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Topological and Experimental Approach to the Pressure-Temperature-Composition Phase Diagram of the Binary Enantiomer System d- and l-Camphor Ivo B. Rietveld,*,† Maria Barrio,‡ Philippe Espeau,† Josep Lluis Tamarit,‡ and Ren
e C
eolin†,‡ †

Universit
e Paris Descartes, Facult
e de Pharmacie, D
epartement de Physico-chimie du M
edicament, EAD Physico-chimie Industrielle du M
edicament (EA4066), 4 Avenue de l’Observatoire, 75006 Paris, France ‡ Grup de Caracteritzaci
o de Materials (GCM), Departament de Física i Enginyeria Nuclear, Universitat Politecnica de Catalunya, ETSEIB, Diagonal 647, 08028 Barcelona, Spain ABSTRACT: In 1981, Jacques, Collet, and Wilen already put forward the idea to use pressure to influence equilibria in binary enantiomer systems in analogy with temperature (Jacques et al. Enantiomers, Racemates and Resolutions; John Wiley & Sons: New York, 1981). Whereas temperature is used routinely to study phase equilibria, pressure is an all but forgotten parameter. This is therefore possibly the first paper on the influence of pressure on a binary enantiomer system: d- and lcamphor. The study consists of two parts, a topological approach, which uses data obtained from routine measurements (differential scanning calorimetry, X-ray diffraction), and the experimental determination of phase transitions as a function of pressure and temperature. This has resulted in two topological pressuretemperature phase diagrams of the pure enantiomer d-camphor and of the racemic mixture dl-camphor; both have been verified by the experiments as a function of pressure. In turn, these results have been used to construct part of the pressuretemperature-composition phase diagram of d- and l-camphor. A method to obtain the excess Gibbs energy from these binary phase diagrams as a function of pressure is proposed.

1. INTRODUCTION The present paper is likely the first of its kind to present pressure-temperature (p-T) phase behavior of binary mixtures of enantiomers, even though Jacques et al. remarked in 1981 that pressure may be an important variable with the capability of changing equilibria between pure enantiomers and their racemic mixture.1 The paper includes p-T data of dl-camphor, of which no high-pressure data have ever been published. Phase behavior of camphor as a function of pressure has not received any attention since Bridgman published the high-pressure phases of d-camphor in 1916.2 Previously, the dependence of d-camphor’s melting point on pressure had been published in 1899 by Hulett.3 Pressure-temperature-composition (p-T-x) is an important issue for the pharmaceutical industry because drugs processing occurs frequently at increased pressure. As an example, the term “tablet” translates in French into “comprim
e” (=compressed). Knowledge about high-pressure phase behavior will facilitate the prediction of structural modifications that can occur during processing, modifications that may have implications for storage or bioavailability of the drug. Moreover, the study of p-T phase behavior allows verification and prediction of polymorphism at room temperature, in particular concerning the existence of metastable phases, which can be metastable extensions of highpressure phases.4,5 Through the topological approach of p-T phase diagrams, a chart is obtained of the stability hierarchy of known phases, and this chart allows inference of phase changes in stability with r 2011 American Chemical Society

pressure and temperature. Nevertheless, it is important to verify topological results with p-T measurements. Simply for reasons of proper scientific practice, the topological approach and experimental data points provide a reciprocal consistency check. Topology needs to correspond to the experimental fact, and experiment needs to correspond to the thermodynamic basis laid out by Gibbs and his contemporaries. Moreover, topology alone cannot replace the strength of a verified experimental fact. The topological approach, described in previous publications,6-8 entails the determination of all triple points in the p-T phase diagram, leading to an overview of the position and stability hierarchy of the phase transitions in the p-T plane, because all two-phase equilibrium curves have to pass through the triple points shared by their two phases. Once the stability chart has been obtained, one can infer whether certain phases will never be stable, stable under ambient conditions, or stable at higher pressure and/or temperature. In this paper, construction of topological p-T phase diagrams will be presented for d- and dl-camphor. The diagrams will be compared with experimental p-T data for transition III-II and fusion of d- and of dl-camphor. The paper will conclude with the p-T-x diagram of d- and l-camphor, demonstrating the effect of pressure on the T-x equilibria. Received: September 17, 2010 Revised: November 26, 2010 Published: January 31, 2011 1672

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2. EXPERIMENTAL SECTION Many data used in this paper have been published previously with the experimental details.9 New data were obtained with highpressure differential thermal analysis (HP-DTA). HP-DTA measurements were performed by way of an in-house constructed highpressure differential thermal analyzer similar to W€urflinger’s apparatus10 with operating ranges in temperature of 190-470 K and pressure of 0-250 MPa. d- and dl-Camphor were molten and sealed in cylindrical tin pans. Because the presence of air in the pans had to be avoided, the pans were sealed under deliberate loss of liquid camphor. The pressure on the pans, consisting of a ductile metal, was transmitted by a silicon oil. All HP-DTA measurements were performed at a heating rate of 2 K min-1. 3. RESULTS AND DISCUSSION 3.1. Construction of Topological p-T Diagrams. The goal of a topological p-T diagram is to determine the stability hierarchy of known phases as a function of pressure and temperature. Speaking of stability hierarchy, it is understood that stable, metastable, and increased degrees of metastability are all relative notions. Clues for the stability hierarchy can be obtained from DSC measurements. Enantiotropy for a system at “ordinary” pressure, in equilibrium with its own vapor pressure, can be observed as a sequence of endothermic peaks with increasing temperature, indicating the transformation of one stable phase into another stable phase, in accordance with Le Chatelier’s principle. Camphor is a good example with three stable solid phases under ordinary pressure: orthorhombic (III), hexagonal (II), and cubic (I) (cf. ref 9 for the transition temperatures of dand dl-camphor). Exothermic processes during a heating run indicate that a phase is metastable and transforms irreversibly into a more stable phase. Since the transition is exothermic, it is not bound to a specific temperature and the stability domain of the more stable phase can therefore not be determined by DSC measurements alone. The hierarchy and transition data obtained with DSC are the first valuable pieces of information for the construction of a topological p-T diagram. However, thermodynamics is more than only heat measurements; also work has to be considered through differences in density (specific volume), because between two phases, the less dense phase should transform into the denser phase with increasing pressure, in accordance with Le Chatelier’s principle. Thus, through enthalpy and specific volume changes, the slope dp/dT of any two-phase equilibrium curve can be obtained and p-T diagrams can be constructed by way of the Clapeyron equation. 3.2. d-Camphor. For d-camphor, vapor pressure data are available (cf. ref 9) over a considerable temperature range, constituting by its very nature part of the p-T diagram.11-15 Vapor pressure data are important, because they provide a link between the topological diagram and experimental p-T data at ordinary pressure; however, it is not essential to the topological method. The vapor pressure data extends over three phases: liquid (L) and solid phases I and II. The vapor pressures of the different phases were fitted separately with the following expression

lnðpÞ ¼ -

Δvap R H þbvap, R RT

ð1Þ

with p the pressure, Δvap R H the enthalpy of vaporization (R = liquid) or sublimation (R = I, II), considered independent of temperature,

R = 8.314 51 J K-1 mol-1, T the temperature, and bvap,R a constant. Enthalpy is a state function; hence, its change is the same irrespective of the thermodynamic pathway. In other words, the enthalpy of transition between two phases must be equal to the difference between their enthalpies of vaporization at the transition (triple) point. Thus, the enthalpy of sublimation of phase III can be calculated by adding the III-II transition enthalpy to the sublimation enthalpy of phase II, assuming that the specific heat contribution within the temperature domain of phase II can be neglected. The constants bvap,R were obtained by fitting the vapor pressure data. For phase II, the fit was aided by the phase transition enthalpy obtained by DSC and by the fact that the vapor pressure of phase II must be equal to the vapor pressure of phase I at their transition temperature. Once the expression of the vapor pressure for phase II is known, the same procedure can be used to obtain bvap,III for phase III; the vapor pressure of phase II and phase III must be equal at their transition temperature of 242.7 K. Values of Δvap R H and bvap,R are compiled in Table 1. The vapor pressure curves, which represent a two-phase equilibrium between vapor and a condensed phase, pass through three triple points involving the vapor phase: two solid-solid transitions and the fusion of phase I (III-II-v, II-I-v, I-L-v marked 1, 2, and 3, respectively, in Figure 1), defined by the temperature and pressure at the intersection of the two vapor pressure curves. They are the stable phase transitions determined with DSC. The pressure in the diagram is thus solely the partial pressure of camphor (i.e., the pressure of the system) and not the ambient “external” pressure. The DSC transition temperatures are the triple point temperatures under “ordinary pressure”, which in fact is the partial pressure of camphor in a sealed capsule, provided that the heating rate is not too fast for equilibrium to keep up. Three additional triple points sharing the vapor phase are obtained by extrapolation of the vapor pressure curves: II-L-v, III-L-v, and III-I-v (marked 6, 5, and 4, respectively, in Figure 1) of which the first two are metastable melting points of the solid phases II and III and the last is the metastable transition of phase III into phase I. The resulting six triple points are compiled in Table 2, but the positions of four others are still at large, four other triple points because with five phases—III, II, I, L, and v—10 triple points must exist.16 Here the real strength of the topological method is revealed. The subsequent procedure is based on the Clapeyron equation: dp ΔβR S ΔβR H ¼ β ¼ dT ΔR V T R f β ΔβR V

ð2Þ

ΔβRS is the entropy change, ΔβRV the volume change, and ΔβRH the enthalpy change at the R to β transition at temperature TRfβ. With the coordinates of a triple point and the slope resulting from the Clapeyron equation, the position and slope of a twophase equilibrium curve at this triple point are unequivocally defined in the p-T plane. The data in ref 9 provide all the necessary information to calculate the slopes by way of eq 2. Transition enthalpies follow directly from DSC measurements or from the difference in vaporization (or sublimation) enthalpies and are listed in Table 2, as are the volume changes for the different phase transitions. The topological diagram at low pressures is presented in Figure 1. Due to the low vapor pressure values, pressure and temperature data are not properly scaled in order to avoid the 1673

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Table 1. Parameters (eq 1) of the Vapor Pressure (in Pa) of d- and dl-Camphor for Liquids and Solid Phases Found near Ordinary Pressure liquid

cubic (I)

hexagonal (II)

orthorhombic (III)

d-Camphor -1 Δvap ) R H (kJ mol

44.3 ( 0.3

51.1 ( 0.1

51.3 ( 0.3

62.5 ( 0.7

bvap,R methoda

22.61 ( 0.07 fit of p-T data

24.41 ( 0.03 fit of p-T data

24.5 ( 0.1 p-T and DSC data

30.1 ( 0.3 DSC data

dl-Camphor -1

44.3 ( 0.3

50.3 ( 0.3

50.5 ( 0.3

51.7 ( 0.6

bvap,R

22.61 ( 0.07

24.2 ( 0.2

24.3 ( 0.4

25.0 ( 0.9

methoda

equal to d-camphor

DSC data

DSC data

DSC data

Δvap R H

(kJ mol

)

a

DSC data indicate that for calculating the coefficients, the equilibrium point with the neighboring phase was used in combination with DSC data (cf. Table 4 in ref 9).

Figure 1. Topological diagram at low pressures and temperature 200-500 K. Please note that pressure and temperature are not to scale; otherwise, most of the diagram would fall onto the zero-pressure line. Triple points are numbered as follows: 1, III-II-v (242.7 K, 0.4 Pa); 2, II-I-v (370 K, 2.4 kPa); 3, I-L-v (451.8 K, 50 kPa); 4, III-I-v (244 K, 0.5 Pa); 5, III-L-v (295 K, 90 Pa); 6, II-L-v (450 K, 47.5 kPa). Different levels of stability are indicated by the style of the line: bold continuous line, stable; bold broken line, metastable; thin broken line, super-metastable; dotted line, hyper-metastable (=less stable than supermetastable).

collapse of curves onto the temperature axis. The topological approach provides at a glance the hierarchy between the phases following from the vapor pressure inequalities obeying the Ostwald rule (the lower the vapor pressure, the more stable the phase) and thus their character: stable, metastable, supermetastable, and hyper-metastable, in order of decreasing stability. At a triple point, stable and one degree less stable equilibrium curves must alternate. Transition lines in the p-T plane are projections of the intersections of Gibbs energy surfaces in the G(p, T) space; thus, the hierarchy in stability reflects the order in height of the intersecting Gibbs energy surfaces. This implies that a triple point consists of three phases with the same degree of stability; their Gibbs energy must be the same, or the phases cannot coexist. On the other hand, two two-phase equilibrium curves without a common phase crossing each other must do so at different hierarchical levels, because they represent different equilibria at different energy levels. Considering Figure 1, it can be seen that due to the interchanging stability requirement for a triple point, the stability

hierarchy for all triple points is determined. For example, line L-v, which starts on the right of the diagram as a stable equilibrium curve, becomes metastable once it meets triple point I-L-v (3), that is, the melting triple point with the highest temperature. Another phase is stable at the other side of the triple point, but the metastable phase transition L-v still exists and continues to triple point II-L-v (6), where three metastable lines meet. Triple point 6 cannot be stable, because phase I is stable in this region; L-v must become supermetastable in passing triple point 6. The line L-v crosses triple point 5, III-L-v, and must become even more metastable, that is hyper-metastable, because phase II is stable and phase I is the metastable phase in this region. Triple point 5 must therefore consist of interchanging super- and hyper-metastable transitions. In this way, the ranking of all triple points can be determined. In Figure 1 the lines of transitions III-II, III-I, III-L, II-I, III-v, II-L, and I-L have all been calculated with the Clapeyron equation (eq 2). To determine the remaining triple points, these lines are extrapolated leading to several intersections at pressures significantly different from ordinary pressure (cf. Figure 2). The calculated slopes at ordinary pressure merely give an indication of the triple-point positions. For example, lines III-II, III-I, and II-I must intersect at triple point III-II-I. With the calculated slopes, three intersections will be found slightly displaced from each other. Because the transition III-I is metastable at ordinary pressure, the slope has been determined with values obtained from extrapolations, which will increase the uncertainty of the slope. Therefore, the line III-I with the highest uncertainty is adjusted until all three lines intersect at the same point: the triple point. The same procedure is followed for triple point II-I-L by adjusting the slope of the II-L transition. Finally, the triple points III-II-L and III-I-L are found by adjusting the slope of the III-L transition. The coordinates of the triple points are given in Table 2. Thus, the slopes in Table 2 have been calculated with data obtained at ordinary pressure, whereas some of the triple points, in particular those between the least stable phases, have been obtained by adjusting the slopes. Therefore, the slopes of the equilibria between the least stable phases in Figure 2 will deviate from the calculated ones in Table 2, which leads to the explanation for the word “topological” phase diagram. The predicted location of the triple points is approximate, because not only the uncertainty increases with extrapolation but also the transition lines may be curved; however, they have to be 1674

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Table 2. Triple Points and Their Coordinates (T, p) for d- and dl-Camphor triple point

temperature (K)b

pressure (kPa)b

ΔtransitionHb,d (kJ mol-1)

ΔtransitionVb,d (cm3 mol-1)

(dp/dT)calc (MPa K-1)

d-Camphor Stable I-L-v

451.8 ( 0.7a

50

6.1 ( 0.5a

21.7

0.69

II-I-v

370 ( 4a

2.4

0.20a,c

0.64

0.85

III-II-v

242.7 ( 0.6a

0.0004

11.2 ( 0.4a

10.2

4.6

II-L-v

450

47.5

7.0

21

0.74

III-L-v

295

0.09

18

18

3.4

11.4

12.6

3.7

Metastable

III-I-v

244

0.0005

III-II-Ie II-I-Le

213 8.4

-133  103 -307  103

III-I-Le

205

-171  103

205

-170  103

e

III-II-L

dl-Camphor Stable I-L-v

448.0 ( 0.7a

45

6.0 ( 0.3a

18.9

0.70

II-I-v

364 ( 5a

2.0

0.20a,c

0.64

0.86

III-II-v

206.1 ( 0.3a

0.000006

1.3 ( 0.3a

1.15

5.5

II-L-v III-L-v

445 370

41.3 3.7

6.2 7.5

17 1.9

0.82 10.8

1.5

4.5

1.5

Metastable

III-I-v

219

0.00003

III-II-Ie

177

-161  103

II-I-Le

-11

-323  103

167

-197  103

171

-194  103

e

III-I-L

e

III-II-L a

b

From DSC data, ref 9. Most of the data are calculated by extrapolation. The values are therefore approximate only. c The enthalpy of transition II-I is not well-defined, and also the uncertainty in the DSC measurements is large. 200 J mol-1 has been chosen as the rounded-off value. d The values are given for the first two phases listed for the triple point; thus, for I-L-v they represent the transition I-L under the presence of phase v. e Determined by extrapolation (see text).

Figure 2. Topological p-T diagram of d-camphor. For the numbered triple points, see also Figure 1. Bold line, stable; bold broken line, metastable; thin broken line, super-metastable; dotted line, hypermetastable. Inset: schematic close-up of the area with the triple points III-II-I, III-II-L, and III-I-L.

monotonously increasing. Triple points with negative pressure are always metastable, because the vapor phase is the stable phase, whereas positive-pressure triple points may be stable. In the case of d-camphor, the four additional triple points are

located in the negative pressure area (i.e., the area for expanded condensed phases17-20), resulting in enantiotropy at ordinary pressure and above. The stability hierarchy in between the triple points III-II-I, III-II-L, and II-I-L is shown schematically in an inset in Figure 2. The inset in combination with the entire diagram shows how the requirement of interchanging stability levels shapes the stability hierarchy. Because a large part of the triple points can be found in the negative pressure domain, a few words on negative pressure may be appropriate. Although for the topological method, negative pressure is mainly a calculation tool to obtain triple points and phase hierarchy, it is an existing experimental condition, namely, if there is a constant pull on the sample, forcing it to expand against equilibrium conditions (therefore all phases are metastable under negative pressure).17-20 From a physical point of view, there is no difference between push or pull (except for the sign of the force); thus, the Clapeyron equation will remain valid under negative pressure. Measurements in the negative pressure domain would be possible, but because all phase transitions will be between metastable phases, it is at least at present very difficult to control experimental conditions in such a way that useful information about those transitions can be obtained. 1675

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Figure 3. Topological p-T diagram of dl-camphor. Bold line, stable; bold broken line, metastable; thin broken line, super-metastable; dotted line, hyper-metastable. The inset of Figure 2, highlighting the stability hierarchy, applies also to the present figure in the area with the triple points III-II-I, III-II-L, and III-I-L.

3.3. dl-Camphor. For dl-camphor, no vapor pressure data are available; however, the vapor pressures of liquid dl-camphor and liquid d-camphor must be equal, because the difference in interaction between d- and l-camphor is expected to be minimal in the liquid state. Therefore, all triple points can be constructed following the procedure described to obtain the vapor pressure of d-camphor’s phase III. The vapor pressure of dl-camphor phase I must be equal to the vapor pressure of the liquid at the melting point, and the transition enthalpy will be equal to the difference between the two enthalpies of vaporization. Applying this procedure to phase I, then II, and then III using eq 1 leads to the enthalpies of vaporization compiled in Table 1. For dl-camphor, a similar diagram can be constructed as shown in Figure 1. The intersections of the vapor pressure curves, employed to obtain the values in Table 1, are the stable triple points, which are compiled in Table 2, I-L-v, II-I-v, III-II-v. Extrapolation of the curves leads to three additional triple points in which the vapor phase takes part: II-L-v, III-L-v, and III-I-v (Table 2). The differences in the enthalpies of vaporization (sublimation) provide the transition enthalpies between the two condensed phases. The differences in volume can be obtained by extrapolation of the data presented in ref 9. With the use of the Clapeyron equation (eq 2), the slope in the p-T diagram of the “missing” two-phase equilibrium can be calculated. By adjusting the slopes of the most metastable transitions (because the uncertainty in their data is largest), the remaining triple points can be found, exactly in the same way as for d-camphor. The resulting p-T phase diagram can be found in Figure 3. Comparing Figure 3 with Figure 2, p-T behavior of dlcamphor is very similar to that of d-camphor, something that cannot be inferred from the T-x diagram alone. The enantiotropy at ordinary pressure for dl-camphor is evident. 3.4. Experimental p-T Curves. A topological diagram is the result of extrapolation, and the positions of the triple points are only approximate. However, the general trend, whether a triple point is found under positive or negative pressure and at low or high temperature, can be established. The diagram can be verified by comparison of the calculated lines with experimental p-T data (Figure 4). The agreement is very good for the III-II transitions. In the case of the melting transitions, there is more curvature in the measured points. The p-T melting curve for

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Figure 4. Topological results compared with experimental p-T data. Squares, d-camphor; triangles, dl-camphor. Filled symbols, orthorhombic (III)-hexagonal (II) transition; gray symbols (only at ordinary pressure), hexagonal (II)-cubic (I) transition; open symbols, melting. The topological results are given by a solid line for d-camphor and a broken line for dl-camphor.  data for d-camphor determined by Bridgman (III-II),2 * data for d-camphor by Hulett (I-L).3

Table 3. Comparison between Direct Measurement of Volume Differences and Those Obtained Indirectly from p-T Measurement d-camphor

dl-camphor

ΔV (cm3 mol-1) p-T meas.a

ΔV (cm3 mol-1) p-T meas.a

III-II

9.2 ( 0.5

10 ( 2

0.9 ( 0.2

1(1

I-L

18 ( 2

22 ( 8

16 ( 2

18 ( 3

The volume differences marked with “meas.” are obtained with X-ray diffraction and direct observation of the volume (cf. ref 9).

a

d-camphor published by Hulett3 more than a century ago perfectly coincides with the present measurements. The values obtained by Bridgman for the d-camphor III-II transition coincide with the measurements too.2 The transition II-I could not be observed with the high-pressure DTA, due to its weak transition enthalpy; therefore, only the measurement points obtained with DSC are available. The slope obtained by the Clapeyron approach depends on the transition enthalpy, the transition temperature, and the volume change. Because the transition enthalpy is determined with an uncertainty of generally less than 5% and temperature is within an error of (1 K, it is often the uncertainty in the volume change, obtained by X-ray diffraction or direct observation, which is large, the more so because the volumes of the involved phases are similar to start with. Therefore, as a consistency check, the volume changes can be calculated with the measured slopes under the assumption that the transition enthalpy and temperature are precisely known. The measured and the calculated volumes are given in Table 3. As can be seen, the differences are relatively large, but considering the measurement precision, the deviations seen here are not surprising. 3.5. p-T-x Behavior. Finally, using the p-T diagrams for both d- and dl-camphor, a 3D phase diagram can be constructed: p-T-x (cf. Figure 5a). The difference between the melting temperature of the pure enantiomers and the racemic mixture 1676

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Figure 5. (a) p-T-x phase diagram of the d- (x = 0) and l-camphor (x = 1) system for fusion of solid I, based on the experimental data points. The transitions for 0.1, 5, 10, 20, and 30 MPa are shown. The temperature difference between the melting point of d-camphor and the racemic mixture increases with pressure. (b) Same as (a) for the solid II-solid I equilibrium based on the extrapolation of the Clapeyron equation.

increases with pressure in relation to the different slopes calculated with the Clapeyron equation (cf. Table 2). The II-I transition has been calculated based on the transition data at ordinary pressure of d- and dl-camphor presented in ref 9 (cf. Figure 5b). While the II-I transition and I-L melting loops can be unambiguously calculated (assuming that the liquid mixture behaves ideally), the III-II equilibrium is strongly affected by the disordered character of the low-temperature phase III of the racemic compound (cf. ref 9).21,22 The excess Gibbs energy of phase I as a function of pressure can be calculated. Making use of the equal Gibbs energy curve (EGC)23 (see also eqs 2-5 in ref 9), a method for the calculation of excess Gibbs energy as a function of pressure can be obtained, which in principle can be applied to any p-T-x phase system, although this is the simplified version for the case of a binary enantiomer system: IfL ðpÞþ TEGC ðX ¼ 0:5,pÞ ¼ Td-camphor

ΔLI GEEGC ðX ¼ 0:5,pÞ ð3Þ ΔLI Sd-camphor

TEGC (X = 0.5, p) is the temperature of the EGC at mole fraction = 0.5 explicitly depending on the pressure. TIfL d-camphor(p) replaces the first term on the right side of eq 4 in ref 9 and is independent of the mole fraction, because it represents the transition temperature of an ideal mixture between two optical antipodes.ΔLI GEEGC(X = 0.5, p)is the difference in excess Gibbs energy between phase I and phase L at equilibrium. ΔLI Sd-camphor replaces the denominator of the second term in eq 4 of ref 9. This term is independent of the mole fraction because the value is the same for the two pure optical antipodes in the binary mixture. Furthermore, it is assumed that the transition entropy does not change significantly with pressure and temperature for the range under study. Thus for each isobaric T-x diagram, the difference in melting temperature between the equimolar composition and the pure compounds can be used to determine the excess Gibbs energy (The TEGC is equal to the melting temperature of the

equimolar mixture for a binary enantiomer system23.) Under the assumption that the liquid phase, consisting of freely moving optical antipodes, is ideal, the excess Gibbs energy will be a property of phase I. The slopes of the two-phase equilibrium I-L for d- and dlcamphor diverge, which was concluded with the results of the Clapeyron equation (Table 2) and confirmed by independent p-T measurements (Figure 4 and 5a). This supports the general trend of the calculation indicating that the excess Gibbs energy of dl-camphor increases with pressure from about 50 J/mol at ordinary pressure up to about 130 J/mol at 250 MPa. Thus with increasing pressure, the pure enantiomer becomes more stable in comparison to the racemic mixture.

4. CONCLUSIONS With data obtained at ordinary pressure by DSC, X-ray crystallography, liquid density measurements, and vapor pressure measurements, topological p-T phase diagrams figuring all triple points of the known phases can be constructed. The diagrams provide the stability hierarchy of the different phases and phase transitions as a function of pressure and temperature. In the case of camphor, the hierarchy is straightforward and does not change fundamentally between d- and dl-camphor. Their phases behave enantiotropically over the entire pressure range. Nonetheless, the results demonstrate that the topological approach can be applied to enantiomer systems providing a simple method to investigate the influence of pressure on binary mixtures of enantiomers. However, it is not possible to predict the appearance of new phases; thus, if a phase only exists at higher pressure, it cannot be predicted by the topological approach. This is the case with camphor, since Bridgman found three additional phases for dcamphor at higher pressure. Naturally, once the transition temperatures and pressures are known, they can be added to the p-T diagram. 1677

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The Journal of Physical Chemistry B Comparing the topological diagrams with the experimental p-T results, they are consistent. This indicates that the topological approach is correct within experimental uncertainty and that the experimental results are correct within the thermodynamic framework. Thus, within the limitations of experimental uncertainty, the entire binary phase diagram of d- and l-camphor can be constructed in the p-T-x space. The “symmetry” around the equimolar composition enables a straightforward thermodynamic assessment, conducted with the EGC method23 of the binary system between the two optical antipodes. Assuming ideal behavior for the liquid mixtures, the excess Gibbs energy as a function of pressure has been determined to increase with pressure. It is clear from the foregoing that pressure, as a thermodynamic variable, should not be neglected. First, volume changes take place in the pressure domain representing the thermodynamic work of the system. As shown in ref 9, free volume will fill up with vapor of the specimen, leaving less material for the actual phase transitions under study. Second, applying the Clapeyron equation provides a chart of the p-T behavior of all known phases including the likelihood of the appearance of metastable phases. In comparison, one obtains a rather one-dimensional view of a system’s phase behavior if only heat and temperature of transition are measured.

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(18) Imre, A. R.; Drozd-Rzoska, A.; Horvath, A.; Kraska, T.; Rzoska, S. J. J. Non-Cryst. Solids 2008, 354, 4157. (19) Imre, A. R.; Drozd-Rzoska, A.; Kraska, T.; Martinas, K.; Rebelo, L. P. N.; Rzoska, S. J.; Visak, Z. P.; Yelash, L. V. NATO Sci. Ser., II 2004, 157, 177. (20) Imre, A. R.; Drozd-Rzoska, A.; Kraska, T.; Rzoska, S. J.; Wojciechowski, K. W. J. Phys.: Condens. Matter 2008, 20, 244104. (21) Mora, A. J.; Fitch, A. N. J. Solid State Chem. 1997, 134, 211. (22) Nagumo, T.; Matsuo, T.; Suga, H. Thermochim. Acta 1989, 139, 121. (23) Oonk, H. A. J. Phase Theory, The Thermodynamics of Heterogeneous Equilibria; Elsevier: Amsterdam, 1981.

’ AUTHOR INFORMATION Corresponding Author

*Tel. þ33 1 53739675; e-mail [email protected].

’ ACKNOWLEDGMENT R.C. acknowledges an invited position from the Generalitat de Catalunya (2007PIV0011) at the Universitat Politecnica de Catalunya. This work was partially supported by the Spanish grant FIS2008-00837 and by grant 2009SGR-1251 from the Catalan government. The authors express their appreciation for the contribution to this paper by the late Nestor Veglio. ’ REFERENCES (1) Jacques, J.; Collet, A.; Wilen, S. H. Enantiomers, Racemates and Resolutions; John Wiley & Sons: New York, 1981. (2) Bridgman, P. W. Proc. Am. Acad. Art. Sci. 1916, 52, 91. (3) Hulett, G. A. Z. Phys. Chem. (Munich) 1899, 28, 629. (4) Drozd-Rzoska, A.; Rzoska, S. J.; Pawlus, S.; Tamarit, J. L. Phys. Rev. B: Condens. Matter Mater. Phys. 2006, 74, 064201/1. (5) Imre, A. R.; Rzoska, S. J. Adv. Sci. Lett. 2010, 3, 527. (6) Barrio, M.; Espeau, P.; Tamarit, J. L.; Perrin, M.-A.; Veglio, N.; C
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dx.doi.org/10.1021/jp108900v |J. Phys. Chem. B 2011, 115, 1672–1678