# 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

Authors: Yanqin Meng; Libin Yang; Qing Xu; Yafei Guo; Tianlong Deng
Journal: Journal of Chemistry
 (2022)

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## 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 (mA), CsNO3 (mB)|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 θNa,Cs and ψNa,Cs,NO3 were obtained.


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## Body

## 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 θNa,Cs and ψNa,Cs,NO3 are obtained for the first time.


## 2. Experimental

All of the instructions of the chemical reagents used in this work are shown in Table1, 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.
RegentCASGradePurity in mass fractionAnalytical methodSodium nitratea7631-99-4GR0.9999Titration for NO3-Cesium nitratea7789-18-6GR0.9999Titration 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 (mA0)|NO3-ISE(b)
Na-ISE|CsNO3 (mB0)|NO3-ISE(c)
Na-ISE|NaNO3 (mA), CsNO3 (mB)|NO3-ISEAbove galvanic cells contain no liquid junction. Here,mA0 and mB0 are the molalities of NaNO3 and CsNO3 as single salt in water, mA and mB 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 Kpot 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 (yB) 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 ofmA0 from 0.01 to 4.5 mol·kg-1 were selected to determine each corresponding potential (Ea). The Nernst equation for cell (a) can be expressed as:
(1)Ea=E0+2k⋅lnγ±A0⋅m±A0.E0 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 γ±A0 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.
Electrolyteβ0β1CΦImaxσRefCsNO3-0.130040.081690.030181.5000.00057[19]NaNO30.003880.21151-0.0000610.8300.00073[19]To check their linear relationship,Ea was plotted against lnmA0γ±A and shown in Figure 1. By way of this line, E0, 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 1
The response line of Na-ISE and NO3-ISE electrode pair.


### 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 Kpot. The selective coefficient Kpot of electrode Na-ISE for Cs+ can be calculated according to the following equation:
(2)Kpot=expEb−E0/k/mB0⋅γ±B02,whereγ±B0 refers to the activity coefficients of pure CsNO3 in water at 298.15 K, and its value was taken and calculated according to the cited literature [19]. Eb is the EMF value of cell (b) at different mB0 and is shown in Table 3. The mean value of Kpot 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 concentrationsa.
ImCsNO3Ebmol·kg-1mol·kg-1mV0.19881990.1988199-269.60.24830.2483-258.30.39730.3973-229.60.49940.4994-223.20.60000.6000-219.5aStandard uncertainties u are uT =0.02 K, umCsNO3 =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 valuesEm of the ternary system (NaNO3 + CsNO3 + H2O) at different ionic strengths I and mole fractions yB (yB=IB/I=mB/mA+mB). The mean activity coefficients of NaNO3 in the ternary system can be derived from the following Nernst equation and shown in Table 3.
(3)Em=E0+k÷lnγ±A2⋅mA⋅mA+mB+Kpot⋅γ±B2⋅mB⋅mA+mB,where γ±A and γ±B are the mean activity coefficients of NaNO3 and CsNO3, respectively. Because Kpot can be neglected [20] without leading to an appreciable error, we get the simplified form the following equation:
(4)Em=E0+k⋅lnγ±A2⋅mA⋅mA+mB,(5)lnγ±A=Em−E0/2⋅k−1/2lnmA⋅mA+mB.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 2
A plot of the mean activity coefficients of NaNO3 against mole fractions in the system.


### 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:
(6)lnγ±A=lnγ±A0−αAB⋅yB−βAB⋅yB2,where αAB and βAB represent the Harned interaction parameters, dependent on both ionic strength and temperature [10, 22]. γ±A0 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 Ka.
IIyCsNO3mNaNO3mCsNO3Emγ±NaNO3mol·kg-1mol·kg-1mol·kg-1mV0.01000.00000.01000.0000-122.80.90000.01900.00000.01900.0000-89.20.86970.03800.00000.03800.0000-58.80.82940.05050.00000.05050.0000-45.20.81070.08050.00000.80500.000025.70.77720.09990.00000.09990.0000-15.10.76070.20030.00000.20030.000016.10.70350.24860.00000.24860.000027.60.68460.50490.00000.50490.000058.90.61950.59980.00000.59980.000069.20.60300.79510.00000.79510.000081.90.57540.99830.00000.99830.000089.60.55271.50860.00001.50860.0000106.10.51072.10270.00002.10270.0000121.00.47652.50220.00002.50220.0000126.10.45873.09170.00003.09170.0000139.60.43733.50570.00003.50570.0000141.80.42484.49640.00004.49640.0000154.10.40080.10020.10010.09020.0100-15.70.76400.19970.10010.17970.020015.90.70500.50000.10020.44990.050156.80.61970.59860.10010.53870.059964.80.60390.79890.09990.71910.079877.30.57581.00000.10000.90000.100086.90.55361.50110.10001.35110.1500103.90.51182.49690.10002.24720.2498124.60.45873.49030.10003.14130.3490140.00.44164.48340.10004.03510.4483150.00.41690.09930.19980.07950.0199-18.90.76850.19870.19990.15900.039713.00.71060.24960.19990.19970.049923.50.69270.49930.20010.39940.099954.10.62480.59220.20000.47380.118461.30.60520.80000.20000.64000.160074.30.57560.99500.20000.79600.199084.00.55801.50110.20021.20060.3005100.90.51242.48420.20041.98630.4970121.30.45893.50190.20002.80160.7004138.30.45170.01000.39970.00600.0040-134.70.94920.01980.40130.01190.0080-100.90.91650.04000.40140.02390.0160-68.50.84870.04980.40170.02980.0200-57.90.83440.09980.39990.05990.0399-25.70.77460.19920.39990.11950.07976.20.71800.24960.40010.14970.099916.60.70030.49730.40000.29840.198946.60.62680.59440.39990.35670.237754.10.60590.79600.39990.47770.318366.60.57580.99440.40010.59660.397876.90.56231.49970.40010.89960.600193.50.51350.01020.59400.00410.0061-142.60.96710.02010.59910.00810.0120-109.50.93660.04000.60040.01600.0240-78.40.85830.05010.59960.02000.0300-67.90.83790.10000.59950.04000.0599-35.80.77920.19980.60000.07990.1199-3.90.72160.24970.60030.09980.14996.50.70580.49960.60000.19990.299836.70.63130.59790.60030.23900.358943.90.60630.79860.59980.31960.479056.60.57960.99990.60000.39990.599967.00.56580.00990.80570.00190.0080-162.50.98440.01960.80240.00390.0157-128.10.95470.03980.80160.00790.0319-96.30.86560.04970.80360.00980.0399-86.10.84910.10010.80040.02000.0801-53.30.78680.19830.80000.03970.1587-21.90.72650.24990.80010.05000.1999-11.00.71170.49880.79970.09990.398918.90.63400.60110.79990.12030.480826.30.60700.79620.80000.15930.637038.50.57991.00070.79990.20020.800549.50.5704aStandard uncertainties u are uT =0.02 K, u(mNaNO3) =0.0007 mol·kg-1, u(mCsNO3) =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:
(7)lnγM=ZmF2+∑ama2BMa+2CMa+∑cmc2ΦMc+∑amaΨMca+∑a∑amama′ΨMaa′+ZM∑c∑amcmaCca,(8)lnγX=ZXF2+∑cmc2Bc+2CcX+∑ama2ΦXa+∑cmcΨcXa+∑c∑c′mcmc′Ψcc′X+ZX∑c∑amcmaCca,(9)lnγMX=lnγM+lnγX/2,(10)F=AϕI1/2/1+1.2I1/2+2/1.2ln1+1.21/2+∑c∑amcmaBca′+∑c∑c′mcmcΦcc′′+∑a∑a′mama′Φca′′,(11)Φij=θij+θEij,(12)Φij′=θ′Eij,(13)BZA=βCA0+βCA1gα1I1/2+βCA2gα2I1/2,(14)BCA′=βCA1g′α1I1/2+βCA2g′α1I1/2/I,(15)gx=21−1+xexp−x/x2,(16)g′x=−21−1+x+x2/2exp−x/x2,(17)Φ=1+1mNa+mCs+mNO3⋅2−Aϕ⋅I3/2/1+1.2⋅I1/2+mNa⋅mNO3⋅BNaNO3ϕ+ZCNaNO3ϕ+mCs⋅mNO3⋅BCsNO3ϕ+ZCCsNO3ϕ+mNa⋅mCs⋅ΦNa,Csϕ+mNO3⋅ψNa,Cs,NO3,where M, a, and a’ are anions. X, c, and c’ are cations. γM, ZM, and mc are the activity coefficient, valence number, and molar concentration of the cation, respectively. γX, ZX, and ma are the activity coefficient, valence number, and molar concentration of the anion, respectively. Aϕ is called the Debye−Hückel constant. Ψ is the ionic interaction parameters. B is the coefficient in the second dimension. For type 1-1 electrolytes, α1 = 2.0 mol·kg-1, α2 = 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 equationa.
I/mol·kg-1lnγ±A0αABβAB103·RMSD0.01-0.1053-0.15260.05180.90040.02-0.1396-0.14530.03620.00750.04-0.1871-0.06370.01260.87530.05-0.2095-0.07320.02195.28200.10-0.2733-0.04370.00301.92000.20-0.3516-0.05740.02231.81100.25-0.3789-0.06290.01851.01500.50-0.4790-0.03270.00401.22400.60-0.5058-0.01440.00780.61570.80-0.55300.00180.00123.50601.00-0.5929-0.04370.00581.15301.50-0.6719-0.02120.01920.27072.50-0.77930.0050-0.03680.00003.50-0.8561-0.46560.79280.0000aStandard uncertainties u are uT =0.02 K, u(mNaNO3) =0.0007 mol·kg-1, u(mCsNO3) =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 Ka.
I/mol·kg-1θNa,CsψNa,Cs,NO3RMSERef0.1-4.5-0.17500.025610.02This workaStandard uncertainties u are uT =0.02 K, u(mNaNO3) =0.0007 mol·kg-1, u(mCsNO3) =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 systema.
I/mol·kg-1yBγ±CsNO3ΦawGE/KJ·mol-10.100.00.73370.92190.9967-0.09680.10.73300.91820.9967-0.095120.20.73350.91540.9967-0.09270.40.73240.90980.9967-0.09290.60.73150.90580.9967-0.09570.80.73070.90310.9967-0.10020.200.00.65710.90410.9935-0.25410.10.65710.89750.9936-0.25150.20.65740.89170.9936-0.24520.40.65640.88170.9937-0.24580.60.65530.87430.9937-0.25600.80.65540.87020.9938-0.26750.250.00.63030.89820.9920-0.34160.20.62920.88270.9921-0.33300.40.62860.87060.9922-0.33420.60.62790.86190.9923-0.34720.80.62720.85650.9923-0.36870.500.00.53180.87760.9841-0.89220.10.53370.86300.9845-0.88370.20.53440.84990.9848-0.87000.40.53580.82870.9852-0.88420.60.53590.81340.9854-0.92060.80.53690.80500.9856-0.97340.600.00.50560.87210.9813-1.12400.10.50680.85500.9817-1.11840.20.50920.84030.9822-1.10700.40.51030.81550.9827-1.13500.60.51110.79780.9829-1.18770.80.51200.78780.9830-1.26090.800.00.46110.86270.9755-1.63720.10.46220.84090.9760-1.64300.20.46400.82170.9765-1.65410.40.46850.79120.9775-1.67990.60.47180.76990.9780-1.73710.80.47580.75910.9784-1.82461.000.00.42430.85460.9697-2.21480.10.42720.82890.9705-2.21200.20.43120.80670.9714-2.17850.40.43750.77070.9727-2.20250.60.44310.74590.9734-2.29070.80.44920.73350.9738-2.41121.500.00.35860.83870.9553-3.81970.10.36630.80550.9573-3.78600.20.37330.77610.9588-3.78150.40.38760.71000.9623-3.63672.50.00.29110.81800.9288-7.41000.10.30800.77720.9323-7.38370.20.32570.74200.9356-7.25643.50.00.26800.80460.9032-11.48290.10.29920.76690.9079-10.78520.20.33390.73510.9112-10.24714.50.00.27640.79570.8788-15.82820.10.33080.77140.8826-14.8822aStandard uncertainties u are uT =0.02 K, umA =0.0007 mol·kg-1, umB =0.0003 mol·kg-1, uE =0.1 mV, A = NaNO3, B = CsNO3.Figure 3
A plot of the mean activity coefficients of CsNO3 against ionic strengths in the ternary system.Figure 4
A plot of the osmotic coefficients against mole fractions in the ternary system.


### 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]:
(18)GE=R⋅T2mA1−Φ+lnγ±A+2mB1−Φ+lnγ±B,aw=exp−18.0513/500⋅mA+mBΦ.The calculated results are given in Table7 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 5
A plot of water activities against ionic strengths in the ternary system.Figure 6
A plot of excess Gibbs energies against ionic strengths in the ternary system.


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

For cell (a), 13 measurements ofmA0 from 0.01 to 4.5 mol·kg-1 were selected to determine each corresponding potential (Ea). The Nernst equation for cell (a) can be expressed as:
(1)Ea=E0+2k⋅lnγ±A0⋅m±A0.E0 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 γ±A0 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.
Electrolyteβ0β1CΦImaxσRefCsNO3-0.130040.081690.030181.5000.00057[19]NaNO30.003880.21151-0.0000610.8300.00073[19]To check their linear relationship,Ea was plotted against lnmA0γ±A and shown in Figure 1. By way of this line, E0, 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 1
The response line of Na-ISE and NO3-ISE electrode pair.


## 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 Kpot. The selective coefficient Kpot of electrode Na-ISE for Cs+ can be calculated according to the following equation:
(2)Kpot=expEb−E0/k/mB0⋅γ±B02,whereγ±B0 refers to the activity coefficients of pure CsNO3 in water at 298.15 K, and its value was taken and calculated according to the cited literature [19]. Eb is the EMF value of cell (b) at different mB0 and is shown in Table 3. The mean value of Kpot 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 concentrationsa.
ImCsNO3Ebmol·kg-1mol·kg-1mV0.19881990.1988199-269.60.24830.2483-258.30.39730.3973-229.60.49940.4994-223.20.60000.6000-219.5aStandard uncertainties u are uT =0.02 K, umCsNO3 =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 valuesEm of the ternary system (NaNO3 + CsNO3 + H2O) at different ionic strengths I and mole fractions yB (yB=IB/I=mB/mA+mB). The mean activity coefficients of NaNO3 in the ternary system can be derived from the following Nernst equation and shown in Table 3.
(3)Em=E0+k÷lnγ±A2⋅mA⋅mA+mB+Kpot⋅γ±B2⋅mB⋅mA+mB,where γ±A and γ±B are the mean activity coefficients of NaNO3 and CsNO3, respectively. Because Kpot can be neglected [20] without leading to an appreciable error, we get the simplified form the following equation:
(4)Em=E0+k⋅lnγ±A2⋅mA⋅mA+mB,(5)lnγ±A=Em−E0/2⋅k−1/2lnmA⋅mA+mB.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 2
A plot of the mean activity coefficients of NaNO3 against mole fractions in the system.


## 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:
(6)lnγ±A=lnγ±A0−αAB⋅yB−βAB⋅yB2,where αAB and βAB represent the Harned interaction parameters, dependent on both ionic strength and temperature [10, 22]. γ±A0 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 Ka.
IIyCsNO3mNaNO3mCsNO3Emγ±NaNO3mol·kg-1mol·kg-1mol·kg-1mV0.01000.00000.01000.0000-122.80.90000.01900.00000.01900.0000-89.20.86970.03800.00000.03800.0000-58.80.82940.05050.00000.05050.0000-45.20.81070.08050.00000.80500.000025.70.77720.09990.00000.09990.0000-15.10.76070.20030.00000.20030.000016.10.70350.24860.00000.24860.000027.60.68460.50490.00000.50490.000058.90.61950.59980.00000.59980.000069.20.60300.79510.00000.79510.000081.90.57540.99830.00000.99830.000089.60.55271.50860.00001.50860.0000106.10.51072.10270.00002.10270.0000121.00.47652.50220.00002.50220.0000126.10.45873.09170.00003.09170.0000139.60.43733.50570.00003.50570.0000141.80.42484.49640.00004.49640.0000154.10.40080.10020.10010.09020.0100-15.70.76400.19970.10010.17970.020015.90.70500.50000.10020.44990.050156.80.61970.59860.10010.53870.059964.80.60390.79890.09990.71910.079877.30.57581.00000.10000.90000.100086.90.55361.50110.10001.35110.1500103.90.51182.49690.10002.24720.2498124.60.45873.49030.10003.14130.3490140.00.44164.48340.10004.03510.4483150.00.41690.09930.19980.07950.0199-18.90.76850.19870.19990.15900.039713.00.71060.24960.19990.19970.049923.50.69270.49930.20010.39940.099954.10.62480.59220.20000.47380.118461.30.60520.80000.20000.64000.160074.30.57560.99500.20000.79600.199084.00.55801.50110.20021.20060.3005100.90.51242.48420.20041.98630.4970121.30.45893.50190.20002.80160.7004138.30.45170.01000.39970.00600.0040-134.70.94920.01980.40130.01190.0080-100.90.91650.04000.40140.02390.0160-68.50.84870.04980.40170.02980.0200-57.90.83440.09980.39990.05990.0399-25.70.77460.19920.39990.11950.07976.20.71800.24960.40010.14970.099916.60.70030.49730.40000.29840.198946.60.62680.59440.39990.35670.237754.10.60590.79600.39990.47770.318366.60.57580.99440.40010.59660.397876.90.56231.49970.40010.89960.600193.50.51350.01020.59400.00410.0061-142.60.96710.02010.59910.00810.0120-109.50.93660.04000.60040.01600.0240-78.40.85830.05010.59960.02000.0300-67.90.83790.10000.59950.04000.0599-35.80.77920.19980.60000.07990.1199-3.90.72160.24970.60030.09980.14996.50.70580.49960.60000.19990.299836.70.63130.59790.60030.23900.358943.90.60630.79860.59980.31960.479056.60.57960.99990.60000.39990.599967.00.56580.00990.80570.00190.0080-162.50.98440.01960.80240.00390.0157-128.10.95470.03980.80160.00790.0319-96.30.86560.04970.80360.00980.0399-86.10.84910.10010.80040.02000.0801-53.30.78680.19830.80000.03970.1587-21.90.72650.24990.80010.05000.1999-11.00.71170.49880.79970.09990.398918.90.63400.60110.79990.12030.480826.30.60700.79620.80000.15930.637038.50.57991.00070.79990.20020.800549.50.5704aStandard uncertainties u are uT =0.02 K, u(mNaNO3) =0.0007 mol·kg-1, u(mCsNO3) =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:
(7)lnγM=ZmF2+∑ama2BMa+2CMa+∑cmc2ΦMc+∑amaΨMca+∑a∑amama′ΨMaa′+ZM∑c∑amcmaCca,(8)lnγX=ZXF2+∑cmc2Bc+2CcX+∑ama2ΦXa+∑cmcΨcXa+∑c∑c′mcmc′Ψcc′X+ZX∑c∑amcmaCca,(9)lnγMX=lnγM+lnγX/2,(10)F=AϕI1/2/1+1.2I1/2+2/1.2ln1+1.21/2+∑c∑amcmaBca′+∑c∑c′mcmcΦcc′′+∑a∑a′mama′Φca′′,(11)Φij=θij+θEij,(12)Φij′=θ′Eij,(13)BZA=βCA0+βCA1gα1I1/2+βCA2gα2I1/2,(14)BCA′=βCA1g′α1I1/2+βCA2g′α1I1/2/I,(15)gx=21−1+xexp−x/x2,(16)g′x=−21−1+x+x2/2exp−x/x2,(17)Φ=1+1mNa+mCs+mNO3⋅2−Aϕ⋅I3/2/1+1.2⋅I1/2+mNa⋅mNO3⋅BNaNO3ϕ+ZCNaNO3ϕ+mCs⋅mNO3⋅BCsNO3ϕ+ZCCsNO3ϕ+mNa⋅mCs⋅ΦNa,Csϕ+mNO3⋅ψNa,Cs,NO3,where M, a, and a’ are anions. X, c, and c’ are cations. γM, ZM, and mc are the activity coefficient, valence number, and molar concentration of the cation, respectively. γX, ZX, and ma are the activity coefficient, valence number, and molar concentration of the anion, respectively. Aϕ is called the Debye−Hückel constant. Ψ is the ionic interaction parameters. B is the coefficient in the second dimension. For type 1-1 electrolytes, α1 = 2.0 mol·kg-1, α2 = 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 equationa.
I/mol·kg-1lnγ±A0αABβAB103·RMSD0.01-0.1053-0.15260.05180.90040.02-0.1396-0.14530.03620.00750.04-0.1871-0.06370.01260.87530.05-0.2095-0.07320.02195.28200.10-0.2733-0.04370.00301.92000.20-0.3516-0.05740.02231.81100.25-0.3789-0.06290.01851.01500.50-0.4790-0.03270.00401.22400.60-0.5058-0.01440.00780.61570.80-0.55300.00180.00123.50601.00-0.5929-0.04370.00581.15301.50-0.6719-0.02120.01920.27072.50-0.77930.0050-0.03680.00003.50-0.8561-0.46560.79280.0000aStandard uncertainties u are uT =0.02 K, u(mNaNO3) =0.0007 mol·kg-1, u(mCsNO3) =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 Ka.
I/mol·kg-1θNa,CsψNa,Cs,NO3RMSERef0.1-4.5-0.17500.025610.02This workaStandard uncertainties u are uT =0.02 K, u(mNaNO3) =0.0007 mol·kg-1, u(mCsNO3) =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 systema.
I/mol·kg-1yBγ±CsNO3ΦawGE/KJ·mol-10.100.00.73370.92190.9967-0.09680.10.73300.91820.9967-0.095120.20.73350.91540.9967-0.09270.40.73240.90980.9967-0.09290.60.73150.90580.9967-0.09570.80.73070.90310.9967-0.10020.200.00.65710.90410.9935-0.25410.10.65710.89750.9936-0.25150.20.65740.89170.9936-0.24520.40.65640.88170.9937-0.24580.60.65530.87430.9937-0.25600.80.65540.87020.9938-0.26750.250.00.63030.89820.9920-0.34160.20.62920.88270.9921-0.33300.40.62860.87060.9922-0.33420.60.62790.86190.9923-0.34720.80.62720.85650.9923-0.36870.500.00.53180.87760.9841-0.89220.10.53370.86300.9845-0.88370.20.53440.84990.9848-0.87000.40.53580.82870.9852-0.88420.60.53590.81340.9854-0.92060.80.53690.80500.9856-0.97340.600.00.50560.87210.9813-1.12400.10.50680.85500.9817-1.11840.20.50920.84030.9822-1.10700.40.51030.81550.9827-1.13500.60.51110.79780.9829-1.18770.80.51200.78780.9830-1.26090.800.00.46110.86270.9755-1.63720.10.46220.84090.9760-1.64300.20.46400.82170.9765-1.65410.40.46850.79120.9775-1.67990.60.47180.76990.9780-1.73710.80.47580.75910.9784-1.82461.000.00.42430.85460.9697-2.21480.10.42720.82890.9705-2.21200.20.43120.80670.9714-2.17850.40.43750.77070.9727-2.20250.60.44310.74590.9734-2.29070.80.44920.73350.9738-2.41121.500.00.35860.83870.9553-3.81970.10.36630.80550.9573-3.78600.20.37330.77610.9588-3.78150.40.38760.71000.9623-3.63672.50.00.29110.81800.9288-7.41000.10.30800.77720.9323-7.38370.20.32570.74200.9356-7.25643.50.00.26800.80460.9032-11.48290.10.29920.76690.9079-10.78520.20.33390.73510.9112-10.24714.50.00.27640.79570.8788-15.82820.10.33080.77140.8826-14.8822aStandard uncertainties u are uT =0.02 K, umA =0.0007 mol·kg-1, umB =0.0003 mol·kg-1, uE =0.1 mV, A = NaNO3, B = CsNO3.Figure 3
A plot of the mean activity coefficients of CsNO3 against ionic strengths in the ternary system.Figure 4
A plot of the osmotic coefficients against mole fractions in the ternary system.


## 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]:
(18)GE=R⋅T2mA1−Φ+lnγ±A+2mB1−Φ+lnγ±B,aw=exp−18.0513/500⋅mA+mBΦ.The calculated results are given in Table7 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 5
A plot of water activities against ionic strengths in the ternary system.Figure 6
A plot of excess Gibbs energies against ionic strengths in the ternary system.


## 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.



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