Ind. Eng. Chem. Res. 2004, 43, 3205-3215
3205
Modeling the PVTx Behavior of the N-Methylpyrrolidinone/Water Mixed Solvent Begon ˜ a Garcı´a, Santiago Aparicio, Rafael Alcalde, Marı´a J. Da´ vila, and Jose´ M. Leal* Departamento de Quı´mica, Universidad de Burgos, E-09001 Burgos, Spain
An experimental setup for the measurement of densities for the compressed liquid phase of the N-methylpyrrolidinone/water binary solvent from 278.15 to 358.15 K over the full composition range and for pressures up to 60 MPa is reported. Experimental measurements were performed with a recently developed high-pressure vibrating-tube densitometer; the calibration procedure of the apparatus and the reliability and accuracy of the data provided were checked against literature values. The correlation of the density data with pressure and temperature was performed at each composition using the TRIDEN 10-parameter equation. From the experimental data, several derived properties were calculated and analyzed in terms of structural effects and intermolecular interactions. Likewise, the abilities of several cubic equations of state combined with several mixing rules to correlate the measured PVTx data were tested. Introduction Knowledge of the effects of pressure and temperature (PVT behavior) on the density of pure and mixed solvents is an important tool, because such information provides the key data needed for the design of operations and industrial plants involving such solvents, pipelines, and pumps among others; such knowledge is also interesting from a theoretical point of view, as it can provide deeper insight into the structure of liquids and their intermolecular interactions. Over the past several years, a number of chemical processes and separation operations have been designed under highpressure conditions for economic reasons;1 hence, a systematic study of the PVT behavior of the solvents most often used has become a valuable task. However, a detailed survey of the available literature reveals that the majority of the thermodynamic data published so far were reported under atmospheric-pressure conditions and within a somewhat narrow temperature range, which might be insufficient to satisfy the increasing industrial demands for high-pressure data. On the other hand, evaluation of the PVTx behavior of mixed solvents often is a time-consuming task that requires careful handling to yield reliable results; hence, the availability of models and theoretical procedures capable of predicting solvent behavior over wide pressure and temperature ranges also is desirable. Previously, systematic research was reported on the ability of different models to predict the properties of lactam-containing (cyclic amide) mixtures.2,3 N-Methylpyrrolidinone (NMP), the most important lactam, has applications in molecular biology and serves as a model molecule for more complex peptide compounds.4 NMP, pure or mixed with other solvents, often is used in applications such as extractive distillation,5,6 absorption,7 gas desulfurization,8 and coal extraction.9 The NMP/water (W) system also is an important mixed solvent.10,11 For instance, using this solvent system, the interpretation of some properties of peptide compounds * To whom correspondence should be addressed. Tel.: +34 947 258 819. Fax: +34 947 258 831. E-mail:
[email protected].
in aqueous solutions can be performed successfully;12-14 moreover, the critical effect of hydration of the amide group can be studied reliably using this mixed solvent as a polar reaction medium.12 The presence of water has a significant effect on the solvent properties of NMP, particularly its selectivity and efficacy in a number of processes.15 Previous studies on the NMP/W system16 are extended in this work to high-pressure conditions; the density values from 278.15 to 358.15 K were extended up to 60 MPa across the full composition range, and from these readings, the isobaric thermal expansivity, isothermal compressibility, internal pressure, and excess properties were evaluated. These macroscopic properties were analyzed in terms of microscopic effects and intermolecular interactions, mainly H-bonding. Likewise, to properly describe the volumetric behavior of the strongly nonideal NMP/W system, the capabilities of several cubic equations of state (EOS) combined with different mixing rules were also tested. Materials and Methods Reactants and Sample Preparation. NMP (Fluka, 99.9% purity) and hexane (Fluka, >99.9% purity) were stored over Fluka Union Carbide 0.4-nm molecular sieves, out of light. Their purities were assessed by GC with a Perkin-Elmer 990 gas chromatograph, equipped with a Hewlett-Packard 3390A integrator. Ultrapure water (Milli-Q, Millipore, 18.2 mΩ‚cm resistivity) was used without further purification. The pure liquids were degassed with ultrasound before use. To prevent the samples from preferential evaporation, the mixtures (mole fraction of (1 × 10-4) were prepared by syringing amounts, weighed to ∆m ) (1 × 10-5 g with a Mettler AT 261 Delta Range balance, into suitably stoppered bottles. The mixtures were completely miscible over the whole composition range. Instruments and Procedures. The densities were measured with a recently developed automated vibrating-tube densitometer. The reliability of the apparatus was checked by comparing the measured NMP densities with those from the literature. Figure 1 shows the schematic assembly of the apparatus; the central ele-
10.1021/ie049906u CCC: $27.50 © 2004 American Chemical Society Published on Web 05/05/2004
3206 Ind. Eng. Chem. Res., Vol. 43, No. 12, 2004
Figure 1. Diagram of the high-pressure density measurement unit. The continuous lines represent hydraulic connections, and the dashed lines represent electrical connections.
ment of the system is an Anton Paar DMA 512P highpressure cell, which consists of a stainless steel vibrating U-shaped tube and the electronics needed to excite the tube and measure the oscillating period. An Anton Paar DMA58 digital density meter was used as a frequency counter connected to the high-pressure cell. The cell temperature was controlled by a Julabo F32 circulating bath ((0.01 K) using a Pt(100) sensor connected to an AΣΛ F250 thermometer, close to the measuring cell. The pressure of the whole circuit ((0.005 MPa) was controlled by a Ruska 7615 digital pressure controller and measured to within (0.01 MPa with a high-accuracy pressure sensor. The pressurizing fluid was separated from the sample by the Pressurements T3600E high-pressure separator, which contains a Teflon diaphragm for proper pressure transmission. Well-defined and traceable procedures were used to calibrate both the pressure controller and the thermometer. The whole system was fully computercontrolled; specially developed software allows for operations to be performed automatically, thus making feasible the storage of a large number of very accurate data. The specifications of the elements installed in the system enable a working temperature range of 263.15-423.15 K and a working pressure range 0.170 MPa. Calibration. The density measurements, F, were based on the oscillation period, τ, of a vibrating samplefilled U-shaped tube according to the relationship
F ) T1τ2 - T2
(1)
where the instrument constants, T1 and T2, were evaluated using reference fluids of known densities at a given temperature and pressure. These constants strongly depend on temperature and pressure according to the relationships
T1 )
∑i AiT i + ∑j BjPj + CTP
(2)
T2 )
∑i DiT i + ∑j EjPj + FTP
(3)
where i ) 0, 1, 2, 3 and j ) 1, 2.17 Given the range of densities to be measured, hexane and water were taken as well-suited reference liquids and measured from 278.15 to 358.15 K in 5 K steps and from 0.1 to 60 MPa in 1 MPa steps; the densities of the reference liquids were obtained from Lemmon et al. 18 Aside from the uncertainties inherent to the pressure and temperature readings, the main contribution to the quoted (0.0001 g cm-3 uncertainty for density measurements stems from the reference liquids. Experimental Procedure. The experimental readings ranged from 278.15 to 358.15 K in 10 K steps and from 0.1 to 60 MPa in 5 MPa steps. The 45-cm3 samples required to fill the system were syringed through the sample loading port (Figure 1), and the working temperatures were set starting from the lower 278.15 K limit. After thermal stabilization had been reached, the program was run in isothermal scans from the lower 0.1 MPa pressure limit to 60 MPa in 5 MPa steps. The program used is capable of detecting the stability period time, storing the data, and automatically changing to the next pressure step. Once the upper pressure limit was reached, the pressure was reduced to the lower value, and the new temperature (10 K higher than that of the just-completed scan) was fixed, thus starting a new scan. Results and Discussion The densities of pure NMP and of the xNMP/(1 - x)W binary solvent were measured across the full composition range; the reference water densities, used to evaluate the excess properties, were taken from the literature.18 The experimental compressed liquid densities for NMP and for NMP/W were correlated with temperature and pressure at each composition using the TRIDEN 10-parameter equation developed by Ihmehls and Gmehling;17 this approach combines a modified version of the Rackett equation for saturation densities, eq 4, with the Tait equation for isothermal compressed densities, eqs 5 and 6
Ind. Eng. Chem. Res., Vol. 43, No. 12, 2004 3207
F0 )
F)
AR
(4)
BR[1+(1-T/CR)DR] F0
(
V Em )
)
(5)
BT + P 1 - CT ln BT + P0
( )
T T BT ) b0 + b1 + b2 ET ET
2
( )
+ b3
T ET
3
(6)
The reference pressure P0 ) 0.1 MPa was used at all temperatures, and the corresponding reference densities, F0, were correlated with eq 4, the CT Tait equation parameter being treated as temperature-independent. The correlation parameters were deduced using a Levenberg-Marquardt least-squares algorithm, and the optimal fitting was assessed by the absolute average percentage deviation (AAD), given by eq 7
AAD )
100 N Fj,EXP - Fj,CAL
∑| N j)1
Fj,EXP
|
(7)
where N is the number of data pairs. Table 1 summarizes the measured and calculated properties for pure NMP. The data for the xNMP/(1 - x)W mixed solvent are reported as Supporting Information, and the fitting parameters of eqs 4-6 across the full composition range are listed in Table 2; the goodness of the fitting is indicated by the very small deviations (better than 0.03%) between the experimental and the correlated values. Figure 2 shows a plot of the experimental data for pure NMP under high-pressure conditions and the validity of the TRIDEN fitting function as well. The NMP density data were in good agreement with those reported by Ihmehls and Gmehling;19 the deviations with literature values, lower than 0.12%, indicate the suitability of the instrument and demonstrate the performance of the calculations (Figure 3). The PVTx behavior of pure and mixed solvents enable calculation of the following derived thermodynamic properties: isobaric thermal expansivity, RP (eq 8); isothermal compressibility, κT (eq 9); and internal pressure, Pi (eq 10)
κT )
1 ∂F F ∂T P
( )
(8)
1 ∂F F ∂P P
( )
(9)
RP -P κT
(10)
RP ) -
Pi ) T
criteria of Benson and Kiyohara,20 eqs 11-13
Table 1 summarizes the coefficients calculated for pure NMP as a function of temperature and pressure. Figure 4 shows a plot of the effect of temperature on the isobaric properties at the intermediate 30 MPa pressure, the curves shapes being similar at other pressures. Figure 5 shows the effect of pressure at the intermediate 318.15 K temperature. The derived excess properties of the excess molar volume, V Em; the excess isobaric thermal expansivity, REP ; and the excess isothermal compressibility, κET , were evaluated from the measured and calculated properties following the
xMNMP + (1 - x)MW MW MNMP - (1 - x) -x F FNMP FW (11) REP ) RP - φRP,NMP - (1 - φ)RP,W
(12)
κET ) κP - φκT,NMP - (1 - φ)κT,W
(13)
where M is the molar mass and φ is the volume fraction. The non-Gibbsian profile of internal pressure21 made it necessary to further define the ideal reference value; hence, the excess internal pressure, PEi , was calculated according to Marczak22 as
PEi ) Pi - ψPi,NMP - (1 - ψ)Pi,W
(14)
where ψ is defined as
ψ)
xκT,NMP xκT,NMP + (1 - x)κT,W
(15)
Figure 6 shows a plot of the isobaric behavior of these properties at 30 MPa, and Figure 7 shows the isothermal behavior at 318.15 K. Likewise, the effects of pressure on the excess Gibbs energy, ∆GE; excess enthalpy, ∆HE; and excess entropy, ∆SE, were evaluated from the V Em data
∆GE ) GE(x,T,P) - GE(x,T,P0) )
∫PPV Em(x,T,P) dp 0
(16)
∆HE ) HE(x,T,P) - HE(x,T,P0) )
∫PP[V Em - T(∂V Em/∂T)P] dp 0
∆SE )
∆HE - ∆GE T
(17) (18)
where P0 ) 0.1 MPa is the reference pressure. With this in mind and with the reference values at constant temperature, evaluation of the isothermal effect of pressure on these properties is feasible; Figure 8 shows a plot of the ∆GE, ∆HE, and ∆SE values at the reference 298.15 K temperature. Pure NMP can be considered as a rigid, nonassociating molecule in nature; specific self-interactions such as H-bonding are absent, although the high polarity (µ30°C ) 4.09 D)23 brings about a strong reorganization of the solvent structure by dipole-dipole interactions and the high density values determined reveal that the pyrrolidinone ring enables an efficient packing (Figure 2). The strong nonideal behavior of the NMP/W solvent is clearly observed in Figures 4-8; the pronounced nonlinearity of the density, isobaric thermal expansivity, isothermal compressibility, and internal pressure, which display either some maxima or some minima in the water-rich region, reveal the strong restructuring effect of NMP on the surrounding water molecules. NMP consists of a hydrophilic region (the O atom of the carbonyl group) and a hydrophobic tail (the methyl group linked to the N atom); hence, the balance between two such regions in close proximity gives rise to the unusual structuring properties of NMP caused by the presence of water.
3208 Ind. Eng. Chem. Res., Vol. 43, No. 12, 2004 Table 1. Experimental Density, G; Isobaric Thermal Expansivity, rP; Isothermal Compressibility, KT; and Internal Pressure, Pi, for Pure NMP as Functions of Pressure and Temperature T (K) P (MPa)
278.15
288.15
298.15
308.15
318.15
0.10 1.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00 45.00 50.00 55.00 60.00
1.0456 1.0461 1.0480 1.0504 1.0528 1.0551 1.0573 1.0595 1.0617 1.0638 1.0659 1.0680 1.0700 1.0718
1.0370 1.0375 1.0396 1.0420 1.0445 1.0469 1.0492 1.0515 1.0538 1.0560 1.0582 1.0603 1.0624 1.0644
1.0282 1.0287 1.0309 1.0335 1.0361 1.0386 1.0410 1.0434 1.0457 1.0480 1.0503 1.0525 1.0546 1.0567
F (g cm-3) 1.0193 1.0106 1.0198 1.0111 1.0220 1.0135 1.0247 1.0164 1.0275 1.0192 1.0301 1.0219 1.0326 1.0245 1.0351 1.0271 1.0376 1.0297 1.0400 1.0322 1.0423 1.0346 1.0446 1.0370 1.0468 1.0393 1.0490 1.0414
0.10 1.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00 45.00 50.00 55.00 60.00
0.800 0.798 0.790 0.780 0.770 0.760 0.751 0.743 0.734 0.726 0.719 0.711 0.704 0.697
0.819 0.816 0.807 0.796 0.785 0.774 0.764 0.755 0.746 0.737 0.729 0.721 0.713 0.705
0.838 0.835 0.825 0.812 0.800 0.788 0.778 0.767 0.757 0.748 0.738 0.730 0.721 0.713
RP (kK-1) 0.858 0.878 0.855 0.875 0.843 0.862 0.829 0.846 0.815 0.831 0.803 0.817 0.791 0.804 0.779 0.791 0.768 0.779 0.758 0.767 0.748 0.756 0.738 0.746 0.729 0.735 0.720 0.726
328.15
338.15
348.15
358.15
1.0016 1.0022 1.0047 1.0077 1.0106 1.0135 1.0163 1.0190 1.0216 1.0242 1.0267 1.0292 1.0316 1.0338
0.9928 0.9934 0.9960 0.9992 1.0023 1.0053 1.0081 1.0110 1.0137 1.0164 1.0190 1.0216 1.0241 1.0264
0.9837 0.9843 0.9871 0.9904 0.9936 0.9967 0.9998 1.0027 1.0056 1.0084 1.0111 1.0138 1.0164 1.0188
0.9748 0.9754 0.9783 0.9818 0.9852 0.9884 0.9916 0.9947 0.9977 1.0006 1.0034 1.0062 1.0089 1.0115
0.900 0.896 0.881 0.864 0.847 0.831 0.816 0.802 0.789 0.776 0.764 0.752 0.741 0.730
0.922 0.918 0.902 0.882 0.863 0.845 0.828 0.813 0.798 0.784 0.770 0.757 0.745 0.733
0.946 0.942 0.922 0.900 0.878 0.858 0.840 0.822 0.805 0.790 0.775 0.761 0.747 0.734
0.971 0.966 0.943 0.918 0.893 0.871 0.850 0.830 0.811 0.794 0.777 0.761 0.747 0.732
0.10 1.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00 45.00 50.00 55.00 60.00
472.5 470.5 461.7 451.2 441.2 431.6 422.5 413.7 405.4 397.4 389.7 382.3 375.2 368.4
495.8 493.6 483.9 472.4 461.4 450.9 441.0 431.5 422.4 413.7 405.4 397.4 389.8 382.4
521.7 519.2 508.5 495.8 483.7 472.3 461.4 451.0 441.1 431.6 422.6 413.9 405.7 397.7
550.5 547.8 535.9 521.8 508.5 495.8 483.8 472.4 461.6 451.3 441.4 432.0 423.0 414.4
κT (TPa-1) 582.9 579.8 566.6 550.8 536.0 522.0 508.7 496.1 484.2 472.8 462.0 451.7 441.9 432.5
619.4 615.9 600.9 583.3 566.7 551.0 536.3 522.3 509.1 496.6 484.7 473.4 462.7 452.4
660.6 656.7 639.7 619.7 601.0 583.5 567.0 551.4 536.7 522.9 509.7 497.3 485.4 474.2
707.6 703.1 683.7 660.9 639.7 619.9 601.3 583.8 567.4 552.0 537.4 523.6 510.5 498.1
761.5 756.3 733.8 707.7 683.4 660.9 639.8 620.1 601.7 584.4 568.1 552.7 538.2 524.4
0.10 1.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00 45.00 50.00 55.00 60.00
471.1 471.0 470.8 470.6 470.3 469.9 469.6 469.2 468.8 468.4 468.0 467.5 467.0 466.5
475.7 475.7 475.5 475.3 475.0 474.8 474.4 474.1 473.7 473.3 472.9 472.5 472.0 471.5
478.7 478.6 478.5 478.3 478.0 477.8 477.5 477.1 476.8 476.4 475.9 475.5 475.0 474.5
479.9 479.8 479.7 479.5 479.2 478.9 478.6 478.2 477.8 477.4 476.9 476.4 475.9 475.3
Pi (MPa) 479.2 479.2 479.0 478.7 478.4 478.0 477.6 477.2 476.7 476.2 475.7 475.1 474.5 473.8
476.6 476.6 476.3 475.9 475.5 475.0 474.5 474.0 473.4 472.7 472.0 471.3 470.6 469.8
472.0 471.9 471.6 471.0 470.4 469.8 469.1 468.4 467.6 466.8 465.9 465.0 464.0 463.0
465.3 465.2 464.6 463.9 463.1 462.2 461.2 460.2 459.2 458.1 457.0 455.8 454.5 453.2
456.5 456.3 455.5 454.4 453.2 452.0 450.7 449.4 448.0 446.5 445.0 443.5 441.8 440.2
Carver et al.11 reported some molecular dynamics calculations of the dilute NMP/W system; the distribution function provided points to strong H-bonding formation with the oxygen of the CO group (as the first solvation sphere contains 2.5 water molecules and 1.5 H-bonds), whereas a typical distribution of the water molecules around the methyl group results from hydrophobic hydration. So far, there has been no evidence for H-bonding at the sterically hindered N site. The close vicinity of the hydrophobic hydration of the methyl
group and the hydrophilic hydration and H-bonding of the CO oxygen lead to an efficient cooperative effect in the solvation shell; therefore, water tends to become preferentially structured around the NMP units rather than around other water molecules. These conclusions should be emphasized because the interpretation of macroscopic properties in terms of microscopic-level interactions requires careful consideration, and it is not always feasible to trace back which microscopic property is the origin for a particular
Ind. Eng. Chem. Res., Vol. 43, No. 12, 2004 3209 Table 2. Fitting Parameters of the TRIDEN Correlation of Density (g cm-3) with Pressure and Temperature, Eqs 4-6, and AAD Values for the xNMP/(1 - x)W Binary Solventa parameter
x ) 0.0981
x ) 0.2002
x ) 0.3000
x ) 0.4009
x ) 0.5023
x ) 0.6013
x ) 0.7012
x ) 0.8028
x ) 0.9039
x)1
CT b0 (MPa) b1 (MPa) b2 (MPa) b3 (MPa) ET (K) AR (g cm-1) BR CR (K) DR AAD
0.094 568 100.285 99.8591 7.6647 -8.059 87 99.99 7.6055 2.580 776 497.964 -0.117 133 0.0141
0.086 901 332.019 23.8913 -21.4902 -0.035 40 104.39 18.9540 3.888 936 830.476 -0.286 144 0.0295
0.087 261 329.552 21.5725 -21.5074 -0.063 21 104.79 25.2117 4.449 767 833.320 -0.278 111 0.0193
0.086 166 331.200 21.6010 -26.5205 1.326 14 102.45 20.2446 3.988 223 830.752 -0.298 157 0.0130
0.087 272 332.415 22.0757 -29.0353 1.991 77 100.95 17.2340 3.689 948 829.584 -0.310 030 0.0129
0.088 060 331.574 20.9379 -29.8225 2.258 92 100.70 20.4512 4.048 293 731.529 -0.234 212 0.0117
0.085 730 328.211 16.8810 -31.2813 2.896 25 100.53 26.1710 4.589 259 736.418 -0.216 841 0.0175
0.089 098 323.895 13.2610 -28.5835 2.393 43 102.70 33.4520 5.196 614 745.360 -0.204 412 0.0170
0.087 652 312.721 3.4908 -23.8400 1.726 04 105.93 44.3204 5.984 891 764.269 -0.196 821 0.0222
0.089 842 336.899 7.9826 -33.2787 3.592 6 104.46 30.4591 4.953 406 838.680 -0.253 190 0.0157
a
Parameters are applicable within 278.15-358.15 K and 0.1-60 MPa ranges.
Figure 2. Three-dimensional plot of density as a function of pressure and temperature for NMP. (b) Experimental values; shaded surface obtained with eqs 4-6 and parameters from Table 2.
Figure 3. Relative deviations between the experimental NMP density readings and the literature values obtained from the parameters of the fitting equation by Ihmehls and Gmehling19 at temperatures of (b) 278.15, (9) 288.15, (2) 298.15, ([) 308.15, (f) 318.15, (O) 328.15, (0) 338.15, (4) 348.15, and (]) 358.15 K.
observed behavior. The pronounced maxima of the NMP/W densities at xNMP ≈ 0.33 remain essentially unchanged regardless of the changes of pressure and temperature (Figures 4a and 5a); this location reveals the formation of strong 1:2 NMP/W aggregates by H-bonding and allows one to discard a third interaction with the N atom. The minimum in the isothermal compressibility vanishes with increasing temperature (Figure 4c), indicating a weakening of the H-bonding effect and an increase of the free volume with increasing temperature. The effect of pressure on the isothermal compressibility (Figure 5c) reveals that the location of the minimum remains unchanged with increasing pressure. The stoichiometry of the solvation shell also
remains 1:2 and is susceptible to only a slight change due to the more efficient packing the higher the pressure, an effect supported by the shift with temperature of the observed minima (Figure 4c). The excess and mixing properties reported, with strong maxima or minima in the water-rich regions (Figures 6 and 7), confirm the above conclusions. The negative excess molar volumes, with minima at xNMP ≈ 0.33, indicate dense solvation jackets and 1:2 NMP/W H-bonding aggregates; although the strength of the interactions decreases with increasing temperature, the solvation structure remains stable even at higher temperatures. The negative isothermal compressibility values derived indicate that the mixture is less compressible than the pure components; hence, effective solvation and efficient packing follow upon mixing. The formation of strong 1:2 NMP/W aggregates through the two lone electron pairs on the CO oxygen site also was supported by Assarsson and Eirich12 and McDonald et al.24 In the NMP-rich region, the effects were much less pronounced, showing that the NMP dipolar structure remains unaffected by the presence of W. The enthalpy of mixing was found to be highly exothermic; the HEm ) -2744 J mol-1 minimum value reported at xNMP ≈ 0.3324 reveals that the disruption of the water structure by NMP is strongly balanced by the formation of new H-bonds with NMP, with the result being a significant enhancement of the W/W structure in the vicinity of the lactam. The effect of pressure on the thermal properties points to a complex behavior (Figure 8); an increase in pressure gives rise to negative contributions to the excess Gibbs energy, attributed to both the entropic component arising from the efficient packing at higher pressures (Figure 8c) and the solute fitting into the solvent structures (clathrate formation). Nevertheless, this packing effect is less important than the formation of H-bonding; at higher pressures, a positive contribution to the mixing enthalpy appears, (Figure 8b), and therefore, the NMP/W interactions become still more complicated. Applicability of Equations of State Equations of state (EOS) are extremely valuable tools for studying the phase equilibria and PVT behavior of pure and mixed solvents. Among the different types of EOS systems available, cubic EOS are most often used for practical applications in chemical engineering. Because of the fair balance between accuracy, reliability, simplicity, and speed of computation, cubic EOS are the models most often used for computations of phase equilibria in multicomponent mixtures.25 In this section, the abilities of several cubic EOS to properly describe
3210 Ind. Eng. Chem. Res., Vol. 43, No. 12, 2004
Figure 4. Effect of temperature on the isobaric properties at P ) 30 MPa for the xNMP + (1 - x)W binary solvent: (a) density, F; (b) isobaric thermal expansivity, RP; (c) isothermal compressibility, κT; (d) internal pressure, Pi. Symbols as in Figure 3. The data for pure water were obtained from Lemmon et al.18
Figure 5. Effect of pressure on the isothermal properties at T ) 318.15 K for the xNMP + (1 - x)W binary solvent: (a) density, F; (b) isobaric thermal expansivity, RP; (c) isothermal compressibility, κT; (d) internal pressure, Pi. (b) 0.1, (9) 1, (2) 5, ([) 10, (f) 15, (O) 20, (0) 25, (4) 30, (]) 35, (g) 40, (3) 45, (0 ×) 50, (x) 55, and (×) 60 MPa. Data for pure water obtained from Lemmon et al.18
Figure 6. Effect of temperature on the isobaric excess properties at P ) 30 MPa for the xNMP + (1 - x)W binary solvent: (a) excess molar volume, V Em; (b) excess thermal expansivity, REP ; (c) excess isothermal compressibility, κET ; (d) excess internal pressure, PEi . Symbols as in Figure 3.
the PVT behavior of pure NMP and the NMP/W mixed solvent are reported. The predictive abilities of the Soave-Redlich-Kwong (SRK),26 Peng-Robinson (PR),27 Stryjek-Vera modification of Peng-Robinson (PRSV),28 Patel-Teja (PT),29 and Sako-Wu-Prausnitz (SWP) cubic EOS were studied for pure NMP.30 The critical properties needed for calculations were obtained from Gude and Teja31 for NMP and from Wagner and Pruss32 for W; the acentric factor was calculated according to the Lee-Kesler method33 for NMP and according to the method of Reid et al.34 for W. The goodness of the density predictions was estimated by the AAD, eq 7; the AAD values deduced for NMP, specifically, 23.74% (SRK), 15.02% (PR), 6.37% (PRSV), and 14.19% (PT), indicate the poor
predictive capabilities of these EOS for densities and their pressure and temperature dependencies. Predictions by the EOS most often used, SRK and PR, were still worse. The PRSV EOS, which contains an adjustable interaction parameter, gave better predictions but a somewhat low accuracy; the PT EOS contains an additional parameter but did not sufficiently improve the density predictions. Therefore, the most frequently used cubic EOS exhibit serious inadequacies and noticeable deviations that can be ascribed to the complex liquid structure of pure NMP, and these EOS must be discarded for analysis of the NMP/W binary solvent. The cubic EOS by Sako et al.,30 which considers the perturbed hard-chain theory, is applicable to large nonspherical molecules and should, in principle, be well-
Ind. Eng. Chem. Res., Vol. 43, No. 12, 2004 3211
Figure 7. Effect of pressure on the isothermal excess properties at T ) 318.15 K for the xNMP + (1 - x)W binary solvent: (a) excess molar volume, V Em; (b) excess isobaric thermal expansivity, REP ; (c) excess isothermal compressibility, κET ; (d) excess internal pressure, PEi . Symbols as in Figure 5.
Figure 8. Effect of pressure on the isothermal excess properties at T ) 298.15 with respect to their values at 0.1 MPa for the xNMP + (1 - x)W binary solvent: (a) excess Gibbs energy, ∆GE; (b) excess enthalpy, ∆HE; (c) T∆SE contribution. Symbols as in Figure 5.
suited for describing the behavior of strongly nonideal systems or systems with complex structured molecules; this equation contains the additional interaction parameter c, which gives a measure of the external rotational and vibrational degrees of freedom, eq 19
P)
RT(Vm - b + bc) Vm(Vm - b)
a Vm(Vm + b)
(19)
where P, T, Vm, and R stand for pressure, temperature, molar volume and gas constant, respectively. The characteristic parameters a, b, and c account for the attractive forces, molecular size, and external degrees of freedom, respectively. If c tends to the limiting value of c ) 1, then the SWP EOS converts to the SRK EOS, which refers to spherical molecules; these parameters are related to the critical properties by the equations
a ) acR(T) R(T) )
R0(1 - Tr2) + 2Tr2 1 + Tr2
R2Tc2 ac ) f(d) Pc f(d) )
(1 - 2d + 2cd + d2 - cd2)(1 + d2) 3(1 - d)2(2 + d)
(20)
(21)
(22)
(23)
d3 + (6c - 3)d2 + 3d - 1 ) 0 b)
c)-
dRTc 3Pc (d - 1)3 6d2
(24) (25)
(26)
where d and R0 are the characteristic parameters for each compound. Although Elvassore et al.35 developed a group contribution method to compute the a, b, and c parameters of the SWP equation, this method proved to be inadequate because group contribution models such as UNIFAC consider the NMP molecule as a single group that cannot be split into different contributing groups. The d and R0 parameters were evaluated from the saturated liquid densities of NMP and W. The saturation conditions for NMP were obtained from Antoine’s parameters,5 and the saturated liquid densities under such conditions were determined within the range 278.15-358.15 K using the correlation of compressed densities, Table 2; for W, the saturated liquid properties were taken from Wagner and Pruss.32 Figure 9 shows a plot of the correlation with temperature of the saturated liquid densities for NMP, and Table 3 summarizes the correlation parameters that lead to the compressed liquid densities plotted in Figure 10, with AAD ) 0.0841. Hence, the SWP EOS is best able to describe the pure NMP data compared to the above EOS systems. The fine reproduction of the trend of these curves leads us to conclude that the SWP model is
3212 Ind. Eng. Chem. Res., Vol. 43, No. 12, 2004 Table 4. Mixing-Rule Parameters, Eqs 27-38, and AAD Values Obtained with the SWP Equation from the Correlation of Compressed Liquid Density of NMP/W Binary Solvent in the Ranges 278.15-358.15 K and 0.1-60 MPa mixing rule
kij
kji
mij
sij
lij
AAD
quadratic 0.2192 0.0228 0.3900 Panagiotopoulos 0.4757 0.2141 0.0396 0.3522 Adachi 0.3449 -0.1308 0.0396 0.3522 Sandoval -0.0475 0.4757 0.0396 0.3522 Schwartzentruber 0.2192 0.1000 0.1000 0.0228 0.3899 Stryjek 0.6634 0.2697 0.0446 0.3388 Mathias 1.2869 3.1625 0.0379 1.4884
Figure 9. Saturated liquid densities for NMP (b) calculated using eqs 4-6 and parameters in Table 2 under the experimental saturation conditions determined from the Antoine correlation by Hradetzky et al.5 and (;) obtained from the Sako-Wu-Prausnitz equation of state.
(S),39 eq 34; Stryjek and Vera (SV),40 eq 35, and Mathias, Klotz, and Prausnitz (MKP),40 eq 36
aij ) (aiaj)1/2(1 - kij)
(30)
PR PR aij ) (aiaj)1/2[1 - kPR ij + (kij - kij )xi]
(31)
AS aij ) (aiaj)1/2[1 - kAS ij - mij (xi - xj)]
(32)
SD aij ) (aiaj)1/2[1 - kSD ij xi - kij xj SD 0.5(kSD ij + kji )(1 - xi - xj)] (33)
aij ) (aiaj)
Figure 10. Measured compressed liquid densities of NMP and (;) those predicted by the Sako-Wu-Prausnitz equation of state. Symbols as in Figure 3. Table 3. Pure-Component Parameters for the SWP Equation, Eqs 19-26, Deduced from the Correlation of the Saturated Liquid Densities between 278.15 and 358.15 K NMP W
d
R0
AAD
0.1823 0.2091
1.9161 6.0746
0.0154 0.1143
adequate for analyzing the properties of NMP-containing solvents such as NMP/W. Application of cubic EOS to mixed solvents and proper calculation of the EOS parameters requires the use of proper mixing rules. The van der Waals one-fluid mixing rules with quadratic dependences for b and c, eqs 2729, were assumed, except for a in the Mathias-KlotzPrausnitz mixing rule, eq 36
a)
∑i ∑j xixjaij
(27)
b)
∑i ∑j xixjbij
(28)
c)
∑i ∑j xixjcij
(29)
The combining rules selected for the cross-terms are the classical quadratic, eq 30, and those of Panagiotopoulos and Reid (PR),36 eq 31; Adachi and Sugie (AS),37 eq 32; Sandoval (SD),38 eq 33; Schwartzentruber
1/2
(
1-
(
kSij
aij ) (aiaj)1/2 1 a)
-
sSij
mSijxi - mSjixj
)
mSijxi + mSjixj
SV kSV ij kji SV xikSV ij + xjkji
)
)+ ∑i ∑j xixj(aiaj)1/2(1 - kMKP ij ]1/3}3 ∑i xi{∑j xi[(aiaj)1/2mMKP ij
(34)
(35)
(36)
the cross coefficients for b and c being obtained with eqs 37 and 38
1 bij ) (bi + bj)(1 - lij) 2
(37)
1 cij ) (ci + cj)(1 - lij) 2
(38)
The PVTx behavior of the NMP/W system was correlated considering the mixing-rule parameters as being temperature-, pressure-, and composition-independent (Table 4). Figure 11a shows the fair reproduction of the values and the trend of the very complex behavior of the NMP/W densities using a simple mixing rule and only two parameters. For instance, the TRIDEN correlation reported gave better AAD values, but it required 10 parameters for each composition. For a second approach, we obtained more accurate correlations using the SWP equation with the simple quadratic mixing rule. Introduction of additional parameters in the mixing rules did not substantially improve the correlations (Table 4); hence, once the simple quadratic mixing rule was selected, a simple mathematical function was introduced that fairly describes the temperature and composition dependence of the kij and lij pressureindependent parameters (eqs 30, 37, and 38). The
Ind. Eng. Chem. Res., Vol. 43, No. 12, 2004 3213 Table 5. Parameters of Eqs 39-40 for the Composition and Temperature Dependence of the Parameters of the Quadratic Mixing Rule Obtained with the SWP Equation in the Correlation of the Compressed Liquid Density of NMP/ W Binary Solvent in the Ranges 278.15-358.15 K and 0.1-60 MPa kij lij
K1 or L1
K2 or L2
K3 or L3
K4 or L4
K5 or L5
K6 or L6
K7 or L7
K8 or L8
K9 or L9
K10 or L10
-372.0774 3.2361
46.0118 -0.7812
-98.9373 -2.7067
315.6632 1.2792
0.4517 -0.0054
0.0000 0.0000
-851.4067 57.3235
414.7398 -85.5899
523.2381 25.7630
-0.0173 -0.0021
Figure 11. Experimental compressed liquid densities at P ) 30 MPa for the xNMP + (1 - x)W binary solvent. Symbols as in Figure 3 and (;) Sako-Wu-Prausnitz correlations with the following quadratic mixing rules: (a) kij and lij treated as constant parameters and (b) kij and lij treated as temperature- and composition-dependent parameters.
function selected is given by eqs 39 and 40, and the correlation parameters are reported in Table 5, with AAD ) 0.0169
kij )
K1 + K2x + K3x2 + K4x3 + K5T + K6T2
lij )
1 + K7x + K8x2 + K9x3 + K10T L1 + L2x + L3x2 + L4x3 + L5T + L6T2 1 + L7x + L8x2 + L9x3 + L10T
(39)
(40)
Although the number of parameters increased to 20, the accuracy of the correlation was very high (Figure 11b), and the correlation was able to reproduce the trend and the density values with better accuracy than that reported with the TRIDEN method. Figure 12, which shows a three-dimensional plot of the quadratic mixing rule with the parameters of Table 5, makes clear the complex behavior of these functions, related to the maxima in the compressed liquid density for the NMP/ W. Concluding Remarks The compressed liquid densities measured for pure NMP over a wide range of pressures and temperatures were in good agreement with the literature values. The NMP/W binary solvent was studied within the same range of temperatures and pressures, and the thermophysical properties obtained reveal a strongly nonideal system with H-bonding between W and NMP in dilute NMP mixtures. The abilities of several cubic EOS to describe the PVT behavior of this system were analyzed; only the Sako-Wu-Prausnitz equation gave satisfactory results. A simple mixing rule with temperature and composition dependence of the correlation parameters is proposed, giving rise to accurate correlations with relatively few parameters.
Figure 12. Composition and temperature dependence of the kij and lij parameters of the quadratic mixing rule with the SakoWu-Prausnitz equation of state.
Acknowledgment The financial support of Junta de Castilla y Leo´n, Project BU10/03, and Ministerio de Ciencia y Tecnologı´a, Project PPQ2002-02150, Spain, is gratefully acknowledged. Supporting Information Available: Experimental densities of xNMP + (1 - x)water mixed solvent as a function of pressure and temperature (Table 1). This material is available free of charge via the Internet at http://pubs.acs.org. List of Symbols a, b, c, d ) parameters of the Sako-Wu-Prausnitz equation of state Ai, ..., Fi ) parameters of the extended vibrating-tube equation AR, ..., DR ) parameters of the Rackett equation b0, ..., b3 ) parameters of the Tait equation BT, ..., ET ) parameters of the Tait equation kij, mij, sij, lij ) parameters of the different mixing rules K1, ..., K10 ) parameters of the temperature/composition dependence of the kij parameter in the quadratic mixing rule
3214 Ind. Eng. Chem. Res., Vol. 43, No. 12, 2004 L1, ..., L10 ) parameters of the temperature/composition dependence equation of the lij parameter in the quadratic mixing rule M ) molar mass P ) total pressure Pc ) critical pressure Pi ) internal pressure PEi ) excess internal pressure P0 ) reference pressure (0.1 MPa) R ) universal gas constant T ) absolute temperature T1, T2 ) parameters of the vibrating-tube equation Tc ) critical temperature Tr ) reduced temperature Vm ) molar volume V Em ) excess molar volume x ) NMP mole fraction Greek Letters R0 ) parameter of the Sako-Wu-Prausnitz equation of state RP ) isobaric thermal expansivity REP ) excess isobaric thermal expansivity ∆GE ) pressure effect on excess Gibbs energy ∆HE ) pressure effect on mixing enthalpy ∆SE ) pressure effect on excess entropy κT ) isothermal compressibility κET ) excess isothermal compressibility F ) density F0 ) reference density in the Tait equation τ ) oscillation period of the vibrating tube φ ) NMP volume fraction ψ ) fraction for the definition of internal pressure Abbreviations AAD ) absolute average percentage deviation EOS ) equations of state NMP ) N-methyl-2-pyrrolidinone PR ) Peng-Robinson PRSV ) Stryjek-Vera PT ) Patel-Teja SWP ) Sako-Wu-Prausnitz SRK ) Soave-Redlich-Kwong TRIDEN ) fitting equation by Ihmehls and Gmehling17 W ) water Uncertainties δx ) (0.0001 δT ) (0.01 K δP ) (0.01 MPa δF ) (0.0001 g cm-3 δRP ) (0.01 kK-1 δκT ) (1 TPa-1 δPi ) (1 MPa
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Received for review February 2, 2004 Revised manuscript received March 30, 2004 Accepted April 5, 2004 IE049906U
