Molecular Symmetry, Rotational Entropy, and Elevated Melting Points

are three rotational degrees of freedom, and the rota- tional entropy is given by ...... Gases & Liquids; McGraw-Hill: New York, 1987; fourth edition,...
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Ind. Eng. Chem. Res. 1999, 38, 5019-5027

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Molecular Symmetry, Rotational Entropy, and Elevated Melting Points James Wei† School of Engineering and Applied Science, Princeton University, Princeton, New Jersey 08544-5263

In a homologous series of compounds, internal cohesive forces correlate with densities, boiling points, heats of vaporization, refractive indexes, and surface tensions but do not correlate with the puzzling behavior of melting points. Rotational symmetry is the neglected major factor as a determinant of melting points. A symmetrical molecule with a high-symmetry number σ has increased order and reduced rotational entropy by the amount of R ln(σ), which elevates the melting point but has no effect on the boiling point. This opens the prospect that melting points can be predicted by group contribution methods plus molecular symmetry. Introduction In a homologous series of compounds, there is often a regularity of physical and chemical properties.1,2 Within a homologous series, the addition of a substituent that leads to an increase in internal cohesive forces also leads to an increase in densities, boiling points, heats of vaporization, refractive indexes, and surface tensions. The melting points are the outstanding exceptions and behave in idiosyncratic and puzzling manners.1-4 We show here that many outstanding puzzles are due to molecular symmetry. The concept of group contributions is based on the premise that each added substitution group would provide the same increment to molecular properties: for instance, each >CH2 group should add 11.27 K and each -Cl group should add 13.55° to the melting points of organic compounds.4 This is a first-order approximation; as when a group is added to a molecule, there are actually other effects at work: (1) The added group contributes more electrons to the parent molecule to increase the polarizability of the molecule, and thus the London dispersion forces, and possibly also the dipole moment and hydrogen bonds. (2) The added group redistributes electrons among various groups, especially if there is aromatic ring resonance or strong polar groups in adjacent positions. (3) The added group increases or decreases internal strains, including the bond lengths, the bond angles, the torsion angles, and the degree of van der Waals crowding of intramolecular groups. (4) The added group changes molecular symmetry and rotational entropy. The first factor is usually dominant, and the next two factors are well-recognized and often make large contributions and adjustments. The last factor of molecular symmetry has received the least quantitative attention, until it was introduced by Yalkowsky to form part of his group contribution method.5 We discuss here some of the outstanding melting point anomalies and how they are related to molecular symmetry. Notice that the word “group” is used in two different senses here: at one place to mean a molecular residue or substitution † E-mail: [email protected]. Telephone: (609)2582260. Fax: (609)258-6744.

such as CH3- and Cl- and at another place as a set of rotational operation in group theory. Molecular Symmetry A consideration of molecular symmetry starts with three-dimensional molecular architecture, which includes the connectivity of each atom by bonds to other atoms, the bond angles, the torsion angle, and molecular conformation. The point groups of molecular symmetry are covered in many textbooks.6,7 The symmetry number σ was introduced in statistical mechanics to reduce the partition functions of symmetrical molecules,8-11 so that orientations that are not distinguishable would not be counted more than once. Methane has the tetrahedral symmetry of Td and the very high symmetry numbers of σ ) 12, and benzene has the hexagonal symmetry D6h and the same σ ) 12. The molecule SF6 belongs to the cubic point group Oh with σ ) 24, and fullerene belongs to the icosahedron point group Ih with σ ) 60. There are several suggestions on how molecular symmetry leads to elevated melting points. Maitland Jones2 suggested that highly symmetric molecules pack well into crystal lattices, and it takes more energy to break up a well-packed lattice, thus neopentane has a melting point 113 K higher than pentane. No quantitative evaluation of this influence was made. A manifestation of good packing would be a higher ligancy and lower intermolecular distances, leading to a higher density. However, at room temperature, normal pentane has a higher density of 0.621 than neopentane at 0.586. A second suggestion due to Pauling11 is that, for the equilibrium at the melting point, the rate of crystal molecules leaving for the liquid phase is balanced by the rate of liquid molecules approaching and sticking to the crystal phase. The probability of a crystal molecule leaving is dependent on thermal agitation and independent of symmetry. However, the probability of a liquid molecule sticking to the crystal depends on the chance of striking the crystal in a suitable orientation, and a molecule of higher symmetry would have a higher rate of sticking than one of a lower symmetry. No quantitative assessment of the magnitude of this melting point elevation was made. The third suggestion employed here is that, in the liquid phase, a symmetric molecule has greater order and lower rotation entropy than an asymmetric mol-

10.1021/ie990588m CCC: $18.00 © 1999 American Chemical Society Published on Web 11/05/1999

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ecule by the amount of R ln(σ). For an ideal diatomic gas, the partition function for rotation is9,10

qr ) T/σΘr

(1)

where Θr ) h2/8π2Ik is the characteristic rotation temperature and I is the moment of inertia. This equation is valid when T . Θr, which is 185 K for hydrogen and 5 K for oxygen. The value of the symmetry number σ is 1 for a heteronuclear molecule such as HCl and 2 for a homonuclear molecule such as H-H. This symmetry number accounts for the fact that if the homonuclear molecule is rotated by 180°, its orientation is indistinguishable from before. Thus, the partition function of a symmetrical molecule is smaller than that of an asymmetrical molecule, as there are fewer distinguishable states. The rotational entropy is Sr ) R ln(T/Θr) + R - R ln σ. When we have a nonlinear polyatomic molecule, there are three rotational degrees of freedom, and the rotational entropy is given by

S/R ) log

( )

T3 e3/2xπ σ θaθbθc

1/2

(2)

where each rotational degree of freedom has its own characteristic temperature of rotation. Again, a symmetrical molecule has a reduced entropy of rotation by R log(σ). For example, σ ) 12 for metane. However, the symmetry number has no effect on the energy of the system since

E ) kT 2

Tm

Tm′ ) 1+

R ln(σ)Tm Hm

(3)

Let us make the following assumptions: (i) There is no rotation in the solid phase, up to the melting point. (ii) There is free rotation in the liquid phase after melting, so that the rotational entropy of ideal polyatomic gases apply. (iii) The entropy of melting for a symmetrical molecule is smaller by R log(σ), in comparison to a asymmetrical molecule. (iv) There is no difference in the heats of melting between a symmetrical and an asymmetrical molecule. Assumption (i) appears to be valid for most solids, with the exception of solids that undergo solid-solidphase transitions before the melting point, such as cyclohexane which has several transitions with very large heats of transition.12 The solid phase just before melting may have limited rotation.13,14 Assumption (ii) would appear to be more probable: if there is significant volume expansion upon melting so that the molecule can rotate freely, if the molecule is compact rather than extended, and if intermolecular forces are relatively weak such as in the absence of permanent dipole moment and hydrogen bonds. However, if there is on average a significant energy barrier to rotation, the resulting restricted rotation may be similar to the case of internal rotation or torsion of the ethane molecule, so that rotational contributions to entropy would be reduced. This would result in a lower entropy of melting alike for symmetric and asymmetric molecules and a smaller difference in melting points between them. The

(4)

It is expected that Tm′ would correlate with other properties that are associated with cohesive energies, such as density, boiling point, heat of vaporization, refractive index of light, and surface tension. The value of the entropy of melting is of critical importance. Pauling11 suggested that there are three sources for the entropy of melting, and the first is due to the disorder of disrupted crystal, especially in the volume expansion which introduces holes, so that

S ) -R(x1 ln x1 + x2 ln x2)

∂ log q ∂T

∂ log q S ) k log q + kT ∂T

last two assumptions seem to follow from the first two assumptions. A symmetrical molecule has σ > 1, a melting point of Tm, and an entropy of melting of Sm ) Hm/Tm. Let us propose a homomorphic molecule that has been desymmetrized, such as by isotopic exchange, so that σ ) 1. If this symmetrical molecule satisfies all the assumptions above, the desymmetrized homomorphic molecule has a greater rotational entropy in the gas phase as well as in the liquid phase, although there is no rotational degree of freedom in the crystal phase. Thus, the entropy of melting of the symmetrical molecule will be smaller by R ln(σ) than the asymmetrical homomorph, so that Sm′ ) Sm + R ln(σ). The melting point of this asymmetrical homomorph molecule Tm′ will be given by Tm′ ) Hm/Sm′, or

(5)

Here, x2 is the volume fraction of holes so that if the volume expansion is 15%, the contribution to entropy is S ) 3.6 J/(K mol). The second source is due to the smaller ligancy in the liquid, say reduced from 12 to 10, and consequently a smaller characteristic temperature of vibration, which adds from 6 to 13 J/(K mol) to the melting entropy. The third source is the most important one of orientation disorder so that

S ) R ln(W)

(6)

where W is the number of orientations in the liquid, so that if S ) 40, then W ) 120. Dannenfelser and Yalkowsky15,16 have developed methods to estimate the entropy of melting, based on the symmetry number and a flexibility parameter. It should be pointed out that they have revised the definition of “effective symmetry number” from the traditional statistical mechanics literature,8-10 as they have assigned the value of σ ) 10 to linear molecules such as H-CtN and σ ) 100 to methane and all spherical molecules. When a fluorine atom is added to replace an H atom in the tetrahedral methane CH4 with σ ) 12, the resulting CH3F has been desymmetrized to σ ) 3; and when a second F atom is added, the symmetry is further reduced to CH2F2 with σ ) 2. However, if a third F is added, the resulting CHF3 is resymmetrized to σ ) 3; and when a fourth F atom is added, the resulting CF4 has regained the full symmetry of σ ) 12. Thus, a substitution can both upgrade or downgrade the symmetry of a molecule. Consider also the case where three different atoms are substituted into CH4, the resulting CHFClBr has no symmetry and σ ) 1.17

Ind. Eng. Chem. Res., Vol. 38, No. 12, 1999 5021 Table 1. Molecular Rotational Symmetry Groups symmetry group

σ

R ln(σ) (J/(K mol))

Ih Oh Td, D6h D4h D3h, D3d D2h C3v C2v

60 24 12 8 6 4 3 2

34.04 26.43 20.66 17.29 14.90 11.53 9.14 5.76

C1, Cs, Ci

1

molecular examples C60 fullerene SF6 benzene, hexachlorobenzene, CH4, CF4, adamantane cyclobutane 1,3,5-trimethylbenzene, cyclopropane, ethane, cyclohexane, BF3 1,4-dimethylbenzene, naphthalene, 1,2,4,5-tetramethylbenzene NH3, CH3F H2O, methylbenzene, 1,2-dimethylbenzene, 1,3-dimethylbenzene, CH2F2, propane, n-butane, 1,2,3-trimethylbenzene CH2FCl, CHFClBr, isopentane, 1,2,4-trimethylbenzene

0

Table 2. Phase Transition Properties of Small Normal Paraffins molecule

Tm (K)

Tb (K)

Hm (kJ mol)

σ

Tm′ (K)

M

Tm/Tb

methane ethane propane n-butane n-pentane

90.5 89.9 86.0 138.1 143.4

111.4 184.5 230.9 272.6 309.1

0.94 2.86 3.53 4.66 8.42

12 6 2 2 2

30.3 61.2 75.4 118.0 130.6

1.99 0.47 0.14 0.17 0.10

0.812 0.487 0.392 0.507 0.464

A molecule has many degrees of internal thermal motions, such as vibrations and rotations, which has an effect on its symmetry. A molecule can simulate higher symmetry when there is a rapid interconversion of structures in a time scale more rapid than the experiment designed to measure its symmetry. An example is cyclohexane, which is in the nonplanar chair form of group D3d at -100 °C according to the NMR spectrum, but at room temperature there is rapid conversion between the chair and the inverted chair form so that the averaged symmetry result simulates the planar benzene.7 Table 1 gives the symmetry groups for the molecules to be discussed,6,7 their symmetry numbers σ, and their reduction in rotational entropy S ) -R ln(σ).7-10 Note that these are the traditional symmetry numbers in statistical mechanics, and not the revised symmetry numbers of Dannenfelser and Yakowsky. Normal Paraffins The smallest paraffins from methane to n-pentane show a steady and expected progression in boiling points with each additional -CH2- group, but exhibit a strange retrogression in melting points, which reaches a minimum in propane.18-20 Table 2 shows that ethane is actually 0.6 degrees lower in melting point, but 73 degrees higher in boiling point than methane. Propane further prolongs the anomaly of an even lower melting point and higher boiling point than ethane. The highly symmetrical molecule CH4 belongs to the tetrahedral symmetry group Td, with a symmetry number σ ) 12. The entropy of melting is equal to the ratio of the heat of melting to the melting temperature, Sm ) (Hm)/Tm ) 0.94/90.5 ) 10.4 kJ/(K mol), which benefits from a symmetry reduction by R ln(12) ) 20.7. Let us propose a hypothetical molecule that has all the properties of CH4, including the enthalpy of melting, except that its symmetry has been removed. One might expect that an isotopic exchange would be such an example; however, Ubbelohde quoted the results that CH4 has very similar melting points as CH3D and CD4. The symmetry-adjusted entropy of melting for this molecule is Sm′ ) 10.4 + 20.7 ) 31.1, and the adjusted melting point is Tm′ ) 0.94/31.1 ) 30.3 K, which is some 60.2 degrees lower than the experimental value. The value of Tm′ is close to the predicted melting point by group contribution methods. There is a lack of

experimental results on isotope effects upon melting points. The condensed phase isotope effects on many physical properties have been reviewed by Jancuso and van Hook,21 who considered that the electronic structure of atoms and molecules are not affected by the nuclear mass, and the only effect is due to a change in kinetic energy which is mass-dependent. There is no mention of melting points. Kirschenbaum22 compiled many of the properties of heavy water, including the melting point of D2O but does not mention the melting point of desymmetrized HDO. The general formula for this calculation is

Tm ′ )

Tm R ln(σ)Tm 1+ Hm

(4)

Let us designate the parameter RTm ln(σ)/(Hm) as M, which is the ratio of the rotational entropy to the entropy of melting. Thus, eq 4 can be rewritten as

Tm′ ) Tm/(1 + M)

(7)

For methane, the value of M is 1.99, and the ratio of Tm′/Tm ) 0.335 is the lowest of any substance known. At the low temperature of 90 K, the dominant thermal degree of freedom in the solid is lattice vibration, and in the liquid is rotation. This symmetry-induced reduction in rotational entropy also results in an unusual high ratio of Tm/Tb for methane within the homologous series. This ratio is reported to be around 0.65 for many substances by Bondi.13 When a group -CH2- is added, there are two effects to consider: one is the increase in cohesive forces which causes both the melting point and the boiling point to increase and the other is the symmetry effect which affects only the melting point. In the addition of a group, the methane is desymmetrized to form the less symmetrical ethane, of group D3d with σ ) 6, which has a smaller entropy of rotation. Evidently, the effects from the loss in symmetry is greater than the gain in internal cohesion, and the melting point of ethane is smaller than methane. Nevertheless, the ethane molecule still has considerable symmetry, which leads to a significant but smaller R ln(σ) and less adjustment in Tm′, and the value of M is 0.47. Propane, n-butane, and n-pentane all have C2v or C2h symmetry and σ ) 2. Figure 1 gives a plot of the boiling points, the melting points, and the symmetry adjusted Tm′ as a smoothly increasing series with the retrogression removed. Now, Tm′ increases with carbon numbers parallel to Tb. There still remains a strange alternation of melting points between even and odd carbon numbers, which has been explained as a result of crystallographic packing efficiency.23

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Figure 1. The boiling points, melting points and symmetry adjusted melting points of normal paraffins from methane to n-pentane.

The 1-chloroparaffins also follow this pattern. Figure 2 shows that 1-chloroethane (σ ) 1) has an expected higher boiling point, but a much lower melting point, than chloromethane (σ ) 3). As the number of carbons increases above 2, the boiling points increase regularly; the melting points begin to march upward (σ ) 1) despite the alternation between even and odd carbon numbers. Isomers of Pentanes There are three isomers of the paraffin pentane: the five carbon atoms of normal pentane are connected in a zigzag chain, of point group C2v, σ ) 2; isopentane has a branch and no symmetry, σ ) 1; and neopentane is tetrahedral of group Td, σ ) 12. Table 3 shows that there is very little difference in boiling points among them, but there is a great deal of difference in melting points. Neopentane has the lowest density and boiling point, but it has an anomalously high melting point. Figure 3 shows that the boiling points correlate well with the densities, but not the melting points. Another curious difference is the difference Tb - Tm which is 166 degrees for normal pentane, 187 degrees for isopentane, but 26 degrees for neopentane. Neopentane has the same high tetrahedron symmetry as methane, σ ) 12, which causes its melting point to be much elevated over its isomers. When the reduced rotational entropies have been restored by isotope exchange, the symmetry-adjusted melting points of the neopentane, Tm′, is 161 degrees lower than the experimental value, Tm. The ratio Tm′/Tm ) 0.373 comes close to that of methane as one of the lowest. Figure 3 shows that the homomorph Tm′ for the three isomers now increases with density and internal cohesive forces, parallel to Tb. The higher melting point of neopentane is often attributed to better packing in the solid,2 which does not agree with its lower density. An empirical group additivity method to account for the structure and the fusion entropies of isomers is given by Chickos et al.24 They assigned empirical additive coefficients for each primary, secondary, tertiary, and quarternary carbon atoms, with descending values from primary carbons down through quarternary carbons. The result of this is to depress the fusion

Figure 2. The boiling points and melting points of 1-chloroparaffins from chloromethane to chlorodecane.

entropies of compounds with tertiary and quarternary carbon atoms, which tend to be, but are not always, molecules with greater symmetry. If we were to examine the hexanes, we would find a much less dramatic difference among them, as the most symmetrical isomer of hexane is 2,2-dimethylbutane, which does not have a high symmetry group. On the other hand, among the nine carbon nonanes, the highest symmetry of 2,2,3,3-tetramethylpentane have Tm ) 263 K and Tb ) 413 K; but the asymmetrical isomers 2,3,5trimethylhexane has Tm ) 145 K which is 118 degrees lower and Tb ) 404 which is only 9 degrees lower. Cycloparaffins The small cycloparaffins are divided into the more stiff and symmetrical cyclopropane of group D3h, cyclobutane of group D4h, and cyclohexane of group D3d and the more flexible and less symmetrical cyclopentane, cycloheptane, and cyclooctane. Figure 4 shows that their boiling points increase smoothly with the number of carbon atoms, but the melting points exhibit wide deviations: the symmetrical ones with three, four, and six carbons have much higher melting points than the asymmetrical ones with five, seven, and eight carbons, which agrees with their symmetry number σ. Cyclohexane has a symmetry number of σ ) 6 at -100 °C, and σ ) 12 at room temperature and above. It has a boiling point of 350 K and a melting point of 279.7 K. In comparison, methylcyclopentane is an isomer with no rotational symmetry and a melting point of 130.6 K, some 148 degrees lower than cyclohexane. When a methyl group is added to cyclohexane, the result is the less symmetrical methylcyclohexane with a drop in its melting point by 133 degrees to 146.5 K. In fact, Figure 5 shows that when larger alkyl groups are added, up to decylclohexane, the boiling points keep rising to 580 K and the melting point also increases, but does not catch up with cyclohexane.

Ind. Eng. Chem. Res., Vol. 38, No. 12, 1999 5023 Table 3. Phase Transition Properties of Pentanes molecule

Tm (K)

Tb (K)

density (g/mL)

Hm (kJ mol)

σ

Tm′ (K)

M

Tm/Tb

normal pentane isopentane neopentane

143.4 113.3 256.6

309.2 300.9 282.7

0.621 0.616 0.586

8.39 5.15 3.15

2 1 12

130.5 113.3 95.6

0.10 0 1.68

0.464 0.377 0.908

Figure 3. The three isomers of pentane, their boiling points, melting points and symmetry adjusted melting points vs density.

Figure 5. The monoalkyl-substituted cyclohexanes, their boiling points and melting points.

that is 32 degrees higher than cyclopropane, but the melting point is 50 degrees lower.

Figure 4. The cycloparaffins from 3 to 10 carbons, their melting points, boiling points and entropies of rotation.

Cyclobutane has a highly symmetrical structure and σ ) 8, but there is only a small set of compounds that have their properties measured and published in the literature. It has a boiling point of 285.7 K and a melting point of 182.5 K. When a methyl group is substituted, the symmetry is broken, the boiling point rises by 25 degrees but the melting point drops by 71 degrees; and when an ethyl group is substituted instead, the boiling point rises further to 343.9 K but the melting point is only 130.2 K. Methylcyclopropane has a boiling point

Monosubstituted Benzenes Benzene is a planar hexagon, with a symmetry group of D6h and the very high symmetry number σ ) 12. It has six equivalent hydrogen atoms that can be exchanged with a group such as CH3- or Cl-. When one substitution group is added, there is a loss of symmetry as the result has symmetry C2v and σ ) 2. When there are two chlorine atoms added, the two Cl atoms can be in the ortho- or 1,2-position which is 60° apart; in the meta- or 1,3-position which is 120° apart; or in the paraor 1,4-position which is 180° apart, with a higher symmetry than the other two isomers. When there are three chlorine atoms added, they can be in the 1,2,3 or adjacent position, or in the 1,2,4 or iso position which is the least symmetrical, or in the 1,3,5 position which is the most symmetrical. When there are four chlorine atoms added, they can be in the 1,2,3,4 position, which is the counterpart to the ortho position in disubstitution; they can be in the 1,2,3,5 position which is the counterpart to the meta position; and they can be in the 1,2,4,5 position which is the counterpart to the para position. Finally, pentachlorobenzene has the same symmetry as monochlorobenzene and hexachlorobenzene has the same symmetry as benzene. A single-substitution group to benzene leads to an increase in boiling point and often to a decrease in melting point. Table 4 shows that the melting point of

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Table 4. Phase Transition Properties of Monosubstituted Benzenes molecule

Tm (K)

Tb (K)

Hm (kJ/mol)

σ

Tm′ (K)

M

Tm/Tb

benzene toluene ethylbenzene n-propylbenzene n-butylbenzene chlorobenzene bromobenzene nitrobenzene pentamethylbenzene hexamethylbenzene pentachlorobenzene hexachlorobenzene

278.6 178.2 178.1 173.6 185.1 227.8 242.5 278.9 326.1 438.7 357.1 505.0

353.2 383.7 409.4 432.4 456.5 383.7 429.1 483.9 504.1 538.1 504.1 538.1

9.95 6.85 9.18 9.27 11.22

12 2 2 2 2 2 2

176.5 155.0 160.2 156.7 169.0

0.58 0.15 0.11 0.11 0.09

0.789 0.562 0.435 0.401 0.405 0.562 0.565

12.30 20.60

2 12 2 12

282.9 304.7

0.15 0.44

358.4

0.41

0.646 0.815 0.652 0.844

25.5

the internal cohesive forces. It is very pleasing to observe in Table 4 that in the addition of a methyl group to pentamethylbenzene to form hexamethylbenzene, the melting point goes up by 102 degrees, together with a 34 degree increase in the boiling point. This is an example of the reverse process in resymmetrization, where the gain in cohesive forces works together with symmetry, and result in a large increase in melting point. When we add a chlorine to pentachlorobenzene in forming the highly symmetrical hexachlorobenzene, both these effects work together again so that the melting point goes up by 148 degrees as the boiling point goes up by 34 degrees. The ratio of Tm/Tb is also much higher in these high-symmetry molecules. Polychlorinated Benzenes

Figure 6. The monoalkyl-substituted benzenes, their boiling points and melting points.

toluene is 100 degrees lower than benzene, but the boiling point is 30 degrees higher. Once again, the loss in symmetry is greater than the gain in internal cohesion. The effect is quite dramatic, but less dramatic in comparison with the case of methane, as Tm′/Tm ) 0.634. This is probably due to the fact that the heat of fusion of benzene is very high and the value of M is only 0.58. Another manifestation of this effect is that the ratio Tm/Tb is much higher in benzene than in toluene. Figure 6 shows the effects of the addition of methyl, ethyl, n-propyl up to 14 carbon alkyl groups. The boiling points go up with carbon numbers smoothly, but the melting points make a severe drop with the methyl group and remained lower than the parent benzene molecule until the tetradecane with 14 carbon atoms. This is a similar effect to that shown in Figure 5 for monoalkylcyclohexanes. The melting point of chlorobenzene is 51 degrees lower than that of benzene, but the boiling point is 30 degrees higher. Bromobenzene also has a lower melting point than benzene, but the contributions to internal cohesive energy is higher than chlorobenzene’s. However, nitrobenzene has a higher melting point than benzene, as its contribution to internal cohesive forces are sufficient to overpower the effects of symmetry entropy reduction. When more and more methyl groups are added, the melting point eventually increases due to increases in

Among the dichlorobenzenes, there is only a difference of 7 degrees in boiling points between the highest boiling ortho- or 1,2-form and the lowest boiling meta- or 1,3form, which is shown in Table 5. The para-form has the lowest density, but the highest melting point by 69 degrees. It also has a higher symmetry group D2h than the ortho- and the meta-forms, both in the C2v symmetry group. Among the four isomers of tetrachlorobenzene, the isomer 1,2,4,5-tetrachloro is the counterpart of the 1,2-dichloro form, with the lowest density but the highest melting point. Among the trichlorobenzenes, the 1,2,4-form is totally without any symmetry, in group CS, and the 1,3,5-form has the highest symmetry, in group D3h. When all the chlorinated benzenes are plotted in Figure 7, there is a smooth increase of the boiling points with the number of chlorine atoms, with minor differences within each set of the same number of chlorine atoms, which can be attributed to differences in packing efficiencies and densities. There is a wild gyration in the melting points, which can be interpreted as the sum of an orderly increase parallel to the boiling points, and a symmetry-adjusted elevation. The largest elevations are with the molecules with symmetry number σ ) 4, 6, and 12, which leads to rotation entropies of R ln(σ) ) 11.5, 14.9, and 20.7. The computation of adjustments requires the heats of melting, but it appears that the excess melting points correlate very well with the entropies. Substituted Naphthalenes Naphthalene is two benzene rings fused together and belongs to point group D2h with σ ) 4. It has eight hydrogen atoms that can be substituted, and they are numbered clockwise from the top as 1, 2, 3, ..., 8. The positions 1, 4, 5, and 8 are in the inner position next to

Ind. Eng. Chem. Res., Vol. 38, No. 12, 1999 5025 Table 5. Phase Transition Properties of Polychlorinated Benzenes molecule

Tm (K)

Tb (K)

Hm (kJ/mol)

σ

Tm′ (K)

M

Tm/Tb

1,2-dichlorobenzene 1,3-dichlorobenzene 1,4-dichlorobenzene 1,2,3-trichlorobenzene 1,2,4-trichlorobenzene 1,3,5-trichlorobenzene 1,2,3,4-tetrachlorobenzene 1,2,3,5-tetrachlorobenzene 1,2,4,5-tetrachlorobenzene

256.4 248.3 325.8 326.6 290.1 336.6 320.6 327.6 412.6

453.11 446.1 447.1 491.6 486.6 481.1 527.1 519.1 517.6

12.59 12.66 18.20

2 2 4 2 1 6 2 2 4

229.8 223.6 271.3

0.11 0.11 0.20

0.565 0.556 0.729 0.664 0.596 0.699 0.608 0.631 0.797

15.56

290.1

Figure 7. The polychlorinated benzenes, their boiling points, melting points, and rotational entropies.

the joint between the two rings; the positions 2, 3, 6, and 7 are in the outer position. These two sets of substitution positions are not equivalent, although they both have σ ) 1. When there are two substituents, they can be arranged in six positions of homotopic (both inner or both outer positions) symmetry of C2v with σ ) 2: the 1,4 and 1,5 and the 1,8 arrangements have the two methyl groups in balance; and so are the 2,3 and 2,6 and the 2,7 arrangements. The two groups can also be arranged in four heterotopic (one inner and one outer position) asymmetrical positions 1,2 and 1,3 and 1,6 and 1,7 with σ ) 1. Naphthalene has a melting point of 353 K, and a boiling point of 491 K, with σ ) 4. A single substitution of methyl in the 1-position will plunge the melting point to 243 K, but a substitution in the 2-position has a much higher melting point of 307 K. A single chloro substitution to the 1-position has a melting point of 271 K. Figure 8 shows that the boiling points of all 10 isomers of dimethyl-substituted naphthalenes are almost identical so that the arrangement of the substitution positions are irrelevant. But when we examine the melting points of the methylsubstituted isomers, we found very large differences. The four asymmetrical 1,2 and 1,3 and 1,6 and 1,7 isomers have melting points less than 275 K. Five of the symmetrical arrangements have high melting points that are some 120 degrees higher than the four asymmetrical arrangements. The 1,4-arrangement has the same symmetry, but has both methyl groups on the 1-position, which has been demonstrated earlier

Figure 8. The disubstituted naphthalenes and their boiling points and melting points. The substitution groups are methyl, amino, chloro, bromo, and hydroxy.

to lead to a lower melting point. The disparity between the symmetrical arrangements and the asymmetrical arrangements are also observed for the amino, bromo, chloro, and hydroxy substitutions. Yalkowsky5 pointed out that anthracene and phenanthrene are both formed by the fusion of three benzene rings, with the formula of C14H10; both boil at 613 K, but the straight and more symmetrical anthracene has σ ) 4 and a melting point of 489 K, but the bent and less symmetrical phenanthrene has σ ) 2 and a melting point of 374 K. Aldehydes, Ketones, Dibasic Acids, and Others The aldehydes and ketones have the structure of

We have an aldehyde when R1 is H and R2 is an alkyl group, and we have a ketone when both R1 and R2 are alkyl groups. Table 6 shows that there is a steady increase of boiling point from formaldehyde to acetaldehyde to dimethyl ketone (or acetone), but the sym-

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Ind. Eng. Chem. Res., Vol. 38, No. 12, 1999

Table 6. Phase Transition Properties of Aldehydes and Ketones name

R1

R2

Tm (K)

Tb (K)

Tm/Tb

formaldehyde acetaldehyde dimethylketone diethylketone methylpropylketone butyaldehyde

H H CH3 C2H5 CH3 H

H CH3 CH3 C2H5 C3H7 C4H9

181.1 150.1 178.3 234.1 196.2 181.1

254 293.2 329.1 375.0 375.3 376.8

0.713 0.512 0.542 0.624 0.523 0.481

temperature, which results in

ln(a) ) (Sm/R) ln(T/Tm)

(8)

so that for a molecule with a high melting point, the solubility at T is reduced. Consider adamantane, which has the same tetrahedron group Td as methane, a molecular weight of 136.24, and a melting point of 541 K. The addition of a bromine atom to make bromoadamantane results in a much lower melting point of 390 K, which is 151 degrees below that of adamantane.15 The molecule SF6 has a melting point of 222.45 K and a heat of fusion of 5.024 kJ/mol so that it has an entropy of fusion of 22.585. It belongs to the octahedron group Oh with σ ) 24; thus, it has Sm ) 26.4, and it would have an adjusted melting entropy of Sm′ ) 22.6 + 26.4 ) 49.0. We predict that a isotope substituted version with the symmetry removed would have a melting point of Tm′ ) 102.5 K, which is 120 degrees lower than the parent molecule; the value of M ) 1.17 and the ratio Tm′/Tm ) 0.461 are both very unusual. The molecule fullerene C60 belongs to the icosahedron group Ih and is an even better test case for the prediction that high-molecular symmetry causes elevated melting points. Isotope exchange produces much fewer alterations in internal cohesive forces in comparison with more powerful substitution groups so that the effects of symmetry downgrading can be measured unaltered by changes in internal cohesive forces. Water has a melting point of 273 K and Hm ) 6.009, and D2O has a melting point of 276.8 K. We predict that DHO should have a very depressed melting point of 216.3 K or -57 °C. However, it remains to be seen whether water, with its volume contraction upon melting and its hydrogen bond, satisfies the assumptions in deriving eqs 4. Linear and Monatomic Molecules

Figure 9. The carboxylic dibasic acids from two to 10 carbons, their melting points and solubility in water as g/100 g at 25 degrees C.

metrical formaldehyde and dimethyl ketone have melting points 30 degrees higher than the asymmetrical acetaldehyde. When we compare the isomers diethyl ketone, methylpropyl ketone, and valeraldehyde, we find that their boiling points are within 2 degrees of each other, but there is a big difference in melting points from the symmetrical diethyl ketone to the other two asymmetrical isomers. The homologous series of dibasic carboxylic acids range from oxalic acid COOH-COOH, to malonic acid COOH-CH2-COOH, to succinic acid COOH-(CH2)2COOH, up to the 10 carbon sebacic acid and have a curious seesaw pattern of melting points, which is also counter-matched by their solubility in water shown in Figure 9. The stable conformation is the symmetrical trans form where all the C and O atoms are on the same plane, with the two COOH groups on opposite sides of the central paraffinic chain and a 2-fold rotational symmetry of C2v. But when the number of carbons is odd, the stable conformation is the asymmetrical trans form with σ ) 1. Thus, the even number of carbon dibasic acids have much higher melting points than the odd number acids. The cause for the lower solubility for the higher melting points was derived by Hildebrand,25 by taking the partial derivative of solid activity by

A special problem is posed by linear molecules such as O2 and CO2 that have very little moments of inertia in the axial direction, and only 2 degrees of rotational freedom, and monatomic molecules such as neon that have no rotational freedom. The noble gases belong to the continuous rotational group K of the highest symmetry.26 The noble gases have very small values for Tb - Tm, and Tm/Tb > 0.9 from neon to xenon, with the exception of helium with its special quantum effects. The linear molecules are divided into the conical symmetry of C∞v with examples such as CO and HCl, with infinite group order but σ ) 1, and the cylindrical symmetry of D∞h with examples such as O2 and acetylene HCtCH in possession of a C2 axis at the center, with infinite group order but σ ) 2. Figure 10 shows that when a methyl group is added to acetylene or ethyne, the result is propyne HCtCCH3 with a higher boiling point but a lower melting point. Actually, acetylene sublimates at 191.3 K and the triple point is at 189.7 K. When an ethyl group is added, the resulting 1-butyne HCtCC2H5 has an even higher boiling point and lower melting point. However, if two methyl groups are added to both sides of ethyne, the result is the symmetric 2-butyne CH2CtCCH3 with a higher boiling point and a much higher melting point. Conclusion Molecular symmetry reduces the rotational entropy of a molecule and elevates the melting point, but it has

Ind. Eng. Chem. Res., Vol. 38, No. 12, 1999 5027 Oh ) octahedral or cubic point group Tb ) boiling temperature, K Td ) tetrahedral point group Tm ) melting temperature, K Tm′ ) adjusted melting temperature, Tm/(1 + M) σ ) symmetry number of point group

Literature Cited

Figure 10. The linear acetylenes: ethyne HCtCH, propyne HCtCCH3, 1-butyne HCtC2H5, 2-butyne CH3CtCCH3. The triple point for ethyne is plotted instead of the melting point.

no effect on the boiling point. Substitutions that increase cohesive forces normally lead to raised boiling points as well as raised melting points. However, substitutions that downgrade symmetry would lead to a reduction of the melting point which are often sufficient to overcome the effects of an increase in cohesion forces but has no effect on the boiling point. It has been demonstrated by numerous examples that symmetry plays a large role in determining melting points. For molecules that have no solid rotations but have free liquid rotations, a quantitative evaluation of the elevation in melting point can be made, based on the symmetry number σ and the heat or the entropy of melting. There is the prospect that when molecular symmetry is taken into consideration, the melting point can be correlated with cohesion forces, on the same basis as densities, boiling points, heats of vaporization, refractive indexes, and surface tensions. Nomenclature C1 ) point group with no symmetry Cn ) n-fold rotation Cnh ) cyclic point group, with one horizontal reflection Cnv ) cyclic point group, with n vertical reflections Dnd ) n-fold dihedral point group, with n C2 perpendicular axes Dnh ) n-fold dihedral point group, with horizontal reflection E ) identity element in point group h ) point group order Hm ) heat of melting, kJ/mol Ih ) icosahedral point group M ) ratio of rotational entropy to melting entropy, RTm ln(σ)/Hm Sm ) entropy of melting, Hm/Tm, J/(K mol) Sm′ ) adjusted entropy of melting, Sm + R ln(σ)

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Received for review August 5, 1999 Revised manuscript received September 27, 1999 Accepted September 29, 1999 IE990588M