On the Determination of Partial Molar Polarizations and Dipole

13064

J. Phys. Chem. B 2007, 111, 13064-13074

On the Determination of Partial Molar Polarizations and Dipole Moments of Solutes from Multicomponent Solutions Alone: Experimental and Model Development Using Deutero-Labeled Organic Compounds Martin Tjahjono†,‡ and Marc Garland*,‡ Department of Chemical and Biomolecular Engineering, 4 Engineering DriVe 4, National UniVersity of Singapore, Singapore 117576, and Institute of Chemical and Engineering Sciences, 1 Pesek Road, Jurong Island, Singapore 627833 ReceiVed: June 27, 2007; In Final Form: July 29, 2007

A general inverse problem methodology is introduced to determine the partial molar polarizations and the dipole moments of individual solutes from multicomponent solutions alone. A model quaternary system consisting of three deuterated solutes, for example, acetone-d6, acetonitrile-d3, and dimethylformamide-d7 in cyclohexane at 298.15 K and 0.1013 MPa, was studied. Following an experimental design protocol, multicomponent solutions in the range of concentration 0.0006 < xsolute i < 0.0085 were prepared using a semi-batch procedure by injecting one solute at a time. In situ FTIR spectroscopic measurements of these quaternary solutions were performed together with simultaneous condensed-phase bulk measurements of density, refractive index, and relative permittivity. Three different numerical approaches were used to determine the individual limiting solute molar polarizations from the multicomponent solutions. These limiting molar polarizations were then used to calculate the individual solute dipole moments using the Debye formula. In addition, direct dipole moment calculations were performed using the Guggenheim-Smith formula where individual solute parameters were obtained from multivariate analysis of the multicomponent solution data. Response surface models played a central role in many of the inverse problems. The results of the various methods are compared. In general, the dipole moments of all solutes from multicomponent solutions were in good agreement with those determined from independent binary experiments. Additionally, numerical sensitivity analysis was performed in order to identify the significant contributions to dipole moment uncertainty. The general approach introduced in the present contribution can be applied to a wide range of systems.

Introduction Many bulk physicochemical properties of multicomponent solutions (where the number of solutes g2) can be readily measured. However, the corresponding physicochemical properties of the individual solutes cannot be so easily determined. This situation is rather unfortunate since such information is important in order to characterize individual solutes and to better understand solute-solute and solute-solvent interactions. In particular, such information is important for elucidating solute size,1 packing density,2 dipole moments, and hence conformation and structure,3-5 polarizability,1,3 solubility,6 and so forth. Progress has been made concerning the direct analysis of multicomponent data and the corresponding determination of the associated physicochemical properties of the individual solutes. This progress has been achieved by a combination of careful experimental design, high precision measurements, and a response-surface model based framework for solving the corresponding inverse mathematical problems. Examples include (1) the measurement of solution density and the determination of solute partial molar volumes for nonreactive7,8 and reactive systems,8 (2) the measurement of solution density and refractive index and the determination of solute partial molar volumes, partial molar refractions, electronic polarizabilities, and molec* Corresponding author. Phone: (65) 6796-3947. Fax: (65) 6316-6185. E-mail: [email protected]. † National University of Singapore. ‡ Institute of Chemical and Engineering Sciences.

ular radii,9 and (3) the measurement of solution density and the determination of solute partial molar volumes and binary and higher-order volumes of interaction.10 For a variety of reasons, it is important to determine individual solute properties from the direct analysis of multicomponent data alone, without recourse to independent pure component or binary measurements. These reasons include the possibility of determining higher order interactions in complex systems, as well as the possibility that a particular solute is unstable or nonisolatable (and therefore, an independent measurement is not accessible). The later is frequently encountered in the study of kinetically labile systems, possessing spectroscopically observable and quantifiable intermediates, such as in organometallic chemistry and homogeneous catalysis.11 The determination of higher order interactions in complex systems and the determination of the physicochemical properties of non-isolatable intermediates provide rich opportunities for further understanding solution chemistries. Another important bulk solution property is relative permittivity. In principle, this bulk property can be combined with density and refractive index measurements, in order to determine individual solute partial molar polarizations and dipole moments. This advance would represent a significant extension of multicomponent inverse analysis. Indeed, such information would be useful in estimating the total molecular polarizability of a solute and in the accurate determination of solute conformation (especially when comparing to first-principle estimates of new

10.1021/jp0750046 CCC: $37.00 © 2007 American Chemical Society Published on Web 10/20/2007

Partial Molar Polarizations and Dipole Moments molecules to the experimentally obtained dipole moments). Relative permittivity is acutely sensitive to dipole-dipole interactions, and this presents a considerable challenge. The incorporation of relative permittivity in an inverse problem framework for the analysis of dilute multicomponent solutions will also require exceedingly precise dielectric measurements. A three-terminal capacitance cell which is orders of magnitude more sensitive than currently available commercial cells was designed and constructed.12 Three different modeling approaches can be used to determine the individual limiting solute molar polarizations from the multicomponent solutions. The first method (method 1) is based on the extended relationships of relative permittivity, refractive index, and density versus mass (or mole) fractions as introduced by Hedestrand,13 Halverstadt-Kumler,14 and GuggenheimSmith15 for multicomponent solution systems. The second method (method 2) employs response surface models for total molar polarization. The third method (method 3) uses modified response surface models which are particular useful for analyzing ultra dilute solutions.8 By using the solute limiting molar polarizations obtained, the corresponding dipole moments of the respective solutes can be subsequently determined using the Debye formula. Alternatively, a direct dipole moment calculation employing Guggenheim-Smith formula can also be performed. It should be mentioned that in situ spectroscopic measurements play an important role in the analysis of multicomponent mixtures, since they help to ensure the proper interpretation of inverse problem results. Indeed, it is important to confirm (a) that no observable reaction/degradation of solutes has occurred during the experimental period or (b) that the new observable solutes are properly and quantitatively included in the physicochemical model. In this regard, a nondestructive and quantitative spectroscopy, such as FTIR, is needed for in situ measurements of the liquid phase. Furthermore, in situ spectroscopic measurements can help to ensure that the multicomponent solution is thoroughly mixed and homogeneous and that representative bulk physicochemical measurements are performed. In the present contribution, a quaternary system consisting of three deuterated solutes, namely, acetone-d6, acetonitrile-d3, dimethylformamide-d7 in cyclohexane was studied at 298.15 K and 0.1013 MPa. Multicomponent solutions in the range of concentration 0.0006 < xsolute i < 0.0092 were prepared using a semibatch procedure by injecting one solute at a time following a pre-experimental design protocol.7 Simultaneous measurements of density, refractive index, relative permittivity, and IR absorbance spectra are performed. A series of numerical inverse problems were solved to determine the individual solute limiting partial molar volumes, limiting partial molar refractions, mean electronic polarizabilities, effective molecular radii, limiting molar polarizations, and dipole moments directly from the multicomponent solutions alone. The values obtained from the present deutero-labeled compounds are in agreement with independent binary experiments, and the in situ IR measurements confirm that no observable reaction has taken place in the quaternary system. In addition, some of these deutero-labeled values are compared with binary literature information on the nondeutero-analogs, and close agreement is found. Parametric sensitivity analysis is performed in order to ascertain the most important sources of error and the corresponding magnitude of these errors. The present nested-inverse problem approach appears applicable to a wide range of multicomponent solutions in nonpolar and slightly polar solvents.

J. Phys. Chem. B, Vol. 111, No. 45, 2007 13065 TABLE 1: Experimentally Measured Densities and Refractive Indices of the Pure Components Used in This Study at 298.15 K F/(g‚cm-3) component

exptl

toluene-d8 acetone-d6 acetonitrile-d3 dmf-d7 cyclohexane

0.93951 0.86941 0.83397 1.03770 0.77399

a

literature 0.868a 0.77389b 0.7738c

nD exptl 1.49138 1.35398 1.33892 1.42503 1.42341

literature

1.42354b 1.42359c

Reference 16. b Reference 17. c Reference 18.

Experimental Section Materials. The solvent cyclohexane (Sigma-Aldrich, Chromasolv, 99.7%+) was refluxed over Na/K under argon (Soxal, 99.999%). The argon was purified prior to use by passage through a column containing 100 g of reduced BTS-catalyst (Fluka AG Buchs, Switzerland) and 100 g of 4 Å molecular sieves to adsorb trace oxygen and water, respectively. The solutes toluene-d8 (Cambridge Isotope Laboratories Inc., 99.5% D, CAS: 2037-26-5), acetone-d6 (Cambridge Isotope Laboratories Inc., 99.9% D, CAS: 666-52-4), acetonitrile-d3 (Cambridge Isotope Laboratories Inc., 99.8% D, CAS: 2206-26-0), and dimethylformamide-d7 (Cambridge Isotope Laboratories Inc., 99.5% D, CAS: 4472-41-7) were dried over molecular sieves type 4 Å (Aldrich) and kept under argon. Experimental values of density and refractive index of the pure chemicals are compared to literature values (if any) in Table 1. Equipment. A three-terminal high-precision capacitance cell designed and constructed by Scientifica (Princeton, NJ), and an ultrahigh accuracy and stable Andeen-Hagerling model 2550A capacitance bridge (Cleveland, OH) were used for capacitance measurements. During the measurements, the cell was kept isothermal at 298.15 K in an ISOTECH parallel tube liquid bath, model 915-MWE Neptune, with temperature stability ( 0.001 K at room temperature. The relative permittivity values r (equal to C/Co) were determined from the ratio of the capacitance of the cell with liquid C to that of empty cell under vacuum Co. The measured capacitance of the empty cell under vacuum Co at 298.15 K was 93.83029 pF. This instrumentation provided dielectric measurements with accuracy of better than 10-4 and resolution of better than 10-5. The detailed design of this specially constructed and unusually sensitive capacitance cell has been reported elsewhere.12 Density measurements were performed at 298.15 K in an Anton-Paar DMA 5000 (accuracy of (5 × 10-6 g‚cm-3, resolution of (10-6 g‚cm-3) vibration tube densitometer thermostatically controlled to within (0.001 K. Refractive index measurements were made at 298.15 K with a flow-through Abbemat-HP (accuracy of (2 × 10-5 nD, resolution of (10-6 nD) automatic refractometer, thermostatically controlled to within (0.01 K. Mid-infrared spectra were collected using a Varian 7000 FT-IR spectrometer. The experimental setup consisted of a glass reactor (Aceglass) equipped with a magnetic stirrer and a Teflon membrane pump (Cole-Parmer). The fluid was pumped in a closed recycle loop system under isobaric (Argon, 1.0125 × 105 Pa) and isothermal conditions from the reactor through the pump (temperature control using Polyscience 9105, with temperature stability (0.05 K) to the densitometer, refractometer, high-pressure infrared cell (SS316) situated in a Varian 7000 FT-IR spectrometer, then capacitance cell with recycle back to the reactor. Connections for vacuum and argon were provided. A 2.000 bar piezo-

13066 J. Phys. Chem. B, Vol. 111, No. 45, 2007

Tjahjono and Garland

Figure 1. Experimental setup for simultaneous density, refractive index, relative permittivity, and IR absorbance measurements. Legend: (1) argon tank; (2) argon purification column; (3) pressure transducer; (4) jacketed continuous stirrer tank reactor (CSTR); (5) hermetically sealed Teflon pump; (6) Anton Paar DMA 5000 densitometer; (7) Abbemat-HP refractometer; (8) FTIR with high-pressure flow through cell; (9) data acquisition; (10) capacitance cell; (11) ISOTECH parallel tube liquid bath, model 915-MWE; (12) Andeen-Hagerling, model 2550A capacitance bridge.

transducer (PAA-27W, Keller AG, Switzerland) was used throughout for pressure measurements. The high-pressure infrared cell was constructed at the ETH Zu¨rich of SS316 steel. The CaF2 single-crystal windows used (Korth Monokristalle, Kiel, Germany) had dimensions of diameter 40 mm by thickness 15 mm. Two sets of Viton and Silicone gaskets provided sealing, and Teflon spacers were used between the windows. The construction of the flow through cell is a variation on the design by Noack (1968)19 and differs in some respect from other high-pressure infrared cells.20 The cell chamber was purged with purified nitrogen (Soxal, Singapore, 99.999%). The resolution was set to 4 cm-1. A schematic diagram of experimental setup is shown in Figure 1. Experimental Aspects. A typical experiment consisted of initially transferring the anhydrous solvent cyclohexane (ca. 200 mL) to the reactor under argon. The solution composition was changed according to a predetermined experimental design (provided in Table S2, Supporting Information) by injecting one solute at a time using gas-tight syringes (Hamilton) through a rubber septum. The mass of the small amount of solute injected was determined by using a balance (GR-200, A&D, Japan) with a precision of (10-4 g. In this study, the difference of the masses of the syringes before and after the injection was considered as the mass of the solute injected. The experimental setup provided a closed system that prevented loss of both solutes and solvent. Extra care was taken to exclude moisture and air during all chemical purifications, equipment preparations, transfers of all chemicals, and all other experimental steps. Standard Schlenk techniques were used throughout.21 Because of the liquid volumes of the stirred tank (∼110 mL), capacitance cell (∼80 mL), and other components (∼10 mL), as well as pumping rates used, non-negligible mixing times were observed. Under the experimental flow-rate used (ca. 12 mL/ min), numerical simulations indicated that a time of 45 min

was needed to achieve 99.99+% homogeneity for the fluid elements. The relative permittivity, density, refractive index, as well as IR absorbance measurements, confirmed that 45-60 min were required after a perturbation in order to achieve reproducible measurements. To reduce error, while actually measuring the density, refractive index, dielectric constant, and IR absorbance of the solution, the pump was turned off. Thus, measurements were performed under static and not flow conditions. An approximately 10 min temperature-equilibration time was typically required before measurements could be taken.12 After the measurements, the pump was turned on, and the solution was circulated again throughout the system. Concerning the repeatability of the measurements, the density, refractive index, and relative permittivity of the solution were measured at least 6, 12, and 12 times, respectively. The results reported herein are the average values of these measurements. It should be noted that the maximum deviations of the density, refractive index, and relative permittivity of the solution from the reported average values were approximately (2 × 10-6 g‚cm-3, (2 × 10-6 nD, and (5 × 10-5, respectively. Results and Discussion 1. Dilute Binary Systems. Four different semi-batch dilute binary systems of toluene-d8, acetone-d6, acetonitrile-d3, and dimethylformamide-d7 in cyclohexane (solvent) were studied. The binary solution of toluene-d8 and cyclohexane was measured in order to investigate the sensitivity and accuracy of the physicochemical property determinations. Indeed, toluene-d8 is only slightly polar, and its study in a dilute concentration range (i.e., mole fraction less than 0.008) is especially instructive. The other three binary solutions were measured in order to determine the limiting partial molar volumes, limiting partial molar refractions, limiting partial molar polarizations, as well as dipole

Partial Molar Polarizations and Dipole Moments

J. Phys. Chem. B, Vol. 111, No. 45, 2007 13067

Figure 3. Relative concentration versus actual concentration for 2, toluene-d8; [, acetone-d6; b, acetonitrile-d3; and 9, dmf-d7 from the binary experiments. Solid line (s) represents the regression line.

wavenumber channels, respectively. Figure 2. Pure component spectra of (i) toluene-d8 (ii) acetone-d6 (iii) acetonitrile-d3, and (iv) dimethylformamide-d7 in the C-D stretch region, obtained from the binary experiments in cyclohexane using BTEM.

moments of the relevant deuterated solutes. Subsequently, the results of the three binary experiments were compared to the associated values determined from the more complex multicomponent solution systems. Densities, refractive indices, relative permittivities, and IR absorbance spectra of these four binary solutions were simultaneously measured. Several different concentrations at 298.15 K and 0.1013 MPa were examined for each binary system. The experimentally obtained bulk densities, refractive indices, and relative permittivities were used for determining the limiting partial molar quantities as well as dipole moments of the individual solutes (see Supporting Information, Table S1, for bulk data). The associated infrared absorbance spectra of the binary systems were analyzed for two reasons, to confirm that completely homogeneous and well mixed solutions were achieved and to obtain pure component spectral references for the solutes. Infrared Spectroscopic Analysis. The pure component FTIR spectra of the solutes toluene-d8, acetone-d6, acetonitrile-d3, and dimethylformamide-d7 were deconvoluted from respective binary cyclohexane solutions using the band-target entropy minimization (BTEM) algorithm.22 The IR spectra were analyzed in the wavenumber range of 1900-2300 cm-1, where all primary C-D stretching vibrations of the solutes could be observed. The BTEM results of the pure component spectra of toluene-d8, acetone-d6, acetonitrile-d3, and dimethylformamided7 are shown in Figure 2. These dilute pure component spectra are consistent with literature references of the pure substances.23-26 No further spectral patterns could be reconstructed from the solutions using BTEM. The relative concentrations for each solute were calculated from the matrix form of the Lambert-Beer-Bouguer law as given in eq 1. In eq 1, C ˆ k×s is the estimated relative concentra+ tions, Ak×V is the experimental absorbance spectra, and aˆs×V is the pseudo-inverse of the normalized pure component spectra of species s obtained from BTEM. Subscripts k and V denote the number of absorbance spectra collected and the number of

+ C ˆ k×s ) Ak×V aˆs×V

(1)

The calculated relative concentrations of each solute are compared to their actual substance concentrations (determined from injected mass) as illustrated in Figure 3. The results in Figure 3 show that very linear relationships are obtained for all solutes, and hence, the molar absorptivities of the solutes (proportional to slopes) are essentially constant. These results also indicate that (i) measurements were performed on wellmixed homogeneous solutions, (ii) no significant self-association occurs between solute molecules in the range of concentrations used, and (iii) the low signal-to-noise observed for acetonitriled3 and acetone-d6 (seen in Figure 2) is due to their relatively low molar absorptivities. Although weak dimers of acetone,27 acetonitrile-d3,25 and dimethylformamide28 have been reported at high solute concentrations, neither BTEM analysis nor Figure 3 indicated significant dimer formation. Determination of Limiting Partial Molar Properties. An apparent molar property Yφ,i is commonly determined from the difference of the bulk solution molar property Ym and the solvent molar property Yos as defined in eq 2. In eq 2, xi and xs denote mole fractions of the solute i and the solvent. In the current study, the general property Y can represent volume (V), refraction (R), and polarization (P) which are defined in molar units in eqs 3, 4, and 5, respectively.

Ym - xsYos Yφ,i ) xi

Y ) V, R, or P

(2)

where

mT FnT

(3)

(nD2 - 1) M h 2 F (n + 2)

(4)

(r - 1) M h (r + 2) F

(5)

Vm )

Rm )

D

Pm )



TABLE 2: Limiting Partial Molar Volumes and Limiting Partial Molar Refractions and Their Linear-Fitting Coefficients of the Present Deuterated Solutes in Cyclohexane at 298.15 K and 0.1013 MPa, with Comparison to Literature Values of the Limiting Partial Molar Volumes for the Corresponding Non-Deuterated Solutes solute-i toluene-d8 acetone-d6 acetonitrile-d3 dmf-d7

mole fraction rangea

∞ Vφ,i (cm3‚mol-1)

0.0019-0.0080 (5) 0.0007-0.0067 (6) 0.0009-0.0082 (5) 0.0021-0.0070 (5)

107.92 ( 0.36 (108.9 , 109.6 ) 80.33 ( 0.25 (80.4d, 80.49e) 59.02 ( 0.52 (58.87f) 82.62 ( 0.56 (82.4g) b

c

bV,i (cm3‚mol-1)

∞ (cm3‚mol-1) Rφ,i

bR,i (cm3‚mol-1)

73.9 ( 70.4 -112.6 ( 62.9 22.3 ( 102.3 99.7 ( 118.1

30.52 ( 0.05 16.39 ( 0.07 11.69 ( 0.17 19.79 ( 0.18

22.5 ( 9.8 -46.6 ( 17.5 -10.0 ( 34.9 24.9 ( 37.5

a Values in parentheses show number of semi-batch perturbations (data points). b Toluene-h8 in cyclohexane calculated at xi ) 0.0721, ref 32. Toluene-h8 in cyclohexane calculated at xi ) 0.0589, ref 33. d Acetone-d6 in cyclohexane, ref 34. e Acetone-h6 in cyclohexane, ref 12. f Calculated from limiting partial molar excess volume for acetonitrile-h3, ref 35. g Calculated from limiting partial molar excess volume for dmf-h7, ref 36.

c

As shown in eq 3, the measured bulk density F together with total mass mT and total moles nT provides total molar volume of solution Vm. In eq 4, the experimental bulk refractive index of the sodium D-line nD and density F as well as the mean molar mass M h of the solution as proposed by Lorenz29 and Lorentz30 are used to determine total molar refraction of the solution Rm. In a similar form, the experimental bulk relative permittivity r and density F as well as the mean molar mass M h of the solution are used to determine total molar polarization of solution using the Clausius-Mosotti equation (eq 5).31 The molar properties of the pure solvent can also be determined using eqs 3-5, from the experimental density, refractive index, and/or relative permittivity and the molar mass of solvent at the same temperature and pressure. Using eq 2, the apparent molar volumes, the apparent molar refractions, and the apparent molar polarizations of toluene-d8, acetone-d6, acetonitrile-d3, and dimethylformamide-d7 in cyclohexane were determined from these binary experiments. The ∞ were subsequently deterlimiting partial molar values Yφ,i mined by linear extrapolation to infinite dilution (xi f 0) as shown in eq 6 where the coefficient bY,i physically depends on the solute, solvent, temperature, and pressure. ∞ Yφ,i ) Yφ,i + bY,ixi

Y ) V, R, or P

complement to the limiting partial molar polarizations for the determination of dipole moments. In addition, the obtained limiting partial molar refractions can also be further utilized to calculate mean electronic polarizabilities and effective molecular radii as described elsewhere.9 Other Procedures for Calculating Limiting Partial Molar Polarization. In regard to determining solute limiting partial molar polarization P∞i , other more reliable extrapolation procedures have been proposed by Hedestrand (H)13 (on the basis of the mole fraction xi variant) and Halverstadt and Kumler (HK)14 (on the basis of the mass fraction wi variant). These two extrapolation procedures require a few experimentally determined coefficients from the experimental bulk relative permittivity r, and density F (or specific volume V) as given in eqs 7-10. The expression for the square of refractive index n2 as given in eq 11 will be required in later sections.

(6)

The limiting partial molar volumes and limiting partial molar refractions as well as their linear-fitting coefficients for toluened8, acetone-d6, acetonitrile-d3, and dimethylformamide-d7 in cyclohexane are summarized in Table 2. In addition, the literature values for the limiting partial molar volumes of the non-deuterated solutes are also provided for comparison. A detailed and comparative discussion of the corresponding limiting partial molar polarizations of those solutes is reserved for a later section. Since the limiting partial molar volumes of most of the present deuterated solutes have not been previously reported, the literature for the corresponding non-deuterated solutes are primarily used for comparison. Although isotope effects indeed exist for some physicochemical properties,37-40 the effects on partial molar volumes and molar volumes were found to be very small, typically on the order of 1%.34,37-39 As shown in Table 2, the presently reported limiting partial molar volumes of deuterated solutes are in close agreement with those of the corresponding non-deuterated solutes. These results are consistent with previous conclusions regarding the small isotope effect found for limiting partial molar volumes. Finally, it should also be noted that the literature values of non-deuterated toluene could only be evaluated at relatively high concentration (i.e., xi > 0.05) since the relevant dilute data are not available. To the best of our knowledge, literature values for limiting partial molar refraction are not available for the present deuterated solutes nor the corresponding non-deuterated solutes. The present limiting partial molar refractions are a useful

r ) r,s + ai.xi

(7)

F ) Fs + bi.xi

(8)

r ) ′r,s + Ri.wi

(9)

V ) Vs + βi.wi

(10)

n2 ) ns2 + γi.wi

(11)

The resulting Hedestrand and Halverstadt-Kumler equations for limiting partial molar polarization P∞i are provided in eqs 12 and 13 respectively where Mi and Ms are the molar mass of solute and solvent.

P∞i )

3aiMs (r,s + 2) Fs

P∞i )

2

+

3RiVsMi (′r,s + 2)

2

( ) ( )

+

r,s - 1 (Mi - Msbi/Fs) r,s + 2 Fs ′r,s - 1 M (V + βi) ′r,s + 2 i s

(12)

(13)

Analysis of the density, refractive index, and relative permittivity data for the binary mixtures (Table S1) provides the results reported in Table 3. These results include the slopes obtained for the extrapolation procedures (using eqs 7-11) as well as the values of the limiting partial molar polarization for the deuterated solutes determined from three different methods. It can be seen from Table 3 that the solute limiting molar polarizations determined using the Hedestrand (H) or Halverstadt-Kumler (H-K) equations are very similar. It is also seen that the solute limiting molar polarizations determined using eq 6 differ by less than 2.5%. Dipole Moment Analysis. The dipole moment of a solute µi (Debye units, D) is commonly determined using the Debye formulas (eq 14). The limiting partial molar refractions



TABLE 3: Polarization Dataa of the Deuterated Solutes in Cyclohexane at 298.15 K and 0.1013 MPa According to Hedestrand (H) and Halverstadt-Kumler (H-K) Equations as Well as Eq 6 ∞ P∞i or P,φi (cm3‚mol-1)

solute-i

ai

bi (g‚cm-3)

Ri

βi (cm3‚g-1)

γi

H

H-K

eq 6

toluene-d8 acetone-d6 acetonitrile-d3 dmf-d7

0.3184 7.7318 11.0966 14.5005

0.1499 0.0242 -0.0171 0.1477

0.2675 10.1305 21.1000 15.2219

-0.2099 -0.0529 0.0543 -0.2584

0.1270 -0.2750 -0.3277 -0.0654

33.85 176.56 239.41 314.19

33.86 176.29 238.44 314.09

33.91 ( 0.08 172.45 ( 1.33 235.91 ( 1.33 309.63 ( 0.77

a

See text for all symbols.

TABLE 4: Experimentally Determined Dipole Moment of Some Deuterated Solutes in Cyclohexane at 298.15 K and 0.1 MPa Using Several Methods µi (D) solute-i b

toluene-d8 acetone-d6 acetonitrile-d3 dmf-d7

H

H-K

0.297 2.791 3.332 3.787

eq 6

0.298 2.788 3.325 3.786

literaturea

G-S

0.302 2.773 3.306 3.757

0.406 2.797 3.326 3.789

0.37c 2.78d, 2.750e, 2.75f 3.34g, 3.51 3.68, 3.80h

a Literature values for non-deuterated solutes in cyclohexane at 298.15 K in ref 42. Other conditions are specified. b Very sensitive to Pa + Pe approximation, µ may vary ( 0.1 D for Pa + Pe ) (1.00 to 1.05) R∞i . c Pa is assumed to be 0% of Pe. d At 293.15 K. e Reference 12. f In hexane at 293.15 K. g In hexane. h In heptane at 293.15 K.

R∞i (cm3‚mol-1) and limiting partial molar polarizations P∞i (cm3‚mol-1) of the solutes previously shown in Tables 2 and 3 were used together with the experimental absolute temperature T (K) to calculate solute dipole moment.

µi ) 0.0128((P∞i - 1.05R∞i )T)0.5

(14)

In the present dipole moment analysis, total distortion polarization (contributions from atomic and electronic polarizations) was assumed to be equal to 1.05 times the limiting partial molar refraction R∞i (measured using sodium D-line source), as previously suggested by other researchers (Le Fevre, 1953; Few and Smith, 1949; Exner, 1981; and Tanaka et al., 1996).41 The various limiting solute molar polarization values (as provided in Table 3) were used to calculate the solute dipole moments, and the results are summarized in Table 4. In addition, the solute dipole moment µi (Debye unit, D) could also be directly determined using the Guggenheim and Smith15 (G-S) equation (eq 15) using some of the coefficients found in Table 3 (Guggenheim, 1949, 1951; Smith, 1950). In eq 15, T is the experimental absolute temperature (K). These calculated dipole moments are also provided in Table 4. It is important to note that in the G-S equation, the sum of atomic and electronic polarizations are implicitly assumed to be equal to the limiting partial molar refraction.43

µ ) 0.02218

(

MiVs(Ri - γi) (′r,s + 2)

2

)

0.5

T

(in Debye unit) (15)

As shown in Table 4, the dipole moment of toluene-d8 is very sensitive to the total atomic and electronic polarizations approximation used. The dipole moment of toluene-d8 varies (0.1 D depending on the approximation factor used (1.00 R∞i to 1.05 R∞i ). This illustrates the need for very accurate atomic polarization in order to determine relatively low solute dipole moments (i.e., µ < 1.0). However, the dependency on the atomic polarization approximation becomes less important when rela-

tively high solute dipole moments (i.e., µ > 1 D) are determined (see Discussion for more detailed parametric sensitivity analysis). The current contribution appears to be the first report for the dipole moments of these deuterated solutes determined in a nonpolar solvent. Since it is recognized that the isotope effect on the molecular dipole moment is quite small, only on the order of 0.01 D,44 some literature dipole moment values for the corresponding non-deuterated solutes are also presented in Table 4 for comparison. 2. Multi-Component Systems. Three sets of semi-batch experiments consisting of 10 different compositions each were performed according to a previously developed semibatch experimental design.7 In the current study, 10 perturbations were performed for each of the semibatch experiments. All experiments were started with pure cyclohexane as a major component (solvent) under argon, and subsequently one solute was added at a time to change the solution composition. The constraints imposed for the dilute solution region in the current study were as follows:

0 e xi e 0.0092

for i ) 1, 2, 3

0.98 e x4 e 1

(16) (17)

N

xi ) 1 ∑ i)1

(18)

where N is the total number of components (in this study, N ) 4). The subscript i refers to the solutes acetone-d6 (1), acetonitiled3 (2), dimethylformamide-d7 (3), and the solvent, cyclohexane (4). The compositions obtained from the actual experiments and their corresponding densities, refractive indices, and relative permittivities are provided in Supporting Information (Table S2). In addition to these measurements, the IR absorbance spectra of these multicomponent solutions were also simultaneously taken. DeconVolution of IR Multicomponent Spectra. For each composition, the infrared absorbance spectra of the solution mixtures were measured in the range of 1900-2300 cm-1. These spectra were first pre-conditioned in order to subtract the moisture, CO2, and solvent spectra contributions45 as well as to correct the baseline.46 As an illustration, one of the multicomponent absorbance spectra (taken from step 9 of semibatch run I) before and after preconditioning is given in Figure 4. This figure shows that most of the IR absorbance comes from the solvent. The reconstructions of the pure component spectra were performed using BTEM. The results can be seen in Figure 5. In order to compare the spectral similarity of these spectral estimates with the corresponding references (see Figure 2iiiv), the inner products between the unit vectors were calculated. The resulting inner products between the reconstructed pure component spectra of acetone-d6, acetonitile-d3, and dimethyl-


Tjahjono and Garland TABLE 5: Limiting Partial Molar Volumes of Acetone-d6, Acetonitrile-d3, and Dmf-d7 in Cyclohexane at 298.15 K and 0.1013 MPa Determined from Multicomponent Solutionsa solute-i

V h ∞i (cm3‚mol-1)

R h ∞i (cm3‚mol-1)

acetone-d6 acetonitrile-d3 dmf-d7

80.16 ( 0.23 59.15 ( 0.23 83.15 ( 0.22

16.42 ( 0.07 11.67 ( 0.07 20.02 ( 0.07

a

Vos ) 108.723 ( 0.002 (cm3‚mol-1) and Ros ) 27.7064 ( 0.0006 ‚mol-1)

(cm3

Figure 4. Absorbance spectra of multicomponent system of acetoned6, acetonitile-d3, dmf-d7 in cyclohexane (i) before and (ii) after preconditioning (run I step 9).

will not negatively affect the determination of the present limiting physicochemical values. The present methodological development includes rather general and sufficiently robust numerical constructs to accommodate intermolecular interactions (vide infra). Analysis of Total Molar Volume and Total Molar Refraction. In order to determine the limiting partial molar volumes and limiting partial molar refractions of the individual solutes from multicomponent solutions, the response surface models which have been proposed earlier were used.7,9,10 For this dilute quaternary solution, a simple linear response surface model for total molar volume Vm(T,P,x) and a simple linear response surface model for total molar refraction Rm(T,P,x) as shown in eq 19 were used. The use of more complex response surface models involving higher interaction terms is unnecessary as this can be confirmed by PRESS statistical values.47 The multicomponent data in Table S2, excluding the binary data measurements, were used to calculate total molar volume using eq 3 and total molar refraction using eq 4. The linear response surface models for total molar volume and refraction (eq 19) were regressed for the coefficients δV,i and δR,i, respectively. The limiting partial molar volume and limiting partial molar refraction for all solutes could be subsequently determined from the obtained coefficients using the general relationship shown in eq 20. In addition, the coefficients also provide information for the pure molar volume and pure molar refraction of the solvent as shown in eq 21. The determined solute limiting partial molar volumes and limiting partial molar refractions and the pure molar volume and pure molar refraction of the solvent are provided in Table 5. N

Ym(T,P,x) ) Figure 5. The pure component spectra of (i) acetone-d6 (ii) acetonitriled3 and (iii) dimethylformamide-d7 in cyclohexane obtained from experimental multicomponent absorbance data using BTEM.

formamide-d7 (from the multicomponent solutions) and those obtained from the binary solutions were found to be 0.9810, 0.9855, and 0.9955, respectively. This result indicates that there is a high degree of similarity between these two sets of spectral estimates. However, it should be recognized that the spectrum of a solute, say acetone-d6 in cyclohexane, should be slightly different than the spectrum of acetone-d6 in cyclohexane with two additional solutes present since the amplitudes and positions of infrared vibrations are very sensitive to local compositions. Since the pure component spectra of only three solutes were obtained, the analysis suggests that there is no observable reaction between the constituents on the time scale of these experiments. However, it should be noted that there exists an apparent shoulder at 2255 cm-1 for acetone-d6 in these multicomponent solutions (in comparison to acetone-d6 in cyclohexane), and this may be due to a more specific type of intermolecular interaction. If such an interaction does exist, it

δY,ixi ∑ i)1

Y ) V or R

(19)

Y h ∞i (T,P) ) δY,i

(20)

Yos (T,P) ) δY,4

(21)

It can be seen from Table 5 that the all limiting partial molar volumes and limiting partial molar refractions obtained from the multicomponent solutions are consistent with those determined separately from binary solutions (Table 2). Analysis of Mean Electronic Polarizabilities and EffectiVe Molecular Radii. Further, the solute or solvent mean electronic polarizabilities and effective molecular radii can also be determined from the limiting partial molar refractions of the solute or pure molar refraction of the solvent following calculations which have been described elsewhere.9 Using a similar treatment, the mean electronic polarizabilities and the effective molecular radii of each solute i and solvent were calculated, and the results are presented in Table 6. It is important to note that both mean electronic polarizabilities and effective molecular radii are important properties that indicate



TABLE 6: Molar Refraction Roi and Limiting Partial Molar ∞,e Refractions R h ∞i , Mean Electronic Polarizabilities Ro,e i and Ri , o ∞ and Effective Molecular Radiia ai and ai of Component i at 298.15 K and 0.1 MPa Calculated from Multicomponent System component i

h ∞i Roi or R (cm3‚mol-1)

Roi ,e or R∞i ,e × 1023 (cm3)

acetone-d6 acetonitrile-d3 dmf-d7 cyclohexane

16.42 ( 0.07 11.67 ( 0.07 20.02 ( 0.07 27.7064 ( 0.0006

0.651 ( 0.003 0.463 ( 0.003 0.794 ( 0.003 1.098 ( 0.000

aoi or a∞i (Å) 1.867 ( 0.001 1.666 ( 0.001 1.994 ( 0.001 2.222 ( 0.000

the ability of the molecular orbitals to be deformed under an electrical field48 and the size of molecule, respectively. Analysis of Total Molar Polarization. In order to determine the limiting partial molar polarization of each solute from the multicomponent data, three different methods (methods 1-3) are proposed. Method 1 is based on the extension of the relationships of relative permittivity, square of refractive index, and density versus mass (or mole) fractions from the binary case to the multicomponent case. Method 2 is based on the response surface models for total molar polarization. Method 3 is the modified forms of the response surface models. Method 1. In dilute binary solution (typically for xi less than 0.005), the relationships of relative permittivity, density or specific volume, and square of refractive index versus solute mass (or mole) fraction (in eqs 7-11) are typically very linear, unless strong association occurs.43 It is therefore reasonable to assume additivity and thereby extend these physical property relationships versus both mass and mole fractions for multicomponent systems. These extended formalisms are presented in eqs 22-26. It should be mentioned that similar relationships for relative permittivity versus mass fraction (eq 24) in multicomponent systems was previously used by Jenkins and Smith (1970)49 to address monomer-dimer formation as well as acidbase complex formation.

r ) r,s +

∑ ai.xi

(22)

solute i

∑

F ) Fs +

N

Pm(T,P,x) )

δP,i xi ∑ i)1

N

Pm(T,P,x) )

(27)

N-1 N

δ′P,i xi + ∑ ∑ δP,ij xixj ∑ i)1 i)1 j>i

(28)

The multicomponent data from Table S2 (Supporting Information), excluding the binary data, were used to first calculate total molar polarization of the multicomponent solutions using eq 5. Subsequently, both response surface models for total molar polarizations (eqs 27 and 28) were regressed to obtain coefficients δP,i from the linear model as well as coefficients δ′P,i and δP,ij from the linear-bilinear model. These coefficients were used to determine the limiting solute molar polarization using eq 29 and eq 30 for the linear and linear-bilinear model, respectively. Details of the derivations of eqs 29 and 30 from the total molar polarization, which are analogous to the derivation using total molar volume, can be found elsewhere.7 The limiting partial molar polarizations for each solute derived from the linear and linear-bilinear models are reported in Table 8 and denoted as method 2A and 2B, respectively.

bi.xi

(23)

∑ Ri.wi

(24)

∑ βi.wi solute i

(25)

P h ∞i (T,P) ) δP,i

(29)

(26)

P h ∞i (T,P) ) δ′P,i + δP,i4

(30)

solute i

r ) ′r,s +

solute i

V ) Vs +

or the Halverstadt-Kumler equation (eq 13). The solute limiting partial molar polarizations determined from multicomponent solutions are summarized in Table 8. Comparison of these two different extrapolation procedures shows that the results calculated using the Hedestrand equation are slightly more consistent with an average relative error of approximately 2% , whereas those calculated using the Halverstadt-Kumler equation have errors of approximately 2.4%. Both relative errors were calculated with respect to the solute limiting partial molar polarization given in Table 3. Method 2. The multicomponent response surface models previously used for total molar volume and total molar refraction (references) are presently re-formulated for total molar polarization Pm(T,p,x). In this regard, two response surface models, namely, a linear (eq 27) and linear-bilinear (eq 28) response surface model were used to describe the surface maps for total molar polarization in this dilute quaternary system. In both eq 27 and eq 28, extensions to the number of components N (including solvent) are used. For dilute systems, an additive or linear relationship for total polarization (equivalent to eq 27) has been previously used.50

n2 ) ns2 +

∑ γi.wi solute i

The multicomponent data in Table S2 (Supporting Information), excluding the binary data, were analyzed using eqs 2226. All obtained coefficients are provided in Table 7. Subsequently, the respective coefficients for each solute and the corresponding intercepts were used to calculate solute limiting molar polarization using either the Hedestrand equation (eq 12)

In addition to the partial molar polarization for the solutes, the pure molar polarization for the solvent can also be determined using this approach and obtained from the coefficients δP,4 in eq 27 and δ′P,4 in eq 28. The corresponding values were 27.6337 ( 0.0135 cm3‚mol-1 and 27.5062 ( 0.0134 cm3‚mol-1, respectively. The PRESS statistical values for linear and linear-bilinear response surface models were 0.0848 and

TABLE 7: Polarization Data of Some Deuterated Solutes in Cyclohexane at 298.15 K from Multicomponent Data solute i

ai

bi (g‚cm-3)

Ri

βi (cm3‚g-1)

γi

acetone-d6 acetonitrile-d3 dmf-d7

7.662 ( 0.052 10.703 ( 0.053 14.227 ( 0.051

0.019 ( 0.002 -0.016 ( 0.002 0.146 ( 0.002

9.985 ( 0.071 20.243 ( 0.104 14.888 ( 0.056

-0.042 ( 0.004 0.050 ( 0.005 -0.254 ( 0.003

-0.261 ( 0.002 -0.323 ( 0.003 -0.061 ( 0.003

a Intercept data: r,s ) 2.0182 ( 0.0004 (eq 21); Fs ) 0.77410 ( 0.0001 g‚cm-3 (eq 22); ′r,s ) 2.0186 ( 0.0005 (eq 23); Vs ) 1.29182 ( 0.00002 cm3‚g-1 (eq 24); n2s ) 2.02600 ( 0.00001 (eq 25).



TABLE 8: Limiting Partial Molar Polarization of Some Deuterated Solutes in Cyclohexane at 298.15 K Determined from Multicomponent Solutiona P∞i solute i

method

acetone-d6

1

acetonitrile-d3

2A 2B 3A 3B 1

or

P h ∞i

H

H-K 173.96 (-1.32)

1

extrapolation

163.86 (-4.98) 170.62 (-1.06) 172.99 (0.31) 173.88 (0.83) 229.08 (-3.93)

2A 2B 3A 3B dmf-d7

217.69 (-7.73) 247.15 (4.76) 224.91 (-4.66) 239.59 (1.56) 308.45 (-1.83)

2A 2B 3A 3B

∆P ) Pm(T,P,x) - Pos xs )

(cm3‚mol-1)

175.08 (-0.84)

231.19 (-3.44)

tion are given eqs 31 and 32, respectively.

307.40 (-2.13) 291.22 (-5.94) 322.10 (4.02) 297.85 (-3.81) 310.32 (0.22)

a Value in parentheses is the % relative error compared to the limiting solute molar polarization from binary system (Table 3) obtained using the same extrapolation procedure.

0.0188, respectively. The lower PRESS statistical value of the linear-bilinear surface model suggests that the total molar polarization of this dilute quaternary system should be better represented by this model rather than by using the linear surface model. The use of a better surface representation results in more reliable extrapolation values. As expected, the solute limiting partial molar polarizations derived using the linear-bilinear model are more accurate than those derived using a simple linear model. Method 3. The above-mentioned response surface models can be rewritten in terms of the solute contributions alone. Such an approach has been previously used to describe total volume in multicomponent dilute solutions.8 The resulting linear and linear-bilinear response surface models for total molar polariza-

∑

φP,i xi

(31)

solute i

∆P ) Pm(T,P,x) - Pos xs )

∑

φ′P,i xi +

solute i

∑ ∑

φP,ij xixj (32)

solute i solute j>i

It should be noted that these modified forms assume that the molar polarization of the solvent Pos remains constant or changes only slightly when composition is changed. The molar polarization of the solvent can be calculated using eq 5 from the solvent density and initial relative permittivity measurement taken at the beginning of each semibatch experiment. The multicomponent data from Table S2 (Supporting Information), excluding the binary data, were used to calculate total polarization contributed by the solutes ∆P as shown by the left hand side (LHS) of eqs 31 and 32. Next, both modified response surface models were regressed to obtain the coefficients φP,i from the linear model as well as the coefficients φ′P,i and φP,ij from the linear-bilinear model. The solute limiting partial molar polarizations are now equal to the coefficients φP,i of the modified linear model as well as the coefficients φ′P,i of the modified linear-bilinear model. The solute limiting partial molar polarizations obtained from these modified linear and linearbilinear models are provided in Table 8 and denoted as the results of method 3A and 3B, respectively. Dipole Moment Analysis. The dipole moment of each solute was subsequently calculated using the Debye equation (eq 14). The total atomic and electronic polarizations were set equal to 1.05 of the limiting partial molar refractions (also determined from the multicomponent solution analysis in Table 5). The various limiting solute molar polarization values determined using the three different methods (as summarized in Table 8) were used accordingly to calculate the respective solute dipole moments. In addition, the dipole moment determination using the Guggenheim and Smith equation (eq 15) was also performed. Some coefficients provided in Table 7 were used for this calculation. All dipole moment results are summarized in Table

TABLE 9: Dipole Moment of Some Deuterated Solutes in Cyclohexane at 298.15 K Determined from Multicomponent Solutions Using Various Methodsa µi (D) solute i acetone-d6

acetonitrile-d3

dmf-d7

a

methodb

H

H-K

1 2A 2B 3A 3B B 1 2A 2B 3A 3B B 1 2A 2B 3A 3B B

2.778 (-0.48)

2.768 (-0.74)

2.791 3.271 (-1.82)

3.332 3.748 (-1.02)

3.787

2.788 3.256 (-2.09)

3.325 3.741 (-1.18)

3.786

extrapolation

G-S 2.774 (-0.82)

2.677 (-2.81) 2.738 (-0.60) 2.759 (0.16) 2.767 (0.45) 2.773 3.169 (-4.15) 3.388 (2.49) 3.224 (-2.49) 3.334 (0.82) 3.306 3.634 (-3.28) 3.836 (2.09) 3.678 (-2.10) 3.760 (0.08) 3.757

2.797 3.258 (-2.06)

3.326 3.745 (-1.15)

3.789

Value in parentheses is the % relative error compared to the corresponding dipole moment from binary system, B (Table 4), according to the same method of calculations. b Methods 2A and 2B represent the dipole moment calculated using limiting solute partial molar polarization derived from eqs 27 and 28, respectively. Similarly, methods 3A and 3B represent the dipole moment calculated using limiting solute partial molar polarization derived from eqs 31 and 32, respectively



TABLE 10: Sensitivity Analysis of Some Factors Involved in the Dipole Moment Calculation of Acetone-d6 in Cyclohexane at 298.15 K from a Multicomponent System Hedestrand

Halverstadt-Kumler ∆µi

r,s ( 2σ ai ( 2σ Fs ( 2σ bi ( 2σ R h ∞i ( 2% factor (1.0-1.1)a total error µi (D) error in percentage a

(0.00048 (0.01855 (5.6 × 10-5 (0.00102 (0.00304 (0.00723 0.0303 2.778 1.09

Guggenheim-Smith ∆µi

′r,s ( 2σ Ri ( 2σ Vs ( 2σ βi ( 2σ R h ∞i ( 2% factor (1.0-0.1)a

(0.00049 (0.01916 (4.8 × 10-5 (0.00102 (0.00305 (0.00726 0.0310 2.768 1.11

∆µi ′r,s ( 2σ Ri ( 2σ Vs ( 2σ n2s ( 2σ γi ( 2σ

(0.00031 (0.01910 (4.3 × 10-5 (6.9 × 10-6 (0.00061 0.0201 2.774 0.72

Factor represents Pa + Pe ) (1.0 to 1.1)‚ R h ∞i approximation.

9 and compared to those determined from previous binary experiments. It can be seen from Table 9 that all dipole moments determined from the multicomponent solution are, in general, consistent with those determined from the binary solutions (B). Comparing their relative errors, the dipole moments derived from the modified linear-bilinear response surface model (method 3B) clearly outperform those determined by other methods. This demonstrates that the modified forms of the response surface models are useful and effective for dilute multicomponent solution systems. The results in Table 9 also show that the dipole moments calculated from method 1 are slightly more accurate than those determined using both linear (method 2A) and linear-bilinear response surface (method 2B) models. Although the dipole moments obtained from method 1 are slightly less accurate than those calculated from the modified linear-bilinear model (method 3B), it should be noted that method 1 actually offers two favorable practical aspects. These are that method 1 (i) permits direct analysis using the raw experimental density, refractive index, and relative permittivity data and , more importantly, (ii) avoids the use of nonlinear model terms. The latter could, in turn, significantly reduce the number of experimental measurements needed and thus simplify the experimental design required. 3. Parametric Sensitivity Analysis in Dipole Moment Determination. An analysis was performed to identify the dominant source of numerical sensitivity in the Hedestrand, the Halverstadt-Kumler, and the Guggenheim and Smith methods for dipole moment determinations. Errors or uncertainties in the experimental measurements, the multivariate analyses, and the approximation for atomic polarization were examined. As an illustration, sensitivity analysis was performed for the dipole moment of acetone-d6 from a multicomponent system. The results are given in Table 10. On the basis of the factors considered in Table 10, it is apparent that an upper bound for the dipole moment uncertainty of approximately 0.7-1.1% can be expected using the various equations for method 1. Furthermore, it is clear that the largest single contributing factor to dipole moment error is the relative permittivity dependency on concentration, that is, Ri or ai. This term alone accounts for approximately 60-95% of the uncertainty, depending on the model used. The second (ca. 25%) and third (ca. 10%) most important terms affecting the numerical sensitivity are the approximation factor used in determining total distorted polarization (i.e., factor in approximating Pa + Pe) and the determination of the limiting partial molar refraction R h ∞i , respectively. The above analysis emphasizes the need for extremely sensitive dielectric measurements for dilute solution studies and

the usefulness of the specially designed and constructed threeterminal capacitance cell used in this study. The capacitance cell used in this study had an accuracy of better than 10-4 for relative permittivity, and this value significantly out-performs the accuracy of approximately 10-3 of many commercial instruments. Conclusion Simultaneous measurements of density, refractive index, relative permittivity, and IR spectra were performed in this study using an on-line experimental setup. In the numerical analysis, multiple inverse problems in volumetric, optical, and electrical properties were successfully solved using multicomponent data alone, without recourse to any pure or binary experimental data. As a result, accurate determination of several thermodynamic and physicochemical solute properties, such as partial molar volume, partial molar refraction, mean electronic polarizability, effective molecular radius, partial molar polarization, and dipole moment were achieved. The values obtained from the present deutero-labeled compounds are in agreement with independent binary experiments, and the in situ IR measurements confirm that no observable reaction has taken place in the quaternary system. In addition, some of these deutero-labeled values were compared with binary literature information on the nondeuteroanalogs, and close agreement is found. The multicomponent spectroscopic data were also analyzed and successfully deconvoluted to obtain each pure component solute spectrum using BTEM, a spectral reconstruction algorithm. This latter result suggests that in the future, the physicochemical solute properties of non-isolatable species can be determined from multicomponent solutions as well. Acknowledgment. M.T. thanks the Singapore Millennium Foundation for an SMF scholarship. Funding and resources for the present research was provided under the Advanced Reaction Engineering, Process Analytics and Chemometrics program of ICES. Supporting Information Available: Experimental data of densities, refractive indices, and relative permittivities for each corresponding solute mole fractions of the binary solutions and the multicomponent solutions measurements. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Glasstone, S. Textbook of physical chemistry; Van Nostrand: London, 1946. (2) King, E. J. J. Phys. Chem. 1969, 73, 1220-1232.

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On the Determination of Partial Molar Polarizations and Dipole

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