Boiling Point of Alkanes and Alkenes - From Graph Eccentricity

 

M. Yamuna*, T. Divya

Department of Mathematics, SAS, VIT University, Vellore, Tamilnadu, India, 632 014.

*Corresponding Author E-mail: myamuna@vit.ac.in

 

ABSTRACT:

There is a relationship between the physical properties and the molecular structure of chemicals. This relation is well established by the determination of a power formula for determining the boiling point of chemicals using Wiener index. A power formula relating this is developed by Harry Wiener. Calculation of Wiener index requires determination of distance between every pair of vertices in the chemical graph representation of the molecular structure. Eccentricity is a graph property that requires only the longest distance calculation from every vertex. If the physical properties can be developed using eccentricity only, then it reduces manual calculation of distance between all pair of vertices. In this paper a power formula to determine the boiling point of alkanes and alkenes is developed. The boiling point determined using this power formula is verified with the values developed from the existing one and is seen to be in good correlation with the existing formula.

 

KEYWORDS: Molecules, Molecular structure, Isomeric group, Alkanes, Alkenes, Wiener Index, Distance matrix, Eccentricity, Boiling Point.

 

 


INTRODUCTION:

The introduction of Wiener index and Wiener polarity index by Harry Wiener in his paper titled “Structural Determination of Paraffin Boiling Points” opened scope of studying and determining physical properties from graph structures [1]. He stated that the boiling points of organic compounds as well as their physical properties depend functionally upon the number kind and structural arrangement of atoms in a molecule. He further stated that within a group of isomers both the number and the kind of atoms are constant and variations in physical properties are due to change in structural inter-relationships alone. Further in this paper, he has developed a relationship between the boiling point, Wiener index and Wiener polarity index given by the equation  t = (98/n2) w + 5.5p, Where t  is the boiling point of group of isomers. w = w0 – w, where w0 = (1/6) (n-1) (n) (n+1), n- vertices and w is the path number i.e., the sum of the distances between any two carbon atoms in a molecule, in terms of carbon-carbon bonds. p = p0- p, where p0 = n-3, p the polarity number

 

defined as the number of pairs of carbon atoms which are separated by three carbon-carbon bonds. He has provided a list of the detailed results obtained by applying this equation to the 37 paraffins from C4H10 to C8H18. He further extended this method for the boiling point data available for the nonanes and decanes. After Wiener numerous researchers had developed various techniques to determine physical and chemical properties from topological indices. In [2] boiling point  model from the topological indices is developed and compared with the existing models. In [3] the boiling point is predicted with the weighted sum of the generalized Zagreb index, the Second Zagreb index, Wiener index for vertex weighted graphs. In [4] a property relationship study for prediction of boiling point of aliphatic alkenes is determined. In [5] a formula for computing the Wiener index in terms of Binary Hamming labeling of G is determined. In [6] a method of determining Wiener index of trees from its subtree is determined.

 

To calculate the Wiener index of a graph with n vertices, a n×n distance matrix is required. This means that distance between n2 pair of vertices should be determined. A reduction in this calculation would reduce the manual difficulty involved in physical property determination. Eccentricity is a graph property involving distance. Eccentricity is determined only for vertices. This means that for a chemical graph with n vertices, only n calculations are required. So physical properties can be determined from eccentricities only. Then a enormous manpower require for distance calculation may reduce. In this paper we propose method of determining the boiling point of alkanes and alkenes is obtained.

 

PRELIMINARIES:

In this section we provide the basic details of Wiener index and chemical graph that is required in the proposed calculation.

 

Molecules:

A molecule is an electrically neutral group of two or more atoms held together by chemical bonds [ 7 ]. Molecules are distinguished from ions by their lack of electrical charge. However, in quantum physics, organic chemistry, and biochemistry, the term molecule is often used less strictly, also being applied to polyatomic ions. In the kinetic theory of gases, the term molecule is often used for any gaseous particle regardless of its composition. According to this definition, noble gas atoms are considered molecules as they are in fact mono atomic molecules. (Snapshot 1) shows the example of a molecule- Sucrose molecule [ 8 ].

 

 

 

Snapshot 1: Sucrose molecule

 

Molecular structure:

The arrangement of chemical bonds between atoms in a molecule (or in an ion or radical with multiple atoms), specifically which atoms are chemically bonded to what other atoms with what kind of chemical bond, together with any information on the geometric shape of the molecule needed to uniquely identify the type of molecule. (Snapshot 2) gives an example of the molecular structure of a molecule [ 9 ].

 

 

Snapshot 2: Molecular Structure

Isomeric  group:

An isomer is a molecule with the same molecular formula as another molecule, but with a different chemical structure. (Snapshot 3) provides the structural formulas for an isomer [10]

 

 

Snapshot 3: Structural formulas for C4H10O isomers

 

Boiling Points:

The boiling point of a substance is the temperature at which the substance boils, or enters a state of rapid evaporation. For pure water this is 100° Celsius or 212° Fahrenheit. This is measured at one atmosphere, that is, the air pressure at sea level. The boiling point of a liquid depends on the pressure of the surrounding air [11].

 

Compound

Boiling Point (°C)

Compound

Boiling Point (°C)

Benzyl Alcohol

205

Ethyl Benzote

213

Glycerol

290

Methyl salicylate

223

Ethylene Glycol

197

Nitrobenzene

211

Phenol

182

Aniline

184

o-Cresol

191

o-Toluidine

200

Benzaldehyde

178

Chlorobenzene

132

Acetophenone

202

Bromobenzene

156

Phenyl Acetate

196

Benzoyl Chlorid

197

 

Snapshot 4: Boiling point of different compounds

Alkanes:

In organic chemistry, an alkane, or paraffin (a historical name that also has other meanings), is an acyclic saturated hydrocarbon. In other words, an alkane consists of hydrogen and carbon atoms arranged in a tree structure in which all the carbon-carbon bonds are single. Alkanes have the general chemical formula CnH2n+2 [12]. (Snapshot 5) gives the examples of Alkanes [13].

 

 

Snapshot 5: Example for Alkanes and its structure

 

Alkenes:

In organic chemistry, [14] an alkene is an unsaturated hydrocarbon that contains at least one carbon–carbon double bond. The words alkene and olefin are often used interchangeably (see nomenclature section below). Acyclic alkenes, with only one double bond and no other functional groups, known as mono-enes, form a homologous series of hydrocarbons with the general formula CnH2n. (Snapshot 6) shows the examples of Alkene molecules [15].

 

 

Snapshot 6: Example for Alkenes and its structure

 

Graph:

A graph G is an ordered triple V(G), E(G), x(G) consists of non-empty set  of vertices called V(G),  a non-empty set of edges called E(G),  and an incident function (G)  which associates with each edges of G and unordered pair of vertices. (Figure 1) shows the example of a graph [16].

 

Fig. 1: Example of a Graph

 

Path:

A path in a graph is a finite or infinite sequence of edges which connect a sequence of vertices which are all distinct from one another [ 17 ]. Examples of paths with n = 1, 2, 3, 4 vertices are shown in (Figure 2).

 

Fig. 2: Examples for  Paths

 

Distance:

In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path (also called a graph geodesic) connecting them. This is also known as the geodesic distance [ 18 ]. Notice that there may be more than one shortest path between two vertices.[ 19 ] If there is no path connecting the two vertices, i.e., if they belong to different connected components, then conventionally the distance is defined as infinite. The following example (Figure 3) shows the distance of two vertices is 5.

 

 

Fig. 3: A Graph showing distance  between two vertices is 5.{\displaystyle d(u,v)}{\displaystyle d(v,u)}

 

Eccentricity:

The eccentricity of any vertex v in a graph G is defined as the length of the longest possible from the vertex to any other vertex in the graph.

 

 

Fig. 4: A Graph with eccentricity 4, 3, 4, 4, 3

 

For example for the graph in (Figure 4) the length of the longest possible path is to vertex 5 namely 1 to 2 to 3 to 4 to 5. So the eccentricity of vertex 1 is 4. Similarly the eccentricity of the vertices 2, 3, 4 and 5 are 3, 4, 4, 3 respectively.

 

{\displaystyle \epsilon (v)}{\displaystyle v}{\displaystyle v}Wiener index:

In chemical graph theory, the Wiener index (also Wiener number) is a topological index of a molecule, defined as the sum of the lengths of the shortest paths between all pairs of vertices in the chemical graph representing the non hydrogen atoms in the molecule [ 6 ]. The distances dij form the so-called distance matrix D(G) = [ dij] of the graph G. The Wiener index of G is the number

 

W(G) =  = ½

 

 

Wiener index from distance matrix

 

Fig. 5: Distance matrix of Tree T

 

The Wiener index of a path with n – vertices is n (n2-1) / 6. [ 20 ]

 

PROPOSED METHOD OF BOILING POINT DETERMINATION:

The boiling point of organic compounds give importance to many physical properties and structural characteristics.

 

Harry Wiener in his paper on the structural determination of paraffin boiling point establish that the boiling point can be determined based on the structural arrangement of atoms in a molecule. He has given a relation that boiling points of paraffins are given by the linear formula TB = aW + b P + c where a,b and c are constants for a given isomeric groups while P and W are structural variables determined by Wiener and later labeled as Wiener index and Wiener polarity index respectively. The regular procedure to determine Wiener index uses distance between every pair of vertices in any chemical structure ( here after we shall refer to them as chemical graphs or simply graphs ).

 

Eccentricity for every vertex is defined as the distance of the longest possible path from the vertex. In a distance matrix, the distance between all pair of vertices is required, while to determine the eccentricity of any vertex only longest possible path is required. If a method can be determined to find the boiling point using eccentricity then lot of manual calculation gets reduced. Specifically as the size of graph increases, we attempt to determine the boiling point using eccentricity values only.

 

An equation of the form B = α W β relating the boiling point B and Wiener index W is approximated by B = 181 W 0.1775 [ 21 ] . We attempt to determine similar formula using eccentricity of any given vertex. The chemical graph structure 2,2 – dimethyl pentane is seen in (Figure 6).

 

 

Fig. 6: Chemical graph structure of  2, 2- dimethyl pentane

 

The eccentricity of the vertices is represented as label of vertex. (Figure 7) shows the eccentricity of above Fig 6  in which the sum of the eccentricity = 24.

 

 

Fig. 7: A graph showing the eccentricities of Fig. 6

We shall define the eccentricity of any graph G to be the sum of eccentricities of all the vertices in G. The graph in (Figure 8) displays a graph of the boiling point eccentricity index data.


 

 

Fig. 8: A Scatter diagram with boiling point and eccentricity index data

 


We observe that there is a strong correlation. The boiling point Vs eccentricity of the chemical structure can be closely approximated by an increasing curve. Fitting a power equation B = α E β to the data we find that the relation between the boiling point B and the eccentricity E for the data is approximated by B = 178 (E ) 0.21 (Figure 9) gives the graph of fitting the power equation to data for Alkanes.


 

 

Fig. 9: Graph showing the fit of power equation to data for Alkanes

 

(Figure 10) shows the fitting of power formula for Alkenes

 

 

Fig.10: Graph showing the fit of power formula for Alkenes


We observe that the power equation developed can be used to determine the approximate boiling point of any new chemical whose molecular structure [ 22 ]  is known. The power formula developed can be extended to determine an approximate boiling point of other isomeric groups also. An approximate boiling point of alkenes can be determined using this formula. (Table 1) gives a detail calculation of boiling points for alkanes and alkenes using the power formula.


 

Table 1: Calculation of Boiling point for Alkanes and Alkenes using the Power formula

ALKANES

No

Name

Ball and stick representation

Chemical structure with eccentricity values

Eccentricity of the graph

Boiling point using power formula

Original boiling point

1

Ethane

 

 

2

206

184

2

Propane

 

 

 

5

250

233

3

n-n-butane

 

 

10

289

272

4

2-methyl propane

 

 

7

263

261

5

n-pentane

 

 

16

319

309

6

2-methyl butane

 

 

13

305

301

7

n-hexane

 

 

24

347

342

8

2-methyl pentane

 

 

20

334

333

9

3-methyl pentane

 

 

19

330

336

10

2,3-dimethyl butane

 

 

16

319

331

11

2,2-dimethyl butane

 

 

16

319

323

12

n-heptane

 

 

33

371

371

13

2-methyl hexane

 

 

29

361

363

14

3-methyl hexane

 

 

28

358

365

15

2,2-dimethyl pentane

 

 

24

347

352

16

3,3-dimethyl pentane

 

 

22

341

359

17

2,3-dimethyl pentane

 

 

23

344

363

18

2,4-dimethyl pentane

 

 

24

347

354

19

3-ethyl pentane

 

 

23

344

366

20

2,2,3-trimethylbutane

 

 

19

330

354

ALKENES

 

21

Ethene

 

 

 

2

206

169

22

Propene

 

 

5

250

225

23

Butene

 

 

10

289

267

24

Pentene

 

 

16

319

303

25

Hexene

 

 

24

347

336

26

Heptene

 

 

33

371

367

27

Octene

 

 

 

44

394

394

28

Nonene

 

 

56

415

420

29

Decene

 

 

70

434

445

 

Table 2: Calculation depicting the comparison between the boiling points of original and new power formula

Alkanes and Alkenes

Chemical name

Wiener index W

Eccentricity E

Boiling point from original power formula

B = 181( W )0.1775

Boiling point from new power formula

B = 178( E )0.21

Ethane

1

2

181

206

Propane

4

5

231

250

n-butane

10

10

272

289

2-methylpropane

9

7

267

268

n-pentane

20

16

308

319

2-methylbutane

18

13

302

305

2,2-dimethylpropane

16

9

296

282

n-hexane

35

24

340

347

2-methylpentane

32

20

335

334

3-methylpentane

31

19

333

330

2,3-dimethylbutane

29

16

329

319

n-heptane

56

33

370

371

2-methylhexane

52

29

365

361

3-methylhexane

50

28

362

358

2,2-dimethylpentane

46

24

357

347

3,3-dimethylpentane

44

22

354

341

2,3-dimethylpentane

46

23

363

344

2,4-dimethylpentane

48

24

360

347

3-ethylpentane

48

23

360

344

2,2,3-trimethylbutae

42

19

351

330

Ethene

1

2

181

206

Propene

4

5

231

250

Butene

10

10

272

289

Pentene

20

16

308

319

Hexene

35

24

340

347

Heptene

56

33

370

371

Octene

84

44

397

394

Nonene

120

56

423

415

Decene

165

70

448

434

 


 

Fig. 11: Correlation of boiling points of Alkanes and Alkenes

 

 

 

Fig. 12: A Column graph showing the correlation of boiling point of Alkanes and Alkenes

These figures reveal the fact that there is correlation between both the data. They are very close to each other and coincide with each other at many points.

 

CONCLUSION:

A power formula to find the boiling point of Alkanes and Alkenes using eccentricity only is developed. Determination of boiling point from Wiener index requires n2 calculations. Determination of eccentricity of a graph involves only n calculations. This reduces n2 – n manual calculations. Even for small molecular chemical structures with 10 vertices, 90 calculations is reduced. So the proposed method is better than the existing techniques. Also a comparison with the traditional method revels that the values obtained are very close and hence can be used in the place of the traditional methods. Further such formulas can be developed for other physical properties and can be used in the determination of probable values of the physical properties whenever new chemicals are discovered.

 

REFERENCE:

1.       Harry Wiener. Structural determination of Paraffin boiling points. J. Am. Chem. Soc. 1947; 69 (1). Jan 1947: pp. 17–20.

2.       Gerta Rucker. On Topological Indices, Boiling Points, and Cycloalkanes. J. Chem. Inf. Comput. Sci., 1999; 39 (5): pp. 788–802.

3.       Mikhail Goubko. Oleg Miloserdov. Simple Alcohols with the Lowest Normal Boiling Point Using Topological Indices. Match Commun. Math.Comput. Chem., number 1; 2016; Volume 75: pp. 29-56

4.       SumanRawat. and. Sati OP. Quantitative structure- Property relationship study for prediction of Boiling point of Aliphatic Alkanes.  International Journal of Pharmacy and Pharmaceutical Sciences, 2013; Vol 5.

5.       SandiKlavzar. IvanGutman. Wiener number of vertex-weighted graphs and a chemical application. 5 December 1997; Volume 80: pp. 73-81.

6.       Yamuna M. Wiener index of chemical trees from its subtree. Der Pharma Chemica. 2014; 6(5): pp. 235-242.

7.       McNaught AD. and. Wilkinson A. IUPAC, Compendium of Chemical Terminology, 2nd ed. the Gold Book. Blackwell Scientific   Publications; 1997.

8.       http://igoscience.com/wp-content/uploads/sucrose-molecule-ball-stick-C12H22O11-v11.png

9.       http://www.chemistryexplained.com/photos/molecular-structure-3416.jpg

10.     https://www2.chemistry.msu.edu/faculty/reusch/virttxtjml/intro3.htm

11.     https://simple.wikipedia.org/wiki/Boiling_point

12.     "Alkane - Definition from the Compendium of Chemical Terminology". iupac.org. Retrieved 14 June 2016.

13.     http://herb06.weebly.com/uploads/1/2/4/2/12420435/519948675.gif

14.     https://en.wikipedia.org/wiki/Alkene

15.     http://mp3rapalace3.tripod.com/chem/U1balkenes.jpg

16.     http://people.math.sc.edu/lu/petersen2.png

17.     https://en.wikipedia.org/wiki/Path_(graph_theory)

18.     Bouttier. Jérémie. Di Francesco P. Guitter E. Geodesic distance in planar graphs. Nuclear Physics B. doi:10.1016/S0550-3213(03)00355-9; 663 (3): 535–567.

19.     Weisstein. Eric W. Graph Geodesic. MathWorld--A Wolfram Web Resource. Wolfram Research. no. 88.

20.     BaskarBabujee J. and. Senbagamalar J. Applied Mathematical Sciences. no. 88. Vol. 6; 2012: pp. 4387 – 4395.

21.     Colin Adams. Robert Franzosa. Introduction to topology. Pearson Education; 2008.

22.     http://www.tennoji-h.oku.ed.jp/tennoji/oka/OCDB/Hydrocarbon/?C=M;O=A

 

 

 

Received on 26.09.2017           Modified on 17.11.2017

Accepted on 21.12.2017          © RJPT All right reserved

Research J. Pharm. and Tech 2018; 11(5):1962-1970.

DOI: 10.5958/0974-360X.2018.00364.5