Experimental determination of freezing point depressions in the CuSO 4 -H 2 SO 4 -H 2 O system

Limited experimental data of the aqueous ternary CuSO 4 -H 2 SO 4 -H 2 O system hinders the modelling of its thermodynamic properties. Although different mixing rules for predicting the behavior of mixed electrolyte solutions have been proposed in the literature, significant deviations from them can be expected for systems where e.g., changes in speciation occur. In this work, freezing points of the ternary aqueous solutions were measured using a purpose-built apparatus. The measured samples spanned the concentration range 0.0140 to 0.5534 mol ⋅ (kg-H 2 O) – 1 at four different CuSO 4 – to – H 2 SO 4 ratios. The experimental results were compared with Zdanovskii ’ s rule predictions.


Introduction
As a relevant chemical system in numerous industries (Akilan et al., 2006;Sibarani et al., 2022), new thermodynamically consistent experimental data for the CuSO 4 -H 2 SO 4 -H 2 O system can be beneficial in improving the existing models (May et al., 2010).Interestingly, the database of available thermodynamic properties for the title system is rather limited, freezing point depression data being virtually nonexistent.This work aimed to fill that gap.
It is well-known that accurate values of freezing point depression can be used to derive accurate values for osmotic coefficients (Brown and Prue, 1955), even at extreme dilutions where other methods, e.g. the isopiestic method, fail (Rard and Platford, 1991).The inevitable tradeoff is of course that the results are constrained to freezing point curve.That is, they are limited temperatures below the freezing point of pure water and do not correspond to a single isotherm.Due to the absence of any prior experimental work on aqueous CuSO 4 -H 2 SO 4 -H 2 O freezing point, Zdanovskii's rule was applied to create a credible comparator for our experimental data (Stokes and Robinson, 1966;Zdanovskii, 1936).Zdanovskii's rule or semi-ideal mixing rule states that, in an aqueous mixture containing solute 1 and solute 2, where m i is the molality of each solute in the ternary mixture and m i0 is the molality of the endmember in a binary solute-water system with the same water activity as the ternary mixture.As discussed extensively by Rowland and May (2010), when effective interaction between electrolyte components in an aqueous mixture does not develop, the solvent activity shall behave linearly in accordance with this rule.The freezing point depression data in this work is presented also as osmotic coefficients to investigate the consistency more thoroughly, especially when compared to the endmembers.The determination for the Zdanovskii freezing points for the CuSO 4 -H 2 SO 4 -H 2 O system required thermodynamic models of its constituent subsystems, namely CuSO 4 -H 2 O and H 2 SO 4 -H 2 O. Recently, a machine learning technique for analyzing mathematical correlation between various datasets, including physicochemical properties, was introduced (Neumann et al., 2020).However, the focus of current investigation is on adhering to the widely adopted and dedicated model for aqueous systems, i.e., Pitzer equations (Pitzer, 1973;Rowland et al., 2015).
The model for the CuSO 4 -H 2 O system was constructed using Pitzer equation and meticulously assessed in our prior study (Sibarani et al., 2022), as was undertaken for the H 2 SO 4 -H 2 O system by Sippola and Taskinen (2014).Both models demonstrated satisfactory fitting to their respective literature data, employing a concise set of eight terms each.In the present work, the CuSO 4 -H 2 O system was experimentally investigated prior to CuSO 4 -H 2 SO 4 -H 2 O system.The freezing point data from the literature for the CuSO 4 -H 2 O system, assessed in the prior study, was by no means consistent except within the dilute region.Consequently, Pitzer parameters for the CuSO 4 -H 2 O needed to be re-fitted to a data set that is consistent across the majority of the ice curve.It is crucial to emphasize that the re-optimized binary parameters were exclusive for Zdanovskii freezing points calculations in the present investigation and did not supersede our previous work.

Experimental
Sources and purities of the reagents used are given in Table 1.Briefly, a concentrated CuSO 4 (aq) stock solution was prepared by dissolving CuSO 4 ⋅5H 2 O crystals in ultra-pure water (Direct Q 5 UV, MilliPore).The prepared stock solution was vacuum filtered (PTFE membrane, 0.45 µm) to remove any solid particles.Its concentration was determined by evaporative gravimetry.Triplicate analysis resulted in m = (1.0844± 0.0024) mol⋅(kg-H 2 O) −1 , where the uncertainty was calculated as the standard deviation of the determinations.
H 2 SO 4 stock solution was prepared by dilution from concentrated H 2 SO 4 and ultra-pure water.Its concentration was determined by interpolation from a published volumetric dataset of aqueous sulfuric acid solution (Vielma et al., 2021).Density of the stock solution was determined in triplicate with a commercial vibrating tube densimeter, DMA 5000 M (Anton Paar GmbH, Graz, Austria) (Baird et al., 2017;Hu et 2016;Vielma et al., 2021).The device has a built-in thermostat that controls the sample temperature with a ± 0.001 K precision.Each sample was measured four to five times (multiple measurement mode) to check the measurement precision.Repeat measurements agreed generally within ± 0.000010 g⋅cm −3 , and in most cases within ± 0.000003 g⋅cm −3 .
Buoyancy corrections (Le Neindre and Vodar, 1968;Skoog et al., 2004) were applied throughout, thus the density of every sample was required.In case of freezing point samples, they were measured after being kept at room temperature for about an hour and got rid of the air bubbles, which emerged due to the reduction of gas solubility in water at room temperature.Finally, four ternary CuSO 4 (aq)-H 2 SO 4 (aq) stock solutions were prepared by weight from the binary stock solutions (Table 2).

Freezing point depression
Freezing point depressions (FPDs) were determined with a purposebuilt apparatus.The apparatus consisted of a dewar flask as the solution container, a platinum resistance thermometer (ASL/WIKA CTR2000 and Pt100 probe) with digital reading up to 0.001 degrees, capillary tubes for introducing and sampling the solution, an ice bath to reduce heat flow from the surroundings, and a shaking table to thoroughly but gently mix the sample (Fig. 1).
Equal volumes of ice and solution were introduced into the vessel in the beginning, which would significantly shorten the equilibration time (Rüdorff, 1873).Solution and ice were confirmed to reach equilibrium once the thermometer stabilized (or, at higher concentrations and lower temperatures, showed only a minor drift), usually after one day of equilibration.Vacuum lining of the dewar flask served to further inhibit heat exchange between the system and the environment.Shaking table was crucial to significantly shorten the equilibration time of the system.
Prior to, and occasionally in-between, measuring the ternary solution FPDs, the apparatus was run with pure de-ionized water (conductivity 0.05 µS/cm, 298.15 K) to set the zero-degree point of the thermometer.The samples were extracted from the dewar flask by using a disposable syringe to avoid evaporation of the H 2 O from the solution.
One crucial checkpoint during the freezing point equilibration was to prevent any condensation around the container lid, the tube, and the temperature probe, an apparent indication of a sufficiently closed system (Hefter and Tomkins, 2004).
After each sampling (approximately 40 ml of solution), a similar volume of pure de-ionized water was injected into the system to set a new equilibrium point.When the temperature was around 273.10 K after multiple dilutions, it marked the last equilibrium point of one process cycle.The remaining solution inside the dewar flask was disposed along with the ice, and the prepared stock solution (refer to Table 2) was injected to commence a new cycle.The molality uncertainty was determined from triple the standard deviation of the quadruplicates (to achieve 0.99 level of confidence).
Several minor practical details for the experiment, which can be detrimental to the result's precision if neglected, were discovered mostly during the early test with simpler system, e.g., CuSO 4 -H 2 O and pure water.First, sampling must be done in a single drawing without any pause, otherwise an inconsistency in the temperature reading might

Post chemical analysis of solution
Concentrations of the collected samples were determined by evaporative gravimetry in quadruplicate.Each extracted sample was inserted into four glazed porcelain crucibles with lids and heated up to 370-⁰C in a muffle furnace, at which the water and sulfuric acid were completely removed from the sample, leaving the remaining dry CuSO 4 to be weighed (model AB204-S, Mettler Toledo).
The heating was performed stepwise.First, the samples were evaporated to dryness at temperature near 90 • C for a day to avoid splattering at higher temperatures.The temperature was then increased to 230 ⁰C for a day, to remove most of the sulfuric acid.Finally, the samples were dried to constant weight at temperature around 370 ⁰C.This stepwise procedure ensured that no splattering occurred, and all the samples were quantitatively accounted for.The amount of H 2 SO 4 in the sample was calculated from the amount of CuSO 4 and the known CuSO 4 -H 2 SO 4 ratio of the stock solution used.Buoyancy corrections were applied throughout.
Powder XRD analysis (model X'Pert Pro MPD, Malvern Panalytical) was conducted to confirm the chemical formula of the dry sample, which gave the exact CuSO 4 .Tube gave CuKα radiation and set at 40 kV and 40 mA.The powder was ground in an agate mortar to < 50 µm and then spread evenly on a flat sample holder.The dry residues were grouped up based on the cycle and XRD scanned separately.

Results and discussion
For the CuSO 4 -H 2 O system, the agreement between the current experimental data (Table 3) to Brown and Prue (1955), Hausrath (1902) and the model from our previous work (Sibarani et al., 2022) was within ± 0.003 K and/or three times measured molality standard deviation (0.99 level of confidence), at dilute concentration.At moderate concentration, however, the gap between them steadily increases, which is most likely due to poor quality of available literature freezing point depressions (FPD) data that were employed in the model (Fig. 2).Compared to the scattered literature data of the past, the present experimental data show an internal consistency from dilute to neareutectic concentration.The standard uncertainty of temperature by this apparatus was determined to be between 0.002 K and (0.005⋅θ exp ) K, which exhibit the positive correlation between the degree of freezing point depression and its uncertainty.Meanwhile, type A evaluation was applied to standard uncertainties of solution concentration (BIPM et al., 2008a).The experimental FPD from Table 5 were converted to water activities by using the empirical equation by Sippola and Taskinen (2018), from which the stoichiometric osmotic coefficients were calculated (Le Neindre and Vodar, 1968).
For the calculation of Zdanovskii water activities, Pitzer parameters from the previous work (Sibarani et al., 2022) were re-optimized using FactSage software (Bale et al., 2016) and presented in Table 4.The updated data inputs included the FPD, comprising data from the present study and three experimental literatures, specifically the work of Brown and Prue (1955), Hausrath (1902), and Klein and Svanberg (1918).Simultaneously, the errors (input to the optimizer) associated with the water activities at freezing point were significantly reduced to 0.0001 in response to the improved data consistency.In addition, the solubility data of Patrick and Aubert (1896) were excluded from the optimization, while the remaining literature data were retained.
D. Sibarani et al. quadruplicate determinations, and the expanded uncertainty is thrice their standard deviation (0.99 level of confidence).
The osmotic coefficients were then plotted, incorporating the literature data of CuSO 4 -H 2 O and H 2 SO 4 -H 2 O (Hausrath, 1902;Lide et al., 2005;Randall and Scott, 1927) as illustrated in Fig. 4. It reveals that the osmotic coefficients of the ternary mixtures fall between those of the pure binaries.Since osmotic coefficient is the product of water activity and solutes concentration, the uncertainty of the osmotic coefficient was calculated using the "law of propagation of uncertainty" by JCGM-GUM (BIPM et al., 2008b), where c (y) is the combined uncertainty of a property of interest, f is the function to calculate y, and x i are the variables in f that possess an uncertainty, i.e., solutes concentration and temperature.The visually smooth curves of R1 to R4 indicate internally consistent datasets.
Given the absence of prior freezing point data for the ternary system, the present results were compared against predictions made by applying the Zdanovskii's rule.Zdanovskii water activities, which correspond to Zdanovskii freezing points and osmotic coefficients, were calculated using the re-optimized Pitzer parameters of CuSO 4 -H 2 O (refer to Table 4) and H 2 SO 4 -H 2 O (Sippola and Taskinen, 2014).Residuals between the experimental data of this work (ɸ exp ) and the Zdanovskii's rule prediction (ɸ zdn ) are depicted in Fig. 5.The chart displays that the residuals tend to increase as the solutes concentrations decrease, particularly below 0.1 mol⋅(kg-H 2 O) −1 of total sulfate molality, which is deemed highly probable due to osmotic coefficient error increases as a function of 1/m.It is noteworthy that the residuals spread around Δф of −0.01, rather than 0.00, with a range of ± 0.01.
The ternary solution R4, which shows the largest residuals from the Zdanovskii osmotic coefficients, possesses the largest ratio of CuSO 4 -to-H 2 SO 4 .This implies that the prominent deviation of R4 might be attributed to the model of CuSO 4 -H 2 O.Despite so, the conformity of most experimental data to the predicted values by Zdanovskii's rule assert the linearity of the ternary CuSO 4 -H 2 SO 4 -H 2 O system, where its thermodynamic properties could be estimated quite precisely through a mixing rule.

Conclusion
Freezing point depression in the ternary CuSO 4 -H 2 SO 4 -H 2 O system have been investigated experimentally.Four kinds of ternary mixtures were prepared based on the CuSO 4 to H 2 SO 4 mole ratio, with the total sulfate concentration ranging between (0.014 and 0.549) mol⋅(kg-H 2 O) −1 .Plotting of the measured osmotic coefficient over total sulfate molality showed internal consistency among the data and relative to the binary CuSO 4 -H 2 O and H 2 SO 4 -H 2 O.
As no prior literature on the freezing point depression of the ternary system was reported, the present results were compared to Zdanovskii's rule predictions.In essence, the deviations of osmotic coefficient between the experimental and calculated values are generally distributed around −0.01 with a range of ± 0.01, except at total solutes  This work, (Brown and Prue, 1955), (Klein and Svanberg, 1918), (Hausrath, 1902).
concentration below 0.1 mol⋅(kg-H 2 O) −1 .Such profile confirms the agreement between the ternary CuSO 4 -H 2 SO 4 -H 2 O system with the semi-ideal mixing rule.

Table 1
Reagents sources and purities.Concentration of CuSO 4 -H 2 SO 4 -H 2 O stock solutions. .It might unintentionally increase the solution temperature, causing a partial melting of the ice inside thus each extraction must be treated as a separate sample.Second, the shaking table should not be stopped during sample extraction.Third, the capillary tube must be sufficiently small to prevent ice fragments from getting into the sample solution (this work used 1 mm inner diameter), thus rendering any dilution due to the melting ice unlikely.
a major impurity water with around 0.1 mass fraction of ionic impurities b major impurity water, with 4.4 ppm of ionic impurities Table 2 Fig. 1.Static freezing point depression measuring apparatus schematic.D. Sibarani et al. happen

Table 3
Concentration-based freezing point depression θ exp of CuSO 4 -H 2 O with its expanded uncertainty (k = 3) of molality U(m CuSO4 ) and its corresponding stoichiometric osmotic coefficient ф exp and residuals from our re-optimized model Δθ (θ Pitzθ exp ).

Table 4
Re-optimized Pitzer parameters for temperature residuals plotting.