A Two-Parameter Theoretical Model for Predicting the Activity and Osmotic Coefficients of Aqueous Electrolyte Solutions

Abstract

A thermodynamic model of electrolyte solutions is proposed. The model consists of a long-range term expressed by the Pitzer–Debye–Hückel equation (PDH) and a short-range term expressed by the molecular interaction volume model (MIVM). The new model was fitted with 39 different types of single electrolyte systems and was compared with the Pitzer equation, and the mean standard deviation (SD) and the mean average relative deviation (ARD%) are 0.0264, 0.0040 and 2.09%, 0.40%, respectively. Meanwhile, the physical meaning of the two electrolyte-specific interaction parameters ($B_{ca,s}$ and $B_{s,ca}$) of the new model is also discussed. By further comparison with the Pitzer equation and a state-of-the-art model, eUNIQUAC-NRF, the new model exhibits relatively robust extrapolation capability, and also shows the potential ability to predict the activity coefficients of individual ions. In addition, only using binary parameters to predict 29 different types of ternary systems, the overall prediction results of the new model are slightly better than those of the Pitzer equation, and the mean SD and ARD% are 0.0288, 0.0396 and 2.88%, 3.81%, respectively. For some cases involving Rb and Cs, the Pitzer equation needs two ternary adjustable parameters ($\theta$ and $\psi$) to achieve the prediction accuracy of the new model. Furthermore, we also compared the predictions of the new model with the eUNIQUAC-NRF model for several ternary systems; the new model also shows better performance, and its overall prediction accuracy was about twice that of the eUNIQUAC-NRF model, with the average SD and ARD% values being 0.0261, 0.0546 and 2.63%, 5.80%, respectively.

Appendices

Appendix A: Brief Introduction of MIVM

According to statistical thermodynamics, the canonical partition function of pure liquid i is:

$$Q_{i} = Q_{Ti} Q_{pi} /N_{i} !$$

(19)

where $Q_{Ti} = \left[ {\left( {2\pi mkT} \right)^{1/2} /h} \right]^{{3N_{i} }}$ and $Q_{pi}$ are the translation partition function and the configurational partition function of i molecules, respectively. $Q_{pi}$ can be expressed as:

$$Q_{pi} = \int\limits_{{V_{i} }} \ldots \int {\exp \left( { - E_{pi} /kT} \right)} dx_{1} dy_{1} dz_{1} \ldots dx_{{N_{i} }} dy_{{N_{i} }} dz_{{N_{i} }}$$

(20)

where $V_{i}$ is the volume, $E_{pi}$ is the potential energy, and $N_{i}$ is the molecular number of i, $k$ is the Boltzmann constant. Based on the lattice theory of solutions [29], $E_{pi}$ may be chosen as:

$$E_{pi} = \frac{1}{2}Z_{i} N_{i} \varepsilon_{ii}$$

(21)

where $Z_{i}$ is the nearest molecule or first coordination number. $\varepsilon_{ii}$ is the i–i pair potential energy.

The MIVM originates from the physical picture described in the Introduction. Thus, substituting Eq. 21 into Eq. 20 and integrating it, Eq. 20 can be simplified as:

$$\begin{aligned} Q_{pi} &= \left[ {\iiint_{ + \infty } {\exp \left( { - \frac{{Z_{i} \varepsilon_{ii} }}{2kT}} \right)dxdydz}} \right]^{{N_{i} }} \\ &= \left[ {\int\limits_{0}^{\pi } {\sin \theta d\theta \int\limits_{0}^{2\pi } {d\varphi \int\limits_{0}^{{r_{i} }} {r^{2} \exp \left( { - \frac{{Z_{i} \varepsilon_{ii} }}{2kT}} \right)dr} } } } \right]^{{N_{i} }} \\ &= \left( {\frac{{V_{i} }}{{N_{i} }}} \right)^{{N_{i} }} \exp \left( { - \frac{{Z_{i} N_{i} \varepsilon_{ii} }}{2kT}} \right) \\ \end{aligned}$$

(22)

where $r_{i}$ is the average radius of molecular cell of i.

For a C-component system, the partition functions of real and ideal solutions are expressed as follows:

$$Q = \prod\limits_{i = 1}^{C} {Q_{Ti} Q_{p} } /N_{i} !$$

(23)

$$Q^{id} = \prod\limits_{i = 1}^{C} {Q_{i} }$$

(24)

where C is the number of components, and $Q_{p}$ is the configurational partition function, expressed as:

$$\begin{aligned} Q_{p} &= \int\limits_{V} \ldots \int {\exp \left( { - \frac{{\varepsilon_{p} }}{kT}} \right)} \prod\limits_{i = 1}^{C} {\left( {dx_{1} dy_{1} dz_{1} ...dx_{{N_{i} }} dy_{{N_{i} }} dz_{{N_{i} }} } \right)} \\ &= \prod\limits_{i = 1}^{C} {\left[ {\iiint_{ \pm \infty } {\exp \left( { - \frac{{\varepsilon_{p} }}{kT}} \right)}dx_{i} dy_{i} dz_{i} } \right]^{{N_{i} }} } \\ &= \left[ {\iiint_{ \pm \infty } {\exp \left( { - \frac{{\varepsilon_{p} }}{kT}} \right)}dxdydz} \right]^{N} \\ &= \left( \frac{V}{N} \right)^{N} \exp \left( { - \frac{{N\varepsilon_{p} }}{kT}} \right) \\ \end{aligned}$$

(25)

where $V$ and $N$ are the volume and molecular number of the system, respectively, and $\varepsilon_{p}$ is the potential energy function of mixing of its C kinds of molecules. According to the relation between Gibbs energy and partition function:

$$G = kT\left[ {V\left( {\frac{\partial \ln Q}{{\partial V}}} \right)_{T} - \ln Q} \right]$$

(26)

One can get the real and ideal Gibbs energies of the system, respectively:

$$G = kT\left[ {N - N\ln \left( \frac{V}{N} \right) + \frac{{N\varepsilon_{p} }}{kT} - \ln \left( {\prod\limits_{i = 1}^{C} {N_{i} !Q_{Ti} } } \right)} \right]$$

(27)

$$G^{{{\text{id}}}} = kT\left[ {\sum\limits_{i = 1}^{C} {N_{i} } - \sum\limits_{i = 1}^{C} {N_{i} \ln \left( {\frac{{V_{i} }}{{N_{i} }}} \right)} + \sum\limits_{i = 1}^{C} {\frac{{Z_{i} N_{i} \varepsilon_{ii} }}{kT}} - \ln \left( {\prod\limits_{i = 1}^{C} {N_{i} !Q_{Ti} } } \right)} \right]$$

(28)

Therefore, the excess Gibbs energy of the system can be expressed as:

$$\begin{aligned} G^{{{\text{ex}}}} &= G - G^{{{\text{id}}}} \\ &= kT\left[ { - N\ln \left( \frac{V}{N} \right) + \sum\limits_{i = 1}^{C} {N_{i} \ln \left( {\frac{{V_{i} }}{{N_{i} }}} \right)} + \frac{N}{2kT}\left( {2\varepsilon_{p} - \sum\limits_{i = 1}^{C} {\frac{{Z_{i} N_{i} \varepsilon_{ii} }}{kT}} } \right)} \right] \\ &= nRT\left[ {\sum\limits_{i = 1}^{C} {x_{i} \ln \left( {\frac{{V_{\text{m}i} }}{{V_{\text{m}} }}} \right)} + \frac{1}{2kT}\left( {2\varepsilon_{p} - \sum\limits_{i = 1}^{C} {\frac{{Z_{i} N_{i} \varepsilon_{ii} }}{kT}} } \right)} \right] \\ &= nRT\left[ {\sum\limits_{i = 1}^{C} {x_{i} \ln \left( {\frac{{\varphi_{i} }}{{x_{i} }}} \right)} + \frac{{\Delta \varepsilon_{p} }}{2kT}} \right] \\ \end{aligned}$$

(29)

where $n$ is molar number of the system, $V_{\text{m}i}$ and $V_{\text{m}}$ are the molar volumes of i and the system, respectively.$\varphi_{i} = x_{i} V_{\text{m}i} /V_{\text{m}}$ is the molar volume fraction of component i in the system. $\Delta \varepsilon_{p}$ is the excess potential energy function of the system:

$$\Delta \varepsilon_{p} = 2\varepsilon_{p} - \sum\limits_{i = 1}^{C} {Z_{i} x_{i} \varepsilon_{ii} }$$

(30)

According to multi-liquid theory [29], the potential energy function $\varepsilon_{p}$ of mixing of molecules can be chosen as:

$$\varepsilon_{p} = \frac{1}{2}\sum\limits_{i = 1}^{C} {Z_{i} } x_{i} \left( {\sum\limits_{j = 1}^{C} {x_{ji} \varepsilon_{ji} } } \right)$$

(31)

If $\Delta \varepsilon_{p} = 0$, then Eq. 29 may be reduced to:

$$G^{{{\text{ex}}}} = nRT\sum\limits_{i = 1}^{C} {x_{i} \ln \left( {\frac{{\varphi_{i} }}{{x_{i} }}} \right)}$$

(32)

which is the well-known Flory–Huggins equation.

Equation 29 can be considered as the theoretical form of the MIVM; it only contains properties and parameters of the pure components. Please see reference [27] for more detailed information about the MIVM.

Appendix B: The Derivation of the Electrolyte MIVM (eMIVM)

Before deducing eMIVM, let's start with two special features of the MIVM:

• (1) The local compositions are defined by the local coordination numbers [31].

Suppose that in the C system there are C types of molecular cells whose central molecules are the corresponding molecules of C components. Then, the local coordination numbers of the cell i are $Z_{ji}$ and $Z_{ii}$ that are defined as the molecular numbers of components j and i surrounding the central molecule i. They are also proportional to their corresponding Boltzmann’s factor:

$$Z_{ji} = Z_{i} x_{j} \exp \left( { - \frac{{\varepsilon_{ji} }}{kT}} \right),\;Z_{ii} = Z_{i} x_{i} \exp \left( { - \frac{{\varepsilon_{ii} }}{kT}} \right)\;\;\;i = 1, \ldots ,C\;\;j = 1, \ldots ,C$$

(33)

Based on the concept of the local composition of Wilson [28], the local mole fractions of components j and i around central molecule i can be expressed as follows:

$$x_{ji} = \frac{{Z_{ji} }}{{\sum\limits_{j = 1}^{C} {Z_{ji} } }} = \frac{{x_{j} B_{ji} }}{{\sum\limits_{j = 1}^{C} {x_{j} B_{ji} } }},\;x_{ii} = \frac{{Z_{ii} }}{{\sum\limits_{j = 1}^{C} {Z_{ji} } }} = \frac{{x_{i} }}{{\sum\limits_{j = 1}^{C} {x_{j} B_{ji} } }}\;\;\;i = 1, \ldots ,C$$

(34)

where the pair potential energy parameter $B_{ji}$ is defined as:

$$B_{ji} = \exp \left( { - \frac{{\varepsilon_{ji} - \varepsilon_{ii} }}{kT}} \right)$$

(35)

Further, the relationship between $x_{ji}$ and $x_{ii}$ can be obtained:

$$\frac{{x_{ji} }}{{x_{ii} }} = \frac{{Z_{ji} }}{{Z_{ii} }} = \frac{{Z_{i} x_{j} \exp \left( { - \frac{{\varepsilon_{ji} }}{kT}} \right)}}{{Z_{i} x_{i} \exp \left( { - \frac{{\varepsilon_{ii} }}{kT}} \right)}} = \frac{{x_{j} }}{{x_{i} }}B_{ji}$$

(36)

For convenience in representing other local mole fraction ratios, the following expressions are introduced:

$$\frac{{x_{ji} }}{{x_{li} }} = \frac{{x_{j} B_{ji} }}{{x_{l} B_{li} }} = \frac{{x_{j} }}{{x_{l} }}B_{ji,li} ,\;\;l = 1, \ldots ,C$$

(37)

$$B_{ji,li} = \frac{{B_{ji} }}{{B_{li} }} = \exp \left( { - \frac{{\varepsilon_{ji} - \varepsilon_{li} }}{kT}} \right)$$

(38)

Since MIVM is a local composition model, we want to replace the molar volume fraction of component i ($\varphi_{i}$) in Eq. 29 with the local molar volume fraction of component i ($\xi_{i}$). The local molar volume fraction of component i ($\xi_{i}$) can be expressed as:

$$\xi_{i} = \frac{{x_{ii} V_{\text{m}i} }}{{\sum\limits_{j = 1}^{C} {x_{ji} V_{\text{m}j} } }} = \frac{{x_{i} V_{\text{m}i} }}{{\sum\limits_{j = 1}^{C} {x_{j} V_{\text{m}j} B_{ji} } }},\;\;i = 1, \ldots ,C$$

(39)

Similar expressions can also be obtained for cells with a central component j.

• (2) MIVM defines a clear coordination number.

This feature of the MIVM makes the eMIVM also have a clear coordination number. In the eMIVM, we consider three cases, where the coordination number is 6, 8 and 10, respectively, and find that the fitting quality of the model is not sensitive to the coordination number used. A value of 10 for the coordination number yields the minimum value of the deviation for majority of electrolytes. Therefore, in this paper, the coordination numbers of species are all taken as 10.

The extension of MIVM to electrolyte solutions is mainly according to the idea described in Sect. 2.2. In more detail: since this work assumes that the electrolyte dissociates completely in aqueous solution, three kinds of local cells will appear in the solution, namely, cells with a central ion (either an anion or cation) surrounded by other counterions and molecules, and cells with a central molecule surrounded by anions, cations and molecules. Two key hypotheses proposed by Chen [9, 10] are adopted, namely, the like-ion repulsion and the local electroneutrality. The central ion cells satisfy the like-ion repulsion assumption, i.e., no anions are allowed in cells with a central anion, no cations are allowed in cells with a central cation. The central molecule cells satisfy the local electroneutrality assumption, i.e., the net local charge around a central molecule is zero. For the three kinds of local cells, the following normalizing equations hold:

$$\sum\limits_{c^{\prime}} {x_{c^{\prime}s} } + \sum\limits_{a^{\prime}} {x_{a^{\prime}s} } + \sum\limits_{s^{\prime}} {x_{s^{\prime}s} } = 1\;\;\;\;\left( {\text{central}}\, {\text{molecule}} \, {\text{cells}} \right)$$

(40)

$$\sum\limits_{a^{\prime}} {x_{a^{\prime}c} } + \sum\limits_{s^{\prime}} {x_{s^{\prime}c} } = 1\;\;\;\;\left( {\text{central}} \,{\text{cation}}\, {\text{cells}} \right)$$

(41)

$$\sum\limits_{c^{\prime}} {x_{c^{\prime}a} } + \sum\limits_{s^{\prime}} {x_{s^{\prime}a} } = 1\;\;\;\;\;\left( {\text{central}} \, {\text{anion}}\, {\text{cells}} \right)$$

(42)

where $s$ and $s^{\prime}$ denote molecular species, $a$ and $a^{\prime}$ denote anionic species, and $c$ and $c^{\prime}$ denote cationic species.

By combining Eqs. 36 and 37 and Eqs. 40–42, the following expressions for the local mole fractions in term of overall mole fractions may be derived:

$$x_{is} = \frac{{x_{i} B_{is} }}{{\sum\limits_{c^{\prime}} {x_{c^{\prime}} B_{c^{\prime}s} } + \sum\limits_{a^{\prime}} {x_{a^{\prime}} B_{a^{\prime}s} } + \sum\limits_{s^{\prime}} {x_{s^{\prime}} B_{s^{\prime}s} } }}(i = a,c,s)$$

(43)

$$x_{ic} = \frac{{x_{i} }}{{\sum\limits_{a^{\prime}} {x_{a^{\prime}} } B_{a^{\prime}c,ic} + \sum\limits_{s^{\prime}} {x_{s^{\prime}} B_{s^{\prime}c,ic} } }}(i = a,s)$$

(44)

$$x_{ia} = \frac{{x_{i} }}{{\sum\limits_{c^{\prime}} {x_{c^{\prime}} } B_{c^{\prime}a,ia} + \sum\limits_{s^{\prime}} {x_{s^{\prime}} B_{s^{\prime}a,ia} } }}(i = c,s)$$

(45)

To obtain an expression for the excess Gibbs energy of the new model, we need to modify the energy term and the volume term on the right side of Eq. 29. Let’s modify the energy term first. According to the three kinds of local cells in electrolyte solution, Eq. 31 may be changed into:

$$\begin{aligned} \varepsilon_{p} &= \frac{1}{2}\sum\limits_{i = 1}^{C} {Z_{i} x_{i} \left( {\sum\limits_{j = 1}^{C} {x_{ji} } \varepsilon_{ji} } \right)} \\ &= \frac{1}{2}\left[ \begin{gathered} \sum\limits_{s} {Z_{s} x_{s} \left( {\sum\limits_{s^{\prime}} {x_{s^{\prime}s} \varepsilon_{s^{\prime}s} + } \sum\limits_{c^{\prime}} {x_{c^{\prime}s} \varepsilon_{c^{\prime}s} + } \sum\limits_{a^{\prime}} {x_{a^{\prime}s} \varepsilon_{a^{\prime}s} } } \right)} \\ + \sum\limits_{c} {Z_{c} x_{c} \left( {\sum\limits_{s^{\prime}} {x_{s^{\prime}c} \varepsilon_{s^{\prime}c} + } \sum\limits_{a^{\prime}} {x_{a^{\prime}c} \varepsilon_{a^{\prime}c} } } \right)} \\ + \sum\limits_{a} {Z_{a} x_{a} \left( {\sum\limits_{s^{\prime}} {x_{s^{\prime}a} \varepsilon_{s^{\prime}a} + } \sum\limits_{c^{\prime}} {x_{c^{\prime}a} \varepsilon_{c^{\prime}a} } } \right)} \\ \end{gathered} \right] \\ \end{aligned}$$

(46)

The pure component state, taken as the reference state for molecules, and the hypothetical homogeneously mixed, completely dissociated liquid electrolyte mixture is adopted as the reference state for electrolytes. Therefore, the reference potential energy term $\sum\limits_{i = 1}^{C} {Z_{i} x_{i} \varepsilon_{ii} }$ in Eq. 30 may be changed as:

$$\sum\limits_{i = 1}^{C} {Z_{i} x_{i} \varepsilon_{ii} } = \sum\limits_{s} {Z_{s} x_{s} \varepsilon_{ss} } + \sum\limits_{c} {Z_{c} x_{c} \left( {\frac{{\sum\limits_{a^{\prime}} {x_{a^{\prime}} \varepsilon_{a^{\prime}c} } }}{{\sum\limits_{a^{\prime\prime}} {x_{a^{\prime\prime}} } }}} \right)} + \sum\limits_{a} {Z_{a} x_{a} \left( {\frac{{\sum\limits_{c^{\prime}} {x_{c^{\prime}} \varepsilon_{c^{\prime}a} } }}{{\sum\limits_{c^{\prime\prime}} {x_{c^{\prime\prime}} } }}} \right)}$$

(47)

Substituting Eqs. 46 and 47 into Eq. 30, then the excess potential energy function $\Delta \varepsilon_{p}$ based on the three kinds of local cells can be expressed as:

$$\begin{gathered} \Delta \varepsilon_{p} = 2\varepsilon_{p} - \sum\limits_{i = 1}^{C} {Z_{i} x_{i} \varepsilon_{ii} } \\ = \sum\limits_{s} {Z_{s} x_{s} \left[ {\left( {\sum\limits_{s^{\prime}} {x_{s^{\prime}s} \varepsilon_{s^{\prime}s} + } \sum\limits_{c^{\prime}} {x_{c^{\prime}s} \varepsilon_{c^{\prime}s} + } \sum\limits_{a^{\prime}} {x_{a^{\prime}s} \varepsilon_{a^{\prime}s} } } \right) - \varepsilon_{ss} } \right]} \\ + \sum\limits_{c} {Z_{c} x_{c} \left[ {\left( {\sum\limits_{s^{\prime}} {x_{s^{\prime}c} \varepsilon_{s^{\prime}c} + } \sum\limits_{a^{\prime}} {x_{a^{\prime}c} \varepsilon_{a^{\prime}c} } } \right) - \left( {\frac{{\sum\limits_{a^{\prime}} {x_{a^{\prime}} \varepsilon_{a^{\prime}c} } }}{{\sum\limits_{a^{\prime\prime}} {x_{a^{\prime\prime}} } }}} \right)} \right]} \\ + \sum\limits_{a} {Z_{a} x_{a} \left[ {\left( {\sum\limits_{s^{\prime}} {x_{s^{\prime}a} \varepsilon_{s^{\prime}a} + } \sum\limits_{c^{\prime}} {x_{c^{\prime}a} \varepsilon_{c^{\prime}a} } } \right) - \left( {\frac{{\sum\limits_{c^{\prime}} {x_{c^{\prime}} \varepsilon_{c^{\prime}a} } }}{{\sum\limits_{c^{\prime\prime}} {x_{c^{\prime\prime}} } }}} \right)} \right]} \\ \end{gathered}$$

(48)

The next step is to modify the volume term. As stated previously, the molar volume fraction ($\varphi_{i}$) in Eq. 29 is replaced by the local molar volume fraction ($\xi_{i}$), and based on the assumption of the like-ion repulsion, there are no ions of like charge around the central ion (i.e.,$x_{aa} = x_{cc} = 0$); therefore, the local molar volume fraction of the central ion cells (modified Eq. 39) are zero:

$$\zeta_{c} = \frac{{x_{cc} V_{\text{m}c} }}{{\sum\limits_{c^{\prime}} {x_{c^{\prime}c} V_{\text{m}c^{\prime}} + \sum\limits_{a^{\prime}} {x_{a^{\prime}c} V_{\text{m}a^{\prime}} + } \sum\limits_{s^{\prime}} {x_{s^{\prime}c} V_{\text{m}s^{\prime}} } } }} = 0$$

(49)

$$\zeta_{a} = \frac{{x_{aa} V_{\text{m}a} }}{{\sum\limits_{c^{\prime}} {x_{c^{\prime}a} V_{\text{m}c^{\prime}} + \sum\limits_{a^{\prime}} {x_{a^{\prime}a} V_{\text{m}a^{\prime}} + } \sum\limits_{s^{\prime}} {x_{s^{\prime}a} V_{\text{m}s^{\prime}} } } }} = 0$$

(50)

while the local molar volume fraction of the central molecule cell ($\xi_{s}$) is remains in the volume term, and by combining with Eq. 43, $\xi_{s}$ can be obtained as follows:

$$\begin{aligned} \zeta_{s} &= \frac{{x_{ss} V_{\text{m}s} }}{{\sum\limits_{s^{\prime}} {x_{s^{\prime}s} V_{\text{m}s^{\prime}} + \sum\limits_{c^{\prime}} {x_{c^{\prime}s} V_{\text{m}c^{\prime}} + } \sum\limits_{a^{\prime}} {x_{a^{\prime}s} V_{\text{m}a^{\prime}} } } }} \\ & = \frac{{x_{s} V_{\text{m}s} }}{{\sum\limits_{s^{\prime}} {x_{s^{\prime}} B_{s^{\prime}s} V_{\text{m}s^{\prime}} + \sum\limits_{c^{\prime}} {x_{c^{\prime}} B_{c^{\prime}s} V_{\text{m}c^{\prime}} + } \sum\limits_{a^{\prime}} {x_{a^{\prime}} B_{a^{\prime}s} V_{\text{m}a^{\prime}} } } }} \\ & = \frac{{x_{s} V_{\text{m}s} }}{{\sum\limits_{k} {x_{k} B_{ks} V_{\text{m}k} } }} \\ \end{aligned}$$

(51)

Now replace the molar volume fraction ($\varphi_{i}$) in Eq. 29 with the local molar volume fraction ($\xi_{i}$) of Eqs. 49–51, and then substitute Eq. 48 into Eq. 29 and combine it with Eqs. 43–45. Finally, we can obtain the expression of the molar excess Gibbs energy of the short range term of the new model as shown in Eq. 5.

The relationship between the partial molar and molar quantity (RPMQ) at constant temperature and pressure:

$$\overline{G}_{i}^{{{\text{ex}}}} = RT\ln \gamma_{i} = \left[ {\frac{{\partial \left( {nG_{\text{m}}^{{{\text{ex}}}} } \right)}}{{\partial n_{i} }}} \right]_{{T,p,n_{j \ne i} }} = G_{\text{m}}^{{{\text{ex}}}} + \left( {\frac{{\partial G_{\text{m}}^{{{\text{ex}}}} }}{{\partial x_{i} }}} \right)_{{T,P,x_{k \ne i} }} - \sum\limits_{j = 1}^{C - 1} {x_{j} } \left( {\frac{{\partial G_{\text{m}}^{{{\text{ex}}}} }}{{\partial x_{j} }}} \right)_{{T,P,x_{k \ne j} }}$$

(52)

The activity coefficients of any species can be obtained by combining Eqs. 5 and 52. The activity coefficient equations for any species are given in Eqs. 6–8.

Here it is worth noting that based on the conclusion that RPMQ and the Gibbs–Duhem equation is equivalent [105], the thermodynamic consistency of these activity coefficient expressions can be verified and confirmed by using only the sum formula of the molar excess Gibbs energy:

$$G_{\text{m}}^{{\text{ex,MIVM}}} = RT\left( {\sum\limits_{c} {x_{c} \ln \gamma_{c}^{{{\text{MIVM}}}} } + \sum\limits_{a} {x_{a} \ln \gamma_{a}^{{{\text{MIVM}}}} } + \sum\limits_{s} {x_{s} \ln \gamma_{s}^{{{\text{MIVM}}}} } } \right)$$

(53)

Appendix C: Model Parameters

3.1 Binary Parameters

By applying the assumption of local electroneutrality to the central molecule cell, one can get:

$$z_{a} x_{as} = z_{c} x_{cs}$$

(54)

Substituting Eq. 36 into Eq. 54, and because the electrolyte solution is electrically neutral (i.e.,$z_{a} x_{a} = z_{c} x_{c}$), hence the following relationship can be obtained:

$$\varepsilon_{as} = \varepsilon_{cs}$$

(55)

Since the pair potential energies are symmetric (i.e.$\varepsilon_{ji} = \varepsilon_{ij}$), it can be further deduced that:

$$B_{as} = B_{cs} = B_{ca,s}$$

(56)

$$B_{sa,ca} = B_{sc,ac} = B_{s,ca}$$

(57)

For a binary electrolyte system, $B_{ca,s}$ and $B_{s,ca}$ are the only two adjustable electrolyte-specific parameters in the new model.

3.2 Multicomponent Parameters

In the same way, the assumption of local electroneutrality applied to the central molecule cell, and the following relation can be obtained:

$$\sum\limits_{c} {x_{cs} } z_{c} = \sum\limits_{a} {x_{as} } z_{a}$$

(58)

Again, substituting Eq. 36 into Eq. 58, and because solution electroneutrality requires $\sum\limits_{c} {x_{c} } z_{c} = \sum\limits_{a} {x_{a} } z_{a}$, it follows that:

$$\sum\limits_{c} {z_{c} x_{c} B_{cs} } = \sum\limits_{a} {z_{a} x_{a} B_{as} }$$

(59)

because the local composition concept considers only two-body interactions and the like-ion repulsion is assumed. Therefore, Eq. 59 can be generalized to:

$$B_{cs} = \frac{{\sum\limits_{a} {z_{a} x_{a} B_{ca,s} } }}{{\sum\limits_{a^{\prime}} {z_{a^{\prime}} x_{a^{\prime}} } }}$$

(60)

$$B_{as} = \frac{{\sum\limits_{c} {z_{c} x_{c} B_{ca,s} } }}{{\sum\limits_{c^{\prime}} {z_{c^{\prime}} x_{c^{\prime}} } }}$$

(61)

$B_{ca,s}$ in Eqs. 60 and 61 is the binary parameter.

The variables $B_{sa,ca}$ and $B_{sc,ac}$ can be computed from the $B_{is}$(Eqs. 60 and 61):

$$\begin{aligned} B_{sa,ca} & = \exp \left( { - \frac{{\varepsilon_{sa} - \varepsilon_{ca} }}{kT}} \right) = \exp \left( { - \frac{{\varepsilon_{sa} - \varepsilon_{ss} }}{kT} - \frac{{\varepsilon_{ss} - \varepsilon_{ca} }}{kT}} \right) \\ & = \frac{{\exp \left( { - \frac{{\varepsilon_{sa} - \varepsilon_{ss} }}{kT}} \right)\exp \left( { - \frac{{\varepsilon_{sa} - \varepsilon_{ca} }}{kT}} \right)}}{{\exp \left( { - \frac{{\varepsilon_{sa} - \varepsilon_{ss} }}{kT}} \right)}} \\ &= \frac{{B_{as} B_{s,ca} }}{{B_{ca,s} }} \\ \end{aligned}$$

(62)

and

$$B_{sc,ac} = \frac{{B_{cs} B_{s,ca} }}{{B_{ca,s} }}$$

(63)

Furthermore, the salt–salt binary interaction parameters are also required in the new model, and these parameters are handled as follows [10]:

$$B_{ca,c^{\prime}a} = \exp \left( { - \frac{{\varepsilon_{ca} - \varepsilon_{c^{\prime}a} }}{kT}} \right) = \frac{1}{{\exp \left( { - \frac{{\varepsilon_{c^{\prime}a} - \varepsilon_{ca} }}{kT}} \right)}} = \frac{1}{{B_{c^{\prime}a,ca} }}$$

(64)

and

$$B_{ca,ca^{\prime}} = \frac{1}{{B_{ca^{\prime},ca} }}$$

(65)

The salt–salt binary interaction parameter is determined by fitting solubility data or activity and osmotic coefficient data in the ternary system. It can be seen that $B_{ca,s}$,$B_{s,ca}$,$B_{ss^{\prime}}$,$B_{s^{\prime}s}$,$B_{ca,c^{\prime}a}$,$B_{c^{\prime}a,ca}$,$B_{ca,ca^{\prime}}$ and $B_{ca^{\prime},ca}$ are the adjustable binary parameters for the multicomponent system. Specifically, for a ternary system with only one solvent component and a common cation, the $B_{ss^{\prime}}$,$B_{s^{\prime}s}$,$B_{ca,c^{\prime}a}$ and $B_{c^{\prime}a,ca}$ are equal to unity.

Appendix D: The Molar Volume of Species

The molar volumes of species are required in the new model. For the sake of simplicity, the ions are treated as spherical, so the molar volumes of ions can be calculated by the following formula [36]:

$$V_{\text{m}i} = \left( {\frac{{4\pi N_{{\text{A}}} }}{3}} \right)r_{i}^{3} = 2522.5r_{i}^{3}$$

(66)

where $N_{{\text{A}}}$ is the Avogadro constant,$r_{i}$(nm) is the ion radius in aqueous solution, and its values are tabulated in Table 7. The molar volume of pure water ($V_{ms}$) is 18.07 cm³·mol⁻¹ at 298.15 K.

Table 7 Ionic radii for cations and anions^a