Predicting aqueous solubilities from aqueous free ...

JOURNAL OF CHEMICAL PHYSICS

VOLUME 119, NUMBER 3

15 JULY 2003

Predicting aqueous solubilities from aqueous free energies of solvation and experimental or calculated vapor pressures of pure substances Jason D. Thompson, Christopher J. Cramer, and Donald G. Truhlar Department of Chemistry and Supercomputer Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431

共Received 25 February 2003; accepted 10 April 2003兲 In this work, we explore the possibility of making predictions of solubilities from free-energy calculations by utilizing the relationship between solubility, free energy of solvation, and solute vapor pressure. Because this relationship is only strictly valid when all activity and fugacity coefficients are unity, it is not clear when it will hold and when it will break down for a given solute–solvent system. So we have tested the validity of this relationship using a variety of liquid solutes and solid solutes in liquid water solvent. In particular, we used a test set of 75 liquid solutes and 15 solid solutes composed of H, C, N, O, F, and Cl. First we compared aqueous free energies of solvation calculated from experimental solute vapor pressures and aqueous solubilities to experimental aqueous free energies of solvation for the 90 solutes in the test set and obtained a mean-unsigned error 共MUE兲 of 0.26 kcal/mol. Second, we compared aqueous solubilities calculated from experimental solute vapor pressures and aqueous free energies of solvation to experimental aqueous solubilities for the 90 solutes in the test set and obtained a mean-unsigned error of the logarithm 共MUEL兲 of the aqueous solubility of 0.20. These results indicate that the relation has useful accuracy. Using this relationship, we have also investigated the utility of three continuum solvation models, in particular Solvation Model 5.42R implemented at the Hartree–Fock, Becke-3– Lee–Yang–Parr, and Austin Model 1 levels 共SM5.42R/HF, SM5.42R/B3LYP, and SM5.42R/AM1, respectively兲 to predict aqueous solubilities of liquid solutes and solid solutes in water solvent. The SM5.42R solvation model can predict the aqueous free energy of solvation and, given several solvent descriptors, it can also predict the free energy of self-solvation 共which can be converted to a solute vapor pressure兲. We compared aqueous solubilities calculated from experimental solute vapor pressures and SM5.42R aqueous free energies of solvation to experimental aqueous solubilities for the 90 solutes in the test set and obtained an MUEL of the aqueous solubility of 0.40 for SM5.42R/HF, 0.35 for SM5.42R/B3LYP, and 0.43 for SM5.42R/AM1. We also compared aqueous solubilities calculated from SM5.42R aqueous free energies of solvation and SM5.42R vapor pressures to experimental aqueous solubilities for all 75 liquid solutes and the 7 solid solutes for which vapor pressures can be predicted by the SM5.42R solvation model; these computations yielded an MUEL of the solubility of 0.39 for SM5.42R/HF, 0.37 for SM5.42R/B3LYP, and 0.36 for SM5.42R/AM1. © 2003 American Institute of Physics. 关DOI: 10.1063/1.1579474兴

I. INTRODUCTION

structures and theoretical descriptors such as topological indices, molecular-surface properties, calculated partial atomic charges, or even calculated partition coefficients which themselves depend on other properties. The relations need not be linear; in some cases, neural networks have been used to optimize the functional relations.17,18,21,22,24,26,29 Other workers have developed hybrids of these approaches, such as methods that combine categories 1 and 2.30 Nevertheless, existing methods fail to be accurate enough for many practical problems.31 Empirical relationships between thermodynamic properties 共as in category 1 above兲 are sometimes called quasithermodynamic or extrathermodynamic relations. A more fundamental approach to solubility prediction would be based directly on free-energy calculations and true thermodynamic relations. There has been enormous progress in calculating free energies of liquid-phase molecules in recent years,32 as evidenced especially by a number of successful methods for

Solubility is a fundamental physical property,1 and there is a great need, especially in environmental, pharmaceutical, and industrial chemistry, for accurate predictions of solubilities that have not been measured and even of solubilities of compounds that have not yet been synthesized. A great variety of theoretical methods has been proposed for such predictions, but most such methods may roughly be classified into a few general categories: 共1兲 single or multiple linear regressions against other experimental properties2–5 or more general methods of correlating solubility to experimental properties,6,7 where the experimental properties could be partition coefficients, melting points, molar volumes, etc.; 共2兲 group contribution methods;1,8 –15 and 共3兲 quantitative structure–property relationships16 –29 共QSPRs兲, which are similar to the regressions already mentioned except that experimental variables are replaced by molecular structural properties and descriptors, including three-dimensional 0021-9606/2003/119(3)/1661/10/$20.00

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J. Chem. Phys., Vol. 119, No. 3, 15 July 2003

calculating free energies of solvation and partition coefficients,33–35 the latter being equivalent to the differences of the free energy of solvation of a given solute in two different solvents. Despite this progress, there has been very little work reported on the use of free-energy calculations to predict solubility. One prominent exception is the work of Klamt and co-workers,35,36 who successfully used freeenergy methods to predict solubilities of liquids in liquids; they also combined free-energy calculations with linear regressions for the enthalpy of fusion to predict solubilities of solids in liquids. Another exception is the very recent work of Amovilli and Floris,37 who combined free-energy calculations with a simple theory 共mobile order theory兲 of combinatorial mixing entropy to predict solubilities of water in liquid hydrocarbons. In the present work we examine a number of fundamental issues involved in the use of free-energy calculations for the prediction of solubilities of both liquid solutes and solid solutes in liquid solvents. All of our numerical calculations use continuum models33,35– 41 共also called implicit models兲 for the solvent and take the solvent as water, but the questions are more general and many of the conclusions are equally applicable to free-energy calculations based on atomistic molecular dynamics simulations42 or Monte Carlo simulations.43,44 The first question we ask is whether it causes appreciable error to set the activity coefficient equal to unity for a saturated solution. This is very important because most calculations of free energy of solvation are carried out for the limiting case of a single solute molecule, i.e., an infinitely dilute solution. Is this applicable to saturated solutions, which must be modeled to predict solubility? The answer is especially in doubt for strongly interacting solute–solvent systems such as alcohols and aldehydes in water. We test this by starting with a thermodynamic relation between solubility, free energy of solvation for an infinitely dilute solution, and solute vapor pressure that is valid when all activity and fugacity coefficients are unity. Then we test how well this relation holds. Second, we ask whether current methods for calculating free energies of solvation and vapor pressures are accurate enough to compete with quasithermodynamic relationships and other existing empirical approaches for predicting solubilities of liquids in liquids. For example, Klopman and Zhu15 found a mean-unsigned error in the logarithm 共MUEL兲 of the solubility of 0.5 for a group contribution approach trained on 1168 solubilities, which 共they pointed out兲 was superior to other group contribution methods available at that time 共2001兲. 共All logarithms in this paper are to the base 10.兲 More recently, a survey4 of 15 modeling methods characterized their accuracy in terms of the MUEL of the solubility and found values ranging from 0.38 to 0.88 over various training sets and test sets with a median MUEL of 0.53. Our goal in the present work does not involve a careful comparison of methods over a consistent test set, but rather an examination of the theoretical basis for using predictions of free energies of the infinitely dilute solvation to predict solubilities, and therefore we will consider only a small test set representative of common organic functionalities.

Thompson, Cramer, and Truhlar

Finally, we consider the very fundamental question of how useful continuum dielectric methods are for solids and whether we can use them to predict solubilities of solids in liquids with the same accuracy that we can predict solubilities of liquids in liquids. The numerical tests in this paper are based on a test set of 75 liquid solutes and 15 solid solutes that contain a variety of functional groups; the aqueous solubilities of the molecules in this test set span about 7 orders of magnitude. We use this test set to make a series of four critical comparisons: 共1兲 aqueous free energies of solvation calculated from experimental vapor pressures of pure substances 共all vapor pressures in this work refer to pure-substance vapor pressures兲 and solubilities compared to experimental aqueous free energies of solvation, 共2兲 aqueous solubilities calculated from experimental solute vapor pressures and aqueous free energies of solvation compared to experimental solubilities, 共3兲 aqueous solubilities calculated from experimental solute vapor pressures and aqueous free energies of solvation from solvation model 5.42R 共SM5.42R兲41共i–l兲 compared to experimental solubilities, and 共4兲 solubilities calculated from solute vapor pressures and free energies of solvation from the SM5.42R solvation model compared to experimental solubilities. Finally, we compare errors in the present calculated results to errors in solubilities calculated by using the popular UNIFAC 共Refs. 8 –12兲 fragment model. From these various comparisons we will draw conclusions that are useful for the further development of the field of solubility modeling. In the sections that follow, we briefly outline the relationship between aqueous solubility and aqueous free energy of solvation in the case where all activity and fugacity coefficients are set equal to unity. We then describe the SM5.42R solvation model, the test set of 90 liquid solutes and solid solutes used throughout this work, and the computational details of our calculations. We present and discuss our results and then finally offer some concluding remarks.

II. RELATIONSHIP BETWEEN AQUEOUS SOLUBILITY AND AQUEOUS FREE ENERGY OF SOLVATION

The thermodynamics of solvation is treated in detail elsewhere.45– 47 To relate the aqueous free energy of solvation of a liquid solute to that solute’s aqueous solubility, we first consider a liquid species A in equilibrium with its vapor: 共1兲

A共g)↔A共l),

where the parenthetical notation indicates the state of the solute 共e.g., g for gas phase and 1 for liquid phase兲. Assuming a 1 molar standard-state and ideal behavior in both phases, the standard-state free energy of this process, which 48 we will denote as ⌬G 䊊 1 , is ⌬G 䊊 1 ⫽RT ln

䊉 PA l P䊊M A

,

共2兲

where R is the universal gas constant, T is the temperature, 䊉 is the equilibrium vapor pressure of A over pure A, P 䊊 is PA the pressure 共24.45 atm兲 of an ideal gas at 1 molar concenl tration and 298 K, and M A is the equilibrium molarity of the



Predicting aqueous solubilities

pure liquid solution of A, which is obtained from the liquid density of A. Next, we consider the equilibrium between pure liquid A and an aqueous solution of A: 共3兲

A共l)↔A共aq).

Again, assuming that all activity coefficients are unity, the standard-state free energy of this process, which we will denote as ⌬G 䊊 2 , is ⌬G 䊊 2 ⫽⫺RT

ln

aq MA l MA

共4兲

,

aq where M A is the equilibrium aqueous molarity of solute A, i.e., its solubility in molarity units. Combining Eqs. 共1兲 and 共3兲 gives

共5兲

A共g)↔A共aq).

The standard-state free-energy change in this process is the standard-state aqueous free energy of solvation of solute A, ⌬G S䊊 (aq), and we can calculate it by adding Eqs. 共2兲 and 共4兲, which yields ⌬G S䊊共 aq兲 ⫽RT ln

䊉 PA

P䊊

aq ⫺RT ln M A .

共6兲

Equation 共6兲 is based on the assumption that the aqueous solution of A obeys Henry’s law,45,47,49 i.e., that the saturated solution is infinitely dilute. One of the goals of this paper is to see if this is a serious impediment to using Eq. 共6兲 to aq is the aqueous solubility of predict solubilities. Note that M A A in molarity units, and we shall also denote it by S. For a solid solute, Eq. 共1兲 becomes 共7兲

A共g)↔A共s) and Eq. 共3兲 becomes

共8兲

A共s)↔A共aq); then, Eqs. 共2兲, and 共4兲, respectively, become ⌬G 䊊 1 ⫽RT ln

䊉 PA s P䊊M A

⌬G 䊊 2 ⫽⫺RT ln

aq MA s MA

共9兲

,

⫽⫺RT ln

S s MA

,

共10兲

and again we obtain Eq. 共6兲. Our treatment of solid solutes now becomes the same as our treatment for liquid solutes, except that P A䊉 corresponds to the pure-substance vapor press corresponds to the molarity of solid A, sure of solid A, M A which is obtained from the solid density of A, and Eq. 共9兲 corresponds to the process of transferring gaseous A to pure solid A. Note that in the applications below, we are usually using Eq. 共6兲 to predict solubility, i.e., aq ⫽ S⬅M A

冉冊冋䊉 PA

P䊊

exp

⫺⌬G S䊊共 aq兲 RT

册

.

共11兲

The above equations clearly apply 共under the same assumptions兲 to any solvent, not just water.

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III. SM5.42R CONTINUUM SOLVATION MODEL

When Eq. 共11兲 is valid, the aqueous solubility of a liquid or solid solute can be predicted given its aqueous free energy of solvation and its free energy of self-solvation 共which corresponds to the pure-substance vapor pressure of the solute兲. The SM5.42R41共i–l兲 solvation model is an implicit solvation model designed to predict free energies of solvation, and it yields ⌬G S䊊 ⫽⌬E E⫹G P⫹G CDS ,

共12兲

where G P is the electronic polarization energy from mutual polarization of the solute and the solvent 共which is taken to be a continuous medium of dielectric constant ␧兲 and ⌬E E is the change in the solute’s internal electronic energy upon immersing the solute into the dielectric medium. The sum of ⌬E E and G P is the net effect due to bulk electrostatics, and it results from a self-consistent-field molecular orbital calculation that employs the generalized Born 共GB兲 approximation.38,50–52 The final term of Eq. 共12兲, G CDS , is a semiempirical term that accounts for nonbulk contributions, such as interactions of the solute with solvent molecules in its near vicinity, i.e., inner solvation-shell effects. In SM5.42R, the G CDS term is a sum of two terms: 1兴 2兴 G CDS⫽G 关CDS ⫹G 关CDS ,

共13兲

where 3

1兴 ⫽ G 关CDS

兺 S ␦ 兺k 兺j ␴ 关Z1 兴j ␦ f Z j 共兵 Z k ⬘ ,R k ⬘k ⬙其兲 A k共 R兲

␦ ⫽1

k

k

共14兲

and where the index ␦ goes over the solvent descriptors n, ␣, and ␤ 共denoted as S ␦ and discussed below兲, k runs over all atoms in the solute, Z k is the atomic number of atom k, R k ⬘ k ⬙ is the interatomic distance between atoms k ⬙ and k ⬘ , f Z k j ( 兵 Z k ⬘ ,R k ⬘ k ⬙ 其 ) is a geometrical function described elsewhere,41共a兲,共c兲–共h兲 A k (R) is the solvent accessible surface area 共SASA兲 of atom k calculated by the ASA algorithm,53 and ␴ 关Z1 兴j ␦ is an atomic surface tension coefficient 共an optik mized parameter兲. The second term in Eq. 共14兲 is 4

2兴 ⫽ G 关CDS

兺

␦ ⫽1

S ␦ ␴ 关␦2 兴

兺k A k共 R兲 ,

共15兲

where the index ␦ goes over the solvent descriptors ␥, ␤ 2 , ␾ 2 , and ␺ 2 , and ␴ 关␦2 兴 is a molecular surface tension coefficient 共an optimized parameter兲. The SM5.42R solvation model is a universal solvation model, meaning that if the geometry 共needed for the GB and SASA calculations兲, the solvent descriptors 共needed for the computation of G CDS), and the dielectric constant for a given solvent 共needed for the GB computation兲 are known or can be estimated, then the free energy of solvation of a solute in that solvent can be calculated. The solvent descriptors in Eqs. 共14兲 and 共15兲 are n, refractive index at the wavelength of the Na D line; ␣, Abraham’s16,54共a兲–共d兲 hydrogen bond acidity parameter, which he calls 兺 ␣ 2 ; ␤, Abraham’s16,54共a兲–共d兲 hydrogen bond basicity parameter,


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which he calls 兺 ␤ 2 ; ␥ equals ␥ m / ␥ 0 , where ␥ m is the macroscopic surface tension 共in units of cal mol⫺1 Å⫺2兲 at a liquid–air interface at 298 K, and ␥ 0 is 1 cal mol⫺1 Å⫺2; ␾ 2 , square of the fraction ␾ of nonhydrogenic solvent atoms that are aromatic carbon atoms 共aromaticity兲; and ␺ 2 , square of the fraction ␺ of nonhydrogenic solvent atoms that are F, Cl, or Br 共electronegative halogenicity兲. The ␴ 关Z1 兴j ␦ and ␴ 关␦2 兴 parameters were optimized41共i–k兲 with k a training set of 2135 experimental free energies of solvation in 91 solvents 共90 liquid organic solvents⫹water兲 plus 16 additional data41共l兲 for silicon-containing solutes. IV. TEST SET OF LIQUID AND SOLID SOLUTES

We constructed the test set used in this work by first taking the subset of solutes from the SM5.42R aqueous solvation model training set41共a兲,共b兲,共g兲,共j兲,共k兲 that contained at most H, C, N, O, F, and Cl. From this subset, we kept only solutes for which we found experimental aqueous solubilities2,6,7,19,55– 65 and pure-substance vapor pressures.48,60,61,66 –71 This led to a test set of 75 liquid solutes and 6 solid solutes. We added 9 more solid solute data by searching additional sources for experimental aqueous free energies of solvation,34,72 aqueous solubilities,36 and pure-substance vapor pressures.73–75 Note that in order to calculate a solute’s vapor pressure, we need to know not only its solute properties but also its solvent properties. All data l or used in this work, including density 共used to obtain M A s M A), solvent descriptors, and dielectric constants, are given in Table S-I in the supplementary material.76 We note that in creating the test set, we found two errors in our experimental aqueous solvation energy training set41共j兲,共k兲 and in our experimental vapor pressure data set.48 In particular, we previously reported the experimental free energy of solvation of methyl benzoate to be ⫺2.22 kcal/ mol, but in this work we have found it to be ⫺4.58 kcal/mol.77 We also previously reported the vapor pressure of 2,4-dimethylpyridine to be 230.8 Pa, but we have now found it to be 393.3 Pa.78 V. COMPUTATIONAL DETAILS

For each solute in our test set, we optimized its gasphase geometry using the hybrid density functional theory mPW1PW91 共Refs. 79 and 80兲 with the MIDI! 共Ref. 81兲 basis set and using the semiempirical molecular orbital theory Austin Model 1 共AM1兲.82,83 We used the GAUSSIAN 98 共Ref. 84兲 computer package for both of these sets of calculations. 共For the mPW1PW91 calculations, we used a local version of GAUSSIAN 98 in which the error85 in coding this functional in GAUSSIAN 98 is corrected.兲 The SM5.42R model will be applied with ab initio Hartree–Fock 共HF兲 theory86 and the 6-31G共d兲 basis set,87– 89 with the hybrid density functional theory B3LYP 共Refs. 90–93兲 and the MIDI! basis set,81 and with the semiempirical molecular orbital theory AM1.82,83 For both sets of gas-phase geometries, aqueous free energies of solvation and free energies of self-solvation were calculated by SM5.42R/HF, SM5.42R/B3LYP, and SM5.42R/AM1. The SM5.42R/HF and SM5.42R/B3LYP calculations were carried out using MN-GSM,94 which is an


add-on module for GAUSSIAN 98. The SM54.2R/AM1 calculations were carried out using AMSOL.95 We will discuss the results for the more accurate mPW1PW91 geometries in Secs. VI A and VI B and those for the AM1 geometries in Sec. VI C. As mentioned above, we require both solute and solvent descriptors for each solute.48 For the 15 solid solutes in our test set, we found the solvent descriptors needed for Eqs. 共14兲 and 共15兲,54共b兲,96 and we found the required dielectric constants97,98 for naphthalene, anthracene, biphenyl, phenanthrene, pyrene, phenol, and p-cresol. In the absence of macroscopic surface tensions for solid solutes, we used the macroscopic surface tension of liquid benzene for naphthalene, anthracene, biphenyl, phenanthrene, and pyrene because they are all similar in structure to benzene, and we used the macroscopic surface tension of liquid m-cresol for phenol and p-cresol. We also predicted solubilities using the UNIFAC method8 –12 implemented in a computer program99 that is described elsewhere.100 Because the reference state for a UNIFAC solubility calculation is a hypothetical liquid at 1 atm and the temperature of interest, we make a correction recommended by Yalkowsky101 for solid solutes. The correction C to the solubility is101 C⫽exp关 ⫺0.023共 T m⫺298兲兴 ,

共16兲

where T m is the melting point of the solid. The solubility of a solid calculated by the UNIFAC method, S UNIFAC共solid), incorporating this correction is S UNIFAC共solid)⫽CS UNIFAC共liquid),

共17兲

where S UNIFAC共liquid) is the UNIFAC-calculated solubility of the hypothetical liquid. We obtained the melting point of each solute in our test set from the NIST chemistry webbook.102 VI. RESULTS AND DISCUSSION A. Liquid solutes

In Sec. II, we related the free energy of solvation of a solute to its pure-substance vapor pressure and solubility by assuming all activity coefficients are unity. In this subsection, we test the validity of this relationship for liquid solutes in water. First, we compute the mean-unsigned error 共MUE兲, as compared to experiment, of the aqueous free energy of solvation calculated via Eq. 共6兲 using experimental solubilities and pure-substance vapor pressures for various solute classes of our liquid–solute test set. These MUEs are shown in Table I, and they range from 0.06 kcal/mol 共for aromatic hydrocarbons and nitro compounds兲 to 0.41 kcal/mol 共for compounds composed of C, H, and N兲. For all 75 liquid solutes, the MUE is 0.20 kcal/mol, which is quite small; in fact, it is the same as previous estimates41共d兲,103 of the reliability of typical experimental standard-state solvation energies. To put these errors in context, we note that the free energies of solvation of the 75 solutes cover a range of 13.9 kcal/mol 共from ⫺11.0 kcal/mol to ⫹2.9 kcal/mol兲. Table I also compares ⌬G S䊊 (aq) calculated by the SM5.42R/HF, SM5.42R/B3LYP, and SM5.42R/AM1 models 共via eq. 共12兲兲 to experiment for each of the various solute




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TABLE I. Mean-unsigned error 共MUE兲 of the aqueous free energy of solvation, ⌬G S䊊 (aq) 共in kcal/mol兲, of various solute classes calculated by various methods. SM5.42Ra Solute class

No. data

From expt. S and P 䊉

HF

B3LYP

AM1

Nonaromatic hydrocarbons Aromatic hydrocarbons Alcohols and m-cresol Ethers and anisole Aldehydes and ketones Esters C, H, N compounds Nitro compounds All H, C, N, O compounds Halogenated hydrocarbons All liquid solutes

11 6 9 4 11 7 7 5 60 15 75

0.14 0.06 0.34 0.29 0.19 0.13 0.41 0.06 0.21 0.19 0.20

0.38 0.15 0.33 0.29 0.51 0.69 0.79 0.57 0.47 0.40 0.45

0.39 0.30 0.38 0.28 0.30 0.46 0.41 0.36 0.36 0.41 0.37

0.36 0.18 0.42 0.36 0.57 0.83 0.44 0.72 0.48 0.42 0.47

a

Using mPW1PW91 geometries.

classes of our liquid-solute test set. The SM5.42R/HF and SM5.42R/B3LYP models yield similar accuracies across many of the liquid classes, although there are significant differences for aromatic hydrocarbons, carbonyl compounds, and compounds containing nitrogen. Nevertheless, averaged over all liquid solutes, the results are very similar, with MUEs of 0.45 kcal/mol and 0.37 kcal/mol, respectively. For all 75 solutes, the MUE of the free energy of solvation calculated by SM5.42R/AM1 is slightly higher, 0.47 kcal/mol. Next, we evaluate the usefulness of eq. 共11兲 for calculating aqueous solubilities. This requires vapor pressures and free energies of solvation, and our first test, given in Table II, is based on using experimental values for both of these quantities. Table II shows that the resulting mean-unsigned error of the logarithm 共MUEL兲 of the solubility ranges from 0.04 to 0.3 for the various solute classes in our liquid-solute test set, with an overall MUEL of the solubility of 0.15. Table II also gives the MUEL of the solubility calculated by using experimental vapor pressures and SM5.42R free energies of solvation. Of the three continuum solvation models, SM5.42R/B3LYP performs the best, particularly for car-

bonyl compounds, esters, compounds composed of C, H, and N, and nitro compounds. The overall MUEL of the solubility is 0.32 for B3LYP, 0.38 for HF, and 0.41 for AM1. To put these numbers in context, we note that the logarithm of the solubility of the solutes in the present study covers a range of 7.4 log units 共from ⫺6.6 to 0.8兲. Finally, Table II shows the MUEL of the solubility calculated by the UNIFAC method for the various solute classes of our liquid-solute test set. Because the UNIFAC parameters are not available for interactions between unsaturated halogenated hydrocarbons, such as dichloroethene and trichloroethene, and water the 11th and last rows of Table II show the MUEL of the solubility for the subset of halogenated compounds and the subset of all liquid solutes for which UNIFAC parameters are available. The MUEL of the solubility calculated by the UNIFAC method for the 70 solutes for which it has available parameters is 0.53. The errors for the UNIFAC method range from 0.13 for esters to 1.40 for nonaromatic hydrocarbons. Table II shows that the SM5.42R methods are especially superior to UNIFAC for predicting solubilities of nonaromatic hydrocarbons, alcohols, and

TABLE II. Mean-unsigned error of the logarithm 共MUEL兲 of the solubility of various solute classes calculated by various methods. SM5.42Ra Solute class

No. data

From expt. ⌬G S䊊 (aq) and P 䊉

HF

B3LYP

AM1

UNIFAC

Nonaromatic hydrocarbons Aromatic hydrocarbons Alcohols and m-cresol Ethers and anisole Aldehydes and ketones Esters C, H, N compounds Nitro compounds All H, C, N, O compounds Halogenated hydrocarbons Halogenated hydrocarbonsb All liquid solutes Subset of liquid solutesb

11 6 9 4 11 7 7 5 60 15 10 75 70

0.10 0.04 0.25 0.21 0.14 0.10 0.30 0.04 0.15 0.14 0.12 0.15 0.15

0.25 0.14 0.38 0.32 0.44 0.54 0.73 0.37 0.40 0.30 0.30 0.38 0.38

0.25 0.23 0.30 0.34 0.31 0.38 0.55 0.25 0.32 0.33 0.37 0.32 0.33

0.24 0.15 0.51 0.36 0.48 0.65 0.60 0.48 0.43 0.34 0.34 0.41 0.42

1.40 0.17 0.58 0.54 0.31 0.13 0.54 0.20 0.55

Solubilities calculated using SM5.42R calculations 共at mPW1PW91 geometries兲 of vapor pressures. b Subset of this solute class for which UNIFAC parameters are available. a

⌬G S䊊 (aq)

0.39 0.53

and experimental


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TABLE III. Mean-unsigned error of the logarithm 共MUEL兲 of the vapor pressure of various solute classes calculated by various SM5.42R models and using mPW1PW91 geometries. SM5.42R Solute class

No. data

HF

B3LYP

AM1

Nonaromatic hydrocarbons Aromatic hydrocarbons Alcohols and m-cresol Ethers and anisole Aldehydes and ketones Esters C, H, N compounds Nitro compounds All H, C, N, O compounds Halogenated hydrocarbons All liquid solutes

11 6 9 4 11 7 7 5 60 15 75

0.32 0.19 0.28 0.15 0.66 0.34 0.18 0.08 0.32 0.40 0.34

0.30 0.27 0.24 0.15 0.49 0.25 0.19 0.23 0.29 0.40 0.31

0.23 0.16 0.32 0.21 0.84 0.57 0.40 0.33 0.41 0.48 0.43

ethers, while UNIFAC is more accurate than the continuum solvation model for predicting solubilities of esters and compounds composed of C, H, and N. We have shown 共Table I兲 that the SM5.42R model predicts aqueous free energies of solvation reasonably well, so now we investigate the ability of SM5.42R to predict puresubstance vapor pressures for liquid solutes in our test set.60 Table III gives the MUEL of the logarithm of the puresubstance vapor pressure calculated via Eq. 共2兲 with Eq. 共12兲 for ⌬G 䊊 1 . The SM5.42R/HF and SM54.2R/B3LYP models perform similarly, with MUELs of 0.34 and 0.31, respectively, while SM5.42R/AM1 has a MUEL 0.1 logarithm unit higher. Comparison of Tables I and II leads to the conclusion that the three continuum solvation models are slightly more accurate at computing free energies of self-solvation than they are at computing aqueous free energies of solvation. We next consider the prediction of aqueous solubilities of liquid solutes using Eq. 共11兲 by using SM54.2R to calculate not only the free energies of solvation but also the vapor pressure. Table IV gives the MUEL of the solubility calculated this way by SM5.42R/HF, SM5.42R/B3LYP, and

SM5.42R/AM1 for each solute class of our liquid-solute test set. Comparing the SM5.42R results of Table IV to those of Table II 共where the only difference between the two tables is the method of obtaining the vapor pressure兲, we see that using theoretical vapor pressures yields solubilities with approximately the same accuracy as using experimental vapor pressures. When using theoretical vapor pressures, larger MUELs of the solubility are observed for nonaromatic hydrocarbons and ethers than when using experimental vapor pressures. However, the MUELs of the solubility calculated from using theoretical vapor pressures for alcohols and esters are smaller than the corresponding MUELs of the solubility calculated from using experimental vapor pressures 共i.e., there is some cancellation of errors in the two theoretical components兲. For all 75 solutes, the MUELs of the solubility that is computed from aqueous free energies of solvation and vapor pressures calculated by SM5.42R/HF, SM5.42R/ B3LYP, and SM5.42R/AM1 are 0.36, 0.34, and 0.34, respectively. For comparison, Table IV also repeats the MUEL of the solubility calculated by the UNIFAC method. For the 60 H, C, N, and O compounds, the SM5.42R MUELs are 0.17– 0.20 logarithm units lower than the UNIFAC MUEL. In addition, the SM5.42R methods yield smaller MUELs of the solubility than the UNIFAC method for the subset of halogenated hydrocarbons available for calculation by UNIFAC. B. Solid solutes

In this subsection, we consider the prediction of aqueous free energies of solvation and solubilities of solid solutes. Table V gives the MUE of the aqueous free energy of solvation for all solid solutes in our test set calculated by Eq. 共6兲 using experimental vapor pressures and experimental solubilities. Table V also shows the MUE of the aqueous free energy of solvation for the solid solutes in our test set calculated by SM5.42R. When using experimental pure-substance vapor pressures and aqueous solubilities, the MUE of the free energy of solvation for all solid solutes is 0.57 kcal/mol,

TABLE IV. Mean-unsigned error of the logarithm 共MUEL兲 of the solubility of various solute classes calculated by various methods. SM5.42Ra Solute class

No. data

HF

B3LYP

AM1

UNIFAC

Nonaromatic hydrocarbons Aromatic hydrocarbons Alcohols and m-cresol Ethers and anisole Aldehydes and ketones Esters C, H, N compounds Nitro compounds All H, C, N, O compounds Halogenated hydrocarbons Halogenated hydrocarbonsb All liquid solutes Subset of liquid solutesb

11 6 9 4 11 7 7 5 60 15 10 75 70

0.46 0.11 0.26 0.47 0.38 0.26 0.69 0.38 0.38 0.29 0.25 0.36 0.36

0.41 0.08 0.25 0.48 0.40 0.31 0.49 0.35 0.35 0.31 0.26 0.34 0.34

0.37 0.08 0.31 0.48 0.52 0.19 0.57 0.23 0.36 0.29 0.21 0.34 0.33

1.40 0.17 0.58 0.54 0.31 0.13 0.54 0.20 0.55

Solubilities calculated using SM5.42R calculations 共at mPW1PW91 geometries兲 of pressures. b Subset of this solute class for which UNIFAC parameters are available. a

⌬G S䊊 (aq)

0.39 0.53 and of vapor




TABLE V. Mean-unsigned error 共MUE兲 of the aqueous free energy of solvation, ⌬G S䊊 (aq) 共in kcal/mol兲 of all 15 solid solutes calculated by various methods.

three SM5.42R methods all perform considerably better than the UNIFAC method for the 13 solids for which UNIFAC makes predictions. We also used the SM5.42R methods to predict vapor pressures of the 7 solid solutes in our test set for which we have solvent descriptors and a dielectric constant 共naphthalene, anthracene, biphenyl, phenanthrene, pyrene, phenol, and p-cresol兲. The top half of Table VII gives the MUEL of the vapor pressure calculated by SM5.42R/HF, SM5.42R/ B3LYP, and SM5.42R/AM1 for these 7 solutes using only ⌬E E and G P in Eq. 共12兲, i.e., n⫽ ␣ ⫽ ␤ ⫽ ␥ ⫽ ␾ ⫽ ␺ ⫽0, then using a macroscopic surface tension of zero for each solute, i.e., ␥⫽0, and finally including all terms and using an estimated macroscopic surface tension for the 7 solid solutes 共see Sec. V for details of the estimates兲. The MUEL of the vapor pressure decreases by approximately a factor of 2.3–3 when the n, ␣, ␤, ␾, and ␺ terms are included and by another factor of about 2.8 共for all three SM5.42R models兲 when the macroscopic surface tension term of Eq. 共15兲 is included. These results indicate that the atomic surface tension derived CDS terms are meaningful even for self-solvation of solids, i.e., for solid solvents. Next, we predict aqueous solubilities of the solid solutes using Eq. 共11兲 with SM5.42R aqueous free energies of solvation and SM5.42R vapor pressures. The bottom half of Table VII gives the MUEL of the solubility calculated by this way with n⫽ ␣ ⫽ ␤ ⫽ ␥ ⫽ ␾ ⫽ ␺ ⫽0, then with only ␥⫽0, and finally with an estimated value for ␥ 共see Sec. V for details of the estimates兲. In a similar finding to what we found for the MUEL of the vapor pressure, there is a substantial decrease in the MUEL of the solubility when including the n, ␣, ␤, ␾, and ␺ terms. A further decrease in the MUEL of the solubility for all SM5.42R models is observed when the estimated value of ␥ is included. Thus it appears that the nonbulk terms used in the SM5.42R model, which were parametrized only for liquid organic solvents and water, seem to be applicable to systems in which the solvent is a solid. Comparing the results of the bottom half of Table VII to the results of Table IV, we see that the MUEL of the solubility for the solid data is larger than the MUEL of the solubility for the liquid data by about 0.3 logarithm units; this is a measure of the improvement needed to bring solid—solute modeling up to the level of liquid—solute modeling. Note that the use of liquid solvent descriptors for computing the vapor pressure of the solid is equivalent in some sense to assuming a free energy of fusion of zero. That approximation appears to work acceptably for the 7 solids studied here, but we do not assert it

SM5.42Ra From expt. S and P 䊉

HF

B3LYP

AM1

0.57

0.41

0.42

0.59

1667

a

Aqueous free energy of solvation calculated by SM5.42R at mPW1PW91 geometries.

which is 0.37 kcal/mol larger than the corresponding MUE for the 75 liquid solutes. However, we note that the free energy of solvation of benzamide is predicted to be ⫺7.82 kcal/mol, while its experimental aqueous free energy of solvation is ⫺11.00 kcal/mol, and omitting this outlier lowers the solid MUE to 0.38 kcal/mol, which is only 0.18 kcal/mol larger than the liquid value. For SM5.42R/HF and SM5.42R/ B3LYP, the MUE of the aqueous free energy of solvation of the solid solutes is comparable to the corresponding MUE for the 75 liquid solutes in our test set. For SM5.42R/AM1, the MUE of the aqueous free energy of solvation for the solid solutes is 0.59 kcal/mol, while the corresponding MUE for the 75 liquid solutes is 0.47 kcal/mol. Table VI shows the MUELs of the solubility calculated by using experimental vapor pressures and free energies of solvation, and compares them to values calculated from experimental vapor pressures and SM5.42R free energies of solvation and to UNIFAC predictions. Note that UNIFAC parameters are not available for two of our solid solutes 共benzamide and acetamide兲, so we also report the MUEL of the solubility for the 13 data excluding these two solutes. The MUEL of the solubility for the 15 solid solutes calculated from experimental pure-substance vapor pressures and aqueous free energies of solvation are not as accurate as the corresponding MUEL of the solubility of the 75 liquid solutes 共0.42 for the solid-solute data versus 0.15 for the liquid—solute data兲. Removing the two amides from the data set, however, decreases the MUEL of the solubility to 0.26. The MUELs of the solubility calculated from SM5.42R/HF and SM5.42R/B3LYP are comparable to the MUEL of the solubility calculated from experimental vapor pressures and aqueous free energies of solvation for all 15 data and for the 13 data excluding the two amides. In addition, these two models predict the solubility of the solids with about the same accuracy as they do for liquid solutes 共see Table II兲. The SM5.42R/AM1 model is not as accurate as the other SM5.42R models at predicting solubilities, however. The

TABLE VI. Mean-unsigned error of the logarithm 共MUEL兲 of the aqueous solubility of solid solutes calculated by various methods. SM5.42Ra Solute class All solid solutes Subset of solid solutesb

No. data

From expt. ⌬G S䊊 (aq) and P 䊉

HF

B3LYP

AM1

UNIFAC

15 13

0.42 0.26

0.48 0.32

0.47 0.39

0.65 0.54

0.77

Solubilities calculated using SM5.42R calculations 共at mPW1PW91 geometries兲 of vapor pressures. b Subset of this solute class for which UNIFAC parameters are available. a

⌬G S䊊 (aq)

and experimental


1668



TABLE VII. Mean-unsigned error of the logarithm 共MUEL兲 of the vapor pressure and of the solubility of solid solutes calculated by various methods. SM5.42Ra Solute class

No. data

HF

B3LYP

AM1

solutes 共electrostatics only兲 solutes 共no ␥ term兲b solutes

7 7 7

6.03 2.04 0.71

6.10 2.13 0.75

5.32 2.36 0.84

solutes 共electrostatics only兲 solutes 共no ␥ term兲b solutes

7 7 7

5.83 1.89 0.69

5.76 1.85 0.66

6.45 1.88 0.64

UNIFAC

䊉

log P Solid Solid Solid log S Solid Solid Solid

0.53

⌬G S䊊 (aq)

Solubilities calculated using SM5.42R calculations 共at mPW1PW91 geometries兲 of and of vapor pressures. b The macroscopic surface tension is set to zero in the calculation of G CDS 关see Eqs. 共13兲 and 共15兲兴. a

to be necessarily general. Further efforts are underway to better clarify the issue. For comparison to the present approach, Table VII shows the MUEL of the solubility for the 7 solid solutes calculated by the UNIFAC method. The MUEL of the solubility calculated by the SM5.42R models for the solid data are only 0.11–0.16 logarithm units larger than the MUEL of the solubility calculated by the UNIFAC method. C. Sensitivity to geometry

We also investigated the sensitivity of our predicted results to the geometry used for the solutes in the test set. The mPW1PW91 geometries used for the work discussed above should be reasonably accurate, and they represent the results that can be expected when any reasonably accurate geometry is used. However, sometimes a reasonably accurate geometry is not available. If it is too time consuming to compute accurate geometries, one may resort to low-level geometries, such as those calculated by the AM1 method, and therefore it is important to check the sensitivity of the results to the quality of the geometries. Tables S-II–S-VIII, which are located in the supplementary material,76 present data analogous to that in Tables I–VII, but obtained using AM1 geometries instead of mPW1PW91 geometries. The trends observed using AM1 geometries are the same as the trends observed using mPW1PW91, except for carbonyl compounds and esters. For these solute classes, the MUE of ⌬G S䊊 (aq) and the MUEL of the solubility using AM1 geometries are approximately a factor of 2 larger than the corresponding errors from using mPW1PW91 geometries. Furthermore, when experimental vapor pressures are used, the MUEL of the solubilities calculated using AM1 geometries is about 0.1 logarithm unit higher than when using mPW1PW91 geometries. VII. CONCLUSIONS

We have used thermodynamic equations to relate the aqueous free energy of solvation of a solute to its aqueous solubility via the pure-substance vapor pressure. With a data set of 75 liquids and 15 solids with experimentally known aqueous solubilities, aqueous free energies of solvation, and pure vapor pressures, we have shown that these equations

can be used to predict aqueous free energies of solvation and aqueous solubilities to an accuracy comparable to that of previously available models. Using the SM5.42R continuum solvation model, we also show that it is possible to treat solid solvents in the same way that liquid solvents are treated, i.e., as a continuous medium characterized by a bulk dielectric constant with additional terms to account for molecular surface effects beyond bulk electrostatics. Our test set for the present work is relatively small, and we make no claim of a consistent test against existing solubility models 共that is beyond our scope兲, but it is encouraging that our model already gives an MUEL of the solubility 共0.33–0.36 for our relatively small test set兲 that is smaller than many existing methods 共0.38 –0.88; see Introduction and Table IV兲. This result is even more encouraging when we recall that our calculations involve no new parameters determined expressly for predicting solubilities, and in fact the relevant parameters in the model tested here were determined in previous work entirely on the basis of free energies of solvation 共air–water partition coefficients兲 with no use at all of any solubility data or other experimental data such as melting points or vapor pressures 共although our solvent descriptors are derived from experiment兲. Our general conclusion is that models designed for the prediction of free energies of infinitely dilute solvation are applicable to solubility calculations, and this conclusion is relevant not to just the SM5.42R model tested here, but also to other implicit and explicit solvation models. Therefore future solubility models can incorporate successful models developed for calculating free energies, and this should lead to a useful consolidation of efforts and a more unified approach to condensed-phase equilibria. VIII. SUPPLEMENTARY MATERIAL

A full description of the solutes in our test set, including l s and M A ), solvent descriptheir density 共used to obtain M A tors, and dielectric constant are reported in Table S-I in the supplementary material.76 Data analogous to that presented in Tables I–VII, but using AM1 geometries instead of mPW1PW91 geometries are given in Tables S-II–S-VIII in the supplementary material. Finally, for each solute in the test set, both experimental values and various theoretical or



predicted values of ⌬G S䊊 (aq), log S, and log P䊉 are given in Tables S-IX–S-XVI. The supplementary material is available from Electronic Physics Auxiliary Publication Services 共E-PAPS兲. ACKNOWLEDGMENT

This work was supported by a Department of Defense 共DOD兲 Multidisciplinary University Research Initiative 共MURI兲 grant managed by the Army Research Office. 1

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