Acta Chim. Slov. 2000, 47, 293-316. 293 ON TOPOLOGICAL INDICES INDICATING BRANCHING. PART 2. THE SUITABILITY OF SOME PHYSICOCHEMICAL PROPERTIES OF ALKANES AS REFERENCES TO ASSESS BRANCHING A. Perdih Mala vas 12, SI-1000 Ljubljana, Slovenia Received 31-05-2000 Abstract The susceptibility for branching, Si,j, the difference-normalised difference of data for octanes, Anori, as well as a number of additional criteria are used to judge the suitability of 24 physicochemical properties of alkanes as references for branching. The most appropriate as primary references for branching regarding these criteria seem to be DHf °g, Tc/Pc, and BP/Tc. They are to be used simultaneously. The following properties, Tc2/Pc, w, and C are considered less appropriate. The other tested physicochemical properties are considered inappropriate measures of branching. From the data of DHf °g, Tc/Pc, and BP/Tc, reference values for branching indices are constructed, obeying the Methane-based definition as well as for those indices obeying the n-Alkane-based definition of molecular branching. Introduction In the previous paper,1 the suitability of topological indices (further on: indices) J, W, Z, D, MTI, Xu, ID, c, ll 1, EAmax, and l 1 as branching indices, as well as of physicochemical properties (further on: properties) MON, BP, d, Vi, Vm, Vc, Tc, Pc, dc, Zc, ac, DHv, A, B, C, nD, MR, a0, b0, DHf°g, DGf°g, S, R2, and w, as reference properties for branching of alkanes was tested by means of the Principal Component Analysis. It has been observed that most of the properties and indices correlate highly with the carbon number of alkanes, although they are influenced also by branching. Of the influences of branching, assessed separately by the properties and by the indices, the most important is the number of branches, followed by the type of branched structure, i.e. whether the branch bearing carbon is tertiary or quaternary, by the position of branches, i.e. whether they are central or peripheral, distant or adjacent, and the least important is the shape of branched molecules, i.e. whether they are spherical, flat or elongated, as well as symmetric or asymmetric. The properties have been divided into intrinsic and A. Perdih: On ... branching. Part 2. Physicochemical properties of alkanes as references … 294 Acta Chim. Slov. 2000, 47, 293-316. interaction-dependent ones and it has been explained why the latter ones are not suitable as primary references for branching. Two definitions of branching have been presented, the Methane-based definition as a general definition and the n-Alkane-based definition as a special definition more familiar to chemists. In the present paper we discuss additional criteria, which might enable us to assess branching better, in particular criteria to assess the suitability of physicochemical properties as references for branching indices. Data The data for the boiling point (BP) and the quadratic mean radius (R2) were taken from Ren2. The data for the melting point (MP), density (d), the critical data Tc, Pc, Vc, Zc, ac, and dc, as well as the standard enthalpy of formation for the ideal gas (DHf°g), the standard Gibbs energy of formation for the ideal gas (DGf°g), the enthalpy of vaporisation (DHv), the Antoine constants A, B, and C as well as the Pitzer's acentric factor (w) and the refractive index (nD) were taken from CRC Handbook3 or from Lange's Handbook4. The data of vapour pressure (log VP) were taken from Goll and Jurs5. The data for the liquid molar volume (Vm), the intrinsic molar volume (V i), the van der Waals parameters a0 and b0, and the molar refraction (MR) were calculated from data presented in the handbooks. After the first step of analysis, the ratios Tc2/Pc and Tc/Pc are used instead of the van der Waals parameters a0 and b0, which are derived from these ratios. Besides them also BP/Tc is included. Methods The structures of alkanes are presented in shorthand, e.g. 223M5 is 2,2,3-trimethyl pentane or 3Et2M5 is 3-ethyl-2-methyl pentane. Other terms are explained on 2,3- and 2,5-dimethyl hexane (23M6 and 25M6) as examples. The branches (tertiary carbons) in 23M6 are adjacent and those in 25M6 are distant. The branches on carbons No. 2 and 5 are peripheral, that on carbon No. 3 is central (as far as it is possible in this case). 25M6 is symmetric, whereas 23M6 is not. A. Perdih: On ... branching. Part 2. Physicochemical properties of alkanes as references … Acta Chim. Slov. 2000, 47, 293-316. 295 The susceptibility of properties for branching is used as the Method No. 1 to assess the suitability of properties as reference for branching indices. It is defined as the normalised difference of the properties' values, Eq. 1, Si,j = P/Pj –1 (1) where S i,j is the susceptibility to branching, P is a property, i refers to the more branched structure and j refers to the less branched structure. Which one is more branched is concluded by intuition as presented in ref6 as well as by the Methane-based definition and the n-Alkane-based definition1. Two groups of Si,j data are used. In the first group, i refers to the structures having the same number and type of branches as j, but one carbon more. For example, in Soctane,heptane octane means any octane having the same number and type of branches as a heptane. In SOct,Hp i refers to n-octane (Oct) and j to n-heptane (Hp), or in S2m7,2M6 to 2-methyl heptane (2M7) and 2-methyl hexane (2M6), etc. In the second group of Si,j data, i and j refer to alkanes of the same carbon number; for example, Si ,Oct refers to any octane and n-octane, S2M7,Oct refers to 2-methyl heptane and n-octane. To assess the data, the following criterion is applied as the criterion No. 1: A property might be useful as reference, if the sign of all Soctane,heptane values is the same and that of all Si ,Oct values is the same as well. It is not useful if the sign among members within each group differs. The two groups of sign may or may not be equal. As the Method No. 2, the difference-normalised difference of the properties' values for octanes is used, Eq. 2, Anori = (Pi - POct)/(P2233M4 - POct) (2) Criterion No. 2: A property is suitable as reference if all 0 < Anorm i < 1, and not suitable if any Anori < 0 or Anori > 1. The criteria No. 1 and 2 serve to eliminate the properties, which are the most inappropriate references for branching. The other criteria presented below serve to rank the remaining properties. As the Method No. 3 we use the dependence of properties on carbon number expressed in different ways: A. Perdih: On ... branching. Part 2. Physicochemical properties of alkanes as references … 296 Acta Chim. Slov. 2000, 47, 293-316. The Soctane,heptane compared to the corresponding S-values derived from data for carbon number, NC, The correlation coefficient with NC, the rNC. Criterion No. 3: No clear-cut criterion could be set in this case. As the Method No. 4 we use the dependence of properties on branching. As a general dependence it is expressed by the Si ,Oct data as well as by the ratio Si ,Oct/ S octane,heptane . As special indicators we use the following ratios: S234M5,Oct/S2233M4,Oct, S25M6,Oct/S22M6,Oct, and S34M6,Oct/S33M6,Oct compare the susceptibility of a property for the presence of tertiary carbons vs. that of quaternary ones in octanes. The ratio S25M6,Oct/S34M6,Oct indicates whether a property is more susceptible to peripheral or central position of branches. Criterion No. 4 could not be clearly defined yet, but some useful information is given by this method. As the Method No. 5 we use the comparison of the sequences of octanes having the same number of branches and the extent of information that can be derived thereby. Criterion No. 5: As a reference is better the property having the sequences closer to those expected by intuition as well as that one presenting more information on structural details. As the Method No. 6 we use the evenness of Anormi data. It is assessed visually in Fig. 1 and 2, by the differences of Anori, i.e. A(Anormi) max, A(Anori) min, as well as by their relative standard deviation (RSD). Besides the evenness as such, from these data can be deduced also the influence of the number of branches, of the (mutual) position of branches, and of the symmetry of molecules among octanes on the properties considered potentially useful as reference properties. Criterion No. 6: As a reference is better the property having more evenly distributed Anori data. As the Method No. 7 we use the estimate of the linearity of the increase of the values for n-alkanes with carbon number. Criterion No. 7: A property is the better reference the closer to linearity is its increase in n-alkanes with carbon number. A. Perdih: On ... branching. Part 2. Physicochemical properties of alkanes as references … Acta Chim. Slov. 2000, 47, 293-316. 297 Results and discussion A selection of topological indices and physicochemical properties of alkanes from methane to octanes was studied by means of the Principal Component Analysis1. Most of the properties and indices were found to correlate highly with the carbon number of alkanes and the influence of branching on them seemed to be lower. Of the influences of branching, the most important was the number of branches, followed by the type of branched structure, i.e. whether the branch bearing carbon is tertiary or quaternary, the position of branches, i.e. whether they are central or peripheral, distant or neighbouring, and the least important was the shape of branched molecules, i.e. whether they are spherical, flat or elongated, as well as symmetric or asymmetric. The suitability of topological indices as branching indices is usually tested with physicochemical properties of alkanes. This test is performed by binary correlations. Different properties indicate different agreements with indices, so the first question is which of the properties are suitable as references for branching. The answer to this question is approached here in several steps. First, the sign of Si,j as a criterion enables elimination of some properties which are not appropriate as references. In the second step, the remaining properties are tested with Anorm i as a criterion and several of them are eliminated. In the additional steps the properties remaining after the second step are tested with additional methods to rank them for their suitability to serve as references. The methods for the selection of physicochemical properties as references for branching of alkanes Method No. 1: The sign of the susceptibility of a property for branching, S i,,j In the previous paper the Principal Component Analysis was used to study the physicochemical properties of alkanes for their suitability as references for branching1. As another possibility to assess the data we suggest here the susceptibility of properties for branching, Si,,j, defined as the normalised difference of the properties' values. A. Perdih: On ... branching. Part 2. Physicochemical properties of alkanes as references … 298 Acta Chim. Slov. 2000, 47, 293-316. The reason to apply Si,,j as a criterion is as follows. An index is expected to behave consistently, i.e. to increase with carbon number and either to increase or decrease with branching but not to have mixed behaviour. The same must be expected from a physicochemical property, which should serve as a reference property. On increasing carbon number at the same type of branching, the value of a property should in all cases either increase or decrease, so the Si,,j data should be either all positive or all negative. On increasing branching, the same should hold true. If a property gives rise in either case to a mixed result, i.e. if the values for some isomers increase with carbon number or branching whereas those for other isomers decrease, then such a property is considered inappropriate as a reference for branching. These suppositions were tested with Soctane,heptane for the increase of carbon number from 7 to 8, i.e. from n-heptane to n-octane, from 2M6 to 2M7, from 3M6 to 3M7, from 3Et5 to 3Et6, from 22M5 to 22M6, from 23M5 to 23M6 and 34M6, from 24M5 to 24M6 and 25M6, from 33M5 to 33M6, and from 223M4 to 223M5, as well as with Si,Oct for the increase of branching from n-octane to all other octanes for which the data were available. Table 1 presents the grouping of alkanes' properties regarding the sign of Si,j. Table 1. Grouping of 24 physicochemical properties of octanes regarding the sign of their susceptibility to branching. S octane,heptane Si,Oct All + Physicochemical property All + DHf°g All + 0 Mw, Vi All + All - w, BP, BP/Tc, Vc, ac, DHv, A, B, a0, b0 All - All + C, VP All - All - - Some +, some - All + dc Some +, some - All - - All + Some +, some - d, Vm, Tc, nD, MR, DGf°g All - Some +, some - Pc* Some +, some - Some +, gative, S25M6 some -,Oct is 0. Zc Pc* : S2M7,Oct is ne - : none found According to Table 1, among the intrinsic molecular data considered here, only the susceptibility to branching of DHf°g is positive and increases both with carbon number A. Perdih: On ... branching. Part 2. Physicochemical properties of alkanes as references … Acta Chim. Slov. 2000, 47, 293-316. 299 and branching. This is a serious reason to consider it a better primary reference for branching than other properties. Molar weight and the intrinsic molar volume, Vi, increase with carbon number and do not depend on branching. The values of w increase with carbon number and decrease on branching. Among the properties dependent on intermolecular interactions, the values of Si,j of BP, BP/Tc, Vc, ac, DHv, A, B, a0, and b0 are positive on increasing the carbon number and negative on increasing branching. The values of Si,j of C and VP decrease with carbon number and increase with branching. No property gives rise to only negative values of Si,j, i.e. no one decreases with increasing carbon number and branching. The properties Tc, Pc, dc, d, Vm, nD, MR, and especially Zc give rise in the first and/or the second Si,j group to some positive as well as to some negative values and for this reason they seem to be the most inappropriate references for branching among the tested properties. Therefore, they are not considered further. The Soctane,heptane values of C decrease and those of DHf°g, BP/Tc, Vc, w, ac, a0, and b0 increase on insertion of a secondary carbon. The decrease and the increase, respectively, are higher if the insertion is adjacent to an existing secondary carbon. BP and B increase more if the inserted secondary carbon is not adjacent to an existing one. It might be worth mentioning that a0 and b0 are in fact derived from Tc and Pc considered here as inappropriate references for branching. It seems that Tc and Pc contain the information of the influence of branching as well as of packing. In the derivation of the secondary data a0 and b0 the information of packing seems to be cancelled out to an appreciable extent. To stress where they are derived from, the data and designations Tc2/Pc and Tc/Pc are used from this point on instead of a 0 and b0. In the discussion of the measures of branching in molecules several questions may be raised in spite of the thorough treatment of the problem performed until now6-8. The first question is which of the known measures of branching is better. It is a general opinion that the best way to describe something is to do it in a simple and straightforward way in direct relationship to the problem in question. A reverse relationship is acceptable only if a direct one is lacking. The most appropriate reference for branching is in this respects DHf°g, whereas w, BP, BP/Tc, Vc, ac, DHv, A, B, Tc2/Pc, Tc/Pc, and C may be A. Perdih: On ... branching. Part 2. Physicochemical properties of alkanes as references … 300 Acta Chim. Slov. 2000, 47, 293-316. considered in that order as secondary ones, to be used only in those cases when DHf°g does not indicate an unequivocal answer. Method No. 2: The difference-normalised difference, Anornii The 12 properties DHf°g, w, BP, BP/Tc, Vc, a c, DHv, A, B, Tc2/Pc, Tc/Pc, and C, giving, according to Table 1, the same sign in the same group of susceptibilities for branching, were tested also with the seco nd criterio n presented here, the Ano rm i. VP could not be included since its value for the 2233M4 was not available. The least branched octane, the n-octane, has the value of Anormi equal to 0 and the most branched o ctane, the 2233M4 has the value of Anorm i equal to 1 by definition (Eq. 2). Any property having for any octane Anori < 0 or Anori > 1 is considered inappropriate as a reference for judging the indices, since in the former case, it presents the octane in question to be less branched than the n-octane and, in the latter case, as more branched than the 2233M4, which is obviously not the fact. The results of this analysis are presented in Table 2. Table 2. The grouping of 13 properties regarding the criterion 0 < Anori < 1. _________0 < Anori < 1_________ ________Anori > 1 *________ DHf°g, w, C, BP/Tc, Tc2/Pc, Tc/Pc BP (1), AHv (1), R2 (3),Vc (7), _____________________________________________________A (11), B (12), ac (15) * In parenthesis the number of cases out of the 16 possible is given, when Anori > 1. According to this criterion, the properties DHf°g, w, C, BP/Tc, Tc2/Pc, and Tc/Pc seem to be suitable as reference properties. BP, AHv, R2, Vc, A, B, and especially ac, on the other hand, are not suitable as references. In most of the unsuitable properties the data indicate that 2,2,4-trimethyl pentane (224M5) is more branched than 2,2,3,3-tetramethyl butane (2233M4). This is not surprising if we recall that 224M5 in the liquid state has the highest intermolecular "free" space1, i.e. the loosest packing, whereas 2233M4 is packed the most densely of all octanes. The data of 224M5 might serve as a good indicator of whether a property is suitable as a reference of branching or not. A. Perdih: On ... branching. Part 2. Physicochemical properties of alkanes as references … Acta Chim. Slov. 2000, 47, 293-316. 301 Ranking of the 6 remaining properties by a series of additional criteria Method No.3: The dependence on carbon number The properties, which seemed suitable as references to evaluate branching indices, i.e. DHf°g, w, C, BP/Tc, Tc2/Pc, and Tc/Pc, were additionally tested using several criteria. Their dependence on carbon number is presented on the one hand by the mean and the maximum Soctane,heptane, Table 3. As a reference value, Soctane,heptane for carbon number, NC, is included. Table 3. Susceptibilities of properties to the increase in carbon number. Soctane,heptane Mean 0.25 > Tc2/Pc > w ~ Tc/Pc > NC = 0.14 > DHf°g > BP/Tc > 0 > C > -0.04 Max. 0.25 > Tc2/Pc > w > Tc/Pc > NC > DHf°g > BP/Tc > 0 > C > -0.05 RSD 0.2 > C > DHf°g > w > BP/Tc > 0.1 > Tc/Pc > Tc2/Pc > NC = 0 rNC 1.000 > Tc/Pc > Tc2/Pc > DHf°g > BP/Tc > 0.95 > w > 0.90 >>> -0.95 > C According to Table 3, w, Tc2/Pc, and Tc/Pc are more susceptible to the increase in carbon number than the carbon number itself, whereas DHf°g, C, and BP/Tc are less susceptible. On the other hand, the dependence on carbon number can also be presented by rNC. The properties tested here correlate well with carbon number. Method No.4: The dependence on branching As another means to assess the properties, the susceptibility to branching at the same carbon number is presented by the mean and the maximum Si,Oct, Table 4. DHf°g and C increase on branching, whereas w, Tc/Pc, Tc2/Pc, and BP/Tc decrease on branching. The ranking according to the absolute value of the mean susceptibility to branching is w > 0.17 > Tc2/Pc > Tc/Pc > 0.05 > DHf°g > C > 0.02 >BP/Tc. The most susceptible to branching at the same carbon number are w, Tc2/Pc, and Tc/Pc. As the next means, the comparison of the susceptibilities to branching vs. that to the increase of carbon number is presented by the ratio of mean Si,Oct/Soctane,heptane in Table 4. A. Perdih: On ... branching. Part 2. Physicochemical properties of alkanes as references … 302 Acta Chim. Slov. 2000, 47, 293-316. The absolute values of this ratio indicate that only w is somewhat more susceptible to branching than to the increase of the carbon number, C and BP/Tc are slightly less susceptible, whereas DHf°g, Tc/Pc, and Tc2/Pc are more than two times more susceptible to the increase in carbon number than to branching. Table 4. Mean and maximum susceptibilities of properties to the branching. Si,Oct Mean 0.05 > DHf°g > C > 0 > BP/Tc > Tc/Pc > Tc2/Pc > w > -0.2 Max. 0.1 > DHf°g ~ C > 0 > BP/Tc > Tc/Pc ~ Tc2/Pc > w > -0.4 RSD 0.7 > BP/Tc > DHf°g > C ~ w ~ Tc/Pc > Tc2/Pc > 0.4 Si,Oct/Soctane,heptane Mean 0.5 > DHf°g >>> -0.3 > Tc2/Pc> Tc/Pc > -0.5 > BP/Tc > C > -1 > w RSD - Relative standard deviation of the influence of branching. Besides the mean and maximum Si,j values, also the value of the most branched octane, S2233M4,Oct, as well as the ratios of the values of some typical octane structures, e.g. S234M5,Oct/S2233M4,Oct, S25M6,Oct/S22M6,Oct, and S34M6,Oct/S33M6,Oct can present some additional information about the remaining properties. Table 5. The ranking of properties regarding some susceptibilities to branching and their ratios. Criterion Ranking S2233M4,Oct 0.1 >DHf°g>C> 0 >BP/Tc> -0.1 >Tc/Pc>Tc2/Pc> -0.2 >w> -0.4 |S2233M4,Oct| 0.4 >w> 0.2 >Tc2/Pc>Tc/Pc > 0.1 >DHf°g>C> 0.05 > BP/Tc S234M5,Oct/S2233M4,Oct Tc2/Pc > 0.7 > Tc/Pc > BP/Tc > w > DHf°g > 0.5 > C S25M6,Oct/S22M6,Oct 0.9 > C > Tc2/Pc > DHf°g> 0.8 > Tc/Pc > w > BP/Tc > 0.6 S34M6,Oct/S33M6,Oct Tc/Pc > 1 > Tc2/Pc > BP/Tc > w > 0.7 > C > 0.5 > DHf°g > 0.3 S25M6,Oct/S34M6,Oct 3.2 > DHf°g > 1 > C > Tc2/Pc > w = 0.7 > BP/Tc > Tc/Pc > 0.4 The rankings of the properties are presented in Table 5. According to S2233M4,Oct, among the tested properties only DHf°g and C increase with branching, whereas the most susceptible to branching is w, followed by Tc2/Pc. A. Perdih: On ... branching. Part 2. Physicochemical properties of alkanes as references … Acta Chim. Slov. 2000, 47, 293-316. 303 The branches are placed in alkanes on tertiary and/or quaternary carbons. The ability to distinguish between the presence of tertiary and quaternary carbons in the structure of alkanes is presented here as the ratio of susceptibilities to branching, S234M5,Oct/S2233M4,Oct, S25M6,Oct/S22M6,Oct, and S34M6,Oct/S33M6,Oct. This ability depends on the position of branches. If the branches are central, as presented by S34M6,Oct/S33M6,Oct, then the most susceptible to the presence of tertiary carbons is Tc/Pc and to that of quaternary ones it is DHf°g. If the branches are peripheral, as presented by S25M6,Oct/S22M6,Oct, then the property most susceptible to the presence of tertiary structures is C and to quaternary structures it is BP/Tc. This relationship is thus quite complex. The ratio S25M6,Oct/S34M6,Oct indicates that in symmetric structures, DHf°g is more susceptible to peripheral substitution whereas C, Tc2/Pc, w, BP/Tc, and Tc/Pc are more susceptible to central substitution. Other ratios like this contain information of several features at once, such as the symmetry of substitution, the adjacency of branches, and their position from the centre of the molecule, so they are more difficult to interpret unequivocally and therefore they are not presented. Method No.5: The influence of branching of octanes on their properties Since among the influences expressed by branching on properties and indices, the most important is the number of branches, followed by the type of branched structure, i.e. whether the branch bearing carbon is tertiary or quaternary, the position of branches (central or peripheral), (distant or adjacent), and the least important is the shape of branched molecules (spherical, flat or elongated), (symmetric or asymmetric), the properties DHf°g, BP/Tc, C, w, Tc/Pc, and Tc2/Pc were tested in this respect, too, Table 6-11. The most consistent is the dependence on branching of the intrinsic property DHf°g, as well as of the property Tc/Pc, which is derived from interaction dependent properties Tc and Pc. At each number of branches they give the same ranking of the influence of particular characteristics of the branches and the highest number of these characteristics. A. Perdih: On ... branching. Part 2. Physicochemical properties of alkanes as references … 304 Acta Chim. Slov. 2000, 47, 293-316. In this respect they are followed by Tc2/Pc and BP/Tc, whereas w and C present the least information. Table 6. The influence of branching on DHf°g of octanes. Nbr Series Result 1 3Et < 4M < 3M < 2M Et < M, centr. w >BP/Tc > Tc2/Pc > Tc/Pc > DHf°g A(Anori)min___________Tc/Pc > Tc2/Pc > BP/Tc > DHf°g, C, w A. Perdih: On ... branching. Part 2. Physicochemical properties of alkanes as references … 306 Acta Chim. Slov. 2000, 47, 293-316. RSD C > w >BP/Tc > Tc2/Pc > Tc/Pc > DHf°g To illustrate the series presented in Table 12, Fig. 1 presents the Anorm i data for octanes of properties AHf°g, Tc/Pc, and Tc2/Pc, whereas Fig. 2 presents those of C, co, and BP/Tc. The data points are arranged according to the number of branches. Using only data of octanes, the influence of carbon number is excluded. In Table 13 compilation of the content of Fig. 1 and 2 is presented in shorthand. 22M6„ 234M5 33M6 233M5 2233M4 A A4oo,„,A224M5 3Et3M5 '¦ ¦¦"= 34M6 25M6 0 0 O 3Et2M5 23M6 24M6 2M7D3mP ^M7 3Et6 TcA2/Pc 'Oct ¦22M6 224M5A 234M5A A223M54233M5 ¦33M6 ¦ 3Et3M5 2233M4 25M6 O 24M60 O O O 3Et2M5 23M6 34M6 O GD O 2M7 3M7 4M7 3Et6 Tc/Pc 'Oct 233M5 A 234M5 A 223M5 A 2233IW 224M5 A ¦ 3Et3M5^ 33M6 ¦ 22M6^ 3Et2M5 O 34M6<> <>23M6 24M6 O 25M6<> O oo 3Et6 4M7 3M7 "Oct O 2M7 DHf°g D 0,2 0,4 0,6 0,8 Dnorm; 1 Figure 1: The distribution of octanes by Anormi of AHf°g, Tc/Pc, and Tc2/Pc. Empty symbols indicate the presence of tertiary carbons, filled symbols indicate the presence of (also) quaternary carbons in the structure of octanes. According to Fig. 1 and 2, the values of particular alkanes are the most evenly distributed by AHf°g, Tc/Pc, and BP/Tc. Regarding the distribution in Fig. 1 and 2 which is compiled in Table 13, the properties to serve best as a reference for branching would be AHf°g and Tc/Pc, followed by BP/Tc, whereas Tc2/Pc, co, and especially C seem to be A. Perdih: On ... branching. Part 2. Physicochemical properties of alkanes as references … Acta Chim. Slov. 2000, 47, 293-316. 307 less suitable. For Tc2 /Pc it seems that the distortion due to the influence of packing contained in Tc as well as in Pc is still contained to a noticeable degree, whereas in Tc/Pc it seems to be well compensated. The influence of packing might be contained in C to an appreciable extent. The reaso n fo r the wo rse perfo rmance o fco is no t clear at the moment. Oct 223M5 234M5A A 233M5A4224M5 ¦ 22M6 ¦SSMô ¦ 3Et3M5 23M6^O0 3Et2M5 CD O- 4M7 3Et6 3M7 ^M7^2M7 2233M4 224M5 223M5 234M5 a AAA 233M5 + 22M6 +33M6 +3Et3M5 25M6 O 00 O O 3Et2M5 23M6?4Mfi34M6 2M7?00 4M7 3M7 3Et6 Oct 2233M4 w 234M5 224M5^ A ¦ 22M6 ¦ 33M6 25M60 24M60 O O O 3Et2M5 23M6 34M6 2MT 4Mr3M7 3Et6 Oct 2233M4 A223M^233M5 ¦ 3Et3M5 BP/Tc 0,2 0,4 0,6 0,8 a norm/ 1 Figure 2: The distribution of octanes by Anorm i of BP/Tc, co, and C. Empty symbols indicate the presence of tertiary carbons, filled symbols indicate the presence of (also) quaternary carbons in the structure of octanes. In addition to preceding findings, from the Anormi data, cf. Fig. 1 and 2 and Tables 6-11, one can also deduce the influence of the number of branches, Nbr, of the type of branched structure, i.e. the influence of tertiary and quaternary carbons, t/q, of the position of branches, i.e. whether they are central or peripheral, centr./periph, adjacent A. Perdih: On ... branching. Part 2. Physicochemical properties of alkanes as references … 308 Acta Chim. Slov. 2000, 47, 293-316. or distant, adj./dist., as well as of the symmetry of octanes, sym./asym., on the considered properties. The conclusions made here must be viewed with caution, since except for the influence of the number of branches, where 9 pairs of data were available, they are derived from only 4 pairs of data each. The presented conclusions are to be checked with data of nonanes and decanes when they become available. The results are collected in Table 14 as a ranking of properties regarding the extent of influence of particular structural details, and in Table 15 as a ranking of importance of particular structural details for individual properties. Table 13. The sequences acco rding to Ano rm i of octanes having the same number of branches. Given is the position of branches. If not marked Et for ethyl, the branch is methyl. Nbr Sequence Property 1(t ) 2, 3, 4, 3Et 2, 4, 3, 3Et 3, 4, 3Et, 2 AHf°g, Tc/Pc, and Tc2/Pc BP/Tc, co C 2(t ) 25, 24, 23, 34, 3Et2 25, 23, 34, 24, 3Et2 25, 23, 24, 34, 3Et2 23, 25, 24, 34, 3Et2 AHf°g, Tc/Pc, and BP/Tc Tc2/Pc CO C 2(q) 22, 33, 3Et3 All 3(t,q) 224, 223, 234, 233 AHf°g 224, 234, 223, 233 234, 233, 223, 224 234, 224, 223, 233 234, 223, 233, 224 Tc/Pc, BP/Tc Tc2/Pc CO C t - tertiary; q - quaternary DHf°g differs in this respect markedly from other properties, since it is the least of all dependent on the number of branches. This fact is in contrast also to the previous observation on all 24 tested properties as a group1 that the number of branches is the most important feature next to carbon number. Low dependence on the number of branches seems to be a peculiarity of DHf°g, making it less appropriate as a reference A. Perdih: On ... branching. Part 2. Physicochemical properties of alkanes as references … Acta Chim. Slov. 2000, 47, 293-316. 309 property than it would be expected with regard to its other characteristics. For the interaction dependent properties as well as for w, the number of branches is more important than the other influences tested here. Among the other structural details tested here, it is important where the branches are placed, in the centre or on peripheral positions of molecules. The adjacency of branches, as well as the symmetry of molecules, which have a similar degree of influence, seems to be less important than where the branches are placed. Table 14. The influence of the number of branches, of the (mutual) position of branches, and of the symmetry of molecules on DHf°g, Tc/Pc, BP/Tc, Tc2/Pc, w, and C of octanes. Influence of The series of decreasing influence Nbr Tc2/Pc ~ Tc/Pc > w ~ BP/Tc > C ~ DHf°g t/q DHf°g > C > w > Tc/Pc > Tc2/Pc > BP/Tc centr./periph. DHf°g > Tc/Pc > BP/Tc > Tc2/Pc ~ w > C adj./dist. DHf°g > Tc/Pc > BP/Tc > Tc2/Pc > C ~ w, sym./asym. DHf°g > Tc/Pc > BP/Tc > C ~ w ~ Tc2/Pc Table 15. The contribution of the following structural details: the number of branches, the (mutual) position of branches, and the symmetry of molecules on DHf°g, Tc/Pc, BP/Tc, Tc2/Pc, w, and C of octanes. The contribution to Ranking of importance of structural details DHf°g t/q ~ centr./periph. > adj./dist. > sym./asym. > Nbr Tc/Pc Nbr > centr./periph. > sym./asym. ~ adj./dist. > t/q BP/Tc Nbr > centr./periph. > adj./dist. ~ sym./asym. > t/q w Nbr > t/q > centr./periph. > sym./asym. ~ adj./dist. Tc2/Pc Nbr > centr./periph. > t/q > adj./dist. > sym./asym. C Nbr > t/q > centr./periph. ~ sym./asym. ~ adj./dist. Method No. 7: The increase of the values for n-alkanes with carbon number The next question is how DHf°g, Tc/Pc, and BP/Tc increase with the increase of carbon number in n-alkanes. According to the Methane-based definition1, a linear increase would be expected. To see whether this is true, the difference-normalised data of n-alkanes from C1 to C18 are presented in Fig. 3. A. Perdih: On ... branching. Part 2. Physicochemical properties of alkanes as references … 310 Acta Chim. Slov. 2000, 47, 293-316. According to Fig. 3, DHf°g increases from ethane on linearly with carbon number. Such an increase is expected according to the Methane-based definition of branching, since the insertion of an additional CH2 group into an n-alkane presents the same contribution in any case. DHf°g might be in this respect, as well as due to its increase with branching, a prototype reference property to test whether a topological index obeys the Methane-based definition of branching. The sign of tabulated data of DHf°g of alkanes3,4, which is by definition negative in exothermic reactions, is not related to branching. This sign indicates only the direction of heat flow during the reaction. Therefore, absolute values of DHf°g are to be used when branching of molecules is considered. 1 8 15 Figure 3. The increase of DHf °g, Tc/Pc, and BP/Tc with carbon number (NC) in n-alkanes (the data are normalised at n-octane). Tc/Pc, on the other hand, increases slightly faster than carbon number, whereas BP/Tc increases slower than carbon number but faster than BP and Tc. In this respect, BP/Tc and especially Tc/Pc might be useful references for branching at particular carbon numbers. A. Perdih: On ... branching. Part 2. Physicochemical properties of alkanes as references … Acta Chim. Slov. 2000, 47, 293-316. 311 Reference data for judging the branching indices In the preceding parts of this paper has been found that the properties that seem to be the best reference ones for judging the branching indices are DHf°g and Tc/Pc, followed by BP/Tc. Each of them has some peculiarities, which diminish its ability to serve as the single best one. One of the strengths of DHf°g is its linear increase with carbon number in n-alkanes as well as its increase with branching. This behaviour is in accordance with the Methane-based definition of branching and it would be desired that other reference data share it. Its weakness is the low dependence on the number of branches. With Tc/Pc and BP/Tc, the situation is reversed. They do not increase linearly with carbon number but they seem to have an appropriate dependence on the number of branches. Thus, all these properties, DHf°g, Tc/Pc, and BP/Tc are to be used simultaneously as references. Another possibility is to take the best of the reference properties and to construct one or more quasi-ideal references. For example, the principle of linear increase with carbon number is taken from DHf°g and its values are rearranged to serve as reference values. The data for methane is an outlier in the trend of DHf°g values for n-alkanes. In spite of that, the reference values (RVDHf°g) can be based on data for DHf°g of other n-alkanes setting for RVDHf°g of n-alkanes the value NC - 1 by definition to obtain for methane the reference value of 0. The RVDHf°g are then calculated as follows, Eq. 3: RVDHf°g = (NC-1)* DHf°g i /DHf°g n (3) where i stands for any alkane of the same carbon number and n stands for n-alkane of the same carbon number. They are presented in Table 16. In a similar manner the reference values based on Tc/Pc or BP/Tc are constructed, forcing them to align to the linear increase of the values for n-alkanes and setting the value of 0 for methane, Eq. 4: RVX = (NC - 1)* Xn/Xi (4) A. Perdih: On ... branching. Part 2. Physicochemical properties of alkanes as references … 312 Acta Chim. Slov. 2000, 47, 293-316. where X is Tc/Pc or BP/Tc and NC is the carbon number (number of vertices in the graph). These reference values are presented in Table 17. The value of RVTc/Pc for 2M3 is an outlier. To correct it, better data for Tc and Pc of n-butane and 2-methyl propane are needed. Table 16. Reference values for branching indices, derived from DHf°g, obeying the Methane-based definition ( RVDHf°g). CxHy RVDHf°g CxHy RVDHf°g CxHy 3Et6 RVDHf°g 7.0787 M 0 Hp 6 22M6 7.5466 Et 1 2M6 6.2286 23M6 7.1841 Pr 2 3M6 6.1444 24M6 7.3681 Bu 3 3Et5 6.0602 25M6 7.4763 2M3 3.1990 22M5 6.5869 33M6 7.3920 Pe 4 23M5 6.3663 34M6 7.1532 2M4 4.2194 24M5 6.4545 3Et2M5 7.0927 22M3 4.5337 33M5 6.4398 3Et3M5 7.2192 Hx 5 223M4 6.54418 223M5 7.3920 2M5 5.2127 Oct 7 224M5 7.5269 3M5 5.1326 2M7 7.2361 233M5 7.2684 22M4 5.5493 3M7 7.1405 234M5 7.3021 23M4 5.3166 4M7 7.1222 2233M4 7.5859 The correlation coefficients of RVDHf°g, RVTc/Pc, and RVBP/Tc are: rDHf°g,Tc/Pc = 0.993, rDHf°g,BP/Tc = 0.996, and rTc/Pc,BP/Tc = 0.997. A good branching index obeying the Methane-based definition seems to have to fall within this range. Table 17. Reference values for branching indices, derived from Tc/Pc and BP/Tc, obeying the Methane-based definition ( RVTc/Pc and RVBP/Tc). CxHy RVtc/Pc RV BP/Tc CxHy RVtc/Pc RV BP/Tc CxHy 3Et6 RVtc/Pc 7.3669 RV BP/Tc 7.0854 M 0 0 Hp 6 6 22M6 7.3464 7.1017 Et 1 1 2M6 6.0632 6.0265 23M6 7.4498 7.1135 A. Perdih: On ... branching. Part 2. Physicochemical properties of alkanes as references … Acta Chim. Slov. 2000, 47, 293-316. 313 Pr 2 2 3M6 6.1835 6.0520 24M6 7.3752 7.1005 Bu 3 3 3Et5 6.2892 6.0858 25M6 7.2218 7.0621 2M3 2.9999 3.0035 22M5 6.2667 6.0966 33M6 7.5422 7.1620 Pe 4 4 23M5 6.3651 6.1104 34M6 7.5784 7.1423 2M4 4.1011 4.0282 24M5 6.1936 6.0658 3Et2M5 7.6053 7.1582 22M3 4.1159 4.0415 33M5 6.4603 6.1627 3Et3M5 7.7792 7.2295 Hx 5 5 223M4 6.5412 6.1920 223M5 7.7389 7.2207 2M5 5.1360 5.0270 Oct 7 7 224M5 7.5406 7.1688 3M5 5.2256 5.0511 2M7 7.0894 7.0280 233M5 7.8533 7.2565 22M4 5.3314 5.0999 3M7 7.2148 7.0556 234M5 7.6979 7.1900 23M4 5.2811 5.0869 4M7 7.2278 7.0533 2233M4 8.0742 7.3420 From the conclusion made above, that a reference value should be equal to ( NC - 1) for n-alkanes, as well as from the fact that the tested properties are dependent not only on carbon number but also on the number of branches, the type of branched structure, the position of branches, and the symmetry of molecules1, it follows that to compose a branching index obeying the Methane-based definition (BIM), the following general equation is to be used, Eq. 5: BIM = (NC - 1) + f(Nbr, t/q, centr./periph., adj./dist., sym./asym.) (5) where f(Nbr, t/q, centr./periph., adj./dist., sym./asym.) is nothing but the n-Alkane-based definition of branching1 obeying branching index (BIA). Thus, Eq. 6 presents this relation: BIM = (NC - 1) + BIA (6) The reference values to assess BIA can be derived from the reference properties DHf°g, Tc/Pc, or BP/Tc. From the data tabulated in Tables 16 and 17 we can derive the reference values useful for assessing BIA by subtracting from them NC - 1, i.e. the value set for the n-alkane. Based on the findings made in a previous1 and the present work, a general equation (Eq. 7 or 7a) presenting the dependence of physicochemical properties on branching can be set, too: A. Perdih: On ... branching. Part 2. Physicochemical properties of alkanes as references … 314 Acta Chim. Slov. 2000, 47, 293-316. P = f(NC) + f(Nbr, t/q, centr./periph., sym./asym., adj./dist.) + f(packing) (7) or, P = f(NC) + BIA + f(packing) (7a) In the case of reference values for branching indices, f(NC) = NC - 1. For f(packing), f(Vm), e.g. in the form of f(Vm-Vi), might be representative as a reference (cf. the discussion of the density of alkanes in the previous paper1). Conclusions Summing up this analysis, the suitability of the studied alkanes' properties to serve as references to judge the branching indices is as follows: Regarding the criterion No. 1, the sign of Si,j, the following properties might be appropriate: DHf°g, w, BP, Vc, a c, DHv, A, B, C, Tc2/Pc, Tc/Pc, and BP/Tc. Regarding the criterio n No . 2, 0 < Ano rm i < 1, the following ones might be appropriate: DHf°g, w, C, Tc/Pc, Tc2/Pc, and BP/Tc. Regarding their mean dependence on carbon number, they are ranked Tc2/Pc > w > Tc/Pc > DHf°g > BP/Tc > C. Their correlation coefficient with carbon number, rNC, is > 0.95, except for w. - Their susceptibility for branching is w > Tc2/Pc > Tc/Pc > DHf°g > C > BP/Tc. - Regarding the criterion No. 5, they are ranked DHf°g, Tc/Pc > Tc2/Pc, BP/Tc > w, C. Regarding the criterio n No . 6, the evenness o f Ano rm i data, they are ranked as fo Ilo ws: By the relative standard deviatio n o f Ano rm i these properties would be ranked: DHf°g > Tc/Pc > Tc2/Pc > BP/Tc > w > C (> stands for "better than"). Regarding the AAnorri max they would be ranked DHf°g > Tc/Pc > Tc2/Pc > BP/Tc > w> C. Regarding the AAnorrni min they would be ranked Tc/Pc > BP/Tc > Tc2/Pc > DHf°g, w, C. The three best properties are in this respect DHf°g, Tc/Pc, and BP/Tc. Regarding the criterion No. 7, the linearity of the increase of the values for n-alkanes, the three best properties are ranked DHf°g > Tc/Pc > BP/Tc. A. Perdih: On ... branching. Part 2. Physicochemical properties of alkanes as references … Acta Chim. Slov. 2000, 47, 293-316. 315 Regarding the increase on branching (obeying the Methane-based definition of branching) DHf°g would be the most appropriate and C the least appropriate. Regarding the evenness of the distribution of data for alkanes at the fixed value of NC and the fixed value of Nbr, the six best properties would be ranked DHf°g, Tc/Pc > BP/Tc >> Tc2/Pc, w > C. Regarding the contribution of the number of branches to its value, DHf°g is not a good reference, whereas regarding the contribution of the (mutual) position of branches, and of the symmetry of molecules it is one of the best reference properties. As a result of this comparison it can be concluded that no one of the tested properties is a single best reference to judge the branching indices. Among the best ones seem to be DHf°g and Tc/Pc, followed by BP/Tc. All of them are to be considered simultaneously. The properties w, Tc2/Pc, and C do not seem sufficiently good to serve as reference properties, whereas all the other tested properties are not to be used for this purpose unless their peculiarities are taken into account. Based on data of the best three properties, i.e. DHf°g, Tc/Pc, and BP/Tc, there were derived reference values for branching indices obeying the Methane-based rule of branching as well as those for branching indices obeying the n-Alkane-based rule of branching. References 1. A. Perdih, M. Perdih, Acta Chim. Slov. 2000, 47, 231-259. 2. B. Ren, J. Chem. Inf. Comput. Sci. 1999, 39, 139-143. 3. D.R. Lide, CRC Handbook of Chemistry and Physics, 76th Ed., CRC Press, Boca Raton 1995-96. 4. J.A. Dean, Lange's Handbook of Chemistry. McGraw-Hill, New York 1985. 5. E.S. Goll, P.C. Jurs, J. Chem. Inf. Comput. Sci. 1999, 39, 1081-1089. 6. D. Bonchev, N. Trinajstiæ, J. Chem. Phys. 1977, 67, 4517-4533. 7. E.C. Kirby, J. Chem. Inf. Comput. Sci. 1994, 34, 1030-1035. 8. M. Randiæ, Acta Chim. Slov. 1997, 44, 57-77. Povzetek A. Perdih: On ... branching. Part 2. Physicochemical properties of alkanes as references … 316 Acta Chim. Slov. 2000, 47, 293-316. Obèutljivost za razvejanost, Si,j, normalizirane razlike podatkov za oktane, Anorm i, in še vrsta dodatnih kriterijev je bila preizkušena za vrednotenje 24 fizikokemijskih lastnosti alkanov kot referenènih lastnosti pri ugotavljanju razvejanosti. Najprimernejše referenène lastnosti za vrednotenje razvejanosti so videti DHf °g, Tc/Pc (uporabljena namesto b0) in Tvrel./Tc. Naslednje lastnosti: Tc2/Pc (uporabljena namesto a0), w in C, se kažejo kot manj primerne. Druge preizkušene fizikokemijske lastnosti alkanov niso primerna merila razvejanosti. Iz podatkov za DHf °, Tc/Pc in BP/Tc so pripravljene referenène vrednosti za indekse razvejanja, ki ustrezajo metanskemu pravilu, kot tudi za tiste, ki ustrezajo n-alkanskemu pravilu razvejanosti. A. Perdih: On ... branching. Part 2. Physicochemical properties of alkanes as references …