J. Phys. Chem. B 2008, 112, 2063-2069
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Polymorphism of Even-Numbered Carbon Atom n-Alkanes Revisited through Topological P-T Diagrams Philippe Espeau* and Rene´ Ce´ olin Laboratoire de Chimie Physique, EA 4066, Faculte´ de Pharmacie, 4 AVenue de l’ObserVatoire, 75006 Paris, France ReceiVed: May 15, 2007; In Final Form: October 2, 2007
The phase relationships involving the metastable orthorhombic (RI) phase and the stable triclinic (T) phase were established for even-numbered carbon atom n-alkanes in the range of 8-20 carbon atoms. It is shown that the RI phase behaves monotropically whatever the pressure and temperature (i.e., the RI phase exhibits no stable pressure-temperature (P-T) region). Then, the incidence of the overall monotropic behavior of the RI phase on the construction of the temperature-composition (T-x) phase diagrams involving at least one even member was examined through four temperature-molar fraction (T-x) phase diagrams. Discussion on the location of the RI-T-vapor (V) triple points in the P-T diagrams of this series of n-alkanes was determined and extended to higher even members.
1. Introduction Crystalline polymorphism is an old enigmatic phenomenon of science (still qualified as “the nemesis of crystal design” by Desiraju1 after it was recognized first by Mitscherlich,2 who clearly demonstrated that elemental sulfur crystallizes in two systems, thus ending the controversy3 that arose at the end of the 18th century between Hau¨y and Klaproth about the physicochemical differences in the two kinds of natural calcium carbonate (calcite and aragonite). However, the question as to how polymorphs crystallize remained open (for instance, Bergman thought that it depended on external conditions4), and classical thermodynamics was still lacking before the wellestablished science Gibbs founded many years later.5 Later, Lehmann6,7 showed that two kinds of behaviors can be observed upon heating and then cooling crystalline solids capable of existence in more than one crystal structure. Some solids, which transformed upon heating into other crystalline modifications, reverted back to their initial forms upon cooling, while others did not, and Lehmann coined the terms “enantiotropy” and “monotropy” to account for such a difference in the thermal behavior of polymorphs. In addition, pressure was recognized as a parameter capable of inducing similar changes, and experimental investigations into so-called P-T state diagrams were begun by Mallard and Le Chatelier.8 However, experimental data had to fullfill theoretical requirements of classical thermodynamics, and Bakhuis-Roozeboom9 was the first author who compared experimental data with theoretical topological drawings. In particular, regarding dimorphism, this author drew four cases of topological P-T diagrams that still addressed the question as to the temperature and pressure dependences of any crystal structure modification that was somewhat not as simple as the two alternatives inferred from Lehmann’s work. Unfortunately, the conclusions of BakhuisRoozeboom were not fully understood as efficient tools based upon a related geometry, whose purpose was not to give precise P-T values but to provide qualitative information as to the * Corresponding author. E-mail:
[email protected]; tel./ fax: (33) 1-53-73-96-76.
relative positions of phase equilibria with respect to each other,10 inferred from Gibbs’ classical thermodynamics, which takes into account not only heat but also work (through volume changes), as underlined by Maxwell.11 In addition, the fourth case of a P-T diagram discussed by Bakhuis-Roozeboom was virtually never cited in phase diagram textbooks, perhaps because no example was given by Bakhuis-Roozeboom himself. From such a drawing, it can be deduced that one polymorph (over two) exhibits no stable phase region whatever the pressure and temperature, and this is different from cases that assign stable phase regions at higher pressures to polymorphs that are not stable at ordinary pressure. It is worth mentioning that such a difference was previously questioned by Bridgman,12 who did not address the question of polymorphs that become more and more metastable as the pressure increases. With such a historical perspective, the work presented in the following discussion focuses on the dimorphism of some n-alkanes, as part of a more general interest in the topological depiction of the P-T (state) diagrams accounting for polymorphism13-22 as well as an example of polymorphs that should not exhibit stable phase regions in their P-T diagram. The polymorphism of pure n-alkanes through thermal analyses and X-ray diffraction has been largely reported in the literature.23-37 In the range of nC ) 8-21 carbon atoms (nC is the number of C atoms), the even-numbered carbon atom crystalline solids exhibit only a triclinic crystal structure (T) before melting, and the odd members transform from a primitive to a face-centered orthorhombic crystal structure (RI) just below the melting point. Investigations into the temperature-composition (T-x) binary phase diagrams reveal that the RI phase, stable just before melting in binary systems involving at least one evennumbered carbon atom n-alkane, is also stabilized when two even members are involved. When the stability domain of the RI phase in the T-x diagram is wide enough, the RI-L equilibrium can be connected to the metastable RI phase of the pure even members whose thermodynamic data are inferred from the data of the pure odd members. However, the RI-T transition point is not experimentally observed for n-alkanes with nC e 21, even though for C22H46 and C24H50, this transition
10.1021/jp0737348 CCC: $40.75 © 2008 American Chemical Society Published on Web 01/25/2008
2064 J. Phys. Chem. B, Vol. 112, No. 7, 2008
Espeau and Ce´olin
is observed and determined.24,26,28,29 This entails that the RIT-vapor (V) triple point should be located at a temperature higher than that of the T-liquid (L)-V triple point in the pressure-temperature state diagram of each n-alkane with nC e 21. Referring to the work of Bakhuis-Roozeboom9 on the P-T state diagrams for dimorphism, two possible cases describe this kind of behavior of phases in equilibrium with their vapor (i.e., solid-vapor equilibrium curves): either a case of overall monotropy (i.e., no domain of stability for the metastable phase) or a case of monotropic behavior at low pressure that turns into enantiotropic behavior at high pressure.9,13 In this study, the topological P-T diagrams for two n-alkanes, C16H34 and C18H38, were constructed to choose between the two Bakhuis-Roozeboom cases. Then, new insight will be shared concerning already investigated binary phase diagrams from which the dependence of the relative stabilities of the RI and T phases on the number of C atoms (nC) will be established. 2. Topological P-T Diagrams for C16H34 and C18H38 2.1. Construction Method of the P-T Diagrams. General rules for constructing topological P-T diagrams have been previously described13-15 and applied to some cases of polymorphism.13-22 Recently, they have been used to question the hypothetical dimorphism of orpiment20 and to predict the topological P-T diagram accounting for the dimorphism of paracetamol.13 It is worth noting that, for the latter, the topological method was proven to be successful experimentally.38 With these topological P-T diagrams, the stability hierarchy of polymorphs is simply inferred from the relative positions of their sublimation curves. This makes the application of Ostwald’s rule, according to which the more stable phase should exhibit the lower sublimation curve, easy. Following Oonk,39 any two-phase equilibrium curve in a P-T diagram can be topologically drawn as a straight line, that is, the simplest geometrical translation of the Clapeyron equation: dP/dT ) ∆H/T∆V. Thus, for each two-phase equilibrium, the transition enthalpy and the specific volumes of the liquid and of the solid at the transition point have to be determined. Figure 1a,b shows that the construction of the P-T diagram for dimorphism leads to the appearance of four triple points.40 Three of them (points 1-3 in Figure 1a,b) involve the vapor phase. Points 1 and 2 are related to the melting of the T and RI phases (i.e., T-L-V and RI-L-V) and are located on the same liquid-vapor equilibrium curve (segment a-a′ in Figure 1a,b). The third triple point (point 3) corresponds to the transition T-RI under saturating vapor pressure (T-RI-V). This is the crossing point of the sublimation curves of the T (T-V) and RI (RI-V) phases (segments b-b′ and d-d′ in Figure 1a,b). The fourth triple point (point 4) concerns the equilibrium between T, RI, and the liquid (T-RI-L). Regarding the two possible cases that describe the monotropic behavior of any compound in equilibrium with its vapor (Figure 1a,b), the main difference in the two diagrams concerns the location of the third triple point T-RI-L (point 4). This point is located at high pressure in Figure 1a and is stable since it is located on the stable part of the T-L equilibrium. Then, the RI phase presents a stability domain at high pressure and high temperature. On the contrary, in Figure 1b, the T-RI-L triple point is located at low pressure (lower than those of the melting triple points of the T and RI phases), and it is metastable since no stable two-equilibrium curve intersects it. Thus, no stability domain is encountered for the RI phase. From a construction
Figure 1. Topological P-T diagrams describing the monotropy in the case of dimorphism (triclinic T phase and face-centered orthorhombic RI phase) encountered for the even-numbered carbon atom n-alkane. (a) Monotropic behavior at low pressure and (b) overall monotropy. Two-phase equilibrium curves: a-a′ ) L-V, b-b′ ) RIV, c-c′ ) RI-L, d-d′ ) T-V, e-e′ ) T-L, f-f′ ) RI-T. Solid lines are stable parts, dashed lines are metastable parts, and dotted lines are supermetastable parts of the two-phase equilibrium curves. Triple points: point 1 ) T-L-V, point 2 ) RI-L-V, point 3 ) RI-T-V, and point 4 ) RI-T-L. Black points are stable triple points, and white points are metastable triple points.
point of view, this third triple point is the result of the crossing of the solid-liquid equilibrium curves T-L and RI-L (segments c-c′ and e-e′ in Figure 1a,b): this will be found at high pressure (Figure 1a) if the melting curves converge or at low pressure (Figure 1b) if they diverge, as the pressure increases. To answer the question as to the location of the T-RI-L triple point, the melting triple points and, consequently, the melting equilibrium curves have to be located. 2.2. Application to C16H34 and C18H38. The temperature and enthalpy values for the fusion of the stable phase for the n-alkanes25 (T for the even members and RI for the odd members), from nC ) 8 to nC ) 21, are plotted, respectively, in Figure 2a,b. These values are gathered in Table 1. The temperature and enthalpy values for the fusion of the metastable RI phase of the even members have been obtained using the same polynomials as those fitting the temperatures and enthalpies of the stable fusion of the RI phase for the odd members (Figure 2a,b). The inferred values are also reported in Table 1. Vapor pressure values for nC ) 16 and nC ) 18 at the T-L-V triple point were obtained from the reported pressure measurements.41,42 Then, the following values can be obtained: for C16H34: P ) 0.083 Pa at Tfus (T) ) 290.75 K and P ) 0.070 Pa at Tfus (RI) ) 289.21 K and for C18H38: P ) 0.025 Pa at Tfus (T) ) 301.05 K and P ) 0.021 Pa at Tfus (RI) ) 299.47 K. Both triple points (points 1 and 2) are located in the P-T diagrams presented in Figures 3a and 4, respectively, for C16H34 and C18H38. Each two-phase solid-liquid equilibrium curve (assumed to be linear) goes through the corresponding melting triple point. To calculate the slope dP/dT of the two-phase equilibrium curve from the Clausius-Clapeyron equation, the specific volume change at the triple point has to be known. For the stable melting
Polymorphism of Even-Numbered C Atom n-Alkanes
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Figure 2. Melting data of n-alkanes as a function of the number nC of carbon atoms for nC ) 8-21. 9: Even members and 0: odd members. (a) Melting temperatures (in K). Third-order polynomial curve fits: even members: T (K) ) 28.795 + 33.902nC - 1.50644nC2 + 0.0257nC3 and odd members: T (K) ) -34.736 + 43.4874nC - 1.997nC2 + 0.034nC3 (b) Melting enthalpies, in kJ mol-1. Second-order polynomial curve fits: even members: ∆fusH (kJ mol-1) ) -27.232 + 5.6107nC - 0.1nC2 and odd members: ∆fusH (kJ mol-1) ) -4.4107 + 2.91024nC + 0.0367nC2.
TABLE 1: Temperature and Enthalpy of Fusion Values of the T and RI Phases and Calculated Enthalpy Value of Vaporization for the Even-Numbered Carbon Atom n-Alkanes for nC ) 8-20 nC
Tfus (T)a (K)
∆fusH (T)a (kJ/mol)
Tfus (RI)b (K)
∆fusH (RI)b (kJ/mol)
∆vapH (kJ/mol)
8 10 12 14 16 18 20
216.65 243.05 263.15 278.35 290.75 301.05 309.65
21.79 27.65 35.70 42.75 53.00 59.81 68.09
202.76 234.47 258.35 276.05 289.21 299.47 308.44
11.24 18.85 25.65 31.66 36.86 41.26 44.86
43.88 51.39 61.5 71.3 81.4 91.6 110
a Experimental values. b Values inferred from ref 25, plotted in Figure 2a,b.
point (melting of the triclinic phase), the experimental results from ref 43 for nC ) 16 and from ref 44 for nC ) 18 allow to calculate the specific volume changes ∆VT-L (Table 2a). Since the RI phase was never observed for a pure evennumbered carbon atom n-alkane with nC
