Correlation of gas–liquid partition coefficients using a generalized linear solvation energy relationship

Microchemical Journal 80 (2005) 183 – 188 www.elsevier.com/locate/microc

Correlation of gas–liquid partition coefficients using a generalized linear solvation energy relationship Jung Hag Parka,*, Young Kyu Leea, Jin Soon Chaa, Seog K. Kima, Yong Rok Leeb, Chong-Soon Leec, Peter W. Carrd a Department of Chemistry, Yeungnam University, Kyongsan 712-749, South Korea Department of Applied Chemistry, Yeungnam University, Kyongsan 712-749, South Korea c Department of Biochemistry, Yeungnam University, Kyongsan 712-749, South Korea d Department of Chemistry, University of Minnesota, Minneapolis, MN 55455, USA

b

Received 22 July 2004; accepted 25 July 2004 Available online 12 October 2004

Abstract In a number of previous communications, we reported on the utility of the solvatochromic linear solvation energy relationship (LSER) method for the correlation of a number of solute and solvent-dependent properties. In those studies, it was our practice to examine the effect of a variety of solvents on a given solute or a number of solutes in a given solvent. Here we report on a novel generalized LSER in which the solute and solvent were both simultaneously varied so as to assess the validity of the entire LSER concept and define its limits. The Hildebrand solubility parameter, d H, the Kamlet–Taft solvatochromic parameters, p*, a, b and the solute molar volume, V 2, were used as the explanatory variables. The gas–liquid partition coefficient (K) was the property of interest. We have found that the correlation using the generalized linear solvation energy relationship is statistically as good as the previous LSER correlations despite the use of a far smaller number of freely adjustable parameters. Furthermore, the new approach is able to give reasonable predictions of K values of systems not included in the data set upon which the regression is based. D 2004 Elsevier B.V. All rights reserved. Keywords: Linear solvation energy relationship; Gas–liquid partition coefficients; Solvatochromic parameters

1. Introduction In an earlier series of papers, Kamlet, Taft and their coworkers [1] demonstrated that many types of solventdependent properties ( P) are well correlated by equations containing three types of terms: P ¼ Po þ cavity term þ dipolar term þ hydrogen bonding termðsÞ

ð1Þ

The cavity term arises in many concepts of the solution process, including regular solution theory [2], the solvo-

* Corresponding author. Tel.: +82 53 810 2360; fax: +82 53 810 4613. E-mail address: [email protected] (J.H. Park). 0026-265X/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.microc.2004.07.014

phobic theory [3,4], and scaled particle theory [5,6]. It is related to the energy needed to separate the solvent molecules so as to provide a void sufficient to accommodate the solute. Within the context of solubility parameter theory, which was used by Taft et al. [7], the solvent property that governs the cavity term is the square of the solvent Hildebrand solubility parameter (d H)2. From an analysis of a large body of data on gas–liquid partition coefficients for nonpolar gases, Abraham et al. [8] showed that, as expected, the most appropriate measure of cavity size is scaled with the third power of the solute radius. In subsequent work, the solute property that governs the cavity term was taken as V 2, the solute molecular volume, which may be the liquid molar volume, the solute molecular weight divided by its density at 25 8C, or a computer-calculated intrinsic molecular volume, V I [9]. The dipolarity/polarizability term measures

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the exoergic effects of solute-solvent dipole–dipole and dipole-induced dipole interactions. The solute and solvent properties related to this term are p*2 and p*1 , p* being the solvatochromic parameter which measures the ability of a molecule to stabilize a neighboring charge or dipole, and to induce a dipole in a neighboring nondipolar molecule. The hydrogen bonding terms measure the exoergic effects of hydrogen bonding which involves the solvent as donor and the solute as acceptor, measured by a 1 and b m2, and/or the solute as donor and the solvent as acceptor, measured by b 1 and a m2. The solvatochromic parameters a and b are measures of solute and solvent hydrogen bond donor (HBD) acidity and hydrogen bond acceptor (HBA) basicity, respectively. The subscripts 1 and 2 denote the solvent and the solute, and the subscript m indicates that for compounds capable of self-association, the parameter applies to the non-self-associated bmonomerQ solute, rather than the self-associated boligomerQ solvent. For non-selfassociating compounds, a m=a, b m=b. Thus, the generalized linear solvation energy relationship (LSER) describing properties governed by solute–solvent interactions is given by Eq. (2): 2

P ¼ Po þ M V2 ðdH1 Þ þ Sp42 p41 þ Abm2 a1 þ Bam2 b1 :

ð2Þ

When the effect of different solvents on the properties of a single solute is considered, the parameters relating to the solute can be subsumed into the constants in Eq. (2), and the dependence of each term in the resulting Eq. (3) is explicitly on the solvent parameters: P ¼ Po þ hðdH1 Þ2 þ sp41 þ aa1 þ bb1 :

ð3Þ

Similarly, when a series of solutes in a single solvent is being studied, the resulting equation relates specifically to the solute properties: P ¼ Po þ mV2 þ sp42 þ aam2 þ bbm2 :

ð4Þ

Over 600 correlations [1] of solvent effects on many different types of physicochemical properties and reactivity parameters have been performed using Eq. (3) or (4). This includes a large number of systems of immediate relevance to chromatography. For example, we have studied Rohrschneider’s gas–liquid partition coefficients [10], reversedphase liquid chromatography [11,12], and Snyder’s solvent strength parameters for normal phase chromatography [13]. However, the generalized linear free energy relationship, that is, Eq. (2), has not as yet been tested for its ability to correlate properties involving multiple solutes in different solvents. In this study, we will first test the statistical appropriateness of the general model (Eq. (2)) and if it is found to be acceptable, the model will be used to predict, rather than to simply correlate, the properties of solute– solvent systems which were not employed in the correlation. The property examined in this work is the gas–liquid partition coefficients (K) of six prototypical solutes (octane,

2-butanone, ethanol, toluene, p-dioxane and nitromethane) in some 40 solvents whose d H values and solvatochromic parameters are known: K ¼ Cl =Cg

ð5Þ

In this equation, C l is the solute concentration in the liquid and C g is the solute concentration in the gas phase in equilibrium with C l. 1.1. Development of simultaneous multi-solute linear least squares regression Let P ij be a certain property, e.g. the natural logarithm of the partition coefficient (lnK ij ) of solute i in solvent j: Pij ¼ Poi þ M Vi ðdHj Þ2 þ Sp4i p4j þ Abi aj þ Bai bj

ð6Þ

where i denotes the solute (i=1, 2,. . . n) and j denotes the solvent ( j=1, 2,. . . m). For a specific solute 1: P1j ¼ Po1 þ M V1 ðdHj Þ2 þ Sp41 p4j þ Ab1 aj þ Ba1 bj

ð7Þ

Given all the necessary parameters for this solute and the solvents, one can then regress the property, P 1j vs. the parameter products, V 1(d Hj )2,. . . a 1b j . The resulting regression coefficients are P o1, M, S, A and B. If one regresses P 2j vs. V 2(d Hj )2,. . . a 2b j , one will obtain P o2. Even if the model is accurate, a somewhat different set of values for M, S, A and B, due to random experimental error in P 2 and the other parameters, will be obtained. If the model were perfect, one could, in principle, get the coefficients, M, S, A and B, via a single solute regression and then obtain the solute-dependent intercept, P oi for the next solute by simply plugging these coefficients and the necessary parameters into Eq. (6). However, the models are not perfect, nor are the data free of experimental errors, and thus what one has to do is to regress all P ij against all input parameters. To illustrate the approach in a simple fashion, let us suppose we have two solutes and four solvents. This will provide a set of eight equations to be solved so as to minimize the residual sum of squares (RSS) for each solute by choosing five parameters ( P o1, P o2, M, S, A and B). P11 ¼ Po1 þ M V1 ðdH1 Þ2 þ Sp41 p41 þ Ab1 a1 þ Ba1 b1 P14 ¼ Po1 þ M V1 ðdH4 Þ2 þ Sp41 p4 þ Ab1 a4 þ Ba1 b4 P21 ¼ Po2 þ M V2 ðdH1 Þ2 þ Sp42 p41 þ Ab2 a1 þ Ba2 b1 P24 ¼ Po2 þ M V2 ðdH4 Þ2 þ Sp42 p4 þ Ab2 a4 þ Ba2 b4 In general, there are n solutes and m solvents. We have n m equations to be solved to obtain n values of intercept ( P oi ) for the n solutes and four universal regression

J.H. Park et al. / Microchemical Journal 80 (2005) 183–188

coefficients (M, S, A and B) for use in the generalized linear free energy relationship (Eq. (2)). We note here that attempts to develop a scheme in which a single universal intercept was used gave very poor correlations. We believe that this is a consequence of the fact that the LSER approach does not contain an explicit representation of dispersive interactions, nor does it accurately model solute dipole solvent induced dipole processes [14]. Furthermore, it is known that the zero point of the p* scale does not correspond to zero interaction energy.

2. Experimental Regressions based on the above idea have been implemented through a BASIC program in which all the coefficients in the least squares problem are generated and solved by the Gauss elimination method. The experimental gas–liquid partition coefficients used in the correlations were measured by using an automated head space gas chromatographic method in the course of an experimental re-examination of Rohrschneider’s gas–liquid partition coefficients [15]. The solute and solvent parameters are listed in Tables 1 and 2, respectively. In generating the least squares coefficients, the V 2(d H)2 product was scaled by dividing by 105.

3. Results and discussion The result of the simultaneous multi-solute regression of the lnK ij of the solutes in all solvents is given in regression result I in Table 3. The overall correlation coefficient (r=0.875) and total average standard deviation (r=0.44) seem to indicate that the resulting regression equation is not good enough to be used for predicting the K ij . When the residuals of the lnK ij values calculated using the regression result I were inspected, we found that 14 of the calculated lnK ij values, out of a total 252 values, deviated by more than 15% from the experimental values. The 14 outlying numbers are the data for five of the solutes (except ethanol) in ethylene glycol, ethanol and dioxane as solutes in water, Table 1 The solute parameters Solute

Va

d Hb

p*mc

a mc

b mc

Octane 2-Butanone Ethanol Toluene p-Dioxane Nitromethane

163.54 90.17 58.68 106.85 85.71 53.96

7.57 9.30 12.78 8.91 10.13 12.70

0.01 0.67 0.40 0.42d 0.55 0.85

0 0.04 0.33 0 0 0

0 0.48 0.45 0.11 0.37 0.25

a b c d

Molar volume (cm3) at 25 8C. Data from Ref. [18]. Hildebrand solubility parameter (cal/cc)1/2. Data from Ref. [18]. Kamlet–Taft solvatochromic parameters. Data from Ref. [1]. (p mdd) value [19].

185

Table 2 The solvent parameters Solvent

Classa

d Hb

p*c

ac

bc

Cyclohexane Triethylamine Ethyl ether Hexane Isooctane Tetrahydrofuran Isopropyl ether Toluene Benzene p-Xylene Chloroform Tetrachloromethane Butyl ether Dichloromethane Decane Chlorobenzene Bromobenzene Fluorobenzene 1,2-Dichloroethane Ethyl acetate 2-Butanone Anisole Cyclohexanone t-Butanol Pyridine p-Dioxane n-Butanol Isopropanol Acetone Benzonitrile Acetophenone Hexamethylphosphoramide Ethanol Nitrobenzene Dimethylacetamide Methanol Dimethylformamide Acetonitrile Butyrolactone Nitromethane Dimethylsulfoxide Ethylene glycol Water Pentane Heptane Hexadecane

S S S S S S S AR AR AR PC PC S PC S AR AR AR PC S S AR S HB AR NS HB HB S AR AR NS HB AR S HB S S NS S S HB HB S S S

8.20 7.42 7.40 7.27 6.85 9.90 7.06 8.91 9.17 8.77 9.49 8.55 7.76 9.88 7.72 9.68 9.87 9.11 9.78 9.10 9.30 9.50 9.90 10.60 10.62 10.13 11.60 11.50 10.00 8.40 10.58 10.50 12.78 10.00 10.80 14.5 12.10 12.11 12.60 12.70 12.00 14.20 23.53 7.02 7.43 8.00

0 0.14 0.27 0.04 0.04 0.58 0.27 0.55 0.59 0.51 0.58 0.28 0.24 0.82 0.03 0.71 0.79 0.62 0.81 0.55 0.67 0.73 0.76 0.40 0.87 0.55 0.47 0.48 0.71 0.90 0.90 0.87 0.54 1.01 0.88 0.60 0.88 0.75 0.87 0.85 1.00 0.92 1.09 0.08 0.02 0.08

0 0 0 0 0 0 0 0 0 0 0.44 0 0 0.3 0 0 0 0 0 0 0.06 0 0 0.68 0 0 0.79 0.76 0.08 0 0 0 0.83 0 0 0.93 0 0.19 0 0.22 0 0.90 1.17 0 0 0

0 0.71 0.47 0 0 0.55 0.49 0.11 0.10 0.13 0 0 0.46 0 0 0.07 0.06 0.07 0 0.45 0.48 0.22 0.53 1.01 0.64 0.37 0.88 0.95 0.48 0.41 0.49 1.05 0.77 0.39 0.76 0.62 0.69 0.31 0.49 0.25 0.76 0.52 0.18 0 0 0

a

Solvent class: S, select solvent; AR, aromatics; PC, polychlorinated; HB, hydrogen-bonding solvent; NS, non-select aliphatics. b Data from Ref. [18]. c Data from Ref. [16].

octane as the solute in nitromethane and dimethyl sulfoxide, ethanol as the solute in nitrobenzene and hexadecane, nitromethane as the solute in bromobenzene, p-dioxane and hexadecane. Of the above 7 solvents, the basicity parameters of water and ethylene glycol are known to be uncertain [16]. This could be responsible for the large error in the calculated lnK ij values, but the reason why the data in the other solvents are deviant is not clear. The regression excluding these 14 points gave much better results (r=0.926, r=0.238).

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Table 3 Regression results for data set of different solvent class* Intercept for each solutea

IA IB II III IV

Coefficient estimate

O

B

E

T

D

N

M

S

A

B

9.84 10.10 10.03 10.72 10.74

5.94 6.13 6.02 6.37 6.34

4.58 4.58 4.41 4.84 4.39

8.87 8.84 8.42 8.91 8.92

7.48 7.56 7.43 7.83 7.73

5.33 5.51 5.74 5.71 5.81

12.30F0.73 12.97F0.67 12.98F0.93 17.63F1.15 19.10F1.15

5.30F0.24 5.18F0.18 5.49F0.22 5.33F0.18 6.05F0.20

2.94F0.41 2.61F0.34 1.56F0.39 – –

10.70F0.99 11.22F0.74 12.23F0.97 13.30F0.94 13.71F1.18

rb

rc

nd

0.664 0.488 0.467 0.434 0.400

0.875 0.926 0.934 0.946 0.963

252 238 135 169 96

a

Solute-dependent intercept: O, octane; B, butanone; E, ethanol; T, toluene; D, p-dioxane; N, nitromethane. Average standard deviation. c Correlation coefficient. d Number of data points. * Solvent data set: IA, all solvents; IB, all solvents after excluding 14 deviant data points; II, select+hydrogen bond donor+hydrogen bond acceptor solvents; III, select+aromatic+polychlorinated solvents with a=0; IV, select solvents only. b

A requirement that needs to be met before a linear solvation energy relationship deserves any confidence is that chemically distinct subsets should give similar intercepts and coefficients. The results of the correlations for different subsets are given in Table 3. A correlation comprised of only the select and hydrogen bonding solvents did, indeed, give similar intercepts and coefficients (see II, Table 3) to those generated for the full solvent set (IB). Two correlations for solvents with a=0 (regression results III and IV) also give similar intercepts and coefficients for the explanatory variables with the exception of the coefficient of the cavity term. This is probably due to the fact that the Hildebrand solubility parameter is not an accurate model for the cavity term, especially for nonselect solvents. The validity of the multi-solute simultaneous regression was further checked by carrying out a series of correlations after deleting the data for each of the six solutes. The results of these correlations are given in Table 4. All six correlations give similar intercepts and coefficient estimates, except the B coefficient upon deletion of ethanol. The large negative value of B after deleting ethanol is most likely an artifact due to the small value assigned to a m (0.04) for 2butanone. In order to determine if the multi-solute simultaneous regression is statistically as good as the usual single-solute regression [1], an F-test was performed by comparing the mean square error of the residuals obtained in the multi-

2 ) to the sum of the mean square errors solute regression (r M for each of the single-solute regressions (r S2):

FvM ;vS ¼ r2M =r2S ¼ ðRSS M =vM Þ=ðRRSS i =vS Þ ¼ 0:238=0:200 ¼ 1:19 where RSSM represents the residual sum of squares for the multi-solute regression, and RSSi is the residual sum of squares for the ith single solute regression. The degrees of freedom (v) are calculated from the number of data points, minus the total number of adjustable parameters (v M=228, v S=214). The critical value of F is approximately 1.1 (obtained by interpolation from a standard F table) at a probability level of 0.05. Although the calculated F value is greater than the critical value, it appears that the multi-solute regression yields results which are only slightly less reliable than the single-solute regressions. A second F-test was performed by comparing the mean square error of the residuals for the multi-solute regression for each solute (r Mi2) to the mean square error for each single solute regression (r Si2): FvMi ;vSi ¼ r2Mi =r2Si ¼ ðRSS Mi =vMi Þ=ðRSS Si =vSi Þ The results for this comparison are shown in Table 5. In this case, only the results for butanone are significantly degraded when the multi-solute regression procedure is used.

Table 4 Regression results after deleting data set for one solute at a timea Solute deleted

Intercept for each solute* O

B

E

T

D

N

M

S

A

B

None Octane Butanone Ethanol Toluene Dioxane Nitro-methane

10.10 – 10.36 10.23 10.02 10.08 10.02

6.13 6.08 – 6.54 6.04 6.13 6.21

4.58 4.55 4.54 – 4.53 4.57 4.62

8.48 8.39 8.57 8.54 – 8.47 8.51

7.56 7.51 7.58 7.57 7.48 – 7.62

5.51 5.51 5.42 5.49 5.42 5.50 –

12.97F0.67 11.88F0.76 14.62F0.84 13.82F0.64 12.46F0.74 12.88F0.71 12.45F0.69

5.18F0.18 5.06F0.17 5.56F0.95 5.23F0.17 5.31F0.19 5.18F0.20 4.81F0.21

2.61F0.34 2.34F0.34 2.56F0.64 3.42F0.37 2.49F0.36 2.49F0.39 2.67F0.34

11.22F0.74 11.29F0.73 11.62F0.85 15.9F5.61 11.16F0.76 11.26F0.76 11.31F0.70

* See footnotes for Table 3.

Coefficient estimate

rb

rc

nd

0.488 0.483 0.481 0.458 0.507 0.501 0.468

0.926 0.932 0.933 0.906 0.927 0.928 0.926

238 197 198 201 195 197 202

J.H. Park et al. / Microchemical Journal 80 (2005) 183–188

187

Table 5 Results of single-solute regression and comparison with results of multi-solute regression Solute

Intercept

Octane Butanone Ethanol Toluene Dioxane Nitro-methane

10.31 6.42 4.02 8.82 7.60 5.30

a b c

Coefficient estimate M

S

A

B

11.75F1.33 10.43F0.77 6.77F3.71 12.39F1.28 14.19F2.22 12.67F2.17

70.95F24.33 4.62F0.24 6.41F0.64 3.51F0.37 5.28F0.39 5.69F0.34

– 3.00F0.37 1.79F0.70 – 3.17F0.69 0.97F1.24

– 13.57F4.26 10.88F0.74 – – –

r

r

n

RSSSa

RSSMb

Fc

F crit

0.233 0.122 0.233 0.108 0.192 0.338

0.869 0.922 0.963 0.779 0.858 0.930

41 40 37 43 41 36

8.86 4.26 7.47 4.33 7.11 10.83

10.51 9.78 9.72 6.30 7.17 11.97

1.25 2.29 1.30 1.54 1.04 1.14

1.69 1.69 1.79 1.69 1.69 1.79

Residual sum of squares (RSS) obtained in the single-solute regression. Residual sum of squares for a given solute in the multi-solute regression. F=r Mi2/r Si2.

A number of other explanatory variables were tested in the general correlation equation (Eq. (2)) to see if they could improve the regression results. The Onsager reaction field, L(n 2)=(n 21)/(2n 2+1), was included in the correlation in the form of the term L(n 12)L(n 22), so as to better describe dispersive interactions between the solute and the solvent. Such interactions are very important but they are not explicitly represented in the p* parameter. Inclusion of the Onsager reaction field term did not provide statistically better results (r=0.931, r=0.47). Using the (surface tensionvan der Waals molecular area) instead of V 2(d H1)2 as the cavity term gave poorer results (r=0.897, r=0.57). It could also be argued that the p* scale should be offset, so that a p* equal to zero indicates zero interactions. This was done by adding the gas-phase value (p*=1.06) to each of the p* values listed in Tables 1 and 2. A slight decrease in the quality of the regression was observed, but the solutedependent intercepts changed significantly. This may be due to the uncertainty in the location of the zero interaction point of the p* scale, or to the magnitude of the extrapolation involved. Consequently, the correlation results in IB in Table 3 were used in Eq. (2) for the prediction of the lnK ij values of solute–solvent systems which were not employed in the multi-solute regression. Assuming that the vapor phase is ideal, one can compute the gas–liquid partition coefficient of the solute above the pure liquid solute, given the saturated vapor pressure of the pure liquid: K ¼ Cl =Cg ¼ 1000  d  RT =ðMW  po Þ

can then compute the partition coefficient of the solute in any solvent with known d H and solvatochromic parameters, using Eq. (2). The calculated partition coefficients for several solute–solvent systems which were not employed in the multi-solute regression are compared with literature values in Table 6. On the whole the agreement is satisfactory. The multi-solute correlation equation obtained in this work is not good enough to predict very accurate (better than 5%) gas–liquid partition coefficients at this point. We think this is due primarily to the problem in the cavity term [17] represented by V 2(d H1)2 and to a lesser extent, to the inaccuracies in the solvatochromic parameters. Work related to determining the best model for the cavity term is currently in progress in this laboratory. When the model (Eq. (2)) is

Table 6 Comparison of the predicted lnK values with literature data No.

Solute–solvent

Temperature (8C)

lnK ca

lnK eb

Devc

Ref.

1

acetone– methanol acetone– acetonitrile acetone– ethytlene chloride benzene– methanol benzene– acetonitrile benzene– heptane benzene– toluene pentane– toluene acetonitrile– methanol triethylamine– butanol triethylamine– tetrahydrofuran

25

6.69

6.88

0.19

[20]

20

6.72

7.22

0.50

[21]

20

6.98

7.18

0.20

[21]

25

6.50

6.46

+0.04

[22]

25

7.50

7.09

+0.41

[23]

25

6.16

6.75

0.59

[20]

20

7.62

7.50

+0.12

[24]

20

5.03

5.23

0.20

[21]

30

7.74

7.67

+0.07

[25]

25

8.45

8.52

0.07

[26]

25

7.29

7.93

0.64

[26]

2 3

4 5

ð8Þ 6

where d denotes the density, MW the molecular weight, and p o the vapor pressure of the liquid, respectively, and R and T have their usual meaning. From the calculated gas–liquid partition coefficient, the four universal correlation coefficients and the solvatochromic parameters, one can compute the solute-dependent intercept, P o, using Eq. (9): 2

ln K ¼ Po þ MV ðdH Þ þ Spm4pb4 þ Abm ab þ Bam bb

ð9Þ

where the subscript m and b indicate that the parameters apply to the monomer solute and the bulk liquid of the same molecule. Once we have the intercept, P o, for the solute, one

7 8 9 10 11 a b c

Calculated value. Experimental value. Deviation=calculated valueexperimental value.

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J.H. Park et al. / Microchemical Journal 80 (2005) 183–188

better established by using a larger set of experimental data and more accurate parameters, we expect it will produce more accurate predictions of the gas–liquid partition coefficients.

Acknowledgement This work was supported by a Research Grant of Advanced Research Center in Yeungnam University (105096).

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