Thermodynamic Modeling of Lead Nitrate Aqueous Solution

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Cite This: J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Thermodynamic Modeling of Lead Nitrate Aqueous Solution: Pitzer Temperature Dependency Parameters Mouad Arrad*

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Department of Process Engineering, National School of Mines of Rabat, BP 753 Agdal, Rabat, Morocco ABSTRACT: A comprehensive thermodynamic model was elaborated for lead nitrate aqueous solution using the Pitzer framework. The evaluation of the ion interaction parameters for the Pitzer model was carried out using the osmotic coefficient of lead nitrate aqueous solutions collected from the open literature at 298.15 K and also saturated vapor pressure from 283.15 to 373.15 K. A temperature dependency form of Pitzer parameters was selected after several tests of different Pitzer parameter sets. The chosen set of parameters was able to cover wider temperature ranges and consequently calculate system properties from dilution to saturation in the temperature range of 273.15− 373.15 K. An equation for the solubility product for lead nitrate was also proposed, leading to the prediction of solubility data for this system. The temperature dependency of the Pitzer model shows a satisfactory behavior, giving the possibility of using the model to predict the thermodynamic properties. The extrapolation capacity of the model was also tested at temperature above 373.15 K.

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INTRODUCTION Lead is one of the seven metals of antiquity, and it was known to ancient metalworkers for its soft silvery-blue color. Over the centuries, the utility of materials arising from lead chemistry has led to a large variety of applications. Unfortunately, many lead-containing materials are toxic and probably carcinogenic. Concerns about safety have mitigated its use in certain applications, but many research activities are done in order to propose suitable methods for the safe elimination of lead.1In aqueous media, lead can have several forms like lead nitrate, lead acetate, or lead chlorate. Those salts are an exception to the rule that most lead compounds have very low solubility in water. Lead nitrate has the advantages of containing a higher lead content, having a higher solubility, and being the less expensive lead salt. For all these reasons, lead nitrate is the most commonly used source of lead in many industrial processes such as gold ore cyanidation.2 It is also used as a heat stabilizer in nylon and polyesters,3,4 as a coating for photothermographic paper, and in rodenticides. The need to present a consistent thermodynamic modeling study is very high due to the hazardous nature of the salt, and there is a need to enhance the comprehension of the behavior of lead nitrate in water in a large solubility range but also taking into consideration the effect of temperature. The Pitzer formalism,5 which is one of the most frequently used and successful models for electrolyte solutions, was adopted. The model is well-known for its capacity to correlate strong electrolyte solutions and to describe different solution properties and phase equilibria with good flexibility and acceptable accuracy.6−12 In this study, the model was able to correlate osmotic coefficient data of lead nitrate available in the open literature. The obtained set of parameters gives a good representation of © XXXX American Chemical Society

the solution properties from low concentration to the saturation corresponding to the appropriate temperature. The agreement between the calculated properties and the experimental data is very good. In addition to the prediction of the water activity and activity coefficient of lead nitrate solution in the temperature range of 273.15−373.15 K, the expression of solubility constant for lead nitrate is also presented.

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REVIEW OF AVAILABLE EXPERIMENTAL DATA The data available in the chemical literature were evaluated with the purpose to find all the experimental data involving lead nitrate aqueous solutions. As noticed earlier in previous work, nitrate salt11 data belonging to metal nitrate are less abundant in the open literature. Concerning lead nitrate, the only available source of experimental data of the osmotic coefficient comes from Robinson and Stokes,13 and the reported data in the Russian compilation by Mikulin14 was the same set going from 0.1 to 2 molal at 298.15 K. Solubility data available in the compilation of Seidell15 report experimental values from Mulder16 and Kremers17 and one experimental data point from Euler.18 In the Lange handbook,19 the solubility of lead nitrate in water was compiled and given from 273.15 to 373.15 K with a regular interval of 10 K. The origin of the experimental data point used to present these values was not reported, but from the comparison of the values of molality at 273.15 K we can confirm that data from Mulder16 and Kremers17 were used to Received: July 17, 2019 Accepted: August 22, 2019

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DOI: 10.1021/acs.jced.9b00698 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

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propose a recommended set of values of solubility. The most relevant work dealing with solubility of lead compounds was from Clever and Johnston.20 In their work, Clever and Johnston20 were able to collect 46 data points from 14 papers in the temperature range 273.15− 373.15 K. To the best of the author’s knowledge, Clever and Johnston’s list of papers16−18,21−31 is the most exhaustive on lead nitrate solubility. Later, Payne32 measured the solubility data of lead nitrate from 293.65 to 341.05 K. The EMF method enables Mikdaze and Makhov33 to measure the activity coefficient of lead nitrate at 298.15 K from very low concentration to 1 molal and to compare their results with the work of Kielland34 and Harned and Owen.35Randall and Vanselow36 calculated activity coefficients at 273.15 K for very dilute lead nitrate solution, using measured freezing point data. Dingemans37 presents the vapor pressure value and water activity of saturated lead nitrate solution from 283.15 to 403.15 K, but the major problem with his data is there is no indication of the values of saturation molality at each given temperature. In this situation, the use of these data is not useful for modeling without the determination of the corresponding saturation concentration at each given temperature. Table 1 represents the available experimental data for the lead nitrate aqueous solution. From this review of data, the first observation that should be mentioned is the number of experimental data from Mulder’s work was not accurately reported in the Clever and Johnston20

paper. This fact is not very important since solubility data from Mulder16 seem to be very low compared to the general tendency of experimental data; on the other hand, data from Kremers17 are too high, and the same remark is valid for the set of solubility values of Payne.32 Clever and Johnston20 present an equation describing the variation of the solubility as a function of temperature using all the solubility data expect those from Kremers,17 Mulder,16 and Motornoya et al.22 The two set of values (Kremers16 and Mulder17) were excluded from the fit for the same reason cited above. The authors were not able to include data from Motornoya et al.22 as the fitting procedure was already done. In this work another correlation of solubility function of temperature is proposed in order to calculate the saturation molality and to complete the set of experimental data measured by Dingemans.37 Using all the solubility data mentioned in Table 1 (only data points from Kremers,17 Mulder,16 and Payne32 were excluded), the solubility of lead nitrate solution can be expressed using the equation: m = 1.1923 + 0.023361(T − 273.15) + 4.7749 × 10−5(T − 273.15)2

This expression is valid from 273.15 to 373.15 K since the available solubility data are in this temperature range. The representation of data using this expression was satisfactory with a deviation of 0.1% in comparison with the equation given by Clever and Johnston.20 Using eq 1, the saturation molality of each vapor pressure measured by Dingemans37 was affected by the corresponding temperature. Table 2 presents the list of experimental values with their corresponding molalities. Values at 273.15 K and 278.15 K are an extrapolation of the expression of water activity as a function of temperature given by Dingemans37

Table 1. Experimental Data for Lead Nitrate Aqueous Solution data type osmotic coefficient activity coefficient

salt solubility

vapor pressure

temperature (K)

molality

data points

298.15

0.1−2

15

298.15

0.0001−1

9

298.15 298.15

0.0001−0.02 0.001−1

5 9

273.15

0.0001−0.1

9

293.65−341.05

1.889−3.098

20

273.13−373.15 273.15−373.15 273.15−372.95 298.15−313.15

1.100−3.834 1.17−4.191 1.111−3.970 2.168−2.787

11 11 11 3

299.15−313.15 273.15−373.15

1.805−3.330 1.215−3.789

3 3

288.15−293.15

1.482−1.577

2

290.15 293.15

1.593 1.685

1 1

293.15 298.15 298.15 298.15

1.664 1.778 1.846 1.802

1 1 1 1

298.15

1.799

1

283.15−403.15

saturation

49

(1)

reference Robinson and Stokes13 Mikdaze and Mokhov33 Kielland34 Harned and Owen35 Randall and Vaselow36 Payne32

a w = 0.9766 − 0.000946t

(2)

in °C. Equation 2 was used for the conversion of vapor pressure to water activity data. The experimental values presented in Table 2 in addition to osmotic coefficient data of Robinson and Stokes13 were used in the estimation of Pitzer parameters.

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THERMODYNAMIC MODEL: PITZER FRAMEWORK The Pitzer model was used to fit the osmotic coefficients and to calculate the mean ionic molal activity coefficient of the electrolyte γ±. The expressions of the Pitzer model for binary (i.e., salt-water) systems can be written as ÄÅ É ÅÅ I1/2 ÑÑÑ Å ÑÑ + 2mijj vMvX yzzB ϕ Å ϕ = 1 − Aϕz M|z X|ÅÅ Ñ j z MX ÅÅÇ 1 + bI1/2 ÑÑÑÖ k v { ÄÅ É Å (v v )3/2 ÑÑÑ Å ÑÑC ϕ 2Å M X + 2m ÅÅÅ ÑÑ v ÅÅÅÇ ÑÑÑÖ (3)

Mulder16 Kremers17 Kazantsen21 Motornaya et al.22 Ferris23 Glasstone and Sanders24 Fishman and Bulakhova25 Euler18 Le Blanc and Noyes26 Fedotieff27 Malquori28 Fock29 Akerlof and Turek30 Richards and Shumb31 Dingemans37

iv v y ϕ ln γ± = z M|z X|f γ + 2mjjj M X zzz(BMX + BMX ) k v { ÄÅ É Å (v v )3/2 ÑÑÑ Å ÑÑC ϕ 2Å M X + 3m ÅÅÅ ÑÑ ÅÅ ÑÑ v ÅÇ ÑÖ

(4)

where B


ÄÅ ÉÑ ÅÅ ÑÑ I ij 2 yz Å Å f = AϕÅÅ + jj zzln(1 + b I )ÑÑÑÑ ÅÅÇ (1 + b I ) ÑÑÖ kb{

Journal of Chemical & Engineering Data Table 2. Water Activities (aw) and Osmotic Coefficients of Lead Nitrate Aqueous Solutions at Salt Saturation Molalities, from 273.15 to 373.15 K and p = 0.1 MPa T/K

m/mol·kg−1

aw

osmotic coefficient

273.15 278.15 283.15 285.65 288.15 290.65 293.15 295.65 298.15 300.65 303.15 305.65 308.15 310.65 313.15 315.65 318.15 320.65 323.15 325.65 328.15 330.65 333.15 335.65 338.15 340.65 343.15 345.65 348.15 350.65 353.15 355.65 358.15 360.65 363.15 365.65 368.15 370.65 373.15

1.192 1.130 1.431 1.492 1.554 1.616 1.679 1.742 1.806 1.871 1.936 2.002 2.068 2.135 2.203 2.271 2.340 2.409 2.480 2.550 2.621 2.693 2.766 2.839 2.912 2.987 3.061 3.137 3.213 3.290 3.367 3.445 3.523 3.602 3.682 3.762 3.843 3.924 4.006

0.976 0.972 0.967 0.965 0.962 0.960 0.957 0.955 0.953 0.950 0.948 0.945 0.943 0.941 0.938 0.936 0.934 0.931 0.929 0.926 0.924 0.922 0.919 0.917 0.915 0.912 0.910 0.908 0.905 0.903 0.900 0.898 0.896 0.893 0.891 0.889 0.886 0.884 0.882

0.367 0.402 0.432 0.445 0.456 0.467 0.476 0.485 0.493 0.501 0.508 0.514 0.520 0.525 0.530 0.535 0.539 0.543 0.547 0.550 0.553 0.556 0.558 0.561 0.563 0.565 0.567 0.569 0.570 0.572 0.573 0.574 0.575 0.576 0.577 0.578 0.579 0.579 0.580

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γ

(5)

ϕ BMX = β (0) + β (1) exp( −α1I1/2) + β (2) exp( −α1I1/2)

(6)

BMX = β (0) + β (1)g ( −α1I1/2) + β (2)g ( −α1I1/2)

(7)

The function g(x) in eq 7 is 2(1 − (1 + x)e−x) x2

g (x ) =

(8)

In eqs 3 and 4, the symbols zM and zX are the electric charges of the cation (M) and anion (X) in the electrolyte; νM and νX are the stoichiometric coefficients of the ions in the salt, with the notation ν = νM + νX; and I is the ionic strength of the solution. Its relation with the composition coordinate of the electrolyte solution expressed with molality mi is given as I=

1 2

∑ mizi2

(9)

i

where the summation is over all aqueous solute species and (Zi) is the electric charge for species i. The parameters BϕMX and BMX are the concentration dependence of the electrolytespecific terms in eqs 3 and 4, while α1, α2, β(0), β(1), β(2), and Cϕ are the Pitzer parameters given in eqs 3, 4, 6, and 7. Since we deal with electrolytes of the type 1−2, the parameters α2 and β(2) were considered zero in the Pitzer model. The internal parameters of the Pitzer model used in this work were b = 1.2 (kg·mol−1)1/2 and α1 = 2.0 (kg· mol−1)1/2, and they are the original values proposed by Pitzer. Aϕ is the theoretical, limiting Debye−Hückel slope at infinite dilution, calculated according to the correlation given by Pitzer.5

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PARAMETER ESTIMATION The estimation of Pitzer was carried out using an objective function (OF) minimizing the square error between each experimental osmotic coefficient (ϕExp) and that calculated (ϕModel) by the model according to eq 9: OF =

∑ (ϕmodel,i − ϕexp,i)2

(10)

i

Table 3. Pitzer Sets of Parameters in the Form of p = a + b/T for Lead Nitrate Solution from 273.15 to 373.15 K β(0) Pb(NO3)2 set set set set set set set set set set set set

1 2 3 4 5 6 7 8 9 10 11 12

Cφ

β(1)

a

b

a

b

a

b

objective function

ln(K) at 298.15 K

−0.1999 0.6204 0.6322 0.9242 0.0918 0 0.6325 0.1638 0 0 0.9114 0.6836

0 −190.6349 −190.6553 −229.3734 −34.8399 0 −192.7911 −51.5678 −9.1437 −12.5227 −226.1997 −202.8992

1.2591 0 −0.1269 −1.9915 52.2477 −1.347 0 52.0786 0 52.0786 0 −1.2934

−3.7533 16.9449 0 77.1447 −15546.3379 77.1444 0 −15546.3169 −168.5459 −15484.2351 −516.5732 324.6930

−0.0723 −0.0623 −0.0576 0 0.0052 0.009 −0.0608 0 0 0 0 −0.0577

42.5983 18.6123 16.1693 −17.7812 0 −1.121 17.7462 0 3.9412 3.8553 −17.5187 15.0198

0.075 0.026 0.025 0.052 0.003 0.13 0.026 0.006 0.112 0.007 0.048 0.027

−3.53 −3.71 −3.77 −4.12 −3.64 −4.09 −3.72 −3.66 −3.95 −3.60 −4.12 −3.78

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As shown in eq 10, the objective function used in this study does not include any weighting factors. However, some data points, presenting random large deviations from the general trend (more than 5%), were neglected in the parameter optimization, and activity coefficient data33−36 were not used in the fitting but used only for the validation of the model parameters. In order to obtain the best possible set of Pitzer parameters, several combinations were tested. The criteria of choice of the best set are not only the lowest objective function value but also the number of the used parameters, the good representation of the physical properties of the system in the temperature range, and also the capacity of the model to estimate the equilibrium constant at several temperatures. Table 3 presents the obtained values according to a temperature dependency of the form p = a + b/T. According to the thermodynamic data available in the literature,15,19 the estimated value of ln(K) is −3.63 at 298.15 K. After the analysis of the results, sets 5, 8, and 10 seem to be the best options to represent the system due the low objective function values (respectively, 0.003, 0.006, and 0.007). Set 8 is an example to show that low objective function value can be obtained without using the Cφ parameter, but according to Pitzer and Mayorga,6 this parameter is mandatory for the modeling of this system. For this reason, set 8 cannot be adopted. The estimated values of ln(K) by set 5 and the one calculated from the thermodynamic data at 298.15 K are in excellent agreement, better than the estimated value by set 10. However, the main difference is noticed in the capacity of set 5 to estimate the activity coefficient of lead nitrate with higher accuracy than set 10 (MAPE at 298.15 K, 1.3% and 3%, respectively) and also its capacity to predict solubility product for temperature above 298.15 K better than set 10. The lowest value of the objective function and the best representation of the solubility product were the selection criteria to adopt. Set 5 was for the calculation of water activity, activity coefficient, and to calculate the solubility constant for anhydrous lead nitrate from 273.15 to 373.15 K The quality of experimental data was an additional important factor for the determination of the best possible combination of Pitzer parameters with the best estimation of solubility constant coefficients.

Table 4. Estimated Pitzer Parameters for Aqueous Solutions of Lead Nitrate, up to 5 molal at 298.15 K this work Pitzer and Mayorga6 May et al.10

β(0)

β(1)

Cφ

−0.0250 0.0361 0.0058

0.1050 0.2850 −0.1669

0.0052 0.0053 −0.0052

Figure 1. Osmotic coefficient of lead nitrate aqueous solutions at 298.15 K.

calculated curve by Pitzer and Mayorga6 and May et al.10 The Pitzer model gives a satisfactory representation of this quantity. This fact confirms the higher flexibility of the proposed Pitzer parameters to express the temperature dependency since the other models are valid only at 298.15 K. Another important analysis is also the comparison between the calculated molality scale mean ionic activity coefficients of lead nitrate at 298.15 K and calculated values from the model of Pitzer and Mayorga,6 May et al.,10 and Robinson and Stokes.13 The obtained results are shown in Figure2. The experimental data which are not used in the fitting procedure are in excellent agreement with the calculated activity coefficient. Vapor pressure data measured by Dingemans37 are compared to the calculated water activity. The extrapolation of the model parameters until 403.15 K is also presented.

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RESULTS AND DISCUSSION It should be mentioned that rare attempts of modeling lead nitrate aqueous solution are available in the literature. The work of Pitzer6 and recently the paper from May et al.10 are the only sources of Pitzer parameter data given at 298.15 K using the osmotic coefficient from Robinson and Stokes.13 May et al.10 present a different set of parameters using the same set of experimental data but using an objective function composed from several systems at the same time. A significant comment was stated in the same work refereeing that the data are limited for this system. Chen et al.38 also used Robinson and Stokes’s13 osmotic coefficient with their calculated activity coefficient to give eNRTL parameters for lead nitrate solution at 298.15 K. Table 4 shows Pitzer paramaters at 298.15 K with the parameters from Pitzer and Mayorga6 and May et al.10 Figure 1 shows a comparison of the calculated values and the experimental osmotic coefficient at 298.15 K along with the

Figure 2. Mean ionic molal activity coefficient of lead nitrate in aqueous solutions at 298.15 K. D



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As shown in Figure 3, the model was able to reproduce the water activity of lead nitrate solution from 273.15 to 373.15 K

Figure 5. Activity coefficient of dilute lead nitrate solution at 273.15 K.

Figure 3. Water activity of lead nitrate solution from 283.15 to 403.15 K.

The equation of dissolution of lead nitrate in water (eq 11) leads to the calculation of the solubility product (eq 12) using the Pitzer mean activity coefficient

with good accuracy. The capacity of the model to predict the water activity until 403.15 K was also tested. The parameter set of Pitzer parameters was also used in the representation of some properties of the aqueous solutions of the investigated salt. The predicted osmotic coefficient at two temperatures is an example and is shown in Figure 4.

Pb(NO3)2 (s) ↔ Pb2 +(aq) + 2NO−3 (aq)

(11)

2 − KPb(NO3)2 = a Pb2+(aq) × a NO 3 (aq)

(12)

where ai stands for the activity of species i. In this work, we assumed the solid phase to be pure solid with activity 1, and mean activity coefficient was calculated. The fitting procedure using the chosen set of parameters (set 5) leads to the determination of the expression of the solubility product (eq 13): ln(K ) = −456.1639 − 0.1607*T − 15.2713/T + 87.8404 * ln(T )

(13)

The solubility curve of anhydrous lead nitrate, which is the only form of the salt present in the aqueous solution in the temperature range between 273.15 and 373.15 K, is reported by all the experimental solubility data. The Pitzer model shows an excellent agreement with the recommended values by Clever and Johnston.20 The Pitzer model was successful in the representation of solubility with high accuracy. The obtained results confirmed the choice of Pitzer model as a thermodynamic framework to model this system because of the moderate saturation molality (not exceeding 5 molal) which is in the recommended range of molality well described by Pitzer (around 8 molal). Figure 6 shows the representation of experimental solubility data from several authors,16−18,21−31 the recommended data by Clever and Johnston,20 and the calculated solubility with the Pitzer model. The prediction of water activity and activity coefficient using this version of Pitzer parameters was a good confirmation of the possibility of the Pitzer model to give an excellent agreement with available experimental data but also manages to solve the problem of the good estimation of solubility which is based on the calculation of the solubility product at saturation values with vapor pressure data or osmotic coefficient. This situation is not always possible when the calculation of solubility product is more complicated due to the presence of several salt hydrates in the phase diagram. One of the important outcomes of this evaluation of lead nitrate

Figure 4. Predicted osmotic coefficient of lead nitrate solution at 273.15 and 373.15 K.

The presented osmotic coefficient of lead nitrate has the same tendency of soluble 1−2 electrolyte salts in aqueous solution.5−13 Then, the model will be very useful in several industrial applications to avoid handling of toxic and harmful products. The model capacity to predict activity coefficients at low temperature was also tested. Figure 5 shows the calculated lead nitrate activity coefficient at 273.15 K in comparison with the calculated activity coefficient from Randall and Vanselow.36 In their work, Randall and Vanselow36 used the integration method to convert freezing point depression to activity coefficient data. This method was based on the integration of the Gibbs−Duheim equation with an assumed integration constant, and the variation of this constant could lead to a large discrepancy. However, the importance of these data is to show the variation of activity coefficients of very dilute solutions at low temperature. E



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Figure 6. Solubility of lead nitrate in water between 273.15 and 373.15 K. (2) Deschenes, G.; Lastra, R.; Brown, J. R.; Jin, S.; May, O.; Ghali, E. Effect of lead nitrate on cyanidation of gold ores: Progress on the study of the mechanism. Miner. Eng. 2000, 13, 1263−1279. (3) Hsieh, T. Y.; Su, T. S.; Ikegami, M.; Wei, T. C.; Miyasaka, T. Stable and efficient perovskite solar cells fabricated using aqueous lead nitrate precursor: Interpretation of the conversion mechanism and renovation of the sequential deposition. Materials Today Energy 2018, 1−10. (4) Alger, M. S. M. Polymer Science Dictionary, 2nd ed.; Chapman and Hall: London, UK, 1997. (5) Pitzer, K. S., Ed. Activity Coefficients in Electrolyte Solutions, 2nd ed.; CRC Press: Boca Raton, FL, 1991; pp 90−94. (6) Pitzer, K. S.; Mayorga, G. Thermodynamics of Electrolytes, II. Activity and osmotic coefficients with one or both ions univalent. J. Phys. Chem. 1973, 77 (19), 2300−2308. (7) Pitzer, K. S.; Mayorga, G. Thermodynamics of Electrolytes. III. Activity and osmotic coefficients for 2−2 electrolytes. J. Solution Chem. 1974, 3 (7), 539−546. (8) Kim, H. T.; Frederick, W. J. Evaluation of Pitzer ion interaction parameters of aqueous electrolytes at 25 °C. 1. Single salt parameters. J. Chem. Eng. Data 1988, 33, 177−184. (9) Hasan, M.; Partanen, J. I.; Vahteristo, K. P.; Louhi-Kultanen, M. Determination of the Pitzer Interaction Parameters at 273.15 from the Freezing-Point Data available for NaCl and KCl Solutions. Ind. Eng. Chem. Res. 2014, 53, 5608−5616. (10) May, P. M.; Rowland, D.; Hefter, G.; Konigsberger, E. A generic and updatable Pitzer characterization of aqueous binary electrolyte solutions at 1 bar and 25°C. J. Chem. Eng. Data 2011, 56, 5066−5077. (11) Arrad, M.; Kaddami, M.; Sippola, H.; Taskinen, P. Determination of Pitzer parameters for ferric nitrate from freezing point and solubility of ice. J. Chem. Eng. Data 2016, 61, 674−678. (12) Lach, A.; André, L.; Guignot, S.; Christov, C.; Henocq, P.; Lassin, A. A Pitzer Parametrization To Predict Solution Properties and Salt Solubility in the H−Na−K−Ca−Mg−NO3−H2O System at 298.15 K. J. Chem. Eng. Data 2018, 63, 787−800.

solution is the good opportunity to explore the Pitzer model which is more flexible and simple than other types of models in a reasonable molality range with one or two salt hydrates.

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CONCLUSIONS The available experimental data from the literature were critically analyzed before least-squares fitting and parametrization of the lead nitrate system. The Pitzer model was used to calculate the water activities, mean activity coefficients of the dissolved salt, and calculated solubilities, showing very good results. The prediction of osmotic coefficient and activity coefficient of lead nitrate was also possible, using a small number of adjustable parameters. The solubility branch of lead nitrate calculated by the Pitzer model was in excellent agreement with the recommended solubility data. In the case of a similar electrolyte, the use of this expression of temperature dependency is recommended for a moderate range of concentrations (until 8 molal.)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Mouad Arrad: 0000-0001-9752-0501 Notes

The author declares no competing financial interest.

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REFERENCES

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Thermodynamic Modeling of Lead Nitrate Aqueous Solution

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