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Molecular description of copper (i) oxide and copper (ii) oxide |

Wei Gao I. School of Information Science and Technology, Yunnan Normal University, Kunming, 650500, China Recebido em: 14/03/2018 Endereço para correspondência
*e-mail: gaowei@ynnu.edu.cn RESUMO
The topological index of a molecular structure correlate certain physicochemical properties such as boiling point and stability of that molecular structure. Recently several topological indices have been introduced and studies have shown their association with certain physical and chemical properties of other molecules. Specifically, Zagreb, Atom-bound connectivity (ABC) and Sanskruti indices have been correlated with extent of branching, enthalpy and entropy of some organic molecules. In this study we have calculated Zagreb, Atom-bound connectivity (ABC) and Sanskruti indices (S) for Copper (I) oxide and Copper (II) oxide CuO. Additionally, from the experimentally available data, we calculated entropy and enthalpy per unit cell for both the oxides and compared with Atom-bound connectivity and Sanskruti indices. It appears that these indices have very poor relation with these physical properties in both oxides of copper and they may need some modifications to show considerable compliance as reported in other organic molecules." Palavras-chave: molecular graphs; zagreb indices; atom-bound connectivity index; sanskruti index; copper oxide.
"A graph that shows constituents of a molecule and their connectivity is known as molecular graph and such representation is usually referred as topological representation of molecule. "are usually characterized by different topological indices for correlation of chemical structure of a molecule with biological, chemical or physical properties. Studies have reported several applications of different topological indices in quantitative structure-activity relationship (QSAR) and quantitative structure-property relationship (QSPR), virtual screening and computational drug designing. "So far, many different topological indices have been developed, and most of them are only graph descriptors. ^{22} However, later M and _{1}M were associated with extent of branching in a molecule. They are described by equation (1) & (2) for a graph G=(V,E), where V to be the vertex set and E to be the edge set of G. The degree d_{2}_{p} of the vertex p is the quantity of edges of G incident with p. The length of a most limited path in a graph G is a distance d(p,q) amongst p and q.""Another topological index is atom-bound connectivity (ABC) that was introduced by "Another molecular descriptor that was correlated with thermodynamic properties is Sankruti index. This index was introduced in 2016 by S. M. Hosamani and it shows considerable correlation with entropy of some organic molecules. "This study is designed to calculate Zagreb, ABC and Sanskruti indices for two oxides of & CuO). It will allow us to compare these indices with the experimentally determined entropy, enthalpy and other physical properties of Cu_{2}O and CuO."Cu_{2}O
^{26-28} Crystal structure of is composed on small cubic unit cells based on Cu and O atoms (Figure. 1a). Analysis of the crystal lattice showed that each Cu atom is linked with two O atoms; in turn every O atom is connected with four Cu atoms (Figure 1b)"Cu_{2}O
Figure 1. Copper(I) oxide . A) Unit cell of . B) Three dimensional crystal lattice of Cu and O are shown in blue and red spheres, respectively
Here, we have considered monolayer of we choose the settings of this graph as; Cu_{2}O ≅ G[m,n] be the chemical graph of Cu_{2}O with (Cu_{2}Om × n) unit cells in the plane. More preciously, m represents the number of unit cells in row and n represents the number of unit cells in column. Also for our convenience we represents [m,n] as a graph G. The Figure 1(b) is representing Cu_{2}O[2,2]. Also Cu_{2}OFigure 2 represent one sheet view of Copper(I) oxide [4,4]. Computational analysis showed that the quantity of vertices and edges of Cu_{2}O[m,n] are 7mn+2m+2n+2 and 8mn, respectively. In Cu_{2}O[m,n], the number of zero degree vertices is 4, the number of one degree vertices is 4m+4n-4, the number of two degree vertices is 6mn-2m-2n+2 and the number of four degree vertices is mn (Table 1)."Cu_{2}O
Figure 2. One sheet view of copper(I) oxide [4, 4]
In the next Theorem, we have computed the exact result of first and second Zagreb index for the chemical graph m, n]."
1, then its first and second Zagreb index is equal to,"
"By using Table 1 and Equation 1 the second Zagreb index are computed as below:" "In the next Theorem, we have computed the exact result of ABC index for the chemical graph
with m, n ≥ 1, then its ABC index is equal to,"Cu_{2}O[m,n]
"The Table 2 shows partition of edges of the chemical graph
"The next Theorem shows the exact value of Sanskruti index of
[m,n] with m, n ≥ 2, then its Sanskruti index S(G) is equal"Cu_{2}O
The crystal structure of Copper(II) oxide ( for convenience. To find the indices we choose the settings of this graph as; let CuO ≅ G[m,n] be the chemical graph of CuO with (m × n) unit cells in the plane. The Figure 3b is representing the lattice of CuO[4,4]. Computational analysis showed that number of vertices and edges of CuO[m,n] are 8mn+2m+2n and 12mn, respectively. In CuO[m,n] the number of one degree vertices are 2n, the number of two degree vertices are 2mn+4m+2n, the number of three degree vertices are 4mn-2n and the number of four degree vertices are 2mn-2m."CuO
Figure 3. Copper (II) oxide CuO: (a) is the unit cell with blue and red circles represents copper and oxygen atom, respectively (b) shows the lattice CuO[4, 4]
In the next Theorem, we have computed the exact value of first and second Zagreb index for the chemical graph m, n]."
[m, n]. The first Zagreb index is computed as:CuO"In the next Theorem, we have computed the exact result of ABC index for the chemical graph
"The Table 4 shows partition of edges of the chemical graph
The next Theorem shows the exact value of Sanskruti index of
"In this section we have compared the first (M1) and second Zegrab (M2) indices of Copper(I) oxide which is shown in Table 5. Analysis of the data showed that both the indices are higher in CuOCuO compared to those of . For instance, M1(CuO)/M1() for [1,1] cell is 1.44 and the ratio increases to 1.943 for [7,7] and then it decrease to 1.87 for [10,10] cell. On the other hand, M2(CuO)/M2() for [1,1] is 1.40 and the ratio gradually increases to 2.413 for [10,10] cell."
"Comparison of ABC indices for both the oxides of Copper showed higher values for
"In the Table 7, we gave a comparison of the Sanskruti indices of Copper(I) oxide . Analysis of the data showed that host higher Sanskruti index compared to CuOCuO. The S(CuO)/S() for [2,2] is 3.825 and it increases to 5.34 for [10,10]. This shows its increasing trend with the increase of cells."
A topological index is a numerical value that is calculate form molecular graph for explaining the relationship of chemical structure with physiochemical properties. So far, several applications of different indices have been reported for organic molecules, such as entropy and enthalpy determination, chirality identification, . Additionally, we compared these indices with the physical properties these oxides as described in Pubchem (https://pubchem.ncbi.nlm.nih.gov/)."CuO"Zagreb indices ( at any cubic level i.e. at any equal value of (m × n) (Table 5). It suggests more branching in Cu_{2}O compared to CuO. It may cause more compactCu_{2}O (Density: 6.315 g/cmCuO^{3}) structure compared to Cu2O (6.0 g/cm^{3}). Moreover, higher melting point of (1599 K) compared to that of CuO (1505 K) may also be associated to the higher branching."Cu_{2}O
were calculated at different unit cells as shown in Table 6. Analysis of the data showed higher CuOABC indices for compared to those of CuO (Table 6). Figure 4 shows that Cu_{2}OABC indices increase exponentially with number of cells (m × n) for both the oxides of copper. On the other hand, a linear relation of ABC index was observed with formula units (Figure 5). Since, ABC index has previously been linked with the thermodynamic properties" of different alkanes,^{7,11} "therefore, we determined the heat of formation per different units cells of both the oxides from the reported real molar enthalpy of formations (Table 6). For example, experimental molar enthalpy for is -156 kJ molCuO^{-1} so for one formula unit it will be -156/6.022×10^{-23} kJ. The obtained value then multiplied with the number of formula units present in each cell to get the enthalpy for the cell. Similarly, enthalpy for calculated considering standard molar enthalpy -170 kJ molCu_{2}O^{-1} (Table 6). Analysis of the data showed that ABC index is not strongly associated with enthalpy of formation for the oxides of copper as reported in case of alkanes.^{6,13}" Therefore we determined a mathematical relationship, by using equation generator (http://www.1728.org/threepts.htm), between the ABC indices and the heat of formation of both the oxides of copper as below:
Figure 4. ABC indices of Cu_{2}O and CuO showing exponential increase with number of units cells (at x-axis)
Figure 5. ABC indices of Cu_{2}O and CuO showing linear increase with number of formula units (at x-axis)
These equations can be used for the transformation of Figure 6 shows a linear relation of Sanskruti indices with number of formula units and analysis showed higher indices for . Previous studies have shown an association of Sanskruti indices with entropy of octance.CuO^{15} Here we calculated the entropy of both the oxides at different cells from the standard molar entropy. For instance, standard molar entropy for is 93 J molCuO^{-1} K^{-1} so for one formula unit it will be 93/6.022×10^{-23} J K^{-1}, and for each cell we just multiplied the obtained value with the no. of formula units present in the cell. Similarly entropy for calculated considering standard molar entropy 43 J molCu_{2}O^{-1} K^{-1} (Table 7). Analysis of the data showed a considerable difference between the Sanskruti indices and entropies for the oxides of Cu (Table 7). Therefore, we determined a mathematical relation for the transformation of Sanskruti indices into approximate entropies of both the oxides of copper as below:
Figure 6. Sanskrit indices of Cu_{2}O and CuO showing linear increase with number of formula units (at x-axis)
In conclusion, ABC and Sanskuriti indices of showed considerable difference with the experimentally reported entropy and enthalpy of these oxides. On the other hand, Zagreb index and all these indices showed exponential increase with the number of unit cells. On the other hand, these indices showed a linear behavior with number of formula units, suggesting that we may need a slight transformation of these indices for better explanation of physical properties of crystalline compounds."CuO
The authors are grateful to the anonymous referees for their valuable comments and suggestions that improved this paper. This research is supported by the Start-up Research Grant 2016 of United Arab Emirates University (UAEU), Al Ain, and United Arab Emirates via Grant No. G00002233 and UPAR Grant of UAEU via Grant No. G00002590.
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