Aqueous Binary Lanthanide(III) Nitrate Ln(NO3)3 Electrolytes

Article pubs.acs.org/jced

Aqueous Binary Lanthanide(III) Nitrate Ln(NO3)3 Electrolytes Revisited: Extended Pitzer and Bromley Treatments Sayandev Chatterjee, Emily L. Campbell, Doinita Neiner, Natasha K. Pence, Troy A. Robinson, and Tatiana G. Levitskaia* Pacific Northwest National Laboratory, Richland, Washington 99354, United States

Downloaded by UNIV OF CALIFORNIA SAN DIEGO on September 13, 2015 | http://pubs.acs.org Publication Date (Web): September 11, 2015 | doi: 10.1021/acs.jced.5b00392

S Supporting Information *

ABSTRACT: To date, only limited thermodynamic models describing activity coefficients of the aqueous solutions of lanthanide ions are available. This work expands the existing experimental osmotic coefficient data obtained by classical isopiestic technique for the aqueous binary trivalent lanthanide nitrate Ln(NO3)3 solutions using a combination of water activity and vapor pressure osmometry measurements. The combined osmotic coefficient database for each aqueous lanthanide nitrate at 25 °C, consisting of literature available data as well as data obtained in this work, was used to test the validity of Pitzer and Bromley thermodynamic models for the accurate prediction of mean molal activity coefficients of the Ln(NO3)3 solutions in wide concentration ranges. The new and improved Pitzer and Bromley parameters were calculated. It was established that the Ln(NO3)3 activity coefficients in the solutions with ionic strength up to 12 mol kg−1 can be estimated by both Pitzer and singleparameter Bromley models, even though the latter provides for more accurate prediction, particularly in the lower ionic strength regime (up to 6 mol kg−1). On the other hand, for the concentrated solutions, the extended three-parameter Bromley model can be employed to predict the Ln(NO3)3 activity coefficients with remarkable accuracy. The accuracy of the extended Bromley model in predicting the activity coefficients was greater than ∼ 95 % and ∼ 90 % for all solutions with the ionic strength up to 12 mol kg−1 and 20 mol kg−1, respectively. This is the first time that the activity coefficients for concentrated lanthanide solutions have been predicted with such a remarkable accuracy.

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INTRODUCTION The closing of the nuclear fuel cycle remains a cornerstone issue for the successful advancement of nuclear power production in the United States. This necessitates separation of the fission products including lanthanides from the used nuclear fuel prior recycling; however, there currently are no approved methods for industrial reprocessing of the used nuclear fuel in the United States.1 Development of industrially viable methods to separate the trivalent lanthanide fission products from the minor actinide elements americium and curium is driven by the prospect of reducing the long-term radiological hazard associated with used nuclear fuel.2,3 Despite the decades-long scientific efforts, separation of the trivalent lanthanides from the chemically similar trivalent minor actinides remains a challenging step in closing the nuclear fuel cycle.4 The high ionic strength of the reprocessing solutions results in nonideal solution behavior and complicates design of a liquid−liquid extraction separation process to achieve controlled selective separation of the species of interest. Development of the successful separation schemes and their translation to the industrial scale requires thorough understanding of the extraction mechanisms and reliable predictive modeling capabilities, which, in turn, rely on the availability of the accurate thermodynamic frameworks over a wide range of concentrations and conditions for both aqueous electrolyte and organic extractant solutions. To date, only limited thermodynamic data for the aqueous solutions of lanthanide and actinide ions are © XXXX American Chemical Society

available, which makes the development of predictive modeling challenging. The objective of this work was to address this gap and to expand existing experimental osmotic coefficient data for the binary trivalent lanthanide nitrate solutions and use them in combination with the available literature data to improve prediction of the trivalent lanthanide activity coefficients using extended Pitzer and Bromley thermodynamic models. Literature describes many methods for the estimation of the activity coefficients of strong aqueous electrolytes.5 In this work, Bromley6,7 and Pitzer8−11 models were further developed and tested for the prediction of the activity coefficients of the trivalent lanthanide nitrates over a wide range of concentrations. This choice was based on the consideration that Pitzer and Bromley methods provide for the benefit of expanding quantitative evaluation of the activity coefficients from the binary electrolyte solutions to the complex multicomponent mixture and therefore can be developed for the characterization of the radioactive streams generated in the used nuclear fuel reprocessing and quantification of the nonideality effects in any aqueous electrolyte solution applicable to fuel cycle separations. Both models incorporate the Debye−Hückel12,13 term treating the long-range electrostatic interactions taking place Received: May 5, 2015 Accepted: August 27, 2015

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DOI: 10.1021/acs.jced.5b00392 J. Chem. Eng. Data XXXX, XXX, XXX−XXX


Journal of Chemical & Engineering Data

Article

osmotic coefficient data sets, substantially improved the accuracy of the calculated activity coefficients in the corresponding binary and Nd(NO3)3/HNO3 ternary systems with the ionic strength up to 6 mol kg−1. For the solution with ionic strengths higher than 6 mol kg−1, the error of prediction rapidly increases due to the exponential nature of the Bromley function, and further modification of the Bromley model is needed.19,21 In his work, Bromley had suggested the use of a three parameter model for strong electrolytes where two additional terms with the 3/2 2 and D(3) higher power of ionic strength (I) C(3) MXI MXI are added to his single parameter model to account for the enhanced ion interactions in the concentrated solutions.7 The objective of this work was in part to test this approach and to develop threeparameter Bromley model for the lanthanide nitrate series so that it may be effectively used to determine lanthanide activity coefficient data in the complex nitrate solutions over the wide concentration range.

between the ions of opposite charge. To account for the shortrange forces, both Pitzer and Bromley treatments add higher terms depending on the solution ionic strength. The semiempirical ion-interaction model of Pitzer is based in part on a physicochemical approach built on statistical mechanical calculations and is one the most widely used models for predicting strong electrolyte activity coefficients. This model employs a set of equations describing experimental osmotic coefficients values of an electrolyte solution in terms of a virial expansion of the excess Gibbs free energy. Virial coefficients are given by a linear combination of parameters, which can then be used to quantify the interactions among ions and the solvent and to obtain activity coefficients. Using original experimental osmotic coefficient thermodynamic data, Pitzer had employed his model to determine a set of parameters for the lanthanide nitrates, chlorides, and perchlorates.10 Rard and co-workers had subsequently expanded thermodynamic osmotic coefficient experimental data for the entire lanthanide nitrate salts with the exception of Ce(NO3)3 and Pm(NO3)3 and have evaluated the corresponding set of Pitzer parameters.14−17 This complete set of parameters describing short-range interactions of the ion pairs, i.e., β(0) and β(1) parameters, and triplets, that is, C(0) have been summarized by Pitzer.11 Although to date this remains the most comprehensive thermodynamic model for the aqueous solutions of trivalent lanthanide nitrates, it is limited to the solutions with ionic strength less than 6 mol kg−1−12 mol kg−1, which is a general constraint of the Pitzer treatment.11 The other peculiar consideration is that these Pitzer parameters were calculated using an assumption that the β(1) parameter does not exhibit systematic trend across the LnX3 series, and therefore, it was set to an identical value for each salt of the same anion (perchlorate, chloride, or nitrate) throughout the entire lanthanide series.10 Pitzer himself noted the potential inconsistency of this approach since the individual differences in ∂β(1)/∂T for each lanthanide salt appeared to be significant. It was of our interest to obtain a refined set of the Pitzer parameters by simultaneously adjusting three β(0), β(1), and C(0) variables for each lanthanide nitrate and to explore whether the determined parameters exhibit systematic trend across the lanthanide series leading to the improved estimates of the activity coefficients. Furthermore, in order to extend applicability of the Pitzer treatment to the concentrated electrolyte solutions, we obtained an expanded set of Pitzer parameters including β(0), β(1), β(2), C(0), C(1), C(2), and C(3) based on a modified model proposed by Archer adding the ionic strength dependence into the third virial coefficients18 and explored the accuracy of this expanded model for the prediction of activity coefficients. For the complex aqueous electrolyte solutions with high ionic strength typical for the used nuclear fuel dissolved in the concentrated nitric acid, the empirical Bromley model can potentially offer an accurate prediction of the activity coefficients for the trivalent lanthanides for the solutions with the ionic strength greater than 6 mol kg−1. The Bromley method expands the Debye−Hückel approximation12,13 by adding higher empirical terms depending on the solution ionic strength. This approach has been extensively used in calculations of the activity coefficients in aqueous solutions of 1:1, 2:2, 1:2, and 2:1 electrolytes,7,19,20 but applied only scarcely for 3:1 lanthanide salts,7,19,21 for which significant differences were observed between the calculated and experimental activity coefficients. Recent work by Lalleman et al. has demonstrated that new Bromley single parameters obtained for aqueous Nd(NO3)3 and NdCl3 binary systems by utilizing the expanded experimental

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EXPERIMENTAL SECTION Materials. The inorganic salts used to prepare aqueous electrolyte solutions were La(NO3)3·6H2O, 99.999 %; Ce(NO3)3· 6H2O, 99.999 %; Nd(NO3)3·6H2O, 99.9 %; Sm(NO3)3·6H2O, 99.9 %; Eu(NO3)3·6H2O, 99.9 %; Gd(NO3)3·6H2O, 99.9 %; Tb(NO3)3·5H2O, 99.9 %; Dy(NO3)3·xH2O, 99.9 %; Ho(NO3)3· 5H2O, 99.9 %; Er(NO3)3·5H2O, 99.9 %; Tm(NO3)3·5H2O, 99.9 %; Yb(NO3)3·6H2O, 99.9 %; and Lu(NO3)3·5H2O, 99.9 % obtained from Sigma-Aldrich and Pr(NO3)3·6H2O, 99.99 % obtained from Alfa Aesar. All aqueous solutions were prepared using deionized quartz-pure water; a quartz double distillation system was used to purify the water used in these experiments (procedure can be found at http://www.avexports.net/water-distiller-quartz.html). The solutions were prepared by weighing the salts and the water, and concentrations were recorded in molality. Density measurements were conducted at 25 °C and used to determine the salt concentrations using 10 mL AASHTO standard T-74 picnometer (Thomas Scientific) equipped with a calibrated thermometer. Each solution was measured at least five times to provide a measurement precision of ± 0.0002 g cm−3 for density and ± 0.03 °C for temperature. The literature reported density values for the lanthanide nitrate solutions were then used to verify the solution concentrations.22−24 Sample Description Table chemical name

source

initial mole fraction purity

La(NO3)3·6H2O Ce(NO3)3·6H2O Pr(NO3)3·6H2O Nd(NO3)3·6H2O Sm(NO3)3·6H2O EuNO3)3·6H2O Gd(NO3)3·6H2O Tb(NO3)3·5H2O Dy(NO3)3·xH2O Ho(NO3)3·5H2O Er(NO3)3·5H2O Tm(NO3)3·5H2O Yb(NO3)3·6H2O Lu(NO3)3·6H2O

Sigma-Aldrich Sigma-Aldrich Alfa Aesar Sigma-Aldrich Sigma-Aldrich Sigma-Aldrich Sigma-Aldrich Sigma-Aldrich Sigma-Aldrich Sigma-Aldrich Sigma-Aldrich Sigma-Aldrich Sigma-Aldrich Sigma-Aldrich

99.999 % 99.999 % 99.99 % 99.9 % 99.9 % 99.9 % 99.9 % 99.9 % 99.9 % 99.9 % 99.9 % 99.9 % 99.9 % 99.9 %

Methods. Water activity (aw) measurements for the solutions with aw < 0.97 were performed using a commercial hydrometer, AquaLab 4ETV Water Activity Meter (Novasina AG, Lachen B




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for the solutions in the entire available concentration range. These databases corresponding to the aqueous binary Ln(NO3)3 solutions with the concentration up to 2 mol kg−1 were also used to calculate and evaluate the new and improved Pitzer and Bromley parameters, which can be utilized to determine the Ln(NO3) 3 activity coefficients in the complex mixtures in future. The calculated corresponding mean molal activity coefficient values were then calculated utilizing Pitzer and Bromley treatments and cross-validated using osmotic coefficient data independently obtained in this work. The comparison with the Pitzer and Bromley parameters obtained in this work with the previous literature values is also presented. Experimental Activity Coefficients. To calculate the experimental mean molal activity coefficients, the approach successfully employed by Rard and co-workers for the rare-earth binary electrolyte solutions was applied.14,15,17,24 The osmotic coefficients of the lanthanide nitrate solutions in the entire available concentration range were fitted by the empirical polynomial relation 314,15,17,24

Switzerland) consisting of a Lab Master controller unit coupled with three Lab Partners units. The water activity meter was calibrated using six standards provided by Novasina, SAL-T consisting of saturated salts solutions of known relative humidities: 0.973 ± 0.003 (K2SO4, > 25 %), 0.843 ± 0.003 (KCl, > 25 %), 0.753 ± 0.003 (NaCl, > 25 %), 0.576 ± 0.003 (NaBr, > 25 %), 0.328 ± 0.003 (MgCl2, > 25 %), and 0.113 ± 0.003 (LiCl, > 25 %). In addition, quartz pure water was used as a standard with a water activity aw = 1. All water activity measurements were performed at 25.0 ± 0.1 °C and each chamber was protected by the Novasina EVC-21 filter. At least three replicate measurements were conducted for each sample. The sample was placed in the measurement chamber and allowed to equilibrate at 25 °C for 30 min prior to each measurement. The free water present in the sample humidifies or dehumidifies the air volume inside the chamber until the partial pressure of water vapor saturation is zero. The instrument temperature and pressure changes inside the chamber are monitored via a computer interface. If these parameters remain stable over 1 h to 2 h (adjustable by the user), then the instrument converts that into aw. The osmotic coefficients of the Ln(NO3)3 solutions were calculated from the water activity by eq 1 φ=

−1000ln(a w ) νmAX M H2O

φ=1−

⎛ A ⎞ 1/2 ⎜ ⎟m + ⎝3⎠

∑ A i mr

i

(3)

i

where A = 8.6430 is the Debye−Hückel limiting slope for 3:1 electrolytes, where Ai is the least-squares coefficient and ri is empirical power of molalities. Using the values of Ai and ri obtained through a least-squares fit of eq 3, mean molal activity coefficients γ± were determined using the relation 4:

(1)

where aw is the water activity, φ is the osmotic coefficient, ν is the stoichiometric number of ions in electrolyte sample AX, m is the molality, and MH2O is the molecular weight of water. The osmotic coefficients of the dilute lanthanide nitrate solutions with the water activity aw > 0.97 were determined from the osmolality (mosm) values measured using a vapor pressure osmometer (VPO) Vapro 5520 (Wescor, Inc., Logan, UT) at 25 °C. The VPO instrument was placed in a glovebox enclosure maintained at 25.0 ± 0.2 °C using a 75 W incandescent light bulb controlled by a JKEM Scientific Model 270 temperature controller. Cooling was achieved by a Neslab CFT 25 recirculating chiller connected to a copper coil heat exchanger. A fan was used to circulate air in the incubator to maintain a constant temperature. The fan was turned off momentarily during sample injection into the instrument to avoid evaporation of the analyzed sample. The instrument was calibrated using the manufacturer supplied osmolality calibration standards (100, 290, 1000 mmol kg−1 NaCl in water). The standard error of the VPO measurements was determined to be < 3 % from the replicate (five or greater) measurements. The solution osmotic coefficients were determined from the measured osmolality according to eq 2 mosm φ= 1000νmAX (2)

ln γ± = −Am1/2 +

⎛ ri + ⎝ ri

∑ Ai ⎜ i

1 ⎞ ri ⎟m ⎠

(4)

The refined polynomial coefficients Ai and ri as well as calculated experimental γ± were found to be consistent with the values previous reported by Rard and co-workers for all Ln(NO3)3; the corresponding data for Ce(NO3)3 were obtained for the first time. Pitzer Model. The detailed description of the Pitzer treatment of the strong electrolyte solutions can be found elsewhere,9,11,28 and only the set of relations utilized in this work is described below. According to the Pitzer model, osmotic coefficient is related to the binary electrolyte molality by the eq 511,25 ⎡ 2(ν ν )3/2 ⎤ ⎛ 2ν ν ⎞ φ φ M X ⎥CMX φ − 1 = |z Mz X|f φ + m⎜ M X ⎟BMX + m2⎢ ⎝ ν ⎠ ⎢⎣ ⎥⎦ ν (5)

where m is the stoichiometric molality, ν is the total number of ions formed upon complete electrolyte dissociation, νM and νX are the stoichiometric number of anions and cations formed upon dissociation, and zM and zX are the charges of the cation and anion, respectively. Function f φ is defined as

The water activity was calculated using relationship 1. Data Treatment and Computational Methods. Approach. The experimental water activity and osmotic coefficient data sets obtained in this work by the combination of VPO measurements at low concentration regimes and water activity measurements at high concentration regimes exhibited excellent agreement with literature results obtained using classical isopiestic and other techniques (Figure 1).14−17,23,25−27 On the basis of this agreement, a combined experimental osmotic coefficient database, which included literature-reported values and the results obtained in this work, was created for each lanthanide nitrate and used to calculate experimental mean molal activity coefficients

fφ = −

AφI1/2 (1 + bI1/2)

(6) 10

where b is a constant set to 1.2, and Aφ is the Debye−Hückel solvent-dependent limiting slope for osmotic coefficient (for the aqueous electrolyte solutions the solvent is water) and is defined by eq 7 3/2 1/2 1 ⎛ 2πN0d w ⎞ ⎛ e 2 ⎞ ⎟ ⎜ Aφ = ⎜ ⎟ 3 ⎝ 1000 ⎠ ⎝ εkT ⎠

C

(7) DOI: 10.1021/acs.jced.5b00392 J. Chem. Eng. Data XXXX, XXX, XXX−XXX



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Figure 1. Comparison of the experimental (blue diamonds) and previously reported (red line)14,15,17,23,24 water activity and osmotic coefficient data for the aqueous binary Ln(NO3)3 electrolytes as a function of molality and molality1/2, respectively, at 25 °C.

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3 φ CMX (16) 2 To extend applicability of the Pitzer treatment to the concentrated electrolyte solutions, Archer proposed a modified model by adding the ionic strength dependence into the third virial coefficient.18 In a later work, Pitzer et al.30 developed a generalized extended ion-interaction model. The detailed description of the extended Pitzer treatment and its application to the aqueous solutions of lanthanide chlorides, nitrates, and perchlorates as well as their mixtures can be found elsewhere,31,32 the corresponding set of the molality-based equations utilized in this work is described below.33 Pitzer’s original ion-interaction equation for molality based osmotic coefficient φ of a binary solution of a general electrolyte MX may be written in the general extended form in the highly soluble and highly unsymmetrical 3:1 electrolytes as shown in eq 1730,31

where N0 is Avogadro’s number, dw is density of water at temperature T, e is electronic charge, k is Boltzmann’s constant, ε is the dielectric constant or relative permittivity of water. For water at 25 °C, the Aφ value is equal to 0.391. Solution ionic strength I is given by eq 8 1 I = ∑ mizi2 (8) 2 and for a 3:1 electrolyte systems I = 6m. Parameters BφMX and CφMX in eq 5 are defined by eqs 9 and 10, respectively φ BMX = β (0) + β (1)exp( −αMXI1/2)

(9)

φ CMX = C(0) + C(1)exp( −ωMX I1/2) −1/2

(10)

where α = 2 kg mol for 3:1 electrolytes, ω = 2.5 kg mol−1/2, (0) (1) (0) and β , β , C , and C(1) are the Pitzer parameters; β(0) is the pairwise short-range interaction parameter, β(1) is the combination of the pairwise short and long-range interactions, and C(0) and C(1) are the triple interaction parameters. Combining eqs 9 and 10 with the eq 5 results in eq 11, which relates the experimental osmotic coefficient data to the Pitzer parameters 1/2


γ CMX =

φ−1=−

|z Mz X|AφI1/2 1 + bI

1/2

1/2

⎛ I1/2 ⎞ ⎛ 2ν ν ⎞ ⎟ + ⎜ M X ⎟m φ − 1 = −|z Mz X|Aφ⎜ ⎝ 1 + bI1/2 ⎠ ⎝ ν ⎠ × {β (0) + β (1)exp( −α B1I1/2) ⎛ 2(ν ν )3/2 ⎞ φ M X ⎟⎟m2CMX + β (2)exp(−α B2I1/2)} + ⎜⎜ ν ⎝ ⎠

+ m(2νMνX)

⎡ (ν ν )3/2 ⎤ φ ⎥CMX × [β (0) + β (1)exp( − αMXI1/2)] + m2⎢2 M X ⎢⎣ ⎥⎦ ν (11)

(17) φ CMX = 2[C(0) + C(1)exp( −αC1I ) + C(2)exp( −αC2I )

+ C(3)exp(−αC3I )]

Incorporating the eq 8 and values for b, Aφ, ν, νM, νX, zM, and zX in the eq 11 and setting CφMX = C(0) MX based on the Pitzer recommendation,10 the Pitzer equation for 3:1 electrolyte system can be represented as φ−1=−

The corresponding expression for the molality-based mean activity coefficient is given by eq 19. ⎫ ⎧⎛ I1/2 ⎞ ⎛ 2 ⎞ 1/2 ⎜ ⎟ln(1 + bI ⎬ ⎟ ln γ± = −|z Mz X|Aφ⎨⎜ ) + 1/2 ⎩⎝ 1 + bI ⎠ ⎝ b ⎠ ⎭

2.8732·m1/2 (1 + 2.9394·m1/2)

⎛3 ⎡3 ⎤ 3 ⎟C(0) + m·⎢ β (0) + β(1)exp( −4.8988.m1/2)⎥ + m2⎜ ⎣2 ⎦ 2 ⎝ 2 ⎠ (12) 3/2 ⎞

+

where (3/2)β(0), (3/2)β(1), and (33/2/2)C(0) are the three Pitzer coefficients for 3:1 electrolyte. Although in the previous calculations the (3/2)β(1) parameter was fixed at 7.7 for lanthanide nitrates,10,17,24,29 in this work, all three Pitzer parameters are refined simultaneously using osmotic coefficient data for the solutions with the ionic strength up to 12 mol kg−1. The Pitzer parameters are then used to calculate the mean molal activity coefficients via eq 13, which are compared with the corresponding experimental activity coefficients.

⎪

⎪

⎪

⎪

⎡ ⎛ 2β (1) ⎞ ⎛ 2νMνX ⎞ (0) ⎟m × ⎢2β + ⎜⎜ 2 ⎟⎟ ⎢ ⎝ ν ⎠ ⎝ α B1 I ⎠ ⎣

⎜

⎧ ⎫ ⎛ α B 2I ⎞ ⎪ ⎪ × ⎨1 − ⎜⎜1 + α B1I1/2 − 1 ⎟⎟exp( −α B1I1/2)⎬ ⎪ ⎪ 2 ⎠ ⎝ ⎩ ⎭ ⎧ ⎛ ⎛ 2β (2) ⎞⎪ α B 2I ⎞ + ⎜⎜ 2 ⎟⎟⎨1 − ⎜⎜1 + α B2I1/2 − 2 ⎟⎟ ⎪ 2 ⎠ ⎝ α B 2 I ⎠⎩ ⎝ × exp( −α B2I

⎡ 2(ν ν )3/2 ⎤ ⎛ 2ν ν ⎞ γ γ M X ⎥C MX ln γ± = |z Mz X|f + m⎜ M X ⎟BMX + m2⎢ ⎝ ν ⎠ ⎢⎣ ⎥⎦ ν γ

1/2

⎫⎤ ⎛ 3(νMνX)3/2 ⎞ 2 φ ⎟⎟m CMX )⎬⎥ + ⎜⎜ ⎪⎥ ν ⎠ ⎭⎦ ⎝ ⎪

(19)

Bromley Model. The set of Bromley relations utilized in this work is described below, for the detailed description of the Bromley treatment of the strong electrolyte solutions readers are referred to the original publications.6,7 According to Bromley, the mean molal activity coefficients of strong electrolytes in aqueous solutions can be described by the modified form of the Debye− Hückel eq 20 with added higher terms for the solution ionic strength.

(13)

The ionic strength-dependent variables in eq 14 are defined by the relations 14−16. ⎡ I1/2 ⎤ 2 1/2 ⎥ f γ = − Aφ ⎢ + ln(1 + bI ) b ⎣ 1 + bI1/2 ⎦

(18)

(14)

⎛ 2β (1) ⎞⎡ ⎛ 1 ⎞ −αI1/2 ⎤ γ ⎟e BMX = 2β (0) + ⎜⎜ 2 ⎟⎟⎢1 − ⎜1 + αI1/2 − ⎥ ⎝ ⎦ 2α 2I ⎠ ⎝ α I ⎠⎣

log γ± = −A m |z Mz X|

(15)

Ic 1 + Bc a ̇ Ic

+ βmI + CI 2 + DI 3 + ... (20)

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The constants βm, C, D, ... are empirical and are not assigned any physical meaning. Am is the Deby-Hückel constant for an activity coefficient and has a value 0.5098 kg1/2 mol−1/2 at 25 °C. This model makes no distinction among ions of the same charge and cannot be applied for a solution with ionic strength > 1 mol kg−1 and to multicomponent systems. To address these drawbacks, Bromley later proposed an advanced single-parameter model, which takes into account the specificity of every ion present in the medium to evaluate the activity coefficient of an electrolyte in a binary solution up to 6 mol kg−1.7


log γ± = −A m |z Mz X| +

I I

1+

+

(1) (0.06 + 0.6BMX )I

(

1+

1.5 I |z M z X |

2

)

(1) BMX I

Bromley parameters for the entire Ln(NO3)3 series. It was found that the obtained parameters were able accurately estimate the Ln(NO3)3 activity coefficients for even more concentrated solutions with the ionic strength up to 20 mol kg−1 as described below.

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RESULTS AND DISCUSSION Experimental Determination of Activity Coefficients. The water activity and osmotic coefficient data measured in this work for the binary Ln(NO3)3 solutions in wide concentration range were plotted as a function of molality and the square root of molality, respectively (Figure 1, blue diamonds). Comparison with the previously reported data collected using classical isopiestic method (Figure 1, red line)14,15,17,23,24 indicates that the technique based on the combination of the water activity and VPO measurements utilized in this work is in an excellent agreement with the isopiestic method. The values of osmotic coefficients are also in good agreement with those calculated by Libus et al. for Nd(NO3)3, Sm(NO3)3, and Gd(NO3)3.35 To obtain experimental mean molal activity coefficients, the combined osmotic coefficient databases, which included previous literature and obtained in this work measurements for each Ln(NO3)3, were fitted to the eq 3. Based on this expression, the least-squares coefficients Ai for each value of the empirical power of molality ri were determined (Supporting Information Table S1) and found to be very similar to the values originally reported by Rard and co-workers.14,15,24 The polynomial coefficients for Ce(NO3)3 are reported in this work for the first time. The obtained polynomial coefficients were used to calculate the mean molal activity coefficients by eq 4 (Table S1). For all dilute Ln(NO3)3 solutions, the activity coefficients initially decreased as the ionic strength increased (Figure 2 and Supporting Information Figure S2, green circles). For the solutions with the molality greater than 1 mol kg−1 (I = 6 mol kg−1), the dependence of the activity coefficients on the ionic strength is drastically different for the early and late lanthanide nitrates. For La3+ through Tb3+ nitrates, the activity coefficients increase gradually and remain less than 1 even for the extremely concentrated solutions. For the late Ln(NO3)3, the activity coefficients undergo steep increase as the solution ionic strength becomes greater than 4 mol kg−1. These systematic trends can be attributed to the structure of the lanthanide aqua ions and their ability to incorporate water and nitrate anion in the inner coordination sphere. In the dilute aqueous solutions, the first coordination sphere of Ln3+ exhibits the hydration number changing from nine for the light ions (La3+ to Sm3+) to eight for the heavy ions (Dy3+ to Lu3+) in accord with the contraction of the lanthanide ionic radii in the same order. For the middle Eu3+ to Tb3+ region of the lanthanide (III) series, intermediate (between 9 and 8) hydration numbers are proposed.36 Monte Carlo simulations of Ln3+(aq) ions proposed a dissociative substitution mechanism and very fast exchange between the bulk and hydrated water during the change in coordination number from 9 to 8 in mid lanthanide series.37 The second coordination sphere of Ln3+ contains as many as 17 to 18 water molecules. As the aqueous concentration of the lanthanide salt increases, the average number of the water molecules in the second coordination sphere decreases due to the preferential coordination of anions. This process is more pronounced for the late Ln3+ with the small ionic radii and responsible for the gradual reduction of the water activity and increase of the osmotic coefficient from the early to the late lanthanides. Pitzer Model. Three-Parameter Pitzer Model. Pitzer parameters for each binary Ln(NO3)3 electrolyte were obtained

| z Mz X |

(21)

B(1) MX

The is an empirical constant for a given electrolyte MX, which signifies the overall electrolyte contribution to the activity coefficient and is described by a combination of the empirical constants for contributions of each of the individual ions (BM, BX) and their interactions(δM, δX) eq 22 (1) BMX = BM + BX + δMδ X

(22)

The individual ion contributions BM and δM have been previously reported for the lighter lanthanides (La to Eu)7 but not for the heavier lanthanides (Gd to Lu), making the applicability of the model to the entire lanthanide series difficult. Further, even for the cases where the B(1) MX values and the individual contributions are reported, they are in poor agreement with the experimental data. This problem is in part due to the fact that only limited experimental data existed at the time of Bromley’s work, and the B(1) MX values were determined using isopiestic data collected over small data range mostly at low ionic strength (< 1 mol kg−1) resulting in large overestimation of activity coefficients at high ionic strengths.7,19 Lalleman et al.19 has recently shown that a wide range of the osmotic coefficient and activity coefficient experimental data collected over an wider range of ionic strengths can be used to obtain new and improved B(1) MX values for the binary 3:1 electrolyte systems, which are able to accurately predict activity coefficients for Nd(NO3)314 and NdCl334 within the range 0 mol kg−1 to 6 mol kg−1. In this work, the single-parameter Bromley method was utilized to determine B(1) MX for the entire series of lanthanide nitrates (except Pm). For the electrolyte solutions with ionic strength greater than 6 mol kg−1, the Bromley model based on the single interaction parameter significantly overestimates the activity coefficient values. In order to address this problem, Bromley recommended 3/2 2 addition of the new empirical terms C(3) and D(3) MXI MXI to eq 21 to account for the ionic interactions at the higher concentrations.7 The expanded three parameter Bromley equation is log γ± = −A m |z Mz X|

I 1+

I

+

(3) (0.06 + 0.6BMX )I

(

1+

(3) (3) 3/2 (3) 2 + BMX I + CMX I + DMX I

1.5 I |z M z X |

2

)

| z Mz X |

(23)

B(3) MX

where is the modified empirical B constant for the three parameter model. In this work, eq 23 was used to fit the experimental activity coefficients for the solutions with the ionic (3) (3) strength up to 12 mol kg−1 and to determine B(3) MX, CMX, and DMX F



Article

by fitting osmotic coefficient data as a function of the square root of molality according to eq 12. In the fitting, β(0), β(1), and C(0) Pitzer parameters were refined simultaneously (Supporting Information Figure S1). The contribution of the fourth C(1) term


was found insignificant, and therefore, it was not considered for the refinement. Osmotic coefficient data for the solutions with molality up to 2 mol kg−1 were used in the fitting and the obtained results are shown in Supporting Information Table S1.

Figure 2. continued G




Article

Figure 2. A plot of activity coefficients at 25 °C as a function of ionic strength for the binary Ln(NO3)3 solutions. Symbols: experimental activity coefficients, this work. Lines represent estimated activity coefficients using the following treatments. Solid blue line: three-parameter Pitzer model eq 13 using literature values of β(0), β(1), Cφ parameters taken from Pitzer, 199138 (Table 1). Dashed red line: three-parameter Pitzer model eq 13 using values of β(0), β(1), Cφ parameters reported in Table 1, this work. Dashed green line: extended seven-parameter Pitzer model (eq 19) using literature values of β(0), β(1), β(2), C(0), C(1), C(2), and C(3) parameters taken from Wang et al.31 (Table 2). Solid brown line: extended seven-parameter Pitzer model (eq 19) using literature values of β(0), β(1), β(2), C(0), C(1), C(2), and C(3) parameters reported in Table 2, this work.

Figure 3. Variation of the three Pitzer parameters for the aqueous binary Ln(NO3)3 electrolytes with ionic radii of Ln3+: red symbols, this work; black symbols, literature values reported by Pitzer;38 green symbols, values of the (3/2)β(0) parameters for LnCl3 taken from Pitzer et al.38 Ln3+ radii taken from Marcus.41

light and heavy lanthanides, respectively. This finding suggests the important contribution of the (3/2)β(1) parameter to the second virial coefficient for Ln(NO 3)3. The (33/2/2)C(0) parameter exhibits only minor variation across the lanthanide series indicating similar extend of the triple ion interactions for the entire series of lanthanide nitrates. Although physical meaning and interpretation of the Pitzer treatment in terms of interionic forces is not trivial, a qualitative discussion is possible. In the Pitzer treatment,9,28 the BφMX virial coefficient is defined by (3/2)β(0) and (3/2)β(1) parameters and represents a weighted mean of the interactions of pairs of ions with charges (+)(−), (+)(+), and (−)(−). Parameter (3/2)β(0) primarily reflects short-range (+)(−) interactions, which dominate in the entire range of the solution ionic strength but is especially important in the dilute solutions. As the solution ionic strength increases, the relative contribution of the (+)(+) and (−)(−) interactions and therefore of the (3/2)β(1) parameter to the second virial coefficient becomes more significant. The third virial coefficient is usually small and sometimes negligible.

The simultaneous refinement of the β(0), β(1), and C(0) parameters adopted in this work resulted in the smaller values of the (3/2)β(0) parameter for all lanthanide nitrates as compared with the literature values obtained by fixing the value of the (3/2)β(1) parameter at 7.70. On the other hand, the refined (3/2)β(1) values were on most occasions greater than the fixed 7.70 value. The newly obtained (33/2/2)C(0) values were found to be slightly greater than the literature values. To examine whether the determined Pitzer parameters exhibit systematic trend across the lanthanide series, they were plotted against the lanthanide radius (Figure 3). It was observed that the values of the (3/2)β(0) parameter remain nearly identical within experimental error for the light Ln(NO3)3 as the ionic radius decreases from La3+ to Eu3+ (Table 1). This is followed by steep increase of (3/2)β(0) from Gd3+ to Tm3+. Subsequently, from Tm3+ to Lu3+ the (3/2)β(0) values are plateaued. Very similar trend was observed for the previous literature (3/2)β(0) values. As the lanthanide radius decreases, the newly obtained values of the (3/2)β(1) parameter exhibit gradual and steep increase for the H



Article

Table 1. Comparison of the Fitting Parameters of Three-Parameter Pitzer Model Obtained in This Work and Reported in the Literature for the Aqueous Binary Ln(NO3)3 Electrolytes at 25°C38


(3/2)β(0)

(3/2)β(1)

103σ

(33/2/2)C(0)

Ln(NO3)3

literature

this work

literature

this work

literature

this work

this work

La(NO3)3 Ce(NO3)3 Pr(NO3)3 Nd(NO3)3 Sm(NO3)3 Eu(NO3)3 Gd(NO3)3 Tb(NO3)3 Dy(NO3)3 Ho(NO3)3 Er(NO3)3 Tm(NO3)3 Yb(NO3)3 Lu(NO3)3

0.737 NR 0.724 0.702 0.701 0.713 0.776 0.838 0.848 0.876 0.938 0.952 0.948 0.926

0.687 0.681 0.718 0.699 0.678 0.682 0.739 0.762 0.786 0.818 0.842 0.874 0.876 0.867

7.70 NR 7.70 7.70 7.70 7.70 7.70 7.70 7.70 7.70 7.70 7.70 7.70 7.70

7.23 8.00 7.27 7.73 7.81 7.99 7.91 8.43 8.35 8.24 8.65 9.01 8.74 9.32

−0.198 NR −0.173 −0.142 −0.131 −0.125 −0.170 −0.202 −0.181 −0.185 −0.226 −0.222 −0.208 −0.174

−0.143 −0.106 −0.151 −0.118 −0.108 −0.101 −0.128 −0.131 −0.138 −0.143 −0.147 −0.159 −0.161 −0.134

6.4 5.6 7.6 6.5 8.1 7.4 5.6 7.0 4.3 8.1 7.2 4.1 7.3 8.6

compared to the chlorides. Similarly, comparison of the (3/2)β(0) parameters for the 2:1 electrolytes also indicates that the (3/2)β(0) values are generally smaller for nitrate than for chloride salts regardless of the cation nature11 suggesting a higher tendency of the divalent metal nitrate salts toward ion pair formation. The new (3/2)β(0) values obtained in this work for Ln(NO3)3 follow this anticipated trend being smaller than the corresponding (3/2)β(0) values for LnCl3. Because the (3/2)β(0) and (3/2)β(1) parameters depend on the same properties of ions in a given solvent, the correlation between them is expected. Pitzer9 demonstrated that for 1:1 type electrolytes this correlation has tight boundaries, whereas 2:1 and 3:1 electrolytes exhibit significantly scattered but nevertheless apparent trends. It was of our interest to test whether there is a correlation between β(0) and β(1) parameters obtained in this work for Ln(NO3)3 and how it agrees with other 3:1 electrolytes. The corresponding plot constructed using data reported by Pitzer for 3:1 and 1:3 electrolytes and obtained in this work (Figure 4) demonstrates obvious interdependence between the (3/2)β(0) and (3/2)β(1) parameters, which can be approximately described by the linear correlation. Further, the (3/2)β(0) and

As for the activity coefficients, the observed systematic trends for the (3/2)β(0) and (3/2)β(1) parameters across lanthanide series can be attributed to the interplay among the lanthanide radius and coordination number, hydration and nitrate ion pair formation similarly as described before for the systematic trends of the activity coefficients across the series of Ln(NO3)3. The difference in the trends of the (3/2)β(0) and (3/2)β(1) values for the La3+ to Sm3+ and the rest of the group is due to the fact that the lighter lanthanides are nine coordinated in water (La3+ to Sm3+) while the remainder ones are eight coordinated.39 For the first four lanthanides, the lanthanide coordination sphere and the ionic radii allow an equal number of nitrate ions to be accommodated within the inner sphere with equal ease resulting in the little dependence of the (3/2)β(0 and (3/2)β(1) values on the lanthanide radius. Subsequently, from Sm3+ to Tm3+, as the ionic radius and the coordination number decreases, the ease of accommodating nitrate ions within the inner sphere becomes increasingly difficult which is reflected in the marked rise in these parameters. Seemingly, the ability of the heavier octacoordinated lanthanides to accommodate the nitrate ions within the inner sphere reach maximum capacity for Er3+, which still shows nitrate ions bound in bidentate manner, the total coordination number of Er3+ being 8. Further decrease of the cation radius creates progressive difficulty for the nitrate anion to replace water molecules in the inner sphere of the trivalent lanthanide cation leading to the formation of the outer-sphere type ion pairs.39 Raman studies of aqueous lanthanide nitrate solutions in the glassy state40 suggested that for Tm3+, Yb3+, and Lu3+ the inner-sphere lanthanide coordination is entirely comprised of water molecules. This extensive hydration of the late lanthanides is reflected in the high values of the (3/2)β(0 and (3/2)β(1) parameters. Another interesting observation arises from the comparison of the short-range pairwise interaction parameter (3/2)β(0) for the trivalent lanthanide nitrates with those of the chlorides.10 In the original Pitzer model, which utilized fixed (3/2)β(1) values, the (3/2)β(0) values for the last four lanthanide nitrates (Er3+, Tm3+, Yb3+, and Lu3+) are found to be greater than the respective chlorides as shown in Figure 3. This contradicts the wellestablished postulation that Ln3+ exhibit a greater tendency toward coordination of the NO3− anion in the inner-shell complex formation than of Cl−, which is expected to result in lower values of the (3/2)β(0) interaction parameters for nitrates

Figure 4. Relationship between the (3/2)β(0) and (3/2)β(1) Pitzer parameters for the aqueous binary 3:1 and 1:3 electrolytes at 25 °C: blue diamonds, data taken from Pitzer et al.;38 red diamonds, data obtained in this work for Ln(NO3)3; line represents the regression analysis of all data. Equation of the line: 3/2 β(1) = 4.6(3/2 β(0)) + 4.1; R2 = 0.7. I



Article

Table 2. Comparison of the Fitting Parameters of Extended Pitzer Model Obtained in This Work and Reported in the Literature for the Aqueous Binary Ln(NO3)3 Electrolytes at 25°Ca


(3/2)β(0)

La Ce Pr Nd Sm Eu Gd Tb Dy Ho Er Tm Yb Lu

(3/2)β(1)

(3/2)β(2)

(33/2)C(0)

(33/2)C(1)

(33/2)C(2)

(33/2)C(3)

103σ

ref 31

this work

ref 31

this work

ref 31

this work

ref 31

this work

ref 31

this work

ref 31

this work

ref 31

this work

this work

0.3129 NR 0.3342 0.3436 0.4200 0.4117 0.4159 0.4084 0.4396 0.4412 0.4597 0.4562 0.4310 0.5055

0.3429 0.3510 0.3621 0.3736 0.3990 0.4517 0.4020 0.3744 0.4284 0.4392 0.4427 0.3681 0.4310 0.5191

8.15 NR 7.80 7.32 7.10 7.00 7.47 7.82 8.00 8.17 8.41 8.45 8.38 8.11

9.5 9.8 8.9 8.25 8.9 7.9 9.6 9.9 8.54 8.34 9.08 10.8 9.2 9.9

−5.13 NR −4.07 −2.51 −2.20 −1.84 −3.48 −4.59 −5.45 −6.03 −7.15 −7.01 −6.50 −6.16

−5.17 −4.65 −4.07 −2.31 −2.20 −1.54 −5.00 −4.38 −6.23 −6.03 −6.35 −7.01 −6.50 −6.23

−0.01028 NR −0.01166 −0.01230 −0.02185 −0.01829 −0.01696 −0.01490 −0.01976 −0.01982 −0.02212 −0.02126 −0.01589 −0.02748

−0.02228 −0.02210 −0.02166 −0.02030 −0.02185 −0.02629 −0.01500 −0.00490 −0.01816 −0.02009 −0.02012 −0.00946 −0.01089 −0.02998

0.3138 NR 0.3420 0.3763 0.2559 0.3281 0.3718 0.4307 0.3330 0.3689 0.3805 0.5143 0.7046 0.5300

0.3178 0.3558 0.3420 0.3463 0.3359 0.2681 0.4318 0.4723 0.3540 0.3743 0.4265 0.6743 0.6246 0.5388

−0.3629 NR −0.4790 −0.5732 −0.2974 −0.5093 −0.6084 −0.7417 −0.3694 −0.3555 −0.2866 −0.5969 −1.0594 −0.7551

−0.2929 −0.3194 −0.4690 −0.5932 −0.2574 −0.4793 −0.6684 −0.6117 −0.3982 −0.3642 −0.2696 −0.4479 −0.8894 −0.7482

1.1857 NR 1.2981 1.3968 0.8970 1.1994 1.3051 1.6006 1.2190 1.2416 1.1361 1.4200 1.8758 1.4712

1.1857 0.9370 1.5157 1.4768 0.9370 1.2894 1.3851 1.6206 1.6760 1.2216 1.4791 1.4211 2.1558 1.5312

1.8 3.1 1.2 1.9 2.4 1.6 2.5 2.4 1.9 2.1 1.7 1.8 2.1 2.4

Constants used in literature:31 αB1 = 1.8 kg1/2 mol1/2, αB2 = 6.0 kg1/2 mol1/2, αC1 = 0.15 kg mol1/2, αc2 = 0.25 kg mol1/2, αc3 = 0.35 kg mol1/2. Constants used in this work based on Pitzer’s recommendations30,42 other than αB2 which is based on αB1 = 2.0 kg1/2 mol1/2, αB2 = 12.0 kg1/2 mol1/2, αC1 = 0.15 kg mol1/2, αc2 = 0.25 kg mol1/2, αc3 = 0.35 kg mol1/2. NR = not reported. a

(3/2)β(1) parameters obtained for Ln(NO3)3 in this work exhibit similar trend. Overall these findings directly support the simultaneous refinement and determination of three Pitzer parameters applied in this work. The activity coefficients for the Ln(NO3)3 solutions were calculated applying three-parameter Pitzer model eq 13 using both previously reported Pitzer coefficients as well as those obtained in this work listed in Table 1. Figure 2 compares the corresponding calculated activity coefficients (dark blue and dashed red lines, respectively) with the experimental (green circles) activity coefficients. The observed deviations of the activity coefficients calculated using Pitzer parameters obtained in this work from the experimental values obtained in this work are listed in Supporting Information Table S1. It is apparent that the new Pitzer parameters obtained in this work do not result in the consistent improvement of the activity coefficients estimated using original Pitzer parameters. Interestingly, both sets of Pitzer parameters predict an S-shaped behavior of the activity coefficient as a function of ionic strength, which is not shown by the experimental data within the ionic strength regime studied. Both sets of Pitzer parameters underestimate the activity coefficients for all Ln(NO3)3 electrolytes at low solution ionic strength up to about ∼ 6 mol kg−1 by 6 % to 13 %. At higher ionic strengths, there is a narrow ionic strength region (I ∼ 5 mol kg−1 to 9 mol kg−1) where the Pitzer parameters are observed to estimate the activity coefficients fairly accurately (∼ 95 % accuracy) for all lanthanides with the exception of Ce3+, Ho3+, Tm3+, and Lu3+. For Ce3+ and Ho3+, Pitzer parameters obtained in this work tend to slightly underestimate activity coefficients (deviation ∼ 4 % to 6 %), whereas for Tm3+ and Lu3+, they tend to slightly overestimate activity coefficients within the same ionic strength range (deviation ∼ 5 % to 12 %). Subsequently, there appears to be a higher ionic strength region (> 9 mol kg−1) where the Pitzer parameters obtained in this work are observed to slightly overestimate the activity coefficients; this region seems to be progressively broadening from lighter to heavier lanthanides. At even higher ionic strengths (>12 mol kg−1), the activity coefficients show a progressively negative deviation for all

lanthanides with the exception of Lu3+, which continues to show a positive deviation. Taken together, these results suggest that the three-parameter Pitzer model can only be used for a qualitative estimation of mean molar activity coefficients. Extended Pitzer Model. Seven adjustable parameters of the extended Pitzer model were simultaneously obtained by fitting osmotic coefficient data using eq 17 (Table 2) using values of constants αB1 = 2.0 kg1/2 mol1/2,30 αB2 = 12.0 kg1/2 mol1/2,42 αC1 = 0.15 kg mol1/2,30 αc2 = 0.25 kg mol1/2,30 αc3 = 0.35 kg mol1/2 30 based on recommendations from Pitzer and Perez-Villasenor.30,42 As it is seen from the fitting standard deviations σ = 0.0012 to 0.0031, which were found to be within the experimental error for all binary Ln(NO3)3 solutions, and Supporting Information Figure S1, the extended Pitzer model is able to accurately present variation of osmotic coefficients with solution molality. The extended Pitzer model parameters reported in the literature and obtained in this work were applied to calculate Ln(NO3)3 activity coefficients using eq 19 (Figure 2). Comparison of the performance of the three-parameter and the extended Pitzer models indicates that the latter provides for more accurate estimates of the activity coefficients for the dilute solutions but significantly underestimates activity coefficients for all Ln(NO3)3 solutions at the ionic strength 6 mol kg−1 to 8 mol kg−1 and higher. The same result was obtained for the Ln(NO3)3 activity coefficients calculated using literature reported extended Pitzer parameters obtained using different set of constants αB1 = 1.8 kg1/2 mol1/2, αB2 = 6.0 kg1/2 mol1/2, αC1 = 0.15 kg mol1/2, αc2 = 0.25 kg mol1/2, αc3 = 0.35 kg mol1/2.31 The deviations from the experimental values become progressively greater as solution ionic strength increases (Supporting Information Table S2). This limits practical application of the extended Pitzer model. Bromley Model. Single-Parameter Bromley Model. Single parameter Bromley model represented by eq 21 can be utilized for the predicting activity coefficients in the electrolyte solutions with the ionic strength up to 6 mol kg−1. The overall electrolyte 7 contribution B(1) Ln(NO3)3 was reported only for La(NO3)3/H2O. However, reported in the same work individual ion contributions BLn3+ and δLn3+ for the other trivalent lanthanides (Ce3+ J



Article

through Eu3+) and corresponding parameters for nitrate, i.e., BNO−3 = −0.025 and δNO−3 = 0.27 can be used to obtain the overall electrolyte contributions for the respective Ln(NO3)3 eq 22. So, determined B(1) Ln(NO3)3 values shown in Table 3 were used to

Extended Bromley Model. The extended three-parameter Bromley model represented by eq 23 containing the addi(3) tional parameters C(3) Ln(NO3)3 and DLn(NO3)3 reflecting the interionic interaction at higher concentrations was utilized to calculate Ln(NO3)3 activity coefficients. The three Bromley coefficients were simultaneously obtained by a nonlinear least-squares fitting of the experimental activity coefficients versus ionic strength. (3) (3) The obtained values for the B(3) Ln(NO3)3, CLn(NO3)3 and DLn(NO3)3 ionic contributions are given in Table 4. Even though in the refinement the data for the solutions with the ionic strength up to 12 mol kg−1 were used, the activity coefficients were calculated for the entire available concentration range for each Ln(NO3)3 (Figure 5 and Supporting Information Figure S2, solid black lines). The mean molal activity coefficients obtained from the three parameter Bromley model demonstrate remarkable agreement with experimental data within the ionic strength range up to about 20 mol kg−1. The observed deviations of the activity coefficients calculated using extended Bromley model from the experimental values are listed in Supporting Information Table S3. For all measured lanthanide nitrates except Ce(NO3)3 this model is able to estimate the activity coefficients with deviations not exceeding 3.5 % up to an ionic strength of ∼ 13 mol kg−1 to 17 mol kg−1 and exhibits slightly greater deviations for the solutions with the ionic strength up to ∼ 20 mol kg−1 (Supporting Information Table S3, Figure 5, black trace). For Ce3+, the extended Bromley model predicts the activity coefficient fairly accurately with the deviations within 5 % up to the ionic strength of 14 mol kg−1. This deviation is increased to 8.58 % for the Ce(NO3)3 solution with the ionic strength of 17.235 mol kg−1. It is anticipated that the extended Bromley model for Ce(NO3)3 can be improved by the further expansion of the available experimental database. For the concentrated Ln(NO3)3 solutions with the ionic strength above 20 mol kg−1, the extended Bromley model predicts activity coefficients fairly accurately for Sm3+, Eu3+, Gd3+, Tb3+, and Er3+. For other lanthanide nitrates, significant deviations are observed (Supporting Information Figure S2). Overall to date, the extended three-parameter Bromley model provides for the most accurate estimation of the Ln(NO3)3 activity coefficients for the binary Ln(NO3)3 electrolyte solutions at 25 °C. To evaluate if the first Bromley coefficient corresponding to the three-parameter model exhibit systematic trend across the lanthanide series, they were plotted versus the ionic radii of the lanthanides (Figure 6). For the light lanthanides, the observed trend has bow-like shape with the minimum value appeared for Pr3+. It is observed that the B(3) Ln(NO3)3 values decreasing slightly as the ionic radius decreases from Gd3+ to Lu3+. The variation of BLnNO3 parameters with ionic radii follows a very similar trend to the variation of the Pitzer β(0) and β(1) parameters (Figure 3), which suggests that the empirical B(3) Ln(NO3)3 parameter is also influenced by similar ionic interactions.


Table 3. Individual BLn3+ and δLn3+ Parameters for the Aqueous Binary Ln(NO3)3 Electrolytes Taken from Reference7 and the Corresponding Overall Electrolyte B(1) Ln(NO3)3 Contributions Calculated Using Equation 22 at 25 °Ca

a

Ln

BLn3+

δLn3+

B(1) Ln(NO3)3 (Bromley)

La Ce Pr Nd Sm Eu Gd to Lu

0.036 0.035 0.034 0.035 0.039 0.041 NR

0.27 0.27 0.27 0.27 0.27 0.27 NR

0.0868 0.0829 0.0819 0.0829 0.0869 0.0889 NR

NR = not reported.

calculate for the binary Ln(NO3)3 solutions by eq 22 (Figure 5, light blue lines). It is observed that utilization of the original Bromley’s ion contributions given in Table 3, considerably overestimates the activity coefficients for all the lanthanide nitrates in general, and more so in the high ionic strength range (3 to 6 mol kg−1). This is presumably due to the fact that contributions BLn3+ and δLn3+ were established by Bromley at 25 °C for Ln(NO3)3 electrolyte solutions in the limited range of ionic strength using data published before the year 1973.28,43,44 Lalleman et al.19 has recently demonstrated that the significant − − improvement of B(1) MX parameters for NdX3 (where X is NO3 or − Cl ) is possible by utilizing the experimental activity coefficient data corresponding to the binary solutions in the wide range of ionic strengths. The improved values of B(1) MX obtained for Nd(NO3)3 system were able to predict the activity coefficients accurately for ionic strengths up to 6 mol kg−1 (Figure 5, dashed orange trace). A similar refinement of B(1) MX was attempted by Ge et al. for La(NO3)3, Pr(NO3)3, Eu(NO3)3, and Lu(NO3)3.21 However, their obtained B(1) MX parameters were observed to significantly underestimate the activity coefficients values at the entire range of ionic strengths (Figure 5, dashed green traces). In this work, similar refinement of the single parameter Bromley model using a comprehensive database of the experimental activity coefficients assembled in this work for all Ln(NO3)3 was performed. The obtained B(1) Ln(NO3)3 values listed in Table 4 accurately predict activity coefficients for the entire Ln(NO3)3 series up to solution ionic strength of 6 mol kg−1, however progressively overestimates above that (Figure 5, dashed gray trace, Supporting Information Table S3). This upper boundary is the inherent property of the single parameter Bromley model, which suffers from the similar limitations as Pitzer treatment. Pitzer described Bromley’s method as a simplification of his own treatment in which “the third virial coeff icient is omitted, and the second virial coeff icient is modif ied into a form, still dependent on ionic strength, but with a single parameter B,” that is in effect a combination of Pitzer’s β(0) and β(1) parameters.9 The single Bromley parameter B(1) Ln(NO3)3 is unable to account for the ion−ion interaction at ionic strengths higher than 6 mol kg−1, resulting in deviation of predicted activity coefficient values from experimental activity coefficient values.

■

SUMMARY The extensive water activity measurements were conducted for binary lanthanide nitrate Ln(NO3)3 solutions in wide concentration range for the entire trivalent lanthanide series (La to Lu with the exception of Pm) using the combination of two techniques, a variable-impedance hygrometer referred to as a water activity meter and vapor pressure osmometry. It was observed that the obtained osmotic coefficients are in the excellent agreement with the traditional isopiestic technique measurements. New Pitzer interaction parameters for the binary Ln(NO3)3 solutions were obtained and compared with previously reported values. The observed interdependence and systematic trends K



Article

of the β(0) and β(1) parameters across the series of Ln(NO3)3 suggest that the Pitzer parameters obtained in this work present an accurate thermodynamic description of the ionic interactions in the binary Ln(NO3)3 solutions. However, it results only in


marginal improvement in the prediction of the experimental activity coefficients. The Pitzer model can be applied to calculate the Ln(NO3)3 activity coefficients in the solutions with the ionic strength up to approximately 12 mol kg−1. To extend

Figure 5. continued L




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Figure 5. Plot of activity coefficients as a function of ionic strength for the aqueous binary Ln(NO3)3 electrolytes at 25 °C. Symbols: experimental activity coefficients, this work. Lines represent estimated activity coefficients using the following treatments. Solid black line: extended Bromley model (3) (3) (eq 23), parameters B(3) Ln(NO3)3, CLn(NO3)3, and DLn(NO3)3 reported in Table 4, this work. Dashed gray line: single parameter Bromley model (eq 21), reported in Table 4, this work. Solid light blue line: single parameter Bromley model using literature values for B(1) parameters B(1) Ln(NO3)3 Ln(NO3)3 parameters (Table 3) taken from Bromley.7 Dashed orange line: single parameter Bromley model using literature values for B(1) Ln(NO3)3 parameter taken from 21 Lalleman et al.19 Dashed green line: single parameter Bromley model using literature values for B(1) Ln(NO3)3 parameter reported by Ge et al. (3) (3) Table 4. Tabulation of B(1) Ln(NO3)3, BLn(NO3)3, CLn(NO3)3, and D(3) Contributions for the Aqueous Binary Ln(NO3)3 Ln(NO3)3 Electrolytes Calculated by Fitting Bromley’s Model to Activity Coefficient Data Obtained from a Combination of Literature Data and Data Obtained in This Work at 25 °C

Ln(NO3)3

B(1) Ln(NO3)3

B(3) Ln(NO3)3

C(3) Ln(NO3)3

D(3) Ln(NO3)3

104σ

La(NO3)3 Ce(NO3)3 Pr(NO3)3 Nd(NO3)3 Sm(NO3)3 Eu(NO3)3 Gd(NO3)3 Tb(NO3)3 Dy(NO3)3 Ho(NO3)3 Er(NO3)3 Tm(NO3)3 Yb(NO3)3 Lu(NO3)3

0.0527 0.058 0.0495 0.0497 0.0503 0.052 0.0563 0.0621 0.0651 0.0679 0.0714 0.0738 0.0743 0.0751

0.0631 0.0561 0.0535 0.0555 0.0578 0.0586 0.0619 0.0661 0.0721 0.0755 0.0767 0.0782 0.0792 0.0806

−0.0056 −0.00185 −0.003 −0.0025 −0.003 −0.003 −0.0022 −0.0022 −0.0031 −0.0033 −0.00255 −0.0021 −0.0021 −0.0021

−0.0003 −0.00042 −0.0002 −0.00042 −0.0001 −0.00015 −0.00042 −0.0004 −0.00044 −0.00043 −0.00042 −0.00042 −0.0004 −0.00042

3.1 1.8 0.9 1.1 1.9 2.2 0.8 3.1 2.2 1.1 1.2 2.1 1.9 1.3

Figure 6. A plot of B(3) Ln(NO3)3 (obtained using eq 23) for the aqueous binary Ln(NO3)3 electrolytes versus Ln3+ radii taken from Marcus.41

comprehensive database of the experimental activity coefficients significantly improved performance of the original Bromley models available for the light lanthanide nitrates, which was able to accurately predict activity coefficients for each Ln(NO3)3 solution up to solution ionic strength of 6 mol kg−1. For the solutions with higher ionic strengths, the single parameter model was observed to progressively overestimate the activity coefficients, due to the inability of the single Bromley parameter B(1) Ln(NO3)3 to account for the high order ion to ion interaction in the concentrated solutions. This drawback of the single parameter model was overcome by application of the extended three-parameter Bromley model, which estimates the experimental activity coefficients for trivalent lanthanide nitrates with remarkable accuracy over a wide ionic strength range of 0 mol kg−1 to 20 mol kg−1. These results suggest that the extended Bromley model can be further applied to actinide nitrate solutions to determine activity coefficients in the concentrated radioactive solutions relevant to the dissolved nuclear fuel.

applicability of the Pitzer treatment to the highly concentrated electrolyte solutions, an expanded sets of Pitzer parameters β(0), β(1), β(2), C(0), C(1), C(2), and C(3) were obtained for the entire series of trivalent lanthanide nitrates. Comparison of the performance of the extended seven-parameter Pitzer model with the three-parameter Pitzer model indicates that extended model provides for more accurate estimates of the activity coefficients for the dilute solutions; however, it significantly underestimates activity coefficients for all Ln(NO3)3 solutions at the ionic strength 6 mol kg−1 to 8 mol kg−1 and higher. A single parameter Bromley model was obtained for the entire Ln(NO3)3 series. It was observed that the utilization of the M



Article

The observed systematic correlations of the Pitzer β(0) and β(1) parameters and Bromley B(3) Ln(NO3)3 parameter with the ionic radii of the trivalent lanthanide ions provides for the pathway to predict corresponding parameters for Pm(NO3)3.

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(8) Pitzer, K. S.; Mayorga, G. Thermodynamics of Electrolytes.2. Activity and Osmotic Coefficients for Strong Electrolytes with One or Both Ions Univalent. J. Phys. Chem. 1973, 77, 2300−2308. (9) Pitzer, K. S. Thermodynamics of Electrolytes. 1. Theoretical Basis and General Equations. J. Phys. Chem. 1973, 77, 268−277. (10) Pitzer, K. S.; Peterson, J. R.; Silvester, L. F. Thermodynamics of Electrolytes. 9. Rare-Earth Chlorides, Nitrates, and Perchlorates. J. Solution Chem. 1978, 7, 45−56. (11) Pitzer, K. S. Activity Coefficients in Electrolyte Solutions, 2nd ed.; CRC Press: Boca Raton, 1991. (12) Harris, D. C. Quantitative Chemical Analysis, 6th ed.; W. H. Freeman & Company: New York, 2003. (13) Debye, P.; Huckel, E. The Theory of Electrolytes I. The Lowering of the Freezing Point and Related Occurrences. Phys. Z. 1923, 24, 185− 206. (14) Rard, J. A.; Miller, D. G.; Spedding, F. H. Isopiestic Determination of the Activity Coefficients of Some Aqueous Rare Earth Electrolyte Solutions at 25°C. 4. Lanthanum Nitrate, Praseodymium Nitrate, and Neodymium Nitrate. J. Chem. Eng. Data 1979, 24, 348−354. (15) Rard, J. A.; Shiers, L. E.; Heiser, D. J.; Spedding, F. H. Isopiestic Determination of the Activity Coefficients of Some Aqueous Rare Earth Electrolyte Solutions at 25°C. 3. The Rare Earth Nitrates. J. Chem. Eng. Data 1977, 22, 337−347. (16) Rard, J. A.; Spedding, F. H. Isopiestic Determination of the Activity-Coefficients of Some Aqueous Rare-Earth Electrolyte-Solutions at 25°C. 6. Eu(NO3)3, Y(NO3)3, and YCl3. J. Chem. Eng. Data 1982, 27, 454−461. (17) Rard, J. A.; Spedding, F. K. Isopiestic Determination of the Activity-Coefficients of Some Aqueous Rare-Earth Electrolyte-Solutions at 25°C. 5. Dysprosium trinitrate, holmium trinitrate, and lutetium trinitrate. J. Chem. Eng. Data 1981, 26, 391−395. (18) Archer, D. G. Thermodynamic Properties of the NaCl+H2O System 0.2. Thermodynamic Properties of NaCl(Aq), NaCl.2H2O(Cr), and Phase-Equilibria. J. Phys. Chem. Ref. Data 1992, 21, 793−829. (19) Lalleman, S.; Bertrand, M.; Plasari, E.; Sorel, C.; Moisy, P. Determination of the Bromley Contributions to Estimate the Activity Coefficient of Neodymium Electrolytes. Chem. Eng. Sci. 2012, 77, 189− 195. (20) Zematis, J. F.; Clark, D. M.; Rafal, M.; Scrivner, N. C. Handbook of Aqueous Electrolyte Thermodynamics; Wiley-Interscience: New York, 1986. (21) Ge, X. L.; Zhang, M.; Guo, M.; Wang, X. D. Correlation and Prediction of Thermodynamic Properties of Some Complex Aqueous Electrolytes by the Modified Three-Characteristic-Parameter Correlation Model. J. Chem. Eng. Data 2008, 53, 950−958. (22) Spedding, F. H.; Shiers, L. E.; Brown, M. A.; Baker, J. L.; Guitierrez, L.; Mcdowell, L. S.; Habenschuss, A. Densities and Apparent Molal Volumes of Some Aqueous Rare-Earth Solutions at 25 Degrees.3. Rare-Earth Nitrates. J. Phys. Chem. 1975, 79, 1087−1096. (23) Ruas, A.; Simonin, J. P.; Turq, P.; Moisy, P. Experimental Determination of Water Activity for Binary Aqueous Cerium(III) Ionic Solutions: Application to an Assessment of the Predictive Capability of the Binding Mean Spherical Approximation Model. J. Phys. Chem. B 2005, 109, 23043−23050. (24) Rard, J. A.; Spedding, F. H. Isopiestic Determination of the Activity Coefficients of Some Aqueous Rare-Earth Electrolyte Solutions at 25°C. 6. Europium Trinitrate, Yttrium Nitrate, Yttrium Chloride. J. Chem. Eng. Data 1982, 27, 454−461. (25) Rard, J. A.; Palmer, D. A.; Albright, J. G. Isopiestic Determination of the Osmotic and Activity Coefficients of Aqueous Sodium Trifluoromethanesulfonate at 298.15 and 323.15 K, and Representation with an Extended Ion-Interaction (Pitzer) Model. J. Chem. Eng. Data 2003, 48, 158−166. (26) Ruas, A.; Moisy, P.; Simonin, J. P.; Bernard, O.; Dufreche, J. F.; Turq, P. Lanthanide Salts Solutions: Representation of Osmotic Coefficients within the Binding Mean Spherical Approximation. J. Phys. Chem. B 2005, 109, 5243−5248. (27) Ruas, A.; Guilbaud, P.; Den Auwer, C.; Moulin, C.; Simonin, J. P.; Turq, P.; Moisy, P. Experimental and Molecular Dynamics Studies of

ASSOCIATED CONTENT


S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jced.5b00392. A plot of osmotic coefficients at 25 °C as a function of molality1/2 for the aqueous solutions of the lanthanide nitrates for the entire molality range along with the threeparameter and the seven-parameter Pitzer fits are shown in Figure S1. The agreement of the three-parameter Bromley model with the experimental values of activity coefficients for the entire ionic strength range is shown in Figure S2. Powers and coefficients of molalities for osmotic coefficient polynomials at 25 °C for the empirical polynomial eq 3 are shown in Table S1. Tabulation of water activities, osmotic coefficients, and activity coefficients obtained using Pitzer’s formulations are shown in Table S2. Tabulation of water activities, osmotic coefficients, and activity coefficients obtained using Bromley’s formulations are shown in Table S3. (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. Funding

This research was supported by the Separations and Waste Forms Campaign within the U.S. Department of Energy’s Fuel Cycle Research and Development Program and conducted at the Pacific Northwest National Laboratory, operated by Battelle for the U.S. Department of Energy under Contract DE-AC0576RL01830.

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Aqueous Binary Lanthanide(III) Nitrate Ln(NO3)3 Electrolytes

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