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Solvation Entropy Made Simple Alejandro J. Garza J. Chem. Theory Comput., Just Accepted Manuscript • Publication Date (Web): 26 Mar 2019 Downloaded from http://pubs.acs.org on March 27, 2019

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Journal of Chemical Theory and Computation

Solvation Entropy Made Simple Alejandro J. Garza∗ The Dow Chemical Company, 1776 Building Midland, Michigan 48674, United States E-mail: [email protected]

Abstract The entropies of molecules in solution are often calculated using gas phase formulae. It is assumed that, because implicit solvation models are fitted to reproduce free energies, this is sufficient for modeling reactions in solution. However, this procedure exaggerates entropic effects in processes that change molecularity. Here, computationally efficient (i.e., having similar cost as gas phase entropy calculations) approximations for determining solvation entropy are proposed to address this issue. The Sω , S , and Sα models are nonempirical and rely only on physical arguments and elementary properties of the medium (e.g., density and relative permittivity). For all three methods, average errors as compared to experiment are within chemical accuracy for 110 solvation entropies, 11 activation entropies in solution, and 32 vaporization enthalpies. The models also make predictions regarding microscopic and bulk properties of liquids which prove to be accurate. These results imply that ∆Hsol and ∆Ssol can be described separately and with less reliance on parametrization by a combination of the methods presented here with existing, reparametrized implicit solvation models.

Introduction

factory as it inevitably underestimates ∆Sreac effects (a recent publication 10 discusses this and related approaches to estimate entropies in solution from gas phase formulas). Another workaround is to scale gas phase entropies by a rule-of-thumb factor of ≈ 0.65. 1,11–14 While scaling gas phase entropies may be reasonable for small molecules, it is not justified for larger molecules for which the vibrational and cavity, rather than translational and rotational, entropy terms dominate (vibrational entropy does not change drastically upon solvation 15 ). Implicit solvation models are often fitted to reproduce experimental free energies under standard conditions, 2,16–20 so one could expect them to improve ∆Greac . However, unless a model designed to account for temperature effects is used and appropriate derivatives are taken (see, e.g., refs. 21,22), the fact that the T ∆Sreac term in ∆Greac relies on gas phase entropies will result in an incorrect temperature dependence of the reaction. Furthermore, popular implicit solvation models often do not improve the too-high

For practical reasons, it is common in computational chemistry to calculate entropic contributions to reaction free energies from gas phase formulas. 1–14 Gas phase entropies are well described by analytical formulas of statistical mechanics, but there is no similarly efficient way of estimating entropies in solution rigorously. However, the use of gas phase entropies to model processes that change molecularity (e.g., adsorption and binding) in solution often exaggerates entropy changes during the reaction, ∆Sreac . 1–8 In condensed media, the translational and rotational motions of a molecule are hindered, reducing the entropy as compared to the gas phase. Rearrangement of the solvent to form a cavity for the solute further lowers the entropy of the system. For bimolecular reactions in solution, typical errors in ∆Sreac are so large that some authors have suggested discarding translational and/or rotational entropy. 5,6,9 This approach is, however, unsatis-

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binding free energies caused by the use of gas phase entropies. 1–9 Molecular dynamics and alchemical free energy methods can in principle determine free energies in solution without relying on gas phase formulas. 23–25 Nonetheless, such calculations require significant additional work from the user and are computationally demanding, which makes them unsuitable for routine or high-throughput calculations. They also suffer from inherent reproducibility issues due to the sensitivity of Newtonian dynamics to initial conditions and the lingering effects of such conditions if sampling is insufficient. 24,25 The purpose of this work is to formulate an efficient approximation for calculating molecular entropy in solution based only on physical and geometric arguments. Three models, Sω , S , and Sα , are derived that rely only on such arguments—no empirically fitted parameters— and elementary solvent properties (e.g., mass density). The methods differ only in how the cavitation entropy is calculated: Sω , S , and Sα utilize, respectively, the Pitzer acentric factor (ω), the relative permittivity (r ), and r as well as the isobaric thermal expansion coefficient (α). This establishes a connection between ω (a microscopic property), r , and α (macroscopic properties). The cost of evaluating entropy with these models is comparable to the cost of calculating gas phase entropies with the ideal gas/rigid rotor/harmonic approximations. Their accuracy is tested by benchmarking against a database of 110 experimental solvation entropies and 11 activation entropies in solution; average errors are in the range of 2–3 cal/mol-K, which is comparable to the accuracy of gas phase entropies and within chemical accuracy (≤ 1 kcal/mol at 300 K). Additionally, the models make testable predictions regarding molecular and bulk properties of liquids that prove to be in agreement with experiment.

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and the solvent cavity: S = Sv + St + Sr + Sc .

(1)

The vibrational entropy can be computed from the harmonic oscillator approximation. 15,33,34 It is well known, however, that this approximation yields unphysically large contributions to the entropy from low-frequency modes. To avoid this issue, Sv is calculated with the method proposed by Grimme, 26 which can be seen as a quasi hindered rotor that interpolates between the harmonic oscillator and free rotor entropy formulas. We assume that Svgas ≈ Svsol . This approximation is adequate for the systems studied here. 15 Translational Entropy Spectroscopic evidence for translational motion in liquids has been observed for mixtures of rare gases and H2 in Ar, where the rotational and translational signals from H2 can be clearly separated. 27,28 The idea of an effective volume in which a molecule in a liquid can move has also been used to provide a statistical-mechanical basis for Trouton’s rule. 29–32 We build upon this idea of the solute moving in an effective volume to approximate St in solution. The contributions from St in terms of the translational partition function are 33,34 ∂ ln (qt ) , (2) St = k ln (qt ) + k + kT ∂T V where qt is approximated by the familiar expression obtained from the eigenenergies of a particle of mass m confined in a volume V : qt =

2πmkT h2

3/2 V.

(3)

All quantities in eq. 3 are unambiguously defined except for V . For an ideal gas V = kT /P , but in condensed media V will depend on properties of the medium such as, e.g., its density and particle volume. Here, we define V in terms of the volume of the solute cavity, vc , as well as

Theory In our model, the total entropy of an atom or molecule in solution is decomposed into contributions from vibrations, translations, rotations,


2

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n times is xn , Nc may be approximated as

the average number of accessible cavities Nc , V = Nc vc ,

(4)

Nc = 1 + Nx

so that we can evaluate V based on a physical interpretation of Nc and vc . In our model, vc is the volume of a sphere with a radius equal to the sum of the spherical equivalent radii of the solute and the volume of free space per solvent particle. This definition is equivalent to 3 1/3 1/3 vc = VM + Vfree (5) with Vfree =

MwS − VS , NA ρ

2/3

x=

2/3

= 1 + Nx

(6)

.

1 −1 . 1−x

(9)

Typically, Nc ≈ 1, but the hopping terms can be important in cases of small solutes in bulky or low density solvents. Note that previous works have also utilized a cavity volume to determine St . 36 However, the definition of vc and consideration of the possibility of hopping distinguish the present approach from previous ones. Although we have written in eq. 1 separate terms for St , Sr , and Sc , these are actually intertwined. As we see next, the definition of St is important to determine Sr , and Sr is in turn used to derive an approximation for Sc . The way to think about St to more easily understand Sr as conceptualized here is simply as the entropy of a point particle in a box, as opposed to the entropy of an object that has rotations. Rotational Entropy We define Sr in terms of the contributions from the rigid rotor approximation and the translational entropy lost by virtue of acquiring a gyration radius while being confined to V = Nc vc . Assuming that rotation is fast, the radius rc of the cavity in which the centroid of a linear or spherically symmetric rigid rotor can move freely is effectively reduced by its radius of gyration rg ,

2/3

2/3

xk

k=1

where VS/M is the volume of a solvent/solute molecule, ρ the mass density of the medium, NA Avogadro’s number, and Mw the molecular weight (throughout this document, subscripts/superscripts “M” and “S” denote solute and solvent, respectively). Here, VM and VS are determined from van der Waals radii, 35 though vc is relatively insensitive to how molecular volumes are defined (vide infra). To determine Nc , we define the probability per solvent particle of “hopping” to an adjacent cavity, x, based on the solvent, solute, and free volumes as max{Vfree − VM , 0}

k=∞ X

(7)

Vfree + VS

That is, the solute can only hop if the cross sectional area of VM is smaller than Vfree (given that all cross sectional areas are defined identically in terms of volume, regardless of shape). Furthermore, assuming effective spherical shapes for each volume and introducing rc = [3vc /(4π)]1/3 as the radius of the cavity, there will be 2/3 rc2 4π Nx = 4 (8) 2/3 2/3 3 Vfree + VS

rg2

=

1 Natoms

NX atoms

(rk − rmean )2 .

(10)

k=1

For nonsymmetric rotors, the reduction by of rc by rg is also assumed as an averaged radius is necessary to preserve rotational invariance. Hence, the rotational entropy is ∂ ln (qr ) Sr =k ln (qr ) + kT + ∂T V St (T, rc − rg ) − St (T, rc ), (11)

sites for hopping per cavity. Because there will be at least one cavity available for the solute, and considering that the probability of hopping


3


where St (T, r) is the translational entropy at temperature T and volume V = Nc 4πr3 /3, and qr the rigid rotor rotational partition function. For a nonlinear molecule 33,34 3/2 π 1/2 8π 2 IkT qr = , (12) σr h2

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Cavity Entropy: Acentric Factor Approximation Dionis´ıo et al. 38 provide an interpretation of Sc based on a reference ideal liquid (i.e., one that is spherical and nonpolar). Because there are no correlations between the solute and the surrounding molecules, Sc is set to zero for the reference ideal liquid. The cavity term is then identified with the difference in vaporization entropy of the ideal and real versions of a pure liquid. For substances in a standard state that obey a three-parameter corresponding states principle 39 (i.e., those well described by introducing an acentric factor, in addition to reduced temperature and pressure, to correct for nonideal behavior), this difference can be conveniently evaluated in terms of the Pitzer acentric factor, ω, of the solvent as 38

where I = (Ix Iy Iz )1/3 is the average moment of inertia and σr the rotational symmetry number. A possible issue with eq. 11 is that, in the case of an extremely large and nonspherical solute in a dense solvent, we could have rc < rg . Such a situation does not occur in any of the systems studied here, but this issue is resolved by a physical interpretation of the model: when 1/3 rc − rg < (3/4π)2/3 Vfree , one should replace 1/3 rc − rg in eq. 11 with (3/4π)2/3 Vfree . The reason for this being that the solute in this situation will be surrounded in each direction by molecules with an associated free volume Vfree . Furthermore, if rc < rg rotation inside the cavity will not be free and qr should be revised accordingly. In the Appendix, we provide an approximate expression for qr for use in this situation. The description of rotational entropy in this manner is thus often close to that of the free rotor. This is in agreement with the observation of discrete rotational transitions in the Raman spectrum of small molecules in liquids which do not differ significantly from the gas-phase transitions. 27,28 Previous studies have also noted that rotational contributions to the entropy of a solute are often close the their gas-phase values. 32,37 One may ask why have we not chosen to define St → St (T, rc − rg ) and Sr → Srgas given that such a choice would allow to write St and Sr in terms of qt and qr in a more straightforward manner. The reason for our choice here is that, as shown next, the definition of Sr in eq. 11 allows for a convenient way to evaluate Sc from the Pitzer acentric factor of the solvent.

ideal ∆Hvap ω Tc = −5.365ωk.

ideal real ∆Svap − ∆Svap =−

(13)

The 5.365 factor arises due to the fact that an ideal liquid obeying the standard corresponding states theorem will have a constant ∆Hvap /Tc = 5.365k independent of temperature. 40,41 Acentric factors for common solvents are readily available; they relate molecular shape and polarity to non-ideal behavior. The more polar and nonspherical a molecule is, the larger its acentric factor is expected to be. There are two issues with equating Sc to eq. 13. The first one is that, because an ideal liquid experiences changes only in translational motions during a phase transition, eq. 13 contains losses to rotational entropy upon condensation. One must therefore disentangle from eq. 13 the loss in rotational entropy from the entropy changes due to rearrangement of solvent molecules around the solute. We write thus a first draft for Sc that subtracts the loss of rotational entropy from eq. 13: gas→sol ω Sc,draft = −5.365ωk − ∆Sr,S

(14)

= −5.365ωk − St (T, rcS − rgS ) + St (T, rcS ). Eq. 14 applies to a pure substance; it accounts ACS Paragon Plus Environment

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ric objects (e.g., ellipsoids and cuboids). Like volumes, surface areas are here calculated based on van der Waals radii. 35 The second issue that needs to be addressed to calculate Sc from the relation in eq. 13 is that the entropy of cavity formation of an ideal liquid is not zero. The probability of finding an empty site that can fit and ideal solute in a medium of hard spheres is less than one, and thus we must have Sc < 0 even for the ideal liquid. The entropy contributions arising from this situation can be estimated as follows: the probability arising from statistical fluctuations of finding an unoccupied volume VM in a fluid of number density nS = NA ρ/MwS is 44 W (VM , nS ) p0 = exp − , (18) kT

for entropy lost due to correlations between the the dissolved and surrounding molecules. For a mixture, we must consider the differences in shape and size of the solute and solvent as the entropy loss will be greater the more solvent molecules coordinate around the solute. Thus, we write gas→sol ω Sc,draft = −(5.365ωk + ∆Sr,S )G(RM , RS ), (15) where G(RM , RS ) is a function of the molecular geometries of the solute RM and the solvent RS satisfying G(RX , RX ) = 1. The function G must account for the number of solvent molecules that coordinate around the solute relative to the solvent. This is a packing problem and such problems are, in general, combinatorial NP-hard and have no analytical solution (much work relevant to liquids has been done on the packing problem, see, e.g., refs. 42,43 for reviews). For the sake of having a practical method to compute Sc , we introduce an approximation here. Suppose we are packing cubes in a box-shaped cavity. An analytical solution is then straightforward: G(RM , RS ) = AM /AS , with AX being the surface area of X. However, this expression does not take into account curvature, which may lead to inaccurate estimates of the packing. For example, if we interchanged the cubes packed around the box by spheres of diameter equal to the length of a side of the cubes, we would overestimate the G by a factor of 6/π because of the larger volume to surface area ratio of the sphere. If we introduce the following shape factor

φX = AX /Abox X

where W (VM , nS ) is the reversible work required to create the cavity in the fluid. Since we are dealing with an ideal version of the solvent, we neglect surface tension terms and write W as volume work only W (VM , nS ) = VM P (VS , nS ).

(19)

To determine the pressure in the medium, P (VS , nS ), we use the exact relation for hard spheres 45 p0 (VM ≤ VS ) = 1 − VM nS ,

(20)

so that the pressure in the solvent is P (VS , nS ) = −

(16)

kT ln (1 − VS nS ) . VS

(21)

Hence, the cavity entropy in the reference ideal liquid is

where AX is the surface area of X and Abox X the surface area a box that exactly encloses the volume of X (i.e., the minimum bounding box of VX ). Then we can construct G to be exact for cubes and spheres (simply packed on a surface) as A M φS . (17) G(RM , RS ) = A S φM

Sc0 =

kVM ln (1 − VS nS ) . VS

(22)

And the final expression for Scω becomes gas→sol Scω = Sc0 − (5.365ωk + ∆Sr,S )G.

(23)

All of the terms in eq. 1 are now defined and we can see that the acentric factor approximation Sω has the following characteristics: (1)

While approximate, this choice for G also provides better estimates than the area ratio AM /AS for packing of other dissimilar geomet-


5


parameters 19 adjusted (“scaled”) to reproduce known solvent properties. Here, we estimate them based on the same kind of information used to develop the acentric factor approximation and commonly available solvent properties. The ratio R is thus calculated as 1/3 VM R= . (27) VS

It does not require significantly more computational resources than the calculation of gas phase entropies; (2) it is nonempirical in the sense that it does not employ adjustable parameters based on experimental data; and (3) it only requires knowledge of the mass density of the solvent and its acentric factor, both of which are readily available for common solvents. It is worth noting that ω can be estimated from the boiling point (Tb ), critical temperature (Tc ), and critical pressure (Pc ) of a substance as 38 Pc Tb ln − 1. (24) ω= 5.365(Tc − Tb ) 1 atm

Since R is a ratio quantity, we can expect eq. 27 to be reasonably accurate as long as the volumes VM and VM are computed in a consistent manner (here using van der Waals volumes). To define y in a way that is suitable for scaled particle theory, we employ the polarizability-based definition of molecular volume, Vp = αp /(4π0 ), and the Clausius–Mossotti equation, nS αp /3 = 0 (r − 1)/(r + 2), which yields 3 r − 1 . (28) y= 4π r + 2

Quantitative structure-activity relationships can also be used to estimate ω for certain classes of compounds. 46 Cavity Entropy: Scaled Particle Theory An alternative way to describe the thermodynamics of cavity formation is provided by scaled particle theory; a statistical mechanical model based on hard spheres of radii defined such that macroscopic properties are reproduced (for reviews on the subject, see refs. 19,44). The free energy of cavity formation in scaled particle theory is determined from the probability of inserting a cavity center in a liquid composed of spheres of certain volume and number density. The resulting free energy expression that is often used in polarizable continuum models is 19,44 3 R+ Gc =kT − ln(1 − y) + 1−y " # # 2 3y 9 y + R2 , (25) 1−y 2 1−y

In other words, we chose the scaled radius of the solvent RS to be consistent with its relative permittivity. The cavity entropy can now be obtained via Maxwell’s relations, ∂Gc α , (29) Sc = − ∂T P evaluating the partial derivative by application of the chain rule, remembering that nS depends on the temperature. If we let f = Gc /(kT ), then ∂Gc Gc ∂f ∂y = + kT (30) ∂T P T ∂y ∂T P with

where R = RM /RS is the ratio of the scaled radii of the solute and solvent and y is reduced number density of the solvent: y = (4π/3)RS3 nS = (4π/3)RS3 NA

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∂y ∂T

P

∂y = ∂nS

∂nS ∂T

= −αy

(31)

P

where α = (1/V )(∂V /∂T )P is the isobaric volumetric thermal expansion coefficient of the solvent. Frequently, |Gc /T | |αkT y(∂f /∂y)p | so we explore the possibility of computing Sc from

(26) ρS . MwS

Eq. 25 is highly sensitive to the value of y, 47 and hence the effective radii are normally treated as ACS Paragon Plus Environment

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from the Minnesota Solvation Database. 51 A few other sources were also used. 38,52,53 Geometries and vibrational frequencies were computed with the GFN-xTB method. 54 The calculated entropies assume that the gas-phase geometries and vibrational entropies do not change upon solvation (an assumption that has been used in solvation free energy models 18 and is supported by simulations 15 ). For molecules such as noctane and n-hexanol for which configurational entropy becomes important, a configurational entropy term of 1.8 cal/mol-K per non-terminal carbon 55 is included in the total entropy. It is also assumed that this term is identical for the gas-phase and dissolved species. Table 1 shows representative ∆Ssol data from the SI as well as error (calculated − experimental) statistics for the full database. The mean absolute errors (MAEs) for ∆Sω◦ (2.4 ◦ cal/mol-K), ∆S◦ (2.3 cal/mol-K), and ∆Sα (2.2 cal/mol-K) are comparable to the MAE for the calculated gas phase entropy (1.8 cal/molK). At 300 K, these errors (≈ 0.7 kcal/mol) are within what is normally considered chemical ac◦ curacy (≤ 1 kcal/mol). The errors for Ssol = ◦ Sgas + ∆S are also within chemical accuracy for all three approximations Sω◦ (3.0 cal/mol◦ (2.5 cal/molK), S◦ (2.8 cal/mol-K), and Sα K) at 300 K. Figure 1 shows the experimental vs calculated entropies for the complete dataset in the SI. It is seen that the large majority of the calculated total and solvation entropies fall in the range of chemical accuracy with respect to experiment. There is excellent correlation between the predicted and experimental data; ◦ the lower R2 and slope for calculated ∆Ssol as ◦ compared to Ssol is simply a consequence of the fact that the relative uncertainly in the experimental values of the former is larger: a typical solvation entropy is −20 cal/mol-K, but chemical accuracy is only about 3–4 cal/mol-K. For clarity, Sα entropies are omitted in this figure as they are very similar to S entropies (slope = 0.981, intercept = 0.1 cal/mol-K, R2 = 0.975). When dissolving a noble gas in itself, ∆S ≈ ∆St − kT ln(csol /cgas ). The good accuracy with which the models predict ∆S for the Ne–Xe series (error ≈ 1 cal/mol-K) thus indicates that the way in which St is calculated is well-

nS and r only as Sc = Gc /T.

(32)

The rest of the contributions to the entropy can be computed as described before. Thus, the scaled particle theory (SPT) approximations Sα and S , differ from Sω only in the definition of Sc . Like Sω , Sα and S do not employ empirical parameters but require knowledge of elementary properties of the medium: dielectric constant, mass density, and the thermal expansion coefficient in the case of Sα . Thus, a connection between a molecular property, ω, and two macroscopic properties, α and r , is established through the different ways proposed here to calculate Sc . Indeed, as shown later, reasonable estimates of ω can be obtained from r by solving for ω such that Sc = Scω and vice versa. Standard States and Concentration In calculating absolute and solvation entropies, we adopt the following conventions: standard states of gases are defined based on the ideal gas equation at 1 bar and 298.15 K (standard conditions); for pure liquids, it is the state of the substance at standard conditions; for mixtures the solute concentration is 1 M. Entropy changes due to changes in concentration are estimated with the usual relation ∆Sconc = k ln (ci /cf ) ,

(33)

where ci and cf are, respectively, the concentrations in the initial and final state. Thus, the entropy penalty for bringing a gas at standard conditions to a 1 M concentration is ∆Sconc = −6.4 cal/mol-K.

Benchmarks Experimental and computed gas phase and solution entropies for 110 pure substances and binary mixtures are given in the Supporting Information (SI). The reference data were compiled from the NIST Standard Reference Database Number 69, 48 published Henry Law data, 49 and cross-referencing enthalpies from the Acree Enthalpy of Solvation Dataset 50 and free energies


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Table 1: Experimental and calculated gas phase and solvation entropies (cal/mol-K) for various substances and mixtures. ◦ ◦ ◦ Solvent Sgas,exp Sgas,calc ∆Ssol,exp Helium 8.1 9.0 -4.7† Argon 37.0 37.0 -18.4‡ Benzene 64.3 64.7 -22.9 n-Pentane 83.5 80.1 -20.5 n-Octane 111.6 111.5 -25.3 Chloroform 70.6 73.6 -25.2 Acetaldehyde 59.8 60.1 -31.8 Acetone 70.6 66.6 -22.7 Acetic Acid 67.6 69.1 -29.8 Water 45.1 45.1 -28.4 Water 67.6 69.1 -25.4 Water 67.6 64.9 -31.6 Water 86.5 83.0 -35.1 Water 37.0 37.0 -23.0 Water 51.1 50.3 -26.5 Water 71.2 70.9 -31.6 Ethanol 71.6 64.9 -17.1 Ethanol 111.6 111.5 -19.7 Ethanol 71.7 67.4 -17.5 Butanol 57.3 56.6 -20.8 Butanol 111.6 111.5 -22.9 Benzene 95.9 92.2 -20.8 Benzene 71.2 70.9 -16.4 Toluene 67.6 64.9 -17.6 Toluene 111.6 111.5 -16.8 Cyclohexane 62.4 59.7 -14.5 Cyclohexane 105.0 100.9 -17.7 Cyclohexane 76.7 78.2 -18.0 n-Hexane 67.6 64.9 -12.7 n-Hexane 105.0 100.9 -16.7 n-Hexane 71.7 67.5 -14.8 n-Hexane 64.3 64.7 -17.7 Chloroform 67.6 64.9 -19.4 Chloroform 111.6 111.5 -18.9 Mean Error§ -1.1 Mean Absolute Error§ 1.8 Median Error§ 0.0 Median Absolute Error§ 1.5 Mean Absolute Deviation§ 1.8 Minimum Error§ -7.1 Maximum Error§ 4.9 † ‡ S 4.2 K. 87.3 K. Errors are for the 110 solvation entropies in

Solute Helium Argon Benzene n-Pentane n-Octane Chloroform Acetaldehye Acetone Acetic Acid Water Acetic Acid Ethanol Butanol Argon Carbon Dioxide Cyclohexane 1,4-Dioxane n-Octane Nitromethane Methanol n-Octane Pentanol Cyclohexane Ethanol n-Octane Acetonitrile Hexanol Toluene Ethanol Hexanol Nitromethane Benzene Ethanol n-Octane


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∆Sω◦ 0.1 -21.0 -23.5 -22.5 -23.0 -24.3 -26.1 -25.1 -27.9 -31.1 -30.2 -30.3 -35.5 -22.5 -24.9 -36.1 -25.1 -32.8 -23.3 -18.2 -26.5 -19.3 -18.6 -16.8 -20.5 -16.2 -19.7 -18.8 -15.5 -18.9 -15.2 -16.8 -18.3 -22.7 -0.7 2.4 -0.5 2.1 2.8 -13.1 5.7 the SI.

◦ ∆S◦ ∆Sα -3.4 -3.4 -19.8 -19.6 -23.6 -23.2 -21.8 -21.7 -22.3 -22.0 -24.9 -23.6 -27.9 -24.9 -27.1 -24.6 -26.7 -25.6 -32.4 -31.9 -29.0 -28.1 -28.9 -28.0 -32.2 -31.0 -23.9 -23.3 -26.1 -25.4 -32.4 -31.1 -23.5 -20.9 -27.0 -23.6 -22.2 -20.2 -18.7 -17.9 -24.3 -22.6 -19.5 -19.0 -18.6 -18.1 -17.2 -16.8 -20.1 -19.6 -16.4 -16.1 -19.3 -18.9 -18.6 -18.2 -15.7 -15.4 -18.6 -18.2 -15.4 -15.1 -16.7 -16.4 -19.0 -18.0 -22.4 -20.8 -0.6 0.2 2.3 2.2 -0.6 0.1 1.9 1.7 2.7 2.2 -8.5 -6.7 6.6 7.7

Page 9 of 19

Table 2: Experimental and calculated entropies of activation (cal/mol-K) at 1 M concentration for various reactions in the gas phase and aqueous solution. ‡ ‡ Reaction ∆Sgas,exp ∆Sgas,calc H + CH3OH H2 + CH2OH -15.4 -16.7 H + CH3CH2OH H2 + CH3CHOH -16.0 -16.5 -17.3 -16.1 H + (CH3)2CHOH H2 + (CH3)2COH H + CD3CD2OH HD + CD3CDOH -16.0 -16.4 HD + (CD3)2COH -17.2 -16.1 H + (CD3)2CDOH H + CH2(OH)2 -12.4 -14.9 H2 + CH(OH)2 H + (CH2OH)2 H2 + HOCH2CHOH -16.5 -17.5 CH3SH + CH3 -22.1 -18.6 CH3S + CH4 CH3S + CH3OH -23.9 -26.4 CH3SH + CH2OH CH3SH + (CH3)2COH CH3S + (CH3)2CHOH -25.5 -29.7 -0.6 Mean Error Mean Absolute Error 1.8

Calculated Sosol (cal/mol-K)

Sω 80

Sε

60 40 yω = 0.96x+0.3 (R 2 =0.97) yε = 0.98x-0.1 (R 2 =0.97)

20

Chemical Accuracy 0

20

40

60

80

100

Experimental Sosol (cal/mol-K) 0 yω = 0.83x-4.4 (R 2 =0.73)

-5

yε = 0.75x-6.0 (R 2 =0.76)

-10 -15 -20 -25 -30 -35 -35

-30

-25

-20

Experimental

ΔSosol

-15

-10

-5

‡ ∆Saq,exp -7.3 -7.8 -6.3 -5.5 -5.6 -3.1 -10.4 -13.9 -18.8 -16.0

∆Sω‡ ∆S‡ -6.0 -3.6 -4.8 -3.0 -4.5 -2.6 -4.8 -3.0 -4.5 -2.6 -3.9 -1.4 -6.0 -3.9 -7.6 -4.7 -14.6 -12.0 -17.8 -14.6 2.0 4.3 2.5 4.3

‡ ∆Sα -3.8 -3.3 -2.9 -3.3 -2.8 -1.7 -4.2 -5.0 -12.3 -15.1 4.0 4.0

grounded. Entropies in solvents that can form hydrogen bonds such as water are also accurate, even though the models make no special consideration for such interactions. This suggests that, in most cases, the information necessary to determine solvation entropy is encoded in ρ, ω, and r . However, the largest error occurs for octane in ethanol with the Sω method. This method estimates the loss of entropy due to solute-solvent correlations based on the acentric factor of the solvent. Thus, we expect larger, negative errors in cases of nonpolar solutes in solvents having strong interactions (such errors will be exacerbated as the size of the solute increases; however, errors will increase with system size for any approximation when calculating properties which are not intensive 56 ). Therefore, Sω is better suited for describing solutes that have interactions with the solvent that are similar to the solvent-solvent interactions. Note, for example, that the ∆Sω◦ error for octane is dramatically reduced in butanol or toluene (≈ −3.6 cal/molK) as compared to ethanol (−13.1 cal/mol-K). Ethanol also has the largest ω = 0.644 in the set of solvents studied, which suggests greater deviations from ideal behavior and the threeparameter corresponding states principle. 39 We also caution that one should not draw strong conclusions based on errors of ≈ 4 cal/mol-K or less. Particularly when it comes to mixtures, experimental solvation entropies are not as precise as free energies (presumably due to

100

Calculated ΔSosol (cal/mol-K)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0

(cal/mol-K)

Figure 1: Experimental vs calculated entropies for a set of 110 pure substances and mixtures. The shaded region represents the range of chemical accuracy (expt. ± 3.3 cal/mol-K). Results from Sα are very similar to S and are omitted for clarity.


9


the use of extrapolation to determine the former). To give an example, for butanol in aqueous solution, the standard deviation in ∆Ssol determined from various sources of Henry Law data 49 is 3.8 cal/mol-K. Despite the fact that typical errors are within the uncertainty of the experimental techniques employed to determine ∆Ssol , we warn that the Sω and SPT approximations are based on a physical picture that does not consider complexation. Hence, the approximations may fail for ions or molecules that form coordination complexes with the solvent. Table 2 provides additional benchmarks in the form of entropies of activation (∆S ‡ ) for ten bimolecular reactions in gas phase and aqueous solution. These reactions have been studied before in the context of solvation entropy and the experimental data are reasonably accurate. 2 The initial and transition state geometries were computed in the gas phase at the ωB97X-D 57 /6-31G(d) level in Gaussian; 58 frequencies were scaled by the recommended factor for this level of theory (0.949). 59 The av‡ are similar to those for erage errors in ∆Saq ‡ ∆Sgas . The ≈ 1.5 cal/mol-K larger average errors in the SPT methods as compared to Sω arise due to error in the entropy of the hydrogen atom, which is involved in seven of the ten re◦ for the hydrogen actions. The experimental Saq atom is 10.5 cal/mol-K, whereas Sω , S , and Sα give 8.8, 7.0, and 7.3 cal/mol-K, respectively. ‡ Note that, in average, ∆Sgas,calc underestimates ‡ ∆Saq,exp by −10.1 cal/mol-K even though we have adjusted for 1 M concentration and the molecularity of the initial and final states is the same for all reactions in Table 2. This is so because the transition state still experiences a loss of (mostly translational) entropy. The ≈ −10 cal/mol-K of the gas phase formulas is rather typical: it is also seen for the Diels–Alder reaction of cyclpentadiene with methyl acrylate in toluene (Fig. 2). All three Sω , S , and Sα methods estimate ∆S ‡ within 1 cal/mol-K of its experimental value 60 (−29.7 cal/mol-K) for this archetypal Diels–Alder reaction.

Page 10 of 19

∆Sexp = -29.7 ∆Sgas = -40.6 + CO2Me

CO2Me

∆Sω = -30.1 ∆Sε = -29.3 ∆Sεα = -29.6

Figure 2: Experimental and calculated activation entropies (cal/mol-K at 298 K and 1M concentrations) for the Diels–Alder reaction of cyclpentadiene with methyl acrylate in toluene.

Additional Remarks and Testable Predictions The entropy models proposed here make testable predictions that go beyond solvation entropies. Some of these predictions are discussed here along with other features of the models such as their dependence on the definition of molecular volume. As is the case for other solvation models, the solvent cavity is central to our approximations. Here, vc depends on the solvent density and on the volumes VM and VS . There are multiple ways to define molecular volumes apart from the union van der Waals atomic volumes used here (e.g., isodensity surfaces and Bader volumes 61 ). However, the dependence of S on how volumes are defined is small as long as these are reasonable and used consistently. This is illustrated in Fig. 3 using noble gases as an example. A change in VS as large as 270% does not change vc by more than about 20% (Fig. 3A). Likewise, the change in translational entropy— the largest component to the entropy for small molecules—is modest: only about 3 cal/molK differences in the same range of VS variations (Fig. 3B). Volume definitions that make VS larger also make Vfree smaller, partially offsetting changes in vc . Thus, our approximations are relatively insensitive to the definition of molecular volumes. This means that one could use, e.g., Bader volumes 61 or volumes from coupled cluster calculations 62 to apply the model ab initio to species for which the van der Waals radius is not available (e.g., ions, heavy elements).


10

Page 11 of 19

Cavity Volume (Å3 )

300

A)

280

estimating ∆Hvap is provided by Trouton’s rule, 29 which states that ∆Hvap = 10.5kTb , or, more accurately, by the Trouton–Hildebrandt– Everett’s (THE) rule 63,64

Xe

260 240

Kr

THE ∆Hvap = [4 + ln(Tb /K)]kTb ,

220 200

15

which works well for simple liquids that do not from strong interactions such as H-bonds. Figure 4 compares experimental and calculated vaporization enthalpies for 32 liquids with eq. 34. Eq. 34 gives a MAE of 1.4 kcal/mol for ∆Hvap ; Sω , S , and Sα bring the error down to chemical accuracy with MAEs of 0.7, 0.9, and 0.7 kcal/mol, respectively (see SI).

20

25

30

35

40

32 30 28

Vaporization Enthalpy (kcal/mol)

26 24

B) 22

(34)

Ar

180

Entropy (cal/mol-K)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


15

20

25

30

35

40

Atomic Volume (Å3 )

Figure 3: Dependence of the (A) cavity volume and (B) translational entropy on VS for liquid noble gases at their boiling point.

●

Expt.

■

Sω

8

◆ ■

◆ Sε

6

■ ● ● ■◆ ● ●◆ ◆● ◆ ● ■ ● ■ ◆ ◆ ■ ● ● ■◆ ■ ■■ ◆ ● ◆

◆ ●●● ■ ● ■ ■◆ ◆ ●

◆

● ●

■

● ●

4

THE

■ ■◆ ◆ ●

2

■ ◆ ●

● ■ 0 ◆

■ ◆ ●

● ■ ◆

0

100

200

300

400

Temperature (K)

Note also from Fig. 3B that the model predicts a discontinuity in the derivative of S with respect to the volume. This discontinuity corresponds to the point at which VS = Vfree and arises due to the hopping term Nc , which gives the solute a nonzero probability of escaping its cavity if VS < Vfree . Since ρ determines Vfree and depends on the temperature, (∂G/∂T )P will be exhibit a discontinuity as the liquid expands and reaches VS = Vfree . The discontinuity may thus be associated with a liquid-gas phase transition below Tc . Under this interpretation, VS corresponds to the discontinuity point at Tb . Thus, our model predicts the radii of the Ne, Ar, Kr, and Xe to be 1.50, 1.81, 1.88, and 2.07 ˚ A, respectively. This compares well with the van der Waals radii of these elements: 1.54, 1.88, 2.02, and 2.16 ˚ A, in the same order as 35 above. That is, we have obtained molecular volumes from the density of a substance at their boiling point. Relatedly, ∆Hvap (Tb ) can be determined from ∆Svap if Tb is known. A simple way of

Figure 4: Experimental and calculated vaporization enthalpies of 32 compounds compared to the THE rule (eq. 34). From here other predictions of the model can be derived. Consider an ideal liquid that obeys Trouton’s rule. For such a liquid, the vaporization entropy does not change with temperature and ∆Svap ≈ ∆St so that we must have ∂ ln(kT /P vc ) ∂∆Svap =k (35) ∂T ∂T 1 =k − α(1 + VS /Vfree ) = 0. T Per the above discussion, VS = Vfree at Tb and hence 1 α(Tb ) = . (36) 2Tb This relation predicts α = 18.5 × 10−3 K−1 for neon and α = 5.7 × 10−3 K−1 for argon, in agreement with their respective experimental 65,66 values of 15.4 and 4.8 × 10−3 K−1 . More


11


Page 12 of 19

Table 3: Experimental and predicted thermal expansion coefficients α (10−3 K−1 ), acentric factors ω, and dielectric constants r at standard conditions. Mean absolute errors, median absolute errors, and Pearson correlation coefficients are also given. Substance αexpt α(T, ρS ) ωexpt ω(r ) † 16.59 -0.029 -0.099 Neon 15.40 Argon† 4.80 4.72 0.001 -0.118 Ethylene† 2.40 3.13 0.089 0.089 Cyclohexane 1.21 1.47 0.212 0.188 Benzene 1.25 1.48 0.212 0.223 Toluene 1.08 1.46 0.263 0.257 m-Xylene 0.99 1.46 0.325 0.28 n-Pentane 1.58 1.69 0.251 0.185 n-Hexane 1.41 1.64 0.299 0.277 n-Octane 1.14 1.55 0.398 0.334 Chloroform 1.27 1.43 0.218 0.272 Dioxane 1.12 1.81 0.307 0.218 Ethyl Ether 1.60 1.66 0.281 0.378 Acetaldehyde 1.69 1.64 0.303 0.469 Acetone 1.43 1.60 0.304 0.495 Acetonitrile 1.36 1.60 0.278 0.498 Water 0.21 1.65 0.344 0.466 MAE 0.41 0.080 Median AE 0.24 0.066 Pearson Correlation 0.994 0.852 † At T = Tb .

expt r 1.5 1.5 1.0 2.0 2.3 2.4 2.4 1.4 1.9 2.0 4.8 2.3 4.3 21.1 20.7 37.5 78.5

r (ω) 2.5 3.4 1.0 2.4 2.1 2.5 3.2 2.2 2.2 3.1 3.4 4.0 2.3 5.1 4.3 4.5 11.3 3.3 2.8 0.933

cantly larger than |Sc |, which makes r → ∞). The relative errors in predicted ω values are large for certain substances due to the fact that for pure liquids Sc is typically in the range of 0–4 cal/mol-K. Thus, a discrepancy of only 1 cal/mol-K between Scω and Sc results in a large relative error in ω. Despite these caveats, the fact that the predicted constants are reasonable and correlate to experiment is further evidence that the theories presented here are wellgrounded. It is worth discussing here the dependence of the entropy on the solvent constants ω, r , and α in the context of continuum models. In continuum models, the free energy of a solute is typically broken down into electrostatic and non-electrostatic terms. For the entropy models presented here, most of the contributions can be identified as non-electrostatic. However, the cavity terms may introduce electrostatic effects. Evidence of this is the fact that ω is typically larger for more polar/polarizable molecules. Likewise, even though the formulas of scaled particle theory are non-electrostatic,

generally, we can write α(T, ρ) = T −1 (1 + VS /Vfree )−1 .

ω(r , α) -0.121 -0.139 0.088 0.153 0.179 0.216 0.243 0.168 0.242 0.302 0.155 0.179 0.26 0.184 0.256 0.243 0.424 0.069 0.063 0.927

(37)

Furthermore, by equating Scω to Scα , ω can be estimated analytically from r and α as 1 Sc0 − Sc (r , α) − ∆Srgas→sol , 5.365k (38) or just from r by making Scω = Sc = Sc (r , 0). Similarly, r can be estimated from ω by this same relation, but solving a nonlinear equation to obtain the former instead. Table 3 compares experimental α, ω, and r values with those predicted from eqns. 37 and 38. Reasonable estimates of α, ω, and r are obtained that correlate well with their experimental values as measured by the Pearson correlation coefficient (> 0.85). As could be expected, the calculated constants are more precise for weakly-polar solvents that do not form H-bonds. In fact, alcohols that have large ω values are excluded because Scω = Sc () does not have a numerically stable solution in these cases (|Scω | is signifiω=


12

Page 13 of 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Acknowledgements

the dependence on r and α implicitly introduces electrostatic effects. The relative importance of the electrostatic terms will thus depend on the size of the solute relative to the solvent, and on the polarity or strength of interaction of the solvent as reflected by the ω, r , and α constants. This observation is relevant for geometry optimization: for larger, flexible solutes, the cavity term may vary significantly with the geometry, favoring conformations with lower surface area. Consideration of the cavity entropy may therefore be important in determining the geometry of larger molecules, with the caveat that a complementary cavity enthalpy needs to be considered too. We have thus provided ample evidence that— without the need of empirical parameters—the Sω and SPT approximations make sound predictions regarding properties of liquids and provide solvation entropies with average errors that are within chemical accuracy. The fact that solvation entropies can be determined in a fast and accurate manner with these methods has implications on the development of implicit solvation models. If the entropy is computed with any of the Sω , S , or Sα models, then the enthalpy can be calculated with a complementary implicit solvation method such as, e.g., polarizable continuum models 19 or joint density functional theory. 67 Since the enthalpy contributions to solvation are largely electrostatic, such techniques should be able to provide an accurate ∆Hsol . Thus, we would have a model for ∆Gsol that correctly describes both ∆Hsol and ∆Ssol , and that at the same time is less reliant on parametrization than most existing solvation methods. The Sω and SPT approximations can also be used in a standalone manner to estimate ∆G values in solution in cases when ∆Hsol ≈ ∆Hgas (not an uncommon occurrence, especially in nonpolar solvents). Therefore, the methods presented here offer a practical alternative to drastically reduce the problem of inaccurate ∆Sreac terms in solution. This can be of substantial value in catalyst and drug discovery, where processes that change molecularity are ubiquitous, accurate free energies are critical in determining activity, and a high volume of calculations is often inevitable.

I would like to thank Prof. Stefan Grimme (Universität Bonn) for providing Dow Chemical with a test license the XTB program developed by his research group. I would also like to thank Dr. Peter Margl, Dr. Steven Arturo, Dr. Ivan Konstantinov, and Dr. Marc Coons (Dow Chemical) for helpful discussions.

Supporting Information Available Solvent constants and all of the experimental and calculated entropies and enthalpies.

Appendix Here we provide an educated guess for the rotational partition function qr of extremely nonspherical solutes for which rc < rg . Let us begin with the Schrödinger equation for the quantum pendulum 68 h ¯ 2 d2 ψ − U (θ)ψ(θ) = Eψ(θ). − 2I dθ2

(39)

For U = 0 one recovers the free √ rotor limit with 0 imθ eigenfunctions ψm (θ) = e / 2π and eigenenergies E0 (m) = h ¯ 2 m2 /2I. The partition function under the usual integral approximation is Z ∞ 2 2 e−¯h m /2I dm (40) q0 = 0

=

πIkT 2¯h2

1/2 .

(41)

Suppose now that we constrain the pendulum inside a “cavity” by a potential ( 0, if |θ| ≤ θ0 /2 U (θ) = (42) ∞, otherwise. This is simply the particle in a box problem. Acceptablepeigenfunctions is this situation are U ψm (θ) = 2/θ0 cos(mπθ/θ0 ) and the energy spectrum is EU (m) = h ¯ 2 m2 π 2 /2Iθ02 . The partition function from integration is therefore qU =


13

IkT 2π¯ h2

1/2 θ0 .

(43)


The ratio of qU /q0 is qU /q0 = θ0 /π,

Weinberg, N.; Kim, C.K.; Hydration of the Carbonyl Group. Further Evidence for a Cooperative Mechanism from Experimental and Theoretical Studies of the Hydrations of Formaldehyde, Acetaldehyde, Acetone, and Cyclohexanone. Can. J. Chem. 1998, 76, 114–124.

(44)

in agreement with the interpretation of the partition function as the available phase space volume of the system. It is not possible to derive analytically a similar relation for a general rotation (the rigid rotor does not have analytical solutions for Ix 6= Iy 6= Iz ). However, the above analysis suggests that a reasonable partition function for a prolate or oblate, nonlinear solute with rc < rg may be written as qr =

qrgas

θ0 π

2

=

8πIkT h2

3/2

θ02 ,

Page 14 of 19

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(6) Liu, C. T.; Maxwell, C. I.; Edwards, D. R.; Neverov, A. A.; Mosey, N. J.; Brown, R. S. Mechanistic and Computational Study of a Palladacycle-Catalyzed Decomposition of a Series of Neutral Phosphorothioate Triesters in Methanol. J. Am. Chem. Soc. 2010, 132, 16599–16609.

which recovers Eq. 12 when θ02 = π except for the 1/σr factor. Whether or not one should divide qr by a symmetry factor would depend on θ0 and the rotational symmetry of the solute. A reasonable choice for θ based on our model would be   rg , θ0 = 2 arccos  q (46) 2 2 rg + rfree

(7) Garza, A.J., Bell, A.T. and Head-Gordon, M., Is Subsurface Oxygen Necessary for the Electrochemical Reduction of CO2 on Copper? J. Phys. Chem. Lett. 2018, 9, 601–606.

with rfree = [3Vfree /(4π)]1/3 . References on how to determine symmetry numbers in various situations are available in the literature. 69,70

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