Determination of Pitzer Parameters for Ferric Nitrate from Freezing

Article pubs.acs.org/jced

Determination of Pitzer Parameters for Ferric Nitrate from Freezing Point and Solubility of Ice Mouad Arrad,*,† Mohammed Kaddami,† Hannu Sippola,‡,§ and Pekka Taskinen‡ †

Laboratory “Physical Chemistry of Processes and Materials” Faculty of Science and Technology, University Hassan 1, Settat, Morocco ‡ Department of Materials Science and Engineering, Metallurgical Thermodynamics and Modeling, Aalto University School of Chemical Technology, Espoo, Finland § FCG Design and Engineering, Osmontie 34, FI-00601, Helsinki, Finland ABSTRACT: A calculation method is presented in this work to evaluate the ion interaction parameters for the Pitzer model from freezing points of aqueous solutions of pure electrolytes and solubility. To show the reliability of this method, one system that was not covered by the open literature was chosen. Using a small number of experimental data we were able to determine Pitzer parameters for this system reaching a concentration up to 1.8 mol−1 kg−1 Using this set of parameters thermodynamic properties such as osmotic coefficient, water activity, and mean activity coefficient were estimated at low temperatures for ferric nitrate−water system.

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INTRODUCTION Electrolyte solutions are present in the nature and in several industrial processes ranging from metal making to biochemistry. The thermodynamic properties of the solutions including such salts are characterized by high interactions between the ions. Freezing point depression is also a very important colligative property of the solvent and its accurate determination has been proved essential in several fields of applications, such as geochemistry, petroleum engineering, and chemistry.1,2 Unfortunately, there is a huge lack concerning the aqueous metal nitrate data in the literature. Only scattered data are available for trivalent metal nitrates. Jones et al.3 have presented one of these rare pieces of data in several published papers and books.3−5 A recent series of work by Goundali el al.6 have presented a new set of experimental data dealing with ferric nitrate in aqueous solutions. As stated by Hasan et al.,7 the freezing point depression as a thermodynamic property is not exploited to its full potential. An accurate determination of its value enables us to give a sound prediction of activity of the liquid solvent over typically large concentration range, starting from dilute systems, as well as activity of the electrolyte through the Gibbs−Duhem equation when using an activity model or by direct numerical integration. Because no reliable calorimetric, isopiestic, or electromotive force (EMF) data are available for theses salts, we can conclude that the available freezing point measurements in addition to the solubility of ice data are very important in evaluating the thermodynamic properties of this technically important salt, © XXXX American Chemical Society

such as water activity or mean activity coefficient of the electrolyte. We choose for the modeling purpose one of the most frequently used and relatively simple models, the Pitzer formalism, known for its wide applications in the electrolyte solutions and covering all of the essential properties of solutions, such as osmotic coefficient, mean activity coefficient, and calorimetric data. The present study determines the parameters values for the Pitzer model at temperatures below 273.15 K, based on the available literature data of ferric nitratewater solutions.3,4,6 An important outcome of this work is a thermodynamic model able to represent accurately the aqueous solution properties at low temperatures, from its dilute regions up to the saturation of the salt, using only the available freezing point data and with the possibility of extension the model to temperatures higher than 273.15 K

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THEORETICAL BACKGROUND The equilibrium between water and pure ice, H2O(s), in the presence of a salt solution is described by the general equation for a two-phase system Ice ⇄ H 2O(l) K (T ) =

a H2O(l) a H2O(s)

(1)

Received: September 25, 2015 Accepted: December 8, 2015

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DOI: 10.1021/acs.jced.5b00822 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

Article

The freezing equilibrium condition between solid ice and an aqueous solution is expressed by the equality of chemical potentials of the components in the both phases present at the freezing temperature Tf. This condition with reference to water can be written as μH 0(l) = μo H 0(l) + RTf ln(a) = μo H 0(s) 2

2

g (x ) =

where μ refers to the chemical potential and superscript to pure phase. In eq 2, a is activity of water in the electrolyte solution and it assumes that solid ice is a pure substance without any solubility of the electrolyte.8 We obtain from the above equality (eq 2) for the activity of (liquid) water in the electrolyte solution along the phase boundary with ice saturation o

ln(a) = −

I = 0.5Σmizi 2

(3) −1

−1

where R = 8.314 J·mol ·K denotes the universal gas constant, ΔHo = 6009.5 J·mol−1 is the enthalpy of fusion of pure water at its melting point in 1 atm, ΔCp = 37.87 J·K·mol−1 is the difference between the heat capacities at constant pressure of the pure water in the liquid phase and ice at the freezing temperature, and Tof = 273.15 K is the freezing point of pure water. The osmotic coefficient of solution ϕ is in general related to its activity of water aH2O(l) with the following relation ϕ=

−(1000) ln(a H2O(l)) n M H2O(∑i = 1 mi)

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where the molar mass of water is MH2O = 18.01528 g·mol−1, and mi is molality of the species i dissolved in the solution. The Pitzer equations9 were used to fit the osmotic coefficient of the solution and to calculate the mean activity coefficient of the electrolyte γ±, which can be written as

(5)

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

(6)

⎡ ⎤ ⎛2⎞ I f γ = −Aϕ⎢ + ⎜ ⎟ln(1 + b I )⎥ ⎝ ⎠ ⎣ (1 + b I ) ⎦ b

(7)

ϕ BMX = β (0) + β (1) exp( −α1I1/2) + β (2) exp( −α2I1/2)

(8)

BMX = β (0) + β (1)g (α1I1/2) + β (2)g (α2I1/2)

EXPERIMENTAL DATA

The freezing point depression occurs when the activity of the solvent is lowered by additions and dissolution of another substance in it. A solution has a lower freezing point than that of the pure solvent. There are two types of experimental data in the literature about freezing points of electrolyte solutions. The first set of data comes from direct, dynamic, or static freezing temperature measurements.3−5 The experimental technique was described, for example, in early papers by Jones et al.4,5 A recent study from Haghighi et al.2 that reports the experimental measurement of freezing points for selected electrolyte solutions and shows that this method can be used to get accurate values of activity of the solvent? The second experimental set of methods gives the solubility of ice in the binary solution using, for example, the conductometric method.6 The fundamental principles of both methods consist of reaching the solid−liquid equilibrium for a given solution. Thus, in the direct freezing point method at a given concentration the freezing temperature of the solution is measured using the conductometric method at a fixed temperature and looking for the saturation point of the solution by diluting a prepared solution of the salt with known concentration until a variation of the conductivity versus composition slope is observed. In that condition, saturation by ice is reached and the value of the freezing concentration of the solute at this temperature is determined. For more details on the conductometric method, see, for example, ref 6. Basically, the freezing point data may be obtained in a dynamic run by a DTA-type equipment10 or by a static technique, where solid and liquid are allowed to equilibrate until no reactions any more occur.3−5 Within this group of techniques, the static methods are more reliable11 and the dynamic data may contain severe undercooling effects. In our present case, experimental data for the investigated salt obtained by both methods are in good agreement. For consistency, the experimental solubility data were converted from weight percent to the molality scale and the results are presented in Table 1.

(4)

⎤ ⎡ ⎛ν ν ⎞ ϕ I1/2 ⎥ + 2m⎜ M X ⎟BMX ϕ = 1 − Aϕz M|z X|⎢ 1/2 ⎝ ν ⎠ ⎣ (1 + bI ) ⎦ ⎡ (ν ν 3/2) ⎤ ⎥C ϕ + 2m2⎢ M X ⎢⎣ ⎥⎦ ν

(11)

where the summation is over all aqueous solute species and (zi) is the electric charge for i, and BϕMX, BMX are the concentration dependence of the electrolyte specific terms in eqs 5 and 6, and β(0),β(1) and Cφ are the Pitzer parameters as given in eqs 5 and 6 and 8 and 9. Since we have only one electrolyte of the type 1−3 in our scrutiny, the parameters α2 and β(2) were not included in the Pitzer model expressions used in this work. 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 for 1−3 electrolytes and they are the original values proposed by Pitzer.9 Aϕ is the theoretical, limiting Debye−Hückel slope at infinite dilution.

ΔCp ⎡ (Tf − Tf o) ΔH o ⎡ 1 1 ⎤ ⎢ ⎢ − o⎥ + R ⎣ Tf Tf ⎦ R ⎢⎣ Tf

⎛ T ⎞⎤ − ln⎜ fo ⎟⎥ ⎝ Tf ⎠⎥⎦

(10)

In eqs 5−10, the symbols zM and zX are the electric charges of the cation and anion 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 stoichiometric ionic strength of the solution. Its relation with the composition coordinate of the electrolyte solution mi is given as

(2)

2

2(1 − (1 + x)e−x) x2

(9)

where the variable g(x) in eq 9 is B



Article

T. Table 2 presents the studied sets of Pitzer parameters tested with the obtained objective function values. The decision concerning the best set of parameters was based on the obtained objective function, the number of adjustable parameters used, and its capacity to represent accurately the thermodynamic properties. Thus, a set of parameters that has the lowest value of the objective function is not the best one if the value of objective function is not significantly lower than that of a set with fewer terms. After this analysis, we were able to choose the best set of Pitzer parameters for ferric nitrate system that has an acceptable objective function value but was able to represent all the thermodynamic properties in a convenient manner. For example, set #1 has the lowest value of the objective function, and despite of the number of Pitzer parameters used in this set (#6) the predicted activity coefficient has an inacceptable value at high concentrations. After all these series of tests, set #3 is the best set of parameters for this system. Table 3 shows the values of the selected set of parameters at 0 and 25 °C.

Table 1. Calculated Osmotic Coefficients for Fe(NO3)3 at the Freezing Temperature Obtained from the Available Experimental Dataa T/K

m/mol·kg‑1

ϕExp

reference

248.15 251.75 255.89 258.15 261.71 266.41 269.72 271.07 272.20 272.48 272.67 268.15°

1.789 1.496 1.346 1.305 1.047 0.748 0.449 0.299 0.149 0.105 0.075 0.444

2.2361 2.2311 1.9418 1.7129 1.5878 1.2676 1.0505 0.9461 0.8611 0.8578 0.8619 1.5449

6 3 3 3 6 3 3 3 3 3 3 6

Estimated standard uncertainties u are u(T) = 0.1 °C, u(m) = 0.005, u(ϕ) = 0.01250 for ref 6. Estimated standard uncertainties u are u(T) = 0.1 °C, u(m) = 0.010, u(ϕ) = 0.02500 for ref 3. bData point from Goundali et al.6 was excluded due to an error in the measurement of temperature. a

Table 3. Values of the Pitzer Parameters for Ferric Nitrate at 0 and 25 °C Valid up to 1.8 Molal Solutionsa

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RESULTS AND DISCUSSIONS The fitting of Pitzer parameters was done using an objective function (OF) that minimizes the square error between experimental osmotic coefficient and the calculated one given by Pitzer model, eq 12, as follows

β(0)

a

Cφ

β(1)

0 °C

25 °C

0 °C

25 °C

0 °C

25 °C

0.8295

0.7599

5.5527

5.0871

−0.0551

−0.0505

The values at 25°C are extrapolated.

n

OF =

∑ (ϕPitzer − ϕexp)2

The most important parameter is not β(1) due to its large value in comparison with the two other parameters β(0) and Cφ but the most influent parameters is β(0) because a small modification could give a noticeable change on the calculated properties. Therefore, the chosen set of Pitzer parameters (set #3) was used for the determination of the aqueous solutions properties, such as osmotic coefficient, activity of water, and the mean activity coefficient of the salt. Before determining those properties, validation of the obtained model was carried out using aluminum chloride as an example of 1−3 electrolytes. Freezing point data available in ref 4 were used to determine Pitzer parameters for aluminum chloride system, system, the obtained parameters extrapolated at 298.15 K were able to represent water activity, good agreement with the experimental data from Mason14 and Robinson and Stokes15 was obtained; Figure 1 shows the plot of calculated water activity from this study with experimental data.

(12)

i=1

As shown in eq 12, the objective function does not include any weighting factor as the parameter. Some data points that present random, large deviations from the general trend however were neglected in the parameter optimization. Table 1 presents the experimental data points accepted and used in the subsequent model parameter assessments. Only one data point from ferric nitrate system was excluded in the mathematical treatment in order to get the least-squares objective function values. In this study, the correlation of the Aϕ term based on the work of Akinfiev et al.12 was used. It reproduces the temperature variation of the Debye−Hückel term at low temperatures, below room temperature and down to 233.15 K.12 Following Sippola13 several set of Pitzer parameters were tested using a temperature dependency of the form p = a + b/

Table 2. Pitzer Model Values for Several Combination of Temperature Dependency for Fe(NO3)3 Solutions β(0) Fe(NO3)3 Set Set Set Set Set Set Set Set Set

1 2 3 4 5 6 7 8 9

Cφ

β(1)

a

b

a

b

a

b

objective function

0.8135 −0.186278 0 −4.0120 0.8130 0.8004 −0.4360 0 0

0 5341.4395 226.5857 1369.6128 0 0 347.7786 226.70841 248.1248

5.6695 −0.3038 0 4.5078 4.7537 5.7578 0 0.8773 0

0 1531.8914 1516.73120 0 250.0009 0 1529.7433 1277.0898 1277.0946

−0.0284 2.0425 0 −0.2498 −0.0282 0.0337 0 0 −0.2793

0 −681.3660 −15.0650 0 0 −14.4389 −19.1819 −15.0952 47.4613

0.0204 0.0019 0.0189 0.0160 0.0204 0.0204 0.0183 0.0189 0.0203

C



Article

Figure 3. Calculated water activity aw of Fe(NO3)3-H2O solutions plotted against molality at temperatures from 273.15 down to 248.15 K.

Figure 1. Predicted water activity of aluminum chloride against experimental data from Mason14 and Robinson and Stokes15

One of the important findings of this work is the effect of temperature on the calculated activity coefficients using the Pitzer equations. Figure 4 shows that the calculated activity

Table 4 shows the obtained parameters for aluminum chloride at 25 °C Table 4. Pitzer Parameters for Aluminum Chloride at 25°C β(0)

β(1)

Cφ

0.5477

3.8381

0.0555

Using the obtained Pitzer parameters, we were able to test the model for the determination of several properties of the aqueous solutions of the investigated salt starting from very dilute solutions at selected temperatures below 273.15 K. The first tested property to be evaluated was the osmotic coefficient at selected temperatures. Figure 2 shows the calculated osmotic coefficients for ferric nitrate and Figure 3 depicts its water activities against the molality of the solution at several temperatures as well. Figure 4. Calculated mean activity coefficient of Fe(NO3)3 solutions plotted against molality at temperatures from 273.15 down to 248.15 K.

coefficient at the same molality decreases while temperature increases from 248.15 to 273.15 K. This fact should be tested at temperatures above 273.15 K in order to check the characteristic behaviors of these salts at higher temperatures.

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CONCLUSION The available experimental freezing point data and the solubility of ice in the electrolyte solution below 273.15 K were used to determine Pitzer parameters at low temperatures for aqueous ferric nitrate solutions. The original Pitzer model predicts well the aqueous solutions properties, such as the osmotic coefficients of the solutions and the activity coefficients of the selected metal salt. The thermodynamic model presented in this study enables further studies on the nitrate-water solutions, can be used as an initial point for the prediction on the phase diagram at temperatures above 273.15 K, and will be the object of deeper researches and investigations. Another fact is that the model fitted in this work is easy to handle and could be useful for a detailed analysis of freezing crystallization processes, using a simple set of adjustable parameters. However, new solubility data and calorimetric measurements are recommended to enhance the level of control of the

Figure 2. Calculated osmotic coefficient ϕ of Fe(NO3)3 solutions plotted against molality at temperatures from 273.15 down to 248.15 K.

The calculated osmotic coefficients of ferric nitrate have a similar trend of experimental osmotic coefficient generally measured at 298.15 K for several aqueous electrolyte solutions.15 Because no experimental data for ferric nitrate are available, these calculated values could be considered as a good prediction of their osmotic coefficients at low temperatures. We also believe that new experimental data at higher temperatures and higher molalities are needed in order to confirm the accuracy of the current model and parameter set. D



Article

investigated salts in wider concentration and temperature domains of industrial importance.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

(1) Schreiber, D. R.; Dewyse, K.; Shreiber, L. C.; Tung, T. Enthalpies of Dilution of Some Aqueous Transition Metal Sulfate Solutions at 25°C. J. Solution Chem. 1999, 28, 567−573. (2) Haghighi, H.; Chapoy, A.; Tohidi, B. Freezing Point Depression of Electrolyte Solutions: Experimental Measurements and Modeling Using the Cubic-Plus-Association Equation of State. Ind. Eng. Chem. Res. 2008, 47, 3983−3989. (3) Jones, H. C.; Getman, F. H.; Bassett, H. P.; McMaster, L. Hydrates in Aqueous Solution: Evidence for the Existence of Hydrates in Solution. Their Approximate Composition and Certain Spectroscopic investigation bearing upon the hydrate problem; Carnegie Institution of Washington: Baltimore, MD, 1907. (4) Jones, H. C.; Getman, F. H. Ü ber das Vorhandensein von Hydraten in konzentrierten wässerigen Lösungen. Z. Phys. Chem. 1904, 49, 385−455. (5) Jones, H. C.; Getman, F. H. On the Nature of Concentrated Solutions of Electrolytes. Am. Chem. J. 1904, 31, 303−359. (6) El Goundali, B.; Kaddami, M. The ternary system: H2OFe(NO3)3-Co(NO3)2 isotherms −15°C and −25 °C. J. Alloys Compd. 2008, 460, 544−548. (7) 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. (8) Gärtner, R. S.; Genceli, F. E.; Trambitas, D. O.; Witkamp, G. J. Impurity gradients in solution-frown ice and MgSO4·12H2O crystals measured by cryo-laser ablation and high-resolution-induced-coupled plasma mass spectrograph. J. Cryst. Growth 2005, 275, e1773−1778. (9) Pitzer, K. S.: Activity Coefficient in Electrolyte Solutions; CRCPress: Boca Raton, FL, 1991. (10) Toner, J. D.; Catling, D. C.; Light, B. The formation of supercooled brines, viscous liquids, and low-temperature perchlorate glasses in aqueous solutions relevant to Mars. Icarus 2014, 233, 36−47. (11) Toner, J. D.; Catling, D. C.; Light, B. A revised Pitzer model for low-temperature soluble salt assemblages at the Phoenix site, Mars. Geochim. Cosmochim. Acta 2015, 166, 327−343. (12) Akinfiev, N. N.; Mironenko, M. V.; Grant, S. A. Thermodynamic Properties of NaCl Solutions at Subzero Temperatures. J. Solution Chem. 2001, 30, 1065−1080. (13) Sippola, H. Critical evaluation of the 2nd dissociation constants for aqueous sulfuric acid. Thermochim. Acta 2012, 532, 65−77. (14) Mason, C. M. The Activity and Osmotic Coefficients of Trivalent Metal Chlorides in Aqueous Solution from Vapor Pressure Measurements at 25°. J. Am. Chem. Soc. 1938, 60, 1638−1647. (15) Robinson, R. A., Stokes, R. H.: Electrolyte Solutions, 2nd ed.; Butterworths Scientific Publications: London, 1959.

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Determination of Pitzer Parameters for Ferric Nitrate from Freezing

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