also to rare-earth nitrates and perchlorates.

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Moreover, setting the structure of the ionic crystal as the reference lattice for the salt solution is motivated by some experimental evidence about local preservation of the solid lattice structure and space group in aqueous solutions (in this connection, a review on experimental work is given in Ref. [28]). X-ray diffraction data on highly concentrated aqueous solutions of 1:1 salts yielded patterns showing some elements of a dispersion curve of a liquid with “near-neighbour” distances corresponding to the near sites in a FCC (Face Centred Cubic) structure, and analogous fluorite-structure patterns were observed in very concentrated 2:1 and 1:2 electrolyte solutions[28]. Concerning rare-earth halide aqueous solutions, there exists a much scarcer experimental literature, however few investigations of aqueous solutions of LaCl3 (3.03 mol/L) and LaBr3 (2.73 mol/L) with X-ray, Raman spectroscopy, and inelastic neutron scattering are suggesting the existence of a medium range order between the cations[40]. Some computational complexity of QRL is compensated by the immediate availability of results at the experimental level, that is, without need for converting from the McMillan-Mayer level of description, where most PM theories have been developed, to the LewisRandall one, where experiments are performed[4]. In this connexion, approximate conversion equations are available[41], which are set consistently with the Gibbs-Duhem Equation (GD)[4] when going from φ to γ± or vice-versa. This conversion procedure bypasses, though without solving, the intrinsic ambiguity on γ± (or φ) when calculated by means of different thermodynamic routes (within the PM frame), which allows only for a partial use of the powerful formalism offered by the Kirkwood-Buff Theory (KB)[33].

3 Results Simulations were concerned with the computation of Eq. (11) for aqueous solutions (at 25°C, the relative dielectric constant of water is 78.36) of rare-earth chlorides and bromides for which clim values are experimentally known. The numerical integration of Eq. (11) was performed satisfactorily with elementary methods and using reasonable integration steps. Results were found accurate within ±10–4, such an accuracy being definitely acceptable in all the considered cases, though a little inferior compared to that found for symmetric and 2:1 electrolytes. Moreover, the approximation done by using α=4.9 and k=24 in Eq. (8) for all the considered electrolytes was found largely adequate in practice. However, Eq. (11) is computationally more expensive than previous equations concerning symmetric and 2:1 electrolytes[1–3]. In general, more computational issues are to afford with high charge-asymmetry systems, indeed the numerical convergence of equations like Eq. (11)

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may become prohibitive and numerical faults may also occur if integration steps are not properly managed. Concerning the osmotic coefficients Φ, they were calculated using the simulation results of Eq. (11) and according to the GD Equation given in the form below. Φ = 1 + ln γ ± −

1 m

m

∫ ln γ

±

( m )d m

(15)

0

In Eq. (15) the molal scale (m) is used for the concentration, converted into molar scale according to standard formulae. Solution densities for conversion purposes can be obtained using semi-empirical equations, however, in this work, elementary interpolations of data from literature[16] were sufficient in order to fill the concentration ranges of interest for all but for bromides, whose solution densities were calculated with formulae in Ref. [16] extended up to 1.6 m, for lack of experimental information about non-dilute solutions. The integral involved in Eq. (15) was performed numerically. The computational costs were substantially due to the computation of the integrand function (that is, Eq. (11)), and only moderately due to the number of integration steps to use. Note, the procedure[1–3] evaluated both U+ and U−, which must be proportional to UD when c≤climexp(–k+) and c≤climexp(–k–) respectively, the proportionality factors 1 1 and . With the P6 /m one being πα + / 2

πα − / 2

Fig. 1 Mean activity coefficient of LaCl3 (plot of mean (molal) activity coefficient (γ±) versus molal concentration (m). For LaCl3, clim=1.99 mol/L (2.14 mol/kg). Black (continuous) line: this work (QRL); dashed line: DHX; grey line: DH; asterisks: experimental values)

3

has α+ ≈6.9 and α− ≈0.95 for all the considered electrolytes, yielding k+≈30.6 and k–≈5.9. This means that the condition U+=U– can be satisfied only at a concentration higher than climexp(–k–)≈0.002clim. If k is set equal to k+ (that is, if one puts U=U+), then U−=U+ will be obtained at c≈0.015clim, and U+=U−=U may be imposed for c/clim ≥0.015. Conversely, if k is set equal to k– no concentration can be found at which U−=U+. This result is analogous to that previously found for 2:1 electrolytes, and would point out some predominant role of positive carriers. However, because of their analytical form, Eqs. (4) and (8) are poorly affected numerically if one uses k+= 30.6 in place of k=24, and this was confirmed also by the simulation results concerned with the osmotic and the mean activity coefficients. The experimental work about rare-earth halide solutions, used for sake of comparison with the simulation results, is that given in Refs. [13–17]. Osmotic coefficients from isopiestic measurements were reproduced according to formulae in Refs. [15, 17], which involve up to 7 fitting parameters. In this work, simulations were not concerned with Ce3+ due to disagreement among experimental data series in the non-dilute range[42]. Figs. 1 and 2 show a comparison between calculated and experimental γ± (Fig. 1) and φ (Fig. 2) concerning LaCl3. The experimental value of clim is 1.99 mol/L, that is, 2.14 m, while saturation occurs at 3.899 m[15]. Results

Fig. 2 Osmotic coefficient of LaCl3 (plot of osmotic coefficient (φ) versus molal concentration (m). For LaCl3, clim=1.99 mol/L (2.14 mol/kg). Black (continuous) line: this work (QRL); dashed line: DHX; grey line: DH; points: experimental values)

from DH and DHX models are also reported. The best-fit treatment of these models was extended up to 2.14 m, for comparison with QRL, yielding best-fit diameters 0.555 nm (DH), and 0.48 nm (DHX), which are in reasonable agreement with the literature range of 0.50–0.66 nm, suggested by Integral Poisson-Boltzmann Equation (IPBE)[13,14], Replica Integral Equation (RIE)[43], Binding Mean Spherical Approximation (BiMSA)[22], and Speeding Interpolation[15]. The following mean errors were obtained, with respect to experimental γ± and φ: (mean absolute error) 1.8%, 5.7% (QRL); 3.5%, 5.9% (DHX); 3.8%, 10.4% (DH); (mean relative error) 5.4%, 4.6% (QRL); 6.6%, 4.4% (DHX); 10.0%, 8.0% (DH). Moreover, the numerical evaluation of the partial derivative of ln(γ±)

Elsa Moggia, Application of the quasi-random lattice model to rare-earth halide solutions for the computation of …

with respect to c (at given pressure and temperature) yielded the following mean absolute errors: 17% (QRL), 23% (DHX) and 30% (DH). This comparison outlines the competitive potentialities of QRL without need for best- fit treatment in all the cases here considered. Further comparison can be done with models using two or more adjustable parameters. Two fitting diameters are used according to the Smaller-Ion Shell Theory (SiS)[44] for calculating the mean activity coefficient of several electrolytes. In general, in the cited work the agreement with experimental values seems noticeable only for dilute solutions whereas, at medium-high concentrations, deviations from measurements are quite large in various cases. Concerning LaCl3, the agreement is globally inferior to that from QRL, considering also that the reported best- fitting diameters (0.325 and 0.287 nm) look definitely out of literature range for LaCl3. With BiMSA[22], three to four fitting parameters are used for calculating γ± and φ of LaCl3 (and other salts) from infinite dilution up to 3.1 m. The parameters account for concentration-dependent diameters, ion association and effective dielectric constants. However, the physical meaning of the effective dielectric constants is unclear and not comparable with measurements[45] which suggest quite different values: at c=0.17 mol/L, the relative dielectric constants are 77.5[22] and 71[45]; at c=0.35 mol/L, the relative dielectric constants are 76.7[22] and 64[45]. Analogous results to the LaCl3 case were obtained for all the other tri-univalent salts here considered. Corresponding plots are reported in Figs. (3–8).

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Fig. 4 Osmotic and mean activity coefficients of GdCl3 (Plot of logarithm of mean (molal) activity coefficient (γ±) versus molal concentration (m). For GdCl3, clim=1.84 mol/L (1.95 mol/kg). Continuous line: this work (QRL); asterisks: experimental values. Plot of osmotic coefficient (φ) versus molal concentration (m): dashed line, this work (QRL); points, experimental values)

4 Conclusions The agreement between calculated and experimental mean activity coefficients (analogously for the osmotic Fig. 5 Osmotic and mean activity coefficients of NdCl3 (Plot of logarithm of mean (molal) activity coefficient (γ±) versus molal concentration (m). For NdCl3, clim=1.97 mol/L (2.09 mol/kg). Continuous line: this work (QRL); asterisks: experimental values. Plot of osmotic coefficient (φ) versus molal concentration (m): dashed line, this work (QRL); points, experimental values)

Fig. 3 Osmotic and mean activity coefficients of EuCl3 (Plot of logarithm of mean (molal) activity coefficient (γ±) versus molal concentration (m). For EuCl3, clim=1.89 mol/L (2.01 mol/kg). Continuous line: this work (QRL); asterisks: experimental values. Plot of osmotic coefficient (φ) versus molal concentration (m): dashed line, this work (QRL); points, experimental values)

coefficients) is of the same quality for all the tri-univalent systems investigated, and similar to that found for other electrolytes in previous works. Experimental and theoretical curves appear close to each other almost everywhere, the largest deviations (though acceptable) occurring at intermediate concentrations, as usual with QRL since, at such concentrations, the strongest approximations are done in the model[1]. Chlorides show clim values close to 2 mol/L, while for bromides clim values are about 1.6 mol/L. At c=clim, the

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Fig. 6 Osmotic and mean activity coefficients of PrCl3 (Plot of logarithm of mean (molal) activity coefficient (γ±) versus molal concentration (m). For PrCl3, clim=1.97 mol/L (2.10 mol/kg). Continuous line: this work (QRL); asterisks: experimental values. Plot of osmotic coefficient (φ) versus molal concentration (m): dashed line, this work (QRL); points, experimental values)

Fig. 7 Osmotic and mean activity coefficients of LaBr3 (Plot of logarithm of mean (molal) activity coefficient (γ±) versus molal concentration (m). For LaBr3, clim=1.57 mol/L (1.65 mol/kg). Continuous line: this work (QRL); asterisks: experimental values. Plot of osmotic coefficient (φ) versus molal concentration (m): dashed line, this work (QRL); points, experimental values)

mean inter-ionic distances from Eq. (1) range between 0.593 and 0.608 nm for chlorides, while they are about 0.64 nm for bromides. It might be noted that such distances agree with literature concerning size-based parameters[15,22]. This trend was already observed with other electrolytes[3], however its explanation is not immediate. Indeed, although carriers corresponding to (solvated) ions are plausible in the strong-dilution range, nevertheless it is not trivial to relate the statistical properties of effective carriers, measured from fixed space points, with

JOURNAL OF RARE EARTHS, Vol. 32, No. 10, Oct. 2014

Fig. 8 Osmotic and mean activity coefficients of PrBr3 (Plot of logarithm of mean (molal) activity coefficient (γ±) versus molal concentration (m). For PrBr3, clim=1.58 mol/L (1.66 mol/kg). Continuous line: this work (QRL); asterisks: experimental values. Plot of osmotic coefficient (φ) versus molal concentration (m): dashed line, this work (QRL); points, experimental values)

ionic diameters defined, conversely, from reference-ion positions. Most pseudo-lattice models are local-composition models, that is, they consider locally ordered (or partially ordered) structures in the solution. With QRL, local structures (microscopic scale) are implicitly modelled, as previously said, so short-range interactions are let to assume a generic expression, which allows for a wider flexibility of the model. However, at present this also limits the available knowledge from a microscopic point of view. These limitations will be explored in future, with particular emphasis on the radial distribution functions and their implications through the KB Integrals[33], though conscious that Structural Analysis and/or sophisticated Molecular Dynamics[46] seem currently the best strategies for a true microscopic-scale investigation. During decades, potentialities offered by single-parameter theories have been more or less retrenched while attempting the exploration of the whole set of thermodynamic functions concerned with thermal and volumetric properties of electrolyte solutions. This will be, of course, a great challenge also for QRL, encouraged, however, by a well-defined parametrization with experimental validation available in a relevant number of cases. Acknowledgements: The author is greatly indebted with Prof. Bruno Bianco, DITEN (Department of Naval, Electric, Electronic and Telecommunications Engineering), Faculty of Engineering, University of Genoa, Italy, for his precious comments and remarks.

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