At present, thermodynamic models still are important methods to solve problems relating to thermodynamic properties of liquid alloys and silicate melts, such as chemical interaction in Cu-Zr melts,[1] interfacial energies in the fcc Au-Ni, liquid Ga-Pb, and liquid Al-Bi,[2] mixing functions in binary silicate and aluminate melts,[3] element partitioning between plagioclase and melt,[4] and so forth. Especially, the molecular interaction volume model (MIVM)[5] has been successfully applied to some practical systems.[6–10] But in the model calculation it is often difficult to find out the coordination numbers and molar volumes of some components in liquid state, such as P, C, Ta, TiO2, V2O5, Cu2S, FeS, NaCl, MgCl2, CaSiO3, and so on. This would hinder application of the MIVM to many practical systems. For this reason, the author tried to simplify the MIVM reasonably. In its simplification process, however, the incorrect expressions thermodynamically were found in its molar enthalpy of mixing and excess entropy.[11–14] So it is necessary to correct them in public.

For a C-component system, the molar excess Gibbs energy \( {G_{\text{m}}^{\text{E}} } \) of the MIVM[5] can be expressed as

$$ {\frac{{G_{\text{m}}^{\text{E}} }}{RT} = \sum\limits_{i = 1}^{C} {x_{i} \ln \frac{{V_{{{\text{m}}i}} }}{{\sum\limits_{j = 1}^{C} {x_{j} V_{{{\text{m}}j}} B_{ji} } }}} - \frac{1}{2}\sum\limits_{i = 1}^{C} {Z_{i} x_{i} \left( {\frac{{\sum\limits_{j = 1}^{C} {x_{j} B_{ji} \ln B_{ji} } }}{{\sum\limits_{j = 1}^{C} {x_{j} B_{ji} } }}} \right)} }, $$
(1)

where R(J/mol K) is the gas constant, T(K) is the absolute temperature, C is the component number, x i , V mi (cm3/mol), and Z i are the molar fraction, molar volume, and coordination number of component i, respectively, and B ji and B ij are the pair potential parameters of the i−j binary system which are defined as

$$ {B_{ji} = \exp [ - N_{\text{A}} (\varepsilon_{ji} - \varepsilon_{ii} )/RT]}\quad B_{ij} = \exp [ - N_{\text{A}} (\varepsilon_{ij} - \varepsilon_{jj} )/RT], $$
(2)

where \( \varepsilon_{ii} \) (J) is the i−i pair potential energy and \( N_{\text{A}} \) (1/mol) is the Avogadro constant. It can be seen that the temperature dependence of the MIVM is determined by Eq. [2], that is, if both \( \theta_{ji} = N_{\text{A}} (\varepsilon_{ji} - \varepsilon_{ii} )/R \) and \( \theta_{ij} = N_{\text{A}} (\varepsilon_{ij} - \varepsilon_{jj} )/R \) are constant, the binary parameters B ji and B ij gradually approach unity as temperature increases. It indicates that the non-ideality of solutions will diminish at high temperatures. This case of the MIVM is consistent with the tendency described by the rule of Lupis and Elliott (LE rule) which was reformulated and rationalized by Kaptay[15] (LE rule: “Real solid, liquid and gaseous solutions (and pure gases) gradually approach the state of an ideal solution (perfect gas) as temperature increases at any fixed pressure and composition.”). At limited temperatures, if real solutions want to approach the state of an ideal solution, then the necessary conditions from the MIVM are as follows: all of B ji are equal to a same constant and all of V mi are equal to a same constant.

In Eq. [1], the first term involving molar volumes represents a contribution of entropy to the non-ideality and the second term involving coordination numbers represents a contribution of enthalpy to the non-ideality. Especially, when all of the pair potential energy are equal to a same constant or \( B_{ji} = B_{ij} = 1 \), Eq. [1] would be reduced to the famous Flory–Huggins equation[16]: \( G_{\text{m}}^{\text{E}} = RT\left[ {\sum\limits_{i = 1}^{C} {x_{i} \ln (\phi_{i} /x_{i} } )} \right], \) where \( \phi_{i} = x_{i} V_{{{\text{m}}i}} /V_{\text{m}} \) is the volume fraction of component i and \( V_{\text{m}} = \sum\limits_{j = 1}^{C} {x_{j} V_{{{\text{m}}j}} } \) is the molar volume of the system. It means that the MIVM could be equivalent to the contribution of excess entropy of the system or \( {G_{\text{m}}^{\text{E}} = - TS_{\text{m}}^{\text{E}} } \) under this condition. However, the incorrect expressions of molar enthalpy of mixing \( \Delta H_{\text{m}}^{\text{M}} \) and the molar excess entropy \( S_{\text{m}}^{\text{E}} \) from the MIVM were originated from comparing the \( {G_{\text{m}}^{\text{E}} } \) of Eq. [1] with its thermodynamic definition: \( G_{\text{m}}^{\text{E}} = H_{\text{m}}^{\text{E}} - TS_{\text{m}}^{\text{E}} = \Delta H_{\text{m}}^{\text{M}} - TS_{\text{m}}^{\text{E}} \), that is[11–14]

$$ \Delta H_{\text{m}}^{\text{M}} = - \frac{RT}{2}\sum\limits_{i = 1}^{C} {Z_{i} x_{i} \left( {\frac{{\sum\limits_{j = 1}^{C} {x_{j} B_{ji} \ln B_{ji} } }}{{\sum\limits_{j = 1}^{C} {x_{j} B_{ji} } }}} \right)} $$
(3)
$$ S_{\text{m}}^{\text{E}} = - R\sum\limits_{i = 1}^{C} {x_{i} \ln \frac{{V_{{{\text{m}}i}} }}{{\sum\limits_{j = 1}^{C} {x_{j} V_{{{\text{m}}j}} B_{ji} } }}}. $$
(4)

Both of them cannot satisfy the thermodynamic partial derivatives of \( {G_{\text{m}}^{\text{E}} } \):

$$ \left( {\frac{{\partial (G_{\text{m}}^{\text{E}} /T)}}{\partial T}} \right)_{P,x} = - \frac{{\Delta H_{\text{m}}^{\text{M}} }}{{T^{2} }} $$
(5)
$$ \left( {\frac{{\partial G_{\text{m}}^{\text{E}} }}{\partial T}} \right)_{P,x} = - S_{\text{m}}^{\text{E}} \;{\text{or}}\;S_{\text{m}}^{\text{E}} = \left( {\Delta H_{\text{m}}^{\text{M}} - G_{\text{m}}^{\text{E}} } \right)/T, $$
(6)

where the subscripts P and x represent the pressure and composition of the system, respectively. So, substituting Eq. [1] into Eqs. [5] and [6], the correct expressions of \( \Delta H_{\text{m}}^{\text{M}} \) and \( S_{\text{m}}^{\text{E}} \) can be derived as follows: for the i−j binary system, its \( {G_{\text{m}}^{\text{E}} } \) can be written from Eq. [1] as

$$ {\frac{{G_{\text{m}}^{\text{E}} }}{RT} = x_{i} \ln \left( {\frac{{V_{{{\text{m}}i}} }}{{x_{i} V_{{{\text{m}}i}} + x_{j} V_{{{\text{m}}j}} B_{ji} }}} \right) + x_{j} \ln \left( {\frac{{V_{{{\text{m}}j}} }}{{x_{j} V_{{{\text{m}}j}} + x_{i} V_{{{\text{m}}i}} B_{ij} }}} \right)} - \frac{{x_{i} x_{j} }}{2}\left( {\frac{{Z_{i} B_{ji} \ln B_{ji} }}{{x_{i} + x_{j} B_{ji} }} + \frac{{Z_{j} B_{ij} \ln B_{ij} }}{{x_{j} + x_{i} B_{ij} }}} \right). $$
(7)

Here assume that the Z i and Z j in Eq. [7] \( \theta_{ji} = N_{\text{A}} (\varepsilon_{ji} - \varepsilon_{ii} )/R \) and \( \theta_{ij} = N_{\text{A}} (\varepsilon_{ij} - \varepsilon_{jj} )/R \) in Eq. [2] are independent of temperature, and one can obtain the partial derivatives \( (\partial B_{ji} /\partial T)_{P} = - (1/T)B_{ji} \ln B_{ji} \) and \( (\partial B_{ij} /\partial T)_{P} = - (1/T)B_{ij} \ln B_{ij} \). Then, substituting Eq. [7] into Eq. [5], the \( \Delta H_{\text{m}}^{\text{M}} \) of the 1–2 binary system can be derived as

$$ \begin{aligned} \frac{{\Delta H_{\text{m}}^{\text{M}} }}{RT} & = \frac{1}{2}\left\{ {Z_{1} x_{1} \left[ {\left( {\frac{{x_{2} B_{21} \ln B_{21} }}{{x_{1} + x_{2} B_{21} }}} \right)^{2} - \frac{{(1 + \ln B_{21} )x_{2} B_{21} \ln B_{21} }}{{x_{1} + x_{2} B_{21} }}} \right] + Z_{2} x_{2} \left[ {\left( {\frac{{x_{1} B_{12} \ln B_{12} }}{{x_{1} B_{12} + x_{2} }}} \right)^{2} - \frac{{(1 + \ln B_{12} )x_{1} B_{12} \ln B_{12} }}{{x_{1} B_{12} + x_{2} }}} \right]} \right\} \\ & \quad - \frac{{x_{1} x_{2} V_{{{\text{m}}2}} B_{21} \ln B_{21} }}{{x_{1} V_{{{\text{m}}1}} + x_{2} V_{{{\text{m}}2}} B_{21} }} - \frac{{x_{2} x_{1} V_{{{\text{m}}1}} B_{12} \ln B_{12} }}{{x_{1} V_{{{\text{m}}1}} B_{12} + x_{2} V_{{{\text{m}}2}} }}. \\ \end{aligned} $$
(8)

Extending the binary 1–2 to the C-component system, one can obtain its \( \Delta H_{\text{m}}^{\text{M}}: \)

$$ \frac{{\Delta H_{\text{m}}^{\text{M}} }}{RT} = \frac{1}{2}\sum\limits_{i = 1}^{C} {Z_{i} x_{i} } \left[ {\left( {\frac{{\sum\limits_{j = 1}^{C} {x_{j} B_{ji} \ln B_{ji} } }}{{\sum\limits_{j = 1}^{C} {x_{j} B_{ji} } }}} \right)^{2} - \frac{{\sum\limits_{j = 1}^{C} {(1 + \ln B_{ji} )x_{j} B_{ji} \ln B_{ji} } }}{{\sum\limits_{j = 1}^{C} {x_{j} B_{ji} } }}} \right] - \sum\limits_{i = 1}^{C} {x_{i} } \left( {\frac{{\sum\limits_{j = 1}^{C} {x_{j} V_{{{\text{m}}j}} B_{ji} \ln B_{ji} } }}{{\sum\limits_{j = 1}^{C} {x_{j} V_{{{\text{m}}j}} B_{ji} } }}} \right). $$
(9)

Similarly, the \( S_{\text{m}}^{\text{E}} \) expressions of the 1–2 binary and C-component systems can also be derived, respectively, as

$$ \begin{aligned} \frac{{S_{\text{m}}^{\text{E}} }}{RT} & = \frac{1}{2}\left\{ {Z_{1} x_{1} \left[ {\left( {\frac{{x_{2} B_{21} \ln B_{21} }}{{x_{1} + x_{2} B_{21} }}} \right)^{2} - \frac{{x_{2} B_{21} (\ln B_{21} )^{2} }}{{x_{1} + x_{2} B_{21} }}} \right]} \right. + Z_{2} x_{2} \left. {\left[ {\left( {\frac{{x_{1} B_{12} \ln B_{12} }}{{x_{1} B_{12} + x_{2} }}} \right)^{2} - \frac{{x_{1} B_{12} (\ln B_{12} )^{2} }}{{x_{1} B_{12} + x_{2} }}} \right]} \right\} \\ & \quad - x_{1} \left( {\ln \frac{{V_{{{\text{m}}1}} }}{{x_{1} V_{{{\text{m}}1}} + x_{2} V_{{{\text{m}}2}} B_{21} }} + \frac{{x_{2} V_{{{\text{m}}2}} B_{21} \ln B_{21} }}{{x_{1} V_{{{\text{m}}1}} + x_{2} V_{{{\text{m}}2}} B_{21} }}} \right) - x_{2} \left( {\ln \frac{{V_{{{\text{m}}2}} }}{{x_{1} V_{{{\text{m}}1}} B_{12} + x_{2} V_{{{\text{m}}2}} }} + \frac{{x_{1} V_{{{\text{m}}1}} B_{12} \ln B_{12} }}{{x_{1} V_{{{\text{m}}1}} B_{12} + x_{2} V_{{{\text{m}}2}} }}} \right) \\ \end{aligned}, $$
(10)
$$ \frac{{S_{\text{m}}^{\text{E}} }}{R} = \frac{1}{2}\sum\limits_{i = 1}^{C} {Z_{i} x_{i} } \left[ {\left( {\frac{{\sum\limits_{j = 1}^{C} {x_{j} B_{ji} \ln B_{ji} } }}{{\sum\limits_{j = 1}^{C} {x_{j} B_{ji} } }}} \right)^{2} - \frac{{\sum\limits_{j = 1}^{C} {x_{j} B_{ji} (\ln B_{ji} )^{2} } }}{{\sum\limits_{j = 1}^{C} {x_{j} B_{ji} } }}} \right] - \sum\limits_{i = 1}^{C} {x_{i} } \left( {\ln \frac{{V_{{{\text{m}}i}} }}{{\sum\limits_{j = 1}^{C} {x_{j} V_{{{\text{m}}j}} B_{ji} } }} + \frac{{\sum\limits_{j = 1}^{C} {x_{j} V_{{{\text{m}}j}} B_{ji} \ln B_{ji} } }}{{\sum\limits_{j = 1}^{C} {x_{j} V_{{{\text{m}}j}} B_{ji} } }}} \right). $$
(11)

Then, based on the classical thermodynamic relations between partial molar quantity and molar quantity

$$ \frac{{\overline{G}_{i}^{\text{E}} }}{RT} = \ln \gamma_{i} = \frac{{G_{\text{m}}^{\text{E}} }}{RT} + \left( {\frac{{\partial (G_{\text{m}}^{\text{E}} /RT)}}{{\partial x_{i} }}} \right)_{T,P,x[i,C]} - \sum\limits_{l = 1}^{C - 1} {x_{l} } \left( {\frac{{\partial (G_{\text{m}}^{\text{E}} /RT)}}{{\partial x_{l} }}} \right)_{T,P,x[l,C]} $$
(12)
$$ \frac{{\Delta \overline{H}_{i} }}{RT} = \frac{{\Delta H_{\text{m}}^{\text{M}} }}{RT} + \left( {\frac{{\partial (\Delta H_{\text{m}}^{\text{M}} /RT)}}{{\partial x_{i} }}} \right)_{T,P,x[i,C]} - \sum\limits_{l = 1}^{C - 1} {x_{l} } \left( {\frac{{\partial (\Delta H_{\text{m}}^{\text{M}} /RT)}}{{\partial x_{l} }}} \right)_{T,P,x[l,C]} , $$
(13)

where \( \overline{G}_{i}^{\text{E}} \) and \( \Delta \overline{H}_{i} \) are the partial molar excess Gibbs energy and the partial molar enthalpy of mixing of component i, respectively, the subscript symbol \( x[l,C] \) represents that x l and x C are two variables of the partial derivative and \( x_{C} = 1 - \sum\limits_{l = 1}^{C - 1} {x_{l} } \) is a subordinate variable which needs special attention in the deducing processes of the partial derivatives in Eqs. [12] and [13].

Substituting Eq. [7] into Eq. [12], the expressions of \( \ln \gamma_{i} \) for the 1–2 binary system can be, respectively, derived as

$$ \ln \gamma_{1} = 1 + \ln \frac{{V_{{{\text{m}}1}} }}{{x_{1} V_{{{\text{m}}1}} + x_{2} V_{{{\text{m}}2}} B_{21} }} - \frac{{x_{1} V_{{{\text{m}}1}} }}{{x_{1} V_{{{\text{m}}1}} + x_{2} V_{{{\text{m}}2}} B_{21} }} - \frac{{x_{2} V_{{{\text{m}}1}} B_{12} }}{{x_{1} V_{{{\text{m}}1}} B_{12} + x_{2} V_{{{\text{m}}2}} }} - \frac{{x_{2}^{2} }}{2}\left( {\frac{{Z_{1} B_{21}^{2} \ln B_{21} }}{{(x_{1} + x_{2} B_{21} )^{2} }} + \frac{{Z_{2} B_{12} \ln B_{12} }}{{(x_{1} B_{12} + x_{2} )^{2} }}} \right), $$
(14)
$$ \ln \gamma_{2} = 1 + \ln \frac{{V_{{{\text{m}}2}} }}{{x_{1} V_{{{\text{m}}1}} B_{12} + x_{2} V_{{{\text{m}}2}} }} - \frac{{x_{1} V_{{{\text{m}}2}} B_{21} }}{{x_{1} V_{{{\text{m}}1}} + x_{2} V_{{{\text{m}}2}} B_{21} }} - \frac{{x_{2} V_{{{\text{m}}2}} }}{{x_{1} V_{{{\text{m}}1}} B_{12} + x_{2} V_{{{\text{m}}2}} }} - \frac{{x_{1}^{2} }}{2}\left( {\frac{{Z_{2} B_{12}^{2} \ln B_{12} }}{{(x_{1} B_{12} + x_{2} )^{2} }} + \frac{{Z_{1} B_{21} \ln B_{21} }}{{(x_{1} + x_{2} B_{21} )^{2} }}} \right). $$
(15)

Both Eqs. [14] and [15] are consistent with the Gibbs–Duhem equation \( x_{1} d\ln \gamma_{1} + x_{2} d\ln \gamma_{2} = 0 \).

Extending the binary 1–2 to the C-component system, one can obtain the \( \ln \gamma_{i} \) of component i:

$$ \ln \gamma_{i} = 1 + \ln \frac{{V_{{{\text{m}}i}} }}{{\sum\limits_{j = 1}^{C} {x_{j} V_{{{\text{m}}j}} B_{ji} } }} - \sum\limits_{j = 1}^{C} {\frac{{x_{j} V_{{{\text{m}}i}} B_{ij} }}{{\sum\limits_{t = 1}^{C} {x_{t} V_{{{\text{m}}t}} B_{tj} } }}} - \frac{1}{2}\left[ {\frac{{Z_{i} \sum\limits_{j = 1}^{C} {x_{j} B_{ji} \ln B_{ji} } }}{{\sum\limits_{t = 1}^{C} {x_{t} B_{ti} } }} + \sum\limits_{j = 1}^{C} {\frac{{Z_{j} x_{j} B_{ij} }}{{\sum\limits_{t = 1}^{C} {x_{t} B_{tj} } }}} \left( {\ln B_{ij} - \frac{{\sum\limits_{t = 1}^{C} {x_{t} B_{tj} \ln B_{tj} } }}{{\sum\limits_{t = 1}^{C} {x_{t} B_{tj} } }}} \right)} \right]. $$
(16)

And substituting Eq. [8] into Eq. [13], the expressions of \( \Delta \overline{H}_{i} /RT \) for the 1–2 binary system can be, respectively, derived as

$$ \begin{aligned} \frac{{\Delta \overline{H}_{1} }}{RT} & = \frac{{Z_{1} }}{2}\left[ {\left( {\frac{{x_{2} B_{21} \ln B_{21} }}{{x_{1} + x_{2} B_{21} }}} \right)^{2} - \frac{{(1 + \ln B_{21} )x_{2} B_{21} \ln B_{21} }}{{x_{1} + x_{2} B_{21} }}} \right] - \frac{{x_{2} V_{{{\text{m}}2}} B_{21} \ln B_{21} }}{{x_{1} V_{{{\text{m}}1}} + x_{2} V_{{{\text{m}}2}} B_{21} }} \\ & \quad + \frac{{Z_{1} x_{1} x_{2} }}{2}\left[ \begin{aligned} &- \frac{{2x_{2} (B_{21} \ln B_{21} )^{2} }}{{(x_{1} + x_{2} B_{21} )^{2} }} - \frac{{2(1 - B_{21} )(x_{2} B_{21} \ln B_{21} )^{2} }}{{(x_{1} + x_{2} B_{21} )^{3} }} \\ &+ \frac{{(1 + \ln B_{21} )B_{21} \ln B_{21} }}{{x_{1} + x_{2} B_{21} }} + \frac{{(1 - B_{21} )(1 + \ln B_{21} )x_{2} B_{21} \ln B_{21} }}{{(x_{1} + x_{2} B_{21} )^{2} }} \\ \end{aligned} \right] \\ & \quad + \frac{{Z_{2} x_{2}^{2} }}{2}\left[ \begin{aligned} &\frac{{2x_{1} (B_{12} \ln B_{12} )^{2} }}{{(x_{1} B_{12} + x_{2} )^{2} }} - \frac{{2(B_{12} - 1)(x_{1} B_{12} \ln B_{12} )^{2} }}{{(x_{1} B_{12} + x_{2} )^{3} }} \\ &- \frac{{(1 + \ln B_{12} )B_{12} \ln B_{12} }}{{x_{1} B_{12} + x_{2} }} + \frac{{(B_{12} - 1)(1 + \ln B_{12} )x_{1} B_{12} \ln B_{12} }}{{(x_{1} B_{12} + x_{2} )^{2} }} \\ \end{aligned} \right] \\ & \quad + x_{1} x_{2} \left[ {\frac{{V_{{{\text{m}}2}} B_{21} \ln B_{21} }}{{x_{1} V_{{{\text{m}}1}} + x_{2} V_{{{\text{m}}2}} B_{21} }} + \frac{{(V_{{{\text{m}}1}} - V_{{{\text{m}}2}} B_{21} )x_{2} V_{{{\text{m}}2}} B_{21} \ln B_{21} }}{{(x_{1} V_{{{\text{m}}1}} + x_{2} V_{{{\text{m}}2}} B_{21} )^{2} }}} \right] \\ & \quad - x_{2}^{2} \left[ {\frac{{V_{{{\text{m}}1}} B_{12} \ln B_{12} }}{{x_{1} V_{{{\text{m}}1}} B_{12} + x_{2} V_{{{\text{m}}2}} }} - \frac{{(V_{{{\text{m}}1}} B_{12} - V_{{{\text{m}}2}} )x_{1} V_{{{\text{m}}1}} B_{12} \ln B_{12} }}{{(x_{1} V_{{{\text{m}}1}} B_{12} + x_{2} V_{{{\text{m}}2}} )^{2} }}} \right] \\ \end{aligned}, $$
(17)
$$ \begin{aligned} \frac{{\Delta \overline{H}_{2} }}{RT} & = \frac{{Z_{2} }}{2}\left[ {\left( {\frac{{x_{1} B_{12} \ln B_{12} }}{{x_{1} B_{12} + x_{2} }}} \right)^{2} - \frac{{(1 + \ln B_{12} )x_{1} B_{12} \ln B_{12} }}{{x_{1} B_{12} + x_{2} }}} \right] - \frac{{x_{1} V_{{{\text{m}}1}} B_{12} \ln B_{12} }}{{x_{1} V_{{{\text{m}}1}}B_{12} + x_{2} V_{{{\text{m}}2}} }} \\ & \quad + \frac{{Z_{1} x_{1}^{2} }}{2}\left[ {\frac{{2x_{2} (B_{21} \ln B_{21} )^{2} }}{{(x_{1} + x_{2} B_{21} )^{2} }} - \frac{{2(B_{21} - 1)(x_{2} B_{21} \ln B_{21} )^{2} }}{{(x_{1} + x_{2} B_{21} )^{3} }} - \frac{{(1 + \ln B_{21} )B_{21} \ln B_{21} }}{{x_{1} + x_{2} B_{21} }} + \frac{{(B_{21} - 1)(1 + \ln B_{21} )x_{2} B_{21} \ln B_{21} }}{{(x_{1} + x_{2} B_{21} )^{2} }}} \right] \\ & \quad + \frac{{Z_{2} x_{1} x_{2} }}{2}\left[ \begin{aligned} &- \frac{{2x_{1} (B_{12} \ln B_{12} )^{2} }}{{(x_{1} B_{12} + x_{2} )^{2} }} + \frac{{2(B_{12} - 1)(x_{1} B_{12} \ln B_{12} )^{2} }}{{(x_{1} B_{12} + x_{2} )^{3} }} \\ &+ \frac{{(1 + \ln B_{12} )B_{12} \ln B_{12} }}{{x_{1} B_{12} + x_{2} }} - \frac{{(B_{12} - 1)(1 + \ln B_{12} )x_{1} B_{12} \ln B_{12} }}{{(x_{1} B_{12} + x_{2} )^{2} }} \\ \end{aligned} \right] \\ & \quad + x_{1}^{2} \left[ {\frac{{V_{{{\text{m}}2}} B_{21} \ln B_{21} }}{{x_{1} V_{{{\text{m}}1}} + x_{2} V_{{{\text{m}}2}} B_{21} }} - \frac{{(V_{{{\text{m}}2}} B_{21} - V_{{{\text{m}}1}} )x_{2} V_{{{\text{m}}2}} B_{21} \ln B_{21} }}{{(x_{1} V_{{{\text{m}}1}} + x_{2} V_{{{\text{m}}2}} B_{21} )^{2} }}} \right] \\ & \quad + x_{1} x_{2} \left[ {\frac{{V_{{{\text{m}}1}} B_{12} \ln B_{12} }}{{x_{1} V_{{{\text{m}}1}} B_{12} + x_{2} V_{{{\text{m}}2}} }} + \frac{{(V_{{{\text{m}}2}} - V_{{{\text{m}}1}} B_{12} )x_{1} V_{{{\text{m}}1}} B_{12} \ln B_{12} }}{{(x_{1} V_{{{\text{m}}1}} B_{12} + x_{2} V_{{{\text{m}}2}} )^{2} }}} \right] \\ \end{aligned}. $$
(18)

Both Eqs. [17] and [18] are consistent with the Gibbs–Duhem equation \( x_{1} d\Delta \overline{H}_{1} + x_{2} d\Delta \overline{H}_{2} = 0 \).

Similarly, extending the binary 1–2 to the C-component system, one can obtain the \( \Delta \overline{H}_{i} /RT \) of component i:

$$ \begin{aligned} \frac{{\Delta \overline{H}_{i} }}{RT} & = \frac{{Z_{i} }}{2}\left[ {\left( {\frac{{\sum\limits_{j = 1}^{C} {x_{j} B_{ji} \ln B_{ji} } }}{{\sum\limits_{j = 1}^{C} {x_{j} B_{ji} } }}} \right)^{2} - \frac{{\sum\limits_{j = 1}^{C} {(1 + \ln B_{ji} )x_{j} B_{ji} \ln B_{ji} } }}{{\sum\limits_{j = 1}^{C} {x_{j} B_{ji} } }}} \right] - \frac{{\sum\limits_{j = 1}^{C} {x_{j} V_{{{\text{m}}j}} B_{ji} \ln B_{ji} } }}{{\sum\limits_{j = 1}^{C} {x_{j} V_{{{\text{m}}j}} B_{ji} } }} \\ & \quad + \frac{1}{2}\sum\limits_{t = 1}^{C} {Z_{t} x_{t} } \left[ \begin{aligned} &\frac{{2\sum\limits_{j = 1}^{C} {x_{j} B_{jt} \ln B_{jt} } }}{{\left( {\sum\limits_{j = 1}^{C} {x_{j} B_{jt} } } \right)^{2} }}\left( {B_{it} \ln B_{it} - \sum\limits_{j = 1}^{C} {x_{j} B_{jt} \ln B_{jt} } - \frac{{\left( {B_{it} - \sum\limits_{j = 1}^{C} {x_{j} B_{jt} } } \right)\sum\limits_{j = 1}^{C} {x_{j} B_{jt} \ln B_{jt} } }}{{\sum\limits_{j = 1}^{C} {x_{j} B_{jt} } }}} \right) \\ &- \frac{{(1 + \ln B_{it} )B_{it} \ln B_{it} - \sum\limits_{j = 1}^{C} {x_{j} (1 + \ln B_{jt} )B_{jt} \ln B_{jt} } }}{{\left( {\sum\limits_{j = 1}^{C} {x_{j} B_{jt} } } \right)^{2} }} + \frac{{\left( {B_{it} - \sum\limits_{j = 1}^{C} {x_{j} B_{jt} } } \right)\sum\limits_{j = 1}^{C} {(1 + \ln B_{jt} )x_{j} B_{jt} \ln B_{jt} } }}{{\left( {\sum\limits_{j = 1}^{C} {x_{j} B_{jt} } } \right)^{3} }} \\ \end{aligned} \right] \\ & \quad - \sum\limits_{t = 1}^{C} {x_{t} } \left( {\frac{{V_{{{\text{m}}i}} B_{it} \ln B_{it} - \sum\limits_{j = 1}^{C} {x_{j} V_{{{\text{m}}j}} B_{jt} \ln B_{jt} } }}{{\sum\limits_{j = 1}^{C} {x_{j} V_{{{\text{m}}j}} B_{jt} } }} - \frac{{\left( {V_{{{\text{m}}i}} B_{it} - \sum\limits_{j = 1}^{C} {x_{j} V_{{{\text{m}}j}} B_{jt} } } \right)\sum\limits_{j = 1}^{C} {x_{j} V_{{{\text{m}}j}} B_{jt} \ln B_{jt} } }}{{\left( {\sum\limits_{j = 1}^{C} {x_{j} V_{{{\text{m}}j}} B_{jt} } } \right)^{2} }}} \right) \\ \end{aligned}. $$
(19)

Here, note that the partial derivative in Eq. [13] for the binary 1–2 system is

$$ \begin{aligned} \left( {\frac{{\partial (\Delta H_{\text{m}}^{\text{M}} /RT)}}{{\partial x_{1} }}} \right)_{T,P,x[1,2]} & = \frac{1}{2}\left\{ {\begin{array}{*{20}c} {Z_{1} \left[ {\left( {\frac{{x_{2} B_{21} \ln B_{21} }}{{x_{1} + x_{2} B_{21} }}} \right)^{2} - \frac{{(1 + \ln B_{21} )x_{2} B_{21} \ln B_{21} }}{{x_{1} + x_{2} B_{21} }}} \right]} \\ { - Z_{2} \left[ {\left( {\frac{{x_{1} B_{12} \ln B_{12} }}{{x_{1} B_{12} + x_{2} }}} \right)^{2} - \frac{{(1 + \ln B_{12} )x_{1} B_{12} \ln B_{12} }}{{x_{1} B_{12} + x_{2} }}} \right]} \\ { + Z_{1} x_{1} \left[ { - \frac{{2x_{2} (B_{21} \ln B_{21} )^{2} }}{{(x_{1} + x_{2} B_{21} )^{3} }} + \frac{{(1 + \ln B_{21} )B_{21} \ln B_{21} }}{{(x_{1} + x_{2} B_{21} )^{2} }}} \right]} \\ { + Z_{2} x_{2} \left[ {\frac{{2x_{1} (B_{12} \ln B_{12} )^{2} }}{{(x_{1} B_{12} + x_{2} )^{3} }} + \frac{{(1 + \ln B_{12} )B_{12} \ln B_{12} }}{{(x_{1} B_{12} + x_{2} )^{2}}}} \right]} \\ \end{array} } \right\} \\ & \quad - \left( {\frac{{x_{2} V_{{{\text{m}}2}} B_{21} \ln B_{21} }}{{x_{1} V_{{{\text{m}}1}} + x_{2} V_{{{\text{m}}2}} B_{21} }} - \frac{{x_{1} V_{{{\text{m}}1}} B_{12} \ln B_{12} }}{{x_{1} V_{{{\text{m}}1}} B_{12} + x_{2} V_{{{\text{m}}2}} }}} \right) \\ & \quad + x_{1} \left[ {\frac{{V_{{{\text{m}}2}} B_{21} \ln B_{21} }}{{x_{1} V_{{{\text{m}}1}} + x_{2} V_{{{\text{m}}2}} B_{21} }} + \frac{{(V_{{{\text{m}}1}} - V_{{{\text{m}}2}} B_{21} )x_{2} V_{{{\text{m}}2}} B_{21} \ln B_{21} }}{{(x_{1} V_{{{\text{m}}1}} + x_{2} V_{{{\text{m}}2}} B_{21} )^{2} }}} \right] \\ & \quad - x_{2} \left[ {\frac{{V_{{{\text{m}}1}} B_{12} \ln B_{12} }}{{x_{1} V_{{{\text{m}}1}} B_{12} + x_{2} V_{{{\text{m}}2}} }} - \frac{{(V_{{{\text{m}}1}} B_{12} - V_{{{\text{m}}2}} )x_{1} V_{{{\text{m}}1}} B_{12} \ln B_{12} }}{{(x_{1} V_{{{\text{m}}1}} B_{12} + x_{2} V_{{{\text{m}}2}} )^{2} }}} \right]. \\ \end{aligned} $$
(20)

Similarly, extending the binary 1–2 to the C-component system, one can obtain the first partial derivative of component i in Eq. [13] as follows:

$$ \begin{aligned} \left( {\frac{{\partial (\Delta H_{\text{m}}^{\text{M}} /RT)}}{{\partial x_{i} }}} \right)_{T,P,x[i,C]} & = \frac{1}{2}\left\{ {\begin{array}{l} {Z_{i} \left[ {\left( {\frac{{\sum\limits_{j = 1}^{C} {x_{j} B_{ji} \ln B_{ji} } }}{{\sum\limits_{j = 1}^{C} {x_{j} B_{ji} } }}} \right)^{2} - \frac{{\sum\limits_{j = 1}^{C} {(1 + \ln B_{ji} )x_{j} B_{ji} \ln B_{ji} } }}{{\sum\limits_{j = 1}^{C} {x_{j} B_{ji} } }}} \right] - Z_{C} \left[ {\left( {\frac{{\sum\limits_{j = 1}^{C} {x_{j} B_{jC} \ln B_{jC} } }}{{\sum\limits_{j = 1}^{C} {x_{j} B_{jC} } }}} \right)^{2} - \frac{{\sum\limits_{j = 1}^{C} {(1 + \ln B_{jC} )x_{j} B_{jC} \ln B_{jC} } }}{{\sum\limits_{j = 1}^{C} {x_{j} B_{jC} } }}} \right]} \\ { + \sum\limits_{t = 1}^{C} {Z_{t} x_{t} } \left[ {\frac{{2\sum\limits_{j = 1}^{C} {x_{j} B_{jt} \ln B_{jt} } }}{{\left( {\sum\limits_{j = 1}^{C} {x_{j} B_{jt} } } \right)^{2} }}\left( {B_{it} \ln B_{it} - B_{Ct} \ln B_{Ct} - \frac{{(B_{it} - B_{Ct} )\sum\limits_{j = 1}^{C} {x_{j} B_{jt} \ln B_{jt} } }}{{\sum\limits_{j = 1}^{C} {x_{j} B_{jt} } }}} \right) - \frac{{(1 + \ln B_{it} )B_{it} \ln B_{it} - (1 + \ln B_{Ct} )B_{Ct} \ln B_{Ct} }}{{\left( {\sum\limits_{j = 1}^{C} {x_{j} B_{jt} } } \right)^{2} }} + \frac{{(B_{it} - B_{Ct} )\sum\limits_{j = 1}^{C} {(1 + \ln B_{jt} )x_{j} B_{jt} \ln B_{jt} } }}{{\left( {\sum\limits_{j = 1}^{C} {x_{j} B_{jt} } } \right)^{3} }}} \right]} \\ \end{array} } \right\} \\ & \quad - \left( {\frac{{\sum\limits_{j = 1}^{C} {x_{j} V_{{{\text{m}}j}} B_{ji} \ln B_{ji} } }}{{\sum\limits_{j = 1}^{C} {x_{j} V_{{{\text{m}}j}} B_{ji} } }} - \frac{{\sum\limits_{j = 1}^{C} {x_{j} V_{{{\text{m}}j}} B_{jC} \ln B_{jC} } }}{{\sum\limits_{j = 1}^{C} {x_{j} V_{{{\text{m}}j}} B_{jC} } }}} \right) \\ & \quad - \sum\limits_{t = 1}^{C} {x_{t} } \left( {\frac{{V_{{{\text{m}}i}} B_{it} \ln B_{it} - V_{{{\text{m}}C}} B_{Ct} \ln B_{Ct} }}{{\sum\limits_{j = 1}^{C} {x_{j} V_{{{\text{m}}j}} B_{jt} } }} - \frac{{(V_{{{\text{m}}i}} B_{it} - V_{{{\text{m}}C}} B_{Ct} )\sum\limits_{j = 1}^{C} {x_{j} V_{{{\text{m}}j}} B_{jt} \ln B_{jt} } }}{{\left( {\sum\limits_{j = 1}^{C} {x_{j} V_{{{\text{m}}j}} B_{jt} } } \right)^{2} }}} \right) \\ \end{aligned}. $$
(21)

And the second partial derivative \( \left( {\frac{{\partial (\Delta H_{\text{m}}^{\text{M}} /RT)}}{{\partial x_{l} }}} \right)_{T,P,x[l,C]} \) is same as in Eq. [21] in which the subscript i is replaced by the subscript l correspondingly. It can be seen that Eqs. [17] through [21] are complex, especially the last one, but they are correct indeed for the MIVM thermodynamically.

Similarly, the partial molar excess entropy of the MIVM can be also derived from Eq. [22]:

$$ \overline{S}_{i}^{\text{E}} = S_{\text{m}}^{\text{E}} + \left( {\frac{{\partial S_{\text{m}}^{\text{E}} }}{{\partial x_{i} }}} \right)_{T,P,x[i,C]} - \sum\limits_{l = 1}^{C - 1} {x_{l} } \left( {\frac{{\partial S_{\text{m}}^{\text{E}} }}{{\partial x_{l} }}} \right)_{T,P,x[l,C]} \;{\text{or}}\;\overline{S}_{i}^{\text{E}} = \frac{{\Delta \overline{H}_{i} - \overline{G}_{i}^{E} }}{T}, $$
(22)

where \( \overline{G}_{i}^{\text{E}} = RT\ln \gamma_{i} \) can be obtained from Eq. [16].

Next, the author tries to simplify the MIVM reasonably. In a recent work,[11] the author used two different coordination numbers of phosphorus, Z P = 3.04 from Eq. [24] in Reference 11 and Z P = 8.96 from Eq. [26] in Reference 11, to examine how they affect the fitting accuracy of component activities in the M-P (M = Cr, Fe, and Mn) alloys[17–19] and the predicted accuracy of component activities in the Fe-Cr-P[20] in the temperature range from 1471 K to 1817 K (1198 °C to 1544 °C) and Fe-Mn-P[21] from 1302 K to 1678 K (1029 °C to 1405 °C). The results are listed in Tables I and II.

Table I Fitting Errors and Binary Parameters in Binary Liquid Alloys Containing Phosphorus for Z P = 3.04 and Z P = 8.96
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Table II Average Errors Between Predicted Values of MIVM and Experimental Data of Component Activities in Ternary Liquid Alloys Fe-Cr-P[20] and Fe-Mn-P[21] for Z P = 3.04 and Z P = 8.96
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In Table I, it can be seen that the binary parameters B ji and B ij have different values for Z P = 3.04 and Z P = 8.96 in the same binary alloy, but the fitting errors of two component activities are close each other in the M-P (M = Cr, Fe and Mn) alloys, and the errors for Z P = 8.96 are a little smaller than those for Z P = 3.04. Here note that \( S_{i} = \frac{100}{n}\sum\limits_{i = 1}^{n} {\left| {\frac{{a_{i,\exp } - a_{{i,{\text{pre}}}} }}{{a_{i,\exp } }}} \right|} \) is the average relative error, where a i,exp and a i,pre are the experimental data and the predicted values of activity of component i, respectively, and n is the number of experimental data; the binary parameters of the Ti-P at 1998 K (1725 °C) and the V-P at 1973 K (1700 °C) are supplement to their data listed in Table II in Reference 11.

In Table II, it can be seen that the average errors for Z P = 3.04 and Z P = 8.96 are close to each other by comparing the predicted values of the MIVM with the experimental data of all component activities in the ternary liquid alloys Fe-Cr-P and Fe-Mn-P, and the errors for Z P = 8.96 are a little smaller than those for Z P = 3.04 in general, which are similar to the binary M-P (M = Cr, Fe, and Mn) alloys. Here, it should be noticed that in the Fe-Mn-P, although the errors of 59 and 44 pct for P seem to be large, they are acceptable by means of comparing them with the error of 48 pct for P calculated[11] from the unified interaction parameter formalism (UIPF)[22,23] at current knowledge level.

The above results indicate that the difference of component coordination numbers, such as that (5.92) of phosphorus coordination numbers 3.04 and 8.96 in Tables I and II, would almost not affect on the accuracies of fitting and predicting component activities. It is inferred that the differences of coordination numbers in the MIVM could be neglected.

In fact, the coordination numbers of molecules have the same constant in the lattice theory of solutions and may be a value between 6 and 12 depending on the type of packing, i.e., the way in which the molecules are arranged in three-dimensional space; empirically, for typical liquids in ordinary conditions, Z is close to 10.[16] Therefore, one can suppose Z i  = Z j  = Z = 10 in the MIVM for simplifying its calculation.

On the other hand, since the difference between liquid and solid densities of a substance is often small, one can suppose that the molar volume V mi of component i in liquid state can be replaced by its molar volume V i in solid state.

Consequently, the molar excess Gibbs energy of the MIVM in Eq. [1] can be simplified as

$$ \frac{{G_{\text{m}}^{\text{E}} }}{RT} = \sum\limits_{i = 1}^{C} {x_{i} \ln \frac{{V_{i} }}{{\sum\limits_{j = 1}^{C} {x_{j} V_{j} B_{ji} } }}} - \frac{Z}{2}\sum\limits_{i = 1}^{C} {x_{i} \left( {\frac{{\sum\limits_{j = 1}^{C} {x_{j} B_{ji} \ln B_{ji} } }}{{\sum\limits_{j = 1}^{C} {x_{j} B_{ji} } }}} \right)} $$
(23a)
$$ \frac{{G_{\text{m}}^{\text{E}} }}{RT} = \sum\limits_{i = 1}^{C} {x_{i} \ln \frac{{V_{i} }}{{\sum\limits_{j = 1}^{C} {x_{j} V_{j} B_{ji} } }}} - 5\sum\limits_{i = 1}^{C} {x_{i} \left( {\frac{{\sum\limits_{j = 1}^{C} {x_{j} B_{ji} \ln B_{ji} } }}{{\sum\limits_{j = 1}^{C} {x_{j} B_{ji} } }}} \right)}. $$
(23b)

And the other expressions of the MIVM, such as Eqs. [16] and [19], can be also simplified as the similar forms in Eqs. [23a] and [23b]. It can be seen that Eq. [23b] does not explicitly involve the coordination number of pure component and so it is more convenient for use than before.

Now, let us use the simplified form of the MIVM to test the consistency of Eqs. [7], [8], [17], and [18]. Let Z i  = Z j  = Z = 10, \( V_{{{\text{m}}i}} = V_{i}, \) and \( V_{{{\text{m}}j}} = V_{j} \) and choose the 4 binary liquid alloys Au-Cu at 1400 K (1127 °C), Cd-Zn at 700 K (427 °C), Ca-Zn at 1150 K (877 °C), and Ni-Pb at 1900 K (1627 °C)[24], which exhibit negative or positive deviation from ideality. As shown in Figures 1 and 2 and Table III, it can be seen that the fitting curves of molar quantities \( \Delta G_{\text{m}}^{\text{M}} \) and \( \Delta H_{\text{m}}^{\text{M}} \) are in good agreement with the literature data[24] and the former is better than the later. The average errors of the partial molar enthalpies of mixing, 94 pct for \( \Delta \overline{H}_{i} \) and 43 pct for \( \Delta \overline{H}_{j} \), are larger than those of the activities, 5.6 pct for a i and 5.9 pct for a j . But the fitting curves of \( \Delta \overline{H}_{i} \) and \( \Delta \overline{H}_{j} \) may be still in reasonable agreement with the literature data.[24]

Fig. 1
figure 1

In the Au-Cu at 1400 K (1127 °C) and Cd-Zn at 700 K (427 °C), comparison of fitting curves (solid line) with literature data (black or empty square and circle)[24] of (a) activity, (b) molar Gibbs energy of mixing, (c) partial molar enthalpy of mixing, and (d) molar enthalpy of mixing, standard states: pure liquid metals

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Fig. 2
figure 2

In the Ca-Zn at 1150 K (877 °C) and Ni-Pb at 1900 K (1627 °C), comparison of fitting curves (solid line) with literature data (black or empty square and circle)[24] of (a) activity, (b) molar Gibbs energy of mixing, (c) partial molar enthalpy of mixing, and (d) molar enthalpy of mixing, standard states: pure liquid metals

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Table III Fitting Errors of Thermodynamic Properties and Parameters in Binary Liquid Alloys Au-Cu at 1400 K (1127 °C), Cd-Zn at 700 K (427 °C), Ca-Zn at 1150 K (877 °C), and Ni-Pb at 1900 K (1627 °C)[24]
Full size table

The results preliminarily indicate that the correct expressions of the MIVM are mutually coordinated with the same binary parameters B ji and B ij in the four binary alloys. Further systematic investigations are necessary because the good fittings in the case do not always appear. Inversely, some bad fittings are arising in the Al-Be at 1600 K (1327 °C), B-Nd at 3000 K (2727 °C), Ir-Rh at 2800 K (2527 °C), Co-In at 1800 K (1527 °C),[24] and so forth.

With regard to the strength and weakness of the MIVM, according to the experience of 15 years of working on its predictions of thermodynamic properties of liquid alloys and silicate melts, the author has empirically realized that its main points are as follows: (1) the smaller the fitting errors of the related binary systems are, the higher the predicted accuracies of the corresponding multicomponent systems are, but the necessary condition is that the binary experimental data are reliable indeed; (2) for the binary system exhibiting both positive and negative deviations from ideality, it is difficult to fit reasonably its two activity–composition curves from the positive to the negative experimentally; and (3) the MIVM itself does not show the effect of chemical interaction on the molar and partial molar excess Gibbs energies.

In conclusion, it would lead to those incorrect expressions to compare empirically the \( {G_{\text{m}}^{\text{E}} } \) of the MIVM with its thermodynamic definition \( G_{\text{m}}^{\text{E}} = \Delta H_{\text{m}}^{\text{M}} - TS_{\text{m}}^{\text{E}} \), such as Eqs. [3] and [4]. Only based on the thermodynamic partial derivatives in Eqs. [5] and [6], the correct expressions of \( \Delta H_{\text{m}}^{\text{M}} \) and S Em as well as their partial molar enthalpy of mixing and excess entropy can be deduced from the MIVM in Eq. [1]. Their simplified forms are consistent thermodynamically and would be easily used to predict the thermodynamic properties of a multicomponent liquid system.