# On Analysis of Topological Properties for Terbium IV Oxide via Enthalpy and Entropy Measurements

Authors: Muhammad Kamran Siddiqui; Sana Javed; Lubna Sherin; Sadia Khalid; Muhammad Mathar Bashir; Amir Hassan; Mlamuli Dhlamini
Journal: Journal of Chemistry
 (2021)

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## Abstract

A relation between topological indices and thermodynamics properties of terbium IV oxide has been established by using a rational method as it was found the most efficient method based on mean squared error (MSE). Terbium IV oxide has huge application as an insulator in modern technologies such as microelectronics, gas detectors, and luminiferous owing to mechanical and thermal stability, high dielectric constant, radiation resistance, and variable electrical conductivity. The chemical graph and topological indices have attracted the research community due to their potential application in discrete mathematics, biology, and chemistry. Our commitment is to investigate topological indices and thermodynamic properties of terbium IV oxide that depend on an innovative data utilitarian. Moreover, a relationship between topological indices and curve fitting has been established as an application point of view. All curve fittings have been found using MATLAB software.


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## Body

## 1. Introduction

Terbium, a rare earth metal, is a member of the 4f series of the periodic table called lanthanides and has electronic configuration [Xe]4f96s2. It is found between the n−1d and ns block elements and has properties identical to d-block elements. Due to unfilled f orbital, electrons are added to the n−2 level’s ‘f ’ suborbitals. It is silvery white soft metal with a silvery appearance whose melting point ranges from 1000 to 1200 degrees Celsius and is an excellent heat and electricity conductor. Except for promethium, lanthanides are nonradioactive in nature [1, 2]. The synthesis of terbium IV oxide conventionally uses the precipitation approach, but newly designed self-propagation high-temperature synthesis (SHS) provides the high yield of weakly agglomerated nanosized powder of terbium IV oxide. The flow sheet of this process is given in Figures 1 and 2 [3].Figure 1
Schematic illustration of terbium IV oxide synthesis of by SHS.Figure 2
Unit cell structure of terbium oxide [4].Terbium IV oxide films have huge application as an insulator in microelectronics, gas detectors, and luminophores due to unique properties such as radiation resistance and very small leakage current density besides variable electrical conductivity in different gaseous fluids [4, 5]. It is found that various oxidation states cause changes in the stoichiometry of terbium IV oxide that predetermines variations in its optical properties, thus making it useful for the aforementioned applications as well as in Fresnel lenses, pigments, antireflection layers, and photoelastic films [6, 7]. Terbium compounds are brightly fluorescent, and most terbium supplies are used to produce green phosphorous worldwide that allow trichromatic lighting [8]. It is also often used as a dopant for fuel-cell materials and crystalline solid-state devices. The large surface basicity, rapid oxygen ion mobility, and interesting catalytic characteristics of earth oxides are well known. These are the qualities which make them a good chemical sensor [9].The bulk structure ofTbO2 is crystallized as a Tb4O8. In a crystal of terbium oxide, oxygen atoms are located in the cubic closed packed terbium atoms (Figure 1) [10].Metal oxide redox reactions are currently regarded being one of the most favorable long-term methods for producing renewableH2 for immediate use in fuel cells. Terbium IV oxide is a promising candidate for thermochemical production of hydrogen through solar thermochemical water splitting (Tb-WS) cycle. The first step of the cycle includes the thermal reduction of TbO2 into Tb and O2, while the second step involves oxidation of Tb through a water splitting reaction to produce H2. The unit cell structure of terbium oxide is depicted in Figure 2.Step 1.
Step 1:(1)TbO2⟶Tb+O2.
Equation (1) is the endothermic reduction.Step 2.
(2)Tb+HO2⟶TbO2+2H2.
Equation (2) is the endothermic reduction.


## 2. Degree-Based Topological Indices

LetG=V,E be a graph where V is the vertex set and E is the edge set of G. The degree I˜a of a vertex a is the number of edges of G incident with a.In 2013, Shirdel et al. [11] introduced the “Hyper-Zagreb index”:(3)HM=HMG=∑ab∈EGI˜a+I˜b2.In 2012, Ghorbani and Azimi [12] defined multiple Zagreb indices as(4)PM1G=∏ab∈EGI˜a+I˜b,PM2G=∏ab∈EGI˜a+I˜b.For more details about these indices, see [13, 14].In 1972, Furtula and Gutman [15, 16] presented the forgotten topological index which was characterized as(5)FG=∑ab∈EGI˜a2+I˜b2.Furtula et al. [17] introduced the “augmented Zagreb index”:(6)AZIG=∑ab∈EGI˜a×I˜bI˜a+I˜b−23.The Balaban index [18, 19] is a topological index based on order n and size m of graph G:(7)JG=mm−n+2∑ab∈EG1I˜a×I˜b,where I˜a,I˜b are the degrees of the vertices a,b∈VG.For more details about these indices, see [20–22].The redefined version of the Zagreb indices was defined by Ranjini et al. [23].(8)ReZG1G=∑ab∈EGI˜a+I˜bI˜a×I˜b,ReZG2G=∑ab∈EGI˜a×I˜bI˜a+I˜b,ReZG3G=∑ab∈EGI˜a×I˜bI˜a+I˜b.For more details about these indices, see [24–31].


## 3. Results for Terbium OxideTbO2

The number of vertices and edges of the structure of terbium oxide denoted byTbO2 are 22mn and 32mn, respectively. There are three type of vertices in TbO2, namely, the vertices of degree 1, 2, and 4, respectively. The vertex partition of the vertex set TbO2 is presented in Table 1. Also, the edge partition of TbO2 based on degrees of end vertices of each edge is depicted in Table 2.(i)
The Hyper-Zagreb index:The hyper Zagreb index is computed by using Table2 as follows:(9)HMG=∑ab∈EGI˜a+I˜b2,=∑ab∈E1GI˜a+I˜b2+∑ab∈E2GI˜a+I˜b2+∑ab∈E3GI˜a+I˜b2,=HMG=12mn−2n−2m1+42+4mn−2m−2n2+42+16mn+4m+4n4+42,=HMG=4540mn+902m+902n.(ii)
The first and second multiplicative Zagreb index:The first multiplicative Zagreb index is computed as(10)PM1G=1+412mn−2n−2m⋅2+44mn−2m−2n⋅4+416mn+4m+4n,PM1G=512mn−2n−2m⋅64mn−2m−2n⋅816mn+4m+4n,PM1G=∏ab∈EGI˜a+I˜b=∏ab∈E1GI˜a+I˜b+∏ab∈E2GI˜a+I˜b+∏ab∈E3GI˜a+I˜b.The second multiplicative Zagreb index is computed as(11)PM2G=∏ab∈EGI˜a+I˜b=∏ab∈E1GI˜a+I˜b+∏ab∈E2GI˜a+I˜b+∏ab∈E3GI˜a+I˜b,PM2G=1+412mn−2n−2m⋅2+44mn−2m−2n⋅4+416mn+4m+4n,PM2G=412mn−2n−2m⋅84mn−2m−2n⋅1616mn+4m+4n.(iii)
The first and second multiplicative Zagreb index:The numerical representation of the above computed results is presented in Table3.(iv)
The forgotten index:The forgotten index is computed as(12)FG=sumab∈EGI˜a2+I˜b2,=FG=∑ab∈E1I˜a2+I˜b2+∑ab∈E2I˜a2+I˜b2+∑ab∈E3I˜a2+I˜b2,=12mn−2n−2m12+42+4mn−2m−2n22+42+16mn+4m+4n42+42,=12mn−2n−2m17+4mn−2m−2n20+16mn+4m+4n32796mn+54m+54n.(v)
The Augmented Zagreb index:The Augmented Zagreb index is computed as follows:(13)AZIG=∑ab∈EGI˜a×I˜bI˜a+I˜b−23,AZIG=∑ab∈E1I˜a×I˜bI˜a+I˜b−23+∑ab∈E2I˜a×I˜bI˜a+I˜b−23+∑ab∈E3I˜a×I˜bI˜a+I˜b−23,=∑ab∈E11×41+4−23+∑ab∈E22×42+4−23+∑ab∈E34×44+4−23,=642712mn−2n−2m+84mn−2m−2n+5122716mn+4m+4n,=9824mn27+496m9+496n9.(vi)
The Balaban index:The Balaban index is computed as(14)JG=qq−p+2∑ab∈EG1I˜a×I˜b,JG=qq−p+2∑ab∈E11I˜a×I˜b+∑ab∈E21I˜a×I˜b+∑ab∈E31I˜a×I˜b,=qq−p+2∑ab∈E111×4+∑ab∈E212×4+∑ab∈E314×4,=22mn10mn−2×1212mn−2n−2m+1224mn−2m−2n+1416mn+4m+4n.The numerical representation of the abovecomputed results is presented in Table4.(vii)
The redefined Zagreb indices:Table 1
Vertex partition ofTbO2 based on the degree of vertex.
I˜aFrequencySet of vertices14mn+4V122n+2m−4V2418mn−2m−2nV3Table 2
Edge partition ofTbO2.
I˜a,I˜bFrequencySet of edges(1, 4)12mn−2n−2mE1(2, 4)4mn−2m−2nE2(4, 4)16mn+4m+4nE3The redefined Zagreb indices are computed as(15)ReG1G=∑ab∈EGI˜a+I˜bI˜a×I˜b,ReG1G=∑ab∈E1I˜a+I˜bI˜a×I˜b+∑ab∈E2I˜a+I˜bI˜a×I˜b+∑ab∈E3I˜a+I˜bI˜a×I˜b,=∑ab∈E11+41×4+∑ab∈E22+42×4+∑ab∈E34+44×4,=5412mn−2n−2m+344mn−2m−2n+1216mn+4m+4n,ReG2G=∑ab∈EGI˜a+I˜bI˜a×I˜b,ReG2G=∑ab∈E1I˜a+I˜bI˜a×I˜b+∑ab∈E2I˜a+I˜bI˜a×I˜b+∑ab∈E3I˜a+I˜bI˜a×I˜b,=∑ab∈E11×41+4+∑ab∈E22×42+4+∑ab∈E34×44+4,=4512mn−2n−2m+434mn−2m−2n+216mn+4m+4n,ReG3G=∑ab∈EGI˜aI˜bI˜a+I˜b,ReG3G=∑ab∈E1I˜aI˜bI˜a+I˜b+∑ab∈E2I˜aI˜bI˜a+I˜b+∑ab∈E3I˜aI˜bI˜a+I˜b,=2012mn−2n−2m+484mn−2m−2n+12816mn+4m+4n,=2480mn+376m+376n.Table 3
Comparison of indicesHMG,PM2,PM2G.
m,nHMGPM1GPM2G[1, 1]63441.844×10275.192296859×1033[2, 2]217682.699×101064.332296397×10127[3, 3]462723.132×102375.808659799×10281[4, 4]798562.883×104201.251505764×10496[5, 5]1225202.105×106554.333002103×10770[6, 6]1742641.219×109422.410705053×101105[7, 7]2350885.600×1012802.155253650×101500Table 4
Comparison of indices forFG,AZIG,JGTbO2.
m,nFGAZIGJG1,1904474.0727.52,234001675.8599.183,374883605.33221.594,4131686262.52394.265,5204409647.41617.166,62930413760890.287,73976018600.31213.63The numerical representation of the above computed results is presented in Table5.Table 5
Comparison ofReG1G,ReG2G,ReG3G indices for TbO2.
m,nReG1GReG2GReG3G1,12254.432322,296202.67114243,3222444.8245764,4400780.8426885,56301210.67657606,69121734.4937927,712462352126784


## 4. Heat of Formation and Entropy of Terbium IV Oxide

The topological indicesHMG,PMG,PM2G,FG,AZIG,JG,ReG1G,ReG2G,andReG3G were calculated for various numbers of unit cells of terbium IV oxide. The thermodynamic properties of terbium IV oxide, such as heat of formation or enthalpy and entropy, are related to these indices HMG, PMG, PM2G,FG, AZIG, JG,ReG1G, ReG2G, and ReG3G. Terbium IV oxide has a standard molar enthalpy of −972.2kJmol−1, and the standard molar enthalpy for one formula unit was calculated by dividing it by Avogadro’s number. The enthalpy of a cell was determined by multiplying the acquired value by the number of formula units within the cell. The enthalpy of terbium IV oxide is directly proportional to its crystal size and increases as the number of unit cells increases, according to these findings. The entropy of terbium IV oxide was calculated using the same process. The molar standard entropy of terbium IV oxide is 86.9Jmol−1K−1. The result was then determined by multiplying the number of formula units in a single unit cell. If the number of cells gets exponential, the entropy value decreases. The downward trend is the complete opposite of the heat of formation. The graphical representation is depicted in Figures 3–5.Figure 3
HMG,PM1G,PM2G indices vs. unit cell.Figure 4
Comparison of indices F(G), AZI(G), and J(G) forTbO2.Figure 5
Comparison ofReG1G, ReG2G, and ReG3G indices for TbO2.The values for entropy and heat of formation of terbium oxide (TbO2) for 1≤m≤7 and 1≤n≤7 corresponding to different formula units are computed in Table 6.Table 6
Values of heat of formation and entropy for different formula units ofTbO2.
m,nFormula unitsHeat of formation×10−21kJEntropy×10−22kJ1,14−6.4565.7682,216−0.2580.23073,332−0.5160.51914,464−0.01030.92285,5100−0.06140.01446,6144−0.02320.02077,7196−0.03160.0286


## 5. A Mathematical Description of Heat of Formation and Entropy of Terbium Oxide in Terms of Topological Indices

Computational approaches integrated with other disciplines of science provide a coherent way to understand a scientific problem more intensely. Usually, it is not apprehending to understand a problem based on just one science discipline, so adding some computational approaches to the study might provide a clear picture which helps to investigate the underlying phenomenon more deeply and clearly. Developing a mathematical model to describe the dynamics of objects or components involved in a study provides a very convenient mode to tackle and analyze the matter of concern. At present, software technology is playing a vital role in conducting such studies, as they provide more efficient programs to convert an experimental study into a mathematical problem and analyze it.In this section, we have developed mathematical models to represent the thermodynamic properties, namely, heat of formation and entropy, in the form of topological indices of terbium oxide. This might provide us an efficient way to understand the molecular structure of terbium oxide based on its chemical graph structure properties. We have used the software of MATLAB to estimate such models. There are several built-in methods to fit curves between two variables. We tried all of them and found the rational method as the most efficient one as it was providing the least residuals between empirical and fitted values. Tables7 and 8 contain root mean squared error (RMSE), sum of squared error (SSE), and R2, where ratij represents rational fit with numerator degree i and denominator degree j.Table 7
Goodness of fit for heat of formation vs. indices forTbO2.
IndexFit typeSSER2RMSEHMGrat030.01210.99960.0635FGrat230.0035840.99990.05987AZIGrat230.00016961.0000.01302JGrat230.0014811.0000.03849ReG1Grat130.0030940.99990.03933ReG2Grat030.0050140.99990.04088ReG3Grat030.011940.99970.06308Table 8
Goodness of fit for entropy vs. indices forTbO2.
IndexFit typeSSER2RMSEHMGrat130.0010711.0000.02314FGrat130.0010651.0000.02307AZIGrat030.0010781.0000.01895JGrat050.0010311.0000.0321ReG1Grat140.0011181.0000.03344ReG2Grat050.00049061.0000.02215ReG3Grat130.0010691.0000.02312


### 5.1. General Models for Indices vs. Heat of Formation

In this section, a mathematical framework has been developed between each topological index and heat of formation (HoF) of terbium oxide. All the fitted curves are shown in Figures6–12, whereas the estimated parametric values are provided in Tables 9–15.(16)HoFHM=p1HM3+q1×HM2+q2×HM+q3.Figure 6
HoFHM.(17)HoFF=p1×F2+p2×F+p3F3+q1×F2+q2×F+q3.Figure 7
HoFF.(18)HoFAZI=p1×AZI2+p2×AZI+p3AZI3+q1×AZI2+q2×AZI+q3.Figure 8
HoFAZI.(19)HoFJ=p1×J2+p2×J+p3J3+q1×J2+q2×J+q3.Figure 9
HoFJ.(20)HoFReG1=p1×ReG1+p2ReG13+q1×ReG12+q2×ReG1+q3.Figure 10
HoFReG1.(21)HoFReG2=p1ReG23+q1×ReG22+q2×ReG2+q3.Figure 11
HoFReG2.(22)HoFReG3=p1ReG33+q1×ReG32+q2×ReG3+q3.Figure 12
HoFReG3.Table 9
HoF vs. HM.
piCIqiCIi=1178.2(−9.679 e + 06, 9.679 e + 06)1.176 e + 04(−6.386 e + 08, 6.386 e + 08)i=2——1.94 e + 04(−1.053 e + 09, 1.053 e + 09)i=3——7138(−3.876 e + 08, 3.876 e + 08)Table 10
HoF vs. F.
piCIqiCIi=12.805(−6.115 e + 05, 6.115e + 05)688.5(−1.482 e + 08, 1.482e + 08)i=2−20.61(−4.445 e + 06, 4.445 e + 06)1156(−2.493 e + 08, 2.493 e + 08)i=3−9.782(−2.11 e + 06, 2.11 e + 06)444.4(−9.587 e + 07, 9.587 e + 07)Table 11
HoF vs. AZI.
piCIqiCIi=1−0.05294(−0.3091, 0.2032)1.691(−1.61, 4.992)i=2−0.03912(−0.4201, 0.3419)0.6454(−4.759, 6.05)i=3−0.005199(−0.08769, 0.07729)−0.008204(−1.979, 1.962)Table 12
HoF vs. J.
piCIqiCIi=1−190.1(−1.44 e + 08, 1.44e + 08)1.333 e + 04(−1.009 e + 10, 1.009e + 10)i=2−630(−4.771 e + 08, 4.771e + 08)2.228 e + 04(−1.687 e + 10, 1.687 e + 10)i=3−236.6(−1.792 e + 08, 1.792 e + 08)8555(−6.479 e + 09, 6.479 e + 09)Table 13
HoF vs.ReG1
piCIqiCIi=1−175.5(−2.397 e + 06, 2.396e + 06)3781(−5.159 e + 07, 5.16 e + 07)i=2−112.8(−1.541 e + 06, 1.54 e + 06)6438(−8.788 e + 07, 8.79 e + 07)i=3——2553(−3.485 e + 07, 3.486 e + 07)Table 14
HoF vs.ReG2.
piCIqiCIi=1−0.003307(−0.02053, 0.01391)2.406(1.359, 3.452)i=2——1.881(0.1678, 3.595)i=3——0.4865(−0.1403, 1.113)Table 15
HoF vs.ReG3.
piCIqiCIi=1172.1(−9.52 e + 06, 9.52 e + 06)1.171 e + 04(−6.476 e + 08, 6.476 e + 08)i=2——1.93 e + 04(−1.067 e + 09, 1.067 e + 09)i=3——7118(−3.937 e + 08, 3.937 e + 08)Coefficients (with 95% confidence interval (CI)) are given in Table9.Coefficients (with 95% confidence bounds) are given in Table10.Coefficients (with 95% confidence bounds) are given in Table11.Coefficients (with 95% confidence bounds) are given in Table12.Coefficients (with 95% confidence bounds) are given in Table13.Coefficients (with 95% confidence bounds) are given in Table15.


### 5.2. General Models for Indices vs. Entropy

This section establishes a mathematical framework between each topological index and entropy of terbium oxide. All the fitted curves are shown in Figures13–19 while the estimated parametric values are given in Tables 16–22.(23)EntropyHM=p1×HM+p2HM3+q1×HM2+q2×HM+q3.Figure 13
EntropyHM.(24)EntropyF=p1×F+p2F3+q1×F2+q2×F+q3.Figure 14
EntropyF.(25)EntropyAZI=p1AZI3+q1×AZI2+q2×AZI+q3,Figure 15
EntropyAZI.(26)EntropyJ=p1J5+q1×J4+q2×J3+q3×J2+q4×J+q5.Figure 16
EntropyJ.(27)EntropyReG1=p1×ReG1+p2ReG14+q1×ReG13+q2×ReG12+q3×ReG1+q4.Figure 17
EntropyReG1.(28)EntropyReG2=p1ReG25+q1×ReG24+q2×ReG23+q3×ReG22+q4×ReG2+q5.Figure 18
EntropyReG2.(29)EntropyReG3=p1×ReG3+p2ReG33+q1×ReG32+q2×ReG3+q3.Figure 19
EntropyReG3.Table 16
Entropy vs. HM.
piCIqiCIi=10.001132(−0.04439, 0.04666)1.842(1.762, 1.922)i=20.01288(−0.03159, 0.05736)0.9484(0.8749, 1.022)i=3——0.1427(0.1092, 0.1761)Table 17
Entropy vs.F.
piCIqiCIi=10.001791(−0.04354, 0.04713)1.845(1.766, 1.925)i=20.01254(−0.03159, 0.05667)0.9604(0.8889, 1.032)i=3——0.1465(0.1139, 0.1792)Table 18
Entropy vs.AZI.
piCIqiCIi=10.01144(0.007304, 0.01558)1.845(1.812, 1.877)i=2——0.9537(0.9148, 0.9925)i=3——0.1431(0.1392, 0.147)Table 19
Entropy vs.J.
piCIqiCIi=1−110.9(−2.497 e + 07, 2.497 e + 07)2563(−5.765 e + 08, 5.765 e + 08)i=2——−3952(−8.901 e + 08, 8.901 e + 08)i=3——−1.355 e + 04(−3.052 e + 09, 3.052 e + 09)i=4——−8056(−1.814 e + 09, 1.814 e + 09)i=5——−1311(−2.951 e + 08, 2.951 e + 08)Table 20
Entropy vs.ReG1.
piCIqiCIi=11.041(−1.206 e + 06, 1.206 e + 06)−1145(−1.341 e + 09, 1.341 e + 09)i=2−10.05(−1.175 e + 07, 1.175 e + 07)−2126(−2.487 e + 09, 2.487 e + 09)i=3——−1121(−1.311 e + 09, 1.311 e + 09)i=4——−170.7(−1.996 e + 08, 1.996 e + 08)Table 21
Entropy vs.ReG2.
piCIqiCIi=1103.9(−2.399 e + 06, 2.399 e + 06)−3508(−8.104 e + 07, 8.103 e + 07)i=2——−384.3(−8.831 e + 06, 8.83 e + 06)i=3——7836(−1.81 e + 08, 1.81 e + 08)i=4——5312(−1.227 e + 08, 1.227 e + 08)i=5——909.8(−2.101 e + 07, 2.101 e + 07)Table 22
Entropy vs.ReG3.
piCIqiCIi=10.001368(−0.04409, 0.04682)1.843(1.763, 1.923)i=20.01276(−0.0316, 0.05712)0.9527(0.8799, 1.025)i=3——0.144(0.1109, 0.1772)Coefficients (with 95% confidence bounds) are given in Table16.Coefficients (with 95% confidence bounds) are given in Table17.Coefficients (with 95% confidence bounds) are given in Table18.Coefficients (with 95% confidence bounds) are given in Table19.Coefficients (with 95% confidence bounds) are given in Table20.Coefficients (with 95% confidence bounds) are given in Table21.Coefficients (with 95% confidence bounds) are given in Table22.


## 5.1. General Models for Indices vs. Heat of Formation

In this section, a mathematical framework has been developed between each topological index and heat of formation (HoF) of terbium oxide. All the fitted curves are shown in Figures6–12, whereas the estimated parametric values are provided in Tables 9–15.(16)HoFHM=p1HM3+q1×HM2+q2×HM+q3.Figure 6
HoFHM.(17)HoFF=p1×F2+p2×F+p3F3+q1×F2+q2×F+q3.Figure 7
HoFF.(18)HoFAZI=p1×AZI2+p2×AZI+p3AZI3+q1×AZI2+q2×AZI+q3.Figure 8
HoFAZI.(19)HoFJ=p1×J2+p2×J+p3J3+q1×J2+q2×J+q3.Figure 9
HoFJ.(20)HoFReG1=p1×ReG1+p2ReG13+q1×ReG12+q2×ReG1+q3.Figure 10
HoFReG1.(21)HoFReG2=p1ReG23+q1×ReG22+q2×ReG2+q3.Figure 11
HoFReG2.(22)HoFReG3=p1ReG33+q1×ReG32+q2×ReG3+q3.Figure 12
HoFReG3.Table 9
HoF vs. HM.
piCIqiCIi=1178.2(−9.679 e + 06, 9.679 e + 06)1.176 e + 04(−6.386 e + 08, 6.386 e + 08)i=2——1.94 e + 04(−1.053 e + 09, 1.053 e + 09)i=3——7138(−3.876 e + 08, 3.876 e + 08)Table 10
HoF vs. F.
piCIqiCIi=12.805(−6.115 e + 05, 6.115e + 05)688.5(−1.482 e + 08, 1.482e + 08)i=2−20.61(−4.445 e + 06, 4.445 e + 06)1156(−2.493 e + 08, 2.493 e + 08)i=3−9.782(−2.11 e + 06, 2.11 e + 06)444.4(−9.587 e + 07, 9.587 e + 07)Table 11
HoF vs. AZI.
piCIqiCIi=1−0.05294(−0.3091, 0.2032)1.691(−1.61, 4.992)i=2−0.03912(−0.4201, 0.3419)0.6454(−4.759, 6.05)i=3−0.005199(−0.08769, 0.07729)−0.008204(−1.979, 1.962)Table 12
HoF vs. J.
piCIqiCIi=1−190.1(−1.44 e + 08, 1.44e + 08)1.333 e + 04(−1.009 e + 10, 1.009e + 10)i=2−630(−4.771 e + 08, 4.771e + 08)2.228 e + 04(−1.687 e + 10, 1.687 e + 10)i=3−236.6(−1.792 e + 08, 1.792 e + 08)8555(−6.479 e + 09, 6.479 e + 09)Table 13
HoF vs.ReG1
piCIqiCIi=1−175.5(−2.397 e + 06, 2.396e + 06)3781(−5.159 e + 07, 5.16 e + 07)i=2−112.8(−1.541 e + 06, 1.54 e + 06)6438(−8.788 e + 07, 8.79 e + 07)i=3——2553(−3.485 e + 07, 3.486 e + 07)Table 14
HoF vs.ReG2.
piCIqiCIi=1−0.003307(−0.02053, 0.01391)2.406(1.359, 3.452)i=2——1.881(0.1678, 3.595)i=3——0.4865(−0.1403, 1.113)Table 15
HoF vs.ReG3.
piCIqiCIi=1172.1(−9.52 e + 06, 9.52 e + 06)1.171 e + 04(−6.476 e + 08, 6.476 e + 08)i=2——1.93 e + 04(−1.067 e + 09, 1.067 e + 09)i=3——7118(−3.937 e + 08, 3.937 e + 08)Coefficients (with 95% confidence interval (CI)) are given in Table9.Coefficients (with 95% confidence bounds) are given in Table10.Coefficients (with 95% confidence bounds) are given in Table11.Coefficients (with 95% confidence bounds) are given in Table12.Coefficients (with 95% confidence bounds) are given in Table13.Coefficients (with 95% confidence bounds) are given in Table15.


## 5.2. General Models for Indices vs. Entropy

This section establishes a mathematical framework between each topological index and entropy of terbium oxide. All the fitted curves are shown in Figures13–19 while the estimated parametric values are given in Tables 16–22.(23)EntropyHM=p1×HM+p2HM3+q1×HM2+q2×HM+q3.Figure 13
EntropyHM.(24)EntropyF=p1×F+p2F3+q1×F2+q2×F+q3.Figure 14
EntropyF.(25)EntropyAZI=p1AZI3+q1×AZI2+q2×AZI+q3,Figure 15
EntropyAZI.(26)EntropyJ=p1J5+q1×J4+q2×J3+q3×J2+q4×J+q5.Figure 16
EntropyJ.(27)EntropyReG1=p1×ReG1+p2ReG14+q1×ReG13+q2×ReG12+q3×ReG1+q4.Figure 17
EntropyReG1.(28)EntropyReG2=p1ReG25+q1×ReG24+q2×ReG23+q3×ReG22+q4×ReG2+q5.Figure 18
EntropyReG2.(29)EntropyReG3=p1×ReG3+p2ReG33+q1×ReG32+q2×ReG3+q3.Figure 19
EntropyReG3.Table 16
Entropy vs. HM.
piCIqiCIi=10.001132(−0.04439, 0.04666)1.842(1.762, 1.922)i=20.01288(−0.03159, 0.05736)0.9484(0.8749, 1.022)i=3——0.1427(0.1092, 0.1761)Table 17
Entropy vs.F.
piCIqiCIi=10.001791(−0.04354, 0.04713)1.845(1.766, 1.925)i=20.01254(−0.03159, 0.05667)0.9604(0.8889, 1.032)i=3——0.1465(0.1139, 0.1792)Table 18
Entropy vs.AZI.
piCIqiCIi=10.01144(0.007304, 0.01558)1.845(1.812, 1.877)i=2——0.9537(0.9148, 0.9925)i=3——0.1431(0.1392, 0.147)Table 19
Entropy vs.J.
piCIqiCIi=1−110.9(−2.497 e + 07, 2.497 e + 07)2563(−5.765 e + 08, 5.765 e + 08)i=2——−3952(−8.901 e + 08, 8.901 e + 08)i=3——−1.355 e + 04(−3.052 e + 09, 3.052 e + 09)i=4——−8056(−1.814 e + 09, 1.814 e + 09)i=5——−1311(−2.951 e + 08, 2.951 e + 08)Table 20
Entropy vs.ReG1.
piCIqiCIi=11.041(−1.206 e + 06, 1.206 e + 06)−1145(−1.341 e + 09, 1.341 e + 09)i=2−10.05(−1.175 e + 07, 1.175 e + 07)−2126(−2.487 e + 09, 2.487 e + 09)i=3——−1121(−1.311 e + 09, 1.311 e + 09)i=4——−170.7(−1.996 e + 08, 1.996 e + 08)Table 21
Entropy vs.ReG2.
piCIqiCIi=1103.9(−2.399 e + 06, 2.399 e + 06)−3508(−8.104 e + 07, 8.103 e + 07)i=2——−384.3(−8.831 e + 06, 8.83 e + 06)i=3——7836(−1.81 e + 08, 1.81 e + 08)i=4——5312(−1.227 e + 08, 1.227 e + 08)i=5——909.8(−2.101 e + 07, 2.101 e + 07)Table 22
Entropy vs.ReG3.
piCIqiCIi=10.001368(−0.04409, 0.04682)1.843(1.763, 1.923)i=20.01276(−0.0316, 0.05712)0.9527(0.8799, 1.025)i=3——0.144(0.1109, 0.1772)Coefficients (with 95% confidence bounds) are given in Table16.Coefficients (with 95% confidence bounds) are given in Table17.Coefficients (with 95% confidence bounds) are given in Table18.Coefficients (with 95% confidence bounds) are given in Table19.Coefficients (with 95% confidence bounds) are given in Table20.Coefficients (with 95% confidence bounds) are given in Table21.Coefficients (with 95% confidence bounds) are given in Table22.


## 6. Conclusions

A connection between topological indices and thermodynamic properties of terbium IV oxide has been developed. This study helps to understand the chemical structure of terbium IV oxide based on the graphical properties of its underlying graph more deeply as this was economical and more efficient. Curve fitting techniques have been utilized to establish such a relation among indices and heat of formation and entropy. The rational fitting approach was selected based on its efficacy. This direct connection might help to explore the dynamical properties of this terbium IV oxide.



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