Interionic Interactions Interpreted Using Friedman’s Parameters and Their Contribution to the Excess Gibbs Energy of Mixing in Ternary Phosphate Aqueous Solutions at 298.15 K

Abstract

This study examines interactions in aqueous electrolyte solutions using the equations of the Scatchard and Friedman models. The six mixing parameters of the Scatchard model, b_AB⁽⁰¹⁾; b_AB⁽⁰²⁾; b_AB⁽⁰³⁾; b_AB⁽¹²⁾; b_AB⁽¹³⁾ and b_AB⁽²³⁾, were obtained from the literature and estimated by processing experimental results measured by the isopiestic method for osmotic coefficients of three-component systems: {yKCl + (1 − y)K₂HPO₄} (aq), {yKBr + (1 − y)K₂HPO₄} (aq), {yKNO₃ + (1 − y)K₂HPO₄} (aq), {yK₂SO₄ + (1 − y)K₂HPO₄} (aq), {yKH₂PO₄ + (1 − y)K₂HPO₄} (aq) and {yNaH₂PO₄ + (1 − y) K₂HPO₄} (aq) at 298.15 K. The Friedman parameters were calculated from the adopted Scatchard parameters as functions of ionic strength. The effects of pair, triplet, and quadruplet interactions on the excess Gibbs energy of mixing were analyzed, and the total Gibbs energy of the solutions was determined. In the systems {yKCl + (1 − y)K₂HPO₄} (aq), {yKBr + (1 − y)K₂HPO₄} (aq), {yKNO₃ + (1 − y)K₂HPO₄} (aq), interactions between different anions of the same charge predominate. Triplet interactions dominate in the system {yK₂SO₄ + (1 − y)K₂HPO₄} (aq). The strongest contributions of triplet and quadruplet interactions are observed in the {yKH₂PO₄ + (1 − y)K₂HPO₄} (aq) system, whereas pair interactions between the same ion pairs are dominant in the {yNa₂HPO₄ + (1 − y)K₂HPO₄} (aq) system.

1. Introduction

The study of the thermodynamic properties of multicomponent electrolyte solutions is essential for describing the behavior of ternary systems, which is influenced by complex ion–ion and ion–solvent interactions across a wide range of concentrations, from very dilute to saturated. In the co-sphere theory, ions are classified based on their effect on the solvent: structure-breaking ions disrupt the local water structure, while structure-making ions organize it [1]. Generally, two ions will be attracted to each other if they exert similar effects on the solution structure or if their tendency to orient water molecules is comparable. Additionally, these ions can be distinguished by their affinity for the solvent; hydrophilic species attract water molecules, whereas hydrophobic species do not. Notably, hydrophobic ions tend to stabilize the overall solvent structure. This work focuses on the study of the thermodynamic properties of electrolyte systems containing phosphate ions: {yKCl + (1 − y)K₂HPO₄} (aq), {yKBr + (1 − y)K₂HPO₄} (aq), {yKNO₃ + (1 − y)K₂HPO₄} (aq), {yK₂SO₄ + (1 − y)K₂HPO₄} (aq), {yKH₂PO₄ + (1 − y)K₂HPO₄} (aq), {yNa₂HPO₄ + (1 − y)K₂HPO₄} (aq), at T = 298.15 K [2,3,4,5,6,7,8]. Phosphate systems in which possible interactions of pairs, triplets and quadruplets of ions were investigated contain K⁺(aq), Na⁺(aq), Cl⁻(aq), Br⁻(aq), NO₃⁻(aq), SO₄²⁻(aq), H₂PO₄⁻(aq) and HPO₄²⁻(aq). In a previous study, where the influence of different ions on the solvent (water) structure was monitored, ions were classified as follows: structure-breakers K⁺(aq), Cl⁻(aq), Br⁻(aq), NO₃⁻(aq), SO₄²⁻(aq)), borderline ions Na⁺(aq), H₂PO₄⁻(aq), and structure-formers HPO₄²⁻(aq) [9]. Some authors, however, categorize the SO₄²⁻(aq) ion as a structure stabilizer. This classification refers to effects observed through macroscopic measurements in dilute solutions. The influence of ions can also be evaluated through the hydration spheres or shells that form in solution, as described by [10]. Variations in electrolyte concentration can alter the number of water molecules within the primary and secondary hydration spheres, thereby influencing the probability of ion pair or triplet formation. For K⁺(aq), the hydration shell is only moderately affected by concentration, with the hydration number decreasing from 6.0 to 4.8. In contrast, for the Cl⁻(aq) ion, the hydration number remains constant at low concentrations, though minor reductions occur at higher concentrations, with the number of water molecules in the anionic hydration shell decreasing from approximately 6.1 ± 1.1 to 5.6 ± 1.6. Another study investigated the structure of phosphate solutions using Raman spectroscopy [11]. Their analysis identified two primary types of hydration: interactions between the phosphate oxygen and the hydrogen of a water molecule, and interactions between the hydrogen of the phosphate hydroxyl group and the oxygen of a surrounding water molecule. The study also suggests the formation of dimers through hydrogen bonding, as well as the potential influence of cations or other anions on the formation of solvent-separated ion pairs, and the authors propose that while hydrogen bonds between water molecules are disrupted, new hydrogen bonds form between phosphate ions, leading to the creation of dimers or polymers in solutions containing H₂PO₄⁻(aq) ions. Furthermore, based on the Raman spectra of KH₂PO₄(aq) solutions, it was concluded that the K⁺(aq) ion has a negligible effect on spectral shifts, providing no evidence for the formation of K⁺(aq)-H₂PO₄⁻(aq) ion pairs. Recent research involving KH₂PO₄(aq), K₂HPO₄(aq) and K₃PO₄(aq) systems at T = 298.15 K utilized dielectric relaxation spectroscopy to investigate hydration and ion association across a wide concentration range [12]. According to the authors, the phosphate anions in these solutions are extensively hydrated; the hydration numbers at infinite dilution are approximately 11 (for H₂PO₄⁻), 20 (for HPO₄²⁻) and 39 (for PO₄³⁻). These high values indicate the presence of a secondary hydration shell around the HPO₄²⁻ and PO₄³⁻ ions. Furthermore, all three salts exhibit significant ion-pair formation. The standard association constants for the 1:1 species increase in the order: KH₂PO₄⁰(aq) < KHPO₄⁻(aq) < KPO₄²⁻(aq). The study also confirmed the existence of both solvent-shared ion pairs and solvent-separated ion pairs.

Based on the relations proposed for Friedman parameters, the values of g₀, g₁ and g₂ were calculated for all investigated phosphate systems. The analysis is based on the principle that changes in solution configuration result from ion interactions involving pairs, triplets, and quadruplets. In one publication, Wen analyzed experimental values of osmotic coefficients in the Mixing Solutions of Alkali Halides and Symmetrical Tetraalkylammonium Halides using Scatchard’s and Friedman’s formalisms to yield the excess Gibbs energy and the free energy interaction parameter g₀. The obtained g₀ values were subsequently discussed in terms of Friedman’s ionic solution theory to gain insights into the nature of ion interactions [13]. In the study by Amdur, the analysis was conducted based on the same principle. The excess Gibbs energy changes on mixing aqueous solutions of tetrapropyl-ammonium chloride and sodium chloride were determined at 298.15 K using the gravimetric isopiestic method. The experimental osmotic coefficients were treated by the Scatchard and Friedman procedures to yield the excess Gibbs energy of mixing ${∆G}^{ex}$ and the free energy interaction parameters g₀ and g₁ [14]. Specifically, the parameter g₀ represents pair interactions, g₁ represents triplet interactions, and g₂ represents quadruplet interactions. The dependence of these Friedman parameters (g₀, g₁ and g₂) on the ionic strength of the solution was plotted, along with the excess Gibbs energies for pairs, triplets, and quadruplets as a function of the ionic strength fraction. The total excess Gibbs energy for these phosphate systems was calculated using both the Friedman parameters and the Scatchard model, the latter of which incorporates all six mixing parameters. Both methods yielded identical values for the Gibbs energy, confirming the consistency of the models. Gurney’s theory was applied to analyze the influence of ionic interactions on the excess Gibbs energy of the investigated phosphate systems at various ionic strengths [1]. This analysis is based on the principle that ionic attraction provides a negative contribution to the excess Gibbs energy, while repulsion results in a positive contribution. According to Gurney’s model, two hydrophobic ions will attract each other, as will two hydrophilic ions of opposite charges. Conversely, two hydrophilic ions of the same charge, or a combination of a hydrophilic and a hydrophobic ion, will experience repulsion. These interactions collectively determine the thermodynamic behavior of the solution across the studied concentration range.

2. Data Sources

This paper uses literature data on the mixing parameters of the Scatchard model, taken from references [2,3,4,5,6,7,8]. These parameters were estimated using the NLMNK method, based on experimental values of osmotic coefficients in three-component systems: {yKCl + (1 − y) K₂HPO₄}(aq), {yKBr + (1 − y)K₂HPO₄}(aq), {yKNO₃ + (1 − y)K₂HPO₄}(aq), {yK₂SO₄ + (1 − y)K₂HPO₄}(aq), {yKH₂PO₄ + (1 − y)K₂HPO₄}(aq) and {yNa₂HPO₄ + (1 − y)K₂HPO₄}(aq). These values were determined by the isopiestic method at a temperature of 298.15 K. The method is based on the equalization of solvent activities in the reference and test solutions when the system is in equilibrium at a given temperature and pressure. The procedure for applying the method is explained in detail in references [2,3,4,5,6,7,8].

The combination of all six mixing parameters of the Scatchard model, b_AB⁽⁰¹⁾; b_AB⁽⁰²⁾; b_AB⁽⁰³⁾; b_AB⁽¹²⁾; b_AB⁽¹³⁾; and b_AB⁽²³⁾, for the systems {yKCl + (1 − y)K₂HPO₄} (aq), {yKBr + (1 − y)K₂HPO₄} (aq), {yKNO₃ + (1 − y)K₂HPO₄} (aq), {yK₂SO₄ + (1 − y)K₂HPO₄} (aq), {yKH₂PO₄ + (1 − y)K₂HPO₄} (aq), and {yNa₂HPO₄ + (1 − y)K₂HPO₄} (aq) with the corresponding values of the standard deviation u(ϕ) are listed in

Table 1. Mixing parameters of the Scatchard model, b_AB⁽⁰¹⁾; b_AB⁽⁰²⁾; b_AB⁽⁰³⁾; b_AB⁽¹²⁾; b_AB⁽¹³⁾; and b_AB⁽²³⁾, for the systems {yKCl + (1 − y)K₂HPO₄} (aq), {yKBr + (1 − y)K₂HPO₄} (aq), {yKNO₃ + (1 − y)K₂HPO₄} (aq), {yK₂SO₄ + (1 − y)K₂HPO₄} (aq) [6], {yKH₂PO₄ + (1 − y)K₂HPO₄} (aq), {yNa₂HPO₄ + (1 − y)K₂HPO₄} (aq), with the corresponding values of the standard deviation u(ϕ) [2,3,4,5,6,7,8].

System	b(01)·(m0)	b(02)·(m0)2	b(03)·(m0)3	b(12)·(m0)2	b(13)·(m0)3	b(23)·(m0)3	u (ϕ)
KCl + K2HPO4 + H2O [2,3]	−0.1095	0.0422	−0.0039	0.0271	−0.0047	−0.0012	0.005
KBr + K2HPO4H2O [4]	−0.0729	0.0177	−0.0010	0.01317	−0.0014	−0.00027	0.0022
KNO3 + K2HPO4H2O [5]	−0.1730	0.0781	−0.0081	0.0047	−0.0015	−0.00071	0.0028
K2SO4 + K2HPO4 + H2O [6]	−0.0027	−0.0065	0.00391	−0.0261	0.0141	−0.00043	0.0005
KH2PO4 + K2HPO4 + H2O [7]	−0.0673	0.1010	−0.0209	0.0193	−0.0137	−0.0131	0.0011
Na2HPO4 + K2HPO4 + H2O [8]	0.6794	−1.2997	0.6285	−0.1418	0.1773	−0.04733	0.0031

m⁰ = 1 mol·kg⁻¹.

.

The adopted parameters of the Scatchard model are valid within the molality range for which experimental data exist. The molality range of validity for the parameters is as follows: for the systems {yKCl + (1 − y)K₂HPO₄}(aq) from 2.370 to 11.250 mol·kg⁻¹, {yKNO₃ + (1 − y)K₂HPO₄}(aq) from 2.4958 to 6.0801 mol·kg⁻¹, {yKBr + (1 − y)K₂HPO₄}(aq) from 2.5452 to 10.0418 mol·kg⁻¹, {yK₂SO₄ + (1 − y)K₂HPO₄}(aq) from 1.3167 to 1.9587 mol·kg⁻¹, {yKH₂PO₄ + (1 − y)K₂HPO₄}(aq) from 0.0 to 2.0 mol·kg⁻¹ and {yNa₂HPO₄ + (1 − y)K₂HPO₄}(aq) from 0.0 to 2.0 mol·kg⁻¹ all at the temperature of T = 298.15 K [2,3,4,5,6,7,8]. As a reference solution, KCl(aq) was used in the systems {yKNO₃ + (1 − y)K₂HPO₄}(aq), {yK₂SO₄ + (1 − y)K₂HPO₄}(aq), {yKH₂PO₄ + (1 − y)K₂HPO₄}(aq) and {yNa₂HPO₄ + (1 − y)K₂HPO₄}(aq) [5,6,7,8]. In the system {yKCl + (1 − y)K₂HPO₄}(aq), CaCl₂(aq) was used as the reference solution, while in the system {yKBr + (1 − y)K₂HPO₄}(aq), both CaCl₂(aq) and KCl(aq) were used as references for two series of measurements. All measurements were performed at 298.15 K. In this work, for the systems {yKCl + (1 − y)K₂HPO₄}(aq), {yKBr + (1 − y)K₂HPO₄}(aq), {yKNO₃ + (1 − y)K₂HPO₄}(aq), all calculations were performed in the ionic strength range from 0 to 4 mol·kg⁻¹, while for the systems {yK₂SO₄ + (1 − y)K₂HPO₄}(aq), {yKH₂PO₄ + (1 − y)K₂HPO₄}(aq) and {yNa₂HPO₄ + (1 − y)K₂HPO₄}(aq), all calculations were performed in the ionic strength range from 0 to 2 mol·kg⁻¹ [2,3,4,5,8].

3. Models

The Pitzer and Scatchard models are most frequently used to analyze experimental activity coefficient data for multicomponent electrolyte solutions [15,16,17,18]. The Pitzer model incorporates two mixing parameters, whereas the Scatchard model includes six, which can be correlated with the Friedman parameters. For this study, we selected all six of Scatchard’s mixing parameters because they provided the best fit between the experimental and calculated values.

3.1. Calculation of the Excess Gibbs Energy by Scatchard’s Model Equation

The equation of the Scatchard model, as described by Scatchard and Rard and Wijesinghe, is used to predict and calculate the thermodynamic properties of mixed electrolyte solutions [18,19]. In this approach, electrolytes A and B are treated as neutral components. The excess Gibbs energy of the mixture is defined as the sum of the contributions from the pure solutions of A and B, plus the contribution from mixing effects. These mixing effects are expressed as relatively simple functions of the total ionic strength (I_m) and the ionic strength fractions (y_A and y_B,), where b_AB represents the Scatchard mixing parameters. The b_AB parameters are determined using the least-squares method, applied to osmotic coefficient data obtained from the mixed electrolyte systems.

$${\displaystyle \frac{∆G_m^{\mathrm{e}\mathrm{x}}}{RT}}=I_m{y}_{\mathrm{A}}{y}_{\mathrm{B}}\left[B_{\mathrm{A}\mathrm{B}}^{\left(0\right)}+\left(y_{\mathrm{A}}-y_{\mathrm{B}}\right)B_{\mathrm{A}\mathrm{B}}^{\left(1\right)}+{\left(y_{\mathrm{A}}-y_{\mathrm{B}}\right)}^2{B}_{\mathrm{A}\mathrm{B}}^{\left(2\right)}+\dots +{\left(y_{\mathrm{A}}-y_{\mathrm{B}}\right)}^n{B}_{\mathrm{A}\mathrm{B}}^{\left(n\right)}\right]$$

(1)

where terms B_AB represent:

$$B_{\mathrm{A}\mathrm{B}}^{\left(n\right)}={\displaystyle {\sum }_{k=n+1}^{k\prime }}{\displaystyle \frac{I^k}{k}}{b}_{\mathrm{A}\mathrm{B}}^{\left(n,k\right) } \mathrm{f}\mathrm{o}\mathrm{r} n \ge 0$$

(2)

In Equation (2), parameter n represents the exponent of the ionic strength fraction difference (y_A − y_B)ⁿ, while parameter k represents the exponent to which the total solution ionic strength I_m is raised.

In the case when we observe interactions of pairs, triplets and quadruplets of ions in the system, functions $B_{\mathrm{A}\mathrm{B}}^{\left(0\right)}$, $B_{\mathrm{A}\mathrm{B}}^{\left(1\right)}$ and $B_{\mathrm{A}\mathrm{B}}^{\left(2\right)}$ are given by the following relations:

$$B_{\mathrm{A}\mathrm{B}}^{\left(0\right)}=I{b}_{\mathrm{AB}}^{(0,1)}+\left(1/2\right)I^2{b}_{\mathrm{AB}}^{(0,2)}+\left(1/3\right)I^3{b}_{\mathrm{AB}}^{(0,3)}$$

(3)

$$B_{\mathrm{A}\mathrm{B}}^{\left(1\right)}=\left(1/2\right)I^2{b}_{\mathrm{AB}}^{(1,2)}+\left(1/3\right)I^3{b}_{\mathrm{AB}}^{(1,3)}$$

(4)

$$B_{\mathrm{A}\mathrm{B}}^{\left(2\right)}=\left(1/3\right)I^3{b}_{\mathrm{AB}}^{(2,3)}$$

(5)

When we substitute Equations (3)–(5) into Equation (1), the total Gibbs energy in the analyzed three-component systems can be expressed in terms of all six Scatchard mixing parameters using Equation (6).

$$\begin{gathered} ∆G_m^{\mathrm{e}\mathrm{x}}/RT=y_{\mathrm{A}}{y_{\mathrm{B}}I}^2{b}_{\mathrm{AB}}^{(0,1)}+\left(1/2\right)y_{\mathrm{A}}{y_{\mathrm{B}}I}^3\left(b_{\mathrm{AB}}^{(0,2)}+\left(y_{\mathrm{A}}-y_{\mathrm{B}}\right)b_{\mathrm{AB}}^{(1,2)}\right) \\ +\left(1/3\right)y_{\mathrm{A}}{y_{\mathrm{B}}I}^4\left(b_{\mathrm{AB}}^{(0,3)}+\left(y_{\mathrm{A}}-y_{\mathrm{B}}\right)b_{\mathrm{AB}}^{(1,3)}+{\left(2y_A-1\right)}^2{b}_{\mathrm{AB}}^{(2,3)}\right) \end{gathered}$$

(6)

3.2. Calculation of Friedman’s Interaction Parameters and the Excess Gibbs Energy by Friedman’s Parameters

When interactions are restricted to multiplets within the Venn–Rowlinson–Mayer (VRM) framework, the j-th Friedman interaction parameter (g_j) as defined by Wigent and Leifer can be decomposed into contributions from pairs, triplets, and higher-order multiplets (g_j,pairs, g_j,triplets, g_j,VRM) [20,21]. These individual parameters, which describe the multiplet interactions, are constants independent of both the total ionic strength and the ionic strength fraction of the electrolyte. However, they depend on temperature, pressure, and the specific types of bonds formed within the mixed electrolyte solution. The Friedman interaction parameters, g_j, can be expressed by the following equations:

$$g_0=g_{0 \mathrm{p}\mathrm{a}\mathrm{i}\mathrm{r}\mathrm{s}}+I{g}_{0 \mathrm{t}\mathrm{r}\mathrm{i}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{s}}+I^2{g}_{0 \mathrm{q}\mathrm{u}\mathrm{a}\mathrm{d}\mathrm{s}}+\dots +I^n{g}_{0\mathrm{V}\mathrm{R}\mathrm{M}}$$

(7)

$$g_1=I{g}_{1 \mathrm{t}\mathrm{r}\mathrm{i}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{s}}+I^2{g}_{1 \mathrm{q}\mathrm{u}\mathrm{a}\mathrm{d}\mathrm{s}}+\dots +I^n{g}_{1 \mathrm{V}\mathrm{R}\mathrm{M}}$$

(8)

$$g_2=I^2{g}_{2 \mathrm{q}\mathrm{u}\mathrm{a}\mathrm{d}\mathrm{s}}+\dots +I^n{g}_{2 \mathrm{V}\mathrm{R}\mathrm{M}}$$

(9)

$$g_n=I^n{g}_{n \mathrm{V}\mathrm{R}\mathrm{M}}$$

(10)

Also, the interaction parameters g_n depend on the Scatchard mixing parameters b_AB via the following relation:

$$g_n={\displaystyle {\sum }_{k=n+1}^k}{\displaystyle \frac{I^{k-1}}{k}}{b}_{\mathrm{A}\mathrm{B}}^{\left(n,k\right)} 3\mathrm{a} n\ge 0$$

(11)

When Equations (7)–(11) are compared, the Scatchard mixing parameter b_AB is found to have the following relationship with g_n pairs, g_{n triplets}, g_{n quadruplets}, as given by the equations:

$$g_{n \mathrm{p}\mathrm{a}\mathrm{i}\mathrm{r}\mathrm{s}}=b_{\mathrm{A}\mathrm{B}}^{\left(n,1\right)} \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} \mathrm{i}\mathrm{s} n = 0$$

(12)

$$g_{n \mathrm{t}\mathrm{r}\mathrm{i}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{s}}=b_{\mathrm{A}\mathrm{B}}^{\left(n,2\right)}/2 \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} \mathrm{i}\mathrm{s} n=\left[0,1\right]$$

(13)

$$g_{n \mathrm{q}\mathrm{u}\mathrm{a}\mathrm{d}\mathrm{s}}=b_{\mathrm{A}\mathrm{B}}^{\left(n,3\right)}/3 \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} \mathrm{i}\mathrm{s} n=[0,1,2]$$

(14)

The Friedman parameter g₀ for pair interactions is:

$$g_{0 \mathrm{p}\mathrm{a}\mathrm{i}\mathrm{r}\mathrm{s}}=b_{\mathrm{A}\mathrm{B}}^{\left(\mathrm{0,1}\right)}$$

(15)

The Friedman parameters g₀ and g₁ for triplet interactions are:

$$g_{0 \mathrm{t}\mathrm{r}\mathrm{i}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{s}}=\left(1/2\right)b_{\mathrm{A}\mathrm{B}}^{\left(\mathrm{0,2}\right)}$$

(16)

$$g_{1 \mathrm{t}\mathrm{r}\mathrm{i}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{s}}=\left(1/2\right)b_{\mathrm{A}\mathrm{B}}^{\left(\mathrm{1,2}\right)}$$

(17)

and the Friedman parameters g₀, g₁ and g₂ for quadruplet interactions are:

$$g_{0 \mathrm{q}\mathrm{u}\mathrm{a}\mathrm{d}\mathrm{s}}=\left(1/3\right)b_{\mathrm{A}\mathrm{B}}^{\left(\mathrm{0,3}\right)}$$

(18)

$$g_{1 \mathrm{q}\mathrm{u}\mathrm{a}\mathrm{d}\mathrm{s}}=\left(1/3\right)b_{\mathrm{A}\mathrm{B}}^{\left(\mathrm{1,3}\right)}$$

(19)

$$g_{2 \mathrm{q}\mathrm{u}\mathrm{a}\mathrm{d}\mathrm{s}}=\left(1/3\right)b_{\mathrm{A}\mathrm{B}}^{\left(\mathrm{2,3}\right)}$$

(20)

If we insert the values of the Friedmann parameters for pairs, triplets and quadruplets into Equations (7)–(9), we obtain the values of the Friedmann parameters g₀, g₁ and g₂, which correspond to the total interactions of pairs, triplets and quadruplets in the systems.

$$g_0=b_{\mathrm{A}\mathrm{B}}^{\left(\mathrm{0,1}\right)}+\left(1/2\right)I{b}_{\mathrm{A}\mathrm{B}}^{\left(\mathrm{0,2}\right)}+\left(1/3\right)I^2{b}_{\mathrm{A}\mathrm{B}}^{\left(\mathrm{0,3}\right)}$$

(21)

$$g_1=\left(1/2\right)I{b}_{\mathrm{A}\mathrm{B}}^{\left(\mathrm{1,2}\right)}+\left(1/3\right)I^2{b}_{\mathrm{A}\mathrm{B}}^{\left(\mathrm{1,3}\right)}$$

(22)

$$g_2=\left(1/3\right){I^{2}b}_{\mathrm{A}\mathrm{B}}^{\left(\mathrm{2,3}\right)}$$

(23)

The excess Gibbs mixing energy can be decomposed into the individual contributions of pairs, triplets, quadruplets, and multiplets as follows:

$$\left({\displaystyle \frac{∆G_m^{ex}}{RT}}\right)={\left({\displaystyle \frac{∆G_m^{ex}}{RT}}\right)}_{\mathrm{p}\mathrm{a}\mathrm{i}\mathrm{r}\mathrm{s}}+{\left({\displaystyle \frac{∆G_m^{ex}}{RT}}\right)}_{\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{s}}+{\left({\displaystyle \frac{∆G_m^{ex}}{RT}}\right)}_{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{d}\mathrm{s}}+\dots +{\left(∆G_m^{ex}/RT\right)}_{\mathrm{H}\mathrm{O}\mathrm{M}}$$

(24)

where the supplementary Gibbs mixing energy for individual contributions of pairs, triplets, quadruplets, or multiplets is given by Equations (25)–(28).

$${\left(∆G_m^{ex}/RT\right)}_{\mathrm{p}\mathrm{a}\mathrm{i}\mathrm{r}\mathrm{s}}=I^2{y}_{\mathrm{A}}{y}_{\mathrm{B}}{g}_{0 \mathrm{p}\mathrm{a}\mathrm{i}\mathrm{r}\mathrm{s}}$$

(25)

$${\left(∆G_m^{ex}/RT\right)}_{\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{s}}=I^3{y}_{\mathrm{A}}{y}_{\mathrm{B}}\left[g_{0 \mathrm{t}\mathrm{r}\mathrm{i}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{s}}+\left(y_{\mathrm{A}}-y_{\mathrm{B}}\right)g_{1 \mathrm{t}\mathrm{r}\mathrm{i}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{s}}\right]$$

(26)

$${\left(∆G_m^{ex}/RT\right)}_{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{d}}=I^4{y}_{\mathrm{A}}{y}_{\mathrm{B}}\left[g_{0 \mathrm{q}\mathrm{u}\mathrm{a}\mathrm{d}}+\left(y_{\mathrm{A}}-y_{\mathrm{B}}\right)g_{1 \mathrm{q}\mathrm{u}\mathrm{a}\mathrm{d}}+{\left(y_{\mathrm{A}}-y_{\mathrm{B}}\right)}^2{g}_{2 \mathrm{q}\mathrm{u}\mathrm{a}\mathrm{d}}\right]$$

(27)

$${\left(∆G_m^{ex}/RT\right)}_{\mathrm{H}\mathrm{O}\mathrm{M}}=I^{n+2}{y}_{\mathrm{A}}{y}_{\mathrm{B}}\left[g_{0 \mathrm{H}\mathrm{O}\mathrm{M}}+\left(y_{\mathrm{A}}-y_{\mathrm{B}}\right)g_{1 \mathrm{H}\mathrm{O}\mathrm{M}}+\dots +{\left(y_{\mathrm{A}}-y_{\mathrm{B}}\right)}^n{g}_{n \mathrm{H}\mathrm{O}\mathrm{M}}\right]$$

(28)

If only pairwise interactions exist in a mixed solution, the total excess free energy equals (ΔG_m^ex/RT)_pairs, and b_AB⁽⁰¹⁾ is the only parameter needed to fit the data. If b_AB⁽⁰¹⁾ alone is insufficient to fit the data, triplet interactions must be present in the solution. In this case, b_AB⁽⁰¹⁾, b_AB⁽⁰²⁾, and b_AB⁽¹²⁾ must all be used, and the excess free energy of mixing will be given by two terms, (ΔG_m^ex/RT)_pairs and (ΔG_m^ex/RT)_triplets. If fitting the data requires pair, triplet, and quadruplet parameters b_AB⁽⁰³⁾, b_AB⁽¹³⁾, and b_AB⁽²³⁾, then the excess free energy of mixing will include an additional term reflecting all possible interactions.

$${\left(∆G_m^{ex}/RT\right)}_{\mathrm{p}\mathrm{a}\mathrm{i}\mathrm{r}\mathrm{s}}=I^2{y}_{\mathrm{A}}{y}_{\mathrm{B}}{b}_{\mathrm{A}\mathrm{B}}^{\left(\mathrm{0,1}\right)}$$

(29)

$${\left(∆G_m^{ex}/RT\right)}_{\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{s}}=\left(1/2\right)I^3{y}_{\mathrm{A}}{y}_{\mathrm{B}}\left[b_{\mathrm{A}\mathrm{B}}^{\left(\mathrm{0,2}\right)}+\left(y_{\mathrm{A}}{-y}_{\mathrm{B}}\right)b_{\mathrm{A}\mathrm{B}}^{\left(\mathrm{1,2}\right)}\right]$$

(30)

$${\left(∆G_m^{ex}/RT\right)}_{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{d}\mathrm{s}}=\left(1/3\right)I^4{y}_{\mathrm{A}}{y}_{\mathrm{B}}\left[b_{\mathrm{A}\mathrm{B}}^{\left(\mathrm{0,3}\right)}+\left(y_{\mathrm{A}}{-y}_{\mathrm{B}}\right)b_{\mathrm{A}\mathrm{B}}^{\left(\mathrm{1,3}\right)}+{\left(y_{\mathrm{A}}{-y}_{\mathrm{B}}\right)}^2{b}_{\mathrm{A}\mathrm{B}}^{\left(\mathrm{2,3}\right)}\right]$$

(31)

$${\left(∆G_m^{ex}/RT\right)}_{\mathrm{H}\mathrm{O}\mathrm{M}}=\left(I^{\left(n+2\right)}/\left(n+1\right)\right)y_{\mathrm{A}}{y}_{\mathrm{B}}\left[b_{\mathrm{A}\mathrm{B}}^{\left(0,\left(n+1\right)\right)}+\left(y_{\mathrm{A}}-y_{\mathrm{B}}\right)b_{\mathrm{A}\mathrm{B}}^{\left(1,\left(n+1\right)\right)}+\dots +{\left(y_{\mathrm{A}}-y_B\right)}^n{b}_{\mathrm{A}\mathrm{B}}^{\left(n,\left(n+1\right)\right)}\right]$$

(32)

4. Results and Discussion

4.1. Friedman Parameter Analysis

In this chapter, the analysis of possible ionic interactions in the systems {yKCl + (1 − y)K₂HPO₄} (aq), {yKBr + (1 − y)K₂HPO₄} (aq), {yKNO₃ + (1 − y)K₂HPO₄} (aq), {yK₂SO₄ + (1 − y)K₂HPO₄} (aq), {yKH₂PO₄ + (1 − y)K₂HPO₄} (aq), {yNa₂HPO₄ + (1 − y)K₂HPO₄} (aq) is presented based on the calculated values of the Friedmann parameters g₀, g₁ and g₂ using Equations (21)–(23) and and the Scatchard model parameter values for these systems from Table 1 at T = 298.15 K. Pair interactions are the main contributors to the parameter g₀, triplet interactions are the main contributors to g₁, and quadruplet interactions are the main contributors to g₂.

Figure 1a shows the dependence of the Friedman parameter g₀ on the ionic strength of aqueous solutions: {yKCl + (1 − y)K₂HPO₄} (aq), {yKBr + (1 − y)K₂HPO₄} (aq), {yKNO₃ + (1 − y)K₂HPO₄} (aq), {yK₂SO₄ + (1 − y)K₂HPO₄} (aq), {yKH₂PO₄ + (1 − y)K₂HPO₄} (aq). Figure 1b also includes values of the Friedman parameter g₀ for the system {yNa₂HPO₄ + (1 − y)K₂HPO₄} (aq) at T = 298.15 K.

From Figure 1a,b, a varying influence of ion-pair interactions is observed across the investigated systems as a function of ionic strength. In the following systems, {yKCl + (1 − y)K₂HPO₄} (aq), {yKBr + (1 − y)K₂HPO₄} (aq), {yKNO₃ + (1 − y)K₂HPO₄} (aq) and {yKH₂PO₄ + (1 − y)K₂HPO₄} (aq), the g₀ parameter remains negative throughout the entire ionic strength range, suggesting that interactions between different ions of the same charge dominate over interactions between identical ions. For the K₂HPO₄ with KCl and KBr, the g₀ parameter shows similar variation, with the chloride ion being more hydrophilic than the bromide ion. The behavior changes significantly for phosphate solutions containing nitrate ions, as well as for mixtures of potassium hydrogen phosphate and potassium dihydrogen phosphate. The influence of ion pairing becomes more pronounced as ionic strength increases. The g₀ parameter increases much more prominently in the {yKH₂PO₄ + (1 − y)K₂HPO₄} (aq) system. In the {yK₂SO₄ + (1 − y)K₂HPO₄} (aq) system, there is almost no change in the Friedman g₀ parameter with increasing concentration, suggesting that ion-pair interactions are not pronounced in this system. In the {yNa₂HPO₄ + (1 − y)K₂HPO₄} (aq) system, the g₀ values are positive, indicating the dominance of interactions between different ions of the same charge.

From Figure 2a, the Friedman parameter g₁ is positive for the KCl, KBr, and KNO₃ systems, indicating negligible triplet effects on nonideality. The K₂SO₄ and KH₂PO₄ systems, however, are more complex: g₁ is negative for the sulfate system up to 3 mol·kg⁻¹, indicating dominant triplet interactions between different ionic species, and turns positive thereafter with increasing influence of triplets composed of ions of the same charge type SO₄²⁻(aq)-H₂PO₄⁻(aq)-H₂PO₄⁻(aq). For the phosphate–phosphate system, positive g₁ near zero suggests ion pairs/dimers, while negative values signal phosphate triplet formation. In Figure 2b, the Na₂HPO₄–K₂HPO₄ system shows negative g₁ up to 1.5 mol·kg⁻¹, consistent with mixed triplets Na⁺(aq)-HPO₄⁻(aq)-K⁺(aq), Na⁺(aq)-Na⁺(aq)-K⁺(aq) or K⁺(aq)-K⁺(aq)-Na⁺(aq), beyond which same-species phosphate trimers become more influential.

Figure 3a,b shows the dependence of the Friedman parameter g₂ on the ionic strength, I_m, in the three-component systems {yKCl + (1 − y)K₂HPO₄} (aq), {yKBr + (1 − y)K₂HPO₄} (aq), {yKNO₃ + (1 − y)K₂HPO₄} (aq), {yK₂SO₄ + (1 − y)K₂HPO₄} (aq), {yKH₂PO₄ + (1 − y)K₂HPO₄} (aq) and {yNa₂HPO₄ + (1 − y)K₂HPO₄} (aq) at T = 298.15 K.

Figure 3a presents the Friedman parameter g₂ as a function of ionic strength for the systems containing KCl, KBr, KNO₃, K₂SO₄, and KH₂PO₄ with K₂HPO₄, while Figure 3b shows the corresponding data for the Na₂HPO₄–K₂HPO₄ system. In all cases, g₂ values are negative and become more negative with increasing ionic strength, indicating that quadruplet interactions grow stronger as concentration rises. Negative g₂ values signify attraction between different anions, pointing to a thermodynamic preference for mixed quadruplets, which enhance stability through ion mixing. Because HPO₄²⁻ readily forms hydrogen bonds, these quadruplets are not purely electrostatic; they frequently incorporate water molecules from the hydration shell, which stabilize interionic bridges. In the chloride- and bromide-containing systems, the monovalent Cl⁻ and Br⁻ anions naturally repel the divalent HPO₄²⁻ ion. The negative g₂ values suggest that K⁺ acts as a bridging cation, enabling Cl⁻ and Br⁻ to approach HPO₄²⁻ more closely than electrostatic repulsion would otherwise allow. The dominant quadruplet combinations are therefore [K₂Cl(HPO₄)]⁻ and [K₂Br(HPO₄)]⁻. For the sulfate system, both SO₄²⁻ and HPO₄²⁻ are divalent, resulting in strong electrostatic repulsion. The negative g₂ values indicate that K⁺ similarly links the two anions to form clusters such as [K₂SO₄(HPO₄)]²⁻. In the phosphate–phosphate system, H₂PO₄⁻ and HPO₄²⁻ share similar structures and form stable hydrogen-bonded aggregates; for example, [H₂PO₄·HPO₄]³⁻. Owing to these strong hydrogen bonds, the dominant quadruplet is [K₂(H₂PO₄)(HPO₄)]⁻. Finally, in the sodium–potassium phosphate system, negative g₂ values show that the mixed quadruplet is thermodynamically favored over the average of the pure sodium and potassium quadruplets. The dominant species is [NaK(HPO₄)₂]²⁻, in which one Na⁺ and one K⁺ share two HPO₄²⁻ anions.

Figure 4a–f shows the dependence of Friedman’s parameters g₀, g₁ and g₂ on the ionic strength of the following solutions: {yKCl + (1 − y)K₂HPO₄} (aq), {yKBr + (1 − y)K₂HPO₄} (aq), {yKNO₃ + (1 − y)K₂HPO₄} (aq), {yK₂SO₄ + (1 − y)K₂HPO₄}(aq), {yKH₂PO₄ + (1 − y)K₂HPO₄} (aq) and {yNa₂HPO₄ + (1 − y)K₂HPO₄}(aq) at a temperature of T = 298.15 K.

In the {yKCl + (1 − y)K₂HPO₄}(aq) and {yKBr + (1 − y)K₂HPO₄}(aq) systems in Figure 4a,b, g₀ and g₁ both increase with concentration, reflecting growing ion pair and triplet contributions. Despite positive g₁ values, triplets remain significant because g₀ also captures triplet effects. Possible interactions include Cl⁻(aq)–Cl⁻(aq), Br⁻(aq)–Br⁻(aq), Cl⁻(aq)–HPO₄²⁻(aq), Br⁻(aq)–HPO₄²⁻(aq), phosphate–phosphate, and solvent-separated K⁺(aq)–Cl⁻(aq) or K⁺(aq)–Br⁻(aq) pairs; K⁺(aq)–K⁺(aq) contacts are unlikely, and triplet configurations such as K⁺(aq)–HPO₄²⁻(aq)–Cl⁻(aq) and K⁺(aq)–HPO₄²⁻(aq)–Br⁻(aq) may also occur. Because Cl⁻(aq) is a stronger structure breaker than Br⁻(aq), the {yKBr + (1 − y)K₂HPO₄}(aq) system exhibits weaker pair and triplet interactions, resulting in a lower absolute g₀. The g₂ parameter remains near zero, indicating negligible quadruplet effects.

From Figure 4d, it can be observed that for the {yK₂SO₄ + (1 − y)K₂HPO₄}(aq) system, the Friedman parameters g₀, g₁ and g₂ all exhibit negative values. Although close to zero, the negative values of g₀ suggest the formation of ion pairs between different species, specifically interactions of the type SO₄²⁻(aq)-HPO₄²⁻(aq), as well as K⁺(aq)-SO₄²⁻(aq) and K⁺(aq)-HPO₄²⁻(aq). The distinct minima observed in the g₀ and g₁ functions indicate a potential shift in the type and distribution of the formed ion pairs and triplets between different species, occurring at an ionic strength of approximately 1.3 mol·kg⁻¹. The g₂ values remain negative and close to zero up to an ionic strength of about 1.25 mol·kg⁻¹.

Figure 4e shows that, for the {yKH₂PO₄ + (1 − y)K₂HPO₄}(aq) system, the g₀ parameter increases from −0.065 to 0.005. This trend indicates dominance of ion-pair interactions of the type HPO₄²⁻(aq)–H₂PO₄⁻(aq). The shift from negative toward zero can also be attributed to the emergence of homocomplex dimers, specifically HPO₄²⁻(aq)–HPO₄²⁻(aq) and H₂PO₄⁻(aq)–H₂PO₄⁻(aq). The g₁ values are positive and remain close to zero. In contrast, g₂ values are negative and increase in absolute magnitude as ionic strength rises from 1 to 2 mol·kg⁻¹, suggesting that quadruplet interactions in this system become more pronounced than in any of the other investigated systems.

Figure 4f presents the {yNa₂HPO₄ + (1 − y)K₂HPO₄}(aq) system, where both g₀ and g₁ are positive. This indicates a predominance of interactions between like-charged ion pairs: Na⁺(aq)–K⁺(aq), Na⁺(aq)–Na⁺(aq), and K⁺(aq)–K⁺(aq). According to previous research, the probability of forming hydrophilic ion pairs such as Na⁺(aq)–Na⁺(aq) and K⁺(aq)–K⁺(aq), which experience mutual repulsion, is lower than the probability of phosphate ion dimerization and trimerization [22]. The influence of these pairs on deviations from ideality is reflected in the positive g₀ values. As ionic strength increases, changes in solution structure and interionic interactions occur. When g₀ reaches its minimum at I_m = 1.5 mol·kg⁻¹, the g₁ function begins to increase moderately, indicating that triplet interactions, most likely involving phosphate ions, become more significant. The g₂ parameter remains negative across the entire ionic strength range (0 to 2 mol·kg⁻¹), indicating the existence of quadruplet interactions, which are particularly significant above 1.2 mol·kg⁻¹.

4.2. Gibbs Free Energy Analysis

The excess Gibbs energy of a solution, which is influenced by various interactions, can change significantly in both magnitude and sign as the solution concentration varies. This behavior is reflected in the values of the Friedman parameters (g₀, g₁ and g₂). It is common for the curves showing the dependence of excess Gibbs energy on the ionic strength fraction to exhibit a maximum or minimum. These extrema may shift toward higher relative concentrations of one of the electrolytes as the total ionic strength of the solution increases, in accordance with Gurney’s theory. This shift indicates that the parameters g₁ and g₂ are nonzero and must be considered when analyzing interactions within the solution.

Figure 5a–f shows the functional dependence of the excess Gibbs energy for pairs of ions on the fraction of the ionic strength of component, y_A, where A is (KCl, KBr, KNO₃, K₂SO₄, KH₂PO₄) in the systems {yKCl + (1 − y)K₂HPO₄}(aq), {yKBr + (1 − y)K₂HPO₄}(aq), {yKNO₃ + (1 − y)K₂HPO₄}(aq), {yK₂SO₄ + (1 − y)K₂HPO₄}(aq) and {yKH₂PO₄ + (1 − y)K₂HPO₄}(aq), respectively, for total ionic strengths of the solution of I_m = 0.5; 1.0; and 1.5 mol·kg⁻¹ at a temperature of 298.15 K.

Figure 5a–f shows the excess Gibbs energy of mixing for ion pairs as a function of the ionic strength fraction y_A (from 0 to 1), where A represents KCl, KBr, KNO₃, K₂SO₄, KH₂PO₄, and Na₂HPO₄ in their respective systems with K₂HPO₄. The data are given at ionic strengths I_m = 0.5, 1.0, and 1.5 mol·kg⁻¹ and T = 298.15 K. For the systems {yKCl + (1 − y)K₂HPO₄}(aq), {yKBr + (1 − y)K₂HPO₄}(aq), {yKNO₃ + (1 − y)K₂HPO₄}(aq), {yK₂SO₄ + (1 − y)K₂HPO₄}(aq), and {yKH₂PO₄ + (1 − y)K₂HPO₄}(aq), the excess Gibbs energy values are negative. This indicates that ion-pair interactions between unlike ions of the same charge are dominant and more probable in these mixtures. The exception is the {yNa₂HPO₄ + (1 − y)K₂HPO₄}(aq) system, where the situation is reversed, in agreement with the conclusions from the previous chapter. In the five systems mentioned above in Figure 5a–e, the most stable ion pairs are formed at a total ionic strength fraction y_A = 0.5 (where A refers to KCl, KBr, KNO₃, K₂SO₄, or KH₂PO₄) at 298.15 K. The excess Gibbs energy of ion pairs as a function of the ionic strength fraction of KNO₃ is shown in Figure 5c and has the same shape as in the KCl and KBr systems. Based on the excess Gibbs energy values, it can be concluded that ion pairs in the nitrate system are more stable than those in the two previously considered systems. This is most evident when comparing the minimum values of $\left(∆G_m^{ex}/RT \right)$ pairs at y_A = 0.5: in the nitrate system, the value is −0.1, while in the KCl and KBr systems, the values are −0.065 and −0.04, respectively.

The nitrate ion, much like Cl⁻(aq) and Br⁻(aq), is hydrophilic and acts as a structure-breaker (destabilizer) of the solution structure. What distinguishes the {yKNO₃ + (1 − y)K₂HPO₄}(aq) system is the slight decrease in excess Gibbs energy as the KNO₃ fraction increases. This suggests that triplets, although present at low concentrations, may become more stable at higher concentrations, a trend opposite to that observed in the chloride and bromide systems. The {yKH₂PO₄ + (1 − y)K₂HPO₄}(aq) system, containing two different phosphate anions, is highly specific. Given the negative values of the g₀ parameter, dominance of HPO₄²⁻(aq)–H₂PO₄⁻(aq) pair interactions is expected. The HPO₄²⁻(aq) ion is a structure-maker (stabilizer), whereas H₂PO₄⁻(aq) is a borderline ion; thus, their interactions depend heavily on the local environment and the presence of cations. Mutual interactions between such ions typically lead to a decrease in the Gibbs energy of the pairs. From Figure 5e, we see that the excess Gibbs energy values for the pairs are negative in the system {yK₂SO₄ + (1 − y)K₂HPO₄}(aq), indicating that the dominant interactions are between different ions, SO₄²⁻(aq)-HPO₄²⁻(aq), K⁺(aq)-SO₄²⁻(aq) and K⁺(aq)-HPO₄²⁻(aq). In the system {yNa₂HPO₄ + (1 − y)K₂HPO₄}(aq) shown in Figure 5f, the repulsion between ion pairs of the type Na⁺(aq)-K⁺(aq), which are hydrophilic, gives a positive contribution to the Gibbs free energy of the pairs. This contribution shows a clear maximum that becomes more pronounced as the ionic strength of the solution increases from I_m = 0.5 mol·kg⁻¹ to I_m = 1.5 mol·kg⁻¹.

Figure 6a–f shows the functional dependence of the excess Gibbs energy for triplets of ions on the fraction of the ionic strength of component, y_A, where A is (KCl, KBr, KNO₃, K₂SO₄, KH₂PO₄) in the systems {yKCl + (1 − y)K₂HPO₄}(aq), {yKBr + (1 − y)K₂HPO₄}(aq), {yKNO₃ + (1 − y)K₂HPO₄}(aq), {yK₂SO₄ + (1 − y)K₂HPO₄}(aq) and {yKH₂PO₄ + (1 − y)K₂HPO₄}(aq), respectively, for total ionic strengths of the solution of I_m = 0.5; 1.0; and 1.5 mol·kg⁻¹ at a temperature of 298.15 K.

Figure 6a–d shows that the excess Gibbs energy of triplets is positive in the {yKCl + (1 − y)K₂HPO₄}(aq), {yKBr+ (1 − y)K₂HPO₄}(aq), {yKNO₃+ (1 − y)K₂HPO₄}(aq), and {yK₂SO₄ + (1 − y)K₂HPO₄}(aq) systems, indicating that ion-pair interactions are dominant. In contrast, in the {yNa₂HPO₄ + (1 − y)K₂HPO₄}(aq) system, the excess Gibbs energy of triplets is negative, suggesting that triplet interactions are the predominant form of association.

The {yNa₂HPO₄ + (1 − y)K₂HPO₄}(aq) system contains a common anion, the hydrogen phosphate ion, which is classified as a hydrophobic (structure-making) ion by Desnoyers [23]. The composition dependence of the triplet excess Gibbs energy shows a pronounced minimum that becomes increasingly prominent as the total ionic strength rises from I_m = 0.5 mol·kg⁻¹ to I_m = 1.5 mol·kg⁻¹ and higher. This indicates that triplet interactions are most significant at a Na₂HPO₄ ionic strength fraction of y_A = 0.5. The dominant triplet species are Na⁺(aq)–K⁺(aq)–HPO₄²⁻(aq), as well as triplets with two identical cations: Na⁺(aq)–Na⁺(aq)–HPO₄²⁻(aq) and K⁺(aq)–K⁺(aq)–HPO₄²⁻(aq). Among these, triplets involving more potassium ions are energetically more favorable, because the less hydrated K⁺(aq) can more readily access the interionic space of the triplet than the highly hydrated Na⁺(aq).

In the {yKH₂PO₄ + (1 − y)K₂HPO₄}(aq) system, the sign of the excess Gibbs energy function for triplets varies with composition. At ionic strengths of I_m = 1.0 and 1.5 mol·kg⁻¹, the function initially increases up to a KH₂PO₄ fraction of y_A = 0.2, crosses zero at y_A = 0.4, decreases until y_A = 0.8, and returns to zero at the end of the range. In contrast, at I_m = 0.5 mol·kg⁻¹, the triplet excess Gibbs energy remains near zero throughout. For KH₂PO₄ fractions from y_A = 0 to 0.4, the total triplet Gibbs energy values are positive at the higher ionic strengths 1.0 and 1.5 mol·kg⁻¹, indicating that electrostatic repulsion between the two phosphate anions outweighs the stabilization provided by hydrogen bonding.

Conversely, for y_A values between 0.4 and 1.0, the total triplet Gibbs energy values are negative. This results from chemical similarity and the emergence of specific attractive forces. The formation of strong intermolecular hydrogen bonds between H₂PO₄⁻ and HPO₄²⁻, with K⁺ acting as a bridge, facilitates this stabilization. The anions tend to cluster, reducing electrostatic repulsion. Although both anions are negatively charged, the presence of the H⁺ proton allows them to approach each other more closely than, for example, two sulfate ions. The potassium cation further neutralizes the local negative charge, making the K⁺(aq)–H₂PO₄⁻(aq)–HPO₄²⁻(aq) triplet energetically highly favorable.

Figure 7a–f shows the functional dependence of the excess Gibbs energy for quadruplets of ions on the fraction of the ionic strength of component, y_A, where A is: (KCl, KBr, KNO₃, K₂SO₄, KH₂PO₄) in the systems {yKCl + (1 − y)K₂HPO₄}(aq), {yKBr + (1 − y)K₂HPO₄}(aq), {yKNO₃ + (1 − y)K₂HPO₄}(aq), {yK₂SO₄ + (1 − y)K₂HPO₄}(aq) and {yKH₂PO₄ + (1 − y)K₂HPO₄}(aq), respectively, for total ionic strengths of the solution of I_m = 0.5; 1.0; and 1.5 mol·kg⁻¹ at a temperature of 298.15 K.

Figure 7a–f illustrates the dependence of the quadruplet excess Gibbs energy, which takes negative values for the {yKCl + (1 − y)K₂HPO₄}(aq), {yKBr + (1 − y)K₂HPO₄}(aq), {yKNO₃ + (1 − y)K₂HPO₄}(aq), and {yKH₂PO₄ + (1 − y)K₂HPO₄}(aq) systems. Negative values indicate that quadruplet formation lowers the free energy of the system. The dominant quadruplet combinations are K⁺(aq)–K⁺(aq)–Cl⁻(aq)–HPO₄²⁻(aq), K⁺(aq)–K⁺(aq)–Br⁻(aq)–HPO₄²⁻(aq), K⁺(aq)–K⁺(aq)–NO₃⁻(aq)–HPO₄²⁻(aq), and K⁺(aq)–K⁺(aq)–H₂PO₄⁻(aq)–HPO₄²⁻(aq), in which two potassium cations effectively bridge two different anions. In the phosphate buffer system H₂PO₄⁻(aq)-HPO₄²⁻(aq), this effect is further enhanced by hydrogen bonding, with the quadruplet forming a stable ionic structure that reduces the effective number of particles in the solution and thereby lowers activity coefficients. In the KCl, KBr, and KH₂PO₄ systems, the functions show a pronounced minimum at an ionic strength fraction of approximately y_A = 0.7, whereas for the KNO₃ system, this minimum occurs at y_A ≈ 0.5. In contrast, the {yNa₂HPO₄ + (1 − y)K₂HPO₄}(aq) system exhibits positive total quadruplet Gibbs energy values, indicating that these interactions are energetically unfavorable because this system involves the mixing of two different cations with a common anion. The positive value suggests that the presence of two different cations, Na⁺(aq) and K⁺(aq), within the same cluster around the phosphate anion creates a configurational mismatch. Because their hydration radii differ, these cations cannot optimally approach the HPO₄²⁻(aq) ion simultaneously, resulting in an increase in the total Gibbs energy. The {yK₂SO₄ + (1 − y)K₂HPO₄}(aq) system undergoes a sign change in the function, making it the most complex of the systems studied, as the interaction character depends on composition. At lower sulfate fractions, y_A < 0.4, the quadruplet Gibbs energy values are negative, indicating that the K⁺(aq)–K⁺(aq)–SO₄²⁻(aq)–HPO₄²⁻(aq) quadruplet interaction stabilizes the solution. This is likely due to the formation of mixed clusters in which potassium successfully neutralizes the high negative charge of both divalent anions. One must also consider the ambiguous classification of the sulfate ion; while some authors categorize it as hydrophilic, others classify it as hydrophobic. If sulfate ions behave as hydrophobic (structure-making) species, SO₄²⁻(aq)–SO₄²⁻(aq) interactions, together with HPO₄²⁻(aq)–HPO₄²⁻(aq) interactions, could significantly contribute to the negative quadruplet excess Gibbs energy values. However, at sulfate fractions y_A > 0.4, the quadruplet Gibbs energy values become positive. As the sulfate fraction increases, the ionic atmosphere becomes saturated, and strong electrostatic repulsion between the two divalent anions, SO₄²⁻ and HPO₄²⁻, dominates, which the potassium cations can no longer effectively screen within the quadruplet cluster. In summary, in systems with a common cation and monovalent anions, quadruplets act as stabilizing agents. In the presence of mixed cations, structural tension and positive energy arise. In systems featuring two divalent anions, SO₄²⁻(aq) and HPO₄²⁻(aq), the system balances between stabilization and strong electrostatic repulsion, leading to the observed sign reversal of the function.

4.3. Excess Gibbs Energy of Phosphate Solutions

In addition to analyzing the Friedman parameters and the excess Gibbs energy attributed to the formation of ion pairs, triplets, and quadruplets, it is essential to evaluate the total excess Gibbs energy of the solution as a function of the ionic strength fraction for various total ionic strengths of the phosphate systems. The total excess Gibbs energy values were calculated using the Scatchard model with six mixing parameters, as shown in Table 1. Identical values for the excess Gibbs energy of mixing are obtained when using Equation (24), which defines the Gibbs energy as the sum of individual contributions from pairs, triplets, quadruplets, and higher-order multiplets.

Figure 8a–f shows the functional dependence of the excess Gibbs energy on the ionic strength fraction y_A for different total ionic strengths I_m = (1.0; 2.0; 3.0) mol∙kg⁻¹ systems at T = 298.15 K for all the investigated systems.

Figure 8a–c shows that for the {yKCl + (1 − y)K₂HPO₄}(aq), {yKBr + (1 − y)K₂HPO₄}(aq), and {yKNO₃ + (1 − y)K₂HPO₄}(aq) systems, the excess Gibbs energy remains negative over the entire investigated range, indicating thermodynamic stability. In contrast, for the {yK₂SO₄ + (1 − y)K₂HPO₄}(aq), {yKH₂PO₄ + (1 − y)K₂HPO₄}(aq), and {yNa₂HPO₄ + (1 − y)K₂HPO₄}(aq) systems, the composition-dependent curves vary significantly. At total ionic strengths of I_m = 1.0, 2.0, and 3.0 mol·kg⁻¹, these curves are asymmetric, confirming the presence of higher-order interactions in solution. The total excess Gibbs energy for the first three systems shows a consistent profile as a function of the ionic strength fraction y_A. For the KCl and KBr systems, a minimum appears at an ionic strength fraction slightly below y_A ≈ 0.4 Figure 8a,b. This pronounced minimum corresponds to a balanced influence of the KCl/KBr and K₂HPO₄ electrolytes, representing approximately equal anion concentrations in solution. The nitrate system differs slightly, as its minimum is shifted toward a higher fraction of KNO₃, occurring at y_A ≈ 0.5 in Figure 8c. In all three systems, the total excess Gibbs energy remains negative across the entire composition range, and the minima become more pronounced as the total ionic strength increases from 1 to 3 mol·kg⁻¹. In these systems, interactions between different anions of the same charge dominate, specifically Cl⁻(aq)–HPO₄²⁻(aq), Br⁻(aq)–HPO₄²⁻(aq), and NO₃⁻(aq)–HPO₄²⁻(aq). Ions such as Cl⁻(aq), Br⁻(aq), and particularly NO₃⁻(aq) are structurally weakly hydrated, whereas HPO₄²⁻(aq) is highly charged and strongly hydrated. When these ions coexist, partial release of water molecules from their hydration spheres occurs, leading to the formation of stable dimers that lower the free energy of the system. Furthermore, stabilization through solvent-separated ion pairs of opposite charge, such as K⁺(aq)–Cl⁻(aq) or K⁺(aq)–Br⁻(aq), is also possible.

The structural changes and specific interactions influencing the total excess Gibbs energy in the {yK₂SO₄ + (1 − y)K₂HPO₄}(aq) system are uniquely distinctive because of the coexistence of both phosphate and sulfate ions, which have high charges and compete for water molecules. As shown in Figure 8d, at ionic strengths of I_m = 1.0 and 2.0 mol·kg⁻¹, the total excess Gibbs energy initially increases up to a sulfate fraction of y_A ≈ 0.1. It then decreases, passes through zero at y_A ≈ 0.4, continues to decline until it reaches a minimum at y_A ≈ 0.8, and finally returns to zero at y_A = 1.0. When the sulfate fraction is low, y_A < 0.3, the total excess Gibbs energy is positive, indicating that the hydration of HPO₄²⁻(aq) ions dominates. However, when y_A > 0.3, the increasing sulfate concentration disrupts the solution structure, causing the curve to enter the negative region. The resulting negative excess Gibbs energy values indicate that attractive interactions between oppositely charged hydrophilic ion pairs, such as K⁺(aq)–SO₄²⁻(aq), or like-charged hydrophilic anionic pairs, such as HPO₄²⁻(aq)–HPO₄²⁻(aq), begin to dominate. Notably, at an ionic strength of I_m = 3.0 mol·kg⁻¹, the function exhibits the opposite sign behavior. Up to a sulfate fraction of y_A ≈ 0.1, the function decreases to a total excess Gibbs energy value of $\left(∆G_m^{ex}/RT \right)$ ≈ −0.015. At this low sulfate fraction, the system stabilizes through the formation of mixed triplets, namely K⁺(aq)–SO₄²⁻(aq)–HPO₄²⁻(aq), which lowers the Gibbs energy into the negative domain. Beyond this point, the total Gibbs energy values increase; the function passes through zero at y_A ≈ 0.25 and continues to rise until it reaches a maximum of $\left(∆G_m^{ex}/RT \right)$≈ 0.005 at y_A ≈ 0.7. With further addition of sulfate, y_A > 0.25, intense competition for the remaining water molecules ensues between SO₄²⁻(aq) and HPO₄²⁻(aq). Because there is insufficient water to sustain their individual hydration spheres, the system responds by forming large ionic clusters—quadruplets, as confirmed by the negative values of the g₂ parameter, as well as triplets of the SO₄²⁻(aq)–HPO₄²⁻(aq)–HPO₄²⁻(aq) type. The positive sign of the $\left(∆G_m^{ex}/RT \right)$ function in this region demonstrates that the mixing of these two ions under water-deficient conditions is thermodynamically highly unfavorable, causing the function to rise to positive values with a maximum at y_A ≈ 0.7. From this peak, the function decreases again, returning to zero at y_A = 1.0. The change in the sign of the function when transitioning from I_m = 2.0 to 3.0 mol·kg⁻¹ provides crucial evidence of a shift in the interaction mechanism. At I_m = 3.0 mol·kg⁻¹, there is virtually no free water left in the system, as all water molecules are fully engaged within the hydration spheres of the ions.

The existence of phosphate dimers in aqueous solutions remains a subject of debate in the literature. Certain electromotive force (EMF) measurements have failed to detect dimeric phosphate species, whereas other EMF studies confirm their presence [24,25,26,27,28]. Additional thermodynamic support stems from the work of Childs et al. [29,30], who observed unusually high excess Gibbs energies of mixing for Na⁺, K⁺, Cl⁻, H₂PO₄⁻ salt combinations at 25 °C, pointing toward dimer formation. Similarly, Wood employed the isopiestic method on the NaH₂PO₄ and NaClO₄ system to establish that isopiestic data can effectively quantify dimerization between ions of the same charge sign [31]. Figure 8e, which shows the composition dependence of the total excess Gibbs energy for the {yKH₂PO₄ + (1 − y)K₂HPO₄}(aq) system, clearly demonstrates that the behavior of the $\left(∆G_m^{ex}/RT \right)$function changes significantly with increasing total ionic strength, I_m. At an ionic strength of I_m = 1.0 mol·kg⁻¹, the total excess Gibbs energy is negative. In this moderately dilute solution, interactions between H₂PO₄⁻(aq) and HPO₄²⁻(aq) ions predominate. Since H₂PO₄⁻(aq) is a borderline ion, it exhibits hydrophobic behavior in the presence of the more strongly hydrated HPO₄²⁻(aq) ion, thereby stabilizing the solution structure. This results in attractive interactions and the formation of phosphate dimers and mixed triplets, which contribute negatively to the excess energy, which is consistent with Childs’ claims. For ionic strengths of I_m = 2.0 and 3.0 mol·kg⁻¹, the total excess Gibbs energy becomes positive, and the functional dependence of $\left(∆G_m^{ex}/RT \right)$on the ionic strength fraction shows a maximum at y_A ~ 0.6. As concentration increases, the situation changes markedly as the H₂PO₄⁻(aq) ion alters its character under the influence of the surrounding environment. Strong repulsive interactions between like-charged ions, K⁺(aq)-K⁺(aq), and specific hydrophilic–hydrophobic interactions between HPO₄^2-(aq)–K⁺(aq) begin to dominate. Notably, at I_m = 3.0 mol·kg⁻¹ the values of the g₂ parameter are distinctly negative, indicating that the system is dominated by quadruplet interactions involving two cations and two anions. The formation of these complex associates consumes the remaining free water and induces thermodynamic strain within the mixture, causing the total excess Gibbs energy to reach high positive values.

Figure 8f shows the dependence of the total excess Gibbs energy on the fraction of the system {yNa₂HPO₄+(1 − y)K₂HPO₄}(aq). A pronounced maximum is observed for the highest ionic strength of I_m = 3.0 mol·kg⁻¹ at y_A ≈ 0.5. The total excess Gibbs energy values remain positive across the entire composition range from y_A = 0 to 1. The Na⁺(aq) and K⁺(aq) ions possess different ionic radii and charge densities, which translates to differing hydration capacities. The mixing of two cations with disparate hydration properties disrupts the local structure of water. Because the ions preferentially remain within their native hydration environments rather than undergoing mixing, a repulsive effect occurs in terms of free energy, resulting in positive $\left(∆G_m^{ex}/RT \right)$> 0 values. With an increase in ionic strength, this effect is drastically enhanced due to the severe depletion of free water molecules.

In the {yKCl + (1 − y)K₂HPO₄}(aq), {yKBr + (1 − y)K₂HPO₄}(aq), and {yKNO₃ + (1 − y)K₂HPO₄}(aq) systems, the negative values of the total Gibbs energy within the components’ fraction range y_A from 0 to 1 for KCl, KBr, and KNO₃, which indicates that the mixing of KCl, KBr, and KNO₃ salts with K₂HPO₄ in water is a spontaneous and thermodynamically favorable process. In the {yK₂SO₄ + (1 − y)K₂HPO₄}(aq) system, phosphate and sulfate ions are present, both bearing high charges and competing for water molecules. The negative values of the excess Gibbs energy indicate the dominance of attractive interactions. The positive sign of the function indicates that the mixing of H₂PO₄⁻(aq) and SO₄²⁻(aq) ions under water-deficient conditions is thermodynamically highly unfavorable. The change in the sign of the function at ionic strengths of I_m = 2.0 and I_m = 3.0 mol·kg⁻¹ provides key evidence of a shift in the interaction mechanism. In the {yKH₂PO₄ + (1 − y)K₂HPO₄}(aq) system, at lower ionic strengths, the total Gibbs energy values are negative, dominated by interactions between H₂PO₄⁻(aq) and HPO₄²⁻(aq) ions, which favorably influence the stability of the system. At higher ionic strengths of I_m = 2.0 and I_m = 3.0 mol·kg⁻¹, the total excess Gibbs energy becomes positive so that the system is dominated by quadruplet interactions.

5. Conclusions

In the {yKCl + (1 − y)K₂HPO₄}(aq), {yKBr + (1 − y)K₂HPO₄}(aq) and {yKNO₃ + (1 − y)K₂HPO₄}(aq) systems, the Friedman parameters g₀ and g₂ are negative, while the g₁ parameter remains positive throughout the entire ionic strength fraction range y_A from 0 to 1. These systems are dominated by attractive interactions between different anions of the same charge, specifically Cl⁻(aq)-HPO₄²⁻(aq), Br⁻(aq)-HPO₄²⁻(aq) and NO₃⁻–HPO₄²⁻, respectively.

In the chloride and bromide mixtures, the weakly hydrated Cl⁻(aq) and Br⁻(aq) ions and the strongly hydrated HPO₄²⁻(aq) ion undergo partial dehydration, releasing water molecules from their hydration spheres to form stable dimers. The negative values of both g₂ and the quadruplet contributions confirm the formation of stable mixed clusters, namely [K₂Cl(HPO₄)]⁻, [K₂Br(HPO₄)]⁻ and [K₂NO₃(HPO₄)]⁻. In all three systems, the influence of triplets is negligible, and the total excess Gibbs energy function shows a pronounced minimum across the entire y_A range. In the {yKNO₃ + (1 − y)K₂HPO₄}(aq) system, the partial dehydration effect and dimer stabilization are even more pronounced; the minimum of the function reaches approximately −0.18, in contrast to the chloride and bromide systems where the minima occur at approximately −0.14 and −0.12, respectively. This is attributed to the fact that the nitrate ion acts as a powerful structure-breaking chaotrope in water. Furthermore, in the nitrate-containing system, the minimum is shifted towards a higher nitrate fraction y_A ≈ 0.5 compared to the chloride and bromide systems y_A ≈ 0.4, indicating stronger specific interactions between the nitrate and phosphate anions.

In the {yKH₂PO₄ + (1 − y)K₂HPO₄}(aq) system, the Friedman parameter g₀ is negative, while g₁ and the total excess Gibbs energy change sign depending on the solution composition y_A and the total ionic strength I_m. At I_m = 1.0 mol·kg⁻¹, the total excess Gibbs energy is strictly negative, driven by the dominant attractive forces between H₂PO₄⁻(aq) and HPO₄²⁻(aq) ions. Since H₂PO₄⁻(aq) acts as a borderline ion, it exhibits hydrophobic characteristics in the presence of the more strongly hydrated HPO₄²⁻(aq), thereby stabilizing the solution structure through phosphate dimer formation.

Conversely, at I_m = 2.0 and 3.0 mol·kg⁻¹, the total excess Gibbs energy becomes positive, displaying a maximum at y_A ~ 0.6. Under these higher concentrations, the H₂PO₄⁻(aq) ion changes its solvation character, allowing strong repulsive interactions between like-charged cations K⁺(aq)–K⁺(aq) and specific hydrophilic–hydrophobic interactions, HPO₄²⁻(aq)–K⁺(aq), to prevail. The triplet excess Gibbs energy contribution changes sign at y_A ≈ 0.4; for y_A < 0.4, electrostatic repulsion between the phosphate anions dominates, whereas for y_A > 0.4 an energetically favorable mixed triplet, K⁺(aq)–H₂PO₄⁻(aq)-HPO₄²⁻(aq), is formed. This triplet is stabilized by strong intermolecular hydrogen bonds, with the K⁺(aq) cation acting as an electrostatic bridge to neutralize the local negative charge density. The distinctly negative values of the quadruplet excess Gibbs energy at I_m = 1.5 mol·kg⁻¹ indicate the dominance of [K₂ (H₂PO₄)(HPO₄)]⁻ clusters. The extensive formation of these higher-order associates heavily consumes the remaining free water molecules and induces severe thermodynamic strain within the mixture, macroscopically driving the total excess Gibbs energy into high positive values.

In the [K₂(H₂PO₄)(HPO₄)] system, the g₀ parameter values are close to zero, indicating that pairwise interactions between different ions are remarkably weak. Instead, the quadruplet excess Gibbs energy function changes sign as a function of both composition and total ionic strength. At a lower ionic strength of I_m = 1.0 mol·kg⁻¹, the ions behave according to their primary hydration spheres, so pairs and triplets govern the deviations from ideality. At the higher ionic strength of I_m = 3.0 mol·kg⁻¹, the acute deficiency of free water molecules forces the system to rearrange into higher-order clusters, fundamentally altering the underlying interaction mechanism through quadruplet formation. At lower sulfate fractions of y_A < 0.4, the mixed quadruplets [K₂(SO₄)(HPO₄)]²⁻ are highly stable because the potassium ions successfully neutralize the negative charge of both divalent anions. At higher sulfate fractions of y_A > 0.4, the ionic atmosphere becomes saturated, triggering strong electrostatic repulsion between the SO₄²⁻(aq) and HPO₄²⁻(aq) anions, which shifts the excess function into the positive domain. A negative contribution to the quadruplet energy may partially arise from the dual (hydrophobic/hydrophilic) nature of the sulfate ion via specific SO₄²⁻(aq)-SO₄²⁻(aq) interactions.

In the {yNa₂HPO₄ + (1 − y)K₂HPO₄}(aq) system, the positive Friedman parameter g₀ values indicate the clear dominance of interactions between identical ion pairs. Meanwhile, the g₁ parameter changes sign with increasing ionic strength, revealing the emergence of like-charged triplets at elevated concentrations. The total excess Gibbs energy values remain strictly positive $\left(∆G_m^{ex}/RT \right)$ across the entire composition range for all three investigated ionic strengths, featuring a sharp maximum at y_A ~ 0.5 for I_m = 3.0 mol·kg⁻¹. This common-anion system is structurally dominated by interactions between like-ion pairs: Na⁺(aq)-Na⁺(aq), K⁺(aq)-K⁺(aq), and HPO₄²⁻(aq)-HPO₄²⁻(aq). Because Na⁺(aq) and K⁺(aq) have distinct ionic radii and charge densities, they exhibit vastly different hydration capacities. Mixing these two cations severely disrupts the local hydrogen-bonding network of water. Since the cations preferentially remain within their native hydration environments rather than mixing, a strong free-energy repulsion effect occurs, driving the excess Gibbs energy into the positive domain. This structural rejection is drastically enhanced at high I_m due to the depletion of free water molecules. A notable exception is found in the triplet excess Gibbs energy, which uniquely exhibits negative values, thereby favoring the stabilization of mixed Na⁺(aq)-K⁺(aq)-HPO₄²⁻(aq) triplets. Conversely, the quadruplet contribution remains positive due to configurational mismatch; because of the disparate hydration radii, Na⁺(aq) and K⁺(aq) cannot closely approach the HPO₄²⁻(aq)⁻ anion simultaneously.