Structure of Chloric Acid Hydrates and Equilibrium Composition of the HClO₄–H₂O System

Abstract

Composition, structure, energy parameters, and existence regions of hydrate complexes formed in the HClO₄–H₂O system are studied by IR spectroscopy and quantum chemical methods. Three concentration-structural regions are determined. The HClO₄–H₂O solutions diluted to the 1:13.3 molar ratio of components contain $\text{ClO}^{-}_4$ and $\text{H}_5\text{O}^{+}_2$ solvate-separated ions, each hydrated by four water molecules. More concentrated solutions (1:13.3-1:5) contain hydrated ion pairs $\text{ClO}^{-}_4\cdot\text{H}_5\text{O}^{+}_2$ that are contact ion pairs if the [HClO₄]:[H₂O] ratio falls within the region 1:9-1:5. As the acid concentration increases further up to the transition of the HClO₄–H₂O system into the solid phase, pair complexes are formed. The structure of the latter is formed by a cycle of two contact ion pairs $\text{ClO}^{-}_4\cdot\text{H}_5\text{O}^{+}_2$ connected by two H-bonds. The solid phase consists of interacting polymer chains formed by $\text{ClO}^{-}_4\cdot\text{H}_5\text{O}^{+}_2$ and $\text{ClO}^{-}_4\cdot\text{H}_3\text{O}^{+}$ ion pairs.

INTRODUCTION

In aqueous perchlorate solutions, $\text{ClO}^{-}_4$ anions undergo negative hydration (according to the terminology by Samoilov [1]), i.e. water molecules in the anion′s hydration shell form H-bonds with its oxygen atoms, and the bonds are weaker than those between water molecules in the pure solvent. The IR spectra of such solutions contain an intense band caused by vibrations of those OH bonds of water molecules that interact with anions. The maximum of the band is at ~3500 cm^–1 [2] (2630 cm^–1 in D₂O solutions [3]). According to the melting diagram [4], NaClO₄ aqueous solutions contain NaClO₄·6H₂O complexes in a wide range of component concentrations, while significant excess of water leads to complete dissociation of the salt into ions and formation of $\text{ClO}^{-}_4$ tetrahydrate anions [2] and Na⁺ hexahydrate cations [5].

In our previous work (JSC, 2023, 112758), we established the equilibrium composition of the NaClO₄–H₂O system, determined the structure of NaClO₄·6H₂O complexes, and quantitatively explained changes in the IR spectra in the regions of bending and stretching vibrations of water molecules and the overtone at 6900 cm^–1. It was shown that only complexes with the 1:6 composition can participate in some stages of acid catalyzed hydrolysis of esters by enhancing the nucleophilic assistance and thereby increasing the reaction rate [6, 7].

According to the melting diagram, the HClO₄–H₂O system undergoes numerous structural changes manifested as a number of maxima corresponding to crystal hydrates with compositions 1:1 (49.9 °C), 1:2 (–20.65 °C), 1:2.5 (–32.1 °C), 1:3 (–40.2 °C), 1:3.5 (–45.6 °C), 1:4 (–58.0 °C), where digits in parentheses indicate melting points of the complexes [8]. The data reported in [8] were used in the calculations of the HClO₄–H₂O system.

The IR spectra of the HClO₄–H₂O system were analyzed and interpreted using a comprehensive experimental and computational approach based on the regularities characterizing the formation of hydrogen-bonded complexes in solutions [9]. This approach was earlier successfully utilized to study intercomponent interactions in various liquid acid-base systems [10-15]. According to these regularities, each molecular complex in the solutions is the most stable one in the series of isomers, has a cyclic or polycyclic structure, contains unstrained H-bonds, and has the most dense molecular packing among all complexes with similar compositions. As the solution concentration changes, the structure of each next complex is formed from the structure of the previous one with minimal energy costs and structural transformations. The solution with the highest content of complexes of the same type also contains a small number of complexes with previous and subsequent compositions.

The purpose of the present work is to determine the equilibrium composition of the HClO₄–H₂O system and to establish the structure of the hydrate complexes formed in the system. To this aim, IR spectra of aqueous HClO₄ solutions were recorded and analyzed, and quantum chemical calculations of perchloric acid hydrates were conducted.

EXPERIMENTAL

In the experiments, perchloric acid (reagent-grade) was used. Aqueous HCl solutions were prepared by saturating distilled water with gaseous HCl. The concentration of initial aqueous HClO₄ and HCl solutions was determined by alkalimetric titration. Solutions with lower acid concentrations were prepared by weight dilution.

The IR spectra were recorded at 30 °C by the method of multiple frustrated total internal reflection (MFTIR) [16] on an attachment with a Ge working element providing a radiation incidence angle of 30°. The effective thickness of the absorbing layer at a frequency 2000 cm^–1 was 3.35 µm. The spectra were measured in the 1300-4000 cm^–1 frequency range at the HClO₄ concentrations of 0-8.69 mol/L ([H₂O]:[HClO₄] = 3.89) and HCl concentrations of 0-11.65 mol/L ([H₂O]:[HCl] = 3.59). The band absorbances (D) were determined from the baselines with an accuracy of 5-7%. The optical density of continuous absorption was measured relative to the absorbance of the empty cuvette.

COMPUTATIONAL DETAILS

Total energies and optimal configurations of HClO₄ molecules, HClO₄·nH₂O (n = 1-9), 2HClO₄·nH₂O (n = 2-12) hydrate complexes, and the polymer chain fragment (HClO₄)₈·(H₂O)₈ were calculated by the density functional theory method (B3LYP/6-31++G(d,p)) using the Gaussian09 program [17]. The structure of each HClO₄·nH₂O (n = 1-9) hydration complex was determined by considering all those possible relative arrangements of their molecules, which are compatible with the laws determining the formation of H-bonded complexes in solutions [9], and then choosing the most stable one. The paired complexes were calculated in the assumption that each of them is formed by two HClO₄·nH₂O complexes or by HClO₄·nH₂O and HClO₄·(n+1)H₂O complexes. The analysis of stationary points on the potential energy surfaces of the studied systems showed that their optimal configurations correspond to energy minima (all elements of the Hessian matrices are positive).

When estimating formation energies (ΔE) of hydrate complexes and polymer chain fragment, their total electronic energies were used under the assumption that a negligibly small restructuring energy is required for a molecule as it is incorporated into an H-bonded complex of any composition and structure. Therefore, the formation energy of each hydrate was calculated by subtracting the energies of its molecules from the total hydrate energy. Similarly, when estimating the formation energy of the 2HClO₄·nH₂O paired complex (ΔE_p) from two complexes, each containing one HClO₄ molecule, their total energies were subtracted from that of the paired complex.

Hydrogen bonds in the figures with optimized geometries of the complexes and fragments of the polymer structure satisfy the following conditions: bond lengths < 3 Å, and their number and directions correspond to those of the lone pairs of oxygen atoms.

RESULTS AND DISCUSSION

Computation results. According to the melting diagram, complexes with stoichiometric compositions 1:2.5 and 1:3.5 are formed in the HClO₄–H₂O system [8]. We assumed that these are stable paired complexes obtained due to the combination of complexes HClO₄·nH₂O and HClO₄·(n+1)H₂O (n = 2, 3). Complexes with close compositions are similarly formed due to the combination of two complexes HClO₄·nH₂O (n = 2-4). This assumption is supported by the fact that the transition from the largest of these paired complexes (2HClO₄·8H₂O) to the HClO₄·5H₂O complex, containing one HClO₄ molecule, proceeds through minimal structural changes and energy costs, in full agreement with the laws discussed in [9, 10, 12-15]. The fact that the HClO₄–H₂O system (in contrast to most of the earlier studied systems containing one paired complex with the composition 2:2 [9, 12-15]) has a range of concentrations where various paired complexes are formed may be explained by the presence of four H-bond forming oxygen atoms in the HClO₄ molecule.

The equilibrium composition of the HClO₄–H₂O system was determined by calculating two series of H-bonded complexes, HClO₄·nH₂O (n = 1-9) and 2HClO₄·nH₂O (n = 2-12), while structural and energetic parameters of polymer chains in the solid phase were studied by calculating the 8HClO₄·8H₂O fragment. Table 1 lists total energies (E) and formation energies (ΔE) of these complexes, as well as the formation energy (ΔE_p) for each of the studied paired complexes composed of two complexes, each containing one HClO₄ molecule.

Fig. 1 and 2 show optimized geometries of the studied hydrates. The first figure shows three complexes with a 1:1 stoichiometric ratio of molecules and complexes containing one molecule of the HClO₄·nH₂O acid (n = 2-9); the second figure shows paired complexes, except for the 2HClO₄·2H₂O heterotetramer.

Structural and energy parameters of complexes HClO₄·nH₂O (n = 1-9). As a result of optimization, the proton of the HClO₄ molecule in the HClO₄·nH₂O complexes with n > 2 passed on to the H₂O molecule, thus forming three conjugated hydrogen bridges inside the complex (Fig. 1e-k). Parameters of two such bridges are identical in the HClO₄·3H₂O hydrate and differ for all three bridges in other complexes (Table 2). Note that the length of the shortest and least asymmetric hydrogen bridge (r(O⋯O) = 2.487-2.560 Å) is in all cases characteristic for the $\text{H}_5\text{O}^{+}_2$ ion whose structure is distorted under the influence of surrounding species. Therefore, it can be stated that complexes HClO₄·nH₂O (n = 3-9) contain $\text{ClO}^{-}_4$ anions and $\text{H}_5\text{O}^{+}_2$ cations. (Despite this structural transformation, the optimized ion-containing complexes we will be referred to below as nHClO₄·mH₂O).

Note that the $\text{ClO}^{-}_4$ anion and the $\text{H}_5\text{O}^{+}_2$ ion in HClO₄·nH₂O complexes (n = 3-9) interact with each other via one or two H-bonds, thus forming a contact ion pair (Fig. 1e-k). Even though the relative arrangement of the counterions and the interaction between them can vary in different complexes, this ionic pair is the basis of the hydrate structure. Therefore, the solvate-separated ion pairs are formed and the HClO₄·nH₂O complexes are separated into hydrated $\text{H}_5\text{O}^{+}_2$ and $\text{ClO}^{-}_4$ at a smaller than 1:9 molar ratio of the HClO₄–H₂O solution components.

The calculation results indicate that $\text{H}_5\text{O}^{+}_2$ and $\text{ClO}^{-}_4$ tend to participate in the interactions between the species in varying degrees. In the HClO₄·nH₂O (n ≥ 4) hydrates, the $\text{H}_5\text{O}^{+}_2$ ion forms four H-bonds, using all its capabilities. At the same time, the $\text{ClO}^{-}_4$ anion has three free lone pairs even for n = 9. The anion forms only three H-bonds in small HClO₄·nH₂O complexes (n = 3 and 4) and four in larger complexes (n = 5-8), while one of its oxygen atoms never participates in the interactions between the species. In the largest calculated hydrate containing one acid molecule (HClO₄·9H₂O), the $\text{ClO}^{-}_4$ anion forms five H-bonds (using all four oxygen atoms), while H₂O molecules are interconnected by six H-bonds. They strongly interact with each other also in other cases (Fig. 1h-j). Thus, in the series of HClO₄·nH₂O complexes containing the $\text{H}_5\text{O}^{+}_2$ ion, it is energetically most favorable for water molecules to form H-bonds with this ion, while the interaction with other water molecules rather than with the $\text{ClO}^{-}_4$ anion is more profitable in the case of its complete hydration. This result agrees with the fact that $\text{ClO}^{-}_4$ anions exhibit negative hydration in perchlorate aqueous solutions [1, 2].

Table 1 Total Energies (E, hartree) and Formation Energies (ΔE, kcal/mol) of HClO·nH₂O and mHClO·nH₂O Hydrate Complexes; Formation Energies of 2HClO·nH₂O Complexes Composed of Two Complexes Containing One HClO₄ Molecule (E_p, kcal/mol)

Structural and energy parameters of 2HClO₄·nH₂O complexes (n = 2-12). The 2HClO₄·2H₂O complex with alternating acid and water molecules (Fig. 1b) contains two hydrogen bridges, in each of which the proton of the HClO₄ molecule approaches the oxygen of the H₂O molecule without passing to it (Table 2). The energy required for this complex to from two HClO₄·H₂O complexes (Fig. 1a) is 10.63 kcal/mol (Table 1). The structure of 2HClO₄·2H₂O complexes allows them to be H-bonded into infinite polymer chains interacting with each other, which is a condition for a solution to transform into the solid phase [18]. Fig. 1c shows the optimized geometry of the chain fragment composed of four 2HClO₄·2H₂O complexes. As can be seen, protons of HClO₄ molecules (except those at the ends of the fragment) in the 8HClO₄·8H₂O fragment move to H₂O molecules to form $\text{ClO}^{-}_4\cdot\text{H}_3\text{O}^{+}$ ion pairs. This result agrees with a high melting point of perchloric acid monohydrate [8].

Fig. 1

Structure of hydrate complexes HClO₄·H₂O (a), 2HClO₄·2H₂O (b), 8HClO₄·8H₂O (c), HClO₄·2H₂O (d), HClO₄·3H₂O (e), HClO₄·4H₂O (f), HClO₄·5H₂O (g), HClO₄·6H₂O (h), HClO₄·7H₂O (i), HClO₄·8H₂O (j), and HClO₄·9H₂O (k). In figures f-k, the $\text{H}_5\text{O}^{+}_2$ ion is at the top.

Fig. 2

Structure of hydrate complexes 2HClO₄·3H₂O (a), 2HClO₄·4H₂O (b), 2HClO₄·5H₂O (c), 2HClO₄·6H₂O (d), 2HClO₄·7H₂O (e), 2HClO₄·8H₂O (f), 2HClO₄·9H₂O (g), 2HClO₄·10H₂O (h), 2HClO₄·11H₂O (i), and 2HClO₄·12H₂O (j).

As the content of water in the HClO₄–H₂O system increases, 2HClO₄·4H₂O complexes are formed after 2HClO₄·2H₂O complexes (within the polymer chains) (Fig. 2b). Each of them consists of two HClO₄·2H₂O complexes, which combined together so that the protons of their HClO₄ molecules passed to the H₂O molecules. As a result, two $\text{ClO}^{-}_4$ anions and two $\text{H}_5\text{O}^{+}_2$ cations (the structure of the latter is distorted compared to that in the gas phase) were formed inside 2HClO₄·4H₂O. At the same time, the bridge of each $\text{H}_5\text{O}^{+}_2$ ion is connected with two unstrained hydrogen bridges (Table 2). The structure of the 2HClO₄·4H₂O complex can be described as a cycle composed of two $\text{H}_5\text{O}^{+}_2\cdot\text{ClO}^{-}_4$ contact ion pairs connected by two H-bonds. Each ion pair also has a cyclic structure where the cation and the anion are connected by two H-bonds.

Table 2 Interatomic Distances in the Bridge of the $\text{H}_5\text{O}^{+}_2$ Ion and in the Conjugated Hydrogen Bridges of Complexes HClO₄·nH₂O (n = 3-9) and 2HClO₄·nH₂O (n = 3-12)

The structure of 2HClO₄·4H₂O complexes allows them to combine with each other and with 2HClO₄·2H₂O complexes into infinite interconnected polymer chains. For the molar component ratio 1:1.5, in the case of equal amounts of 2HClO₄·2H₂O complexes (in the form of $\text{H}_3\text{O}^{+}\cdot\text{ClO}^{-}_4$ ion pairs) and 2HClO₄·4H₂O complexes, the system exists in the solid state (T_m = 26 °C) [8]. As the fraction of 2HClO₄·4H₂O complexes increases (with increasing water concentration), the melting temperature decreases and the system transforms into the liquid phase [8], apparently because $\text{H}_5\text{O}^{+}_2$ ions form weaker H-bonds with the environment than H₃O⁺ ions. Structural features of 2HClO₄·4H₂O complexes suggest that they interact with each other in solutions to form small polymer chains.

Adding one water molecule to the 2HClO₄·4H₂O complex followed by the formation of the 2HClO₄·5H₂O complex (Fig. 2c) does not significantly affect the structure of this compound (the framework of the 2HClO₄·4H₂O complex is maintained). A characteristic feature of the resulting compound is that it is formed at the boundary between two concentration-structural regions. The first of them contains complexes consisting of two contact ion pairs, and the second one contains hydrated contact ion pairs.

The next complex, 2HClO₄·6H₂O (Fig. 2d), consists of two HClO₄·3H₂O complexes that were strongly deformed during the optimization. The complexes are connected by four H-bonds whose total energy (18.83 kcal/mol) exceeds the ΔE_p values of other paired complexes (Table 1). Two of these bonds connect the contact ion pairs to each other; each of the other two H-bonds is formed by a water molecule hydrating two contact ion pairs.

The 2HClO₄·nH₂O complexes (n = 7-12) contain hydrated contact ion pairs with counterions sharing a common hydrogen bond (Fig. 2e-k). Components of the 2HClO₄·7H₂O complex (HClO₄·3H₂O and HClO₄·4H₂O) are connected by four H-bonds (Fig. 2e). In the complexes 2HClO₄·nH₂O (n = 7-8), similarly to the paired ones with a lower water content (n = 3-6), two contact ion pairs are connected by two H-bonds. In larger complexes (n = 9-12), the interaction between ion pairs is much weaker (Table 1). The 2HClO₄·9H₂O complex contains one H-bond, and two contact ion pairs are separated by water molecules beginning from 2HClO₄·10H₂O. Weakening the interaction between the individual parts of paired complexes will lead to their dissociation in the solution and subsequent formation of complexes containing one HClO₄ molecule (Fig. 1g-k).

The above results explain why paired complexes are formed in the HClO₄–H₂O system at the molar component ratio ≥1:4, while HClO₄·nH₂O complexes are formed in more dilute solutions (n = 5-9). The structure of 2HClO₄·nH₂O (n = 4-8) complexes is based on a very strong cycle composed of two $\text{ClO}^{-}_4\cdot\text{H}_5\text{O}^{+}_2$ contact ion pairs connected by two H-bonds. Therefore, in contrast to the complexes with weakly interacting constituent parts 2HClO₄·nH₂O (n ≥ 9), such complexes do not break down in the solution into smaller complexes containing one HClO₄ molecule.

Our calculations indicate that the HClO₄–H₂O system contains the following concentration-structural regions. Components of the most concentrated solutions are partially or completely connected into 2HClO₄·4H₂O cyclic complexes composed of two $\text{H}_5\text{O}^{+}_2\cdot\text{ClO}^{-}_4$·$REMOVE$ contact ion pairs. For lower acid contents, the following paired complexes are formed in solutions: 2HClO₄·5H₂O, 2HClO₄·6H₂O, 2HClO₄·7H₂O, and 2HClO₄·8H₂O. The complexes contain hydrated ion pairs where counterions are still bonded to each other. Even more dilute solutions ([HClO₄]:[H₂O] from 1:5 to 1:9) contain complexes comprising one acid molecule each and also consisting of $\text{ClO}^{-}_4\cdot\text{H}_5\text{O}^{+}_2\cdot n\text{H}_2\text{O}$ hydrated contact ion pairs (n = 5-9).

Thus, there is the following explanation of the melting curve structure [8]. As the solution temperature decreases, paired complexes combined into interacting chain structures characterized by relatively high melting points [8].

Asymmetry degree of O⋯H⋯O hydrogen bridges as a function of their length. Another interesting result was obtained by analysing the parameters of three conjugated hydrogen bridges in the 2HClO₄·nH₂O (n = 4-8) and HClO₄·nH₂O (n = 5-9) complexes observed in the studied solutions. It was established for r(O⋯O) < 2.6 Å that there is a linear dependence between the length of the O⋯H⋯O bridge and its asymmetry degree (Fig. 3) characterized by the coefficient K_asym equal to the ratio between long and short H-bonds of the bridge (r(H⋯O)/r(O⋯H)). Linear nature of this dependence in the considered range of r(O⋯O) values indicates that two main structural parameters (r(O⋯O) and K_asym) in short hydrogen bridges (e.g. in $\text{H}_5\text{O}^{+}_2$ ions) change symbatically with the molecular environment. The deviation of this dependence from a straight line increases with increasing distance between the oxygen atoms. Using these data, the proton position in the $\text{H}_5\text{O}^{+}_2$ ion bridge can be estimated only from its r(O⋯O) length.

IR spectroscopy study. Fig. 4 shows MFTIR spectra of water and aqueous perchloric acid solution with a concentration of 6.6 mol/L (39.8 mol/L H₂O). The differences observed in these spectra over the entire frequency range are due to the formation of $\text{H}_5\text{O}^{+}_2$ hydrated ions with strong quasi-symmetric H-bonds in O⋯H⋯O bridges [19-21] and due to the hydration of $\text{ClO}^{-}_4$ anions. A characteristic feature of the IR spectra of $\text{H}_5\text{O}^{+}_2$ ions is continuous absorption (CA) in the region from 900 cm^–1 to 4000 cm^–1 caused by the presence of a large number of overlapping intense high-order bands arising due to the electro-optical anharmonicity of the O⋯H⋯O bridge and strong kinematic interaction between its coordinates and the ligand coordinates [22]. The spectrum of $\text{H}_5\text{O}^{+}_2$ ions exhibit a number of separate bands on the background of continuous absorption [23].

Fig. 3

Asymmetry coefficient of conjugated hydrogen bridges in 2HClO₄·nH₂O (n = 4-8) and HClO₄·nH₂O (n = 5-9) complexes as a function of their length. Designations: asymmetry coefficients (K_asym) of the $\text{H}_5\text{O}^{+}_2$ bridge (1), first (2), and second (3) conjugated bridges.

Fig. 4

MFTIR spectra of water (1), aqueous perchloric acid solution (6.6 mol/L HClO₄, 39.8 mol/L H₂O) (2) and sodium perchlorate solution (7.3 mol/L NaClO₄, 35.5 mol/L H₂O) (3). The thickness of the absorption layer is 3.35 µm.

The changes in the IR spectra of aqueous NaClO₄ solutions due to the ion hydration are localized near the band maxima at 1640 cm^–1 and 3500 cm^–1 (JSC, 2023, 112758), Fig. 4, spectrum 3. To establish characteristics of $\text{ClO}^{-}_4$ hydration in aqueous HClO₄ solutions, contributions of the $\text{H}_5\text{O}^{+}_2$ absorption spectrum to the total absorbance in these regions should be determined. This was done as follows.

In addition to the contribution from water molecules, the D₃₅₀₀ IR spectra of aqueous HClO₄ solutions have also a contribution of $\text{H}_5\text{O}^{+}_2$ ions. The value of the latter was determined from the IR spectroscopy data obtained for aqueous HCl solutions. The spectral characteristic of $\text{H}_5\text{O}^{+}_2$ ions is continuous absorption that does not depend (according to the mechanism of its formation) on the HA acid anion [22, 24]. We will prove it on the example of HClO₄ and HCl aqueous solutions (Fig. 5) for which CA absorbances and acid concentrations were compared at a frequency of 2000 cm^–1. The contribution of water molecules should be also considered since they exhibit some weak absorption at this frequency. Therefore, the $D^{*}_{2000}$ absorbance caused only by the absorption of $\text{H}_5\text{O}^{+}_2$ ions was searched by subtracting the contribution of water molecules not involved in the formation of these ions from the experimental D₂₀₀₀ values:

$$D^{*}_{2000} = D_{2000} - \varepsilon_{2000}(\text{H}_2\text{O})\cdot l_{2000} \cdot ([\text{H}_2\text{O}]_0 - 2[\text{HA}]_0) = \varepsilon_{2000}(\text{H}_5\text{O}^{+}_2)\cdot l_{2000}\cdot[\text{HA}]_0,$$

(1)

where ε₂₀₀₀(H₂O), $\varepsilon_{2000}(\text{H}_5\text{O}^{+}_2)$ are absorption coefficients for water and $\text{H}_5\text{O}^{+}_2$ ions; l₂₀₀₀ is the effective absorption layer thickness at a frequency of 2000 cm^–1.

Fig. 5

$D^{*}_{2000}$ as a function of HClO₄ (1) and HCl (2) concentration. Symbols A₁ and A₂ denote acid concentrations in HClO₄ and HCl aqueous solutions with equal water:acid ratios equal to 4.09.

In relatively dilute solutions of both acids, the $D^{*}_{2000}$ values exhibit the same dependence: CA absorbance grows linearly up to the values 4.9 mol/L HClO₄ and 5.6 mol/L HCl, corresponding for such HClO₄ and HCl concentrations to equal component ratios [HA]₀:[H₂O]₀ = 1:8.8. The $\varepsilon_{2000}(\text{H}_5\text{O}^{+}_2)$ absorption coefficients for both acids are equal to 180 L/(mol·cm). In more concentrated solutions, the linearity is violated, the $\varepsilon_{2000}(\text{H}_5\text{O}^{+}_2)$ coefficients decrease, and dependences demonstrated by HClO₄ and HCl solutions diverge (Fig. 5). This discrepancy is not due to the anion influence on $\varepsilon_{2000}(\text{H}_5\text{O}^{+}_2)$ and corresponds to the [HA]₀:[H₂O]₀ ratios that depend differently on the concentrations of HClO₄ and HCl acids. We will show that the $\varepsilon_{2000}(\text{H}_5\text{O}^{+}_2)$ values coincide at the same component ratios in HClO₄ and HCl solutions. Fig. 5 shows compositions of HClO₄ (A₁) and HCl (A₂) solutions with equal component ratios ([HA]₀:[H₂O]₀ = 4.1). The dashed line starts from the coordinate origin and passes through both of these points, and its inclination angle determines the $\varepsilon_{2000}(\text{H}_5\text{O}^{+}_2)$ coefficient value at the given component ratio. The following analysis of IR spectra of aqueous HClO₄ solutions was performed under the assumption that the IR spectrum of $\text{H}_5\text{O}^{+}_2$ ions is not affected by the anions.

The contributions to the D₃₅₀₀ absorbance in aqueous HClO₄ solutions with a significant excess of water include the absorption of $\text{H}_5\text{O}^{+}_2$ ions, water molecules, hydrating $\text{ClO}^{-}_4$ anions, and water molecules not involved in the hydration of anions and the formation of $\text{H}_5\text{O}^{+}_2$ ions.

$$D_{3500} = \varepsilon_{3500}(\text{H}_5\text{O}^{+}_2)\cdot l_{3500} \cdot ([\text{HClO}_4]_0 + 4\varepsilon_{hydr}\cdot l_{3500} \cdot [\text{HClO}_4]_0 + \varepsilon l_{3500} \cdot ([\text{H}_2\text{O}]_0 - 6 [\text{HClO}_4]_0).$$

(2)

Equation (2) takes into account that the acid is completely dissociated, the anion is hydrated by four water molecules, and two more water molecules are required to form the $\text{H}_5\text{O}^{+}_2$ ion. Here, $\varepsilon_{3500}(\text{H}_5\text{O}^{+}_2)$, ε_hydr, and ε are absorption coefficients of $\text{H}_5\text{O}^{+}_2$ ions, water molecules hydrating the anions, and water molecules not involved in the anion hydration and the formation of $\text{H}_5\text{O}^{+}_2$ ions, respectively; l₃₅₀₀ is the effective absorbing layer thickness at 3500 cm^–1. For aqueous HCl solutions, (2) is written as follows:

$$D_{3500} = \varepsilon_{3500}(\text{H}_5\text{O}^{+}_2)l_{3500} \cdot [\text{HCl}]_0 + \varepsilon l_{3500} \cdot ([\text{H}_2\text{O}]_0 - 2 [\text{HCl}]_0)$$

and

$$D_{3500} / ([\text{H}_2\text{O}]_0 - 2 [\text{HCl}]_0) = \varepsilon l_{3500} + \varepsilon_{3500} (\text{H}_5\text{O}^{+}_2)l_{3500} \cdot [\text{HCl}]_0 / ([\text{H}_2\text{O}]_0 - 2 [\text{HCl}]_0).$$

(3)

The linear correlation (3) is true throughout the considered concentration range from 0 to 10.6 ([HCl]₀:[H₂O]₀ = 1:4.1) mol/L HCl, $\varepsilon_{3500}(\text{H}_5\text{O}^{+}_2)l_{3500}$ = 0.017 L/mol (Fig. 6). Note that the difference ([H₂O]₀ – 2[HCl]₀) in (3) includes two kinds of water molecules: those which do and do not participate in $\text{H}_5\text{O}^{+}_2$ hydration. Therefore, the obtained linear correlation (3) indicates that the spectral manifestations of these water molecules in the region of OH stretchings are identical.

The $\varepsilon_{3500}(\text{H}_5\text{O}^{+}_2)l_{3500}$ coefficient value equal to 0.017 L/mol was used to determine the contribution of $\text{H}_5\text{O}^{+}_2$ absorption to the absorbance at 3500 cm^–1 in HClO₄ aqueous solutions. This way, we calculated $D^{*}_{3500}$ values equal to $d_{3500}-\varepsilon(\text{H}_5\text{O}^{+}_2)[\text{HClO}_4]_0$, and obtained (4) from (2) for the subsequent experimental data analysis:

$$D^{*}_{3500}/([\text{H}_2\text{O}]_0 - 6 [\text{HClO}_4]_0) = \varepsilon l_{3500} + \varepsilon_{hydr}l_{3500}\cdot4[\text{HClO}_4]_0/([\text{H}_2\text{O}]_0-6[\text{HClO}_4]_0).$$

(4)

The ε_hydrl₃₅₀₀ value calculated from the linear correlation in the coordinates of equation (4) (Fig. 7) is equal to 0.017 L/mol and exceeds the εl₃₅₀₀ value for water molecules in a pure solvent by ~35%. This is significantly less than in NaClO₄ aqueous solutions where ε_hydr exceeds ε more than doubly (JSC, 2023, 112758). This difference is explained as follows. According to the quantum chemical calculations, the lengths of H-bonds between water molecules and oxygen atoms of the $\text{ClO}^{-}_4$ ion vary much stronger for the complexes formed in the HClO₄–H₂O system than in the NaClO₄–H₂O system. This should correspond to a noticeable broadening of the 3500 cm^–1 band in the spectra of acid solutions and to the decrease of the coefficient at its maximum, as was indeed observed in the experiment (Fig. 4). The linear dependence in Fig. 7 is observed in the HClO₄ concentration range from 0 to 5.2 mol/L ([HClO₄]₀/([H₂O]₀ = 1:8.3). The obtained data do not allow separating this range into individual concentration-structural regions with pronounced features of $\text{ClO}^{-}_4$ hydration in HClO₄ aqueous solutions with a varying water excess.

We consider changes in the IR spectra of HClO₄ aqueous solutions in the region of water molecule bendings. The band at 1640 cm^–1 overlaps with a band of scissoring vibrations of terminal groups of $\text{H}_5\text{O}^{+}_2$ ions with a maximum at 1720 cm^–1 [25, 26]. Since the band is symmetrical [26], the $\text{H}_5\text{O}^{+}_2$ absorption is the same at 1800 cm^–1 and 1640 cm^–1. Using this fact, we obtained $D^{*}_{1640}$ values corresponding to the bendings of those water molecules that do not participate in the formation of $\text{H}_5\text{O}^{+}_2$ ions ([H₂O] is the concentration of such molecules):

Fig. 6

Linear correlation in the coordinates of (3) at 3500 cm^–1 in HCl aqueous solutions.

Fig. 7

Linear correlation in the coordinates of (4) at 3500 cm^–1 in HClO₄ aqueous solutions.

$$D^{*}_{1640} = D_{1640} - D_{1800},$$

$$[\text{H}_2\text{O}] = [\text{H}_2\text{O}]_0 - 2 [\text{HClO}_4]_0.$$

It was earlier established that the absorption coefficients of water molecules that do and do not participate in the $\text{ClO}^{-}_4$ hydration in dilute NaClO₄ aqueous solutions are equal at 1640 cm^–1. This is true up to the 1:11 component ratio (JSC 2023, 112758). Fig. 8 shows the dependence of $D^{*}_{1640}$ on [H₂O] for the HClO₄–H₂O system: it is proportional in the 0-3.5 mol/L HClO₄ concentration range approximately up to [HClO₄]₀:[H₂O]₀ = 1:13.3. More concentrated solutions exhibit smaller absorption coefficients, and this range can be divided into two intervals. The absorption coefficient decreases insignificantly by no more than 10% up to the 1:6 molar component ratio and a decreases abruptly in more concentrated solutions so that its value for the 1:3.9 ratio is approximately 2.5 times lower than in dilute solutions.

Fig. 8

$D^{*}_{1640}$ as a function of [H₂O]₀–2[HClO₄]₀.

The obtained data indicate that the system HClO₄–H₂O contains three concentration-structural regions. The first of them is a region of dilute solutions up to [HClO₄]₀:[H₂O]₀ ≈ 1:13.3 containing solvate-separated $\text{ClO}^{-}_4$ and $\text{H}_5\text{O}^{+}_2$ hydrated ions. The acid concentration ranges corresponding to this concentration-structural region obtained in our studies (up to 1:13.3) virtually coincide with the NMR data (up to 1:14) reported in [27, 28]). At a significant excess of water in HClO₄ solutions, two water molecules are involved in the formation of the $\text{H}_5\text{O}^{+}_2$ ion, and four water molecules participate in the formation of each solvation shell of the counterions.

The linear dependence of $D^{*}_{1640}$ on [H₂O] is violated (Fig. 8), apparently due to the fact that some water molecules are coordinated both with the anion and with the cation, thus forming ion pairs in the solutions. The calculation data suggest that the coefficient slightly decreases in solutions containing hydrated contact ion pairs with one HClO₄ molecule (second concentration-structural region) (Fig. 1g-k) and changes significantly due to the formation of paired complexes (third concentration-structural region) (Fig. 2b-f).

CONCLUSIONS

We conducted a comprehensive computational and experimental study of the composition, structure, energy parameters, and existence regions of hydrate complexes formed in the HClO₄–H₂O system. The following concentration-structural regions were determined by comparing the calculation and IR spectroscopy data. Dilute solutions up to the 1:13.3 molar component ratio contain solvate-separated $\text{ClO}^{-}_4$ and $\text{H}_5\text{O}^{+}_2$ hydrated ions. More concentrated solutions (1:13.3-1:5) contain hydrated solvate-separated or contact ion pairs, counterions of the latter being interconnected by one or two H-bonds. Below the 1:5 ratio, complexes containing two acid molecules are formed. Their structure is based on strong cycles composed of two contact ion pairs connected by two H-bonds. Such cyclic complexes with the general formula $2\text{ClO}^{-}_4\cdot2\text{H}_5\text{O}^{+}_2\cdot n \text{H}_2\text{O}$ (n = 4-0) are successively formed over a wide range of HClO₄ concentrations, and non-hydrated ion pairs (n = 0) are also formed after the HClO₄–H₂O system transforms into the solid phase.

It was shown that short hydrogen bridges O⋯H⋯O (r(O⋯O) < 2.6 Å), in particular $\text{H}_5\text{O}^{+}_2$ ion bridges, exhibit a linear dependence between their length and the asymmetry degree. This fact can be used to estimate the proton position in the $\text{H}_5\text{O}^{+}_2$ ion bridge when only its length is known.

The DFT study of structural and energy parameters of the hydrate complexes formed in HClO₄–H₂O solutions and some interpretations of experimental data based on the calculation results were funded by the Ministry of Science and Higher Education of the Russian Federation as part of the State Assignment for Kurnakov Institute of General and Inorganic Chemistry of the Russian Academy of Sciences.