Mean Activity Coefficients of KCl in KCl + K₂B₄O₇ + H₂O Ternary System at 298.15 K Determined by Potential Difference Measurements

Abstract

The mean activity coefficients of KCl in the KCl + K₂B₄O₇ + H₂O ternary systems were determined at total ionic strength ranging from (0.0100 to 1.0000) mol·kg⁻¹ at 298.15 K by potential difference measurements from the electrochemical cell without liquid junction: ${\text{K-ISE}}\left| {{\text{KCl }}\left( {m_{ 1} } \right),{\text{K}}_{ 2} {\text{B}}_{ 4} {\text{O}}_{ 7} \left( {m_{ 2} } \right)} \right|{\text{ Cl-ISE}}$. The mean activity coefficients of KCl were measured at different ionic strength fractions, $y_{\text{B}}$, of K₂B₄O₇ (with $y_{\text{B}} = 0. 8, \, 0. 6, \, 0. 4, \, 0. 2,{\text{ and }}0. 0$). The experimental results showed that K-ISE and Cl-ISE in this work had good Nernstian responses, and the mean activity coefficients of KCl in KCl + K₂B₄O₇ + H₂O mixtures were calculated using the Nernst equation. Mixing interaction parameters of $\theta_{{{\text{Cl}}^{ - } {\cdot} {\text{B}}_{4} {\text{O}}_{5} ({\text{OH}})_{4}^{2 - } }}$ and $\psi_{{{\text{K}}^{ + } {\cdot} {\text{Cl}}^{ - } {\cdot} {\text{B}}_{4} {\text{O}}_{5} ({\text{OH}})_{4}^{2 - } }}$ in Pitzer’s equation were evaluated from the present measurements of the mean activity coefficients of KCl. In addition, the osmotic coefficients, water activity, and the excess Gibbs energy of this system were calculated by Pitzer’s equations.

1 Introduction

Electrolyte solutions commonly found in nature are a medium for many organic and inorganic reactions and are widely used in desalination, brine development, and hydrometallurgy. Thus, there is considerable interest in the thermodynamic properties of electrolyte solutions [1–3]. The activity coefficients of electrolytes are important in thermodynamics, since they reflect the extent of deviations between the real solution and an ideal solution [4–6].

There are many methods to measure activity coefficients in electrolyte solutions, including: gas–liquid chromatography, kinetics, dilute solution colligative method, conductivity, solubility, freezing point depression, equal pressure method, and potential difference measurements [7, 8]. The emergence of ion–selective electrodes is undoubtedly an important advance in potential analysis. These have a wide range of linear response, high sensitivity, good selectivity, high analysis speed and measurements are easy to automate, providing continuous measurement and control, so they have caught people’s attention [9–11].

Sodium, potassium, magnesium, chloride, bromine and boron resources are widely distributed in seawater, underground brines and salt lake brines [12]. Therefore, the study of the thermodynamic properties of salt–water systems is necessary. Research reports about activity coefficients calculated using the Pitzer equation are increasing common; for example, Millero et al. [13, 14] measured and calculated the solubilities of oxygen in aqueous solutions of KCl, K₂SO₄, and CaCl₂ as a function of concentration and temperature, and determined the dissociation of TRIS in NaCl solutions using the Pitzer equations. Roy et al. [15–17] studied the activity coefficients of HCl + GdCl₃ + H₂O system from 278 to 328 K and the thermodynamics of the system HCl + SmCl₃ + H₂O with the application of Harned’s rule and the Pitzer equations, and the thermodynamics of the HBr + NiBr₂ + H₂O system from 278 to 328 K were determined. Sirbu et al. [18], Galleguilos et al. [19, 20] and White et al. [21] studied the activity coefficients of NaCl + Na₂SO₄ + H₂O, KI + KNO₃ + H₂O, and NaCl + Na₂CO₃ + H₂O. Li et al. [22] determined the mean activity coefficients of NaCl and KCl in the NaCl + KCl + H₂O system at 308.15 K.

In our previous work we studied mean activity coefficients of KBr in the KBr + K₂B₄O₇ + H₂O and KBr + K₂SO₄ + H₂O ternary system [23, 24] and NaBr in the NaBr + Na₂B₄O₇ + H₂O system [25] at 298.15 K by potential difference measurements; we also studied multi–temperature phase diagrams in a series of sub–system of the NaCl + NaBr + Na₂SO₄ + Na₂B₄O₇ + KCl + KBr + K₂SO₄ + K₂B₄O₇ + H₂O system for salt–water system, that is, NaCl + Na₂SO₄ + Na₂B₄O₇ + KCl + K₂SO₄ + K₂B₄O₇ + H₂O at 323 K and 298 K [26, 27]; Na₂B₄O₇ + Na₂SO₄ + NaCl + H₂O at 323 K [28]; NaBr + Na₂SO₄ + KBr + K₂SO₄ + H₂O at 323 K [29]; and NaBr + Na₂SO₄ + H₂O at 323 K [30]. Pitzer’s equations can be used to calculate the thermodynamic properties for brines, so we determined the thermodynamic properties of the salt–water system to predict the thermodynamic equilibrium for brine resources.

So far, mean activity coefficients of KCl in the KCl + K₂B₄O₇ + H₂O ternary system at 308.15 K by potential difference measurements have been reported by our group [31], but no report has been found on thermodynamic properties of the KCl + K₂B₄O₇ + H₂O ternary system at 298.15 K. Therefore, in this paper, the activity coefficients of KCl in KCl + K₂B₄O₇ + H₂O ternary system were determined by potential difference measurement at 298.15 K and in the range 0.0100 to 1.0000 mol·kg⁻¹ total ionic strength, and the Pitzer’s ion interaction parameters $\theta_{{{\text{Cl}}^{ - } {\cdot}{\text{B}}_{4} {\text{O}}_{5} ({\text{OH}})_{4}^{2 - } }}$ and $\varphi_{{{\text{K}}^{ + } {\cdot} {\text{Cl}}^{ - } {\cdot} {\text{B}}_{4} {\text{O}}_{5} ({\text{OH}})_{4}^{2 - } }}$ were evaluated from the activity coefficients of KCl in the KCl + K₂B₄O₇ + H₂O ternary system. Then the osmotic coefficients, water activity, and excess Gibbs energies of this system were calculated.

2 Experimental Section

The water for experiments was deionized water, with a conductivity less than 1 × 10⁻⁴ S·m⁻¹. Prior to use, the G.R. grade K₂B₄O₇·4H₂O (mass fraction % >99.5) from Tianjin Kemiou Chemical Reagent Co., Ltd. and KCl salts (mass fraction % >99.5) from Tianjin Guangfu Fine Chemical Research Institute, were placed in an oven under 393 K for 2 h.

Experimental apparatus were as follows: AL104 electronic balance (Mettler–Toledo Group, the smallest error value 0.0001 g); Pxsj-216 ion meter (Leici Precision Scientific Instrument Co., Ltd., accuracy ±0.1 mV); Bilon-HW-05 thermostatic circulating water bath (Beijing Bi-Lang Co., Ltd., accuracy ±0.1 K); JB-1 stirrer (Leici Precision Scientific Instrument Co., Ltd., with automatic speed adjustment); 232-01 calomel reference electrode (Leici Precision Scientific Instrument Co., Ltd.); 401 potassium ion selective electrode (Jiangsu Jiangfen Electroanalytical Instrument Co., Ltd.); PCl-1-01 chloride ion selective electrode (Leici Precision Scientific Instrument Co., Ltd.); 50 mL flasks and other conventional laboratory glass.

3 Method

The K-ISE was soaked for 30 min in 10⁻² mol·L⁻¹ KCl aqueous solution and washed with deionized water to a blank potential around −160 mV. The Cl-ISE was soaked activation for 2 h in 10⁻³ mol·L⁻¹ NaCl aqueous solution. The reference electrode was a double–junction saturated calomel electrode with the salt bridge filled with G.R. grade saturated solution of potassium chloride and the outer salt bridge filled with 0.1 mol·L⁻¹ lithium acetate solution. During the measurements, we used the same ion-selective electrodes. First, the potential difference of each single salt was determined so the electrode response slope of each electrode and the electrode constant were obtained. The salt composition with KCl single cell without liquid junction was:

$$ {\text{K-ISE}}\left| {{\text{KCl}}\left( {m_{0} } \right)} \right|{\text{Cl-ISE}} $$

(a)

whose potential value was:

$$ E_{\text{a}} = E^{{0{\prime }}} + k{ \ln }a_{ + } a_{ - } = E^{{0{\prime }}} + k{ \ln }a_{{0 \pm {\text{KCl}}}} = E^{{0{\prime }}} + 2k{ \ln }m_{0} \gamma_{{0 \pm {\text{KCl}}}} $$

(1)

where $a_{ + }$, $a_{ - }$, $a_{{0 \pm {\text{KCl}}}}$, m ₀ and $\gamma_{{0 \pm {\text{KCl}}}}$, respectively, represent a single positive ion activity, negative ion activity, mean activity, molality, and activity coefficient, E _a indicates the galvanic potential difference, $E^{{0{\prime }}}$ indicates the standard potential, and k indicates the electrode response slope.

For the mixed salt, the cell without liquid junction was:

$$ {\text{K-ISE}}\left| {{\text{KCl }}\left( {m_{1} } \right),{\text{ K}}_{2} {\text{B}}_{4} {\text{O}}_{7} \left( {m_{2} } \right),{\text{ H}}_{2} {\text{O}}} \right|{\text{Cl-ISE}} $$

(b)

whose potential value is:

$$ E_{\text{b}} = E^{{0{\prime }}} + k{ \ln }a_{{{\text{K}}^{ + } }} {\cdot} a_{{{\text{Cl}}^{ - } }} = E^{{0{\prime }}} + k{ \ln }m_{1} \left( {m_{1} + 2m_{2} } \right)\gamma_{{ \pm {\text{KCl}}}}^{2} $$

(2)

where m ₁ and m ₂, respectively, represent the molality of KCl and K₂B₄O₇ in mixed solution, $a_{{{\text{K}}^{ + } }}$ and $a_{{{\text{Cl}}^{ - } }}$, respectively, represent the activity of $\text K^{ + }$ and $\text Cl^{ - }$, $\gamma_{{ \pm {\text{KCl}}}}$ expresses the mean activity coefficient of KCl.

The method and procedure is as follows: the total ionic strength I (I = m ₁ + 3m ₂) ranges from (0.0100 to 1.0000) mol·kg⁻¹; ionic strength fractions $y_{\text{B}},$ $y_{\text{B}} = 3m_{2} /(m_{1} + 3m_{2} )$ of K₂B₄O₇ are 0.8, 0.6, 0.4, 0.2, and 0.0. The appropriate masses of KCl and K₂B₄O₇ were weighed into a beaker and dissolved in 30 mL of deionized water, with stirring, before being placed into the thermostated cell.

Before determining the activity coefficients in the mixture, the potential difference of cell (a) was measured so as to determine the standard potential difference $E^{{0{\prime }}}$ and practical response slope k. Here k = RT/F represents the theoretical Nernst slope. The R, F, and T are the gas constant, Faraday constant, and absolute temperature, respectively. Then the potential difference of cell (b) with different ionic strengths was measured, varying the concentration from low to high.

During measurements, the experimental solution was kept at 298.15 ± 0.1 K until the potential difference was stable to ±0.1 mV for 30 min.

For ionic activity coefficients the corresponding relations are:

$$ \left( {\gamma_{{ \pm {\text{KCl}}}} } \right)^{2} = \gamma_{{{\text{K}}^{ + } }} {\cdot} \gamma_{{{\text{Cl}}^{ - } }} $$

(3)

$$ \left\{ {\gamma_{{ \pm {\text{K}}_{2} {\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} } \right\}^{3} = \left( {\gamma_{{{\text{K}}^{ + } }} } \right)^{2} {\cdot} \gamma_{{{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4}^{2 - } }} $$

(4)

$$ \begin{aligned} { \ln }\gamma_{{{\text{K}}^{ + } }} & = F + m_{\text{Cl}} \left( {2B_{{{\text{K}},{\text{Cl}}}} + ZC_{{{\text{K}},{\text{Cl}}}} } \right) + m_{{{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} \left( {2B_{{{\text{K}},{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} + ZC_{{{\text{K}},{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} } \right) \\ & \quad + \;m_{\text{K}} m_{\text{Cl}} C_{{{\text{K}},{\text{Cl}}}} + m_{\text{K}} m_{{{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} C_{{{\text{K}},{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} \\ & \quad + \;m_{\text{Cl}} m_{{{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} \psi_{{K,{\text{Cl}},{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} \\ \end{aligned} $$

(5)

$$ \begin{aligned} { \ln }\gamma_{{{\text{Cl}}^{ - } }} & = F + m_{\text{K}} \left( {2B_{{{\text{K}},{\text{Cl}}}} + ZC_{{{\text{K}},{\text{Cl}}}} } \right) + m_{\text{K}} m_{\text{Cl}} C_{{{\text{K}},{\text{Cl}}}} + m_{\text{K}} m_{{{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} C_{{{\text{K}},{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} \\ & \quad + \;m_{{{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} \left( {2\Phi_{{{\text{Cl}},{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} + m_{\text{K}} \psi_{{{\text{K}},{\text{Cl}},{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} } \right) \\ \end{aligned} $$

(6)

and

$$ { \ln }\gamma_{{{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4}^{2 - } }} = 4F + m_{\text{K}} \left( {B_{{{\text{K}},{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} + ZC_{{{\text{K}},{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} } \right) + 2m_{\text{K}} m_{\text{Cl}} C_{{{\text{K}},{\text{Cl}}}} + \;2m_{\text{K}} m_{{{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} C_{{{\text{K}},{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} + \;m_{\text{Cl}} \left( {2\Phi_{{{\text{Cl}},{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} + m_{\text{K}} \psi_{{{\text{K}},{\text{Cl}},{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} } \right) $$

(7)

where,

$$ \begin{aligned} F & = - A^{\phi} \left[ {\frac{{I^{1/2} }}{{\left( {1 + bI^{1/2} } \right)}} + \left( {2/b} \right)ln\left( {1 + bI^{1/2} } \right)} \right] + m_{\text{K}} m_{\text{Cl}} B^{\prime }_{{{\text{K}},{\text{Cl}}}} \\ & \quad + \;m_{\text{K}} m_{{{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} B^{\prime }_{{{\text{K}},{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} \\ & \quad + \;m_{\text{Cl}} m_{{{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} \Phi^ {\prime }_{{{\text{Cl}},{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} \\ \end{aligned} $$

(8)

where I is the ionic strength, the constants b = 1.2 ${\text{mol}}^{ - 1/2} {\cdot} {\text{kg}}^{1/2}$, and $A^{\phi } = 0.391475\;{\text{mol}}^{ - 1/2} {\cdot} {\text{kg}}^{1/2}$ is the value of the Debye–Hückel limiting–law slope for an aqueous solution at T = 298.15 K [32, 33]. Values of the Pitzer parameters $\beta_{{{\text{M}},{\text{X}}}}^{\left( 0 \right)}$, $\beta_{{{\text{M}},{\text{X}}}}^{\left( 1 \right)}$ and $C_{\text{MX}}^{\phi }$ for KCl and ${\text{K}}_{2} {\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4}$ at 298.15 K are from references [34, 35].

For the {(1 − y _B) KCl + y _B K₂B₄O₇} (aq) system, the osmotic coefficient equation is:

$$ \phi = 1 + \left( {\frac{2}{{m_{\text{K}} + m_{\text{Cl}} + m_{{{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} }}} \right)\left[ {\left( {\frac{{ - A^{\phi } I^{3/2} }}{{1 + bI^{1/2} }}} \right) + \,m_{\text{K}} m_{\text{Cl}} \left( {B_{{{\text{K}},{\text{Cl}}}}^{\phi } + ZC_{{{\text{K}},{\text{Cl}}}} } \right) +\, m_{\text{K}} m_{{{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} \left( {B_{{{\text{K}},{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }}^{\phi } + ZC_{{{\text{K}},{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} } \right) + m_{\text{Cl}} m_{{{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} \left\{ {\Phi_{{{\text{Cl}},{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }}^{\phi } + m_{\text{K}} \psi_{{{\text{K}},{\text{Cl}},{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} } \right\}} \right] $$

(9)

where Z is given by: $Z = m_{\text{K}} + m_{\text{Cl}} + 2m_{{{\text{B}}_{4} {\text{O}}_{7} }} ;{\text{and }}m_{\text{K}} = m_{1} + 2m_{2} ; m_{\text{Cl}} = m_{1} ; m_{{{\text{B}}_{4} {\text{O}}_{7} }} = m_{2} .$

The quantities $B_{{{\text{M}},{\text{X}}}}^{\phi }$; $B_{{{\text{M}},{\text{X}}}}$ $C_{\text{MX}} ; {\text{and B}}^{\prime }_{{{\text{M}},{\text{X}}}}$ are defined with the following dependences on ionic strength:

$$ B_{{{\text{M}},{\text{X}}}}^{\phi } = \beta_{{{\text{M}},{\text{X}}}}^{\left( 0 \right)} + \beta_{{{\text{M}},{\text{X}}}}^{\left( 1 \right)} { \exp }\left( { - \alpha I^{1/2} } \right) $$

(10)

$$ B_{{{\text{M}},{\text{X}}}} = \beta_{{{\text{M}},{\text{X}}}}^{\left( 0 \right)} + 2\beta_{{{\text{M}},{\text{X}}}}^{\left( 1 \right)} \left\{ {\frac{{\left[ {1 - \left( {1 + \alpha I^{1/2} } \right){ \exp }\left( { - \alpha I^{1/2} } \right)} \right]}}{{\alpha^{2} I}}} \right\} $$

(11)

$$ C_{\text{MX}} = \frac{{C_{\text{MX}}^{\phi } }}{{\left( {2\left| {Z_{\text{M}} Z_{\text{X}} } \right|^{1/2} } \right)}} $$

(12)

$$ B_{{{\text{M}},{\text{X}}}}^{ '} = \frac{{\beta_{{{\text{M}},{\text{X}}}}^{\left( 1 \right)} \left\{ { - \frac{{2\left[ {1 - \left( {1 + \alpha I^{{\frac{1}{2}}} + \frac{{\alpha^{2} I}}{2}} \right){ \exp }\left( { - \alpha I^{{\frac{1}{2}}} } \right)} \right]}}{{\alpha^{2} I}}} \right\}}}{I} $$

(13)

where M denotes K⁺ and X denotes Cl⁻ or $\text B_{4} \text O_{5} \left( \text {OH} \right)_{4}^{2 - }$; $Z_{\text{M}}$ and $Z_{\text{X}}$ are the valences of ions M and X. These mixing functions are related to the mixing parameters by:

$$ \Phi_{{{\text{Cl}},{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }}^{\phi } = \theta_{{{\text{Cl}},{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} + {^{\rm E}\theta_{{\text{Cl},{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }}} + I^{\text{E}} \theta^{\prime }_{{{\text{Cl}},{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} $$

(14)

$$ \Phi_{{{\text{Cl}},{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} = \theta_{{{\text{Cl}},{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} +\; {^{\text{E}}{\theta_{{{\text{Cl}},{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }}}} $$

(15)

and

$$ \Phi^{\prime }_{{{\text{Cl}},{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }}\;=\;{^{\text{E}}{\theta^{\prime }_{{{\text{Cl}},{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4}} }}} $$

(16)

where the superscript E applied to a function identifies it as an electrostatic contribution that does not depend on a specific characteristic of the pair of ions (other than their charges). The quantity $^{\text{E}} \theta_{{{\text{Cl}},{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }}$ and its ionic strength derivative $^{\text{E}} \theta^{\prime }_{{{\text{Cl}},{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }}$ can be calculated, and their values depend only the total ionic strength I and the valences of the ions of like sign, in this case $Z_{\text{Cl}}$ and $Z_{{{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }}$. The equations are the following forms:

$$ ^{\text{E}} \theta_{{{\text{Cl}},{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} = \left( {\frac{{Z_{\text{Cl}} Z_{{{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} }}{4I}} \right)\left[ {{\text{J}}\left( {\chi_{{{\text{Cl}},{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} } \right) - \frac{{J\left( {\chi_{{{\text{Cl}},{\text{Cl}}}} } \right)}}{2} - \frac{{J\left( {\chi_{{{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} ,{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} } \right)}}{2}} \right] $$

(17)

$$ \begin{aligned}^{\text{E}} \theta^{\prime }_{{{\text{Cl}},{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} &= - \left( {\frac{{^{\text{E}} \theta_{{{\text{Cl}},B_{4} O_{5} \left( {\text{OH}} \right)_{4} }} }}{I}} \right) + \left( {\frac{{Z_{\text{Cl}} Z_{{{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} }}{{8I^{2} }}} \right) \\& \quad * \left[ {\chi_{{{\text{Cl}},{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} J^{'} \left( {\chi_{{{\text{Cl}},{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} } \right) - \frac{{\chi_{{{\text{Cl}},{\text{Cl}}}} J^{\prime }\left( {\chi_{{{\text{Cl}},{\text{Cl}}}} } \right)}}{2} - \frac{{\chi_{{{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} ,{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} J^{\prime }\left( {\chi_{{{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} ,{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} } \right)}}{2}} \right]\end{aligned} $$

(18)

where

$$ \chi_{{{\text{Cl}},{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} = 6Z_{\text{Cl}} Z_{{{\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} A^{\emptyset } I^{1/2} $$

(19)

$$ J\left( \chi \right) = \chi \left[ {4 + 4.581\chi^{ - 0.7237} { \exp }\left( { - 0.0120\chi^{0.528} } \right)} \right]^{ - 1} $$

(20)

$$ \begin{aligned} J^{'} \left( \chi \right) & = \left[ {4 + 4.581\chi^{ - 0.7237} { \exp }\left( { - 0.0120\chi^{0.528} } \right)} \right]^{ - 1 } \\ & \quad + \left[ {4 + 4.581\chi^{ - 0.7237} { \exp }\left( { - 0.0120\chi^{0.528} } \right)} \right]^{ - 2} 4.581\chi \\ & { \exp }\left( { - 0.0120\chi^{0.528} } \right)\left( {0.7237\chi^{ - 1.7237} + 0.0120*0.528\chi^{ - 0.472} \chi^{ - 0.7237} } \right) . \\ \end{aligned} $$

(21)

The excess Gibbs energy ($G^{\text{E}}$) and activity of water ($a_{\text{W}}$) are calculated from the following relations:

$$ G^{E} = RT\left[ {2m_{1} \left( {1 - \phi + { \ln }\gamma_{{ \pm {\text{KCl}}}} } \right) + 3m_{2} \left( {1 - \phi + { \ln }\gamma_{{ \pm {\text{K}}_{2} {\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} }} } \right)} \right] $$

(22)

and

$$ a_{\text{W}} = { \exp }\left[ {\left( { - \frac{18.0513}{1000}} \right)\left( {2m_{1} + 3m_{2} } \right)\emptyset } \right] $$

(23)

4 Results and Discussion

Using the values of m ₀ (molality of KCl), E _a (the potential difference value of KCl), and $\gamma_{{0 \pm {\text{KCl}}}}$ (activity coefficients of KCl at 298.15 K), the standard potential $E^{{0{\prime }}}$ and the electrode response slope k were determined by Eq. 1; these are collected in Table 1 and shown in Fig. 1.

Table 1 The values for potential difference $E_{\text{a}}$, activity logarithm ln $a_{{0 \pm {\text{KCl}}}}$, and mean ionic activity coefficient $\gamma_{{0 \pm {\text{KCl}}}}$ of KCl in aqueous solutions for different KCl molalities $m_{0}$ at 298.15 K

Fig. 1

The response curve of K − ISE | KCl (m ₀), H₂O | Cl − ISE galvanic cell versus ln {KCl activity} at 298.15 K

As shown in Fig. 1, $E_{\text{a}}$ increases linearly with ${ \ln }a_{{0 \pm {\text{KCl}}}}$, (R ² = 0.999) showing that the K-ISE and Cl-ISE have a good linear Nernstian responses and the measured potential difference value is reliable. The electrode constants and the electrode response slope are listed in Table 2.

Table 2 The standard potential, $E^{{0{\prime }}}$, and the electrode response slope, k, of the galvanic cell ${\text{K-ISE}}\left| {{\text{KCl }}\left( {m_{0} } \right),{\text{ H}}_{ 2} {\text{O}}} \right|{\text{ Cl-ISE}}$ at 298.15 K

The molal concentrations $m_{1} ,m_{2}$ and the potential difference $E_{\text{b}}$ of the mixed solutions are listed in Table 3. From the electrode constants $E^{0'}$, the electrode response slope k and the potential difference $E_{\text{b}}$, the mean activity coefficient $\gamma_{{ \pm {\text{KCl}}}}$ of mixed solution can be calculated according to Eq. 2, the results are also listed in Table 3. The relationship between the mean activity coefficients $\gamma_{{ \pm {\text{KCl}}}}$ in the mixed solutions and the ionic strength fractions $y_{\text{B }}$ of K₂B₄O₇ is shown in Fig. 2.

Table 3 The values for the total ionic strengths I, stoichiometric ionic strength fraction of K₂B₄O₇ $y_{\text{B}}$, molality of KCl and K₂B₄O₇ $m_{1}$, $m_{2}$, respectively, potential difference $E_{\text{b}}$, and mean activity coefficient of KCl $\gamma_{{ \pm {\text{KCl}}}}$ in the KCl + K₂B₄O₇ + H₂O ternary system at 298.15 K

Fig. 2

Plots of the logarithm of the mean activity coefficient $\ln \gamma_{{ \pm {\text{KCl}}}}$ versus stoichiometric ionic strength fraction of K₂B₄O₇ y _B at different ionic strengths of the KCl + K₂B₄O₇ + H₂O ternary system at 298.15 K

As can be seen from Table 3, in a mixed solution containing K₂B₄O₇, ${ \ln }\gamma_{{ \pm {\text{KCl}}}}$ decreases with increasing I. As shown in Fig. 2, when I is constant, ${ \ln }\gamma_{{ \pm {\text{KCl}}}}$ increases with increase of $y_{\text{B}}$ for I less than 0.1000 mol·kg⁻¹ but decrease with increasing $y_{\text{B}}$ for I greater than 0.1000 mol·kg⁻¹.

According to an early experimental study on the phase equilibrium of potassium borate solutions from 298.15 to 323.15 K, we found that potassium tetraborate crystallized in the form of K₂B₄O₇·4H₂O from its saturated solution [27]. Therefore, this study treats K₂B₄O₇·4H₂O as the structure of ${\text{K}}_{2} {\text{B}}_{4} {\text{O}}_{5} \left( {\text{OH}} \right)_{4} {\cdot}2{\text{H}}_{2} {\text{O}}$ [36].

Values of the Pitzer parameters ($\beta^{\left( 0 \right)}$, $\beta^{\left( 1 \right)}$ and $C^{\phi }$) for pure KCl and pure K₂B₄O₇ are listed in Table 4. According to measured values $E_{\text{b}}$ of the KCl + K₂B₄O₇ + H₂O system and the Pitzer model formula, the Pitzer parameters $\theta_{{{\text{Cl}}^{ - } {\cdot}{\text{B}}_{4} {\text{O}}_{5} ({\text{OH}})_{4}^{2 - } }}$ and $\psi_{{{\text{K}}^{ + } {\cdot} {\text{Cl}}^{ - } {\cdot} {\text{B}}_{4} {\text{O}}_{5} ({\text{OH}})_{4}^{2 - } }}$ were calculated using the linear regression method (using Matlab) on the basis of Eqs. 2–23. The results are shown in Table 5. The osmotic coefficient, water activity, and excess Gibbs energy results for the KCl + K₂B₄O₇ + H₂O ternary system at 298.15 K are listed in Table 6. The relationship of osmotic coefficient and total ionic strength of the KCl + K₂B₄O₇ + H₂O ternary system are shown in Fig. 3.

Table 4 Values of the Pitzer parameters ($\beta^{\left( 0 \right)}$, $\beta^{\left( 1 \right)}$ and $C^{\phi }$) for pure KCl and K₂B₄O₇ at 298.15 K

Table 5 Pitzer parameters $\theta_{{{\text{Cl}}^{ - } {\cdot}{\text{B}}_{4} {\text{O}}_{5} ({\text{OH}})_{4}^{2 - } }}$ and $\psi_{{{\text{K}}^{ + } {\cdot} {\text{Cl}}^{ - } {\cdot} {\text{B}}_{4} {\text{O}}_{5} ({\text{OH}})_{4}^{2 - } }}$ for the KCl + K₂B₄O₇ + H₂O ternary system at 298.15 K

Table 6 The values for the total ionic strengths I, stoichiometric ionic strength fraction of K₂B₄O₇ $y_{\text{B}}$, mean activity coefficient of K₂B₄O₇ $\gamma_{{ \pm {\text{KCl}}}}$, osmotic coefficient $\phi$, water activity $a_{\text{W}}$, and excess Gibbs energy G ^E for the KCl + K₂B₄O₇ + H₂O ternary system at 298.15 K

Fig. 3

Plot of the osmotic coefficients $\phi$ against total ionic strength I of the KCl + K₂B₄O₇ + H₂O ternary system at different y _B at 298.15 K

As is shown in Fig. 3, when $y_{\text{B}}$ is constant, the osmotic coefficient $\phi$ and excess Gibbs energy G ^E show a downward trend with the increase of I; when I is constant, they decrease with the increase of $y_{\text{B}}$, while the water activity $a_{\text{W}}$ increases with increasing $y_{\text{B}}$.

5 Conclusion

The thermodynamics of the KCl + K₂B₄O₇ + H₂O ternary system was studied by the EMF method using K-ISE and Cl-ISE at 298.15 K. The mean activity coefficients of KCl in pure and mixed solution were determined from cells without liquid junction. From the electrode constant and the response slope, the mean activity coefficients of KCl in the KCl + K₂B₄O₇ + H₂O systems were calculated using the Nernst equation. Pitzer ion interaction parameters $\theta_{{{\text{Cl}}^{ - } {\cdot}{\text{B}}_{4} {\text{O}}_{5} ({\text{OH}})_{4}^{2 - } }}$ and $\psi_{{{\text{K}}^{ + } {\cdot} {\text{Cl}}^{ - } {\cdot} {\text{B}}_{4} {\text{O}}_{5} ({\text{OH}})_{4}^{2 - } }}$ were calculated by linear regression. The osmotic coefficients $\phi$, activity of water $a_{\text{W}}$ and the Gibbs energies $G^{\text{E}}$ of the system were calculated using these mixing parameters and Pitzer equations. The results showed that the Pitzer model can be used to describe this aqueous system satisfactorily. The results of Pitzer model indicate that the present investigations in this work can provide basic thermodynamic reference data for further research applications.