Cavity Formation and Dipolar Contribution to the Gauche−Trans

520

J. Phys. Chem. B 2001, 105, 520-526

Cavity Formation and Dipolar Contribution to the Gauche-Trans Isomerization of 1-Chloropropane and 1,2-Dichloroethane Yanira Mele´ ndez-Paga´ n, Brian E. Taylor, and Dor Ben-Amotz* Department of Chemistry, Purdue UniVersity, West Lafayette, Indiana 47907-1393 ReceiVed: August 2, 2000

The effects of pressure and temperature on the gauche-trans conformational equilibrium of 1-chloropropane and 1,2-dichloroethane dissolved in diethyl ether (ethoxyethane) are measured using Raman spectroscopy and analyzed using thermodynamic perturbation theory. The effects of solvation on the isomerization equilibrium, monitored using the C-Cl stretch band intensities of the two conformers, are interpreted as a sum of intermolecular attractive and repulsive solute-solvent interactions. A perturbative analysis of the isomerization enthalpy, ∆H, and volume, ∆V, is used to extract repulsive cavity formation and attractive mean field solvation contributions to the conformational equilibrium. Comparison of the thermodynamics of the two solutes with each other, and with previous gas phase and solution results, are used to confirm the self-consistency of the perturbative modeling strategy and to quantify dipolar contributions to the gauchetrans isomerization process.

I. Introduction Molecular conformational equilibrium is a central concept in the structural chemistry of biomolecules, polymers, and smaller flexible compounds. The thermodynamics of such molecular folding processes depend not only on the internal potential energy surface of the folding molecule but also on solvent perturbations which can shift the delicate balance of structural stability. Numerous previous experimental1-19 and theoretical19-32 studies have focused on the conformational equilibria of small polyatomic molecules in the vapor and liquid phase. The solutes whose gauche-trans conformational isomerizations are studied in this work: 1-chloropropane (CP) and 1,2-dichloroethane (DCE), are of particular interest because of their similar molecular van der Waals volumes and polarizabilities but very different dipolar properties (see Figure 1). Both CP and DCE have a van der Waals volume of, VvdW ≈ 7.4 ( 2 Å3 33 and polarizability of, R ≈ 8 ( 0.5 Å3,34 in both conformational states. However, DCE undergoes a large change in dipole moment upon isomerization (µgauche ) 3.12 D,8 µtrans ≈ 0 D) while CP has the same dipole moment in both isomeric states (µgauche ≈ µtrans ≈ 2 D35). Few experimental systems offer such a unique opportunity to turn on and off strong dipolar solute-solvent coupling, while leaving other molecular parameters essentially unchanged. This work combines experimental pressure and temperature dependent Raman measurements with a theoretical perturbed hard fluid (PHF) analysis to quantitatively probe cavity formation and dipolar solvent contributions to the gauche-trans isomerization process of CP and DCE in ethoxyethane (ether). Measurement of the relative areas of Raman bands unique to each conformer are used to monitor pressure and temperature dependent changes in the gauche-trans population ratio. These not only provide information about the excess solvent contributions to the partial molar enthalpy ∆H and volume ∆V of isomerizations, but also higher order thermodynamic derivatives represented by the pressure and temperature dependence of ∆H

Figure 1. Gauche and trans structures of (a) CP and (b) DCE.

and ∆V. Such changes have not been explained by any previous theoretical or computer simulation studies of such isomerization processes. The PHF molecular modeling strategy used in this work aims to explain the effects of pressure and temperature on isomerization equilibria within a perturbative theoretical framework which may be viewed as a modern generalization of the van der Waals equation of state.1,20,28-30 In particular, we treat the solution of interest as a reference hard sphere fluid plus longrange intermolecular cohesive interactions. The effective hard sphere diameters of the solvent and solute molecules are derived in a self-consistent way from the experimental compressibility of the solvent and the molecular shape of the gauche and trans solute isomers. Cohesive interactions are modeled using the van der Waals mean field approximation, and described by a single temperature and pressure independent parameter. The current variant of the PHF model is somewhat simpler than that used to analyze the gauche-trans isomerization of 1-bromopropane.1 The key difference is that, instead of using the excluded volume anisotropy (EVA) model to derive effective hard sphere diameters of the two isomers,1 in this work we optimize these by fitting isomerization enthalpy data. Furthermore, we

10.1021/jp002781w CCC: $20.00 © 2001 American Chemical Society Published on Web 12/20/2000

Gauche-Trans Isomerization assume that the solutes are sufficiently dilute that they do not interact with each other, and that the structures of the two isomers are pressure and temperature independent. Although the latter assumption amounts to neglecting slight changes in the gauche dihedral angle with pressure,8,19 the good agreement between our PHF and experimental results suggest that such changes do not significantly affect the isomerization thermodynamics of interest. Although the PHF model clearly involves a significant degree of simplification, the results are found to be sufficiently realistic to reproduce all of the observed phenomena, including previously unexplained changes in the isomerization ∆H and ∆V with temperature and pressure. Furthermore, because of the very different dipolar properties of CP and DCE (and their very similar shape, size, and polarizability) our results offer the first quantitative experimental measure of the effects of dipolar solute-solvent interactions on the isomerization equilibrium, independent of the effects of cavity formation and dispersive solute-solvent interactions. II. Experimental Procedure Spectral grade CP (1-chloropropane) and DCE (1,2-dichloroethane) were purchased from Aldrich and ether (ethoxyethane) was purchased from Mallinckrodt, and all were used without further purification. Solutions of CP/ether and DCE/ether (at a concentration of ∼20 wt %) were loaded into a diamond anvil cell (DAC) for both pressure and temperature dependent studies. The Merrill-Basset DAC includes a stainless steel gasket of 500 µm thickness with a 700 µm hole (sample chamber). Pressure measurements were performed using the fluorescence of several ruby chips (10-50 µm) placed inside the DAC in the ether solution.1 The microscope-based Raman system used for both ruby fluorescence and Raman measurements includes a 40 mW HeNe laser (Spectra Physics model 127) (632.8 nm) as the excitation source. The laser was focused onto the sample using a microscope objective (Olympus ×20 long working distance); the backscattered Raman signal was collected using the same microscope objective and detected using a liquid nitrogen cooled CCD detector (Princeton Instruments LN/CCD 1152E) mounted to a spectrograph (ISA HR320 F/4.2) with an 1800 grooves/ mm grating. The exposure time was 300 s per Raman spectrum. Since the frequency shift of the ruby fluorescence depends on both temperature and pressure, the ruby fluorescence shift inside the DAC was determined relative to a ruby chip at 1 atm and at the same temperature as the high pressure solution (both the DAC and the external ruby chip were immersed in the same thermostated oil bath). The sample temperature was determined using a thermocouple placed in the oil bath, near the DAC and continuously monitored to ensure temperature stability of (0.3 °C during each experimental measurement. The intensities of the symmetric C-Cl stretch bands of trans (∼723 cm-1) and gauche (∼651 cm-1) isomers of CP (see Figure 2a), and trans (∼754 cm-1) and gauche (∼677 cm-1) isomers of DCE (see Figure 2b) were obtained, after background subtraction, by fitting the peaks to a Voigt function whose area was taken as the experimental band intensity (using IgorPro by WaveMetrics Inc.). The shoulder on the high-frequency side of the gauche band of DCE (see Figure 2b), which is due to the asymmetric gauche stretch,2,8 was excluded from the measured peak area. This was done by fitting only the low frequency and maximum region of the large gauche band, and then assuming that area under to full best-fit Voigt band represents the gauche symmetric stretch band intensity.

J. Phys. Chem. B, Vol. 105, No. 2, 2001 521

Figure 2. Raman spectra of (a) CP and (b) DCE in ether at 1 atm and high pressure, showing the prominent trans (higher frequency) and gauche (lower frequency) C-Cl stretching bands.

III. Theoretical Modeling and Data Analysis The PHF analysis of isomerization equilibria begins by formally separating the reaction free energy into ideal gas, repulsive (cavity formation) and attractive (cohesive) excess solvation free energies.

∆G ) ∆Gi + ∆Gx ) ∆Gi + ∆Gr + ∆Ga

(1)

All of the above free energies pertain to a standard state in which the number densities (concentrations) of the reactants and products are equal. Since the number of moles of reactant and product molecules are the same in an isomerization process, none of the resulting reaction thermodynamic quantities depend on the numerical value of the standard state concentration (e.g., 1 mol/L or 1 molecule/m3, etc.). The ideal contribution to the free energy ∆Gi corresponds to the reaction of the isolated (nonsolvated) molecules in an ideal gas state. Thus, the repulsive ∆Gr and attractive ∆Ga contributions represent the effect of solute-solvent interactions on the isomerization process. Cavity formation energy differences ∆Gr are predicted using the Boublik-Mansouri-Carnahan-Starling-Leland (BMCSL) mixed hard sphere fluid equation of state.1,36 This makes use of the following BMCSL expression for the chemical potential of a hard sphere solute of diameter σ dissolved (at infinite dilution) in a hard sphere solvent of diameter σs.

3ηd(1 + d - d2) 3ηd2 2ηd3 µ + + ) kT (1 - η)3 (1 - η)2 (1 -η) (1 - 3d2 + 2d3)ln(1 - η) (2) The parameters d and η are the solute-solvent diameter ratio (σ/σs) and the solvent packing fraction (πFσs3/6), respectively. In practice, the solvent number density, F, and the effective hard sphere diameter, σs, are obtained by fitting the experimental compressibility of the pure solvent to the Carnahan-Starlingvan der Waals (CS-vdW) equation of state.37 This same equation of state is also used to determine changes in the number density of the solvent with pressure and temperature.37 The resulting repulsive contribution to the reaction free energy is equated with the difference between the chemical potentials of the gauche and trans conformers, under standard state conditions of equal number density.

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∆Gr ) NA∆µ ) NA[µ(η,dt) - µ(η,dg)]

(3)

The effective diameters of the two conformers dt and dg, which appear in the above expression, are determined as described in the Appendix, and NA is Avogadro’s number. The attractive contribution to the isomerization free energy ∆Ga is modeled using a van der Waals mean field approximation. This implies that ∆Ga is a linear function of solvent density (and is temperature independent)1.

∆Ga ) ∆CaF ) (Ct - Cg)F

()

It Ft ) ∆GCalc ∆GExp ) -RT ln CΩ ) -RT ln Ig Fg

[

[

∆VExp ) -R

] [

∂ ln(It/Ig) ∂(1/T)

P

] [

∂ ln(It/Ig) ∂p

T

)R

(6)

]

(7)

∂(1/T)

P

∂ ln(Ig/It) ∂p

∆UExp ) ∆HExp - P∆VExp

T

P

) ∆Hi + ∆Hx ) ∆Hi + ∆Ha +

[

(∂T∂F) ] - RT [∂(∆µ/kT) ∂T ]

∆Hr ) ∆Hi + ∆Ca F - T ∆VCalc ) ∆Vx )

2

P

) ∆V + ∆V ) (∂∆G ∂P ) ∂F [∆C + N (∂∆µ ∂F ) ](∂P) a

(9)

P

r

T

a

A

∆UCalc ) ∆HCalc - P∆VCalc

T

T

(10) (11)

Note that there is no ideal contribution the isomerization volume because ∆Vi ) ∆n RT/P, and there is no change in the number of molecules upon isomerization (∆n ) 0). The solute and solvent effective hard sphere diameters required by the PHF model may be accurately estimated using pure solvent compressibility data combined with quantum structures of the gauche and trans states (as described in the Appendix). However, since the PHF predictions are very sensitive to small variations in the trans-gauche diameter ratio R, we have chosen to optimize R by fitting the experimental pressure dependence of ∆H for CP/ether. The resulting R value is used in analyzing the remaining CP/ether and DCE/ether results. The cohesive energy parameters ∆Ca for CP/ether and DCE/ether are determined by fitting experimental ∆V and ∆H data to eqs 9 and 10 (as described in section IV), assuming that ∆Ca is pressure and temperature independent (and using the CS-vdW equation of state to predict the required derivative of density with respect to pressure37). Finally, since the gas-phase isomerization enthalpy ∆Hi is not known with sufficient accuracy (as discussed in section IV) this is treated as an additional adjustable parameter in fitting experimental ∆H data to eq 9 (and is found to be in good agreement with previous experimental and theoretical estimates). The resulting solvent and solute PHF parameters are collected in Table 1. IV. Results

]

∂ ln(Ig/It)

)R

]

∂(∆G/T) ∂(1/T)

(5)

Note that the above expression implies that the translational entropies of the gauche and trans isomers behave ideally (in the sense that they scale logarithmically with concentration). The constant CΩ represents the ratio of the effective Raman cross sections of the trans and gauche isomers (including instrumental collection and detection efficiency factors). The value of CΩ is expected to be approximately pressure and temperature independent since solvent induced Raman crosssection changes should scale both the trans and gauche band intensities by the same factor (particularly for vibrational modes involving similar atomic motions).38 Thus, the experimental isomerization enthalpy (∆HExp), volume (∆VExp), and energy (∆UExp) may be determined from the appropriate thermodynamic derivatives of the experimental Raman band intensity ratios.

∆HExp ) -R

[

(4)

The parameter ∆Ca quantifies the difference in cohesive solvation energies between the product (trans) and reactant (gauche) conformers. Since the gauche and trans isomers of CP have virtually identical polarizabilities and dipole moments, little or no attractive solvation contribution to this isomerization is anticipated. On the other hand, DCE should have a significant dipolar solvation contribution as only the gauche state has a dipole moment (and both states have the same polarizability). Thus, for CP ∆Ca is expected to be near zero, while for DCE the magnitude of ∆Ca might be expected to be close to that estimated by calculating the dipole-dipole solute-solvent interaction energy of the gauche conformer, averaged over all solvent configurations (as described in the Appendix). However, since such approximate estimates of ∆Ca may not be sufficiently accurate for direct comparison with experimental measurements, ∆Ca is treated as an adjustable parameter in the current PHF analysis. However, comparisons of ∆Ca values obtain by independent dipolar solvation estimates are used to check the self-consistency of our PHF results. Raman peak area ratios (It/Ig), obtained at a variety of pressures and temperatures, are closely related to the isomerization free energy1.

( )

∆HCalc )

(8)

These experimental values may be compared with corresponding to theoretical expressions obtained from the PHF model.1

The measured peak area ratios Ig/It obtained at various pressures and temperatures for CP/ether and DCE/ether are given in Table 2. The relative trans/gauche Raman cross-section ratios estimated using semiempirical and ab initio Gaussian 98 are CΩ ≈ 1.1 for CP and CΩ ≈ 3.2 for DCE. Although these are not considered to be quantitatively reliable, they do suggest that the cross sections for the two isomers of DCE are quite different, while those for CP are nearly the same. Furthermore, the 2-fold degeneracy of the gauche state tends to increase the population of the gauche conformer relative to the nondegenerate trans state. Thus, apparent differences between the relative intensities of the trans and gauche peaks in CP/ether and DCE/ether (see Figure 2) cannot be taken to directly represent the energy or population differences of the corresponding isomers. However, the T and P dependence of these ratios may nevertheless be used to determine ∆H, ∆V, and ∆U via eqs 6-8. Figure 3a,c show the intensity ratios of the two solutions plotted as a function of inverse temperature, at four pressures. The points in these figures were interpolated from raw experimental data (see Table 2) plotted as a function of pressure, at each temperature. The lines in Figure 3a,c are best fits from which the corresponding isomerization enthalpies ∆HExp were determined using eq 6. Figure 3b,d show the experimental intensity ratios plotted as a function of pressure, interpolated from the data in Figure 3a,c. The best fit lines in this figure yield the isomerization volumes of both molecules ∆VExp,

Gauche-Trans Isomerization

J. Phys. Chem. B, Vol. 105, No. 2, 2001 523

TABLE 1: Solute and Solvent Parameters CS-vdW parametersa

PHF input and output parametersb

ethyl ether σ20 (Å) T(∂σ/∂T)20 (Å) τ20 (K) T(∂τ/∂T)20 (K)

5.388 -0.15 1698 -262

input: dave output: R output: ∆Ca (kJ nm3/mol) output: ∆Hi (kJ/mol)

1-chloropropane

1,2-dichloroethane

0.95 ( 0.1 0.998 ( 0.001 0.012 ( 0.05 -1.02 ( 0.5

0.94 ( 0.1 0.998 ( 0.001 0.321 ( 0.05 -4.91 ( 0.5

a The solvent equation of state parameters were obtained using the ambient density of diethyl ether,34 along with σ20 and eqs 7a and 7b in ref 37 (because the diethyl ether parameters in Table 1 of ref 37 are derived from thermal expansion data only, they are not considered to be as accurate in predicting high-pressure properties as the above estimates). b The uncertainties of the PHF parameters reflect the approximate variations which are consistent with the experimental uncertainties in ∆V and ∆H (when holding all other parameters at their optimal values).

TABLE 2: Experimental Raman Peak Area Ratios 1-chloropropane

1,2-dichloroethane

T (°C)

P (kbar)

ln(Ig/It)

P (kbar)

ln(Ig/It)

25 25 25 25 50 50 50 50 75 75 75 75 100 100 100 100

0.001 1.1 7.1 15.8 0.001 1.4 7.1 16.7 0.001 2.1 7.0 16.7 0.001 2.5 7.6 16.7

0.821 0.892 1.061 1.190 1.042 1.197 1.250 1.381 1.210 1.373 1.475 1.562 1.535 1.539 1.698 1.823

0.001 5.3 14.8 20.5 0.001 5.5 14.7 21.4 0.001 5.5 14.7 21.4 0.001 6.1 15.8 21.1

-0.538 -0.158 0.119 0.325 -0.643 -0.278 -0.041 0.144 -0.784 -0.529 -0.185 0.040 -0.946 -0.592 -0.375 -0.138

determined using eq 7. Note that this interpolation procedure has the effect of eliminating any curvature that may have been present in the raw pressure dependent data. The resulting pressure derivatives represent ∆VExp at a pressure near P ≈ 2.5 kbar for CP/ether and near P ≈ 5 kbar for DCE/ether. This pressure estimate is based on PHF predictions which suggest that the average ∆V obtained from data collected over a pressure range of 1 atm < P < Pmax should be approximately equal to the value of ∆V at a pressure of P ≈ Pmax/4. The experimental enthalpies ∆HExp do not appear to depend significantly on temperature, and so the resulting slopes may be taken to represent ∆HExp throughout the experimental temperature range. Figure 4 shows experimental and theoretical excess partial molar reaction enthalpies, ∆Hx ) ∆HExp - ∆Hi, plotted as a function of pressure, and the excess partial molar reaction volumes, ∆Vx ) ∆VExp, plotted as a function of temperature. Figure 4a,b contain CP/ether results and 4c,d contain the corresponding DCE/ether results. The points in Figure 4 represent the experimental slopes shown in Figure 3 (using ideal gas isomerization enthalpies, ∆Hi, given in Table 1). The solid lines in Figure 4 represent the predicted ∆Hx and ∆Vx obtained using eqs 6 and 7. The dash and dot-dash lines represent attractive and repulsive PHF predictions, respectively. The theoretical results for CP/ether (Figure 4a,b) correspond to a temperature of 50 °C for ∆H, and a pressure of 2.5 kbar for ∆V predictions. The theoretical results of DCE/ether shown in (Figure 4c,d) correspond to a temperature of 50 °C for ∆H, and a pressure of 5 kbar for ∆V predictions. The PHF cohesive mean field parameters, ∆Ca, were determined by fitting the data shown in Figure 4. The experimental error bars in Figure 4 also roughly correspond the degree of variation in the PHF predictions when ∆Ca is changed by ( 0.05 nm3 kJ/mol (with respect to the best fit ∆Ca values given in Table 1).

Figure 3. Logarithm of the smoothed experimental gauche/trans Raman peak intensity ratios of (a, b) CP and (c, d) DCE plotted as a function of inverse temperature (K-1) at four pressures, and as a function of pressure (kbar) at four temperatures.

Figure 4. Comparisons of the experimental and predicted pressure dependence of the excess isomerization enthalpy ∆Hx and temperature dependence of the excess isomerization volume ∆Vx for (a, b) CP and (c, d) DCE in ether. The dot-dash, dash, and solid curves represent the PHF repulsive, attractive, and total solvent excess thermodynamic functions, respectively, obtained from a global fit to the experimental data (see text for details).

The values of the ideal gas enthalpies ∆Hi, obtained from direct gas-phase measurements, quantum calculations, or by fitting our liquid experimental data to the PHF model (eq 9), often vary by (2 kJ/mol. For example, previously published ∆Hi values range from -1.26 kJ/mol e ∆Hi e 1.97 kJ/mol for CP15,16 and from -5.52 kJ/mol e ∆Hi e -4.6 kJ/mol for

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TABLE 3: Excess Isomerization Thermodynamic Functions Derived from a PHF Analysis of Experimental Gauche-Trans Equilibria 1-chloropropane ∆Hx

(kJ/mol) ∆Vx (Å3) ∆Ux (kJ/mol) ∆Hr (kJ/mol) ∆Vr (Å3) ∆Ur (kJ/mol) ∆Ha (kJ/mol) ∆Va (Å3) ∆Ua (kJ/mol)

1,2-dichloroethane

25 °C 1 atm

75 °C 1 atm

25 °C 10 kbar

75 °C 10 kbar

25 °C 1 atm

75 °C 1 atm

25 °C 10 kbar

75 °C 10 kbar

0.4 1.6 0.4 0.3 1.4 0.3 0.1 0.2 0.1

0.3 2.0 0.3 0.2 1.7 0.2 0.1 0.3 0.1

0.8 0.6 0.5 0.7 0.6 0.4 0.1 0.0 0.1

0.6 0.6 0.2 0.5 0.6 0.1 0.1 0.0 0.1

3.0 7.2 3.0 0.3 1.3 0.3 2.7 5.9 2.7

2.9 10.1 2.9 0.2 1.6 0.2 2.7 8.5 2.7

3.5 1.0 2.9 0.7 0.6 0.3 2.8 0.4 2.5

3.3 1.1 2.6 0.5 0.6 0.1 2.8 0.5 2.5

DCE.5,22 Our quantum calculation results obtained using a MP2/ 6-31 g(d) basis set (Gaussian 98), ∆Hi ) -0.38 kJ/mol for CP and ∆Hi ) -5.84 kJ/mol for DCE, are probably no more accurate. The ∆Hi value we have obtained by fitting our liquid experimental data to the PHF model (eq 9), ∆Hi ) -1.02 kJ/ mol for CP, and ∆Hi ) -4.91 kJ/mol for DCE, are within the range of the above estimates. These latter values are the ones we have used to obtain the experimental and PHF ∆Hx values plotted in Figure 4a,c. Table 3 contains the resulting PHF excess isomerization thermodynamic functions for the isomerization of CP and DCE dissolved in ether at two temperatures and pressures. Note that, since ∆Vi ) 0, the ideal internal energy change is ∆Ui ) ∆Hi (with ∆Hi defined above). The corresponding cavity formation and cohesive contributions to these solvent excess functions, extracted using the PHF analysis of the experimental data, are also given in Table 3. V. Discussion The results shown in Figure 4 demonstrate the PHF model’s ability to reproduce not only the magnitudes of ∆H and ∆V at ambient pressure and temperature, but also the pressure derivative of ∆H and temperature derivative of ∆V. Note that the good agreement between the measured and predicted slopes of ∆H and ∆V represent significant confirmations of the PHF model. Further checks of the self-consistency of the PHF results may be obtained by comparing the excess enthalpy ∆Hx and reaction volume ∆Vx estimates obtained by independent experimental and theoretical methods. Our experimental isomerization enthalpy for DCE/ether, ∆H ) -1.9 kJ/mol (at P ) 1 atm), is close to that predicted using MP2 ab initio reaction field calculations, ∆U ) ∆H ) -2.3 kJ/mol,23 but differs by about 1 kJ/mol from that determined using early IR measurements, ∆H ) -2.9 kJ/mol.17,18 These relatively small differences are probably not significant, particularly given the fact that more recent FT-IR band area measurements for DCE in other solvents22 often differed by over 1 kJ/mol from those inferred from the early IR peak height measurements.17,18 The difference between the excess isomerization enthalpies ∆Hx (or energies ∆Ux) of DCE and CP obtained from our PHF/ experimental results is about 2.6 kJ/mol (at 25 °C and 1 atm), and is nearly pressure and temperature independent (see Table 3). Since CP and DCE have virtually identical size, shape and polarizability, this difference represents the first experimentally derived measure of the effects of solute-solvent dipole-dipole (and possibly higher multipolar) interactions on the isomerization process. In particular, our PHF/experimetal attractive excess enthalpy values are ∆Ha ≈ ∆Ua ≈ 0.1 kJ/mol for CP/ether, ∆Ha ≈ ∆Ua ≈ 2.7 kJ/mol for DCE/ether at 25 °C and 1 atm. These PHF/

experimental values agree remarkably well with independent estimates of the solvent configuration averaged dipolar solvation energy difference between the trans and gauche isomers, ∆Ua ≈ 0 kJ/mol for CP/ether and ∆Ua ≈ 2.8 kJ/mol for DCE/ether39 (see Appendix for details). Such agreement is not only a strong confirmation of the PHF model, but also suggests that quadrupolar and higher order multipolar solute-solvent interactions do not significantly influence the excess reaction energy. These conclusions contrast with early dielectric continuum solvation energy calculations for DCE/ether19 which predicted a dipolar isomerization energy of ∆Ua ≈ 6.6 kJ/mol and a quadrupolar contribution of opposite sign and nearly equal magnitude (∼ -4.6 kJ/mol). More recent ab initio reaction field continuum calculations predict ∆Ua ≈ 2.1 kJ/mol for the sum over dipolar and all multipolar contributions to the excess isomerization energy of DCE in ether.23 The good agreement between the latter continuum calculations and our molecular dipolar solvation estimates suggest that dipole-dipole interactions are the primary contributors to ∆Ua (and thus that higher order multipolar contributions are either all small or else effectively cancel each other). Furthermore, since none of the above continuum calculations include cavity formation contributions to the isomerization processes they could not be expected to accurately predict the effect of pressure on the isomerization equilibrium. The previously reported value of ∆V ) ∆Vx ) 1.8 Å3 for pure liquid CP at T ) 25 °C obtained from data spanning a pressure range of P e Pmax ≈ 5 kbar7 is slightly higher than our value of ∆V ) 0.9 ( 0.4 Å3 for CP in ether obtained T ) 25 °C with Pmax ≈ 10 kbar. A likely explanation for this difference is offered by the PHF model, which predicts that ∆Vx should be pressure dependent, in agreement with previous theoretical and experimental studies of other systems.28-30 In particular, when our PHF/experimental results are extrapolated to ambient pressure they predict a reaction volume of ∆Vx ≈ 1.6 Å3 at 25 °C (see Table 3), which is much closer to the above experimental value for pure CP. However, it is also important to note that since the two results pertain to different solvents, there is no reason to expect exact agreement between the corresponding ∆V values. Even more dramatic evidence for the pressure dependence of ∆V is apparent in the isomerization of DCE. Previous experimental values of ∆V for DCE in various solvents, derived from Raman, IR, and ultrasonic measurements performed over a pressure range of Pmax ≈ 0.001 kbar to Pmax ≈ 10 kbar, vary from 4 Å3 < ∆V < 11 Å3 8,9,11,12 at room temperature (and the largest ∆V value was obtained from the lowest pressure ultrasonic measurements). These ∆V values are all significantly larger than our experimental value of ∆V ) 1.4 ( 0.4 Å3 for DCE in ether at T ) 25 °C with Pmax ≈ 10 kbar. Although the above results again pertain to different solvents, the most likely explanation for our smaller value of ∆V is the higher pressure

Gauche-Trans Isomerization range of our experiments. When our PHF/experimental results (derived from a fit to our high-pressure data) are extrapolated to 1 atm they predict a much larger reaction volume, ∆V ) ∆Vx ≈ 7 Å3, at T ) 25 °C (see Table 3). Thus, there is no apparent inconsistency between the large ∆V obtained from previous low-pressure measurements and our smaller ∆V values measured at high pressure. In fact, the agreement between experimental and PHF predictions for the pressure dependence of ∆V is a further important confirmation of the validity of the PHF theoretical modeling strategy. In addition to explaining pressure dependent differences in experimental ∆V value, the PHF model also clarifies the reasons for the larger ∆V for DCE that CP. The PHF analysis of our experimental results clearly indicates that ∆V for CP is largely dictated by cavity formation energetics, ∆Vx ∼ ∆Vr (see Table 3), while for DCE/ether, both cavity formation and cohesive interactions contribute significantly to ∆Vx ) ∆Vr + ∆Va (see Table 3). The reason that ∆Vr and ∆Va have the same sign is that both cavity formation and cohesive interactions favor the gauche conformer. In particular, ∆Vr is positive because of the slightly larger solute-solvent excluded volume of the trans conformer, while ∆Va is positive because the larger dipole moment of the guache conformer tends to contract the solvent about the gauche state, and thus to further reduce its partial molar volume relative to the trans isomer. VI. Conclusions Pressure and temperature dependent Raman measurements and a PHF theoretical analysis have been used to quantitatively determine both cavity formation and cohesive contributions to the gauche-trans isomerizations of CP and DCE dissolved in ether. The key adjustable parameter of the PHF model is the mean field coefficient, ∆Ca, which represents the difference between the cohesive solvation energies of the two isomers. This would in fact be the only parameter adjusted in fitting PHF predictions to isomerization data, if sufficiently accurate values of the gauche-trans diameter ratio R and gas-phase enthalpy ∆Hi were available. Since this is not the case, R and ∆Hi were treated as an additional adjustable parameter. However, the resulting optimized values of R and ∆Hi remained within the range of independent experimental and/or theoretical estimates. The only other parameters in the PHF model (besides the experimental temperature and solvent density) are the solvent effective hard sphere diameter of, σs, and the average solutesolvent diameter ratio for CP and DCE, dave, which are obtained from independent fits to pure liquid equation of state data. Despite the significant approximations made in implementing the PHF model the resulting analysis confirms that this theoretical approach is capable of explaining all of the observed effects of pressure and temperature on the isomerization thermodynamics. These include not only differences between the solvent excess reaction enthalpies ∆Hx and volumes ∆Vx for the two solutes, but also subtle changes in ∆Hx with pressure and ∆Vx with both pressure and temperature. The excess thermodynamic results shown in Table 3, illustrate the sort of molecular mechanistic information which may be derived from a combined experimental/PHF analysis. The positive excess reaction volume of both solutes reflects the increase in partial molar volume upon unfolding from the gauche to the trans state. Note that the molecular (van der Waals) volumes of the gauche and trans isomers are virtually identical, and so the excess isomerization volume is entirely the result of the larger volume of the solvation shell around the trans isomer. Furthermore, additional larger excess reaction volumes of DCE

J. Phys. Chem. B, Vol. 105, No. 2, 2001 525 than CP reflects the additional contraction of the solvent around the more polar gauche conformer of DCE. Since the volumes and shapes of the two solutes are virtually identical, so should be the corresponding cavity formation contributions to all the reaction thermodynamic functions. This is confirmed by our PHF analysis which predicts very similar ∆Vr, ∆Hr, and ∆Ur for the two solutes. However, the corresponding attractive excess thermodynamic functions differ significantly, as expected, given the different dipolar properties of CP and DCE. In particular, ∆Ua for CP is nearly zero, in keeping with the virtually identical polarizability and dipole moment of the gauche and trans isomers of this compound. On the other hand, the value of ∆Ua ≈ 2.7 kJ/mol for DCE reflects the preferential solvation of the more dipolar gauche conformer, and is in very good agreement with independent molecular dipole-dipole solvation energy estimates, ∆Ua ≈ 2.8 kJ/mol. In fact, such dipolar solvation estimates could have been used to predict ∆Ca values for CP/ether and DCE/ether of, 0.0 kJ nm3/mol and 0.33 kJ nm3/mol, respectively, which are well within the experimental uncertainties of the ∆Ca values derived from the PHF analysis (see Table 1). Thus, the experimentally fit values of all three PHF parameters, ∆Ca, R and ∆Hi, are found to closely agree with independent theoretical and/or experimental estimates. Such self-consistency suggests that the approximations made in implementing the PHF model are physically justified. The thermodynamic functions obtained using the PHF analysis represent the first experimentally derived determinations of the solvent configuration averaged repulsive and attractive contributions to the gauche-trans isomerization of CP and DCE (see Table 3). Although these values may depend to some extent on approximations made in implementing the PHF analysis, the differences between CP/ether and DCE/ether, as well as the pressure and temperature dependence of the thermodynamic functions, are likely to be model independent results. Acknowledgment. This work was supported by the National Science Foundation (Grant CHE-9530595). Appendix In the PHF model, cavity formation contributions to the chemical potential of each isomer are determined using a procedure which requires knowledge of the hard sphere diameters of the solvent and solute as well as an independent measure of the excluded volumes of the gauche and trans isomers. In particular, the average hard sphere diameter ratio of the solute and solvent dave is obtained from the ratio of the effective HS diameters of the corresponding pure solvent and solute liquids37 (see Table 1).

dave )

σsolute σsolvent

(A1)

The individual hard sphere diameters of the gauche and trans isomers may be estimated by assuming that these are equal to the diameters of spheres with the same excluded volumes as the true gauche and trans molecules. Specifically, we have used MP2/6-31 g(d) optimized quantum structures (Gaussian 98) and standard atomic van der Waals radii33 to represent the gauche and trans isomers. Excluded volumes are calculated assuming a solvent hard sphere diameter of σs ) 5.388 Å37 using the GEPOL algorithm.40 The solute diameters obtained in this way are used to predict the isomer diameter ratio R.

526 J. Phys. Chem. B, Vol. 105, No. 2, 2001

R)

Mele´ndez-Paga´n et al.

σg dg ) σt d t

(A2)

The resulting value of R ≈ 0.994 ( 0.001 (for both CP and DCE) should represent a lower bound to the optimal effective diameter ratio in the liquid state, since the anisotropy difference between the two isomers is expected to reduce the difference between the corresponding effective hard sphere diameters.1,41 Estimates of the increase in R, performed using the EVA model (given the EVA anisotropies, ζ(trans) ≈ 1.028 and ζ(gauche) ≈ 1.028, and excluded volumes, VE(trans) ≈ 734.6 Å3 and VE(gauche) ≈ 727.9 Å3 41), predict R ≈ 0.996, which is close to the optimal value of R ≈ 0.998, obtained by fitting the experimental pressure dependence of ∆H for CP/ether. To explicitly obtain dt and dg we combine eqs A1 and A2.

dt )

2dave

(A3a)

(1 + R)

dg ) Rdt

(A3b)

The cavity formation (repulsive) chemical potential of each isomer is then obtained from eq 2, using the corresponding reduced solute diameter, d ) dt or dg, and the solvent packing fraction η (obtained from the experimental compressibility of the pure solvent using the CS-vdW equation of state37). The magnitude of the attractive mean field constant ∆Ca may be approximated using the following expression for the spherically averaged dipole-dipole solute-solvent interaction energy difference between the trans and gauche conformers, averaged over all solvent configurations.

∆CaF ≈ ∆Ga ≈ ∆Ua ) 〈Ua〉t - 〈Ua〉g ) Ci =

〈〉 〈〉 Ct r6

-2µ2i µ2s 3(4πo)2kT

-

Cg r6

(A4a)

(A4b)

Note that µi and µs in the above expression are the dipole moments (not the chemical potentials) of the solute (i ) trans or gauche) and solvent molecules, respectively, 0 is the vacuum permittivity, and k is Boltzmann’s constant.42 The solvent configuration averages (〈...〉) in eq A4 may be estimated using hard sphere radial distribution function integrals43 (and assuming for the purpose of these calculations that the solute and solvent have the same effective hard sphere diameter). In particular, since both conformations of CP have the same dipole moment the two terms on the right-hand side of eq A4 cancel to give ∆Ua ≈ 0. For DCE, on the other hand, since the trans conformation of has no dipole moment the following expression may be used to approximate ∆Ua.

〈〉 Cg r6

N AC g 5 =F Jm(Fσ3s )m 3 m)0 σs

∑

(A5)

The coefficients in the above expansion are,43 J0 ) 4.1888, J1 ) 2.8287, J2 ) 0.8331, J3 ) 0.0317, J4 ) 0.0858, and J5 ) -0.0846. When the appropriate solute and solvent dipole moments, solvent diameter and reduced density are used, eq A5 predicts ∆Ua ≈ +2.8 kJ/mol for DCE/ether at 25 °C and 1 atm.39 References and Notes (1) Hu, M.-H. A.; de Souza, L. E. S.; Lee, M.-R.; Ben-Amotz, D. J. Chem. Phys. 1999, 110, 249 and references therein.

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