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.

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] . It is found between the and ns block elements and has properties identical to d-block elements. Due to unfilled f orbital, electrons are added to the 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 of is crystallized as a . 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 renewable 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 into Tb and , while the second step involves oxidation of Tb through a water splitting reaction to produce . The unit cell structure of terbium oxide is depicted in Figure 2.

Step 1. Step 1:Equation (1) is the endothermic reduction.

Step 2. Equation (2) is the endothermic reduction.

2. Degree-Based Topological Indices

Let be a graph where is the vertex set and is the edge set of . The degree of a vertex is the number of edges of incident with .

In 2013, Shirdel et al. [11] introduced the “Hyper-Zagreb index”:

In 2012, Ghorbani and Azimi [12] defined multiple Zagreb indices as

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

Furtula et al. [17] introduced the “augmented Zagreb index”:

The Balaban index [18, 19] is a topological index based on order and size of graph G:where are the degrees of the vertices .

For more details about these indices, see [20–22].

The redefined version of the Zagreb indices was defined by Ranjini et al. [23].

For more details about these indices, see [24–31].

3. Results for Terbium Oxide

The number of vertices and edges of the structure of terbium oxide denoted by are and , respectively. There are three type of vertices in , namely, the vertices of degree 1, 2, and 4, respectively. The vertex partition of the vertex set is presented in Table 1. Also, the edge partition of 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 Table 2 as follows:(ii)The first and second multiplicative Zagreb index: The first multiplicative Zagreb index is computed as The second multiplicative Zagreb index is computed as(iii)The first and second multiplicative Zagreb index: The numerical representation of the above computed results is presented in Table 3.(iv)The forgotten index: The forgotten index is computed as(v)The Augmented Zagreb index: The Augmented Zagreb index is computed as follows:(vi)The Balaban index: The Balaban index is computed as The numerical representation of the abovecomputed results is presented in Table 4.(vii)The redefined Zagreb indices:

Table 1 
Vertex partition of based on the degree of vertex.
Table 2 
Edge partition of .

The redefined Zagreb indices are computed as

Table 3 
Comparison of indices .
Table 4 
Comparison of indices for .

The numerical representation of the above computed results is presented in Table 5.

Table 5 
Comparison of indices for .

4. Heat of Formation and Entropy of Terbium IV Oxide

The topological indices 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 , , , , , , and . Terbium IV oxide has a standard molar enthalpy of , 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 . 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 
indices vs. unit cell.

Figure 4 
Comparison of indices F(G), AZI(G), and J(G) for .

Figure 5 
Comparison of , , and indices for .

The values for entropy and heat of formation of terbium oxide () for and corresponding to different formula units are computed in Table 6.

Table 6 
Values of heat of formation and entropy for different formula units of .

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. Tables 7 and 8 contain root mean squared error (RMSE), sum of squared error (SSE), and , where ratij represents rational fit with numerator degree i and denominator degree j.

Table 7 
Goodness of fit for heat of formation vs. indices for .
Table 8 
Goodness of fit for entropy vs. indices for .
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 Figures 6–12, whereas the estimated parametric values are provided in Tables 9–15.


Figure 6 
.

Figure 7 
.

Figure 8 
.

Figure 9 
.

Figure 10 
.

Figure 11 
.

Figure 12 
.
Table 9 
HoF vs. HM.
Table 10 
HoF vs. F.
Table 11 
HoF vs. AZI.
Table 12 
HoF vs. J.
Table 13 
HoF vs.
Table 14 
HoF vs. .
Table 15 
HoF vs. .

Coefficients (with 95% confidence interval (CI)) are given in Table 9.

Coefficients (with 95% confidence bounds) are given in Table 10.

Coefficients (with 95% confidence bounds) are given in Table 11.

Coefficients (with 95% confidence bounds) are given in Table 12.

Coefficients (with 95% confidence bounds) are given in Table 13.

Coefficients (with 95% confidence bounds) are given in Table 15.

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 Figures 13–19 while the estimated parametric values are given in Tables 16–22.


Figure 13 
.

Figure 14 
Entropy .

Figure 15 
Entropy .

Figure 16 
Entropy .

Figure 17 
Entropy .

Figure 18 
Entropy .

Figure 19 
Entropy .
Table 16 
Entropy vs. HM.
Table 17 
Entropy vs. .
Table 18 
Entropy vs. .
Table 19 
Entropy vs. .
Table 20 
Entropy vs. .
Table 21 
Entropy vs. .
Table 22 
Entropy vs. .

Coefficients (with 95% confidence bounds) are given in Table 16.

Coefficients (with 95% confidence bounds) are given in Table 17.

Coefficients (with 95% confidence bounds) are given in Table 18.

Coefficients (with 95% confidence bounds) are given in Table 19.

Coefficients (with 95% confidence bounds) are given in Table 20.

Coefficients (with 95% confidence bounds) are given in Table 21.

Coefficients (with 95% confidence bounds) are given in Table 22.

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.

Data Availability

The data used to support the findings of this study are cited at relevant places within the text as references.

Conflicts of Interest

The authors declare no conflicts of interest.

Authors’ Contributions

All authors contributed equally to this work.