Osmotic Coefficients of Aqueous Weak Electrolyte Solutions: Influence

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Osmotic Coefficients of Aqueous Weak Electrolyte Solutions: Influence of Dissociation on Data Reduction and Modeling Thomas Reschke, Shahbaz Naeem, and Gabriele Sadowski J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/jp3005629 • Publication Date (Web): 23 May 2012 Downloaded from http://pubs.acs.org on June 5, 2012

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The Journal of Physical Chemistry

Osmotic Coefficients of Aqueous Weak Electrolyte Solutions: Influence of Dissociation on Data Reduction and Modeling Thomas Reschke, Shahbaz Naeem, Gabriele Sadowski* Laboratory of Thermodynamics, Department of Biochemical and Chemical Engineering, Dortmund University of Technology, Emil-Figge-Straße 70, D-44227 Dortmund [email protected] RECEIVED DATE (to be automatically inserted after your manuscript is accepted if required according to the journal that you are submitting your paper to)

The experimental determination and modeling of osmotic coefficients in electrolyte solutions requires knowledge of the stoichiometric coefficient νi. In contrast to strong electrolytes, weak electrolytes exhibit a concentration-dependent stoichiometric coefficient, which directly influences the thermodynamic properties (e.g., osmotic coefficients). Neglecting this concentration dependence leads to erroneous osmotic coefficients for solutions of weak electrolytes. In this work, the concentration dependence of the stoichiometric coefficients and the influence of concentration on the osmotic coefficient data were accounted for by considering the dissociation equilibria of aqueous sulfuric and phosphoric acid systems. The dissociation equilibrium was combined with the ePC-SAFT equation of state to model osmotic coefficients and densities of electrolyte solutions. Without the introduction of any additional adjustable parameters, the average relative deviation between the modeled and the ACS Paragon Plus Environment

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experimental data decreases from 12.82 % to 4.28 % (osmotic coefficients) and from 2.59 % to 0.89 % (densities) for 12 phosphoric and sulfuric systems compared to calculations that do not account for speciation. For easy access to the concentration-dependent stoichiometric coefficient, estimation schemes were formulated for mono-, di-, and triprotic acids and their salts.

Keywords: dissociation equilibrium, speciation, stoichiometric coefficient, PC-SAFT Introduction Electrolyte solutions play an important role in chemical and biochemical engineering. Applications of these systems include wastewater and drinking water treatment, fertilizer production, enhanced oil recovery1, electrolysis2 and wet flue-gas scrubbing3. Electrolytes can be used in separation processes such as reverse osmosis, distillation (as entrainers), precipitation of biomolecules and extraction processes that use aqueous two-phase systems4. Some weak electrolytes are used as buffers in biosystems. The first step in describing electrolyte solutions is usually to classify them as strong or weak electrolytes. This classification has a direct impact on the evaluation of their thermodynamic properties, such as the osmotic coefficient φ φ=

− ln(a solvent ) M solvent ⋅ ∑ ν i ⋅ m i

(1)

which characterizes molecular interactions in such solutions. In this equation, asolvent and Msolvent are the activity and the molecular weight of the solvent, respectively. The parameter νi is the stoichiometric coefficient of the solute i, which is more or less dissociated in the solvent, and mi is the molality of the solute i. The value of νi must be known prior to (I) determining the experimental osmotic coefficients and (II) performing the respective modeling. Because νi is experimentally not directly accessible for many solutes, the value of νi is usually estimated by heuristics. For molecules that do not dissociate in aqueous solution, such as saccharides, νi is equal to one. For components assumed to be fully dissociated (strong electrolytes), νi is set equal to the number of ions in the electrolyte, e.g., two for 2 ACS Paragon Plus Environment

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NaCl. No such heuristics apply to the stoichiometric coefficients of weak electrolytes. During the last few decades, two main strategies have been pursued for the description of weak electrolytes: (I) treating weak electrolytes as if they were strong (complete dissociation) and (II) explicitly accounting for the partial dissociation. A well-examined example with high industrial and biological impact is the phosphoric acid/water system. In his model, Platford5 assumed that phosphoric acid is a strong electrolyte and set the stoichiometric coefficient—independent of concentration—equal to four. In contrast, the partial dissociation of phosphoric acid was taken into account by Pitzer and Silvester6. Based on the Pitzer equation, Pitzer and Silvester6 considered at least the first dissociation step of H3PO4 into H+ and H 2 PO −4 . To further improve the modeling results, more recent investigations have also considered the

formation of H 5 P2 O8− in a second reaction step7. Held and Sadowski8 combined an ion-pair formation mechanism with the ePC-SAFT framework to describe the mean ionic activity coefficients and the densities of weak electrolyte solutions. In all of these approaches, except in the approach of Platford, the speciation of electrolytes as well as their molecular interactions were taken into account for the modeling. However, to the best of our knowledge, the change in the stoichiometric coefficient νi with concentration was never considered when the osmotic coefficients were experimentally determined, which resulted in occasionally doubtful experimental osmotic coefficients and in modeling results that are difficult to evaluate. In this work, the dissociation constants of weak electrolytes, specifically, H2SO4 and H3PO4 and their sodium and potassium salts, are used to calculate the speciation in aqueous solution. In a first step, the resulting concentration-dependent stoichiometric coefficients allow for the recalculation of the experimental osmotic coefficients for these solutions. The dissociation equilibria are combined with ePC-SAFT to model the osmotic coefficients and the solution densities of the solutions.

Theory and Model

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Dissociation Equilibria and Stoichiometric Coefficients. The determination of the stoichiometric coefficients in a solution of a weak electrolyte is based on the dissociation equilibrium in water. For a monoprotic acid HX, the dissociation can be described by the following reaction scheme: Ka HX + H 2 O ← → H3O+ + X −

(2)

The dissociation equilibrium can be expressed by the law of mass action using the thermodynamic equilibrium constant Ka, where ai are the activities of the different species:

Ka =

a H O+ ⋅ a X − 3

a HX ⋅ a H 2O

=

γ H O+ ⋅ γ X − x H O + ⋅ x X − 3 ⋅ 3 γ HX ⋅ γ H 2O x HX ⋅ x H 2O

(3)

To obtain a readily available formula, the activity coefficients γi are combined in Kγ, and the mole fractions xi are converted into molarities ci.

K cacid =

c − Ka ⋅ cH2O = cH O+ ⋅ X 3 Kγ c HX

(4)

The acid constant K cacid is usually measured potentiometrically and extrapolated to infinite dilution. Often, Equation (4) is directly used to determine the species concentration from K cacid under the assumption that K cacid is independent of concentration. To avoid inaccuracies that arise from changing water concentrations, i.e., at higher solute concentrations, and to obtain a formula that is more applicable to electrolytes, the molarities in Equation (4) are replaced (not exclusively) in this work by molalities. This substitution has the advantage that, in contrast to molarity c, the molality m of water stays constant for all solute concentrations. Using molalities, the dissociation equilibrium of a monoprotic acid can be described by the following equation: m K acid =

m H+ ⋅ m X− m HX

=

n H+ ⋅ n X− V = ~ L ⋅ K cacid ~ n HX ⋅ m w m w (5)


4

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~ becomes For infinite dilution, the ratio between the solution volume VL and the mass of water m w m unity so that K acid and K cacid become equal. To extend the model to polyprotic acids, the dissociation

for every dissociation step n has to be considered explicitly:

[H n X ]

[H n − 1X]− ←acid,2  →

K acid,1 ←   → 1 ⋅ H + +

K

K acid, n K ←  → n ⋅ H + + X n −

(6)

A set of n equations describes the equilibrium concentrations of the different species. These equations can be obtained from the following equation by varying j between 1 and n: m K acid ,j =

m H + ⋅ m [H

n − jX

−

m [H n +1− jX ]

]

,j=1…n

(7)

The resulting set of equations contains n+2 variables (molalities of n dissociated species, the molality of the non-dissociated species, and m H + ), and two more equations are needed to determine the respective concentrations. Therefore, the anions (Equation (8)) as well as the cations (Equation (9)) are balanced: m initial = m [H n X ] + m [H

n −1 X

−

] + ... + m [X ]

(8)

n−

n initial ⋅ m initial = n ⋅ m [H n X ] + ( n − 1) ⋅ m [H

n −1 X

−

] + ... + m H

+

(9)

In these equations, mi are the molalities of the species, and n is the number of hydrogen atoms that the respective species can donate. The dissociation equilibrium of acidic electrolytes can be determined by solving Equations (7)-(9). Because Equation (7) considers the dissociation of acids, Equation (7) must be modified for bases. In this case, the ion product of water m K Water = K acid ⋅Km base = m H + ⋅ m OH −

(10)

m is included to replace the acid constant K acid with the base constant K m base , which leads to Equation

(11): Km base , j =

m OH − ⋅ m [H n +1− jX ] m [H

n − jX

−

]

,j=1…n


(11)

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To determine the molality of the hydroxide ions, the cation balance (Equation (9)) is modified, which leads to Equation (12): n initial ⋅ m initial = n ⋅ m [H n X ] + ( n − 1) ⋅ m [H

n −1 X

−

] + ... − m OH

−

(12)

The dissociation equilibrium of basic electrolytes can be determined by solving Equations (8), (11), and (12).

Using the equilibrium concentrations of all species, the concentration-dependent stoichiometric coefficient νcalc can now be determined by dividing the total molality of all dissolved species by the initial molality: ν calc =

∑mi

(13)

m inital

This method can be applied to describe aqueous solutions of salts that exhibit basic or acidic behavior in solution. To calculate the dissociation equilibrium of salts, the salt in a first step is assumed to dissociate completely into its metallic cation and basic or acidic anion:

[M d HX ] → d ⋅ M + + HX d −

(14)

The dissociation equilibrium can then be formulated and solved for the anion as described previously. Whether the anion exhibits acidic or basic behavior in solution can be determined by comparing the acid constants and base constants of the electrolyte. An example in Figure 1 illustrates the dissociation m scheme for phosphoric acid into its corresponding anions along with their respective K acid (arrows

pointing right) and K m base (arrows pointing left) values. Dihydrogen phosphate donates a larger number -12 m of protons than it accepts because K acid (6.17·10-8) is larger than K m base (1.45·10 ). For dihydrogen m phosphate, a larger number of protons are accepted than are donated because K acid (4.79·10-13) is -7 smaller than K m base (1.62·10 ). Consequently, dihydrogen phosphate is an acid, whereas monohydrogen

phosphate is a base.


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Figure 1. Dissociation scheme of phosphoric species. The arrows pointing right and left describe the m K acid values and K m base values, respectively.

All acid constants used in this work are listed in Table 1. The corresponding base constants can be determined from Equation (10).

Table 1. Acid constants of sulfuric acid and phosphoric acid in water at 25°C used in this work. Acid Sulfuric acid Phosphoric acid

m K acid ,1 10³ 6.92 · 10-3

m K acid ,2 1.26 · 10-2 6.17 · 10-8

m K acid ,3 4.79 · 10-13

Reference 9 10

ePC-SAFT Model. For modeling thermodynamic properties (densities, activity/osmotic coefficients) of electrolyte solutions, many different approaches have been developed in recent decades. Electrolyte perturbed-chain SAFT11 (ePC-SAFT) has already been applied to strong as well as weak electrolyte solutions. ePC-SAFT is also used in this work to model 12 sulfuric and phosphoric systems, specifically, aqueous solutions of H3PO4, NaH2PO4, KH2PO4, Na2HPO4, K2HPO4, Na3PO4, K3PO4, H2SO4, NaHSO4, KHSO4, Na2SO4, and K2SO4.

ePC-SAFT is based on a perturbation theory where the hard-chain system is used as the reference system. All other contributions that can be considered independently are considered additive. The residual Helmholtz energy ares can therefore be written as ares = ahc + adisp + aassoc + aion

(15)


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where ahc represents the hard-chain repulsion of the reference system. The terms adisp, aassoc, and aion account for the Helmholtz-energy contributions due to dispersive, associative, and ionic interactions, respectively. A detailed description of the contributions is presented elsewhere11-13.

For the ePC-SAFT modeling, three pure-component parameters are necessary to describe the molecular properties of a non-associating molecule i: the segment number mseg,i, the segment diameter σi, and the dispersion-energy parameter ui/kB. For associating components such as water, two additional parameters are required: the association energy and the association volume parameters AiBi ε AiBi hb / k B and κ hb / k B . For an ion j, the segment number is set to unity (mseg,j =1), which reduces the

number of ion parameters to two. Dispersive interactions between two ions are neglected14.

In contrast to various other electrolyte models, ePC-SAFT uses ion-specific parameters. In the model, an electrolyte is split into its respective ions, which are treated as independent components. The utilization of ion-specific parameters is physically reasonable and allows ePC-SAFT to predict thermodynamic values of salts for which no experimental data are available. The ion parameters for the electrolytes considered within this work are shown in Table 2.

Table 2. ePC-SAFT parameters used in this work. Ion

σj (Å)

uj / kB (K)

Reference Ion

σj (Å)

uj / kB (K)

Reference

H+

2.2740

1616.4939

14

OH-

1.6401

2444.7555

14

Na+

2.4122

646.0504

14

H 2 PO −4

3.7026

0.0000

K+

2.9698

271.0518

14

HPO 24 −

4.4608

0.0000

SO 24 −

2.4538

0.0000

PO 34−

2.2360

1785.31


14

14

14

this work

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Data Reduction Species Distribution. By solving the dissociation equilibria described in the theory, the species distribution for aqueous sulfate (Figure 2) and phosphate (Figure 3) solutions was obtained as a function of the initial solute molality. For simplicity, the concentration dependence of Kγ was not considered at this point because the results obtained by considering Kγ did not improve. For a better comparison of the species distributions at different concentrations, the respective anion molalities are converted into species fractions fi by dividing them by the initial solute molality. fi =

mi m initial

(16)

Figure 2 shows the calculated anion fractions of sulfuric acid (a), bisulfate (b) and sulfate (c) versus the solute molality minitial.


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Figure 2. Calculated fractions fi of all anionic species obtained in aqueous solutions of sulfuric acid (a), bisulfate (b), and sulfate (c) at 25°C as a function of the solute molality.


10

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For sulfuric acid (Figure 2a), the fractions of the dissolved anions change with increasing sulfuric acid molality. At low concentrations, the solution contains large amounts of sulfate ions. The further addition of sulfuric acid decreases the fraction of sulfate ions in favor of the bisulfate ions. Non-dissociated sulfuric acid is negligible over the whole concentration range. A comparison of Figure 2a to 2c reveals that sulfuric acid (Figure 2a) and bisulfate (Figure 2b) show a concentration-dependent change in anion distribution. For sulfate solutions, the fraction of sulfate anions (Figure 2c) is always one over the whole concentration range. The modeling of sulfuric acid and bisulfate solutions requires the consideration of two anionic species; sulfate solutions can be described with only one anionic species.

Figure 3 illustrates the ion speciation of phosphoric acid and its salts as a function of the solute molality.


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Figure 3. Calculated fractions fi of all anionic species obtained in aqueous solutions of phosphoric acid (a), dihydrogen phosphate (b), monohydrogen phosphate (c), and phosphate (d) at 25°C as a function of the solute molality.

For phosphoric acid (Figure 3a) and tribasic phosphate (Figure 3d), a concentration-dependent change of the species present in the solution is observed. For mono- and dihydrogen phosphate, only one species is present over the whole concentration range. As demonstrated for sulfuric acid, an accurate modeling of phosphoric acid and the tribasic phosphate requires two anionic species (H3PO4/ H 2 PO −4 and HPO 24 − / PO 34− , respectively). Because only one anionic species is present in mono- and dihydrogen

phosphate solutions, salts that contain these anions can be well characterized with only one species considered.

Concentration-Dependent Stoichiometric Coefficients. The knowledge of the species distribution in electrolyte solutions allows for direct calculation of the stoichiometric coefficient. Figure 4 illustrates the concentration-dependent stoichiometric coefficients νcalc of sulfate (Figure 4a) and phosphate i (Figure 4b) solutions as a function of the initial (overall) molality determined by Equation (13).


12

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Figure 4. Calculated concentration-dependent stoichiometric coefficients νcalc in aqueous salt solutions at 25°C as a function of the solute molality: (a) sulfate salts, (b) phosphate salts.

As Figure 4 shows, ν icalc depends on the concentration of sulfuric acid, bisulfate, phosphoric acid and phosphate. The parameter ν icalc is concentration-independent for sulfate, dihydrogen phosphate, and monohydrogen phosphate. The first four previously mentioned solutes show an almost similar curve progression. At low molalities, ν icalc reaches its highest value and then decreases with increasing solute molality. In Figure 3, the species distribution was shown to change with increasing solute molality. This change is accompanied by the formation of hydronium or hydroxide ions in the solution, and this change therefore causes a change in the stoichiometric coefficient. Because the species distributions of sulfate, ACS Paragon Plus Environment

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dihydrogen phosphate, and monohydrogen phosphate are concentration-independent, the amount of hydronium or hydroxide ions formed is negligibly low, and the stoichiometric coefficient is concentration-independent.

Recalculated Osmotic Coefficients. As shown in the previous section, the stoichiometric coefficients may depend on concentration. The concentration of dissolved species in a solution is not always proportional to the initial solute concentration. For compounds with concentration-dependent stoichiometric coefficients, the osmotic coefficients determined previously based on Equation (1) must be recalculated. To compare the osmotic coefficients published in the literature φlit with the osmotic coefficient φcalc obtained using a concentration-dependent stoichiometric coefficient presented in this work, the literature data were re-evaluated by:

φ

calc

φ lit ⋅ ν ilit = calc νi

(17)

The values for the osmotic coefficients φlit and φcalc as well as the corresponding values ν icalc for the systems considered in this work are shown in Table 5. The parameters φlit and φcalc are equal or almost equal for some solutes (e.g., Na2SO4, NaH2PO4), whereas for other solutes (NaHSO4, KHSO4), large differences between φlit and φcalc were obtained. As an example, Figure 5 shows φlit and φcalc for sulfuric and phosphoric acid solutions as functions of the acid concentration at 25°C.


14

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Figure 5. Comparison of osmotic coefficients φlit from the literature5,15 (circles) and re-evaluated osmotic coefficients φcalc (squares) at 25°C for (a) sulfuric acid solution and (b) phosphoric acid solution.

For sulfuric acid solutions (Figure 5a), φlit decreases until it reaches a value of 0.65 at a molality of approximately 0.2 before it steadily increases with increasing molality. No obvious zero-concentration limit at unity is expected for osmotic coefficients. In contrast, φcalc does start at unity and then decreases until it reaches a value of 0.94 again at a molality of approximately 0.2 before it increases steadily (analogously to φlit). The difference between φlit and φcalc is approximately 30 %, which is a result of the difference between the estimated stoichiometric coefficients νlit (complete dissociation, i.e., νlit=3) and νcalc (given in Table 5). ACS Paragon Plus Environment

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For phosphoric acid solutions (Figure 3b), the osmotic coefficient φlit found for the most dilute solution at 25°C is 0.325 at 0.1236 m5. Platford5 determined an osmotic coefficient of 0.7 for a dilute phosphoric acid solution by freezing-point depression measurements, i.e., below 25°C, at 7.6·10-5 m. These two values indicate the inaccuracy of concentration-independent calculations because the osmotic coefficient must approach 1 for high dilutions (Equation (1)). Platford5 assumed a constant stoichiometric coefficient of 4 for phosphoric acid. However, considering Figure 4, the stoichiometric coefficient of a dilute phosphoric acid solution is neither constant nor does it reach the assumed value of 4. Instead, the calculated stoichiometric coefficient νcalc varies between 1.20 and 1.05 (see Table 5). The osmotic coefficient determined by Platford does not converge to unity, which, in contrast, is the case for the re-evaluated osmotic coefficient φcalc. The two examples show that the application of concentration-dependent stoichiometric coefficients that reflect the real chemistry in such systems is highly recommended for weak electrolyte solutions. The appendix contains easily accessible estimation schemes for the stoichiometric coefficients in these systems.

Modeling Results Determination of ePC-SAFT Parameters. As shown in the previous section, the dissociation equilibrium of weak electrolyte solutions can strongly influence the data reduction for osmoticcoefficient measurements. To account for speciation in the modeling of thermodynamic properties, the dissociation scheme is combined here with the ePC-SAFT model (hereafter called ePC-SAFT+DE). To appraise the benefit of this approach, classic ePC-SAFT and ePC-SAFT+DE as well as modeled osmotic coefficients and experimental data are compared in the following sections. As shown previously, the characterization of sulfate, dihydrogen phosphate, and monohydrogen phosphate can be performed by considering only one anionic species. The existing ion parameters for these anions given by Held et al.14 can still be applied to ePC-SAFT+DE. The existing ion parameters of Na+, K+, H+, and OH- are still used in ePC-SAFT+DE.


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For an accurate application of the dissociation equilibrium, parameters for H3PO4, H2SO4, HSO −4 , and PO 34− are required. Because H3PO4 forms dimers in aqueous solutions, H3PO4 is not modeled as an ion

but rather as an associating component using the 2B association scheme. Dispersive interactions between H3PO4 and ions are not considered. In contrast, H2SO4 is modeled using only two parameters (mseg, uij) without accounting for association because H2SO4 does not form dimers in aqueous solutions, and its concentration is always very low (Figure 2a). The ion parameters of PO 34− used for the original ePC-SAFT (Table 2) had to be refitted for ePC-SAFT+DE because PO 34− is highly affected by the dissociation equilibrium (see Figure 3d). The new ion parameters were determined by fitting these ion parameters to solution densities and to corrected osmotic coefficients φcalc of aqueous electrolyte solutions. To assure qualitatively correct pure-component ion parameters, the anion parameters were adjusted to experimental data from a concentration region in which a high fraction of the considered ion is present (Figure 2 and Figure 3). The ion parameters determined in this manner for ePC-SAFT+DE are given in Table 3. To compare the modeling results with the classic ePC-SAFT modeling, the PO 34 − parameters for the classical modeling were fitted to densities and osmotic coefficients of aqueous solutions of Na3PO4 and K3PO4 using the classic approach (complete dissociation). In the same manner, NaHSO4 and KHSO4 were assumed to fully dissociate into Na+, H+, and SO 24− and into K+, H+, and SO 24− , respectively, when the classic approach was used.

Table 3. ePC-SAFT parameters used in this work, which are valid only with the parameter set for water by Held et al.14 These parameters can be used only with ePC-SAFT, which accounts for the dissociation equilibrium of electrolytes. Species H 3 PO 4

mseg 5.9502

σj (A) 2.2525

uj / kB (K) 176.1265

ε Ahbi B i / kB

5137.7752


κ Ahbi B i

0.2000

kij -0.1524

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PO 34−

1.0000

1.6116

2523.8919

-

-

0.0000

H 2 SO 4

1.0000

4.2803

0.0000

-

-

0.0000

HSO -4

1.0000

4.0332

42.2902

-

-

0.0000

Comparison of ePC-SAFT and ePC-SAFT+DE. Using the ion-parameter sets for ePC-SAFT (Table 2) and ePC-SAFT+DE (Table 2 and 3), the osmotic coefficients and the solution densities were modeled at 25°C. The average absolute deviations (AADs) and the average relative deviations (ARDs) between the experimental and the modeled value p are calculated with the number of data points (NP) by:

AAD =

1 p exp − p mod ∑ NP

ARD = 100 ⋅

(18)

1 pexp − p mod ∑ pexp NP

(19)

Equations (18) and (19) were used to determine the AAD and the ARD for ePC-SAFT and ePC-SAFT+DE (Table 4) for the densities and the osmotic coefficients. Whereas the originally published osmotic coefficients φlit were described with the original ePC-SAFT (species distribution is considered neither for the experimental data nor for the modeling), ePC-SAFT+DE was compared with the osmotic coefficients recalculated with Equation (17). The respective osmotic coefficients from the experiment and model were determined using the same stoichiometric coefficient used in Equation (1) and are therefore comparable. The ARDs determined for ePC-SAFT and ePC-SAFT+DE are comparable as well because the literature data and the modeled osmotic coefficient contain the stoichiometric coefficient so that the influence of the ν-values cancels: lit ν mod ν ⋅ lit mod calc − φ φcalc − φmod + DE φ ⋅ ν ν calc = φ − φ = lit φcalc φlit φlit ⋅ ν calc ν lit

lit


(20)

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Table 4. Model deviations (ARD and AAD) from the experimental osmotic coefficient and density data calculated with ePC-SAFT+DE and ePC-SAFT at 25°C. NP is the number of data points, and mmax is the highest molality used for the calculation of the AAD and the ARD. Osmotic Coefficient ePC-SAFT+DE Salt

Ref

mmax

H3PO4

5,15,16

NaH2PO4

AAD

ARD

Density ePC-SAFT AAD

ARD

18.54 0.015 1.14

1.486

15

6.50

0.029 3.94

KH2PO4

15

1.80

Na2HPO4

15

K2HPO4

ePC-SAFT+DE mmax

AAD

ePC-SAFT

ARD

AAD

ARD

373.40 18.14 0.04

0.24

258.18

19.76

0.029

3.96

4.63

3.54

0.34

3.49

0.49

0.018 2.49

0.018

2.52

1.84

5.27

0.49

5.27

0.49

1.10

0.048 7.54

0.048

7.54

0.51

19.02 1.82

19.09

1.82

15

1.00

0.036 5.09

0.037

5.16

1.43

28.25 2.56

28.37

2.57

Na3PO4

15

0.70

0.031 6.10

0.076

13.35

0.50

14.91 1.41

13.14

1.24

K3PO4

15

0.70

0.034 5.46

0.072

10.72

1.25

25.51 2.21

23.21

2.01

H2SO4

15

4.00

0.038 3.76

0.126

13.25

24.84 5.17

0.40

191.54

13.30

NaHSO4

17,18

6.32

0.029 3.07

0.274

36.64

3.54

1.91

0.18

24.96

2.17

KHSO4

15,17

2.65

0.036 4.19

0.214

36.65

2.43

10.57 0.89

51.27

4.30

Na2SO4

15

1.80

0.054 8.13

0.054

8.13

1.18

1.45

0.14

1.45

0.14

K2SO4

15

0.70

0.009 1.24

0.009

1.24

0.50

0.86

0.08

0.86

0.08

The application of ePC-SAFT+DE allows an accurate description of 12 aqueous weak-electrolyte systems with overall ARDs of 0.89 % for the densities and 4.28 % for the osmotic coefficients. A comparison of these modeling results with the modeling results obtained without accounting for the speciation in Table 5 reveals an improvement of the determined ARDs of 1.70 % for the densities and 8.54 % for the osmotic coefficients. The calculation of the overall ARD excludes the ARD of phosphoric acid because the deviation for the osmotic coefficients modeled with just ePC-SAFT is almost 375 % and therefore not representative. Figure 3a shows that the aqueous phosphoric acid


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solution contains remarkable amounts of H3PO4 and H 2 PO −4 but no PO 34− . However, when the speciation is not accounted for, ePC-SAFT models the osmotic coefficient of this solution as PO 34− , the only anionic species that obviously leads to high deviations in modeling of the experimental data. As a direct consequence of considering PO 34− as the only phosphoric species, thermodynamic properties were modeled because the system would contain large amounts of hydronium ions (complete dissociation), whereas their real amount in the solution is almost equal to the amount of H 2 PO −4 and is therefore rather low compared to what is assumed in the modeling with just ePC-SAFT. Because ePC-SAFT+DE determines the true species distribution, very accurate calculations can be performed for the osmotic coefficients and densities of aqueous H3PO4 solutions over a wide concentration range. Considerably high deviations (greater than 10 %) for the ePC-SAFT modeling were also found for Na3PO4, K3PO4, H2SO4, NaHSO4, and KHSO4. Figures 2 and 3 show that all of these systems are highly affected by speciation. The other electrolyte solutions that show almost no speciation can be modeled with comparably low deviations for both models. Figure 6 exemplifies the comparison of ePC-SAFT and ePC-SAFT+DE for modeling the osmotic coefficients (a) and the solution densities (b) of NaHSO4. A comparison of the osmotic coefficients shows that both models provide a good fit for the data up to concentrations of 0.3 m. However, for higher concentrations, neglecting the speciation leads to increasing deviations, whereas ePC-SAFT+DE still precisely fits the experimental data. The same trend is observed for solution densities. For concentrations up to 0.5 m, both modeling approaches are in very good agreement with the literature data; for higher concentrations, however, modeling with ePC-SAFT without accounting for the dissociation equilibria reveals an increasing deviation. With respect to the species distribution of sodium bisulfate (Figure 2b), a significant change in the species distribution can be observed. Varying amounts of ions in the solution have different interactions with the solvent. The calculated amount of H+ differs between ePC-SAFT and ePC-SAFT+DE because ePC-SAFT considers NaHSO4 to be fully dissociated. This assumption leads to accurate results when ePC-SAFT is used for solutions with low concentrations, ACS Paragon Plus Environment

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where NaHSO4 is almost fully dissociated, but leads to high inaccuracies in the modeling of the osmotic coefficients and densities for a wider concentration range because H+ has a substantial influence on the thermodynamic properties of the modeled system. By accounting for dissociation, ePC-SAFT+DE determines the true amount of H+, which leads to very accurate osmotic coefficients and densities for NaHSO4. A comparison of the ARDs of ePC-SAFT and ePC-SAFT+DE indicates that the accuracy of the modeled osmotic coefficients and densities, in particular, increase for the systems that show a concentration-dependent change in the species distribution (Figure 2 and Figure 3). Other systems can be modeled with comparable accuracies with both models.

Figure 6. Osmotic coefficients (a) and densities (b) of aqueous solutions of sodium bisulfate at 25°C as a function of the salt molality. Experimental data: (a) osmotic coefficients given in the open literature φlit ACS Paragon Plus Environment

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(circles) and the recalculated osmotic coefficients φcalc (squares). (b) Solution density

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. The lines

represent ePC-SAFT and ePC-SAFT+DE calculations.

Conditions obviously exist where the explicit consideration of all existing species is highly recommended for accurate modeling. These conditions depend (I) on the concentration and (II) on the dissociation constant(s) of the respective electrolytes. To define whether dissociation needs to be accounted for in the modeling, we determined the pKacid value that causes a fraction of the lower concentrated species of at least 3 % as a function of the solute concentration. Only the first dissociation step was considered for this purpose. As a result, Figure 7 shows that the scope of application of ePC-SAFT+DE is located between pK values of -2 and 4. This scope of application applies to both pKacid and pKbase values and therefore applies to both acids and bases. This approach should also be used for better modeling of several organic and inorganic acids, such as oxalic acid, citric acid or iodic acid. For electrolytes with pK values greater than 4 (e.g., NaH2PO4 or Na2HPO4), ePC-SAFT can be used without accounting for dissociation. The same is valid for pK values lower than -2 because the electrolyte is completely dissociated in this region.


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Figure 7. Scope of application for ePC-SAFT+DE and ePC-SAFT as a function of the solute molality. The scope of application of ePC-SAFT+DE is located between pKacid values of -2 and 4. For higher and lower dissociation constants, ePC-SAFT can be used.

Application of ePC-SAFT+DE for acid–base series. For a more detailed study of the sulfate and the phosphate systems, the modeled thermodynamic properties are compared with the experimental data. Because solution densities can be modeled accurately with ePC-SAFT as well as with ePC-SAFT+DE, only osmotic coefficients are considered here. Figures 8a/b show the experimental and modeled osmotic coefficients of solutions that contain sulfuric acid, bisulfate, and sulfate for sodium (Figure 8a) and potassium (Figure 8b) salts. The osmotic coefficient of all 5 solutes can be described satisfactorily with ePC-SAFT+DE over the whole concentration range with an average ARD of 3.91 %. Moreover, ionspecific effects can also be observed in Figure 8a/b. For both sodium and potassium salts, the osmotic calc calc calc coefficients are sequenced in the order φsulfuric acid > φ bisulfate > φsulfate . The osmotic coefficient of

components with hydrogen is greater than the osmotic coefficient of their corresponding salts. In their work, Held et al.14 also observed a similar effect for other aqueous salt solutions; they showed that a smaller cation usually causes higher osmotic coefficients, which also holds true for the bisulfates: the osmotic coefficient of sodium bisulfate is higher than the osmotic coefficient of potassium bisulfate. The same trend is valid for sulfate solutions; however, for these systems, the difference between the osmotic coefficients of sodium and potassium salt solutions is significantly smaller than that for the bisulfate solutions.


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Figure 8. Osmotic coefficient φcalc of aqueous solutions of sulfuric acid, bisulfates, sulfates (a/b), phosphoric acid, dihydrogen phosphates, monohydrogen phosphates, and phosphates (c/ d) at 25°C as a function of the salt molality. (a/c) Sodium salts, (b/d) potassium salts. Experimental data for sulfate systems: Η2SO4 (circles)15, ΗSO4− (squares)17, SO42− (triangles)15. Experimental data for phosphate systems: Η3PO4 (circles)5, Η2PO4− (squares)15, HPO42− (triangles)15, PO43− (stars)15. The lines represent the respective ePC-SAFT+DE calculations.

Figures 8c/d show experimental and modeled osmotic coefficients of solutions that contain phosphoric acid, dihydrogen phosphate, hydrogen phosphate, and phosphate for sodium (Figure 8c) and potassium (Figure 8d) salts. For all seven solutes, ePC-SAFT+DE accurately models φcalc with an average ARD of 4.54 %. Analogous to the sulfuric species, ion-specific influences are observed for the


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different phosphate species. The osmotic coefficient of components with hydrogen is higher than the osmotic coefficient of their corresponding salts. However, a universally valid ion-specific effect (e.g., for the sulfates in Figure 8a/b) cannot be observed for the phosphoric species. Whereas the osmotic coefficient of sodium dihydrogen phosphate is higher than that of its corresponding potassium salt, the opposite behavior is observed for the monohydrogen phosphates and phosphates. The difference between the osmotic coefficients of sodium and potassium solutions increases with increasing numbers of metal cations, comparable to the behavior of the sulfate solutions. This effect is probably based on ion pairing. Held and Sadowski8 showed that ion pairing can influence the ionspecific sequence of osmotic coefficients. Whereas smaller cations cause higher osmotic coefficients in strong electrolyte systems, the opposite is true for systems with ion pairing. This observation could be explained by the higher charge densities of smaller cations leading to higher ion-pairing constants. Both the dissociation-equilibrium method and the ion-pairing mechanism result qualitatively in the same effect: electrolytes with high ion-pairing constants possess low stoichiometric factors, which causes decreased osmotic coefficients.

Conclusion In this study, the dissociation equilibrium of weak acidic and basic electrolytes was considered to determine the species distribution as a function of the initial electrolyte concentration. The species distribution directly yields the stoichiometric coefficient that becomes, in contrast to common practice, a concentration-dependent value. Thermodynamic properties, such as activity coefficients or osmotic coefficients, published in the open literature for components with pK values between -2 and 4 are consequently erroneous because they were calculated assuming erroneous species concentrations. Based on this knowledge, the osmotic coefficients of H3PO4, NaH2PO4, KH2PO4, Na2HPO4, K2HPO4, Na3PO4, K3PO4, H2SO4, NaHSO4, KHSO4, Na2SO4, and K2SO4 were re-calculated because the original values from the literature had been determined assuming complete dissociation. These re-calculated values are physically more reasonable because the osmotic coefficient of diluted phosphoric acid solutions, in contrast to the literature data, converge to unity. Because the concentration-dependent stoichiometric ACS Paragon Plus Environment

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coefficient is needed for the data reduction of osmotic coefficient measurements, accurate estimation schemes for mono-, di-, and triprotic acids and their salts were formulated. To improve the quality of modeled thermodynamic properties of electrolyte solutions, the dissociation equilibrium was also considered in the modeling of osmotic coefficients with ePC-SAFT. The improved approach (ePC-SAFT+DE) determines the equilibrium concentrations of all dissolved species in solution depending on the respective acid constants and the initial electrolyte concentration. This procedure allowed for the fit of the ion parameter to re-calculate the osmotic coefficients and the liquid densities. Without introducing any new or additionally adjustable parameters, the application of the dissociation equilibrium significantly improves the modeling accuracy compared to calculations that do not account for dissociation. The average relative deviation of the 12 phosphoric and sulfuric systems decreased from 12.82 % to 4.28 % (osmotic coefficients) and from 2.59 % to 0.89 % (solution densities).

List of symbols A

Helmholtz energy per number of moles [J/mol]

ai

activity of component i

ci

molarity of component i [mol/L]

D

number of metal cations in a salt

fi

fraction of species i in the solvent-free solution

K

thermodynamic equilibrium constant

Kacid

acid constant

Kbase

base constant

Kγ

ratio of activity coefficients

mi

molality of component i [mol/kg]

~ m i Mi

mass of component i [g] molecular weight of component i [g/mol]

ni

mole number of component i [mol]

N

number of protons that a salt can donate

pH

negative logarithm of cH+

pK

negative logarithm of K


26

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T

temperature [K]

V

volume [L]

uij

dispersion energy [J]

W

mass fraction

X

mole fraction

Greek Letters γ

activity coefficient of component i

εAiBi hb / k B φ

association-energy parameter (K)

ρ

density [kg/m³]

ν

stoichiometric coefficient

κAiBi hb / k B σ

association-volume parameter

osmotic coefficient

temperature-independent segment diameter of molecule i [Å]

Subscripts A

activity

I

component index

J

dissociation index

Ref

reference

Seg

segment

Superscripts Assoc

association

Calc

calculated

Disp

dispersion

Est

estimated

Hc

hard chain

Ion

ionic

Lit

literature

Res

residual

C

based on molarities

M

based on molalities


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Abbreviations AAD

average absolute deviation

ARD

average relative deviation

NP

number of data points

Ref

Reference

Acknowledgements: The authors gratefully acknowledge financial support from the Ministry of Innovation, Science and Research of North Rhine-Westphalia in the frame of CLIB-Graduate Cluster Industrial Biotechnology, contract no: 314 - 108 001 08. The authors would also like to thank Christoph Held for fruitful discussions and for his help during the preparation of this manuscript.

(1) Enick, R. M.; Klara, S. M. SPE Reserv. Eng. 1992, 7, 253-258. (2) Luckas, M.; Krissmann, J. Thermodynamik der Elektrolytlösungen: Eine einheitliche Darstellung der Berechnung komplexer Gleichgewichte; Springer: Berlin, 2001. (3) Chu, H.; Chien, T. W.; Li, S. Y. Sci. Total. Environ. 2001, 275, 127-135. (4) Walter, H.; Johansson, G. Anal. Biochem. 1986, 155, 215-242. (5) Platford, R. F. J. Sol. Chem. 1975, 4, 591-598. (6) Pitzer, K. S.; Silvester, L. F. J. Sol. Chem. 1976, 5, 269-278. (7) Messnaoui, B.; Bounahmidi, T. Fluid Phase Equilibr. 2005, 237, 77-85. (8) Held, C.; Sadowski, G. Fluid Phase Equilibr. 2009, 279, 141-148. (9) Kolthoff, I. M. Treatise on Analytical Chemistry; Interscience Encyclopedia Inc.: New York, 1959. (10) Lide, D. R. CRC Handbook of Chemistry and Physics, 86 ed.; CRC Press Taylor & Francis, 2005. (11) Cameretti, L. F.; Sadowski, G.; Mollerup, J. M. Ind. Eng. Chem. Res. 2005, 44, 33553362. (12) Gross, J.; Sadowski, G. Ind. Eng. Chem. Res. 2001, 40, 1244-1260. (13) Gross, J.; Sadowski, G. Ind. Eng. Chem. Res. 2002, 41, 5510-5515. (14) Held, C.; Cameretti, L. F.; Sadowski, G. Fluid Phase Equilibr. 2008, 270, 87-96. (15) Lobo, V. M. M. Handbook of electrolyte solutions, Part A and B; Elsevier: Amsterdam, 1989. (16) Elmore, K. L.; Mason, C. M.; Christensen, J. H. J. Am. Chem. Soc. 1946, 68, 2528-2532. (17) Stokes, R. H. J. Am. Chem. Soc. 1948, 70, 874-874. (18) Zafarani-Moattar, M. T.; Mehrdad, A. J. Chem. Eng. Data 2000, 45, 386-390.


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Appendix Table 5. Experimental data for osmotic coefficients φlit from the literature and recalculated values φcalc using corresponding values of stoichiometric constants νcalc for phosphoric acid, dihydrogen phosphate, monohydrogen phosphate, phosphate, sulfuric acid, bisulfate and sulfate at 25°C. The osmotic coefficients of the components labeled with * have been evaluated applying Equation (2) using reference data from Lobo15. m 0.1236 0.1964 0.2854 0.3153 0.4186 0.5600 0.6187 0.8557 1.1050 1.1540 1.2870 1.7530 2.8810 2.9800 3.3280 3.7530 4.4900 4.9900 6.2780 6.4030 7.4270 8.7060 8.7290 9.8870 10.658 12.065 13.903 14.864 17.028

φlit

νcalc H3PO45 * 0.3250 1.2103 0.3070 1.1709 0.2934 1.1440 0.2837 1.1376 0.2820 1.1206 0.2796 1.1051 0.2785 1.1003 0.2780 1.0860 0.2795 1.0761 0.2827 1.0745 0.2831 1.0707 0.2936 1.0609 0.3145 1.0478 0.3149 1.0470 0.3220 1.0446 0.3337 1.0420 0.3495 1.0385 0.3632 1.0365 0.3907 1.0326 0.3938 1.0323 0.4163 1.0301 0.4429 1.0278 0.4450 1.0278 0.4698 1.0261 0.4797 1.0252 0.5001 1.0237 0.5195 1.0221 0.534 1.0213 0.5531 1.0200

φν,calc

m

1.0742 1.0488 1.0258 0.9976 1.0066 1.0120 1.0124 1.0240 1.0390 1.0524 1.0576 1.1070 1.2006 1.2030 1.2330 1.2810 1.3462 1.4016 1.5134 1.5259 1.6166 1.7237 1.7319 1.8314 1.8717 1.9542 2.0331 2.0914 2.1691

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 0.1 0.2 0.3 0.4 0.5


φlit

νcalc NaH2PO415 0.911 2.0002 0.884 2.0001 0.864 2.0001 0.847 2.0001 0.832 2.0000 0.819 2.0000 0.808 2.0000 0.798 2.0000 0.789 2.0000 0.780 2.0000 0.765 2.0000 0.751 2.0000 0.739 2.0000 0.729 2.0000 0.721 2.0000 0.705 2.0000 0.696 2.0000 0.691 2.0000 0.691 2.0000 0.694 2.0000 0.699 2.0000 0.706 2.0000 0.713 2.0000 KH2PO415 0.901 2.0002 0.868 2.0001 0.843 2.0001 0.823 2.0001 0.805 2.0000

φν,calc 0.9109 0.8840 0.8640 0.8470 0.8320 0.8190 0.8080 0.7980 0.7890 0.7800 0.7650 0.7510 0.7390 0.7290 0.7210 0.7050 0.6960 0.6910 0.6910 0.6940 0.6990 0.7060 0.7130 0.9009 0.8680 0.8430 0.8230 0.8050 29


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

18.541 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8

0.5635 1.0191 Na2HPO415 0.802 3.0005 0.754 3.0003 0.720 3.0002 0.693 3.0001 0.670 3.0001 0.651 3.0001 0.637 3.0001 0.620 3.0001 0.608 3.0001 0.596 3.0001 0.586 3.0001 Na3PO415 0.678 4.3644 0.618 4.2752 0.579 4.2314 0.550 4.2039 0.527 4.1846 0.508 4.1700 0.492 4.1585 15 H2SO4 * 0.6796 2.1064 0.6652 2.0584 0.6649 2.0402 0.6690 2.0306 0.6751 2.0246 0.6824 2.0205 0.6906 2.0175 0.6995 2.0152 0.7090 2.0133 0.7191 2.0119 0.7408 2.0096 0.7643 2.0078 0.7894 2.0065 0.8159 2.0054 0.8435 2.0045 0.8720 2.0037 0.9013 2.0030 0.9313 2.0024 0.9617 2.0019

2.2117 0.8019 0.7539 0.7200 0.6930 0.6700 0.6510 0.6370 0.6200 0.6080 0.5960 0.5860 0.6214 0.5782 0.5473 0.5233 0.5038 0.4873 0.4732 0.9679 0.9695 0.9777 0.9884 1.0004 1.0132 1.0269 1.0414 1.0565 1.0723 1.1059 1.1420 1.1803 1.2205 1.2624 1.3056 1.3499 1.3953 1.4412

0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.1041 0.1237 0.1482 0.1529 0.2039 0.2823 0.4272 0.6425 0.7774 1.0840 1.1740 1.3400


0.789 0.773 0.760 0.747 0.736 0.716 0.698 0.683 0.669

2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 K2HPO415 0.805 3.0005 0.764 3.0003 0.739 3.0002 0.722 3.0001 0.708 3.0001 0.698 3.0001 0.690 3.0001 0.684 3.0001 0.679 3.0001 0.674 3.0001 K3PO415 0.709 4.3644 0.678 4.2752 0.665 4.2314 0.658 4.2039 0.655 4.1846 0.654 4.1700 0.653 4.1585 NaHSO417 * 0.7093 2.2964 0.7008 2.2759 0.6937 2.2555 0.6919 2.2522 0.6806 2.2226 0.6698 2.1928 0.6570 2.1599 0.6492 2.1325 0.6467 2.1212 0.6467 2.1037 0.6482 2.0998 0.6509 2.0938

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0.7890 0.7730 0.7600 0.7470 0.7360 0.7160 0.6980 0.6830 0.6690 0.8049 0.7639 0.7390 0.7220 0.7080 0.6980 0.6900 0.6840 0.6790 0.6740 0.6498 0.6344 0.6286 0.6261 0.6261 0.6273 0.6281 0.9266 0.9238 0.9227 0.9217 0.9186 0.9164 0.9126 0.9133 0.9147 0.9222 0.9261 0.9326 30

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3.0 3.2 3.4 3.6 3.8 4.0 0.1068 0.1277 0.1361 0.1445 0.2423 0.4216 0.5287 0.7233 0.7538 1.0020 1.1560 1.5020 1.6644 1.8270 2.2590 2.6530 0.1 0.2 0.3 0.4 0.5 0.6


0.9925 2.0014 1.0236 2.0009 1.0548 2.0005 1.0860 2.0001 1.1173 1.9997 1.1484 1.9993 17 KHSO4 * 0.6965 2.2933 0.6908 2.2722 0.6841 2.2650 0.6824 2.2583 0.6554 2.2064 0.6274 2.1609 0.6161 2.1450 0.6006 2.1254 0.5988 2.1230 0.5881 2.1076 0.5809 2.1006 0.5730 2.0888 0.5686 2.0846 0.5668 2.0809 0.5619 2.0730 0.5598 2.0676 K2SO415 0.779 3.0000 0.742 3.0000 0.721 3.0000 0.703 3.0000 0.691 3.0000 0.679 3.0000

1.4877 1.5347 1.5818 1.6290 1.6762 1.7232 0.9111 0.9121 0.9061 0.9065 0.8912 0.8710 0.8617 0.8477 0.8461 0.8371 0.8297 0.8230 0.8183 0.8171 0.8132 0.8123 0.779 0.742 0.721 0.703 0.691 0.679

1.3820 1.6320 2.3980 3.0650 4.0670 4.1910 5.1120 5.2400 5.6100 6.3200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8

0.6527 2.0924 0.6589 2.0854 0.6898 2.0710 0.7198 2.0630 0.7642 2.0550 0.7703 2.0542 0.8023 2.0492 0.8045 2.0486 0.8154 2.0470 0.8316 2.0443 Na2SO415 0.793 3.0000 0.753 3.0000 0.725 3.0000 0.705 3.0000 0.690 3.0000 0.678 3.0000 0.667 3.0000 0.658 3.0000 0.650 3.0000 0.642 3.0000 0.631 3.0000 0.625 3.0000 0.621 3.0000 0.620 3.0000

0.9358 0.9479 0.9992 1.0467 1.1156 1.1250 1.1746 1.1781 1.1950 1.2204 0.793 0.753 0.725 0.705 0.690 0.678 0.667 0.658 0.650 0.642 0.631 0.625 0.621 0.620

Estimation of Concentration-Dependent Stoichiometric Coefficients for Weak Electrolytes. Because the speciation from dissociation equilibria cannot be calculated analytically for components with more than one acidic or basic group, simplified approaches have been developed to estimate the stoichiometric coefficients (hereafter called νest) for these components. In the following paragraphs, these approaches will be presented for mono-, di-, and triprotic acids and their corresponding bases. ACS Paragon Plus Environment

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Monoprotic Acids. For components that can donate or accept only one proton, νest can be derived without further simplifications using Equations (5), (8), (9) and (13). 2  K K  ν est = d + 1 +  − acid / base +  acid / base  + K acid / base ⋅ m initial 2 2    

  ⋅ m −1  initial 

(21)

In this equation, Kacid/base is the dissociation constant for both acids and bases, and d describes the number of metal cations in the solute. The only exceptions where Equation (21) is not applicable are bases that contain hydroxide ions. When these components dissociate in aqueous solution, the hydroxide ion is released. For other bases, the hydroxide is formed by the base accepting a proton from water. The value of the stoichiometric coefficient of bases that contain hydroxide anions is, by unity, smaller than for other bases that do not contain hydroxide anions.

Diprotic Acids. The stoichiometric coefficients of components that can donate or accept two protons cannot be estimated analytically without simplifications. Two independent assumptions were made in the dissociation model, which leads to two equations that can be used to determine νest. For a diprotic acid, Equation (7) leads with n = 2 to:

K acid ,1 =

K acid ,2 =

m H + ⋅ m HX −

(22)

m H2X m H+ ⋅ m X2−

(23)

m HX −

Assuming a complete first dissociation step, only the second dissociation must be considered. Consequently, the concentration of the non-dissociated acid can be neglected, which leads to the following proton and anion balance (derived from Equations (8) and (9)): m HX − = m initial − m X 2 −

(24)

m H + = 2 ⋅ m initial − m HX −

(25)


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The second dissociation step can be described by Equation (26), which can be rearranged to obtain the molality of the fully dissociated anion in Equation (27).

K acid, 2 =

(m initial + m X 2 − ) ⋅ m X 2 −

(26)

m initial − m X 2 −

m X2− = −

m initial + K acid ,2 2

+ K acid , 2 m +  initial 2 

2

  + K acid ,2 ⋅ m initial 

(27)

The molalities of HX- and H+ can be calculated by Equations (24) and (25). Insertion of the resulting molalities into Equation (13) leads to Equation (28), which can be used as long as K1 > 100 mInitial is fulfilled.

ν

est

 (K + m initial )  = d + 2 +  − acid ,2 + 2  

2

 (K acid ,2 + m initial )    + K acid ,2 ⋅ m initial 2  

  −1  ⋅ m initial  

(28)

An alternative equation (Equation (34)) can be derived under the assumption that the hydronium or hydroxide concentration, respectively, is a function of the first dissociation step only. Thus, the second dissociation step can be calculated independently of the concentration of the fully dissociated species. Using this simplification, Equation (22) can be considered to be independent from Equation (23), which leads to the following proton and anion balance: m H 2 X = m initial − m HX −

(29)

m H + = m HX −

(30)

Using these two equations, the first dissociation step can be described by Equation (31). The resulting equilibrium concentration is given in Equation (32).

K acid ,1 =

(m HX − ) 2

(31)

m initial − m HX −

m HX − = −

K acid ,1 2

2

K  +  acid ,1  + K acid ,1 ⋅ m initial  2 


(32)

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The equilibrium concentration of the fully dissociated anion is equal to Kacid,2 because the molalities of H+ and HX- cancel (Equation (33)): K acid , 2 =

m H+ ⋅ m X2− m HX −

= m X2−

(33)

Insertion of the resulting molalities into Equation (13) leads to Equation (34), which can be used as long as Kacid,2 < minitial / 10 is fulfilled: 2  K  K    −1 ν est = d + 1 +  − acid ,1 +  acid ,1  + K acid ,1 ⋅ m initial + K acid ,2  ⋅ m initial 2    2   

(34)

To determine the applicability and accuracy of Equations (28) and (34), respectively, νest was compared with νcalc determined from Equation (13) for varying molalities and dissociation constants. Data points (169,371) within the following constraints were investigated: pKacid,1 ≥ - 4 pKacid,2 ≥ pKacid,1 + 2 0.0001 mol/kg ≤ minitial ≤ 10 mol/kg Of these data points, 99.58 % (168,659) could be described by either Equation (34) or Equation (40). For the remaining 0.42 %, neither condition 1 nor condition 2 was fulfilled. The ARD between νcalc and νest (analogous to Equation (19)) for the data points is 0.010 % with a maximum relative deviation of 0.8 %. For data points where the conditions for both equations were fulfilled, the application of Equation (34) was more accurate.

Triprotic Acids. For components with a third acidic or basic group, the complexity of the system of equations increases strongly. For the estimation of the stoichiometric coefficient, the third dissociation was assumed to be negligible. Consequently, the νest values were determined according to the approach for components with two acidic or basic groups using Equations (28) and (34). For the third dissociation step, two additional constraints were set:


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pKacid,3 > pKacid,2 + 3 pKacid,3 > 6 The introduction of pKacid,3 as a fourth variable would significantly increase the number of possible data points within the constraints. We thus calculated the following case: pKacid,3 = pKacid,2 + 3 pKacid,3 > 6 For this case, the fraction of data points which can be described satisfactorily is the same as for calculations with two dissociation constants. The ARD was 0.012 % and was therefore negligibly higher than for two acidic or basic groups. The maximum relative deviation was 0.8 %.


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Osmotic Coefficients of Aqueous Weak Electrolyte Solutions: Influence

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