Mean Activity Coefficients of NaNO3 and the Mixing Ion-Interaction Parameters in the Ternary System (NaNO3 + CsNO3 + H2O) at 298.15 K by EMF Method

Abstract

Ion-selective electrodes directly respond to the activity of target ions without destroying the existing form of the original electrolyte, so ion-selective electrodes have been widely used in various fields. Mean activity coefficient of NaNO₃ in the ternary system (NaNO₃ + CsNO₃ + H₂O) at 298.15 K was measured by electromotive force (EMF) with the cell: Na⁺ ion-selective electrode (Na-ISE)|NaNO₃ (m_A), CsNO₃ (m_B)|NO₃^- ion-selective electrode (NO₃-ISE) with total ionic strengths from 0.01 to 4.5 mol·kg^-1 at different ionic strength fractions (0, 0.1, 0.2, 0.4, 0.6, and 0.8). The results showed that the Na-ISE and NO₃-ISE have a good Nernst response, and the mean activity coefficients of NaNO₃ are obtained via the Nernst equation. Based on the data of mean activity coefficients of NaNO₃, the relationship diagrams of activity coefficients of NaNO₃ against ion strengths in the ternary system were demonstrated, and the Pitzer mixing ion-interaction parameters ${\theta }_{\text{Na},\text{Cs}}$ and $\psi \text{Na},\text{Cs},\text{N}{\text{O}}_3$ were obtained.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Electrolyte solutions widely exist in salt lake, marine, underground water, oil/gas-field water, and the engineering of inorganic chemistry and hydrometallurgy [1]. The mean activity coefficients of the electrolytes are essential for the design development of processes such as salt chemical industry and desalination. In China, cesium levels in many salt lake brine range from 10 to 20 mg/L [2]. Therefore, it is of great significance to determine the activity coefficients of the solution of cesium salts.

A series of ion-interaction models of electrolytes have been proposed to predict the activity coefficients of solute and osmotic coefficients of the aqueous systems. Pitzer’s ion-interaction model is one of the most commonly used models [3, 4]. The research methods of the thermodynamic properties of aqueous electrolytes involve the isopiestic vapor pressure [5–7], electromotive force method [8–10], volume properties method [11], hygrometric method, and calorimetric method [12, 13]. Compared with other methods, the electromotive force (EMF) method has the advantages of good selectivity, rapid response, and easy to achieve continuous assay.

The mixed ion-interaction parameter of the ternary system provides basic thermodynamic data for the separation and extraction of pure salts from salt lake brines. The thermodynamic properties of aqueous mixed-electrolyte solutions have received considerable attention. Using EMF method in Hu’s research group [14–18], the mean ionic activity coefficients in the following ternary systems (CsCl + Cs₂SO₄+H₂O), (CsF + Cs₂SO₄ + H₂O), (CsF + CsBr + H₂O), (CsF + CsNO₃+H₂O), (CsCl + CaCl₂ + H₂O), and (CsCl + MgCl₂ + H₂O) have been reported systematically, but the activity coefficients of (NaNO₃+CsNO₃+H₂O) system at 298.15 K have not been reported in the literature till now. Therefore, the electromotive forces of the ternary system at 298.15 K were measured by EMF, and the mean activity coefficients of NaNO₃ and the mixing ion-interaction parameters ${\theta }_{\text{Na},\text{Cs}}$ and $\psi \text{Na},\text{Cs},\text{N}{\text{O}}_3$ are obtained for the first time.

2. Experimental

All of the instructions of the chemical reagents used in this work are shown in Table 1, and they were used without further purification. The deionized distilled water (DDW) produced by ULUP-II-10T (Chongqing Jiuyang Co. Lt., China), whose conductivity is less than 1.0 × 10⁻⁴ S·m⁻¹ and pH =6.60 at 298.15 K, was used during the whole experiment.

Table 1

Chemical reagents used in the experiment.

Regent	CAS	Grade	Purity in mass fraction	Analytical method
Sodium nitrate a	7631-99-4	GR	0.9999	Titration for NO3-
Cesium nitrate a	7789-18-6	GR	0.9999	Titration for NO3-

^aFrom the Aladdin Industrial Corporation.

The Na-ISE and NO₃-ISE were purchased from Shanghai Miriam Electric Science Instruments Co., Ltd. Before use, the Na-ISE and NO₃-ISE were activated at least 2 h in a sodium nitrate with a concentration of 0.001 mol/L and purified with deionized water to a blank potential. Both electrodes were calibrated before use, and they had an excellent Nernst response and selectivity. The ion analyzer was PHSJ-4F with an uncertainty of ±0.1 mV.

The double-walled glass bottle was held at a constant temperature at (298.15 ± 0.02) K by water circulation from a thermostat.

The cell arrangements in this work were as following:

(a) Na-ISE|NaNO₃ (m_A0)|NO₃-ISE

(b) Na-ISE|CsNO₃ (m_B0)|NO₃-ISE

(c) Na-ISE|NaNO₃ (m_A), CsNO₃ (m_B)|NO₃-ISE

Above galvanic cells contain no liquid junction. Here, $m_{\text{A}0}$ and $m_{\text{B}0}$ are the molalities of NaNO₃ and CsNO₃ as single salt in water, $m_{\text{A}}$ and $m_{\text{B}}$ are the molalities of NaNO₃ and CsNO₃ in the ternary system, respectively.

Each concentration of the above solutions was prepared by directly weighing the materials using a Sartorius electronic balance whose accuracy was 0.1 mg. Voltage readings were taken as final when they were constant within 0.2 mV for at least 5 min.

The electromotive force of the above three cells was measured at 298.15 K. First, the electromotive force of the cell (a) was measured to determine whether the electrode pair of Na-ISE and NO₃-ISE had a satisfactory Nernst response, which could judge its suitability for this experiment. Cell (b) was used to measure the electromotive force of CsNO₃ solution at different concentrations, and the selectivity coefficient $K^{\text{pot}}$ of electrode Na-ISE to Cs⁺ can be calculated by equation (2). The purpose was to determine the effect of the presence of Cs⁺ on the response of Na-ISE. Finally, the EMF of cell (c) was measured under different ionic strength fractions ($y_{\text{B}}$) of CsNO₃ in the solutions.

3. Results and Discussion

3.1. The Calibration of Electrode Pair of Na-ISE and NO₃-ISE

For cell (a), 13 measurements of $m_{\text{A}0}$ from 0.01 to 4.5 mol·kg^-1 were selected to determine each corresponding potential ($E_{\text{a}}$). The Nernst equation for cell (a) can be expressed as: $$\begin{array}{}\text{(1)}& E_{\text{a}}=E^0+2k\cdot \ln {\gamma }_{\pm \text{A}0}\cdot m_{\pm \text{A}0}.\end{array}$$

$E^0$ is the standard potential of cell (a), $k=RT/F$ represents the theoretical Nernst slope. The $R$, $F$, and $T$ are the gas constant, Faraday constant, and absolute temperature, respectively. The ${\gamma }_{\pm \text{A}0}$ is the mean activity coefficient of pure NaNO₃ in water, whose values were taken and calculated from the literature [19] and listed in Table 2.

Table 2

Values of the Pitzer parameters for CsNO₃ and NaNO₃ at 298.15 K.

Electrolyte	${\beta }^0$	${\beta }^1$	$C^{\Phi }$	$I_{\max }$	$\mathit{\sigma }$	Ref
CsNO3	-0.13004	0.08169	0.03018	1.500	0.00057	[19]
NaNO3	0.00388	0.21151	-0.00006	10.830	0.00073	[19]

To check their linear relationship, $E_{\text{a}}$ was plotted against $\ln m_{\text{A}0}{\gamma }_{\pm \text{A}}$ and shown in Figure 1. By way of this line, $E^0$, $k$, and the linear correlation coefficient ($r$) can be evaluated using a linear regression method, and they are 120.3, 25.96 mV, and 0.9995, respectively. The obtained value of $k$ gets quite close to the theoretical one (25.69 mV) of the Nernst slope. Those results mean that the electrode pair used in this work have a satisfactory Nernst response and are well suitable for our measurements.

[figure omitted; refer to PDF]

3.2. Selective Coefficient of Na-ISE Electrode for Cs Ion

The electromotive force values of CsNO₃ solutions with different concentrations were measured and used to calculate $K^{\text{pot}}$. The selective coefficient $K^{\text{pot}}$ of electrode Na-ISE for Cs⁺ can be calculated according to the following equation: $$\begin{array}{}\text{(2)}& K^{\text{pot}}=\exp E_{\text{b}}-E^0/k/{m_{\text{B}0}\cdot {\gamma }_{\pm \text{B}0}}^2,\end{array}$$

where ${\gamma }_{\pm \text{B}0}$ refers to the activity coefficients of pure $\text{CsN}{\text{O}}_3$ in water at 298.15 K, and its value was taken and calculated according to the cited literature [19]. $E_{\text{b}}$ is the EMF value of cell (b) at different $m_{\text{B}0}$ and is shown in Table 3. The mean value of $K^{\text{pot}}$ is less than 1.0 × 10^-4, which evidenced that the response of the electrode pair to Cs ion can be ignored.

Table 3

Electromotive force values of CsNO₃ solutions at different concentrations^a.

$$I$$	$$m_{\text{CsNO}3}$$	$$E_{\text{b}}$$
mol·kg-1	mol·kg-1	mV
0.1988199	0.1988199	-269.6
0.2483	0.2483	-258.3
0.3973	0.3973	-229.6
0.4994	0.4994	-223.2
0.6000	0.6000	-219.5

^aStandard uncertainties $u$ are $uT$ =0.02 K, $u{m}_{{\text{CsNO}}_3}$ =0.0003 mol·kg^-1, $uE$ =0.1 mV.

3.3. Measurements of the Mean Activity Coefficients of NaNO₃ in the Ternary System

The cell (c) was employed to determine the EMF values $E_{\text{m}}$ of the ternary system (NaNO₃ + CsNO₃ + H₂O) at different ionic strengths $I$ and mole fractions $y_{\text{B}}$ ($y_{\text{B}}=I_{\text{B}}/I=m_{\text{B}}/m_{\text{A}}+m_{\text{B}}$). The mean activity coefficients of NaNO₃ in the ternary system can be derived from the following Nernst equation and shown in Table 3. $$\begin{array}{}\text{(3)}& E_{\text{m}}=E^0+k÷\ln {\gamma }_{\pm \text{A}}^2\cdot m_{\text{A}}\cdot m_{\text{A}}+m_{\text{B}}+K^{\text{pot}}\cdot {\gamma }_{\pm \text{B}}^2\cdot m_{\text{B}}\cdot m_{\text{A}}+m_{\text{B}},\end{array}$$where ${\gamma }_{\pm \text{A}}$ and ${\gamma }_{\pm \text{B}}$ are the mean activity coefficients of NaNO₃ and CsNO₃, respectively. Because $K^{\text{pot}}$ can be neglected [20] without leading to an appreciable error, we get the simplified form the following equation: $$\begin{array}{}\text{(4)}& E_{\text{m}}=E^0+k\cdot \ln {\gamma }_{\pm \text{A}}^2\cdot m_{\text{A}}\cdot m_{\text{A}}+m_{\text{B}},\text{(5)}& \ln {\gamma }_{\pm \text{A}}=E_{\text{m}}-E^0/2\cdot k-1/2\ln m_{\text{A}}\cdot m_{\text{A}}+m_{\text{B}}.\end{array}$$

The mean activity coefficients of NaNO₃ in the aqueous ternary system can be calculated with equation (5). The related results of cell (c) are collected in Table 4 and shown in Figure 2. It can be seen from Figure 2 that when the mole fractions remain constant substantially, the mean activity coefficient of NaNO₃ decreases with the increase of ionic strength.

[figure omitted; refer to PDF]

3.4. Harned Rule

The Harned rule [21] is one of the earliest proposed treatments for strong electrolyte aqueous ternary systems. Concerning the studied ternary system, the Harned rule can be written as the following equation: $$\begin{array}{}\text{(6)}& \ln {\gamma }_{\pm \text{A}}=\ln {\gamma }_{\pm \text{A}0}-{\alpha }_{\text{AB}}\cdot y_{\text{B}}-{\beta }_{\text{AB}}\cdot y_{\text{B}}^2,\end{array}$$where ${\alpha }_{\text{AB}}$ and β_AB represent the Harned interaction parameters, dependent on both ionic strength and temperature [10, 22]. ${\gamma }_{\pm \text{A}0}$ is the mean activity coefficient of NaNO₃ in pure solutions at the same total ionic strengths $I$ as the ternary system. The fitted results are listed in Table 5, indicating that the Harned rule can be applied to describe the ternary system accurately.

Table 4

Measurement results of the mean activity coefficients of NaNO₃ in the ternary system at 298.15 K^a.

I$I$	$y_{{\text{CsNO}}_3}$	$m_{{\text{NaNO}}_3}$	$m_{{\text{CsNO}}_3}$	$E_{\text{m}}$	${\gamma }_{\pm \text{NaN}{\text{O}}_3}$
mol·kg-1		mol·kg-1	mol·kg-1	mV
0.0100	0.0000	0.0100	0.0000	-122.8	0.9000
0.0190	0.0000	0.0190	0.0000	-89.2	0.8697
0.0380	0.0000	0.0380	0.0000	-58.8	0.8294
0.0505	0.0000	0.0505	0.0000	-45.2	0.8107
0.0805	0.0000	0.8050	0.0000	25.7	0.7772
0.0999	0.0000	0.0999	0.0000	-15.1	0.7607
0.2003	0.0000	0.2003	0.0000	16.1	0.7035
0.2486	0.0000	0.2486	0.0000	27.6	0.6846
0.5049	0.0000	0.5049	0.0000	58.9	0.6195
0.5998	0.0000	0.5998	0.0000	69.2	0.6030
0.7951	0.0000	0.7951	0.0000	81.9	0.5754
0.9983	0.0000	0.9983	0.0000	89.6	0.5527
1.5086	0.0000	1.5086	0.0000	106.1	0.5107
2.1027	0.0000	2.1027	0.0000	121.0	0.4765
2.5022	0.0000	2.5022	0.0000	126.1	0.4587
3.0917	0.0000	3.0917	0.0000	139.6	0.4373
3.5057	0.0000	3.5057	0.0000	141.8	0.4248
4.4964	0.0000	4.4964	0.0000	154.1	0.4008
0.1002	0.1001	0.0902	0.0100	-15.7	0.7640
0.1997	0.1001	0.1797	0.0200	15.9	0.7050
0.5000	0.1002	0.4499	0.0501	56.8	0.6197
0.5986	0.1001	0.5387	0.0599	64.8	0.6039
0.7989	0.0999	0.7191	0.0798	77.3	0.5758
1.0000	0.1000	0.9000	0.1000	86.9	0.5536
1.5011	0.1000	1.3511	0.1500	103.9	0.5118
2.4969	0.1000	2.2472	0.2498	124.6	0.4587
3.4903	0.1000	3.1413	0.3490	140.0	0.4416
4.4834	0.1000	4.0351	0.4483	150.0	0.4169
0.0993	0.1998	0.0795	0.0199	-18.9	0.7685
0.1987	0.1999	0.1590	0.0397	13.0	0.7106
0.2496	0.1999	0.1997	0.0499	23.5	0.6927
0.4993	0.2001	0.3994	0.0999	54.1	0.6248
0.5922	0.2000	0.4738	0.1184	61.3	0.6052
0.8000	0.2000	0.6400	0.1600	74.3	0.5756
0.9950	0.2000	0.7960	0.1990	84.0	0.5580
1.5011	0.2002	1.2006	0.3005	100.9	0.5124
2.4842	0.2004	1.9863	0.4970	121.3	0.4589
3.5019	0.2000	2.8016	0.7004	138.3	0.4517
0.0100	0.3997	0.0060	0.0040	-134.7	0.9492
0.0198	0.4013	0.0119	0.0080	-100.9	0.9165
0.0400	0.4014	0.0239	0.0160	-68.5	0.8487
0.0498	0.4017	0.0298	0.0200	-57.9	0.8344
0.0998	0.3999	0.0599	0.0399	-25.7	0.7746
0.1992	0.3999	0.1195	0.0797	6.2	0.7180
0.2496	0.4001	0.1497	0.0999	16.6	0.7003
0.4973	0.4000	0.2984	0.1989	46.6	0.6268
0.5944	0.3999	0.3567	0.2377	54.1	0.6059
0.7960	0.3999	0.4777	0.3183	66.6	0.5758
0.9944	0.4001	0.5966	0.3978	76.9	0.5623
1.4997	0.4001	0.8996	0.6001	93.5	0.5135
0.0102	0.5940	0.0041	0.0061	-142.6	0.9671
0.0201	0.5991	0.0081	0.0120	-109.5	0.9366
0.0400	0.6004	0.0160	0.0240	-78.4	0.8583
0.0501	0.5996	0.0200	0.0300	-67.9	0.8379
0.1000	0.5995	0.0400	0.0599	-35.8	0.7792
0.1998	0.6000	0.0799	0.1199	-3.9	0.7216
0.2497	0.6003	0.0998	0.1499	6.5	0.7058
0.4996	0.6000	0.1999	0.2998	36.7	0.6313
0.5979	0.6003	0.2390	0.3589	43.9	0.6063
0.7986	0.5998	0.3196	0.4790	56.6	0.5796
0.9999	0.6000	0.3999	0.5999	67.0	0.5658
0.0099	0.8057	0.0019	0.0080	-162.5	0.9844
0.0196	0.8024	0.0039	0.0157	-128.1	0.9547
0.0398	0.8016	0.0079	0.0319	-96.3	0.8656
0.0497	0.8036	0.0098	0.0399	-86.1	0.8491
0.1001	0.8004	0.0200	0.0801	-53.3	0.7868
0.1983	0.8000	0.0397	0.1587	-21.9	0.7265
0.2499	0.8001	0.0500	0.1999	-11.0	0.7117
0.4988	0.7997	0.0999	0.3989	18.9	0.6340
0.6011	0.7999	0.1203	0.4808	26.3	0.6070
0.7962	0.8000	0.1593	0.6370	38.5	0.5799
1.0007	0.7999	0.2002	0.8005	49.5	0.5704

^aStandard uncertainties $u$ are $uT$ =0.02 K, u($m_{{\text{NaNO}}_3}$) =0.0007 mol·kg^-1, u($m_{{\text{CsNO}}_3}$) =0.0003 mol·kg^-1, $uE$ =0.1 mV.

3.5. Pitzer Model

In this paper, the Pitzer and its extended Harvie-Weare model was used to fit the experimental data [23]. The mixed ion-interaction parameter can be obtained by substituting the binary interaction parameters and the mean activity coefficient of the aqueous electrolyte into equation (9). For the ternary system studied, the mean activity coefficients and the osmotic coefficients can be given as in the following equations: $$\begin{array}{}\text{(7)}& \ln {\gamma }_{\text{M}}=Z_{\text{m}}{}^{2}F+\sum _{\text{a}}{m}_{\text{a}}2B_{\text{Ma}}+2C_{\text{Ma}}+\sum _{\text{c}}{m}_{\text{c}}2{\Phi }_{\text{Mc}}+\sum _{\text{a}}{m}_{\text{a}}{\Psi }_{\text{Mca}}+\sum _{\text{a}}\sum _{\text{a}}{m}_{\text{a}}{m}_{{\text{a}}^{\prime }}{\Psi }_{\text{Ma}{\text{a}}^{\prime }}+Z_{\text{M}}\sum _{\text{c}}\sum _{\text{a}}{m}_{\text{c}}{m}_{\text{a}}{C}_{\text{ca}},\text{(8)}& \ln {\gamma }_{\text{X}}=Z_{\text{X}}{}^{2}F+\sum _{\text{c}}{m}_{\text{c}}2B_{\text{c}}+2C_{\text{cX}}+\sum _{\text{a}}{m}_{\text{a}}2{\Phi }_{\text{Xa}}+\sum _{\text{c}}{m}_{\text{c}}{\Psi }_{\text{cXa}}+\sum _{\text{c}}\sum _{c^{\prime }}{m}_{\text{c}}{m}_{c^{\prime }}{\Psi }_{\text{c}{\text{c}}^{\prime }\text{X}}+Z_{\text{X}}\sum _{\text{c}}\sum _{\text{a}}{m}_{\text{c}}{m}_{\text{a}}{C}_{\text{ca}},\text{(9)}& \ln {\gamma }_{\text{MX}}=\ln {\gamma }_{\text{M}}+\ln {\gamma }_{\text{X}}/2,\text{(10)}& F=A^{\varphi }{I}^{1/2}/1+1.2I^{1/2}+2/1.2\ln 1+{1.2}^{1/2}+\sum _{\text{c}}\sum _{\text{a}}{m}_{\text{c}}{m}_{\text{a}}{B}_{\text{ca}}^{\prime }+\sum _{\text{c}}\sum _{c^{\prime }}{m}_{\text{c}}{m}_{\text{c}}{\Phi }_{\text{c}{\text{c}}^{\prime }}^{\prime }+\sum _{\text{a}}\sum _{a^{\prime }}{m}_{\text{a}}{m}_{{\text{a}}^{\prime }}{\Phi }_{\text{c}{\text{a}}^{\prime }}^{\prime },\text{(11)}& {\Phi }_{\text{ij}}={\theta }_{\text{ij}}+{{}^E\theta }_{\text{ij}},\text{(12)}& {\Phi }_{\text{ij}}^{\prime }={{}^E{\theta }^{\prime }}_{\text{ij}},\text{(13)}& B_{\text{ZA}}={{\beta }_{\text{CA}}}^0+{{\beta }_{\text{CA}}}^{1}g{\alpha }_1{I}^{1/2}+{{\beta }_{\text{CA}}}^{2}g{\alpha }_2{I}^{1/2},\text{(14)}& B_{\text{CA}}^{\prime }={{\beta }_{\text{CA}}}^1{g}^{\prime }{\alpha }_1{I}^{1/2}+{{\beta }_{\text{CA}}}^2{g}^{\prime }{\alpha }_1{I}^{1/2}/I,\text{(15)}& gx=21-1+x\exp -x/x^2,\text{(16)}& g^{\prime }x=-21-1+x+x^2/2\exp -x/x^2,\text{(17)}& \Phi =1+\frac{1}{m_{\text{Na}}+m_{\text{Cs}}+m_{\text{N}{\text{O}}_3}}\cdot 2-A^{\varphi }\cdot I^{3/2}/1+1.2\cdot I^{1/2}+m_{\text{Na}}\cdot m_{{\text{NO}}_3}\cdot B_{{\text{NaNO}}_3}^{\varphi }+{ZC}_{{\text{NaNO}}_3}^{\varphi }+m_{\text{Cs}}\cdot m_{\text{N}{\text{O}}_3}\cdot B_{{\text{CsNO}}_3}^{\varphi }+{ZC}_{{\text{CsNO}}_3}^{\varphi }+m_{\text{Na}}\cdot m_{\text{Cs}}\cdot {\Phi }_{\text{Na},\text{Cs}}^{\varphi }+m_{\text{N}{\text{O}}_3}\cdot {\psi }_{\text{Na},\text{Cs},\text{N}{\text{O}}_3},\end{array}$$where $M$, $\text{a}$, and ${\text{a}}^{\text{’}}$ are anions. $X$, $\text{c}$, and ${\text{c}}^{\text{’}}$ are cations. ${\gamma }_{\text{M}}$, $Z_{\text{M}}$_, and $m_{\text{c}}$ are the activity coefficient, valence number, and molar concentration of the cation, respectively. ${\gamma }_{\text{X}}$, $Z_{\text{X}}$_, and $m_{\text{a}}$ are the activity coefficient, valence number, and molar concentration of the anion, respectively. $A_{\varphi }$ is called the Debye−Hückel constant. $\Psi$ is the ionic interaction parameters. $B$ is the coefficient in the second dimension. For type 1-1 electrolytes, ${\alpha }_1$ = 2.0 mol·kg^-1, ${\alpha }_2$ = 0.

Pitzer’s mixing ion-interaction parameters are evaluated through equation (10) by using multiple linear regression techniques, and the result is shown in Table 6.

Table 5

The parameter values of Harned equation^a.

$I$/mol·kg-1	$\ln {\gamma }_{\pm \text{A}0}$	${\alpha }_{\text{AB}}$	${\beta }_{\text{AB}}$	103·RMSD
0.01	-0.1053	-0.1526	0.0518	0.9004
0.02	-0.1396	-0.1453	0.0362	0.0075
0.04	-0.1871	-0.0637	0.0126	0.8753
0.05	-0.2095	-0.0732	0.0219	5.2820
0.10	-0.2733	-0.0437	0.0030	1.9200
0.20	-0.3516	-0.0574	0.0223	1.8110
0.25	-0.3789	-0.0629	0.0185	1.0150
0.50	-0.4790	-0.0327	0.0040	1.2240
0.60	-0.5058	-0.0144	0.0078	0.6157
0.80	-0.5530	0.0018	0.0012	3.5060
1.00	-0.5929	-0.0437	0.0058	1.1530
1.50	-0.6719	-0.0212	0.0192	0.2707
2.50	-0.7793	0.0050	-0.0368	0.0000
3.50	-0.8561	-0.4656	0.7928	0.0000

^aStandard uncertainties $u$ are $uT$ =0.02 K, u($m_{{\text{NaNO}}_3}$) =0.0007 mol·kg^-1, u($m_{{\text{CsNO}}_3}$) =0.0003 mol·kg^-1, $uE$ =0.1 mV.

The mean activity coefficients of CsNO₃ and the osmotic coefficients for the ternary system at different ionic strengths are calculated by Pitzer and its extended Harvie-Weare model. These results are given in Table 6 and Figures 3 and 4. From Figure 3, we can see that for the ternary system studied, the mean activity coefficients of CsNO₃ decrease with an increase of total ionic strengths. It can be seen from Figure 4 that when the ionic strengths are constant, the osmotic coefficient of the ternary system is reduced by increasing the mole fractions of CsNO₃ in the system.

Table 6

Values of the mixing ion-interaction parameters of the Pitzer equation at 298.15 K^a.

I/mol·kg-1	${\theta }_{\text{Na},\text{Cs}}$	${\psi }_{\text{Na},\text{Cs},\text{NO}3}$	RMSE	Ref
0.1-4.5	-0.1750	0.02561	0.02	This work

^aStandard uncertainties $u$ are $uT$ =0.02 K, u($m_{{\text{NaNO}}_3}$) =0.0007 mol·kg^-1, u($m_{{\text{CsNO}}_3}$) =0.0003 mol·kg^-1, $uE$ =0.1 mV.

Table 7

The mean activity coefficients of CsNO₃, osmotic coefficients, water activities, and excess free energies in the ternary system^a.

I/mol·kg-1	$y_{\text{B}}$	${\gamma }_{\pm \text{CsN}{\text{O}}_3}$	$\Phi$	$a_{\text{w}}$	GE/KJ·mol-1
0.10	0.0	0.7337	0.9219	0.9967	-0.0968
	0.1	0.7330	0.9182	0.9967	-0.09512
	0.2	0.7335	0.9154	0.9967	-0.0927
	0.4	0.7324	0.9098	0.9967	-0.0929
	0.6	0.7315	0.9058	0.9967	-0.0957
	0.8	0.7307	0.9031	0.9967	-0.1002
0.20	0.0	0.6571	0.9041	0.9935	-0.2541
	0.1	0.6571	0.8975	0.9936	-0.2515
	0.2	0.6574	0.8917	0.9936	-0.2452
	0.4	0.6564	0.8817	0.9937	-0.2458
	0.6	0.6553	0.8743	0.9937	-0.2560
	0.8	0.6554	0.8702	0.9938	-0.2675
0.25	0.0	0.6303	0.8982	0.9920	-0.3416
	0.2	0.6292	0.8827	0.9921	-0.3330
	0.4	0.6286	0.8706	0.9922	-0.3342
	0.6	0.6279	0.8619	0.9923	-0.3472
	0.8	0.6272	0.8565	0.9923	-0.3687
0.50	0.0	0.5318	0.8776	0.9841	-0.8922
	0.1	0.5337	0.8630	0.9845	-0.8837
	0.2	0.5344	0.8499	0.9848	-0.8700
	0.4	0.5358	0.8287	0.9852	-0.8842
	0.6	0.5359	0.8134	0.9854	-0.9206
	0.8	0.5369	0.8050	0.9856	-0.9734
0.60	0.0	0.5056	0.8721	0.9813	-1.1240
	0.1	0.5068	0.8550	0.9817	-1.1184
	0.2	0.5092	0.8403	0.9822	-1.1070
	0.4	0.5103	0.8155	0.9827	-1.1350
	0.6	0.5111	0.7978	0.9829	-1.1877
	0.8	0.5120	0.7878	0.9830	-1.2609
0.80	0.0	0.4611	0.8627	0.9755	-1.6372
	0.1	0.4622	0.8409	0.9760	-1.6430
	0.2	0.4640	0.8217	0.9765	-1.6541
	0.4	0.4685	0.7912	0.9775	-1.6799
	0.6	0.4718	0.7699	0.9780	-1.7371
	0.8	0.4758	0.7591	0.9784	-1.8246
1.00	0.0	0.4243	0.8546	0.9697	-2.2148
	0.1	0.4272	0.8289	0.9705	-2.2120
	0.2	0.4312	0.8067	0.9714	-2.1785
	0.4	0.4375	0.7707	0.9727	-2.2025
	0.6	0.4431	0.7459	0.9734	-2.2907
	0.8	0.4492	0.7335	0.9738	-2.4112
1.50	0.0	0.3586	0.8387	0.9553	-3.8197
	0.1	0.3663	0.8055	0.9573	-3.7860
	0.2	0.3733	0.7761	0.9588	-3.7815
	0.4	0.3876	0.7100	0.9623	-3.6367
2.5	0.0	0.2911	0.8180	0.9288	-7.4100
	0.1	0.3080	0.7772	0.9323	-7.3837
	0.2	0.3257	0.7420	0.9356	-7.2564
3.5	0.0	0.2680	0.8046	0.9032	-11.4829
	0.1	0.2992	0.7669	0.9079	-10.7852
	0.2	0.3339	0.7351	0.9112	-10.2471
4.5	0.0	0.2764	0.7957	0.8788	-15.8282
	0.1	0.3308	0.7714	0.8826	-14.8822

^aStandard uncertainties $u$ are $uT$ =0.02 K, $u{m}_{\text{A}}$ =0.0007 mol·kg^-1, $u{m}_{\text{B}}$ =0.0003 mol·kg^-1, $uE$ =0.1 mV, A = NaNO₃, B = CsNO₃.

[figure omitted; refer to PDF][figure omitted; refer to PDF]

3.6. Excess Gibbs Free Energies and Water Activities of the Ternary System

The excess Gibbs free energies and water activities of the ternary system have been calculated by using the following relation [22]: $$\begin{array}{c}\text{(18)}& G^E=R\cdot T2m_{\text{A}}1-\Phi +\ln {\gamma }_{\pm \text{A}}+2m_{\text{B}}1-\Phi +\ln {\gamma }_{\pm \text{B}},a_w=\exp -18.0513/500\cdot m_{\text{A}}+m_{\text{B}}\Phi .\end{array}$$

The calculated results are given in Table 7 and Figures 5 and 6. It can be seen that water activities and the excess Gibbs free energies decrease with an increase of total ionic strengths for all of the investigated ternary systems, respectively.

[figure omitted; refer to PDF][figure omitted; refer to PDF]

4. Conclusions

In this work, the electromotive force method was used to study the mean activity coefficients of NaNO₃ in the ternary system (NaNO₃ + CsNO₃ + H₂O) at 298.15 K. The electromotive force method plays an essential role in the study of the thermodynamic properties of dilute electrolyte solutions because of its advantages such as simple device and operation, fast determination speed, and no change in the interaction between ions in solution. Using multiple linear regression fitted the Pitzer mixing ion-interaction parameters of the system. This study could provide the basic thermodynamic data to establish the thermodynamic model of the complex salt lake brine system containing cesium. In addition, the mixed salt parameters of NaNO₃ and CsNO₃ are helpful to establish the predictive phase diagram of the ternary system (NaNO₃+CsNO₃+H₂O) to separate the mixtures of sodium and cesium nitrates.

Acknowledgments

The financial supports from the National Natural Science Foundation of China (22073068 and 21773170), Tianjin Key Laboratory of Brine Chemical Engineering and Resource Eco-utilization (BCERE202002), and the Yangtze Scholars and Innovative Research Team in Chinese University (IRT-17R81) are acknowledged.

References

[1] Z. C. Wang, X. P. Li, S. H. Sang, X. C. Ma, "Mean activity coefficients of NaCl in the NaCl–SrCl2–H2O ternary system at 308 K by EMF method," Journal of Chemical and Engineering Data, vol. 64 no. 2, pp. 442-447, DOI: 10.1021/acs.jced.8b00491, 2019.

[2] L. Gao, G. H. Ma, Y. X. Zheng, Y. Tang, G. X. Xie, J. W. Yu, B. X. Liu, J. Y. Duan, "Research trends on separation and extraction of rare alkali metal from salt lake brine: rubidium and cesium," Solvent Extraction and Ion Exchange, vol. 38 no. 7, pp. 753-776, DOI: 10.1080/07366299.2020.1802820, 2020.

[3] K. S. Pitzer, "Thermodynamics of electrolytes. I. Theoretical basis and general equations," Journal of Physical Chemistry, vol. 77 no. 2, pp. 268-277, DOI: 10.1021/j100621a026, 1973.

[4] K. S. Pitzer, J. J. Kim, "Thermodynamics of electrolytes. IV. Activity and osmotic coefficients for mixed electrolytes," Journal of the American Chemical Society, vol. 96 no. 18, pp. 5701-5707, DOI: 10.1021/ja00825a004, 1974.

[5] K. Y. Zhao, Y. F. Guo, D. Li, L. Z. Meng, T. L. Deng, "Isopiestic investigation and phase equilibrium of the high-efficient absorption refrigerants LiBr and SrBr 2 at 288.15 K," Journal of Molecular Liquids, vol. 304,DOI: 10.1016/j.molliq.2020.112741, 2020.

[6] L. Z. Meng, M. S. Gruszkiewicz, T. L. Deng, Y. F. Guo, D. Li, "Isopiestic measurements of thermodynamic properties for the aqueous system LiBr-CaBr 2 -H 2 O at 373.15 K," Journal of Chemical Thermodynamics, vol. 129, pp. 83-91, DOI: 10.1016/j.jct.2018.09.018, 2019.

[7] J. A. Rard, "The isopiestic method: 100 years later and still in use," Journal of Solution Chemistry, vol. 48 no. 3, pp. 271-282, DOI: 10.1007/s10953-019-00848-4, 2019.

[8] X. H. Hao, S. Li, Q. G. Zhai, Y. C. Jiang, M. C. Hu, "Activity coefficients of CsF in (Urea + H 2 O) or ( N -methylformamide + H 2 O) mixed solvents at 298.2 K," Journal of Molecular Liquids, vol. 220, pp. 829-835, DOI: 10.1016/j.molliq.2016.05.030, 2016.

[9] S. Manohar, J. Ananthaswamy, "Thermodynamic properties of electrolyte solutions: an EMF study of the activity coefficients of KCl in the system KCl - KNO 3 - H 2 O at 25, 35 and 45°C," Journal of Solution Chemistry, vol. 19 no. 11, pp. 1125-1135, DOI: 10.1007/BF00649457, 1990.

[10] J. Zhang, X. Y. Huang, S. P. Xia, "Experimental determination and prediction of activity coefficients of RbCl in aqueous (RbCl + RbNO 3 ) mixture at _T_ = 298.15 K," Journal of Chemical Thermodynamics, vol. 37 no. 11, pp. 1162-1167, DOI: 10.1016/j.jct.2005.02.007, 2005.

[11] K. R. Sun, S. Y. Du, Y. Q. Meng, H. Y. Zheng, Y. F. Guo, T. L. Deng, "Thermodynamic properties for the aqueous solutions of cesium borates at 298.15 K and 101 kPa: Experimental and correlation by Pitzer ion-interaction model," Journal of Molecular Liquids, vol. 318,DOI: 10.1016/j.molliq.2020.114272, 2020.

[12] F. Yuan, L. Li, Y. F. Guo, T. L. Deng, "Dilution enthalpies of LiBO 2 and LiB 5 O 8 aqueous solutions at 298.15 K and the application of ion-interaction model," Thermochimica Acta, vol. 685, 2019.

[13] F. Yuan, Y. L. Zhang, J. T. Song, L. Li, Y. F. Guo, S. Q. Wang, T. L. Deng, "Dissolution enthalpies and the thermodynamic properties of sodium metaborates," Journal of Molecular Liquids, vol. 315,DOI: 10.1016/j.molliq.2020.113813, 2020.

[14] M. C. Hu, J. Tang, S. N. Li, S. P. Xia, R. F. Cui, "Activity coefficients of cesium chloride and cesium sulfate in aqueous mixtures using an electromotive force method at 298.15 K," Journal of Chemical and Engineering Data, vol. 52 no. 6, pp. 2224-2227, DOI: 10.1021/je7002435, 2007.

[15] X. T. Huang, S. N. Li, Q. G. Zhai, Y. C. Jiang, M. C. Hu, "Thermodynamic investigation of RbF + Rb 2 SO 4 + H 2 O and CsF + Cs 2 SO 4 + H 2 O ternary systems by potentiometric method at 298.2 K," Fluid Phase Equilibria, vol. 433, pp. 31-39, DOI: 10.1016/j.fluid.2016.11.006, 2017.

[16] Z. D. Dou, S. N. Li, Q. G. Zhai, Y. C. Jiang, M. C. Hu, "Potentiometric investigation of the thermodynamic properties of mixed electrolyte systems at 298.2 K: CsF + CsBr + H 2 O and CsF + CsNO3+ H 2 O," Journal of Chemical and Engineering Data, vol. 63 no. 10, pp. 3801-3808, DOI: 10.1021/acs.jced.8b00383, 2018.

[17] M. C. Hu, W. J. Zhang, S. N. Li, Q. G. Zhai, Y. C. Jiang, Y. Li, J. Wang, N. Chen, "Thermodynamic study of the mixed system (CsCl + CaCl 2 + H 2 O) by EMF Measurements at _T_ = 298.15 K," Journal of Chemical Thermodynamics, vol. 41 no. 9, pp. 1016-1019, DOI: 10.1016/j.jct.2009.04.007, 2009.

[18] M. C. Hu, W. J. Zhang, S. N. Li, Q. G. Zhai, Y. C. Jiang, "Thermodynamic investigation of a ternary mixed electrolyte (CsCl/MgCl2/H2O) system using electromotive force measurement at 298.15 K," Journal of Chemical and Engineering Data, vol. 54 no. 7, pp. 2023-2027, DOI: 10.1021/je800860p, 2009.

[19] H. T. Kim, W. J. Frederick, "Evaluation of Pitzer ion interaction parameters of aqueous electrolytes at 25.degree.C. 1. Single salt parameters," Journal of Chemical and Engineering Data, vol. 33 no. 2, pp. 177-184, DOI: 10.1021/je00052a035, 1988.

[20] M. F. Zhou, S. H. Sang, J. J. Zhang, J. X. Hu, S. Y. Zhong, "Studies on mean activity coefficients of NaBr in NaBr + SrBr 2 + H 2 O ternary system at 298.15 K by EMF method," Journal of Chemical and Engineering Data, vol. 59 no. 11, pp. 3779-3784, DOI: 10.1021/je500655k, 2014.

[21] H. S. Harned, B. B. Owen, C. V. King, "The physical chemistry of electrolytic solutions, third edition," Journal of the Electrochemical Society, vol. 106 no. 1,DOI: 10.1149/1.2427250, 1959.

[22] X. T. Huang, S. N. Li, Q. G. Zhai, Y. C. Jiang, M. C. Hu, "Thermodynamic studies of (RbF + RbCl + H 2 O) and (CsF + CsCl + H 2 O) ternary systems from potentiometric measurements at _T_ = 298.2 K," Journal of Chemical Thermodynamics, vol. 103, pp. 157-164, DOI: 10.1016/j.jct.2016.08.014, 2016.

[23] C. E. Harvie, N. Moller, J. H. Weare, "The prediction in mineral solubilities in natural waters: the Na – K–Mg–Ca – H–Cl - SO 4 – OH-HCO 3 - CO 3 - CO 2 - H 2 O system to high ionic strengths at 25 °C," Geochimica et Cosmochimica Acta, vol. 48 no. 4, pp. 723-751, DOI: 10.1016/0016-7037(84)90098-X, 1984.