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B: Liquids, Chemical and Dynamical Processes in Solution, Spectroscopy in Solution
Dissociation Constants of Perchloric and Sulfuric Acids in Aqueous Solution Alexander V. Levanov, Oksana Ya. Isaikina, Ulkar D. Gurbanova, and Valery V. Lunin J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.8b01947 • Publication Date (Web): 17 May 2018 Downloaded from http://pubs.acs.org on May 18, 2018
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Dissociation Constants of Perchloric and Sulfuric Acids in Aqueous Solution Alexander V. Levanov,* Oksana Ya. Isaikina, Ulkar D. Gurbanova, Valery V. Lunin Department of Chemistry, M.V. Lomonosov Moscow State University, Leninskiye Gory 1, building 3, 119991 Moscow (Russia) * Corresponding Author. E-mail address:
[email protected]; Fax: (+7) 495939-4575; Phone: (+7) 495-939-3685 ABSTRACT The experimental dissociation constants of strong acids are notoriously ill-defined, and it is necessary to rely on theoretical methods for their evaluation. We present a methodology for the theoretical evaluation of the dissociation constants, and the values of Ka for perchloric and sulfuric acids have been estimated. It has been shown that the acid dissociation constant Ka can be expressed as a product of two terms, Ka = K'a × fHA∞, where K'a is the apparent dissociation constant, and fHA∞ is the infinite dilution activity coefficient of undissociated molecule of acid in liquid solution. The values of K'a can be computed from readily available reference data. The limiting activity coefficients fHA∞ for strong acids can be determined by theoretical methods only. The following estimate for the limiting activity coefficients of perchloric and sulfuric acids has been obtained, –2.5 < log10 f HA∞ < –1.3. The ranges of values of the dissociation constants of HClO4 and H2SO4 at 25 °C have been determined, log10Ka(HClO4) = 10.8 - 12.3; log10Ka1(H2SO4) = 4.5 - 8.6. INTRODUCTION The most important and determining characteristics of an acid is the dissociation constant Ka – the equilibrium constant of reaction HA(aq) ⇄ H+(aq) + A–(aq) (1) in aqueous solution. Acid dissociation can take place in other solvents, but water is of exclusive significance. Acid-base equilibria play a remarkable role in various processes in animate and inanimate nature and in human economic activity, and dissociation constants are indispensable for quantitative investigation of these equilibria. As is demonstrated later in this section, analysis of the literature reveals a very large uncertainty of the published values of dissociation constants of perchloric and sulfuric acid, and one can conclude that these quantities remain unknown. The purpose of this work is to determine the reliable values of the dissociation constants of HClO4 and H2SO4 in aqueous solution at 298.15 K. It 1 ACS Paragon Plus Environment
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should be noted that the treatment for sulfuric acid relates to the dissociation at the first stage, H2SO4(aq) ⇄ H+(aq) + HSO4–(aq), that is, the value of Ka1(H2SO4) is to be estimated. Ionization of perchloric and sulfuric acids in aqueous solution has been studied extensively by the methods of Raman spectroscopy and proton magnetic resonance (PMR). In principle, these methods allow one to find the concentration of undissociated acid mHA and the degree of dissociation α (mi represents the concentration of the i-th species in solution in molality units, mol kg–1). However, no exact values of the dissociation constants of HClO4 and H2SO4 have been determined on the basis of the spectroscopic methods. The determination of the thermodynamic dissociation constant Ka from the experimental values of mHA or α is performed in the following way (see e.g. refs 14 ). The dissociation constant (the equilibrium constant of reaction (1)) is defined by the expression Ka =
m H+ γ H+ m A− γ A− , mol kg–1, m HA γ HA
(2)
where mH+, mA–, mHA are the molalities of ions H+(aq), A–(aq) and undissociated molecules HA(aq), mol kg–1, γH+, γA–, γHA are the activity coefficients in the nonsymmetric standard state convention. Expression (2) can be represented in the form
(M HA γ ± )2 M HA (γ ± )2 Ka = = , m HA γ HA (1 − α ) γ HA
(3)
(see e.g. ref 4), where MHA is the stoichiometric acid concentration, mol kg–1, and γ± is the mean ionic activity coefficient of electrolyte HA on assumption of its complete dissociation. By transforming expression (3), the relation is obtained
M HA (γ ± )2 KaγHA = . (4) 1−α The right hand side of (4) can be computed from the experimental values of α and the reference values of γ±, and it determines the dependency of the product (KaγHA) on the stoichiometric concentration of an acid MHA. Thus, in order to find the thermodynamic dissociation constant Ka, it is necessary to construct the concentration dependence of the product (KaγHA), or its logarithm, log10(KaγHA), on the basis of experimental and reference data, and extrapolate it to an infinitely dilute solution: since lim γ HA = 1, then lim(K a γ HA ) = Ka. M HA → 0
M HA → 0
For strong acids, the determination of the dissociation constant Ka by the described procedure is associated with a principal difficulty. Only in concentrated solutions the degree of dissociation differs from unity. Therefore, the result of extrapolation to the region of an infinitely dilute solution can have such a large 2 ACS Paragon Plus Environment
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uncertainty that the dissociation constant can not be found. In the well-known monograph by Bell (1973)5 it is noted that the value of the dissociation constant Ka = 0.2 is the maximum value, which can be determined with acceptable accuracy, and any quantities greater than unity, Ka > 1, should be considered as doubtful. It can be expected that the dissociation constants of perchloric and sulfuric acids are greater than 104, and therefore the task of their correct estimation is extremely arduous. The use of PMR spectroscopy is connected with additional difficulties. Due to the rapid exchange of protons between molecules of undissociated acid HA(aq), water, and hydrated hydrogen ions H+(aq), only one proton signal is observed in the PMR spectra of aqueous solutions of strong acids. Its chemical shift depends on the degree of dissociation, since it is a weighted average of the shifts of the HA(aq) and H+(aq) species.5 However, the chemical shift of H+(aq) is determined by hydration, that is, depends on the solution concentration.6 As a result, the calculated values of the dissociation degree depend on the choice of H+(aq) hydration model. In addition, the dependence of the chemical shift of undissociated acid HA(aq) on the solution composition is possible.5 Dissociation of perchloric acid in aqueous solution was studied by PMR and Raman spectroscopy. The value of pKa(HClO4) = –1.58 (Ka(HClO4) = 38) was determined by PMR spectroscopy method by Hood et al. (1954, 1960).7, 8 However, this relatively small value does not correspond to the general ideas about the strength of perchloric acid, and the preliminary results of the investigation by Raman spectroscopy. Based on the PMR spectra of perchloric acid dissolved in liquid sulfur dioxide or its mixtures with water (in liquid SO2 there is no exchange of protons with a solvent), the value of pKa(HClO4) = –10 was found by Brownstein and Stillman (1959).9 The above-mentioned works 7-9 apparently served as sources of the values of perchloric acid dissociation constant given in the reference books10, 11 (pKa,HClO4 = –1.6) and the textbooks12, 13 (pKa,HClO4 ~ –10). In 1968 Duerst 6 obtained the estimate pKa(HClO4) = –2.7 ± 1.7 (Ka(HClO4) = 55 5500) with the use of a more comprehensive model of H+(aq) hydration, but he noted its dependence on the model adopted. In the final publication by O. Redlich et al. (1968) 14 of a large series of papers on strong acids ionization in aqueous solutions, almost complete dissociation of perchloric acid is mentioned even in very concentrated solutions (8-12 M), and the difficulty of processing of the experimental data is indicated; any numerical value of HClO4 dissociation constant has not been presented at all. The papers15-20 are devoted to the study of aqueous solutions of perchloric acid by the method of Raman spectroscopy. Covington et al. (1965)17 established complete dissociation of the acid at stoichiometric concentrations up to 10 M, and 3 ACS Paragon Plus Environment
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presented an estimate of the dissociation constant Ka(HClO4) > 103. The closeness of the dissociation degree of HClO4 to unity even in concentrated solutions (at stoichiometric concentration of 8-12 M) is noted in refs18, 19, and it is emphasized that, because of this, it is practically impossible to obtain a defined value of the dissociation constant. Detailed analysis and generalization of all known data on the dissociation degree of HClO4 in aqueous solution were performed by Karelin and Tarasenko (2003).21 In particular, they showed that the mole fraction of undissociated HClO4, xHClO4, can be found from its vapor pressure over the solution, pHClO4, by means of Raoult's law, xHClO4 = pHClO4/p*HClO4, where p*HClO4 is the vapor pressure of pure HClO4. Also they pointed out the complete dissociation of the acid, that is, the absence of undissociated HClO4 molecules, in solutions with stoichiometric mole fraction of the acid less than 0.33 (this corresponds to concentrations of less than 12 М). The validity of applying Raoult's law is justified by the fact that the dissociation degree values found with it coincide with the most accurate results determined through the use of PMR6 and IR22 spectroscopy. The mean ionic activity coefficients of aqueous perchloric acid are given by Redlich et al. (1968).23 The concentration ranges, for which dissociation degrees and activity coefficients are known, partially overlap when the stoichiometric mole fraction of HClO4 takes values between 0.33 and 0.42. 15
13
log10(Ka γHClO4)
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11
9
7
5 0
0,2
0,4
0,6
0,8
1
XHClO4
Fig. 1. Dependence of the decimal logarithm of the product (KaγHClO4) on the stoichiometric mole fraction of perchloric acid in solution (XHClO4). ● – the values of (KaγHClO4), calculated from the data of refs 21 and 23 for 25 °C; ■ – reliable values of (KaγHClO4) from ref 6 for 65 °C; dashed lines - extrapolation to XHClO4 → 0. 4 ACS Paragon Plus Environment
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That makes possible the determination of the apparent dissociation constant of perchloric acid (KaγHClO4) on the basis of equation (4). Fig. 1 shows the concentration dependence of the logarithm log10(KaγHClO4) on molal scale, calculated from the data of refs21 and 23 for 25 °C, and also presented by Duerst 6 for 65 °C (Duerst6 notes only a small difference between the results at 25 and 65 °C). The data in Fig. 1 demonstrate a very large value of the thermodynamic dissociation constant, log10Ka(HClO4) ~ 10. At the same time, Fig. 1 clearly shows that the extrapolation of (KaγHClO4) to the region of infinitely dilute solution is characterized by extremely high uncertainty. The dashed lines in Fig. 1 show the results of extrapolation of log10(KaγHClO4) to infinite dilute solution, under different assumptions about the concentration dependence of log10γHClO4. If we assume that log10γHClO4 = a·(MHClO4)n, we obtain the value log10Ka(HClO4) = 11.0; a form of the dependence log10γHClO4 = a·MHClO4 + b·(MHClO4)2 leads to the quantity log10Ka(HClO4) = 7.46; other assumptions can give essentially divergent values of the constant. Thus, the results of works6-9, 14-21 indicate a very large value of the perchloric acid dissociation constant in aqueous solution, log10Ka(HClO4) ~ +10. However, it is not possible to estimate its value, at least to within three orders of magnitude, on the basis of these data. Aqueous solutions of sulfuric acid were most widely studied by Raman spectroscopy; see, for example, Lund Myhre et al. (2003)24 and references therein. The value of the dissociation constant for the first stage, pKa1(H2SO4) = –3, was obtained in the review by Young and Blatz (1949).25 It is this value that appears in the reference books26, 27 and in the textbook.28 The literature data on Raman spectra of sulfuric acid was generalized and the content of SO42–, HSO4–, H2SO4·H2O, H2SO4, H5O2+ and H3O+ in aqueous solutions of sulfuric acid of various concentrations at 25 °C was given by Librovich et al. (1977).29 On the basis of these results, the value of the dissociation constant Ka1(H2SO4) = 310, pKa1(H2SO4) = –2.5 was determined in ref 30. The treatment of literature data on the composition and acidity functions of sulfuric acid aqueous solutions by the method of ‘activity coefficient function’31 gives the value pKa1(H2SO4) = –8.50.32 Various indirect methods have been extensively employed to evaluate the dissociation constants of perchloric and sulfuric acids. Thus, based on the ionization characteristics in non-aqueous solvents (sulfuric and formic acid), the values of pKa(HClO4) = –9.9 and pKa1(H2SO4) = –4.0 for aqueous solution were obtained in monograph.33 Processing by a special method34 of literature data on the dissociation degrees leads to the results pKa(HClO4) < –4, pKa(H2SO4) = –8.3. On the basis of linear relationships between the free energy of hydrolysis of esters and 5 ACS Paragon Plus Environment
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pKa of strong acids, the values of pKa(HClO4) = –5.0, pKa(H2SO4) = –2.8 were established by Guthrie (1978).35 Owing to the insufficient information on the dissociation constants of strong acids (H2SO4, HClO4, etc.), various simplified relationships between the value of dissociation constant and the structure of acid molecules have been widely used for semiquantitative evaluation of the constants. It is clear that such methods can only give a very approximate estimate of the sought quantities. For oxoacids, the Pauling rules36 are well known, linking the strength of an acid and the number of oxygen atoms and hydroxyl groups in its molecule, as well as other similar correlations.37, 38 The application of these rules leads to the estimates of pKa1(H2SO4) ~ –2 - –3, pKa(HClO4) ~ –7 - –8.39 Quantum-chemical calculations of the dissociation constants of strong acids usually includes the representation of the dissociation reaction (1) using thermodynamic cycles. The thorough and comprehensive surveys of this and related subjects are presented in review works40-42 and monograph.43 To be brief, a typical cycle involves the processes of transfer of undissociated acid molecule from solution into gas phase, dissociation in the gas phase, and hydration of proton and acid anion. As a rule, the solvent is considered as a continuous medium, and solvation energies are calculated using various models of interaction of a polar or charged particles with dielectric continuum. Using a polarization continuum model, Trummal et al.44 obtained the estimate pKa(HClO4) = –15.2 ± 2.0. The acidity of HClO4 in the gas phase and in aqueous solution was studied by Zhang et al.45 employing various computational quantum chemical methods and a continuum solvation model. The values of pKa(HClO4) in the range from –7 to –14 were obtained. Zhang et al. (2009)45 recommend the estimate pKa(HClO4) = –10, based on its proximity to the experiment-based value by Brownstein and Stillman (1959).9 Gutowski and Dixon46 used G3 (MP2) quantum-chemical calculations and COSMO solvation model to evaluate the values of the dissociation constants of strong acids in an aqueous solution. The estimates of pKa1(H2SO4) –7 and –9 were obtained. In an earlier paper47, the calculation with CCSD(T) method and fully polarizable continuum model gave the values of pKa1(H2SO4) equal to −3.4, −7.8, −8.5, depending on a parameter of the solvation model; the final estimate of pKa1(H2SO4) was –6 - –8. In the work,48 the expediency of using continuum solvation models in quantum chemical calculations of acid dissociation constants has been exhaustively discussed, and the value of pKa1(H2SO4) = –4.54 - –5.05 was computed. Literature data on dissociation constants of HClO4 and H2SO4 are summarized in Table 1. As can be seen from the Table, the theoretical and 6 ACS Paragon Plus Environment
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experimental estimates of the dissociation constants are characterized by a very large spread, and it is unclear which of them are trustworthy. Table 1. Literature data on dissociation constants of perchloric and sulfuric acids in aqueous solution at 298.15 K. log10Ka(HClO4) Method and reference log10Ka(H2SO4) Method and reference 1.6 (E) 7, 8 2-3 (P) 39 1.6 (R) 10, 11 2.5 (E) 29, 30 6 2.7 ± 1.7 (E) 2.8 (I) 35 >3 (E) 17 3 (E) 25 >4 (E) 34 3 (R) 26-28 5 (I) 35 4 (I) 33 7-8 (P) 39 4.5 - 5 (Q) 48 8 (R) 28 3.4; 7.8; 8.5 (Q) 47 ~7.5 - ~11 (E) This work; ref 21 6-8 (Q) 47 9.9 (I) 33 8.3 (E) 34 10 (E) 9 8.5 (E) 32 10 (R) 12, 13 7; 9 (Q) 46 7 - 14; 10 (Q) 45 15.2 ± 2.0 (Q) 44 (E) Treatment of original or literature experimental data on the composition of the solutions or dissociation degree (Q) Quantum chemical calculations using continuum solvation models (I) Various indirect methods, as discussed in the Introduction (P) According to the Pauling rule (R) The values given in reference book and textbooks In principle, quantum chemical calculations could provide good evaluation of dissociation constants of strong acids. However, at present their results can very much differ from each other, depending on the method. Therefore, they require verification by comparison with reliable reference values, but there are no such values for strong acids. Currently, the results of the calculations44, 45 and 46-48 are quite scattered and cannot give reliable estimates of Ka(HClO4) and Ka(H2SO4). Thus, the available values of the dissociation constants of HClO4 and H2SO4 are very uncertain, and they need to be substantially refined. METHOD OF EVALUATION OF DISSOCIATION CONSTANTS At first, we obtain a thermodynamic relation that links the dissociation constant Ka with the limiting activity coefficient fHA∞. Let us consider aqueous solution of acid HA and the gas phase above this solution. We assume that the 7 ACS Paragon Plus Environment
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solution contains Н+(aq) and А–(aq) ions, and HA (aq) undissociated molecules, as well as water, and the gas phase contains molecules HA(g) and water vapor. The reaction (1) of dissociation of HA acid in aqueous solution can be represented as the difference of the processes of HA(g) dissolution with the formation of H+(aq) and A–(aq) ions, HA(g) ⇄ H+(aq) + A–(aq), (5) and with the formation of undissociated HA(aq) molecules, HA(g) ⇄ HA(aq). (6) Equilibria of dissolution processes (5) and (6) are characterized by the constants m H+ γ H+ m A− γ A− , mol2 kg–2 bar–1, p HA
KH = H=
m HA γ HA , mol kg–1 bar–1, p HA
(7) (8)
where pHA, bar, is the pressure (fugacity) of HA vapor above the solution. In the literature, both the constants KH and H are called Henry’s law constants. Let us find an expression for Henry’s constant H. The mole fraction of HA(aq) undissociated species, xHA, can be expressed using a relation analogous to Raoult's law, xHA =
p HA , f HA p*HA
(9)
where p*HA, bar, is the vapor pressure of HA pure substance, fHA is the activity coefficient of HA(aq) species in the solution in the symmetric standard state convention. The mole fraction of HA (aq) is related to molality by the ratio mHA xHA = . (10) mHA + mH+ + mA− + 55.51 mole/kg Since stoichiometric molality of HA acid MHA = mHA + mA–, and also mH+ = mA– (the autoionization of water is neglected), the relation (10) is transformed into xHA =
mHA , 2MHA − mHA + 55.51
whence we have mHA = (2MHA+55.51)
xHA . 1 + xHA
(11)
Substituting (9) into (11), we obtain the expression mHA = (2MHA+55.51)
pHA , pHA + f HA p*HA
(12)
which makes it possible to determine the molality of undissociated HA(aq) species in the solution from the pressure of HA, pHA, over the solution. Substituting expression (12) in (8), we obtain the formula for Henry's constant 8 ACS Paragon Plus Environment
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H=
(2MHA + 55.51)γ HA . pHA + f HAp*HA
(13)
In infinitely dilute solution, the following limiting relationships are satisfied: MHA → 0, pHA → 0, γHA → 1, fHA → fHA∞, and the Henry’s constant formula takes the form H=
55.51 , f HA ∞ p*HA
(14)
where fHA∞ is the infinite dilution activity coefficient of undissociated HA molecule in the solution. It follows from the relations (2), (7), (8) that the dissociation constant Ka is equal to quotient of the KH and H, Ka = KH / H. (15) Substituting formula (14) in relation (15), we obtain an expression for determining the HA acid dissociation constant: Ka = KH × fHA∞ × p*HA / 55.51. (16) The Henry’s constant KH can be determined from the reference values of the thermodynamic properties of HA(g) and A–(aq), and the vapor pressure over pure HA liquid, p*HA, can also be taken from the reference literature. On the other hand, it can be expected that infinitely dilute solutions of HA (aq) significantly deviate from ideal solutions, and therefore the limiting activity coefficients fHA∞ differ considerably from unity. The evaluation of fHA∞ requires theoretical estimation of the Gibbs free energy of transition of undissociated HA molecules from the pure substance to aqueous solution. In this connection, it is appropriate to introduce the ‘apparent dissociation constant’ of acid HA: K'a ≡ KH × p*HA / 55.51, (17) and to write the expression (16) as a product of two factors: Ka = K'a × fHA∞. (18) The first factor, the apparent constant K'a, can be easily determined from available reference data. To estimate the second factor, fHA∞, special theoretical methods are required. In comparison with conventional quantum-chemical methods of calculating Ka (see e.g. refs40-43, 48), the use of formula (18) offers substantial advantages. First, theoretical estimations are required for evaluation of the coefficient fHA∞ only, and there is no need for more complicated computational study of the process of acid dissociation itself. It is only necessary to consider the transfer of undissociated HA molecules between pure HA substance and aqueous solution. Second, it is rather easy to verify the correctness of fHA∞ estimation, by evaluating the limiting coefficients on semiquantitative level. Indeed, since fHA∞ is an activity coefficient 9 ACS Paragon Plus Environment
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(in the symmetric convention for the standard state), its value is directly connected with the affinity of HA to water (fHA∞ = 1 in ideal solution, fHA∞ > 1 for hydrophobic compounds, and fHA∞ < 1 for hydrophilic substances, such as HClO4 and H2SO4). More definite estimates of fHA∞ can be obtained by comparison with analogous compounds, as explained below. Third, the apparent constant K'a can be determined solely with help of available reference thermodynamic data, which ensures the reliability of its values. In some works (see e.g. refs49, 50), the dissociation constants of strong acids are determined on the basis of formula (15), with the values of the Henry’s constant KH calculated from the available reference data, and theoretical estimates are used only to determine the Henry’s constant H. The calculation costs for this approach are similar to those required when formula (18) is used. However, this approach has a significant drawback: in contrast to the limiting coefficients, the values of the Henry’s constant H are much more difficult to verify by estimating on semiquantitative level, using analogies. To the best of our knowledge, the estimation of strong acids dissociation constants with the help of limiting activity coefficients has not been performed until the present time. An expression for the apparent dissociation constant K'a, analogous to (17), was first given by Y. Marcus,51 based on the works by R.A. Robinson.52, 53 Marcus also drew attention to the fact that the apparent constant differs from the true dissociation constant by a factor, equal to the rational activity coefficient of undissociated acid HA, fHA. Marcus studied solutions of final concentration, and considered the coefficient fHA to be an undefined quantity. In contrast, in the present work the infinite dilute solutions are considered, and the limiting coefficients fHA∞ for strong acids are evaluated for the first time. The advantage of using infinite dilute solutions is that it is possible to estimate the limiting coefficients in them (and these are numbers, not the functions concentration). In addition, reference thermodynamic data, rather than real pressures over solutions, can be used to estimate the apparent constant K'a, which greatly increases the reliability of the estimates. Calculation of the apparent constant K'a. In this paper, the following methods for estimating the value of K'a were used. Method No.1. Application of formula (17). K'a calculation according to the formula (17), requires the values of the Henry’s constant KH and the saturated vapor pressure (or fugacity), p*HA. The Henry’s constant KH can be found from the standard thermodynamic properties of HА(g) gaseous substances and А–(aq) ions in aqueous solution. In the present study, all the values are determined for a temperature of 298.15 K. The Henry’s constant is calculated on the basis of the 10 ACS Paragon Plus Environment
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known relation connecting the equilibrium constant and the Gibbs energy of a chemical reaction, KH = exp(–
1 1 ∆6Gº) = exp(– (∆fGø298(А–,aq) – ∆fGº298(HA,gas))) (19) RT RT
where ∆6Gº = ∆fGø(А–,aq) – ∆fGº(HA,gas) is standard change of Gibbs free energy of reaction (6), ∆fGø298(А–,aq) and ∆fGº298(HA,gas) are standard Gibbs energies of formation of A– ions in aqueous solution and HA molecules in the gas phase at 298.15 K, R = 8.31446 J mol–1K–1 is the universal gas constant, and T is the absolute temperature, K. In this paper the following notation for standard states is used: º – any standard state, or, a gas in an ideal gas state at a pressure of 1 bar, * – pure component at solution temperature and pressure, ø – component in an infinitely dilute solution with concentration 1 mol kg–1. ‘А–,aq’ denotes ion А– in aqueous solution, and ‘X,aq’ means undissociated molecule X in aqueous solution. The saturated vapor pressure of HClO4 was taken from refs.21, 54 The values of vapor pressure of H2SO4 over 100% sulfuric acid can be found in refs,55-57 and they are characterized by some spread. The data of ref 55 were used as a lower estimate of p*H2SO4, and from ref 57 served as a higher estimate. These pressure values were substituted in formula (17), together with the corresponding minimum and maximum values of the Henry’s constant KH,H2SO4, and the range of the apparent constant K'a1,H2SO4 values was obtained. Method No.2. K'a calculation based on the thermodynamic properties of pure HA compounds in liquid state and A– anions in aqueous solution. To calculate the apparent dissociation constant, we can also use the values of standard thermodynamic properties of ClO4–(aq) and HSO4–(aq) ions and the corresponding liquids HClO4(liq) and H2SO4(liq) as input data. This approach does not require information about vapor pressure and thermodynamic properties of the substances in the gas phase. The calculation are performed as follows. Vapor pressure is estimated from the phase equilibrium condition G(HA,liq) = G(HA,gas), (20) where G(HA,liq) and G(HA,gas) are molar Gibbs free energies of pure HA substance in liquid and gas phases, respectively. Using standard Gibbs energies, the equality (20) can be rewritten as follows: p*HA
G*(HA,liq) +
∫1
Vm (HA,liq) dp = Gº(HA,gas) + RT ln p*HA,
(21)
where Vm(HA,liq) is molar volume of liquid HA substance, G*(HA,liq) and Gº(HA,gas) are standard Gibbs energies of pure substance HA in a liquid or solid state at an external pressure of 1 bar, and in a state of ideal gas at a pressure of 1 bar. The standard Gibbs energies G*(HA,liq) and Gº(HA,gas) can be replaced by 11 ACS Paragon Plus Environment
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the standard Gibbs energies of formation, ∆fG*(HA,liq) and ∆fGº(HA,gas), (subtracting from both sides of the equality (21) the standard Gibbs energies of simple substances forming HA): ∆fG*(HA,liq) +
p*HA
∫1
Vm (HA,liq) dp = ∆fGº(HA,gas) + RT ln p*HA. (22)
Transforming (22), we obtain the expression for the vapor pressure p*HA = exp(–
1 (∆fGº(HA,gas) – ∆fG*(HA,liq) – RT
p*HA
∫1
Vm (HA,liq) dp)). (23)
Substituting the expressions for KH (19) and p*HA (23) into the formula (17), we have 1 1 K'a = exp(– (∆fG ø(А–,aq) – ∆fG*(HA,liq) – 55.51 RT
p*HA
∫1
Vm (HA,liq) dp)). (24)
For the substances considered in this paper, the absolute value of the difference ∆fGø(А–,aq) – ∆fG*(HA,liq) exceeds 30 kJ/mole, and the value of the integral p*HA
∫1
Vm (HA,liq) dp is less than 0.01 kJ/mole. Therefore, the integral in formula (24)
can be neglected, and the formula for calculating K'a becomes K'a =
1 1 exp(– (∆fGø(А–,aq) – ∆fG*(HA,liq))). 55.51 RT
(25)
Determination of the limiting coefficients fHA∞ based on reference data. As was mentioned above, the limiting activity coefficients fHA∞ of undissociated molecules of strong acids HClO4 and H2SO4 can only be found by theoretical estimation. At the same time, the limiting activity coefficients of a number of analogous compounds can be calculated on the basis of available reference data. These quantities are necessary to verify the correctness of the theoretical calculations, as well as to determine the possible ranges of values of the strong acids limiting coefficients. In our work, the following substances were employed as analogous compounds and comparison compounds: CO2, SO2, HF, HCOOH, CH3COOH, C2H5COOH, H3PO4, CF3COOH, HNO3. The following methods for estimating the limiting coefficients from the reference data were used. Method No.3. fX∞ calculation based on the values of Henry's constant HX and the saturated vapor pressure p*X of undissociated molecules X. We used the formula, fX∞ =
55.51 , H X p*X
(26)
which immediately follows from relation (14). The values of Henry's constants HX were taken from reference work,58 and in doing so, the data obtained with the help of theoretical methods (such as structure–property relationships, etc.) were not considered. The pressures of p*X were computed with formulas given in 12 ACS Paragon Plus Environment
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handbook.59 For HF, HCOOH, CH3COOH, and CO2, the thermodynamic fugacity was substituted in formula (26) instead of pressure. The fugacities were calculated from the pressures determined by the dependences given in reference book,57 and the fugacity coefficients computed with Redlich-Kwong equation of state,60 the values of the critical constants were taken from ref 57. Method No.4. fX∞ calculation based on the thermodynamic properties of undissociated molecules X in aqueous solution and in the gas phase, and the saturated vapor pressure p*X. Henry's constant HX is calculated by the formula HX = exp(–
1 (∆fGø298(X,aq) – ∆fGº298(X,g))), RT
(27)
where ∆fGø298(X,aq) and ∆fGº298(X,g) are standard Gibbs energies of formation of compounds X (undissociated molecules) in aqueous solution and in the gas phase at 298.15. Pressure or fugacity was determined as stated above in the description method No.3. Method No.5. fX∞ calculation based on thermodynamic properties of undissociated compounds X in the liquid state and in aqueous solution. The limiting activity coefficients can be calculated from the standard thermodynamic properties of substance X in the liquid state and in aqueous solution in undissociated form. This method is convenient for those substances whose saturated vapor pressure is small and is not given in the reference literature. The shortest way to derive the corresponding formula is as follows. We substitute the expressions of saturated vapor pressure (23) and Henry's constant H (27) into relation (26), and obtain expression fX∞ =55.51 × exp(–
1 (∆fG*(X,liq) + RT
p*HA
∫1
Vm (HA,liq) dp – ∆fG (X,aq))). (28) ø
When using expression (28) in the present work, the difference ∆fG*298(X,liq) – ∆fGø298(X,aq) always exceeds 5 kJ / mole, while the value of integral p*HA
∫1
Vm (HA,liq) dp is less than 0.01 kJ / mole, and therefore it can be neglected. The
final formula has the form fX∞ =55.51 × exp(–
1 (∆fG*(X,liq) – ∆fGø(X,aq))). RT
(29)
Formula (29) shows that the limiting activity coefficient is determined by the change in Gibbs energy in the isothermal transfer of a molecule X from pure liquid substance X to infinitely dilute aqueous solution, ∆liq→aqG° = ∆fGø(X,aq) – ∆fG*(X,liq). The multiplier 55.51 is associated with the transition from the standard state "infinitely dilute solution with solute molar fraction equal to 1" to the standard state "infinitely dilute solution with solute molality 1 mol kg–1". The 13 ACS Paragon Plus Environment
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latter standard state is generally accepted, and thermodynamic properties of substances in solution in most reference books refer to this state. In reference books,61, 62 there is no data on the properties of undissociated HNO3 molecule in aqueous solution. Therefore, using methods No.4 and No.5, the Gibbs energy of formation of undissociated HNO3(aq) was calculated from the values of Gibbs energy of formation of nitrate ion in aqueous solution and dissociation constant of HNO3, Ka,298 = 34 – 37 М:4 ∆fGø298(HNO3,aq) = ∆fGø298(NO3–,aq) + RT ln Ka,298(HNO3). The Supporting Information to this article presents the input thermodynamic data for the calculations of apparent dissociation constants and limiting activity coefficients by the above-described methods, and the results of the calculations. Calculation of the limiting coefficients fHA∞ using COSMO-RS method. The theoretical computation of the activity coefficients fHA∞ was performed with the COSMO-RS program (version 2017)63 of the ADF software package.64, 65 With the help of a specially developed model of the interaction of a solute molecule with a continuous solvent medium (COSMO, conductor-like screening model),66 the program calculates the change in Gibbs free energy ∆Gºliq→solute upon transition of a molecule from pure liquid to infinitely dilute solution, and figures out the limiting activity coefficient from this quantity. As shown by Klamt,67 the performance of the COSMO for the computation of solvation energies for neutral and ionic solutes, in particular in highly dielectric solvents, is almost identical to that of dielectric polarized continuum solvation models in their modern implementations. The procedure for the computation the limiting coefficients was the same as recommended in the manual.68 The input data on the geometric structure of solute molecules were taken from the COSMO-RS database or the online database,69 and preliminary optimization of the geometric structure in the gas phase was performed. The limiting coefficients were evaluated at 298.15 K, with the use of four recommended optimized parameters sets given in the program and designated as ADF combi2005,63, 64 ADF combi1998,64, 70 Klamt,70 MOPAC PM6. RESULTS AND DISCUSSION The values of Henry’s constants KH and the apparent dissociation constants of HClO4 and H2SO4, calculated from the reference data, are given in Table 2. Table 3 presents the logarithms of limiting activity coefficients of various acids calculated with the COSMO-RS program, as well as those determined from the reference data. The data of Table 3 reveal that there is a well-expressed qualitative regularity connecting the limiting activity coefficient and the affinity of a substance to water: the greater the affinity, the smaller the coefficient. Affinity to 14 ACS Paragon Plus Environment
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water corresponds to the ability of the molecules of a substance to form intermolecular hydrogen bonds and to penetrate into the three-dimensional structure of water. Table. 2. Henry's law constants, pressure of pure liquids, and apparent dissociation constants of HClO4 and H2SO4 in aqueous solution at 298.15 K. Acid Method KH, mol2 kg–2 bar–1 p*, bar K'a, mol kg–1 log10K'a 16 61 21, 54 13 HClO4 No.1 4.01 × 10 3.40 × 10 0.0471 13.53 16 62, 71 13 No.1 2.57 × 10 0.0471 2.18 × 10 13.34 No.2 ––––– ––––– 3.65 × 1013 61 13.56 16 61 –8 55 7 H2SO4 No.1 3.01 × 10 1.88 × 10 1.02 × 10 7.01 17 62, 72 –8 57 9 No.1 9.25 × 10 7.58 × 10 1.26 × 10 9.10 9 61 No.2 ––––– ––––– 8.66 × 10 9.94 9 62 No.2 ––––– ––––– 6.34 × 10 9.80 Low values of the limiting coefficients (less than unity) are observed for those substances whose molecules can form several strong intermolecular hydrogen bonds. These are HF, as well as molecules containing O atoms and OH groups, which can both be a donor and acceptor of proton. Conversely, high values of the limiting coefficients are characteristic of substances that poorly penetrate the water structure, form weak hydrogen bonds, and have hydrophobic groups. At a qualitative level, the results of calculation in the program are in good agreement with the indicated regularity. However, it is not possible to establish a clearly expressed quantitative correlation between the calculated and experimental values of the limiting coefficients. Based on the comparison of the values of the limiting activity coefficients of the acids HNO3, CF3COOH, H3PO4, HCOOH, CH3COOH and C2H5COOH (Table 3), computed with the COSMO-RS program and determined from the reference data, it can be concluded that the program does not allow to calculate the coefficients with quantitative accuracy for these and similar complex compounds. Therefore, we perform the estimation of the limiting coefficients of strong acids HClO4 and H2SO4 by analogy. We choose the compounds with similar properties for which these coefficients are known, and assume that the sought coefficients of HClO4 and H2SO4 correlate in a certain way with the known coefficients of the selected compounds. A group of substances with high affinity to water comprises HF, H3PO4, CF3COOH, HNO3, HClO4 and H2SO4. Hydrogen fluoride HF has the lowest limiting activity coefficient, because of its ability to form very strong hydrogen bonds in liquid water, due to the maximum electronegativity of fluorine, the minimum size of the HF molecule, and the particular structure of HF (aq).12, 73-75 15 ACS Paragon Plus Environment
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We assume that the limiting coefficients of HClO4(aq) and H2SO4(aq) are less than or equal to the coefficients of H3PO4, CF3COOH, HNO3, but greater than the coefficient of HF. This assumption is qualitatively supported by COSMO-RS calculations. Table. 3. Logarithms of the limiting activity coefficients of undissociated molecules of various acids and oxides in aqueous solution at 298.15 K, determined from the reference data, and calculated with the COSMO-RS program. The four values in column 5 have been calculated using the corresponding optimized parameter sets (ADF combi2005, ADF combi1998 , Klamt, MOPAC PM6) mentioned earlier. Compound pKa11 log10fHA∞ (from Method and log10fHA∞ (calculated reference data) references with COSMO-RS) 57, 58 CO2 1.44 - 1.61 No.3 1.81 1.72 1.67 1.75 1.56 No.4 57, 61 1.57 No.4 57, 62 SO2 1.00 - 1.10 No.3 58, 59 1.80 1.74 1.71 1.66 59, 62 1.06 No.4 HF 3.20 –2.43 No.4 57, 61 –2.68 –2.74 –1.95 –1.85 57, 62 –2.47 No.4 HCOOH 3.751 −0.95 - −0.19 No.3 57, 58 0.055 0.067 0.22 0.11 –0.79 No.4 57, 61 –0.14 No.5 61 –0.17 No.5 62 CH3COOH 4.756 −0.57 - 0.47 No.3 57, 58 0.67 0.69 0.79 0.78 57, 61 –0.59 No.4 –0.51 No.4 57, 62 0.49 No.5 61 0.60 No.5 62 C2H5COOH 4.87 0.23 - 0.84 No.3 58, 59 1.17 1.21 1.30 1.30 61 H3PO4 2.16 –1.31 No.5 –1.67 –1.52 –0.98 –1.52 –2.38 No.5 62 CF3COOH 0.52 −1.36 - −1.17 No.3 58, 59 –0.39 –0.39 0.13 0.29 4 4, 57, 61 HNO3 –1.55 –2.04 No.4 –1.02 –1.08 –0.46 –0.31 4, 57, 62 –1.96 No.4 –2.11 No.5 4, 61 –2.04 No.5 4, 62 H2SO4 ––––– ––––– –4.54 –4.52 –3.05 –3.00 HClO4 ––––– ––––– –1.51 –1.57 –0.75 –0.46 Thus, the following estimate of the values of the limiting coefficients of HClO4 and H2SO4 can be established: 16 ACS Paragon Plus Environment
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–2.5 < log10 f HClO4∞ < –1.3, –2.5 < log10 f H2SO4∞ < –1.3. (30) The intervals (30) are quite wide, and include all known reference values of the limiting coefficients of strong and medium-strong acids. Therefore, the relations (30) are most likely correct, and gross errors are excluded when evaluating the limiting activity coefficients of HClO4 and H2SO4 using them. The logarithms of true dissociation constants of HClO4 and H2SO4 can be determined from formula (18), log10Ka = log10K'a + log10 fHA∞, on the basis of the values of apparent constants in Table 2, and the obtained intervals of the limiting coefficients (30). Their magnitudes are presented in Table 4. The ranges of HClO4 and H2SO4 dissociation constants given in the literature (see Introduction and Table 1) are very wide, log10Ka(HClO4) = 1.6 - 17, log10Ka(H2SO4) = 2 – 9, and our results in Table 4 are much more certain. Table 4. Dissociation constants of perchloric and sulfuric acids an aqueous solution at 298.15 K, estimated in this work. Acid log10K'a log10Ka log10fHA∞ HClO4 13.34 - 13.56 –1.3 - –2.5 10.8 - 12.3 H2SO4 7.01 - 9.94 –1.3 - –2.5 4.5 - 8.6 The values of the dissociation constants derived from modern quantum chemical calculations44-48 should be more reliable than those obtained by other methods. However, the results of theoretical papers44, 45 constitute the range of log10Ka(HClO4) = 7 – 17 (see Introduction and Table 1), that is, the maximum and minimum estimates of Ka(HClO4) are different by 10 orders of magnitude. The values of Ka(HClO4) obtained in the present work (Table 4) are considerably more accurate, their range is only 1.5 orders of magnitude. For sulfuric acid, quantum chemical calculations46-48 give the interval log10Ka(H2SO4) = 3.4 – 9. Our data in Table 4 define the interval more precisely. The maximum and minimum values for the first dissociation constant of H2SO4 in Table 4 are 104.1 times different. This is primarily caused by the significant uncertainty of the apparent dissociation constant of H2SO4, due to the difference in reference values of the thermodynamic properties of H2SO4(gas) in refs,61, 62, 72 and vapor pressure of H2SO4(liq) in refs.55-57
CONCLUSION
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It has been shown in this work that the dissociation constant of an acid, Ka, can be expressed as the product Ka = K'a × fHA∞. The apparent constants K'a of perchloric and sulfuric acids have been calculated on the basis of readily available reference data. The limiting activity coefficient fHA∞ of undissociated molecules of HClO4 and H2SO4 can only be determined by theoretical methods. In this work, estimates of the coefficients of HClO4 and H2SO4 have been found by analogy with compounds of a similar chemical structure. They are confirmed on a qualitative level by the results of calculations in the COSMO-RS program, and correspond to general chemical concepts of the affinity of substances to water. The values obtained for the dissociation constants of HClO4 and H2SO4 seem to be the most justified, in comparison with the available literature data. ASSOCIATED CONTENT Supporting Information. Brief Description of the Methods for Evaluation of Apparent Dissociation Constants and Limiting Activity Coefficients from Reference Data, the Input Data and Results of the Calculations.
1. 2.
3.
4. 5. 6.
7.
8.
9.
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TOC Graphic
HClO4(aq) ⇄ H+(aq) + ClO4–(aq) Ka = 6 × 1010 - 2 × 1012
H2SO4(aq) ⇄ H+(aq) + HSO4–(aq) Ka1 = 3 × 104 - 4 × 108
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