Anal. Chem. 1999, 71, 3503-3512
Prediction of Activity Coefficients in Low-Molecular-Weight Paraffins from Gas Chromatographic Data Andra´s Dallos† and Ervin sz. Kova´ts*,†,‡
Department of Physical Chemistry, University of Veszpre´ m, H-8201 Veszpre´ m, Hungary, and EÄ cole Polytechnique Fe´ de´ rale de Lausanne, De´ partement de Chimie, CH-1015 Lausanne, Switzerland
It is shown that gas-liquid partition coefficients measured by gas chromatography on nonvolatile paraffin stationary phases at high temperatures allow estimation of gasliquid partition coefficients of solutes at ambient temperatures in volatile alkanes as solvents. Extrapolation to low temperatures was successful by use of the Kirchhoff equation for the description of the temperature dependence of the standard chemical potential of the solute related to its molal Henry coefficient. A new equation is proposed for the prediction of partition data in lowmolecular-weight paraffins. It was derived by using the Flory-Huggins model as a guide, and it combines theory with experiment. It is proposed to accept a combinatorial entropy term determined by experiment as a solute property. The necessary experimental information consists of a set of gas chromatographic data measured at at least four temperatures on two pure, high-molecularweight paraffins. Molal Henry coefficients extrapolated to low-molecular-weight solvents allowed calculation of activity coefficients. Predicted and measured data agreed within ( 20%. The reproducibility of activity coefficients by classical experimental methods is of the same order of magnitude. Properly designed gas liquid chromatographic experiments permit a rapid collection of accurate gas/liquid partition data at ideal dilution.1,2 In the chromatographic column the high-molecular-weight solvent (the stationary liquid) is disposed of as a thin film on a support. Therefore, its specific surface area is high, typically of the order of some 5 m2 g-1 (corresponding to a film thickness of 0.2 µm). Hence, already weak adsorption at the liquid/solid (liquid/support) and the gas/liquid interfaces shall influence the measurement and will result in wrong partition data. By the use of an “inactive support”, adsorption at the liquid/ support interface can be eliminated. Adsorption at the liquid/gas interface is roughly proportional to the surface tension of the * Correspondence to: (tel.) 41 21 6933131; (fax) 41 21 312 8283 (also voice); (e-mail)
[email protected] or
[email protected]. † University of Veszpre ´ m. ‡ E Ä cole Polytechnique Fe´de´rale de Lausanne. (1) Laub, R. L.; Pecsok, R. L. Physicochemical Applications of Gas Chromatography. John Wiley & Sons: New York, 1978. (2) Conder, J. R.; Young, C. L. Physicochemical Measurements by Gas Chromatography. John Wiley & Sons: New York, 1979. 10.1021/ac981255m CCC: $18.00 Published on Web 07/03/1999
© 1999 American Chemical Society
solvent.3-5 It cannot be eliminated but can be reduced to a minimum by the adequate choice of solvent and by the use of highly loaded stationary phases, i.e., by reducing the specific surface area of the solvent. Partition data of a solute, j, in a given solvent, sv, as a function of temperature are best presented by calculating the corresponding standard chemical potential, ∆µj,sv (i.e., the partial molar standard Gibbs free energy).6
∆µj,sv ) RT ln κj,sv ) ∆Hj,sv - T∆Sj,sv
(1)
Here, κj,sv, is a partial pressure (fugacity) characterizing partition, R is the universal gas constant, T is the thermodynamic temperature, and the symbols ∆Hj,sv(T) and ∆Sj,sv(T) are for the partial molar enthalpy and entropy differences of the solute between the ideal gas phase and the ideal dilute solution. The relationship of eq 1 shows that the standard chemical potential difference depends both on specific enthalpic interactions between solute and solvent (attraction due to polarity, hydrogen bonding, etc.) and on the entropy. The latter also includes the so-called combinatorial entropy: contributions due to size and shape differences between solute and solvent molecules. In low-molecular-weight solvents gas/liquid partition data are determined by a series of conventional methods such as differential ebulliometry, gas stripping, headspace chromatography, and the like. Experimental results are mostly presented as activity coefficients instead of partial vapor pressures or Henry coefficients. The activity coefficient is the ratio of the vapor pressure of the solute, j, over its solution in a solvent, sv, and the vapor pressure of the solute dissolved in itself at the same concentration. Obviously, for the calculation of the activity coefficient, the vapor pressure of the pure solute must be available. However, this function is preferred for several reasons. First, its temperature dependence is negligible compared with that of partial pressure or partition data. Second, there exist a series of methods of prediction of activity coefficients based on models which consider interaction forces and solute/solvent size and shape differences (UNIFAC, MOSCED).7 (3) Riedo, F.; sz. Kova´ts, E. J. Chromatogr. 1979, 186, 47-62. (4) Fritz, D. F.; Sahil, A.; sz. Kova´ts, E. J. Chromatogr. 1979, 186, 63-80. (5) Eon, C.; Guiochon, G. J. Colloid Interface Sci. 1973, 45, 188. (6) Abraham, M. H.; Grellier, P. L.; Hamerton, I.; McGill, R. A.; Prior, D. V.; Whiting, G. S. Faraday Discuss. Chem. Soc. 1988, 85, 107-115.
Analytical Chemistry, Vol. 71, No. 16, August 15, 1999 3503
The availability of Henry coefficients is more problematic. Determination of gas/liquid partition data of solutes in volatile solvents implies accurate knowledge of the composition of the gas phase in equilibrium with the dilute solution. Determination of this “head space” composition is delicate, especially if the solute has a considerably lower volatility than the solvent. On the other side, partition data in nonvolatile solvents may be determined with high accuracy by using the solvent as the stationary phase in gas chromatography. As a general rule, data are measured at several temperatures (far) above the domain of our interest. Hence, data at room temperature must be calculated by extrapolation. The temperature dependence of gas/liquid partition data is best given via that of the related standard chemical potential difference. By working in high-molecular-weight solvents, the use of the molal Henry coefficient, gj,sv is preferred. The standard states of the related standard chemical potential, ∆µj,sv (cal mol-1), are the ideal dilute one molal solution and the ideal gas state at 1 atm. For successful extrapolation, it has been proposed to fit the Kirchhoff equation to the experimental data12
∆µj,sv ) RT ln(gj,sv/atm kg mol-1) ) ∆Hj,sv - T∆Sj,sv + Figure 1. Illustration of the envisaged extrapolation of gas/liquid partition data of solutes in solvents of the paraffin family, CzH2z + 2, as a function of the inverse molecular weight of the solvent, ζ ) Mpar-1, and the temperature, t (°C). The choice of the variable, ζ (instead of Mpar), is justified later on.
The wealth of data collected by gas chromatography allows for the testing of existing theories and model calculations. In the present paper we propose to examine the prediction of partition data in low-molecular-weight members of the “simplest” solvent family, the paraffins, in which specific interactions are absent. Consequently, we propose to predict activity coefficients of compounds in volatile paraffins at near ambient temperatures on the basis of partition data determined in high molecular weight paraffins at high temperatures as illustrated in Figure 1. THEORETICAL SECTION State of the Art. The definition of the limiting activity ∞ , of a solute, j, in a solvent, sv, at ideal dilution coefficient, γj,sv (concentration of j f 0) as well as some useful relationships are given by eq 21,2,8 ∞ γj,sv )
hj,sv poj
)
1000gj,sv poj Msv
)
RTdsv Kj,svpoj Msv
(2)
where, hj,sv (atm) is the Henry coefficient, p°j (atm) is the vapor pressure of the pure solute, gj,sv (atm kg mol-1) is the molal Henry coefficient, Msv (g mol-1) is the molar mass of the solvent, R (cm3 atm mol-1 K-1) is the universal gas constant, dsv is the density of the solvent, and Kj,sv (1) is the partition coefficient. Abundant literature data are available for the vapor pressure of pure substances as a function of temperature, permitting calculation of vapor pressure data in the domain of our interest, in most cases, by interpolation.9-11 (7) Park, J. H.; Carr, P. W. Anal. Chem. 1987, 59, 2596-2602. (8) Guggenheim, E. A. Thermodynamics; North-Holland: Amsterdam, 1957.
3504 Analytical Chemistry, Vol. 71, No. 16, August 15, 1999
∆CP,j,sv[T - T† - T ln(T/T†)] (3)
whereupon it is supposed that the difference of the partial molar heat capacity of the solute for the two standard states, ∆CP,j,sv (cal mol-1 K-1), is independent of the temperature, T (K). The symbol R (cal mol-1 K-1) is for the universal gas constant. The regression coefficients are interpreted as follows: ∆Hj,sv (cal mol-1) and ∆Sj,sv (cal mol-1 K-1) are for the difference of the partial molar enthalpy and entropy, respectively, at an arbitrary standard temperature, T†. In fact, Kirchhoff’s proposal is the simplest possible logical way to interpret lege artis (i.e., in accordance with thermodynamics) the slightly nonlinear dependence of the standard chemical potential on temperature. Figure 2 illustrates the description of the temperature dependence of the standard chemical potential by the Kirchhoff equation of octane in a branched paraffin, C78H158 {with T† ) (130 + 273.15) K}. The partial molal heat capacity difference becomes more important with increasing interaction forces between solute and solvent in the condensed phase. Hence, the curvature in the ∆µj,sv/T plot is more pronounced for polar solutes in polar solvents.13-16 The most pronounced curvature was (9) Ohe, S. Computer Aided Data Book of Vapor Pressure; Data Book Publishing Company: Tokyo, 1976. On the basis of literature data, a data bank has been constituted at the University of Veszpre´m where experimental results of several Authors have been averaged. Calculations have been performed with the aid of this data bank (See also refs 10 and 11). (10) Lide, D. R., Ed. Handbook of Chemistry and Physics, 77th ed. CRC Press: Boca Raton, Florida, 1996 (See ref 9). (11) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids, 4th ed. McGraw-Hill: New York, 1987 (See ref 9). (12) De´fayes, G.; Fritz, D. F.; Go ¨rner, T.; Huber, G.; de Reyff, C.; sz. Kova´ts, E. J. Chromatogr. 1990, 500, 139-184. (13) Reddy, K. S.; Dutoit, J.-Cl.; sz. Kova´ts, E. J. Chromatogr. 1992, 609, 229259. (14) Reddy, K. S.; Cloux, R.; sz. Kova´ts, E. J. Chromatogr., A 1994, 673, 181209. (15) De´fayes, G.; Reddy, K. S.; Dallos, A.; sz. Kova´ts, E. J. Chromatogr., A 1995, 699, 131-154. (16) Reddy, K. S.; Cloux, R.; sz. Kova´ts, E. J. Chromatogr., A 1995, 704, 387436.
Figure 3. Structure of the branched paraffins, CzH2z + 2, of the “Apolane” series.18 Note that the inverse molecular weights of the compounds, ζ, are approximately equidistant.
solutions.19-20 Following the Flory-Huggins model for polymer solvents and oligomer solutes, the correction function is given by eq 5.
φ(corr) ) -sj,parTζ Figure 2. Example of the description of the temperature dependence of the standard chemical potential difference by the Kirchhoff equation: octane in the branched paraffin C78H158. Partition data were determined by gas chromatography in the temperature range of 90210 °C with an error smaller than the diameter of a dot in the diagram.13 Note that at ambient temperatures the trace of the Kirchhoff equation seriously deviates from a linear approximation, i.e., if it is supposed that ∆CP ) 0, then ∆H and ∆S are temperature independent.
observed for primary alcohols as solutes in a primary alkanol as solvent.13 Even in this case, the residual variance around the regression was not higher than those for the other some 400 examples. Also, the results of Castells et al.17 show that introduction of a linear temperature dependence of the partial molar heat capacity difference is not justified by experiment. Saturated noncyclic hydrocarbons are the “simplest” solvents insofar that they are not able to enter specific interactions with a solute. De´fayes et al. have shown that there is a regularity of partition data in this solvent family.12 Henry coefficients have been determined by gas chromatography in a temperature range of 90210 °C on a series of five columns prepared with pure paraffins as stationary phases and synthesized by Zeltner et al.18 The structure of the paraffins, CzH2z + 2, with carbon numbers z ) 59103, is depicted in Figure 3. The intent has been to summarize data by an equation of the following form:
∆µj,par ) ∆µj,ref + φ(corr)
(4)
In other words, the intent has been to give the standard chemical potential difference of the solute in a given paraffin as a sum of that in a reference paraffin, ∆µj,ref, and a correction function, φ(corr). The choice of the form of the correction function has been guided by the ideas of Flory and Huggins for equithermal (17) Castells, R. C.; Arancibia, E. L.; Nardillo, A. M. J. Chromatogr. 1990, 504, 45-53. (18) Zeltner, P.; Huber, G. A.; Peters, R.; Ta´trai, F.; Boksa´nyi, L.; sz. Kova´ts, E. Helv. Chim. Acta 1979, 62, 2495-2506. (19) Flory, P. J. J. Chem. Phys. 1941, 9, 660-661; 1942, 10, 51-61; 1944, 12, 425-38.
(5)
Here, sj,par, is a solute-specific constant (a combinatorial entropy term) and ζ ) M-1 par is the inverse of the molecular weight of the paraffin solvent. The constant sj,par can be determined with data measured on at least two high-molecular-weight paraffins. For a hypothetical paraffin of infinite molecular weight (infinite molar volume), the correction function is zero; hence, this paraffin has been elected as the reference solvent. In this solvent the standard chemical potential of the solute related to the molal Henry coefficient is given, in analogy to eq 3, as:
∆µj,∞ ) ∆Hj,∞ - T∆Sj,∞ + ∆CP,j,∞[T - T† - T ln(T/T†)] (6) If the condition of equithermality is not satisfied, a correction function should be chosen which also accounts for the dependence of the partial molar enthalpy on the molecular weight of the solvent. The simplest proposal is given in eq 7.
φ’(corr) ) (hj,par - sj,parT)ζ
(7)
Here, hj,par is an enthalpic, solute-specific constant. However, the data set of De´fayes et al.12 (partition data on five high-molecularweight paraffins) did not permit evaluation of both solute-specific constants. Therefore, first the entropic constant, sj,par, has been determined by regression, and then an enthalpic constant was determined as a nonsignificant refinement. The constants, ∆Hj,∞, ∆Sj,∞, ∆CP,j,∞, and sj,par, (and hj,par) are listed in ref 12 for some 140 solutes. In conclusion, in the reference paraffin partition data at room temperature can be calculated with eq 6 by extrapolation out of the experimental temperature range (90-210 °C). In low-molecular-weight paraffins, partition data can also be calculated with eq 5 also by extrapolation out of the experimental carbon number range (z ) 59-103). Let us control the quality of such doubly extrapolated data by comparing them with experimental partition (20) Huggins, M. J. J. Chem. Phys. 1941, 9, 440-441; J. Phys. Chem. 1942, 46, 151-8; Ann. N. Y. Acad. Sci. 1942, 43, 1-32; Ann. N.Y. Acad. Sci. 1943, 44, 431-443.
Analytical Chemistry, Vol. 71, No. 16, August 15, 1999
3505
Figure 4. Logarithmic plot of the experimental gas/liquid partition coefficients, Kj,C16, of diverse solutes in hexadecane at ideal dilution as a function of those calculated by extrapolation by using data listed in ref 12 in eqs 5 and 6 (see also eq 25) (194 data points in the temperature range of 25-60 °C).
coefficients determined at room temperature in a paraffin of not too low molecular weight. In the literature, numerous data are found in hexadecane as solvent.21-25 Data at 25 °C, determined with special care, have been published recently by the research group of Carr.26 With hexadecane as the stationary phase, gas chromatographic data have been determined by using hexadecane-saturated helium as eluent. In Figure 4 the logarithms of partition coefficients Kj,C16 of Carr as well as those from other sources are plotted as a function of data calculated by double extrapolation (194 experimental points). The correlation is excellent as is shown by the correlation eq 8 (calc) (exp) log Kj,C16 ) -0.077 ( 0.015 + (0.9918 ( 0.0045) log Kj,C16
(8)
with a standard deviation around the regression of σ ) ( 0.056. The correlation coefficient is r ) 0.9980. The symbol Kj,C16 is for the partition coefficient in hexadecane (in the literature denoted sometimes as L16).21 The paraffin hexadecane with z ) 16 is near the low carbon number limit of the predictive ability of the equation proposed by De´fayes et al.12 In Figure 5 literature activity coefficients of 38 solutes (226 data points) in low-boiling n-alkanes (from pentane to tetradecane) listed in Table 1 are plotted on a logarithmic scale (21) Abraham, M. H. Chem. Soc. Rev. 1993, 73-83. (22) Abraham, M. H.; Grellier, P. L.; McGill, R. A. J. Chem. Soc., Perkin Trans. 2 1987, 797-803. (23) Scheller, W. A.; Petricek, J. L.; Young, G. C. Ind. Eng. Chem. Fundam. 1972, 11, 53-57. (24) Alessi, P.; Kikic, I.; Alessandrini, A.; Fermaglia, M. J. Chem. Eng. Data 1982, 27, 445-448. (25) Dohnal, V.; Vrbka, P. Fluid Phase Equilib. 1997, 133, 73-87. (26) Zhang, Y.; Dallas, A. J.; Carr, P. W. J. Chromatogr. 1993, 638, 43-56.
3506 Analytical Chemistry, Vol. 71, No. 16, August 15, 1999
Figure 5. Logarithmic plot of experimental limiting activity coefficients, γ∞j,par, of diverse solutes in low-molecular-weight n-paraffins as a function of those calculated by extrapolation by using data listed in ref 12 in eqs 5 and 6 (see also eq 21) (226 data points in the temperature range of 7-80 °C; O, in pentane; 1, in hexane; 3, in heptane; B, in C8-C14 n-alkanes).
as a function of data calculated with the aid of eqs 5 and 6, i.e., by the method of ref 12. The correlation function is given in eq 9. ∞(calc) log γj,sv ) -0.135 ( 0.037 + ∞(exp) (0.7576 ( 0.0465) log γj,sv (9)
The correlation is poor. In fact, by the nature of approximations introduced in the derivation of the correction function in ref 12 it was not expected to be valid for data in low-molar-volume solvents. RELATIONSHIPS In eq 10, relationships between the specific retention volume, Vg,j,sv (cm3 g-1), the partition coefficient, Kj,sv (1), the Henry coefficient, hj,sv (atm), and the molal Henry coefficient, gj,sv (atm mol kg-1), at the same temperature, T (K) are summarized:1,2,8,12
Vg,j,sv ) Kj,svνsv )
RT RT ) Msvhj,sv 1000gj,sv
(10)
Here, R (cm3 atm mol-1 K-1) is the universal gas constant, vsv (cm3 g-1) is the specific volume, and Msv (g mol-1) is the molar mass of the solvent. Equation 10 permits calculation of partition data, preferentially gj,sv, from gas chromatographic measurements which in turn can be used for the calculation of the related standard chemical potential (see eq 3). Measurements at at least 4 temperatures are necessary for the determination of the three regression coefficients, ∆Hj,sv, ∆Sj,sv, and ∆CP,j,sv, by curve fitting. Knowledge of the regression coefficients allows calculation of the standard chemical potential of the solute at a desired temperature. The molal Henry coefficient is then given by eq 11.
Table 1. Limiting Activity Coefficients, γj∞(exp) ,par , of a Series of Solutes in Low-Molecular-Weight Paraffins, CzH2z + 2, Compared with Those Calculated with Eq 28, γj∞(calc) ,par solute (j) pentane
hexane
heptane
octane
nonane
decane
pentane
toluene
1-chlorobutane
dichloromethane trichloromethane
tetrachloromethane
ethyl bromide
nitroethane
solvent (C-z)
temp t/°C
∞(calc) γj,par
∞(exp) γj,par
C-5 C-5 C-5 C-5 C-6 C-6 C-6 C-6 C-7 C-7 C-7 C-7 C-8 C-8 C-8 C-8 C-9 C-9 C-9 C-9 C-10 C-10 C-10 C-10 C-7 C-8 C-8 C-8 C-8 C-8 C-8 C-12 C-12 C-5 C-6 C-6 C-6 C-7 C-10 C-6 C-6 C-6 C-6 C-7 C-8 C-6 C-6 C-6 C-6 C-6 C-6 C-6 C-6 C-7 C-8 C-8 C-8 C-8 C-6 C-6 C-6 C-6 C-7 C-8 C-6 C-6 C-6 C-6 C-6 C-6 C-6 C-7
20.0 25.0 30.0 35.0 20.0 25.0 30.0 35.0 20.0 25.0 30.0 35.0 20.0 25.0 30.0 35.0 20.0 25.0 30.0 35.0 20.0 25.0 30.0 35.0 20.0 20.0 20.0 40.0 55.0 60.0 75.0 25.0 7.0 25.0 25.0 31.7 49.7 25.0 25.0 27.9 42.2 58.9 67.2 20.0 20.0 27.9 28.8 42.2 44.5 58.7 61.6 67.0 67.5 20.0 20.0 20.0 40.0 60.0 27.9 41.9 59.9 67.2 20.0 55.0 27.9 42.2 58.9 67.2 58.9 66.3 66.3 20.0
0.82 0.81 0.80 0.80 0.90 0.89 0.88 0.87 0.95 0.94 0.93 0.93 1.00 0.99 0.98 0.97 0.98 0.98 0.98 1.01 0.93 0.93 0.94 0.94 0.94 0.97 0.97 0.93 0.89 0.88 0.85 0.96 1.01 1.75 1.70 1.67 1.62 1.64 1.44 1.65 1.54 1.45 1.41 2.33 2.22 1.39 1.39 1.34 1.33 1.28 1.28 1.26 1.26 1.40 1.36 1.36 1.29 1.23 0.99 0.96 0.92 0.91 1.02 0.94 1.82 1.68 1.55 1.49 12.7 11.6 11.6 22.5
1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1.00 0.97 0.97 1.00 1.07 1.05 1.06 0.94 0.97 1.94 1.64 1.59 1.46 1.51 1.26 1.52 1.50 1.43 1.40 2.20 2.15 1.58 1.58 1.53 1.54 1.48 1.46 1.39 1.42 1.47 1.69 1.43 1.55 1.43 1.20 1.20 1.19 1.16 1.20 1.11 1.62 1.54 1.37 1.26 12.3 9.00 10.0 19.8
ref
solute (j) hexane
heptane octane
dodecane i-octane 1-pentene cyclopentane cyclohexane
28 29 30 30 31 30 31 32 32 33 33 35 35 33 33 35 35 35 35 28 28 35 36 35 36 35 36 35 36 28 30 28 30 30 35 35 35 35 29 31 35 35 35 35 35 28 35 28
benzene
ethyl bromide methyl iodide ethyl iodide
cyanoethane
nitromethane
nitroethane
1-propanol 1-butanol
solvent (C-z)
temp t/°C
∞(calc) γj,par
∞(exp) γj,par
ref
C-7 C-8 C-8 C-12 C-12 C-12 C-12 C-5 C-6 C-7 C-10 C-6 C-8 C-8 C-7 C-8 C-12 C-12 C-6 C-6 C-6 C-6 C-7 C-12 C-12 C-7 C-7 C-7 C-7 C-8 C-8 C-12 C-12 C-7 C-8 C-7 C-8 C-6 C-6 C-6 C-6 C-7 C-7 C-8 C-6 C-6 C-6 C-6 C-7 C-7 C-7 C-8 C-5 C-6 C-6 C-6 C-6 C-7 C-7 C-7 C-8 C-8 C-8 C-10 C-6 C-6 C-6 C-6 C-12 C-14 C-5 C-6
20.0 55.0 75.0 7.0 25.0 7.0 25.0 25.0 25.0 25.0 25.0 20.0 55.0 75.0 20.0 20.0 7.0 25.0 27.9 42.2 58.9 67.9 20.0 7.0 25.0 58.0 60.0 77.5 93.0 55.0 75.0 7.0 25.0 20.0 20.0 20.0 20.0 24.9 49.8 56.0 67.5 20.0 20.0 20.0 21.9 49.8 59.2 67.8 20.0 20.0 20.0 20.0 25.0 25.0 49.8 59.2 67.8 20.0 20.0 20.0 20.0 40.0 60.0 25.0 25.0 25.0 42.9 58.9 20.0 20.0 35.0 25.0
0.95 0.93 0.90 1.04 1.00 1.07 1.03 0.77 0.87 0.94 1.04 0.59 0.89 0.86 0.99 1.01 0.96 0.93 0.99 0.96 0.94 0.92 1.02 0.98 0.93 1.48 1.47 1.41 1.38 1.43 1.35 1.32 1.27 1.92 1.86 2.33 2.19 2.29 1.99 1.93 1.84 2.27 2.27 2.16 19.8 12.0 10.6 9.58 20.5 20.5 20.5 20.3 36.9 36.7 23.6 19.0 16.9 39.6 39.6 39.6 38.5 26.5 17.9 32.6 21.1 21.1 15.9 12.7 39.3 36.5 40.7 55.7
1.00 1.07 1.04 0.99 0.99 1.08 1.02 1.18 1.08 1.05 1.02 0.96 1.10 1.06 0.98 0.96 0.90 0.87 1.09 1.09 1.07 1.05 0.99 0.93 0.92 1.37 1.35 1.33 1.27 1.39 1.30 1.33 1.29 1.62 1.62 1.94 1.86 2.07 1.87 1.83 1.67 1.85 1.90 1.80 19.2 13.6 10.7 10.0 21.9 23.8 23.9 20.9 46.8 39.7 23.9 19.9 17.8 41.0 48.1 49.3 39.5 24.6 21.4 30.8 18.2 20.7 13.2 10.3 44.3 38.5 32.1 39.6
29 31 31 32 32 32 32 33 33 33 33 34 31 31 28 28 32 32 35 35 35 36 29 32 32 35 37 35 35 31 31 32 32 28 28 28 28 35 35 35 35 28 29 28 35 35 35 35 28 38 39 28 33 33 35 35 35 28 38 39 28 30 30 33 28 35 35 28 25 25 25 25
Analytical Chemistry, Vol. 71, No. 16, August 15, 1999
3507
Table 1 (Continued) solute (j)
1-nitropropane
1-acetoxyethane
1-acetoxybutane 1-propanol
2-butanone
2-pentanone 1,4-dioxane a
solvent (C-z)
temp t/°C
∞(calc) γj,par
∞(exp) γj,par
ref
C-7 C-7 C-8 C-6 C-6 C-6 C-6 C-7 C-7 C-6 C-6 C-6 C-6 C-7 C-7 C-7 C-8 C-8 C-8 C-8 C-7 C-7 C-5 C-6 C-6 C-6 C-6 C-6 C-7 C-8 C-10 C-7 C-7 C-7 C-8 C-8 C-8 C-10 C-10 C-8 C-8 C-5
20.0 20.0 20.0 27.9 42.2 58.9 67.2 20.0 20.0 25.0 42.9 58.9 66.3 20.0 20.0 20.0 20.0 20.0 40.0 60.0 20.0 20.0 35.0 25.0 27.9 42.2 58.7 67.0 25.0 25.0 20.0 40.0 59.8 70.0 20.0 55.0 75.0 25.0 60.0 55.0 75.0 25.0
22.5 22.5 21.8 12.2 10.1 8.21 7.49 13.8 13.8 4.63 3.88 3.39 3.21 4.82 4.82 4.82 4.70 4.70 3.83 3.23 3.25 3.25 31.5 41.4 38.0 25.8 17.4 14.5 40.4 39.1 42.4 4.10 3.68 3.48 4.56 3.70 3.36 4.19 3.41 3.24 2.91 5.93
25.4 25.3 19.1 14.1 11.5 9.60 8.50 16.1 16.3 3.39 2.95 2.60 2.39 3.29 3.67 3.68 3.25 3.09 2.94 2.81 2.80 2.81 36.2 45.6 39.0 26.1 16.6 13.7 42.1 40.9 45.1 4.12 3.27 3.12 4.15 3.79 3.26 4.13 3.50 3.35 2.84 4.76
38 38 28 35 35 35 35 38 39 35 35 35 35 28 38 39 28 30 30 30 38 39 25 25 35 35 35 35 25 25 25 43 43 43 28 31 31 33 23 31 31 33
solute (j)
1-pentanol
1-hexanol 1-octanol 2-butanol 2-butanone
1,4-dioxane pyridine
tetrahydrofuran
solvent (C-z)
temp t/°C
∞(calc) γj,par
∞(exp) γj,par
ref
C-6 C-6 C-6 C-6 C-7 C-8 C-8 C-10 C-12 C-14 C-6 C-7 C-9 C-9 C-7 C-7 C-8 C-5 C-5 C-5 C-5 C-5 C-6 C-6 C-6 C-6 C-6 C-6 C-7 C-7 C-7 C-6 C-7 C-10 C-6 C-6 C-6 C-6 C-6 C-6 C-6 C-8
27.9 42.2 58.7 67.2 25.0 25.0 60.0 20.0 20.0 20.0 35.0 25.0 80.0 80.0 25.0 25.0 20.0 25.0 30.0 40.1 40.0 50.0 24.8 25.0 30.2 42.2 58.9 67.2 25.0 39.7 40.0 25.0 25.0 25.0 23.5 43.8 57.0 67.5 31.7 49.3 67.1 20.0
50.3 31.8 20.2 16.4 53.8 51.8 18.1 57.2 52.6 48.5 27.6 37.1 9.57 9.57 26.8 39.1 31.9 4.53 4.40 4.11 4.11 3.89 4.56 4.56 4.42 4.09 3.73 3.57 4.51 4.10 4.10 5.61 5.30 4.48 7.76 6.41 5.75 5.36 2.27 2.08 1.93 2.16
33.0 22.5 15.1 12.2 38.1 36.6 13.6 39.6 38.9 35.9 28.5 27.1 9.30 9.32 23.8 33.4 26.7 5.47 5.19 3.92 3.98 3.69 4.38 4.53 4.31 3.97 3.60 3.40 4.26 4.23 4.41 4.03 3.81 3.15 5.90 4.83 4.22 4.04 1.65 1.59 1.51 1.50
35 35 35 35 25 25 35 25 25 25 25 25 25 40 25 25 41 25 43 43 43 43 35 25 43 35 35 35 33 43 42 33 33 33 35 35 35 35 35 35 35 28
The activity coefficients of a n-alkane in a n-alkane is by definition equal 1.
gj,sv ) exp
( ) ∆µj,sv RT
(11)
The vapor pressure of the pure solute can be calculated with knowledge of the constants, Aj, Bj, and Cj of the Antoine equation given in eq 12.
ln(p°j/atm) ) Aj -
Bj T + Cj
(12)
The activity coefficient at ideal dilution is then calculated with eq 2. The relationship between the ratio of the molal Henry coefficients and the ratio of the activity coefficients at ideal dilution in two solvents is easily calculated with eq 10. For the following calculations, the most useful logarithmic form is given in eq 13:
gj,sv Msv ∞ ∞ ln ) ln + ln γj,sv - ln γj,ref gj,ref Mref Here, the subscript ref designates a reference solvent. 3508 Analytical Chemistry, Vol. 71, No. 16, August 15, 1999
(13)
At this point, let us insist that the choice of a particular standard state for the standard chemical potential difference does not influence the results. The molal Henry coefficient, gj,sv, is the vapor pressure (fugacity) of the solute in equilibrium with a hypothetical “one molal ideal dilute solution”, whereas the Henry coefficient, hj,sv, is the vapor pressure of a hypothetical pure solute having molecules in the state of ideal dilution. The relationship between the two Henry coefficients, κj ) gj,sv and hj,sv, is given in eq 10. Substitution of this result in eq 1 gives (h) (g) ) ∆µj,sv - RT ln Msv ∆µj,sv
(14)
(h) where ∆µj,sv is the standard chemical potential difference between the standard states of the Henry coefficient, hj,sv, and (g) ∆µj,sv (≡ ∆µj,sv in the present paper) refers to the standard states of the molal Henry coefficient. Comparison of eq 14 with eq 1 shows that, for the standard partial molar entropy, eq 15
(h) (g) ∆Sj,sv ) ∆Sj,sv - R ln Msv
(15)
(g) holds, where ∆Sj,sv ≡ ∆Sj,sv in the present paper. Obviously, the standard partial molar enthalpy (the heat of solution of one mole solute at ideal dilution) is independent of the standard state of the solute in the solvent. In eq 16 the relationship between the standard molar enthalpy and the specific retention volume, Vg,j,sv is given:
∆Hj,sv ) R
(
)
∂(∆µj,sv/RT) ∂(1/T)
)-R
(
)
∂(ln Vg,j,sv) ∂(1/T)
(16)
Experimental Data in Light of the Model of Flory and Huggins. In the model of Flory and Huggins, both the solute and the solvent are members of a polymer family. The molecules are flexible chains composed of repeating units (“segments”) linked by flexible bonds. Every segment occupies a point on a tridimensional lattice. Members of the family dissolve in each other without heat of mixing. According to Flory19 and Huggins,20 the logarithm of the activity coefficient of a solute at infinite dilution in such a solution is given by eq 17. ∞ ln γj,sv )1-
rj rj + ln rsv rsv
(17)
Here rj and rsv, are the molar volume of the solute and of the solvent, respectively. (It was preferred, in eq 17, to use a new symbol, r, for the temperature-independent “molar core volume”, to clearly distinguish it from a temperature-dependent molar volume, V. In the original paper, r was the number of segments, which is obviously proportional to the molar core volume.) Let us choose one of the members of the family as a reference solvent. Combination of eqs 13 and 17 gives the following after rearrangement for the molal Henry coefficient of the solute with respect to that of the reference solvent:
(
)
rj gj,sv rj rref rsv ln )+ ln - ln gj,ref rsv rref Mref Msv
(18)
By choosing the hypothetical polymer with infinite molecular weight (Mref ) ∞, hence, rref ) ∞) as the reference solvent, eq 18 simplifies to give the following:
gj,sv rj r∞ rsv rj r/∞ ln ) - + ln - ln ) - + ln / gj,ref rsv M∞ Msv rsv r
(19)
sv
Here, r* ) r/M is the specific core volume of the solvent. Let us rewrite eq 19 for paraffin solvents as follows:
ln
gj,par rj r/∞ ) - / ζ + ln / gj,∞ r r par
(20)
In fact, this molecular property should be of the order of the specific volume of liquid paraffins or of the order of Bondi’s core volumes. Hence, its dependence on the molecular weight can be given by the simple relationship shown as eq 22
r/par ) r/∞ + a(r)ζ
where r∞* and a(r) are constants. Multiplication of eq 20 by RT gives the expression of eq 23 for the standard chemical potential difference
rj r/∞ ∆µj,par ) ∆µj,∞ - RT / ζ + RT ln / rpar rpar
(
)
∂∆µj,par ∂ζ
ζ)0
(21)
) - RT
rj + a(r) r/∞
(24)
Consequently, for high-molecular-weight paraffin solvents, the standard chemical potential difference of a solute is given by eq 25, as
[ ] r j + ar
(∆µj,par)ζ ) 0 ) ∆µj,∞ - T R
r/∞
(exp) ζ ) ∆µj,∞ - Tsj,par ζ
(25)
This is the expression used in ref 12 in which the entropic term, sj,par, has been determined with the aid of partition data measured on high-molecular-weight paraffins. In the following we (exp) will refer to this entropic term as sj,par . It was already shown that, with the aid of eq 25, standard chemical potential differences can be predicted with a high accuracy in solutions of highmolecular-weight paraffins. Combination of the Flory-Huggins Model and Experiment. Obviously, in eq 25 the expression in brackets gives the interpretation of the experimental entropy term in the light of the Flory-Huggins model. (exp) sj,par )R
rj + a(r) r/∞
(26)
After rearrangement, eq 27 results.
) r(exp) j
rpar r/par ) Mpar
(23)
Substitution of eq 22 in eq 23 permits calculation of the partial derivative of ∆µj,par with respect to the reciprocal molecular weight, ζ. Its value for high-molecular-weight solvents (ζ f 0) is given in eq 24
par
Concerning solvent properties, eq 20 is written in terms of the specific core volume, r/par, of the solvent molecule defined as
(22)
(exp) / sj,par r∞ - a(r) R
(27)
Here, r(exp) is a (partial) molar core volume of the solute j calculated from experimental data. Substitution of this experimental solute property in eq 23 results in the relationship given Analytical Chemistry, Vol. 71, No. 16, August 15, 1999
3509
in eq 28, which (hopefully) permits extrapolations to lowmolecular-weight solvents.
∆µj,par ) ∆µj,∞ - T
[
]
(exp) / sj,par r∞ - a(r)R
r/par
r/∞ ζ + RT ln / (28) rpar
Here, r/par ) r/par(ζ) is calculated with eq 22. The “best” values of the constants, r∞* and a(r), in eq 22 were determined by an iterative procedure. The criterion for optimization was the correlation coefficient between experimental data and activity coefficients calculated by using eq 28. The highest correlation coefficient (r ) 0.9934) was obtained when using r∞* ) 0.717 cm3 g-1 and a(r) ) 15.9 cm3 mol-1. Let us now put forward the question as to whether the value of these coefficients is realistic. In fact, the constants, r∞* and a(r), may also be determined by regression. By using Bondi’s increments27 for the calculation of the specific core volume of the paraffins, the regression coefficients were the following: r∞* ) 0.729 cm3 g-1 and a(r) ) 5.4 cm3 mol-1. Use of specific volumes of liquid n-paraffins at 25 °C as a database gave regression coefficients r∞* ) 1.169 cm3 g-1 and a(r) ) 30.9 cm3 mol-1. CALCULATIONS Procedure. Equation 28 with the new expression for the combinatorial entropy term allows the estimation of activity coefficients in volatile paraffinic solvents at ambient temperature from data measured by gas/liquid chromatography on highmolecular-weight paraffins at high temperatures. The following (exp) solute properties are needed: ∆Hj,∞, ∆Sj,∞, ∆CP,j,∞, and sj,par . If they are not available (e.g., in ref 12) the retention of the solute must be determined on at least two high-molecular-weight paraffins (preferentially on C-67 and C-103) at at least four temperatures in the temperature domain 50-210 °C (at lower temperatures C-103 is a solid). The data permit calculation of the molal Henry coefficient, gj,par, from the specific retention volume and then the related chemical potential difference, ∆µj,par (eq 3). The resulting (eight) ∆µj,par values permit determination of the above listed four solute properties (with a degree of freedom of 3 for the variance around the regression). The necessary solvent property, rpar*, is calculated with eq 22 by using the following constants: r∞* ) 0.717 cm3 g-1 and a(r) ) 15.9 cm3 mol-1. Finally, the Antoine constants, Aj, Bj, and Cj, for the calculation of the vapor pressure of the pure solute, pj°, are needed. In possession of the necessary data proceed as follows: (i) Calculate ∆µj,∞ at the desired temperature, T, by using, ∆Hj,∞, ∆Sj,∞, and ∆CP,j,∞, in eq 6. (ii) Calculate rpar* of the solvent by using ζ ) Mpar-1 and the above listed constants in eq 22. (iii) Calculate ∆µj,par at the same temperature, T, by using (exp) sj,par in eq 28. (iv) With the result calculate the molal Henry coefficient by using eq 11. (v) Calculate the partial vapor pressure of the pure solute, pj°, at the same temperature, T, by using eq 12. ∞ (vi) The activity coefficient, γj,par , can now be calculated with eq 2. (27) Bondi, A. Physical Properties of Molecular Crystals, Liquids and Glasses; John Wiley & Sons: New York, 1968.
3510 Analytical Chemistry, Vol. 71, No. 16, August 15, 1999
Calculation Example. Nitroethane in hexane at 25 °C. Solute properties. Thermodynamic data:12 ∆Hj,∞ ) -5924 cal mol-1; ∆Sj,∞ ) -16.371 cal mol-1 K-1; ∆CP,j,∞ ) 7.3 cal mol-1 K-1; s(exp)j,par ) 226 cal g mol-2 K-1. Constants of the Antoine equation:9 Aj ) 7.5878; Bj ) 1671.3; Cj ) -31.96. Solvent properties. ζ ≡ MC6-1 ) 1/86.18 ) 0.0116 mol g-1. (i) The standard chemical potential difference of the solute in the reference paraffin of infinite molecular weight at T ) 25.0 + 273.15 ) 298.15 K (eq 6): ∆µj,∞ ) -5924 - 298.15(-16.371) + 7.3[298.15 - 403.15 - 298.15 ln(298.15/403.15)] ) -1153 cal‚mol-1 (ii) The specific core volume of hexane (eq 22; ζ ) 1/86.18 ) 0.0116 mol g-1): r/C6 ) 0.717 + (15.9 × 0.0116) ) 0.901 cm3‚g-1 (iii) The standard chemical potential difference of nitroethane between a one molal solution in hexane and the gas phase at 25 °C (eq 28): ∆µj,C6 ) -1153 - [298.15(226 × 0.717 - 15.9 × 1.9872)/0.901]0.0116 + 1.9872(298.15) ln (0.717/0.901) ) -1789 cal‚mol-1 (iv) The molal Henry coefficient of the solute (eq 11): gj,C6 ) exp[-1789/(1.9872 × 298.15)] ) 0.04883 atm‚kg‚mol-1 (v) The vapor pressure of pure nitroethane at 25 °C (eq 12): log(pj°/Torr) ) 7.5878 - 1671/(298.15 - 31.96) ) 1.309, hence pj° ) 20.4 Torr ) 0.0268 atm (vi) The limiting activity coefficient of nitroethane in hexane at 25 °C (eq 2): ∞ γj,C6 )
1000 × 0.04883 ) 21.1 0.0268 × 86.18
∞ In the literature are reported: γj,C6 ) 18.2 ( 1.828 and 35 ) 20.7 ( 1.8.
∞ γj,C6
RESULTS AND DISCUSSION In Table 1 literature limiting activity coefficients of 38 apolar and polar solutes (normal, branched, and cyclic chloro-, bromo-, iodo-, cyano-, nitro-, and acetoxy alkane derivatives, further 1- and 2-alkanols, 2-alkanons, halomethanes, ethers and pyridine derivatives) are compared in volatile paraffins as solvents in the temperature range of 7-80 °C with those calculated with eq 28. The 202 experimental data from refs 28-43 have been determined by various classical experimental techniques. In Table 1, data of (28) Thomas, E. R. Ph.D. Thesis, University of Illinois, Urbana, IL, 1980. (29) Thomas, E. R.; Newman, B. A.; Long, T. C.; Wood, D. A.; Eckert, C. A. J. Chem. Eng. Data 1982, 27, 399-405. (30) Belfer, A. J.; Locke, D. C. Anal. Chem. 1984, 56, 2485-2489. (31) Arnold, D. W.; Greenkorn, R. A.; Chao, K.-C. J. Chem. Eng. Data 1986, 27, 123-125. (32) Letcher, T. M.; Moollan, W. C. J. Chem. Thermodyn. 1995, 27, 1025-1032. (33) Park, J. H.; Carr, P. W. Anal. Chem. 1987, 59, 2596-2602. (34) Kontogeorgis, G. M.; Coutsikos, P.; Tassios, D.; Fredenslund, A. Fluid Phase Equilib. 1994, 92, 35-66. (35) Thomas, E. R.; Newman, B. A.; Nicolaides, G. L.; Eckert, C. A. J. Chem. Eng. Data 1982, 27, 233-240. (36) Trampe, D. M.; Eckert, C. A. S. J. Chem. Eng. Data 1990, 35, 156-162. (37) Twu, C. H.; Bluck, D.; Cunningham, J. R.; Coon, J. E. Fluid Phase Equilib. 1992, 72, 25-39. (38) Milanova´, E.; Cave, G. C. B. Can. J. Chem. 1982, 60, 2697-2706. (39) Afrashtehfar, S.; Cave, G. C. B. Can. J. Chem. 1986, 64, 198-203. (40) Pividal, K. A.; Sandler, S. I. J. Chem. Eng. Data 1990, 35, 53-60. (41) Sagert, N. H.; Lau, D. W. P. J. Chem. Eng. Data 1986, 31, 475-478. (42) Shen, S.; Nagata, I. Thermochim. Acta 1995, 258, 19-31. (43) Dallinga, L.; Schiller, M.; Gmehling, J. J. Chem. Eng. Data 1993, 38, 147155.
Figure 6. Logarithmic plot of experimental limiting activity coefficients, γ∞j,par, of divers solutes in low-molecular-weight n-paraffins as a function of those calculated by extrapolation by using the present procedure with data listed in ref 12 in eq 28 (226 data points in the temperature range of 7-80 °C; O, alkanes; 1, cyclic and aromatic hydrocarbons; 3, alkanols; B, miscellaneous compounds).
alkanes in alkanes are also included. These activity coefficients are equal to 1 by definition (the activity coefficient of any substance dissolved in itself is equal to unity independently of temperature). Calculation with eq 28 does not give the right figure and the results are even temperature-dependent. Regression of data calculated with Eq 28 as a function of experimental data, (where alkane in alkane data were also included) gave the following result ∞(calc) logγj,par ) -0.0035((0.0063) + ∞(exp) 1.0236((0.0080) × logγj,par (29)
with n ) 226, σ ) ( 0.068 and r ) 0.9934. The correlation is satisfactory. Comparison of the plot of Figure 6 with that of Figure 5 illustrates the improvement of the predicted activity coefficients when using eq 28 instead of eq 25 for the calculation. Let us now recall that in the derivation of eq 28, the “molar core volume of the solute”, rj, appearing in eq 23 (the FloryHuggins result) has been substituted by an “experimental core volume”, rj(exp). The latter was calculated from data measured on high-molecular-weight paraffins, and following the model of Flory-Huggins, it may be interpreted as some “partial molar core volume of the solute” in the paraffin solvent. In Figure 7 the parameter rj(exp) is plotted as a function of molar volumes calculated by Bondi’s procedure, rj(Bondi), i.e., as a sum of structural increments. (These molar volumes are known to correlate reasonably with experimental molar volumes.) The correlation shown in Figure 7 is poor (σ ) (15.6 cm3 mol-1; r ) 0.765), indicating that the partial molar core volume of the solute, the key parameter, cannot be calculated by summing Bondi’s group contributions.
Figure 7. Plot of the “experimental partial molar core volume of the solute”, rj(exp), as a function of its molar core volume calculated with Bondi’s increments,27 rj(Bondi).
CONCLUSIONS The Flory-Huggins equation permits prediction of activity coefficients in polymer mixtures by identifying the parameter r as the molar volume of a component. The equation also permits one to relate limiting activity coefficients of a given solute, j, in paraffins as solvents where the paraffins are considered as members of a polymer family. In this case, however, the parameter, rj, must be considered as an experimental characteristic of the solute. Identifying rj with the molar volume of the pure solute or with Bondi’s core volumes gave wrong results. Hence, this solute property had to be calculated from gas/liquid partition data determined by gas chromatography on two high-molecularweight members of the paraffin family. ACKNOWLEDGMENT This paper reports on part of a project financed by the Foundation “Pro Arte Chimica Helveto-Pannonica” (Veszpre´m, Hungary). A fellowship to A. D. is gratefully acknowledged. GLOSSARY Subscripts j
solute
sv
solvent; in particular: ref
a reference solvent
par
a paraffin
C16
hexadecane
∞
paraffin of infinite molecular weight
Superscripts (exp)
experimental values
(calc)
calculated values
(h)
related to the Henry coefficient h
(g)
related to the Henry coefficient g Analytical Chemistry, Vol. 71, No. 16, August 15, 1999
3511
Symbols a(r) (cm3 mol-1)
coefficient in eq 22
Aj (1) Bj (K-1) Cj (K) constants of the Antoine equation (eq 12) ∆CP (cal mol-1 K-1)
partial molar heat capacity difference
γ∞ (1)
activity coefficient at ideal dilution
g (atm kg mol-1)
molal Henry coefficient
h (atm)
Henry coefficient
∆H (cal mol-1)
partial molar enthapy difference
K (1)
partition coefficient
the universal gas constant
∆S (cal mol-1 K-1)
partial molar entropy difference
T and T† (K)
temperature and standard temperature
v (cm3 g-1)
specific volume
V (cm3)
volume
Vg (cm3 g-1)
specific retention volume
ζ (mol g-1)
inverse molecular weight of the paraffin solvent
) g or h
κ M (g
mol-1)
molecular weight
∆µ (cal mol-1)
standard chemical potential difference
p° (atm)
vapor pressure of the pure substance
r (cm3 mol-1)
molar core volume
r*
R (cal mol-1 K-1) or R (cm3 atm mol-1 K-1)
(cm3
g-1)
specific core volume
3512 Analytical Chemistry, Vol. 71, No. 16, August 15, 1999
Received for review November 16, 1998. Accepted April 27, 1999. AC981255M
