Introduction

Thermodynamic properties of a multicomponent dilute solution such as molten steel, copper, manganese are most well practically characterized by partial excess Gibbs energies of solutes: in other words, activity coefficients of solutes. For practical applications, Wagner proposed a Taylor series expansion for the partial excess Gibbs energy, and it has been most widely used in the metallurgical community.[1] Although this formalism is strictly valid only at infinite dilution, thus thermodynamically inconsistent at a finite concentration as discussed by Darken,[2,3] the formalism has been still used at finite concentrations. Depending on the system used, however, the formalism can cause significant errors.

Darken corrected the Wagner’s formalism to be thermodynamically consistent at finite concentration, but it has not been well appreciated. In 1986, Pelton and Bale proposed a modified form of the Wagner’s formalism,[4] which has been further discussed in their subsequent articles.[5,6] The basic idea of their modification is a correction to the Wagner’s formalism by adding activity coefficient of solvent to activity coefficients of all solutes. This makes the Gibbs–Duhem relation for the activity coefficients among the solvent and all the solutes be obeyed, even at finite concentration. They also stated that Maxwell’s relation for partial excess Gibbs energy should also be respected.[4,5,6] Their proposed formalism exactly reduces to Wagner’s formalism at infinite dilution, to Darken’s quadratic formalism at finite concentration, and also could be reduced to Lupis and Elliott’s formalism[7] as well as that of Margules.[8] They named it as a Unified Interaction Parameter Formalism (UIPF), and the formalism has been successfully applied to multicomponent metallic solutions.[9,10,11,12,13]

Some other approaches also have been proposed to make the Wagner’s formalism be thermodynamically consistent.[14,15,16,17,18] Srikanth and Jacob[14] attempted to keep the original expression of Wagner for the activity coefficient of solute, while the activity coefficient of solvent can be obtained by path-dependent integration of the Gibbs–Duhem equation. This is due to the thermodynamic inconsistency of the original expression of Wagner’s formalism. Hajra and coworkers suggested using the activity coefficients of solutes derived from a Maclaurin infinite series of integral excess Gibbs energy.[15,16,17] While the approaches by Srikanth and Jacob[14] and Hajra et al.[15,16,17] keep the original expression of Wagner for the activity coefficient of solute, the proposal by Darken,[2,3] Pelton and Bale[4,5,6] uses corrected forms of the activity coefficient of solutes. All these approaches proposed one to consider the activity coefficient of solvent to represent thermodynamic consistency between all components through the Gibbs–Duhem equation, which was not considered by Wagner. There has also been a debate that the activity coefficient of solvent satisfying the Gibbs–Duhem relation can be obtained, not in a unique way, but in an infinite number of ways.[18] This would make one to hesitate to have a confidence in the value calculated by the UIPF.[19] Somewhat different approaches are also available in which the activity coefficient of non-metallic solute was formulated as a function of the composition of a metallic solvent of two-component.[20,21,22,23]

In the present article, it will be shown that the activity coefficient of solvent can be obtained uniquely, for a given polynomial expression for the activity coefficient of solute, in a thermodynamically consistent manner. To make this discussion more clear, the corrections made previously are briefly reviewed and it will be pointed out what should be explicitly considered to have the thermodynamic consistency. Two approaches will be shown to obtain expression of the activity coefficient of the solvent in a ternary solution: (1) integrating differential equations containing activity coefficients of solutes and (2) differentiating an integral excess Gibbs energy. Similar approaches are expanded for a multicomponent solution, and it will be shown that the same conclusion is valid in general. This discussion will provide a solid background for the use of such formalism with a confidence. As long as the expression of the activity coefficient of solute has a sound theoretical foundation, and the activity coefficient of solvent can be obtained in a unique and thermodynamically consistent way based on the activity coefficients of solutes, the obtained activity coefficient of solvent should have its own physical meaning.

The present analysis has been carried out under constant temperature and constant pressure condition in most cases. Temperature dependence of the activity coefficient is briefly discussed in Section IV

Thermodynamic Consistency of Interaction Parameter Formalism

Let us consider an N-component solution 1–2–3–\(\cdots \)–N where 1 is a solvent and the others are solutes. According to Wagner’s formalism, the activity coefficient of any solute i (\(\gamma _i\)) is expressed as:

$$\begin{aligned} \ln (\gamma _i/\gamma _i^{\circ }) = \sum _{j=2}^N \epsilon _{ij}X_j \qquad (i=2,3,\ldots ,N) \end{aligned}$$
(1)

where \(\gamma _i^{\circ }\), \(\epsilon _{ij}\), and \(X_j\) are the Henrian activity coefficient of solute i, a first-order interaction parameter between solutes i and j (following the notation after Pelton and Bale[4]), and mole fraction of solute j, respectively. In addition, the following reciprocity relation was proposed:

$$\begin{aligned} \epsilon _{ij} = \epsilon _{ji} \end{aligned}$$
(2)

when \(X_1\) approaches to 1.[1] This has been extended to include second-order terms by Lupis and Elliott[7]:

$$\begin{aligned} \ln (\gamma _i/\gamma _i^{\circ }) = \sum _{j=2}^N \epsilon _{ij}X_j + \sum _{j=2}^N \rho _i^jX_j^2 + \sum _{j,k=2}^N\rho _i^{j,k}X_jX_k \end{aligned}$$
(3)

where \(\rho _i^j\), \(\rho _i^{j,k}\) are the second-order interaction parameters.

Darken: Quadratic Formalism

For a ternary 1–2–3 solution, Darken pointed out that application of Eq. [1] is only valid at infinite dilution (\(X_1\rightarrow \)1), and generally, it is not thermodynamically consistent at finite concentration.[3] He proposed that any formalism should satisfy the following condition of thermodynamic consistency between the solutes 2 and 3:

$$\begin{aligned} (1-X_3)\frac{\partial \ln \gamma _2}{\partial X_3} + X_3\frac{\partial \ln \gamma _3}{\partial X_3} = X_2\frac{\partial \ln \gamma _2}{\partial X_2} + (1-X_2)\frac{\partial \ln \gamma _3}{\partial X_2} \end{aligned}$$
(4)

In addition, Raoult’s law and Henry’s law must be obeyed by the solvent and the solutes, respectively, as limiting conditions at infinite dilution. Then, by analyzing several binary and ternary systems, he proposed a quadratic formalism to represent activity coefficients of solutes:

$$\begin{aligned} \ln (\gamma _2/\gamma _2^{\circ })= & {} \epsilon _{22}X_2 + \epsilon _{23}X_3 - \frac{1}{2} \epsilon _{22}X_2^2 - \frac{1}{2} \epsilon _{33}X_3^2 - \epsilon _{23}X_2X_3 \end{aligned}$$
(5)
$$\begin{aligned} \ln (\gamma _3/\gamma _3^{\circ })= & {} \epsilon _{32}X_2 + \epsilon _{33}X_3 - \frac{1}{2} \epsilon _{22}X_2^2 - \frac{1}{2} \epsilon _{33}X_3^2 - \epsilon _{23}X_2X_3 \end{aligned}$$
(6)

Mathematical forms of the Eqs. [5] and [6] are very similar to the Eq. [3], and it reduces to Eq. [1] at infinite dilution. He had shown that \(\epsilon _{23} = \epsilon _{32}\). He also considered the activity coefficient of the solvent to be[3]:

$$\begin{aligned} \ln \gamma _1 = -\frac{1}{2} \epsilon _{22}X_2^2 - \frac{1}{2} \epsilon _{33}X_3^2 - \epsilon _{23}X_2X_3 \end{aligned}$$
(7)

The formalism eliminates the thermodynamic inconsistency of Wagner’s formalism even at finite concentration via Gibbs–Duhem equation at constant T and P:

$$\begin{aligned} X_1d\ln \gamma _1 + X_2d\ln \gamma _2 + X_3d\ln \gamma _3 = 0 \end{aligned}$$
(8)

Therefore, it was considered that the Eqs. [4] and [8] are to be obeyed in the development of an activity coefficient equation in a ternary solution.

In this approach, as seen in the name of the formalism, the quadratic term of composition is considered.

Pelton and Bale: Unified Interaction Parameter Formalism

The Darken’s formalism applied for several ternary solutions has been generalized and extended into a multicomponent solution by Pelton and Bale.[4,5,6] They proposed the following form of \(\gamma _i\) for solutes and \(\gamma _1\) for solvent:

$$\begin{aligned} \ln (\gamma _i/\gamma _i^{\circ })= & {} \ln \gamma _1 + \sum _{j=2}^N \epsilon _{ij}X_j \nonumber \\&+ \sum _{j,k=2}^N \epsilon _{ijk}X_jX_k + \sum _{\begin{subarray}{c} j,k,l \\ =2 \end{subarray}}^N \epsilon _{ijkl}X_jX_kX_l + \ldots \end{aligned}$$
(9)
$$\begin{aligned} \ln \gamma _1= & {} -\frac{1}{2} \sum _{j,k=2}^N \epsilon _{jk}X_jX_k - \frac{2}{3} \sum _{\begin{subarray}{c} j,k,l \\ =2 \end{subarray}}^N \epsilon _{jkl}X_jX_kX_l \nonumber \\&- \frac{3}{4} \sum _{\begin{subarray}{c} j,k,l,m \\ =2 \end{subarray}}^N \epsilon _{jklm}X_jX_kX_lX_m + \ldots \end{aligned}$$
(10)

where they found that \(\epsilon _{ij}=\epsilon _{ji}\), \(\epsilon _{ijk}=\epsilon _{jik}=\epsilon _{kij}=\ldots \), and so on. If only first-order interaction parameters are considered by setting all higher-order parameters \(\epsilon _{ijk}\), \(\epsilon _{ijkl}\), etc. zero, the above equations are reduced to:

$$\begin{aligned} \ln (\gamma _i/\gamma _i^{\circ })= & {} \ln \gamma _1 + \sum _{j=2}^N \epsilon _{ij}X_j \qquad (i=2,3,\ldots ,N) \end{aligned}$$
(11)
$$\begin{aligned} \ln \gamma _1= & {} -\frac{1}{2} \sum _{j,k=2}^N \epsilon _{jk}X_jX_k \end{aligned}$$
(12)

Equations [11] and [12] are identical to Darken’s formalism for a ternary case (Eqs. [5] to [7]). Pelton and Bale noticed that the last three terms of the Eqs. [5] and [6] are identical to those in Eq. [7]. They added the \(\ln \gamma _1\) to the original form of \(\ln \gamma _i\) for all solutes (Eq. [1]) as a corrective function. This results in an obedience of the Gibbs–Duhem relation in the multicomponent solution at constant T and P:

$$\begin{aligned} \sum _{i=1}^N X_id\ln \gamma _i = X_1d\ln \gamma _1 + X_2d\ln \gamma _2 + \cdots + X_Nd\ln \gamma _N = 0 \end{aligned}$$
(13)

and of the following thermodynamic relationship which is equivalent to Darken’s condition of thermodynamic consistency (See Appendix A):

$$\begin{aligned} \frac{\partial \ln \gamma _j}{\partial n_i} = \frac{\partial \ln \gamma _i}{\partial n_j} \end{aligned}$$
(14)

where \(n_i\) is the number of moles of component i. The formalism has been successfully applied several metal solutions.[9,10,11,12] The formalism keeps the simple form of Wagner’s formalism, and it should be stressed that the formalism can utilize already well-tabulated interaction parameters for the Wagner’s formalism.[24]

Srikanth and Jacob: Path-Dependent Integration

Srikanth and Jacob considered the Eqs. [1] and [3] as truncations of the Maclaurin infinite series (Taylor series at zero concentration).[14] Upon the truncation, the activity coefficients of solutes are thermodynamically inconsistent, except at infinite dilution. Therefore, they pointed out that the interaction parameters (\(\epsilon _{ij},\rho _{i}^{j,k}\)) should satisfy special relationships to be thermodynamically consistent through the Eq. [4]. Moreover, as the \(\ln \gamma _i\)’s are inexact due to the truncation, the \(\ln \gamma _1\), which may be obtained by the integration of the Gibbs–Duhem equation (Eq. [13]), does depend upon the path of integration. In this way, they obtained expressions of the \(\ln \gamma _1\) per chosen integration paths, while they kept the original expression for the \(\ln \gamma _i\) to be Eq. [1] for the first-order case.

Their approach keeps the original mathematical form of Wagner’s formalism for solutes without adding any corrective function, although they are thermodynamically inconsistent except at infinite dilution. However, it was possible to obtain \(\ln \gamma _1\) to be consistent with the Gibbs–Duhem equation (Eq. [13]) because of the path-dependent integration. For example, if first-order interaction parameters are considered for solute 2 and 3 in a 1–2–3 solution from the Eq. [1]:

$$\begin{aligned} \ln (\gamma _2/\gamma _2^{\circ })\,=\,& {} \epsilon _{22}X_2 + \epsilon _{23}X_3 \end{aligned}$$
(15)
$$\begin{aligned} \ln (\gamma _3/\gamma _3^{\circ })\,=\,& {} \epsilon _{23}X_2 + \epsilon _{33}X_3 \end{aligned}$$
(16)

along the path \(X_2/X_3=K_{23}\), where \(K_{23}\) is a constant, they obtained:

$$\begin{aligned} \ln \gamma _1 = \left[ \frac{\epsilon _{22}X_2^2 +2\epsilon _{23}X_2X_3+\epsilon _{33}X_3^2}{(X_2+X_3)^2}\right] \left[ (X_2+X_3)+\ln (1-(X_2+X_3))\right] \end{aligned}$$
(17)

through the integration of Eq. [8].

Upon the paths taken for the integration, there could be many numbers of the expression for the \(\ln \gamma _1\). This is due to the thermodynamic inconsistency of the Eqs. [15] and [16]. Although the Gibbs–Duhem relation is satisfied, the thermodynamic consistency equation (Eq. [4]) is not obeyed except at infinite dilute concentration. It is noted that regardless of the way of integration to get the \(\ln \gamma _1\), it is a prerequisite to have the thermodynamically consistent expression of \(\ln \gamma _i\) for all solutes i.

Malakhov: Corrective Function

Malakhov raised an issue regarding a uniqueness of the expression for \(\ln \gamma _1\), even though \(\ln \gamma _2\) and \(\ln \gamma _3\) obey the condition of thermodynamic consistency (Eq. [4]).[18] In his analysis for a ternary 1–2–3 solution, he defined a corrective function \(\varphi (X_2,X_3)\) to be added to inexact forms of \(\ln \gamma _2\) and \(\ln \gamma _3\) (Eqs. [15] and [16]), so the following revised forms are considered:

$$\begin{aligned} \ln (\gamma _2/\gamma _2^{\circ })\,=\,& {} \epsilon _{22}X_2 + \epsilon _{23}X_3 + \varphi _2 \end{aligned}$$
(18)
$$\begin{aligned} \ln (\gamma _3/\gamma _3^{\circ })\,=\,& {} \epsilon _{23}X_2 + \epsilon _{33}X_3 + \varphi _3 \end{aligned}$$
(19)

to satisfy the Eq. [4] for the thermodynamic consistency:

$$\begin{aligned} (1-X_3)\epsilon _{23} + X_3\epsilon _{33} + \frac{\partial \varphi _2}{\partial X_3} = X_2\epsilon _{22} + (1-X_2)\epsilon _{23} + \frac{\partial \varphi _3}{\partial X_2} \end{aligned}$$
(20)

As long as the final form of activity coefficients of components are to be expressed as a polynomial function, the corrective functions \(\varphi _2\) and \(\varphi _3\) must be a form of the quadratic polynomial: \(\varphi _2 = \alpha _{22}X_2^2 + \alpha _{23}X_2X_3 + \alpha _{33}X_3^2\) and \(\varphi _3 = \beta _{22}X_2^2 + \beta _{23}X_2X_3 + \beta _{33}X_3^2\), where all \(\alpha _{ij}\)’s and \(\beta _{ij}\)’s are constant according to the form of the Eq. [20]. Inserting \(\varphi _2\) and \(\varphi _3\) into the Eq. [20] and setting all coefficients of terms involving the independent variables \(X_2\) and \(X_3\) to be zero, the following conditions are obtained:

$$\begin{aligned} \alpha _{ij} = \beta _{ij} \qquad (i,j = 2, 3) \end{aligned}$$
(21)

yielding \(\varphi _2 = \varphi _3\). Therefore, the corrective function must be identical for the two solutes (\(\varphi _2 = \varphi _3 = \varphi \)), and hereafter it is denoted as \(\varphi \)Footnote 1. Furthermore, it is also obtained:

$$\begin{aligned} 2\alpha _{22} - \alpha _{23}\,=\,& {} \epsilon _{23} - \epsilon _{22} \end{aligned}$$
(22)
$$\begin{aligned} 2\alpha _{33} - \alpha _{23}\,=\,& {} \epsilon _{23} - \epsilon _{33} \end{aligned}$$
(23)

Malakhov pointed out that three variables (\(\alpha _{22},\alpha _{23},\alpha _{33}\)) with two equations Eqs. [22] and [23] result in an infinite number of sets for the \(\alpha _{22},\alpha _{23}\), and \(\alpha _{33}\), so does for the corrective function \(\varphi \). If one chooses

$$\begin{aligned} \alpha _{22}=-\frac{1}{2}\epsilon _{22}, \quad \alpha _{33}=-\frac{1}{2}\epsilon _{33}, \quad \alpha _{23}=-\epsilon _{23} \end{aligned}$$
(24)

thus,

$$\begin{aligned} \varphi = -\frac{1}{2}\epsilon _{22}X_2^2 - \epsilon _{23}X_2X_3 -\frac{1}{2}\epsilon _{33}X_3^2 \end{aligned}$$
(25)

then the form of \(\varphi \) becomes identical to the \(\ln \gamma _1\) in Eq. [7], and the Eqs. [18] and [19] become identical to the Eqs. [5] and [6]. Via the Gibbs–Duhem integration, the obtained \(\ln \gamma _1\) will be the same as the Eq. [7], as well as the \(\varphi \) shown in the Eq. [25].

On the other hand, other choices for the \(\alpha _{22},\alpha _{23}\), and \(\alpha _{33}\) satisfying the Eqs. [22] and [23] (thus satisfying the thermodynamic consistency through the Eq. [20]) may be considered. The other form of \(\varphi \) than the Eq. [25] may have the following form[18]:

$$\begin{aligned} \varphi = \frac{\epsilon _{23}-\epsilon _{22}}{2}X_2^2 + \frac{\epsilon _{23}-\epsilon _{33}}{2}X_3^2 \end{aligned}$$
(26)

This alternative form of \(\varphi \) is inserted into Eqs. [18] and [19], then carrying out the Gibbs–Duhem integration, at least numerically or through a special path as was done by Srikanth and Jacob,[14] will result in a different form of \(\ln \gamma _1\). Nevertheless, this other choice still satisfies the thermodynamic consistency of Eq. [20] and the Gibbs–Duhem relation (Eq. [13]), both. Based on this reasoning, Malakhov claimed that limitless numbers of such corrective functions can be generated, thus any correction may not be used with high confidence.[18] Instead of using this formalism with non-unique correction, he suggested using a modern CALPHAD-type approach where the activity coefficient (partial excess Gibbs energy) can be derived from an integral excess Gibbs energy by differentiating it.

This analysis is in line with the correction to Wagner’s formalism previously done by Darken,[2,3] and Pelton and Bale.[4,5,6] However, in the following section, it will be shown that there is one and only one expression for the activity coefficient of solvent for given polynomial expressions of activity coefficients of solutes by revisiting the condition for the thermodynamic consistency proposed by Darken.[3] Examples will be shown by a ternary case, but it will be further shown to be valid in a multicomponent solution.

Correction to Make Interaction Parameter Formalism Be Thermodynamically Consistent

In the discussions and debate made in previous publications regarding thermodynamic consistency of polynomial representations for the activity coefficient of components,[2,3,4,5,6,14,18] two criteria have been employed: Gibbs–Duhem relationship (Eq. [8]) and Darken’s condition of thermodynamic consistency (Eq. [4]) for solutes in a ternary solution (1–2–3 where 1 is a solvent). Although

  1. 1.

    both corrective functions \(\varphi \) in Eq. [25] and in Eq. [26] make the \(\ln \gamma _2\) (Eq. [18]) and \(\ln \gamma _3\) (Eq. [19]) satisfy the condition of thermodynamic consistency (Eq. [4]), and

  2. 2.

    \(\ln \gamma _1\) is subsequently obtained in such a way that the Gibbs–Duhem relation is obeyed through integration,

it is not clear whether the condition of thermodynamic consistency is also obeyed between solute (2 and 3) and solvent (1). As the Gibbs–Duhem relation concerns all components including the solvent, the condition of thermodynamic consistency must also concern the relation between solutes and the solvent, not just between a solute and the other solute.

Thermodynamic Consistency in a Ternary System: From Partial Excess Gibbs Energy

Applying Eq. [14] for solvent 1 and solute 2 in the ternary 1–2–3 solution, the following relationship is obtained:

$$\begin{aligned} \frac{\partial \ln \gamma _1}{\partial n_2} = \frac{\partial \ln \gamma _2}{\partial n_1} \end{aligned}$$
(27)

Taking \(X_2\) and \(X_3\) as independent variables, and using the following derivatives,

$$\begin{aligned} \frac{\partial X_2}{\partial n_1}= & {} \frac{\partial (n_2/n)}{\partial n_1} = \frac{-n_2}{n^2}= -\frac{X_2}{n} \nonumber \\ \frac{\partial X_2}{\partial n_2}= & {} \cdots = \frac{1-X_2}{n},\quad \frac{\partial X_3}{\partial n_1} = -\frac{X_3}{n}, \quad \frac{\partial X_3}{\partial n_2} = -\frac{X_3}{n} \end{aligned}$$
(28)

where n is the total number of moles of all components in the solution, so \(n=n_1+n_2+n_3\) in this example. Using the above equations, it is obtained:

$$\begin{aligned} \frac{\partial \ln \gamma _1}{\partial n_2}= & {} \frac{\partial \ln \gamma _1}{\partial X_2} \left( \frac{\partial X_2}{\partial n_2}\right) + \frac{\partial \ln \gamma _1}{\partial X_3} \left( \frac{\partial X_3}{\partial n_2}\right) \nonumber \\= & {} \frac{1}{n}\left[ \frac{\partial \ln \gamma _1}{\partial X_2} (1-X_2) + \frac{\partial \ln \gamma _1}{\partial X_3} (-X_3)\right] \end{aligned}$$
(29)

Similarly,

$$\begin{aligned} \frac{\partial \ln \gamma _2}{\partial n_1} = \frac{1}{n}\left[ \frac{\partial \ln \gamma _2}{\partial X_2}(-X_2) + \frac{\partial \ln \gamma _2}{\partial X_3}(-X_3)\right] \end{aligned}$$
(30)

Therefore, according to the Eq. [27], the following equation is obtained which represents the conditions of thermodynamic consistency between two components (solvent 1 and solute 2) in the 1–2–3 solution:

$$\begin{aligned} \frac{\partial \ln \gamma _1}{\partial X_2} -X_2\frac{\partial \ln \gamma _1}{\partial X_2} -X_3\frac{\partial \ln \gamma _1}{\partial X_3} = -X_2 \frac{\partial \ln \gamma _2}{\partial X_2} - X_3\frac{\partial \ln \gamma _2}{\partial X_3} \end{aligned}$$
(31)

Following a similar procedure, the thermodynamic consistency condition for solvent 1 and the other solute 3 is obtained as:

$$\begin{aligned} \frac{\partial \ln \gamma _1}{\partial X_3} - X_2\frac{\partial \ln \gamma _1}{\partial X_2}-X_3\frac{\partial \ln \gamma _1}{\partial X_3} = -X_2\frac{\partial \ln \gamma _3}{\partial X_2}-X_3\frac{\partial \ln \gamma _3}{\partial X_3} \end{aligned}$$
(32)

Any polynomial representation for activity coefficients of solvent and solutes must obey the above conditions (Eqs. [31] and [32]), apart from the Eq. [4]. To correct the original Wagner’s formalism to be thermodynamically consistent, a corrective function \(\varphi \) is added to the \(\ln \gamma _2\) and \(\ln \gamma _3\) as seen in Eqs. [18] and [19]. Inserting these equations into Eqs. [31] and [32] gives:

$$\begin{aligned}&(1-X_2)\frac{\partial \ln \gamma _1}{\partial X_2} - X_3\frac{\partial \ln \gamma _1}{\partial X_3}\nonumber \\&\quad = -X_2\left( \epsilon _{22}+\frac{\partial \varphi }{\partial X_2}\right) -X_3 \left( \epsilon _{23}+\frac{\partial \varphi }{\partial X_3}\right) \end{aligned}$$
(33)
$$\begin{aligned}&(1-X_3)\frac{\partial \ln \gamma _1}{\partial X_3} - X_2\frac{\partial \ln \gamma _1}{\partial X_2} \nonumber \\&\quad = -X_2\left( \epsilon _{23}+\frac{\partial \varphi }{\partial X_2}\right) -X_3\left( \epsilon _{33}+\frac{\partial \varphi }{\partial X_3}\right) \end{aligned}$$
(34)

From Eqs. [33] and [34], it is obtained:

$$\begin{aligned} \frac{\partial \ln \gamma _1}{\partial X_2}-\frac{\partial \ln \gamma _1}{\partial X_3} = (\epsilon _{23}-\epsilon _{22})X_2 + (\epsilon _{33}-\epsilon _{23})X_3 \end{aligned}$$
(35)

Then, it can be easily shown that:

$$\begin{aligned} \ln \gamma _1 = -\frac{1}{2}\epsilon _{22}X_2^2 - \epsilon _{23}X_2X_3 - \frac{1}{2}\epsilon _{33}X_3^2 \end{aligned}$$
(36)

which satisfies the above condition and Raoult’s law at infinite dilution. Therefore, the \(\ln \gamma _1\) has been obtained, no matter what the corrective function \(\varphi \) is.

Inserting the \(\ln \gamma _1\) and \(\varphi = \alpha _{22}X_2^2 + \alpha _{23}X_2X_3 + \alpha _{33}X_3^2\) into Eqs. [33] or [34] yields:

$$\begin{aligned} \varphi = -\frac{1}{2}\epsilon _{22}X_2^2 - \epsilon _{23}X_2X_3 - \frac{1}{2}\epsilon _{33}X_3^2 \end{aligned}$$
(37)

Therefore, the derived \(\ln \gamma _1\) is indeed identical to the corrective function \(\varphi \), and to the expression proposed by Darken (Eq. [7]) and by Pelton and Bale (Eq. [12]) for the ternary case. It is evident that the expression for the \(\ln \gamma _1\) in the Eq. [36] also obeys the Gibbs–Duhem relation along with \(\ln \gamma _2\) and \(\ln \gamma _3\) corrected by \(\varphi \) (Eq. [37]). Therefore, the \(\ln \gamma _1\) and the corrective function \(\varphi \) have been obtained as:

  • the corrective function \(\varphi \) is identical to the \(\ln \gamma _1\)

  • \(\ln \gamma _1\), \(\ln \gamma _2\), and \(\ln \gamma _3\) obey the Gibbs–Duhem relation (Eq. [8])

  • \(\ln \gamma _1\), \(\ln \gamma _2\), and \(\ln \gamma _3\) obey the thermodynamic consistency Eqs. [4], [31], and [32].

The Eqs. [31] and [32] are additional requirements to fulfill the thermodynamic consistency, which have not been explicitly described previously. Regardless of the choice of \(\varphi \), the \(\ln \gamma _1\) is determined from the form of \(\ln \gamma _2\) and \(\ln \gamma _3\) before the correction, through the condition of thermodynamic consistency. Other choices are thermodynamically inconsistent at finite concentration.

Alternative Proof in the Ternary System: From Integral Excess Gibbs Energy

Following the approach proposed by Bale and Pelton,[5,6] the above proof can be made more clearly and simply by deriving all partial excess Gibbs energies from a corresponding integral excess Gibbs energy. Suppose an integral molar excess Gibbs energy of the ternary solution is given as a general quadratic polynomial function of composition:

$$\begin{aligned} {g^{\mathrm {ex}}} = X_1X_2\omega _{12} + X_2X_3\omega _{23} + X_1X_3\omega _{13} + C_2X_2 + C_3X_3 \end{aligned}$$
(38)

where \(\omega _{12}\), \(\omega _{23}\), \(\omega _{13}\), \(C_2\), and \(C_3\) are constants and \(C_2\) and \(C_3\) may not be zero because the equation is valid in the solvent-rich region.[2,6] This expression may be rationalized by a regular solution theory.[6] By replacing \(X_1 = 1 - X_2 - X_3\), partial excess Gibbs energies can be obtained by the following relationship:

$$\begin{aligned} g^{\mathrm {ex}}_i = \frac{\partial (n{\times } g^{\mathrm {ex}})}{\partial n_i}\Big )_{n_j \ne n_i} = {\text R}T \ln \gamma _i \end{aligned}$$
(39)

where \(n\times g^{\mathrm {ex}}\) is the total excess Gibbs energy in J. Differentiating the integral excess Gibbs energy gives

$$\begin{aligned} \ln \gamma _1= & {} {[}\omega _{12}X_2^2 + \omega _{13}X_3^2 + (\omega _{12} + \omega _{13} - \omega _{23})X_2X_3{]/\text {R}T} \end{aligned}$$
(40)
$$\begin{aligned} \ln \gamma _2= & {} {[}\omega _{12}X_2^2 + \omega _{13}X_3^2 + (\omega _{12} + \omega _{13} - \omega _{23})X_2X_3 - 2\omega _{12}X_2 \nonumber \\&+ (\omega _{23} - \omega _{12} - \omega _{13})X_3 + \omega _{12} + C_2{]/\text {R}T} \end{aligned}$$
(41)
$$\begin{aligned} \ln \gamma _3= & {} {[}\omega _{12}X_2^2 + \omega _{13}X_3^2 + (\omega _{12} + \omega _{13} - \omega _{23})X_2X_3 - 2\omega _{13}X_3 \nonumber \\&+ (\omega _{23} - \omega _{12} - \omega _{13})X_2 + \omega _{13} + C_3 {]/\text {R}T} \end{aligned}$$
(42)

By setting:

$$\begin{aligned} {[}\omega _{12} + C_2{]/\text {R}T}= & {} \ln \gamma _2^\circ \end{aligned}$$
(43)
$$\begin{aligned} {[}\omega _{13} + C_3{]/\text {R}T}= & {} \ln \gamma _3^\circ \end{aligned}$$
(44)
$$\begin{aligned} {[}-2\omega _{12}{]/\text {R}T}= & {} \epsilon _{22} \end{aligned}$$
(45)
$$\begin{aligned} {[}-2\omega _{13}{]/\text {R}T}= & {} \epsilon _{33} \end{aligned}$$
(46)
$$\begin{aligned} {[}\omega _{23} - \omega _{12} - \omega _{13}{]/\text {R}T}= & {} \epsilon _{23} \end{aligned}$$
(47)

the above Eqs. [40] to [42] change to:

$$\begin{aligned} \ln \gamma _1= & {} -\frac{1}{2}\epsilon _{22}X_2^2 -\frac{1}{2}\epsilon _{33}X_3^2 -\epsilon _{23}X_2X_3 \end{aligned}$$
(48)
$$\begin{aligned} \ln \gamma _2= & {} \ln \gamma _2^\circ + \epsilon _{22}X_2 + \epsilon _{23}X_3 -\frac{1}{2}\epsilon _{22}X_2^2 -\frac{1}{2}\epsilon _{33}X_3^2 -\epsilon _{23}X_2X_3 \nonumber \\= & {} \ln \gamma _2^\circ + \epsilon _{22}X_2 + \epsilon _{23}X_3 + \varphi \end{aligned}$$
(49)
$$\begin{aligned} \ln \gamma _3= & {} \ln \gamma _3^\circ + \epsilon _{33}X_3 + \epsilon _{23}X_2 -\frac{1}{2}\epsilon _{22}X_2^2 -\frac{1}{2}\epsilon _{33}X_3^2 -\epsilon _{23}X_2X_3 \nonumber \\= & {} \ln \gamma _3^\circ + \epsilon _{33}X_3 + \epsilon _{23}X_2 + \varphi \end{aligned}$$
(50)

It is seen that:

  • the corrective function \(\varphi \) is identical for both \(\ln \gamma _2\) and \(\ln \gamma _3\)

  • the corrective function \(\varphi \) in the above example is the same as the \(\ln \gamma _1\)

  • \(\ln \gamma _1\) obtained in the Eq. [48] is the same as the Eq. [36]

Therefore, it can be concluded that the \(\ln \gamma _1\) can be obtained in a unique form per given polynomial form of an integral excess Gibbs energy or partial excess Gibbs energies of solutes (activity coefficients of solutes), no matter how it has been obtained (differentiation or integration), as long as the activity coefficient of solute is thermodynamically consistent. The \(\ln \gamma _1\) does not depend on the form of corrective function nor path of integration. Moreover, the corrective function to the activity coefficient of solutes should be the same as the activity coefficient of the solvent. This ensures all mandatory requirements of thermodynamic principles: Maxwell relations (Eqs. [4], [31], and [32]), and Gibbs–Duhem equation (Eq. [8]).

Multicomponent System

Darken’s condition of thermodynamic consistency between two solutes was shown by himself for a ternary case (Eq. [4]),[3] and it has been extended to a multicomponent solution by Srikanth and Jacob for a relation of activity coefficients between two solutes.[14] As was shown in Sections. III–A and III–B for the ternary case, to examine the condition between solvent and solute in a multicomponent solution 1–2–\(\cdots \)–N where 1 is a solvent, the equation for the condition of thermodynamic consistency is revised in the present study in more general form (See Appendix A):

$$\begin{aligned} \sum _{k=2}^N (\delta _{ik}-X_k)\frac{\partial \ln \gamma _j}{\partial X_k} = \sum _{k=2}^N (\delta _{jk}-X_k)\frac{\partial \ln \gamma _i}{\partial X_k} \end{aligned}$$
(51)

where i and j are 1 (solvent), 2, \(\ldots \), N. If \(N=3\), \(i=2\) and \(j=3\), then the above equation reduces to the original Darken’s equation (Eq. [4]). If \(i\ne 1\) and \(j\ne 1\), then the above equation reduces to the equation derived by Srikanth and Jacob.[14] For the present analysis, if \(i=1\) and \(j\ne 1\), then

$$\begin{aligned} -\sum _{k=2}^N X_k\frac{\partial \ln \gamma _j}{\partial X_k} = \frac{\partial \ln \gamma _1}{\partial X_j} - \sum _{k=2}^N X_k\frac{\partial \ln \gamma _1}{\partial X_k} \end{aligned}$$
(52)

showing the condition of thermodynamic consistency between the solvent 1 and a solute j.

A corrective function \(\varphi (X_2,X_3,\ldots ,X_N)\) is defined, and added to incorrect\(\ln \gamma _j\) of first-order formalism that:

$$\begin{aligned} \ln (\gamma _j/\gamma _j^\circ )\,=\,& {} \sum _{k=2}^N \epsilon _{jk}X_k + \varphi \end{aligned}$$
(53)
$$\begin{aligned} \frac{\partial \ln \gamma _j}{\partial X_i}\,=\,& {} \epsilon _{ji} + \frac{\partial \varphi }{\partial X_i} \end{aligned}$$
(54)

Inserting the Eqs. [54] into [52] yields:

$$\begin{aligned} \frac{\partial \ln \gamma _1}{\partial X_j} - \sum _{k=2}^N X_k\frac{\partial \ln \gamma _1}{\partial X_k} = - \sum _{k=2}^N X_k\left( \epsilon _{jk} + \frac{\partial \varphi }{\partial X_k}\right) \qquad (j = 2, 3, \ldots , N) \end{aligned}$$
(55)

Considering a similar equation, replacing j by i, and subtracting one from the other, one obtains:

$$\begin{aligned} \frac{\partial \ln \gamma _1}{\partial X_i}-\frac{\partial \ln \gamma _1}{\partial X_j} = \sum _{k=2}^N(\epsilon _{jk}-\epsilon _{ik})X_k \qquad (i,j = 2, 3, \ldots , N) \end{aligned}$$
(56)

From the form of the above equation, \(\ln \gamma _1\) should have a formula

$$\begin{aligned} \ln \gamma _1 = \sum _{k=2}^N a_{kk}X_k^2 + \sum _{\begin{subarray}{c} k,l=2 \\ k\ne l \end{subarray}}^Nb_{kl}X_kX_l + \sum _{k=2}^Nc_kX_k \end{aligned}$$
(57)

Using the Eq. [56], it can be shown as (See Appendix B):

$$\begin{aligned} \ln \gamma _1 = -\frac{1}{2} \sum _{j,k=2}^N \epsilon _{jk}X_jX_k \end{aligned}$$
(58)

which is identical to the \(\ln \gamma _1\) proposed by Pelton and Bale[4,5,6] (Eq. [12]). Therefore, the \(\ln \gamma _1\) can be uniquely defined, no matter how the \(\varphi \) is defined. And the corrective function \(\varphi \) can be uniquely obtained through the Gibbs–Duhem equation along with all \(\ln \gamma _i\)’s, which is indeed identical to the \(\ln \gamma _1\).

Expansion to include higher-order interaction parameters (\(\epsilon _{ijk}, \epsilon _{ijkl}\), etc.) are straightforward.

Discussion

In general, partial properties can be obtained by differentiating an expression of the corresponding integral property, as was shown in Section III–B. By this process, the partial properties can be made in an unique way, and this assures thermodynamic consistency. Therefore, it may be said that there are an infinite number of ways of expressing the partial properties of a solution in a thermodynamically consistent way, for each infinite number of expressions for the integral property. However, calling these “correction” to Wagner’s formalism is a stretch since most of them would not resemble Wagner’s formalism at all. The mathematical expression for the integral property should be based on a reasonable solution model. The quadratic formalism, with polynomial expansions to cubic and higher-order terms, satisfies this criterion since it is based on regular solution theory. Wagner’s formalism is based on the quadratic formalism, but is not thermodynamically consistent except in very dilute solutions. The UIPF can be derived from the regular solution theory as the basis but corrects the thermodynamic inconsistency in non-dilute solutions while reducing to Wagner’s formalism at infinite dilution.

The present thermodynamic consistency analysis has been applied to the composition dependence of the activity coefficients of all the components. Since the activity coefficient of a component i in a logarithm scale is obtained by a partial differential of the total excess Gibbs energy by the amount of the component i (see Eq. [39]), partial differentials by the amount of different components yield different excess partial Gibbs energies and corresponding activity coefficients. Nevertheless, these must satisfy the Gibbs–Duehm relation (Eq. [13]) and the thermodynamic consistency equation (Eq. [51]) by the Maxwell relation. This ensures the uniqueness of the correction to the Wagner’s interaction parameter formalism.

Temperature dependence of the activity coefficients of components may be also sought in view of the thermodynamic consistency. It may be tested by applying the Gibbs–Duhem relation under constant pressure (but not under constant temperature) and the Maxwell relation in the following ways:

$$\begin{aligned} \sum _{i=1}^N X_id g_i + s dT = 0 \end{aligned}$$
(59)

where s is the molar entropy of the solution and,

$$\begin{aligned} \frac{\partial }{\partial n_i}\left( \frac{\partial G}{\partial T}\right) = \frac{\partial }{\partial T}\left( \frac{\partial G}{\partial n_i}\right) \end{aligned}$$
(60)

However, since the partial differential to T does not yield any different expression per each component i, these relations do not provide some useful relationship, contrary to the present analysis for the composition dependence of the activity coefficient.

Temperature dependence of parameters in the UIPF, for example, \(\ln \gamma _2^\circ \), \(\ln \gamma _3^\circ \), \(\epsilon _{22}\), \(\epsilon _{33}\), and \(\epsilon _{23}\) in a ternary 1–2–3 system, depend on the temperature dependence of interaction energies (\(\omega _{12}\), \(\omega _{23}\), and \(\omega _{13}\))) and that of \(C_2\) and \(C_3\), as can be seen in the Eq. [38]. By inspecting the Eqs. [43] through [47], it is evident that there is no interrelationship among the temperature dependence of the parameters, in general. This means, if the formalism is intended to be used only in the solvent-rich region, thereby none of \(C_2\) and \(C_3\) is zero, the temperature dependence of the five parameters is independent of each other. However, if the formalism is to be valid over the whole concentration range, therefore \(C_2 = C_3 = 0\), then, the following relationship should be held:

$$\begin{aligned} \epsilon _{22}= & {} -2 \ln \gamma _2^\circ \end{aligned}$$
(61)
$$\begin{aligned} \epsilon _{33}= & {} -2 \ln \gamma _3^\circ \end{aligned}$$
(62)
$$\begin{aligned} \epsilon _{23}= & {} \frac{\omega _{23}}{\text {R}T} - \ln \gamma _2^\circ - \ln \gamma _3^\circ \end{aligned}$$
(63)

which was already well-known relationship.[3,25] Accordingly, temperature dependence among the parameters (Henrian activity coefficients and interaction parameters) should satisfy similar relations which are simply obtained by the partial differential of the above equations (Eqs. [61] through [63]). This is valid when the solution behaves as the regular solution. If the solution’s behavior is interpreted by the quasichemical theory, different relations between Henrian activity coefficients and interaction parameters are found.[25]

By a choice of the temperature dependence of the \(\omega _{ij}\)’s and \(C_i\)’s in Eq. [38], the temperature dependence of Henrian activity coefficients and interaction parameters are determined. While a linear dependence has been most widely accepted (i.e., \(\omega _{ij} = \eta _{ij} - \sigma _{ij}T\)), a number of different approaches are also available.[26,27,28]

Conclusion

Condition of thermodynamic consistency for the representation of excess partial Gibbs energy of multicomponent solution has been discussed, in particular in the course of correction to the Wagner’s formalism. Apart from the Gibbs–Duhem relation and Darken’s condition of thermodynamic consistency for solutes, it has been stressed that a similar condition should also be obeyed between solvent and solute. A general condition of thermodynamic consistency has been derived in the present study.

A recent issue raised for the possibility of non-unique correction to Wagner’s formalism was analyzed.[18] It has been shown in the present article that the correction can be made in a unique way for given functional form of the activity coefficient of solutes (\(\ln \gamma _i\), \(i=2,3,\ldots ,N\)) regardless of the way of the correction. The corrective function \(\varphi (X_2,X_3,\ldots ,X_N)\), as well as the activity coefficient of solvent (\(\ln \gamma _1\)) can be uniquely defined through the Gibbs–Duhem relation and the general condition of thermodynamic consistency. It ensures the use of UIPF with a confidence for multicomponent dilute solution. Temperature dependence of the interaction parameters (\(\epsilon _{ij}\)) and Henrian activity coefficients (\(\ln \gamma _i^\circ \)) was discussed, although it depends on temperature dependence of interaction energies (\(\omega _{ij}\)).