Order and Disorder and Entropies of Fusion D. F. R. Gilson McGill University, 801 Sherbrooke St.,W. Montreal, Quebec H3A 2K6 That a wide variety of different approaches can be used for the interpretation and teaching of entropy has been emphasized by recent papers in this Journal (Id).Many textbooks on physical chemistry use the idealized and abstract concepts of the Carnot cycle and heat engine as the starting point and, for many students, the wnfusion that can occur leads to an additional thermodynamic principlethat talking about entropy increases it. A simple alternative is to interpret the actual entropy changes of wehecognized, reversible physical processes for which the entropy can be considered as a disordering process. For example, the entropy of vaporization of a liquid is discussed in almost allintroductory texts on physical chemistry. The (apparent) constancy of the entropy of vaporization, a t about 88 J K-' mol-', (Trouton's rule, 1884) is due to the larqe contribution from the expansion of the small volume of liquid to the large vapor volume. The corresponding entropy change for the melting of a solid is seldom, if ever, discussed in any detail. When viewed in terms of the different contributions to the entropy of melting of organic molecular crystals (61, it can serve as a worthy teaching device. The volume change on fusion is very much smaller than the expansion of the liquid on vaporization. As a result, other contributions assume greater importance. The total entropy can be separated into contributions arising from translational (positional), rotational (orientational), and conformational disorder, plus the entropy due to any volume change, eq 1. A % . ,
= %.ms
+ Grot + %mf+
G.01
Consider the data in the table, which were selected for a series of molecules containing 10 carbon atoms, plus xenon, which has a similar molar mass. The two aromatic molecules, naphthalene and durene, and tmns-decalin have similar entropies of fusion, but the two decalin isomers are quite different even though their melting temperatures are similar. Adamantane, a rigid cage hydrocarbon, has an entropy of fusion similar to xenon, although there is a difference of 380 K in their melting temperatures. The entropy of fusion of the flexible alkane n-decane is very large, nearly twice that of tmns-decalin. Consider these results with reference to eq 1.The entropy of fusion of the rare gases is reduced by a n amount equivalent to the degrees of freedom for rotation, since there is no rotation for a spherical particle. The low entropy of fusion for adamantane suggests that the fusion process in this "spherically" shaped molecule resembles the rare gas case and rotation, or orientational disorder, occurs prior to melting. nuns-decalin is a rigid molecule, hut the cis isomer is capable of chair-chair interconversion of the cyclohexyl rings and, importantly, can undergo this process in the solid state. The flexibility of n-decane allows additional disordering in the liquid phase compared with the all trans structure of the solid. The remaining three compounds have no special property of disorder, either prior to melting or in the liquid, and, therefore, an entropy of fusion of 50-60 J K-' mol-' would be the normal value for organic molecular crystals. A much larger compilation of the entropies of fusion can be derived from t h e data given in the Handbook of
Melting Properties of Clo Hydrocarbons -
MP
MW
K
Napthaiene Durene cis-Decalin trans-Decalin ~Decane Adamantane Xenon
353 352 230 242 243 54Ia 161
AHtmion
KJ ~ O I - ' 128.2 134.2 128.2 128.2 142.3 136.2 131.3
18.8 21.0 9.5 14.4 28.8 11.3 2.3
AStusion
JK'
~ O I "
53.2 59.7 41.2 59.4 118.5 20.9 14.3
'In a sealed tube
Chemistry and Physics (71,where the melting enthalpies are listed for 358 compounds. These values have been converted to entropy and plotted in Figure 1. Clearly, the entropies of fusion show a central distribution around an average value of about 50 J K-' mol-', with a smaller cluster (26 compounds) below 20 J K-' mol-', and an additional group with values above 91 J K-' mol-' (29 compounds). The spread of the central distribution can be attributed to different causes. The contribution from the volume change is the major variable and an approximate value is given by = CdP)A V
(2)
where a is the coefficient of thermal expansion, P is the compressibility, and AV the volume change. These quantities are not known for many compounds but approximate ranges are
Thus, ASvo~could vary from less than 10 to more than 50 J K-' mol-I. A reduction in t h e entropy of fusion can result if molecular association occurs in the liquid phase. Thus, for molecules which have hydroxyl, amino, or carboxylic acid groups, changes in hydrogen bonding association occur at t h e transition; methanol, ethanol, acetic acid, and methylamine have entropies of fusion of 18.1, 31.6, 39.8, and 34.1 J K-' mol", respectively, Now consider the two extreme classes of compounds, those with very low and very high entropies of fusion. The former are, typically, rigid compact molecules approximately spherical in shape and were termed "molecules globulaires" by Timmermans, who suggested that molecules with entropies of fusion below 2.5R possessed disorder in the solid state arising from a randomization of the orientations of the molecule in the crystal (8).In recognition of the property of flow under stress they also have been called "plastic crystals". The melting temperatures of such orientationally disordered solids tend to be much higher than is usual for molecules of similar molar Volume 69 Number I
January 1992
23
e5
r--fusion
Entropy change
Entropy of fusion
' mo -', calc~laledfromoata g ven n reference (71 The wlorn of eacn oar n the h~stogram1s R 2 The bar a1 The r gnl ncl~ocsall compounds waln entropes of fJson greater than 85 J K' mol-'. F gdre 1 Ensoptes of t-s on J K
mass. The hiah - temperature phases have a high vapor pressure and may even sublime before mehing. The crystal structure prior to melting (the h ~ g htemprrature phase! is c o m m o n l face-centeredcubic, ~ but occasionally bodycentered or hexaaonal-close-packed. These structures are of high symmetrfand molecdes of lower symmetry cannot be arranaed - in the crystal lattice without disorder. A phase transition to a structure with lower symmetry occurs i n t h e solid, with a n associated enthalpy and entropy of transition. A comprehensive compilation of the properties of t h e s e m a t e r i a l s c a n h e found i n t h e monograph by Parsonage and Staveley (9).A histogram (Fig. 2) of the entropies of fusion of all compounds listed in reference (9) with an entropy change of less than 21 J K-' mol-', together with the sum of the entropies of fusion and transition shows that the latter now resembles the plot for the regular compounds, given in Figure 1, the average of the sum of the entropies being 45 J K-' for 52 compounds. Note that contributions from the heat capacity difference are neglected. Guthrie and McCullough (10) proposed that the entropy should reflect the number of distinguishable ways in which the molecule can be arranged in the unit cell, thus
where Nl and Nz are the number of orientations in the high and low t e m ~ e r a t u r e~ h a s e .s .respectivelv. This equation . appeared to give the m~rrcctentropy of transition in inoreanic salts 111I where the hiah temperature phase was not ;isordered but failed for many organic mol&ular crystals in which the high temperature phases are completely disordered since unreasonably high values for N1 a r e obtained. Alternative equations have been proposed by Clark and co-workers (12, 13) in which a n "excess entropy" is added to the Guthrie-McCullough contribution. The excess entropy has been correlated with the temperature range of the disordered phase and with the molecular dimensions, but, perhaps, should be interpreted in terms of the volume change contribution. There is still no quan-
24
Journal of Chemical Education
Figure 2. Entropies of fusion and the sum of the entropies of fusion and transition, in J K' mol-', for orientationally disordered solids (9). titative theory that can account for the entropies of transition in all cases. Other molecules with exceptionally low entropies of fusion are the liquid crystalline materials that are not included in the data used in Fimres 1and 2. The simplest type of thermotropic liquid irystal, the nematic phase, transforms from a crystalline solid to a viscous liquid and then, a t a higher temperature (the clear point), to a normal isotropic liquid. I n p-azoxyanisole (PAA) a n d p methoxybenzylidene-n-butylaniline(MBBA), two of the better known nematic materials, this latter transition has a n entropy change ofless than 1.5 J K-' mol-'. The nematic phase is characterized by translational motion in any direction, but with restricted rotation about axes perpendicular to the long axis of the molecule. Such materials thus retain "order" in the liquid crystal phase, and the "melting" to the isotropic liquid involves a gain in entropy due to the loss of this last remailiing degree of freedom. The entropy of fusion is, accordingly, very small, since the earlier transition involves the major disordering proeesses. Now consider the molecules with large entropies of fusion. All the molecules in this group are long chain alkanes and their derivatives. The thermal behavior of the solid n-alkanes was studied in detail by Broadhurst (14). The entropies of fusion and transition are plotted versus the number of carbon atoms in the chain (up to C ~ Oin) Figure 3. An odd-even alternation is immediately obvious for chain lengths between Cg and C19, but the differences disappear for chain lengths between 20 and 30 carbon atoms. This behavior arises from the different crystal structures of the alkanes. Four solid forms are observed, denendine ., on the chain lenpth .. and temperature, with monoclinic, triclinir, orthorhombic, or hexagonal crystal structures. In thc Inclinicand monoclinic phases the cham axis makes a n angle to the plane of the end methyl groups, b u t in the orthorhombic and hexagonal structures the chains a r e perpendicular. I n the latter structure, the chains rotate about their long axes, thus acquiring a degree of freedom in the solid and reducing the entropy of fusion. Alkanes up to Cg and the even members of the series up to Czoall melt from the triclinic structure. The
-
O
0
V
'
5
10
15
20
25
30
5
0
Chain lpngth
10
15
20
25
30
31
Chain length
Figure3. Entropies of fusion and transition for +alkanes as a function of the number of carbon atoms in the chain.
Figure 4. Sum of the entropies of fusion and transition for the m alkanes.
odd-members of the series between Cq and Cxl ..all have transitions from orthorhombic to hexagonal structures that accounts for their reduced entrow of fusion, but when the chain is between 20 and 30 carbon atoms then the even-membered alkanes also have transitions, in this case from triclinic or monoclinic to hexagonal. To compare the difference between the ordered stable low temperature structures and the disordered molten state, the entropies of all transitions and fusion are added together and plotted versus chain length, Figure 4. The odd-even alternation that remains is due to the difference in packing in the orthorhombic, triclinic, and monoclinic structures, but a steady, almost linear increase in the total entropy change occurs. In flexible molecules. additional demees of freedom are accessible in the liquidphase as the &-trans structure of the ordered solid acauires various eauche conformations that depend upon t
