BIBECHANA Vol. 21, No. 2, August 2024, 83-94 ISSN 2091-0762 (Print), 2382-5340 (Online) Journal homepage: http://nepjol.info/index.php/BIBECHANA Publisher: Dept. of Phys., Mahendra Morang A. M. Campus (Tribhuvan University)Biratnagar Theoretical exploration of thermodynamic characteristics in lead-free liquid alloys: Zn-Bi-In System Sanjay Kumar Sah1,3,∗, Indu Shekhar Jha2, Ishwar Koirala1 1Central Department of Physics, Tribhuvan University, Kirtipur, Kathmandu, Nepal 2Department of Physics, M.M.A.M. Campus, Tribhuvan University, Biratnagar, Nepal 3Dept. of Physics, Birendra Multiple Campus, Tribhuvan University, Bharatpur, Nepal ∗Corresponding author. Email: sanjay.sah@bimc.tu.edu.np Abstract The calculations of Zn activities in the ternary liquid solder alloy Zn-Bi-In at 873 K were conducted through the application of the Molecular Interaction Volume Model (MIVM). The calculated values were compared to experimental data for four different cross-sectional ex- aminations, namely for the bismuth-to-indium ratios (xBi: xIn) of 1:2, 1:1, 2:1, and 9:1. Moreover, the integral excess free energy of these ternary liquid alloys was assessed using the same model parameters in order to test their validity. Then, the values were compared to the relevant experimental data reported in the literature that was already in existence. A satisfac- tory concordance has been observed between the theoretical predictions and the experimental findings. Keywords Activity; Integral excess free energy; Solder alloys; Ternary liquid alloys; Molecular interaction volume model. Article information Manuscript received: November 21, 2023; Revised: December 26, 2023; Accepted: January 14, 2024 DOI https://doi.org/10.3126/bibechana.v21i2.60097 This work is licensed under the Creative Commons CC BY-NC License. https://creativecommons. org/licenses/by-nc/4.0/ 1 Introduction It is possible to investigate the thermodynamic characteristics of different alloys through experi- mental measurements. But the experimental meth- ods are much more costly and time-consuming, given that numerous measurements and advanced instruments are required. Furthermore, experimen- tal data are scarce for alloys with multiple compo- nents. As such, it is difficult to obtain all thermo- dynamic data solely through experimentation [1–4] Therefore, it is more advantageous to obtain the 83 http://nepjol.info/index.php/BIBECHANA sanjay.sah@bimc.tu.edu.np https://doi.org/10.3126/bibechana.v21i2.60097 https://creativecommons.org/licenses/by-nc/4.0/ https://creativecommons.org/licenses/by-nc/4.0/ Sanjay Kumar Sah et al./ BIBECHANA 21 (2024) 83-94 84 data through theoretical means. Lead-based sys- tems have been traditionally preferred for soldering due to their lower cost and lower melting point. However, lead is a toxic metal and has adverse ef- fects on the environment. It can contaminate the soil, water, and air [5–8]. Currently, there is a worldwide agreement to outlaw lead use. Therefore, the development of soldering alternatives without lead has emerged as a crucial topic in the field of electronics [8]. In addition to comparable melting points, the new alloys ought to possess other quali- ties like low cost, non-toxicity, fatigue and corrosion resistance, and wetting qualities. Thus, researchers have attempted to replace lead in solder alloys mul- tiple times [8]. Zinc, bismuth, and indium can be chosen as beneficial materials for lead-free solder alloys due to several advantageous properties and characteris- tics. These are less toxic than lead, making them more environmentally friendly alternatives. Alloys of those materials can have a relatively lower melt- ing point and often exhibit good wetting properties. They also exhibit good thermal fatigue resistance, contributing to the reliability and durability of sol- der joints, especially in applications where the tem- perature may vary. Mainly, zinc imparts corrosion resistance to the solder alloys, and it is a relatively inexpensive material that is readily available [9]. Various alloy systems have been developed as alternatives to traditional Sn-Pb (tin-lead) al- loys, particularly for electronic soldering applica- tions. Here are examples of binaries, ternaries, and higher alloy systems: Tin-Silver (Sn-Ag), Gold-Tin (Au-Sn), Tin-Copper (Sn-Cu), Tin-Silver-Copper (Sn-Ag-Cu), Aluminium-Tin-Zinc (Al-Sn-Zn), Tin- Antimony-Bismuth (Sn-Sb-Bi), Tin-Silver-Copper- Antimony (Sn-Ag-Cu-Sb), and Tin-Silver-Copper- Zinc (Sn-Ag-Cu-Zn), etc. [6–8, 10–15]. Several ge- ometric models are widely used in predicting the thermodynamic properties of binary, ternary, and other systems of higher complexity. These mod- els include Muggianu, Kohler, Chou, Collinet, R-K equations, the Kaptay model, etc. [16–18]. MIVM is superior to other models because of the way it is formulated, which requires fewer fitting parame- ters (only two) [19]. MIVM has the unique capa- bility to simultaneously represent the influences of the microstate number of molecular configuration (entropy) and molecular interaction (enthalpy) on excess Gibbs energy. Over the past two decades, the precision and dependability of MIVM in predicting the properties of liquid alloys have been substan- tiated [20, 21]. MIVM is a model characterized by two parameters. One of the model’s most impor- tant features is its capability to forecast liquid alloy thermodynamic properties in a system of multiple component mixtures using typical physical param- eters of metals in liquid form and binary infinite dilution activity coefficients alone [1]. Thermodynamic activities of the components of ternary liquid alloys Indium-Bismuth-Tin (In- Bi-Sn), Tin-Gold-Copper (Sn-Au-Cu), and Zinc- Indium-Tin (Zn-In-Sn), along with their excess Gibbs free energy of mixing, have been performed by Sah et al. [9, 22]. A comprehensive thermody- namic representation of the complete ternary sys- tem Bi-In-Zn was achieved through CALPHAD modeling, focusing on the Gibbs energy of the liquid phase [23]. Utilizing a Calvet-type microcalorime- ter and the drop calorimetric technique, the inte- gral enthalpies of mixing for liquid Bi-In-Zn alloys at 773 K were determined [24]. But there is only re- stricted information accessible through this system. The thermodynamic characteristics of the Bi-In-Zn ternary system were established utilizing the elec- tromotive force (EMF) method, employing a liquid electrolyte, as outlined by the reference [25]. Never- theless, there is presently no documentation in the literature regarding the computation of their inte- gral excess free energy. Hence, in this study, the molecular interaction volume model was utilized to determine the thermodynamic activities of the Zn component in the ternary liquid alloys Zn-Bi-In at 873 K for the four cross-sections, namely xBi: xIn = 1:2, 1:1, 2:1, and 9:1. Moreover, the theoretical model was employed to compute the integral ex- cess Gibbs free energy of mixing for Zn-Bi-In at 873 K, considering varying concentrations of Zn (xZn) across all those cross-sections. 2 Methodology The MIVM model states that liquid molecules be- have differently from both gas molecules, which move randomly all the time, and solid molecules, which oscillate continuously at one spot but move between molecular cells in a non-random way. The cells are both mobile and identical, and the molecules in the center and those closest to them can be interchanged [26]. MIVM is a fluid model based on statistical thermodynamics that takes into consideration the inherent physical characteristics of the individual pure metals that make up the al- loy. Molar volume (Vm) and the liquid phase’s first coordination number (Z) are specifically taken into account, which are dependent on temperature [19]. The excess molar Gibbs energy of the liquid binary mixture, i-j, is expressed as per Dong Ping Tao in 2000 based on the molecular interaction volume model [27]: Sanjay Kumar Sah et al./ BIBECHANA 21 (2024) 83-94 85 ∆GXS = RT { xi ln ( Vmi xiVmi + xjVmjAji ) + xj ln ( Vmj xjVmj + xiVmiAij ) − xixj 2 ( ZiAji lnAji xi + xjAji + ZjAij lnAij xj + xiAij )} (1) In this context, Vmi and Vmj represent the molar volumes, while xi and xj denote the molar fractions of components i and j, respectively. Aji and Aij are the interaction parameters associated with the pair potential energy for binary liquid alloys. Zi and Zj are the first coordination numbers specific to the ith and jth component metals in binary liquid al- loys. T stands for the absolute temperature and R represents the gas constant. In the case of a binary liquid alloy, the activity coefficients of components i and j, denoted as γi and γj , are expressed as per Tao in 2008 [26] as: ln γi = ln ( Vmi xiVmi + xjVmjAji ) + xj ( VmjAji xiVmi + xjVmjAji − VmiAij xjVmj + xiVmiAij ) − x2 j 2 [ ZiA 2 ji lnAji (xi + xjAji) 2 + ZjAij lnAij (xj + xiAij) 2 ] (2) ln γj = ln ( Vmj xjVmj + xiVmiAij ) − xi ( VmjAji xiVmi + xjVmjAji − VmiAij xjVmj + xiVmiAij ) − x2 i 2 [ ZjA 2 ij lnAij (xj + xiAij) 2 + ZiAji lnAji (xi + xjAji) 2 ] (3) For the system of higher orders, i.e., for a mul- ticomponent mixture, Eq. (1) becomes [27]: ∆GXS = RT { N∑ i=1 xi ln Vmi∑N j=1 xjVmjAji − 1 2 N∑ i=1 Zixi (∑N j=1 xjAji lnAji∑N k=1 xkAki )} (4) Where N is the number of components in a mix- ture. The formula for the activity coefficient of the ith component, γi, is provided in the work of Tao in 2008 [26] by: ln γi = 1 + ln Vmi∑N j=1 xjVmjAji − N∑ k=1 xkVmiAik∑N j=1 xjVmjAjk − 1 2 { Zi ∑N j=1 xjAji lnAji∑N l=1 xlAli + N∑ j=1 ZjxjAij∑N l=1 xlAlȷ × ( lnAij − ∑N t=1 xtAtj lnAtj∑N l=1 xlAlj )} (5) The formula for Zi is presented in a publication by Tao in 2008 [26] by: Zi = 4 √ 2π 3 ( r3mi − r3oi rmi − roi ) ρirmi exp ( ∆Hmi (Tmi − T) ZcRTTmi ) (6) The molecular number density can be deter- mined using the equation, , where Ni = 0.6022 rep- resents the molecular number. The melting tem- perature and melting enthalpy are indicated by Tmi and ∆Hmi, respectively. Zc signifies a close-packed coordination, with a value of 12 for certain liquid metals. roi represents the first peak value of the radial distribution function, and rmi is the initial value of the radial distribution function, whose ex- pressions are as follows: roi = 0.918 dcovi (7) And rmi = σi (8) Here dcovi and σi represent the atomic covalent diameter and atomic diameter, respectively. The interaction parameters associated with pair potential energy are denoted by Aji and Aij, accord- ing to the findings of Tao et al. in 2002 as [33]: Aji = exp ( −εji − εii KT ) ; Aij = exp ( −εij − εji KT ) (9) Here εjior εij, εii and εjj represent the pair po- tential energies for i-j, i-i, and j-j interactions, re- spectively, with K being the Boltzmann constant. The activity coefficients γ∞ i and γ∞ j for compo- nents i and j in binary liquid alloys are given in the limit of infinite dilution [26] as: ln γ∞ i = 1− ln ( VmjAji Vmi ) − VmiAij Vmj − 1 2 (Zi lnAji + ZjAij lnAij) (10) Sanjay Kumar Sah et al./ BIBECHANA 21 (2024) 83-94 86 ln γ∞ j = 1− ln ( VmiAij Vmj ) − VmjAji Vmi − 1 2 (Zj lnAij + ZiAji lnAji) (11) Equations (10) and (11) must be solved using the Newton-Raphson method in order to determine Aji and Aij. These starting values are then adjusted when component activities in liquid binary alloys are calculated using equations (2) and (3). Odusote et al. (2017) [28] explained that once the appropri- ate values of Aji and Aij at a particular temperature is established, their corresponding values at differ- ent temperatures can also be derived. Considering the Zn-Bi-In ternary liquid alloy as the 1-2-3 system, the activity coefficient of the first metal, denoted as γ1, in the system can be deter- mined using equation (5), as explained by Tao in 2001 as [29]: ln γ1 = 1 + ln ( Vm1 x1Vm1 + x2Vm2A21 + x3Vm3A31 ) − x1Vm1 x1Vm1 + x2Vm2A21 + x3Vm3A31 − x2Vm1A12 x1Vm1A12 + x2Vm2 + x3Vm3A32 − x3Vm1A13 x1Vm1A13 + x2Vm2A23 + x3Vm3 − 1 2 ( Z1 (x2A21 + x3A31) (x2A21 lnA21 + x3A31 lnA31) (x1 + x2A21 + x3A31) 2 + Z2x2A12 [(x2 + x3A32) lnA12 − x3A32 lnA32] (x1A12 + x2 + x3A32) 2 + Z3x3A13 [(x2A23 + x3) lnA13 − x2A23 lnA23] (x1A13 + x2A23 + x3) 2 ) (12) Likewise, the surplus Gibbs free energy of mix- ing, ∆GXS for Zn-Bi-In ternary liquid alloys can be derived from equation (4) as: ∆GXS = RT  x1 ln ( Vm1 x1 Vm1+x2 Vm2 A21+x3 Vm3 A31 ) +x2 ln ( Vm2 x1 Vm1 A12+x2 Vm2+x3 Vm3 A32 ) +x3 ln ( Vm3 x1 Vm1 A13+x2 Vm2 A23+x3 Vm3 ) − 1 2 [ Z1x1(x2 A21 lnA21+x3 A31 lnA31) (x1+x2 A21+x3 A31) +Z2x2(x1 A12 lnA12+x3 A32 lnA32) (x1 A12+x2+x3 A32) + Z3x3(x1 A13 lnA13+x2 A23 lnA23) (x1 A13+x2 A23+x3)  (13) 3 Results and Discussion The required input parameters are outlined in Ta- ble 1. The values of the interaction parameters of potential energy (Aji and Aij) for the binary alloys, as well as the first coordination numbers (Zi and Zj) for the components of the binary systems, are pro- vided in Table 2. Coordination numbers have been computed using the formula provided in Eq. (6). By plugging in the values of Vm1, Vm2, Vm3, Z1, Z2, Z3, A12, A21, A13, A31, A23, and A32 into equa- tions (12) and (13), the activities of the Zn com- ponent in the Zn-Bi-In systems at 873 K and the excess Gibbs free energy of mixing, ∆GXS, for those ternary liquid alloys at 873 K were computed for all four cross-sections of xBi: xIn = 1:2, 1:1, 2:1, and 9:1. The results were then compared with the cor- responding experimental values [25], as illustrated in Table 3 and Table 4 and depicted in Figures 1, 2, 3, and 4, respectively. Table 1: Certain input parameters [30]. Metal ∆Hmi [KJ/mol] roi [×10−8 cm] rmi [×10−8 cm] Vmi [cm3/mol] Zn 7.322 2.16 2.66 9.94(1 + 1.50× 10−4(T − 693)) Bi 11.30 2.78 3.34 20.80(1 + 1.17× 10−4(T − 544)) In 3.263 2.70 3.14 16.30(1 + 0.97× 10−4(T − 430)) Table 2: Values of Aij , Aji, Zi, and Zj computed for a range of temperature conditions in binary alloys. i-j T [K] Aij Aji Zi Zj Bi− Zn 873 1.1106 0.4125 8.1043 8.9699 In− Zn 873 1.008 0.7123 9.1631 8.9699 Bi− In 873 1.3774 0.7545 8.1043 9.1631 Sanjay Kumar Sah et al./ BIBECHANA 21 (2024) 83-94 87 Table 3: Experimental and theoretical data on Zn activities in Zn-Bi-In ternary liquid alloys at 873 K. xZn xBi xIn aZn, Th. E(mV) aZn, Exp.∗ |aZn, Th.− aZn, Exp. ∗ | xBi : xIn = 1 : 2 0.00000 ——- ——- 0.00000 ——- ——- ——- 0.09900 0.30033 0.60067 0.30081 43.3253 0.31605 0.01524 0.20410 0.26530 0.53060 0.53972 23.2751 0.53859 0.00113 0.29920 0.23360 0.46720 0.69326 14.7078 0.67636 0.01690 0.40160 0.19946 0.39894 0.80280 11.456 0.73744 0.06536 0.49120 0.16960 0.33920 0.86063 6.6841 0.83719 0.02344 0.59730 0.13423 0.26847 0.89566 4.584 0.88526 0.01040 0.69580 0.10140 0.20280 0.90833 3.3761 0.91415 0.00582 0.79760 0.06746 0.13494 0.91601 3.0964 0.92097 0.00496 0.89730 0.03423 0.06847 0.93611 1.987 0.94854 0.01243 1.00000 0.00000 0.00000 1.00000 ——- ——- ——- xBi : xIn = 1 : 1 0.00000 ——- ——- 0.00000 ——- ——- ——- 0.09940 0.45030 0.45030 0.30020 44.5091 0.30626 0.00606 0.2017 0.39915 0.39915 0.54240 22.7301 0.54645 0.00405 0.30240 0.34880 0.34880 0.71736 13.1409 0.70513 0.01223 0.39980 0.30010 0.30010 0.83199 7.88080 0.81097 0.02102 0.50040 0.24980 0.24980 0.90211 5.05520 0.87424 0.02787 0.59860 0.20070 0.20070 0.93336 3.4939 0.91129 0.02207 0.70060 0.14970 0.14970 0.93997 2.8001 0.92826 0.01171 0.80020 0.09990 0.09990 0.93665 2.1225 0.94513 0.00848 0.89930 0.05035 0.05035 0.94450 1.5482 0.95967 0.01517 1.00000 0.00000 0.00000 1.00000 ——- ——- ——- xBi : xIn = 2 : 1 0.00000 ——- ——- 0.00000 ——- ——- ——- 0.10030 0.59980 0.29990 0.29176 45.3836 0.29922 0.00746 0.19910 0.53394 0.26696 0.52902 24.8022 0.51716 0.01186 0.30060 0.46627 0.23313 0.71723 14.2952 0.68382 0.03341 0.40010 0.39994 0.19996 0.84696 8.6099 0.79540 0.05156 0.49980 0.33347 0.16673 0.92624 4.5123 0.88695 0.03929 0.60090 0.26607 0.13303 0.96254 4.5873 0.88518 0.07736 0.70020 0.19987 0.09993 0.96658 2.8197 0.92777 0.03881 0.80210 0.13194 0.06596 0.95496 1.6984 0.95585 0.00089 0.90070 0.06620 0.03310 0.95209 1.451 0.96215 0.01006 1.00000 0.00000 0.00000 1.00000 ——- ——- ——- xBi : xIn = 9 : 1 0.00000 ——- ——- 0.00000 ——- ——- ——- 0.10040 0.80964 0.08996 0.26752 49.751 0.26642 0.00110 0.20050 0.71955 0.07995 0.50356 28.031 0.47462 0.02894 0.30040 0.62964 0.06996 0.69851 16.1431 0.65104 0.04747 0.39800 0.54180 0.06020 0.84287 9.2727 0.78151 0.06136 0.50050 0.44955 0.04995 0.94218 4.7572 0.88119 0.06099 0.60020 0.35982 0.03998 0.98945 1.8141 0.95291 0.03654 0.70020 0.26982 0.02998 0.99628 1.6721 0.95651 0.03977 0.80020 0.17982 0.01998 0.97816 1.2784 0.96658 0.01158 0.90010 0.08991 0.00999 0.96181 0.90330 0.97627 0.00108 1.00000 0.00000 0.00000 1.00000 ——- ——- ——- *Experimental [25] Si = ± 3.10 %, S∗ i = ± 0.0302* Sanjay Kumar Sah et al./ BIBECHANA 21 (2024) 83-94 88 Table 4: Experimental and theoretical data on Zn-Bi-In ternary liquid alloys’ excess Gibbs free energy mixing at 873 K. xZn xBi xIn ∆GXSTh. [J/mol] ∆GXSExp.* [J/mol] xBi : xIn = 1 : 2 0.0000 0.3333 0.6667 -1549.8 -1793.0 0.1000 0.2250 0.6750 -373.4 -143.0 0.2000 0.2667 0.5333 355.8 692.0 0.3000 0.2333 0.4667 1133.4 1429.0 0.4000 0.2000 0.4000 1763.4 2025.0 0.5000 0.1667 0.3333 2216.0 2417.0 0.6000 0.1333 0.2667 2453.4 2566.0 0.7000 0.1000 0.2000 2427.1 2460.0 0.8000 0.0667 0.1333 2073.4 2103.0 0.9000 0.0333 0.0667 1304.8 1464.0 1.0000 0.0000 0.0000 0.0 0.0 xBi : xIn = 1 : 1 0.0000 0.5000 0.5000 -1635.1 -1916.0 0.1000 0.4500 0.4500 -633.3 -291.0 0.2000 0.4000 0.4000 281.1 560.0 0.3000 0.3500 0.3500 1087.3 1337.0 0.4000 0.3000 0.3000 1758.5 1984.0 0.5000 0.2500 0.2500 2260.8 2426.0 0.6000 0.2000 0.2000 2550.1 2618.0 0.7000 0.1500 0.1500 2567.7 2544.0 0.8000 0.1000 0.1000 2233.4 2209.0 0.9000 0.0500 0.0500 1433.8 1568.0 1.0000 0.0000 0.0000 0.0 0.0 xBi : xIn = 2 : 1 0.0000 0.6667 0.3333 -1368.1 -1547.0 0.1000 0.6750 0.2250 -230.3 -39.0 0.2000 0.5333 0.2667 443.2 746.0 0.3000 0.4667 0.2333 1224.6 1482.0 0.4000 0.4000 0.2000 1887.7 2094.0 0.5000 0.3333 0.1667 2396.0 2501.0 0.6000 0.2667 0.1333 2700.8 2658.0 0.7000 0.2000 0.1000 2734.2 2562.0 0.8000 0.1333 0.0667 2401.5 2227.0 0.9000 0.0667 0.0333 1563.3 1619.0 1.0000 0.0000 0.0000 0.0 0.0 xBi : xIn = 9 : 1 0.0000 0.9000 0.1000 -512.1 -504.0 0.1000 0.8100 0.0900 266.3 811.0 0.2000 0.7200 0.0800 1003.7 1414.0 0.3000 0.6300 0.0700 1679.0 2001.0 0.4000 0.5400 0.0600 2263.9 2506.0 0.5000 0.4500 0.0500 2719.8 2851.0 0.6000 0.3600 0.0400 2993.4 2981.0 0.7000 0.2700 0.0300 3007.9 2871.0 0.8000 0.1800 0.0200 2650.2 2506.0 0.9000 0.0900 0.0100 1744.6 1812.0 1.0000 0.0000 0.0000 0.0 0.0 Sanjay Kumar Sah et al./ BIBECHANA 21 (2024) 83-94 89 Table 4 indicates that the theoretical and ex- perimental values [25] of the excess free energy of mixing in Zn-Bi-In liquid alloys at 873 K exhibit reasonable agreement, albeit with some discrepan- cies. The highest errors are observed at xZn = 0.1 for all four cross-sections, specifically xBi: xIn = 1:2, 1:1, 2:1, and 9:1. From the experimental work of Knott et al. [25], the activity of the component Zn in the Zn-Bi-In ternary liquid alloy can be determined by a rela- tion given in Eq. (14). ∆GZn = RT ln aZn = −zFE (14) where ∆GZn is the change in Gibbs free en- ergy for the Zn component, z is the valency of Zn, F represents Faraday’s constant (i.e., 96486 Coulomb/mol), R represents the molar gas constant and E represents the measured EMF of the cell, whose values for all the cross-sections of Bi and In have been taken from the reference [25]. Based on the information in Table 3, the mean relative error (Si) and mean standard deviation (S∗ i ) were computed to precisely determine the ex- tent of the difference between the experimental and predicted data. To calculate the average relative error, equation (15) can be used. Si = ±100 n n∑ i=1 ∣∣∣∣ai,Th. − ai,Exp.* ai,Exp.* ∣∣∣∣ (15) Eq. (16) can be used to get the mean standard deviation. S∗ i = ± [ 1 n n∑ i=1 (ai, Th. − ai, Exp. ∗)2 ] 1 2 (16) In this study, 36 experimental data points (n) were considered. Table 3 reveals that according to MIVM, the average relative error and mean stan- dard deviation values are ± 3.10% and 0.0302, re- spectively. These values are notably lower, indicat- ing the model’s precision in predicting the activity of the Zn component in the Zn-Bi-In ternary liquid alloys at 873 K. Table 3 makes it clear that there are some dis- crepancies between the theoretical and experimen- tal values [25] of the Zn component’s activities in Zn-Bi-In liquid alloys at 873 K, but overall there is reasonable agreement. For the cross-sections xBi: xIn = 1:2, 8.86% at xZn = 0.40160, for xBi: xIn = 1:1, 3.18% at xZn = 0.50040, for xBi: xIn = 2:1, 8.73% at xZn = 0.60090, and for xBi: xIn = 9: 1, 7.85% at xZn = 0.39800 exhibits the highest error measurement, respectively. Table 4 indicates that the theoretical and ex- perimental [25] values of the excess free energy of mixing in Zn-Bi-In liquid alloys at 873 K exhibit reasonable agreement, albeit with some discrepan- cies. The highest errors are observed at xZn = 0.1 for all four cross-sections, specifically xBi: xIn = 1:2, 1:1, 2:1, and 9:1. Figure 1(a) demonstrates that there is a signif- icant positive departure from ideality in Zn’s theo- retical and experimental activity (aZn). Up to xZn = 0.2992, the deviation between the theoretical and experimental results is essentially unchanged. The experimental and theoretical values exhibit a small dispersion in the concentration range of 0.2992 ≤ xZn ≤ 0.4912. Again, from xZn = 0.4912 to 1.0, the deviation between them is minor. In particu- lar, these plots show that, as xZn increases within the parameter range of 0.0 ≤ xZn ≤ 0.4016, there is an increasing positive activity deviation from ideal Raoult’s law. On the other hand, a decrease in the positive activity deviation is noted in the range 0.4016 ≤ xZn ≤ 1.0. These observations are partic- ular to cross-sections with xBi: xIn = 1:2 as their defining ratio. For Zn-Bi-In liquid alloys at 873 K, the integral excess free energy of mixing ∆GXS versus xZn for the cross-section xBi: xIn =1:2 was plotted in Fig- ure 1(b). For this cross-section, the data indicates that there were both positive and negative values found for ∆GXS. Under the same concentration, xZn = 0.6, the maximum positive values were 2453.4 J/mol (Th.) and 2566.0 J/mol (Expt.). The appar- ent discrepancy between the theoretical and exper- imental values of ∆GXS is more noticeable at lower xZn concentrations, especially in the concentration range of 0.0 ≤ xZn ≤ 0.5. On the other hand, the observed difference between them decreases within the concentration range of 0.5 ≤ xZn ≤ 1.0, sug- gesting that the two datasets are convergent. Zn’s theoretical and experimental activity shows a notable positive departure from ideality, as shown in Figure 2(a). The difference between the theo- retical and experimental results is nearly constant up to xZn = 0.3024. In the concentration range of 0.3024 ≤ xZn ≤ 0.5986, there is a slight dis- persion between the theoretical and experimental values. Again, there is not much of a difference be- tween xZn = 0.5986 and 1.0. These plots indicate that as xZn increases, there is a noticeable increase in the positive activity deviation of Zn from the ideal Raoult’s law within the parameter range of 0.0 ≤ xZn ≤ 0.3998. In contrast, there is a notice- able decline in the positive Zn’s activity deviation in the range of 0.3998 ≤ xZn ≤ 1.0 at the cross- section, xBi: xIn = 1:1. The plot of ∆GXS versus xZn for the cross- section xBi: xIn = 1:1, as shown in Figure 2(b), Sanjay Kumar Sah et al./ BIBECHANA 21 (2024) 83-94 90 Figure 1: Theoretical and experimental variations of (a) Zn activity with respect to xzn and (b) excess free energy of mixing with respect to xzn presented for Zn-Bi-In liquid alloys at 873 K at the cross-section xBi:xIn =1:2, expt. data taken from the ref. [25]. Figure 2: Theoretical and experimental variations of (a) Zn activity with respect to xzn and (b) excess free energy of mixing with respect to xzn presented for Zn-Bi-In liquid alloys at 873 K at the cross-section xBi:xIn =1:1, expt. data taken from the ref. [25]. Figure 3: Theoretical and experimental variations of (a) Zn activity with respect to xzn and (b) excess free energy of mixing with respect to xzn presented for Zn-Bi-In liquid alloys at 873 K at the cross-section xBi:xIn =2:1, expt. data taken from the ref. [25]. Sanjay Kumar Sah et al./ BIBECHANA 21 (2024) 83-94 91 Figure 4: Theoretical and experimental variations of (a) Zn activity with respect to xzn and (b) excess free energy of mixing with respect to xzn presented for Zn-Bi-In liquid alloys at 873 K at the cross-section xBi:xIn =9:1, expt. data taken from the ref. [25]. Figure 5: Comparison of excess free energy of mixing with respect to xZn presented for Zn-Bi-In liquid alloys at 873 K at all the cross-sections, i.e., xBi: xIn =1:2, 1:1, 2:1, and 9:1. Sanjay Kumar Sah et al./ BIBECHANA 21 (2024) 83-94 92 indicates that there were also both positive and neg- ative values found for ∆GXS. Under the concentra- tions, xZn = 0.7, the maximum positive theoretical value was 2567.7 J/mol, and the maximum positive experimental value was 2618.0 J/mol at xZn = 0.6. The difference between the theoretical and experi- mental ∆GXS values is more noticeable at lower xZn concentrations in the range of 0.0 ≤ xZn ≤ 0.5. On the other hand, convergence occurs in the concen- tration range 0.5 ≤ xZn ≤ 1.0. Figure 3(a) reveals a notable positive depar- ture from ideality in Zn’s theoretical and exper- imental activity. The deviation remains consis- tent up to xZn = 0.1991. The dispersion between them is slightly larger in the concentration range of0.1991 ≤ xZn ≤ 0.4998. At xZn = 0.6009, the difference between them is the maximum. Again, from xZn = 0.6009 to 1.0, the deviation between them becomes smaller. According to these plots, there is a noticeable increase in the positive activ- ity deviation from the ideal Raoult’s law as xZn increases within the specified parameter range of 0.0 ≤ xZn ≤ 0.4001. On the other hand, at cross- sections xBi: xIn = 2:1, a decrease in positive activ- ity deviation is observed within the following range of 0.4001 ≤ xZn ≤ 1.0. As seen in Figure 3(b), the plot of ∆GXS ver- sus xZn for the cross-section xBi: xIn = 2:1 shows that ∆GXS was also found to have both positive and negative values. The maximum positive values were 2734.2 J/mol (Th.) at xZn = 0.7 and 2658.0 J/mol (Expt.) at xZn = 0.6. The disparity in ∆GXS values is pronounced at lower xZn concentrations, notably at 0.0 ≤ xZn ≤ 0.4. Conversely, the dif- ference diminishes at 0.4 ≤ xZn ≤ 0.6, increases at 0.6 ≤ xZn ≤ 0.8, and decreases again toward the concentration range’s end. Figure 4(a) illustrates a notable positive depar- ture from ideality in Zn’s theoretical and experi- mental activity. The deviation remains consistent up to xZn = 0.2005. In the concentration range of 0.3004 ≤ xZn ≤ 0.7002, the dispersion between them becomes larger. Again, after that, in the range of 0.7002 ≤ xZn ≤ 1.0, the deviation be- tween them becomes smaller. According to these plots, there is a noticeable increase in the positive activity deviation from the ideal Raoult’s law as xZn increases within the specified parameter range of 0.0 ≤ xZn ≤ 0.398 . On the other hand, at cross- section xBi: xIn = 9:1, a decrease in positive activity deviation is observed within the following range of 0.398 ≤ xZn ≤ 1.0. As can be seen in Figure 4(b), the plot of ∆GXS versus XZn for the cross-section xBi: xIn = 9:1 shows that ∆GXS was also found to have both positive and negative values. The maximum positive val- ues were 3007.9 J/mol (Th.) at xZn = 0.7 and 2981 J/mol (Expt.) at xZn = 0.6. The appar- ent discrepancy between the theoretical and exper- imental values of ∆GXS is more noticeable at lower xZn concentrations, especially in the concentration range of 0.1 ≤ xZn ≤ 0.3. On the other concentra- tions of xZn, the observed difference between them is smaller. The graphical plots of excess Gibbs free energy of mixing Zn-Bi-In liquid alloys at 873 K for all four cross-sections, i.e., xBi: xIn =1:2, 1:1, 2:1, and 9:1, have been shown in Figure 5. On comparison, we found that when the molar the ratio of Bi and In increases from xBi: xIn =1:2 to 9:1, the max- imum positive values of ∆GXS increase. Further- more, keeping the concentration of Zn constant, we observe that in concentrations greater than 0.4, i.e., xZn = 0.5 to 0.9, in every case, as the ratio (xBi: xIn) increases from 1:2 to 9:1, then the positive values of ∆GXS also increase. For concentrations xZn below 0.5, i.e., for each case of xZn = 0.1, 0.2, 0.3, and 0.4, as the ratio xBi:xIn increases from 1:2 to 1:1, either the negative value of ∆GXS increases or the positive values of ∆GXS decrease, and afterward, as the ratio xBi: xIn increases from 1:1 to 9:1, ei- ther the negative ∆GXS values decrease or positive ∆GXS values increase. For example, in a particular case, at xzn = 0.1, the negative ∆GXS values have decreased and finally shifted to positive as xBi: xIn increases from 1:1 to 9:1. A positive deviation from Raoult’s Law, and thus a positive excess Gibbs free energy for all four cross-sections, indicates that the interactions be- tween the components in the mixture are weaker than expected based on ideal behavior. In the con- text of liquid alloys, a positive deviation in the ac- tivity of a component like Zn suggests that the in- teractions between zinc atoms and the other com- ponents (Bi and In in this case) are weaker in the mixture than one would predict from assuming ideal behavior. It also indicates that the partial vapor pressures of the component Zn in Zn-Bi-In alloys at 873 K are higher than their corresponding va- por pressures in the case of an ideal mixture. Fur- thermore, the positive values of ∆GXS for all three cross-sections indicate that after mixing, the aver- age distance between the molecules of the three liq- uid metals increases, i.e., the volume of the mixture expands relative to the ideal mixture. 4 Conclusion The positive deviation of the activities of the com- ponent Zn (for the Zn-Bi-In system) at all four cross-sections of Bi and In, i.e., 1:2, 1:1, 2:1, and 9:1 from ideal mixing, has been successfully explained theoretically using MIVM. There is a significant positive departure from ideality in Zn’s activity in Zn-Bi-In liquid alloys at all four cross-sections of Bi and In. Similarly, both the negative and positive Sanjay Kumar Sah et al./ BIBECHANA 21 (2024) 83-94 93 values of the excess Gibbs free energy of mixing for the Zn-Bi-In system for all four cross-sections of xBi/xIn have also been successfully explained by the same theoretical model. 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