Solvation Entropy Made Simple - Journal of Chemical Theory and

Article Cite This: J. Chem. Theory Comput. XXXX, XXX, XXX−XXX

pubs.acs.org/JCTC

Solvation Entropy Made Simple Alejandro J. Garza* The Dow Chemical Company, 1776 Building, Midland, Michigan 48674, United States

J. Chem. Theory Comput. Downloaded from pubs.acs.org by AUCKLAND UNIV OF TECHNOLOGY on 05/07/19. For personal use only.

S Supporting Information *

ABSTRACT: The entropies of molecules in solution are often calculated using gas phase formulas. 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, Sϵ, and Sϵα 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 ΔHsol and ΔSsol can be described separately and with less reliance on parametrization by a combination of the methods presented here with existing, reparametrized, implicit solvation models.

■

in ΔGreac 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 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, Sϵ, and Sϵα, 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, Sϵ, and Sϵα utilize, respectively, the Pitzer acentric factor (ω), the relative permittivity (ϵr), and ϵr as well as the isobaric thermal expansion coefficient (α). This establishes a connection between ω (a microscopic property), ϵr, 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

INTRODUCTION 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, ΔSreac.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 ΔSreac are so large that some authors have suggested discarding translational and/or rotational entropy.5,6,9 This approach is, however, unsatisfactory as it inevitably underestimates ΔSreac effects (a recent publication10 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 solvation15). Implicit solvation models are often fitted to reproduce experimental free energies under standard conditions,2,16−20 so one could expect them to improve ΔGreac. However, unless a model designed to account for temperature effects is used and appropriate derivatives are taken (see, e.g., refs 21 and 22), the fact that the TΔSreac term © XXXX American Chemical Society

Received: March 1, 2019 Published: March 26, 2019 A

DOI: 10.1021/acs.jctc.9b00214 J. Chem. Theory Comput. XXXX, XXX, XXX−XXX

Article

Journal of Chemical Theory and Computation 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.

Vfree =

THEORY In our model, the total entropy of an atom or molecule in solution is decomposed into contributions from vibrations, translations, rotations, and the solvent cavity: (1)

x=

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 sol assume that Sgas v ≈ Sv . 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 are33,34 ij ∂ln(qt) yz zz St = k ln(qt) + k + kT jjjj zz ∂ T k {V

2/3

rc2 2/3 + VS2/3 Vfree

(8)

i 1 y − 1zzz k1 − x {

∑ x k = 1 + Nxjjj

k =∞

Nc = 1 + Nx

k=1

(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 = Ncvc. 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)

(4)

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 1/3 1/3 3 vc = (VM + Vfree )

(7)

sites for hopping per cavity. Because there will be at least one cavity available for the solute, and considering that the probability of hopping n times is xn, Nc may be approximated as

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 the average number of accessible cavities Nc V = Ncvc

2/3 Vfree + VS2/3

i 4π y Nx = 4jjj zzz k 3 {

3/2

V

2/3 2/3 max{Vfree , 0} − VM

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

where qt is approximated by the familiar expression obtained from the eigenenergies of a particle of mass m confined in a volume V: i 2πmkT yz zz qt = jjj 2 k h {

(6)

where VS/M is the volume of a solvent/solute molecule, ρ is the mass density of the medium, NA is Avogadro’s number, and Mw is 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

■

S = Sv + St + Sr + Sc

M wS − VS NAρ

rg2

=

1 Natoms

Natoms

∑ k=1

(rk − rmean)2

(10)

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

(5)

with B


Article

Journal of Chemical Theory and Computation

ij ∂ln(qr) yz zz + S (T , r − r ) − S (T , r ) Sr = k ln(qr) + kT jjjj zz t c g t c ∂ T k {V

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:

(11)

where St(T, r) is the translational entropy at temperature T and volume V = Nc4πr3/3, and qr is the rigid rotor rotational partition function. For a nonlinear molecule33,34 π 1/2 ijj 8π 2IkT yzz qr = j z σr jk h2 z{

3/2 gas → sol ω Sc,draft = −5.365ωk − ΔSr,S

(12)

= −5.365ωk − St(T , rcS − rgS) + St(T , rcS)

where I = (IxIyIz) 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 rc − rg < (3/4π)2/3V1/3 free, one should replace rc − rg in eq 11 with (3/4π)2/3V1/3 free. 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 → Sgas r given that such a choice would allow us 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. Cavity Entropy: Acentric Factor Approximation. Dionisió et al.38 provided 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 threeparameter corresponding states principle39 (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 as38 1/3

ideal real ΔSvap − ΔSvap =−

ideal ΔH vap

Tc

ω = −5.365ωk

(14)

Equation 14 applies to a pure substance; it accounts for entropy lost due to correlations between 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 + ΔSr,S ).(R M, R S)

(15)

where .(R M, R S) is a function of the molecular geometries of the solute RM and the solvent RS satisfying .(RX , RX) = 1. The function . 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 and 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 boxshaped cavity. An analytical solution is then straightforward: .(R M, R S) = 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 . 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 /AXbox

(16)

Abox X

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

AM ϕS ASϕM

(17)

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

(13)

The 5.365 factor arises due to the fact that an ideal liquid obeying the standard corresponding states theorem will have a constant ΔHvap/Tc = 5.365k independent of temperature.40,41 Acentric factors for common solvents are readily available; they relate molecular shape and polarity to nonideal behavior. The more polar and nonspherical a molecule is, the larger its acentric factor is expected to be. C


Article

Journal of Chemical Theory and Computation arising from statistical fluctuations of finding an unoccupied volume VM in a fluid of number density nS = NAρ/MSw is44 ÅÄÅ W (V , n ) ÑÉÑ Å M S Ñ ÑÑ p0 = expÅÅÅ− ÅÅ ÑÑÑ kT (18) ÅÇ ÑÖ

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: ρ y = (4π /3)R S3nS = (4π /3)R S3NA SS Mw (26)

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

Equation 25 is highly sensitive to the value of y,47 and hence, the effective radii are normally treated as parameters19 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

W (VM , nS) = VMP(VS , nS)

(19)

ji V zy R = jjj M zzz j VS z k {

To determine the pressure in the medium, P(VS, nS), we use the exact relation for hard spheres45 p0 (VM ≤ VS) = 1 − VMnS

1/3

(20)

Since R is a ratio quantity, we can expect eq 27 to be reasonably accurate as long as the volumes VM and VS 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/(4πϵ0), and the Clausius−Mossotti equation, nSαp/3 = ϵ0(ϵr − 1)/(ϵr + 2), which yields

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

kT ln(1 − VSnS) VS

(21)

Hence, the cavity entropy in the reference ideal liquid is Sc0 =

kVM ln(1 − VSnS) VS

(22)

y=

and the final expression for Sωc becomes gas → sol Scω = Sc0 − (5.365ωk + ΔSr,S ).

3 ijj ϵr − 1 yzz j z 4π jjk ϵr + 2 zz{

(28)

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,

(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) 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. (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 as38 Tb i P y ω= lnjjj c zzz − 1 5.365(Tc − Tb) k 1atm {

(27)

i ∂G y Scϵα = −jjj c zzz k ∂T {P

(29)

evaluating the partial derivative by application of the chain rule, remembering that nS depends on the temperature. If we let f = Gc/(kT), then G ij ∂Gc yz ∂f i ∂y y jj zz = c + kT jjj zzz ∂y k ∂T {P T k ∂T {P

(30)

jij ∂y zyz = ∂y ijj ∂nS zyz = −αy j z j z ∂nS k ∂T {P k ∂T {P

(31)

with

(24)

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 and 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 is19,44 ÅÄÅ ÅÅ 3 Gc = kT ÅÅÅÅ−ln(1 − y) + R ÅÅ − 1 y ÅÅÇ ÄÅ ÉÑ ÉÑÑ 2Ñ ÅÅ ÅÅ 3y 9 ijj y yzz ÑÑÑ 2 ÑÑÑÑ + ÅÅÅ + jj z ÑR j 1 − y zz ÑÑÑÑ ÑÑÑÑ ÅÅ 1 − y 2 k { ÖÑ ÑÑÖ (25) ÇÅ

where α = (1/V)(∂V/∂T)P is the isobaric volumetric thermal expansion coefficient of the solvent. Frequently, |Gc/T| ≫ |αkTy(∂f/∂y)p| so we explore the possibility of computing Sc from nS and ϵr only as Scϵ = Gc /T

(32)

The rest of the contributions to the entropy can be computed as described before. Thus, the scaled particle theory (SPT) approximations Sϵα and Sϵ, differ from Sω only in the definition of Sc. Like Sω, Sϵα, and Sϵ 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 Sϵα. Thus, a connection between a molecular property, ω, and two macroscopic properties, α and ϵr, is established through the different ways proposed here to calculate Sc. Indeed, as shown later, reasonable estimates of ω can be obtained from ϵr by solving for ω such that Sϵc = Sωc and vice versa. D


Article


Table 1. Experimental and Calculated Gas Phase and Solvation Entropies (cal/(mol K)) for Various Substances and Mixtures 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 mean errorc mean absolute errorc median errorc median absolute errorc mean absolute deviationc minimum errorc maximum errorc

solvent

Sgas,exp °

Sgas,calc °

ΔSsol,exp °

ΔSω°

ΔSϵ°

ΔSϵα °

helium argon benzene n-pentane n-octane chloroform acetaldehyde acetone acetic acid water water water water water water water ethanol ethanol ethanol butanol butanol benzene benzene toluene toluene cyclohexane cyclohexane cyclohexane n-hexane n-hexane n-hexane n-hexane chloroform chloroform

8.1 37.0 64.3 83.5 111.6 70.6 59.8 70.6 67.6 45.1 67.6 67.6 86.5 37.0 51.1 71.2 71.6 111.6 71.7 57.3 111.6 95.9 71.2 67.6 111.6 62.4 105.0 76.7 67.6 105.0 71.7 64.3 67.6 111.6

9.0 37.0 64.7 80.1 111.5 73.6 60.1 66.6 69.1 45.1 69.1 64.9 83.0 37.0 50.3 70.9 64.9 111.5 67.4 56.6 111.5 92.2 70.9 64.9 111.5 59.7 100.9 78.2 64.9 100.9 67.5 64.7 64.9 111.5 −1.1 1.8 0.0 1.5 1.8 −7.1 4.9

−4.7a −18.4b −22.9 −20.5 −25.3 −25.2 −31.8 −22.7 −29.8 −28.4 −25.4 −31.6 −35.1 −23.0 −26.5 −31.6 −17.1 −19.7 −17.5 −20.8 −22.9 −20.8 −16.4 −17.6 −16.8 −14.5 −17.7 −18.0 −12.7 −16.7 −14.8 −17.7 −19.4 −18.9

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

−3.4 −19.8 −23.6 −21.8 −22.3 −24.9 −27.9 −27.1 −26.7 −32.4 −29.0 −28.9 −32.2 −23.9 −26.1 −32.4 −23.5 −27.0 −22.2 −18.7 −24.3 −19.5 −18.6 −17.2 −20.1 −16.4 −19.3 −18.6 −15.7 −18.6 −15.4 −16.7 −19.0 −22.4 −0.6 2.3 −0.6 1.9 2.7 −8.5 6.6

−3.4 −19.6 −23.2 −21.7 −22.0 −23.6 −24.9 −24.6 −25.6 −31.9 −28.1 −28.0 −31.0 −23.3 −25.4 −31.1 −20.9 −23.6 −20.2 −17.9 −22.6 −19.0 −18.1 −16.8 −19.6 −16.1 −18.9 −18.2 −15.4 −18.2 −15.1 −16.4 −18.0 −20.8 0.2 2.2 0.1 1.7 2.2 −6.7 7.7

a

4.2 K. b87.3 K. cErrors are for the 110 solvation entropies in the SI.

■

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 ΔSconc = k ln(ci /c f )

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 crossreferencing enthalpies from the Acree Enthalpy of Solvation Data set50 and free energies 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 simulations15). For molecules such as n-octane and n-hexanol for which configurational entropy becomes important, a configurational entropy term of 1.8 cal/(mol K) per non-

(33)

where ci and cf are, respectively, the concentrations in the initial and final states. Thus, the entropy penalty for bringing a gas at standard conditions to a 1 M concentration is ΔSconc = −6.4 cal/(mol K). E


Article


the Ne−Xe series (error ≈1 cal/(mol K)) thus indicates that the way in which St is calculated is well-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 ϵr. 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 intensive56). 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 ΔSω° error for octane is dramatically reduced in butanol or toluene (≈−3.6 cal/(mol K)) 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 three-parameter 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 the use of extrapolation to determine the former). To give an example, for butanol in aqueous solution, the standard deviation in ΔSsol determined from various sources of Henry Law data49 is 3.8 cal/(mol K). Despite the fact that typical errors are within the uncertainty of the experimental techniques employed to determine ΔSsol, 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 (ΔS‡) 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-D57/6-31G(d) level in Gaussian;58 frequencies were scaled by the recommended factor for this level of theory (0.949).59 The average errors in ΔS⧧aq are similar to those for ΔS⧧gas. 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 reactions. The experimental S°aq for the hydrogen atom is 10.5 cal/(mol K), whereas Sω, Sϵ, and Sϵα give 8.8, 7.0, and 7.3 cal/ ⧧ (mol K), respectively. Note that, in average, ΔS gas,calc ⧧ underestimates ΔSaq,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 cyclopentadiene with methyl acrylate in toluene (Figure 2). All three Sω, Sϵ, and Sϵα methods estimate ΔS⧧ within 1 cal/(mol K) of its experimental value60 (−29.7 cal/(mol K)) for this archetypal Diels−Alder reaction.

terminal carbon55 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 ΔSsol data from the SI as well as error (calculated−experimental) statistics for the full database. The mean absolute errors (MAEs) for ΔSω° (2.4 cal/(mol K)), ΔSϵ° (2.3 cal/(mol K)), and ΔSϵα ° (2.2 cal/(mol K)) are comparable to the MAE for the calculated gas phase entropy (1.8 cal/(mol K)). At 300 K, these errors (≈0.7 kcal/ mol) are within what is normally considered chemical accuracy (≤1 kcal/mol). The errors for Ssol ° = Sgas ° + ΔS are also within chemical accuracy for all three approximations S°ω (3.0 cal/ (mol K)), S°ϵ (2.8 cal/(mol K)), and S°ϵα (2.5 cal/(mol K)) at 300 K. Figure 1 shows the experimental vs calculated entropies

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

for the complete data set 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 ΔSsol ° as compared to S°sol 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, Sϵα entropies are omitted in this figure as they are very similar to Sϵ entropies (slope = 0.981, intercept = 0.1 cal/(mol K), R2 = 0.975). When dissolving a noble gas in itself, ΔS ≈ ΔSt − kTln(csol/ cgas). The good accuracy with which the models predict ΔS for F


Article


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

ΔS⧧gas,exp

ΔS⧧gas,calc

ΔS⧧aq,exp

ΔS⧧ω

ΔS⧧ϵ

ΔS⧧ϵα

H + CH3OH → H2 + CH2OH H + CH3CH2OH → H2 + CH3CHOH H + (CH3)2CHOH → H2 + (CH3)2COH H + CD3CD2OH → HD + CD3CDOH H + (CD3)2CDOH → HD + (CD3)2COH H + CH2(OH)2 → H2 + CH(OH)2 H + (CH2OH)2 → H2 + HOCH2CHOH CH3SH + CH3 → CH3S + CH4 CH3SH + CH2OH → CH3S + CH3OH CH3SH + (CH3)2COH → CH3S + (CH3)2CHOH mean error mean absolute error

−15.4 −16.0 −17.3 −16.0 −17.2 −12.4 −16.5 −22.1 −23.9 −25.5

−16.7 −16.5 −16.1 −16.4 −16.1 −14.9 −17.5 −18.6 −26.4 −29.7 −0.6 1.8

−7.3 −7.8 −6.3 −5.5 −5.6 −3.1 −10.4 −13.9 −18.8 −16.0

−6.0 −4.8 −4.5 −4.8 −4.5 −3.9 −6.0 −7.6 −14.6 −17.8 2.0 2.5

−3.6 −3.0 −2.6 −3.0 −2.6 −1.4 −3.9 −4.7 −12.0 −14.6 4.3 4.3

−3.8 −3.3 −2.9 −3.3 −2.8 −1.7 −4.2 −5.0 −12.3 −15.1 4.0 4.0

Figure 2. Experimental and calculated activation entropies (cal/(mol K) at 298 K and 1 M concentrations) for the Diels−Alder reaction of cyclopentadiene 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 volumes61). 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 Figure 3 using noble gases as an example. A change in VS as large as 270% does not change vc by more than about 20% (Figure 3A). Likewise, the change in translational entropythe largest component to the entropy for small moleculesis modest: only about 3 cal/(mol K) differences in the same range of VS variations (Figure 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 volumes61 or volumes from coupled cluster calculations62 to apply the model ab initio to species for which the van der Waals radius is not available (e.g., ions, heavy elements). Note also from Figure 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

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

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 Å, respectively. This compares well with the van der Waals radii of these elements: 1.54, 1.88, 2.02, and 2.16 Å, in the same order as above.35 That is, we have obtained molecular volumes from the density of a substance at their boiling point. Relatedly, ΔHvap(Tb) can be determined from ΔSvap if Tb is known. A simple way of estimating ΔHvap is provided by Trouton’s rule,29 which states that ΔHvap = 10.5kTb, or, more accurately, by the Trouton−Hildebrandt−Everett’s (THE) rule63,64 G


Article

Journal of Chemical Theory and Computation THE ΔH vap = [4 + ln(Tb/K)]kTb

(34)

α(Tb) =

1 2Tb

(36) −3

−1

This relation predicts α = 18.5 × 10 K for neon and α = 5.7 × 10−3 K−1 for argon, in agreement with their respective experimental65,66 values of 15.4 and 4.8 × 10−3 K−1. More generally, we can write α(T , ρ) = T −1(1 + VS/Vfree)−1

(37)

Furthermore, by equating Sωc to Sϵα c , ω can be estimated analytically from ϵr and α as ω=

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. Equation 34 gives an MAE of 1.4 kcal/mol for ΔHvap; Sω, Sϵ, and Sϵα bring the error down to chemical accuracy with MAEs of 0.7, 0.9, and 0.7 kcal/mol, respectively (see the SI). 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 ΔSvap ≈ ΔSt so that we must have ∂T

=k

(38)

or just from ϵr by making Sωc = Sϵc = Sc(ϵr, 0). Similarly, ϵr 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 ϵr values with those predicted from eqs 37 and 38. Reasonable estimates of α, ω, and ϵr 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 Sωc = Sc(ϵ) does not have a numerically stable solution in these cases (|Sωc | is significantly larger than |Sϵc|, which makes ϵr → ∞). 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 Sωc and Sϵc 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 well-grounded. It is worth discussing here the dependence of the entropy on the solvent constants ω, ϵr, and α in the context of continuum models. In continuum models, the free energy of a solute is

Figure 4. Experimental and calculated vaporization enthalpies of 32 compounds compared to the THE rule (eq 34).

∂ΔSvap

1 (Sc0 − Sc(ϵr , α) − ΔSrgas → sol) 5.365k

∂ln(kT /Pvc) i1 y = k jjj − α(1 + VS/Vfree)zzz = 0 ∂T kT {

(35)

Per the above discussion, VS = Vfree at Tb and hence

Table 3. Experimental and Predicted Thermal Expansion Coefficients α (10−3 K−1), Acentric Factors ω, and Dielectric Constants ϵr at Standard Conditionsa substance b

neon argonb ethyleneb cyclohexane benzene toluene m-xylene n-pentane n-hexane n-octane chloroform dioxane ethyl ether acetaldehyde acetone acetonitrile water MAE median AE Pearson correlation

αexpt

α(T, ρS)

ωexpt

ω(ϵr)

ω(ϵr, α)

ϵexpt r

ϵr(ω)

15.40 4.80 2.40 1.21 1.25 1.08 0.99 1.58 1.41 1.14 1.27 1.12 1.60 1.69 1.43 1.36 0.21

16.59 4.72 3.13 1.47 1.48 1.46 1.46 1.69 1.64 1.55 1.43 1.81 1.66 1.64 1.60 1.60 1.65 0.41 0.24 0.994

−0.029 0.001 0.089 0.212 0.212 0.263 0.325 0.251 0.299 0.398 0.218 0.307 0.281 0.303 0.304 0.278 0.344

−0.099 −0.118 0.089 0.188 0.223 0.257 0.28 0.185 0.277 0.334 0.272 0.218 0.378 0.469 0.495 0.498 0.466 0.080 0.066 0.852

−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

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

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

a

Mean absolute errors, median absolute errors, and Pearson correlation coefficients are also given. bAt T = Tb. H


Article


Suppose now that we constrain the pendulum inside a “cavity” by a potential

typically broken down into electrostatic and nonelectrostatic terms. For the entropy models presented here, most of the contributions can be identified as nonelectrostatic. 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 nonelectrostatic, the dependence on ϵr 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, ϵr, 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, Sϵ, or Sϵα models, then the enthalpy can be calculated with a complementary implicit solvation method such as, e.g., polarizable continuum models19 or joint density functional theory.67 Since the enthalpy contributions to solvation are largely electrostatic, such techniques should be able to provide an accurate ΔHsol. Thus, we would have a model for ΔGsol that correctly describes both ΔHsol and ΔSsol, 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 ΔG values in solution in cases when ΔHsol ≈ ΔHgas (not an uncommon occurrence, especially in nonpolar solvents). Therefore, the methods presented here offer a practical alternative to drastically reduce the problem of inaccurate ΔSreac 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.

l o o 0, if |θ| ≤ θ0/2 U(θ ) = m o∞ , otherwise o n

This is simply the particle in a box problem. Acceptable eigenfunctions is this situation are ψmU (θ ) =

i IkT yz z qU = jjj 2z k 2π ℏ {

1/2

∞

i πIkT y =jjj 2 zzz k 2ℏ {

2

e−ℏ m

2

/2I

dm

(43)

qU /q0 = θ0/π

(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 ≠ Iy ≠ 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 3/2 iθ y i 8πIkT y qr = qrgasjjjj 0 zzzz = jjj 2 zzz θ02 k h { kπ{ 2

(45)

θ20

which recovers eq 12 when = 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 ij rg jj θ0 = 2arccosjjj jjj r 2 + r 2 free k g

yz zz zz zz zz {

(46)

1/3

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

■

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jctc.9b00214. Solvent constants and all of the experimental and calculated entropies and enthalpies (PDF)

■

AUTHOR INFORMATION

Corresponding Author

(39)

*E-mail: [email protected].

For U = 0 one recovers the free rotor limit with eigenfunctions ψm0(θ ) = eimθ / 2π and eigenenergies E0(m) = ℏ2m2/2I. The partition function under the usual integral approximation is

∫0

θ0

The ratio of qU/q0 is

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 pendulum68

q0 =

2/θ0 cos(mπθ /θ0)

and the energy spectrum is EU(m) = ℏ2m2π2/2Iθ20. The partition function from integration is therefore

■

ℏ2 d2ψ − − U (θ )ψ (θ ) = E ψ (θ ) 2I dθ 2

(42)

ORCID

Alejandro J. Garza: 0000-0001-6995-2833 Notes

The author declares no competing financial interest.

■

(40)

ACKNOWLEDGMENTS 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

1/2

(41) I


Article


Aqueous Solvation Free Energies of Neutrals, Ions, and Solute-Water Clusters. J. Chem. Theory Comput. 2005, 1, 1133−1152. (19) Tomasi, J.; Mennucci, B.; Cammi, R. Quantum Mechanical Continuum Solvation Models. Chem. Rev. 2005, 105, 2999−3094. (20) Skyner, R. E.; McDonagh, J. L.; Groom, C. R.; Van Mourik, T.; Mitchell, J. B. O. A Review of Methods for the Calculation of Solution Free Energies and the Modelling of Systems in Solution. Phys. Chem. Chem. Phys. 2015, 17, 6174−6191. (21) Klamt, A.; Eckert, F. COSMO-RS: A Novel and Efficient Method for the A Priori Prediction of Thermophysical Data of Liquids. Fluid Phase Equilib. 2000, 172, 43−72. (22) Eckert, F.; Klamt, A. Fast Solvent Screening via Quantum Chemistry: COSMO-RS Approach. AIChE J. 2002, 48, 369−385. (23) Ratkova, E. L.; Palmer, D. S.; Fedorov, M. V. Solvation Thermodynamics of Organic Molecules by the Molecular Integral Equation Theory: Approaching Chemical Accuracy. Chem. Rev. 2015, 115, 6312−6356. (24) Bhati, A. P.; Wan, S.; Hu, Y.; Sherborne, B.; Coveney, P. V. Uncertainty Quantification in Alchemical Free Energy Methods. J. Chem. Theory Comput. 2018, 14, 2867. (25) Loeffler, H. H.; Bosisio, S.; Duarte Ramos Matos, G.; Suh, D.; Roux, B.; Mobley, D. L.; Michel, J. Reproducibility of Free Energy Calculations Across Different Molecular Simulation Software. J. Chem. Theory Comput. 2018, 14, 5567. (26) Grimme, S. Supramolecular Binding Thermodynamics by Dispersion-Corrected Density Functional Theory. Chem. - Eur. J. 2012, 18, 9955−9964. (27) McLennan, J. C.; McLeod, J. H. The Raman Effect with Liquid Oxygen, Nitrogen, and Hydrogen. Nature 1929, 123, 160. (28) Ewing, G. E. Spectroscopic Studies of Molecular Motion in Liquids. Acc. Chem. Res. 1969, 2, 168−174. (29) Trouton, F. On Molecular Latent Heat. Philos. Mag. 1884, 18, 54. (30) McLachlan, D., Jr; Marcus, R. J. The Statistical-Mechanical Basis of Trouton’s Rule. J. Chem. Educ. 1957, 34, 460−462. (31) Logan, S. R. The Rationale of Trouton’s Rule and of the Trouton-Hildebrand-Everett Rule. Z. Naturforsch., A: Phys. Sci. 1996, 51, 1247−1248. (32) Mammen, M.; Shakhnovich, E. I.; Deutch, J. M.; Whitesides, G. M. Estimating the Entropic Cost of Self-Assembly of Multiparticle Hydrogen-Bonded Aggregates Based on the Cyanuric Acid-Melamine Lattice. J. Org. Chem. 1998, 63, 3821−3830. (33) McQuarrie, D. Statistical Mechanics; University Science Books, 2000. (34) Jinnouchi, R.; Anderson, A. B. Electronic Structure Calculations of liquid-solid interfaces: Combination of Density Functional Theory and Modified Poisson−Boltzmann Theory. Phys. Rev. B: Condens. Matter Mater. Phys. 2008, 77, 245417. (35) Bondi, A. Van der Waals Volumes and Radii. J. Phys. Chem. 1964, 68, 441−451. (36) Ardura, D.; López, R.; Sordo, T. L. Relative Gibbs Energies in Solution through Continuum Models: Effect of the Loss of Translational Degrees of Freedom in Bimolecular Reactions on Gibbs Energy Barriers. J. Phys. Chem. B 2005, 109, 23618−23623. (37) Murray, C. W.; Verdonk, M. L. The Consequences of Translational and Rotational Entropy Lost by Small Molecules on Binding to Proteins. J. Comput.-Aided Mol. Des. 2002, 16, 741−753. (38) Dionísio, M. S.; Ramos, J. J. M.; Gonçalves, R. M. The Enthalpy and Entropy of Cavity Formation in Liquids and Corresponding States Principle. Can. J. Chem. 1990, 68, 1937−1949. (39) Mansoori, G. A.; Patel, V. K.; Edalat, M. The Three-Parameter Corresponding States Principle. Int. J. Thermophys. 1980, 1, 285−298. (40) Moura-Ramos, J. J.; Dionísio, M. S.; Gonçalves, R. M.; Diogo, H. P. A Further View on the Calculation of the Enthalpy of Cavity Formation in Liquids. The Influence of the Cavity Size and Shape. Can. J. Chem. 1988, 66, 2894−2902. (41) Rowlinson, J. S.. Liquids and Liquid Mixtures, 2nd ed.; Butterworths: London, 1971.

thank Dr. Peter Margl, Dr. Steven Arturo, Dr. Ivan Konstantinov, and Dr. Marc Coons (Dow Chemical) for helpful discussions.

■

REFERENCES

(1) Soroush, M. Computational Quantum Chemistry: Insights into Polymerization Reactions; Elsevier Science, 2018; pp 43−45. (2) Leung, B. O.; Reid, D. L.; Armstrong, D. A.; Rauk, A. Entropies in Solution from Entropies in the Gas Phase. J. Phys. Chem. A 2004, 108, 2720−2725. (3) Wolfe, S.; Kim, C. K.; Yang, K.; Weinberg, N.; Shi, Z. Hydration of the Carbonyl Group: A Theoretical Study of the Cooperative Mechanism. J. Am. Chem. Soc. 1995, 117, 4240−4260. (4) Wolfe, S.; Shi, Z.; Yang, K.; Ro, S.; 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. (5) Sumimoto, M.; Iwane, N.; Takahama, T.; Sakaki, S. Theoretical Study of Trans-Metalation Process in Palladium-Catalyzed Borylation of Iodobenzene with Diboron. J. Am. Chem. Soc. 2004, 126, 10457− 10471. (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. (7) Garza, A. J.; Bell, A. T.; Head-Gordon, M. Is Subsurface Oxygen Necessary for the Electrochemical Reduction of CO2 on Copper? J. Phys. Chem. Lett. 2018, 9, 601−606. (8) Garza, A. J.; Pakhira, S.; Bell, A. T.; Mendoza-Cortes, J. L.; Head-Gordon, M. Reaction Mechanism of the Selective Reduction of CO2 to CO by a Tetraaza [CoIIN4H]2+ Complex in the Presence of Protons. Phys. Chem. Chem. Phys. 2018, 20, 24058−24064. (9) Tanaka, R.; Yamashita, M.; Chung, L. W.; Morokuma, K.; Nozaki, K. Mechanistic Studies on the Reversible Hydrogenation of Carbon Dioxide Catalyzed by an Ir-PNP Complex. Organometallics 2011, 30, 6742−6750. (10) Besora, M.; Vidossich, P.; Lledós, A.; Ujaque, G.; Maseras, F. Calculation of Reaction Free Energies in Solution: A Comparison of Current Approaches. J. Phys. Chem. A 2018, 122, 1392−1399. (11) Spickermann, C. Entropies of Condensed Phases and Complex Systems: A First Principles Approach; Springer, 2011. (12) Yu, Z. X.; Houk, K. N. Intramolecular 1, 3-Dipolar Ene Reactions of Nitrile Oxides Occur by Stepwise 1, 1-Cycloaddition/ Retro-Ene Mechanisms. J. Am. Chem. Soc. 2003, 125, 13825−13830. (13) Liang, Y.; Liu, S.; Xia, Y.; Li, Y.; Yu, Z. X. Mechanism, Regioselectivity, and the Kinetics of Phosphine-Catalyzed [3 + 2] Cycloaddition Reactions of Allenoates and Electron-Deficient Alkenes. Chem. - Eur. J. 2008, 14, 4361−4373. (14) Plata, R. E.; Singleton, D. A. A Case Study of the Mechanism of Alcohol-Mediated Morita Baylis−Hillman Reactions. The Importance of Experimental Observations. J. Am. Chem. Soc. 2015, 137, 3811− 3826. (15) Ribeiro, R. F.; Marenich, A. V.; Cramer, C. J.; Truhlar, D. G. Use of Solution-Phase Vibrational Frequencies in Continuum Models for the Free Energy of Solvation. J. Phys. Chem. B 2011, 115, 14556− 14562. (16) Barone, V.; Bencini, A.; Cossi, M.; Matteo, A. D.; Mattesini, M.; Totti, F. Assessment of a Combined QM/MM Approach for the Study of Large Nitroxide Systems in Vacuo and in Condensed Phases. J. Am. Chem. Soc. 1998, 120, 7069−7078. (17) Andzelm, J.; Kölmel, C.; Klamt, A. Incorporation of Solvent Effects Into Density Functional Calculations of Molecular Energies and Geometries. J. Chem. Phys. 1995, 103, 9312−9320. (18) Kelly, C. P.; Cramer, C. J.; Truhlar, D. G. SM6: A Density Functional Theory Continuum Solvation Model for Calculating J


Article

Journal of Chemical Theory and Computation (42) Torquato, S.; Stillinger, F. H. Jammed Hard-Particle Packings: From Kepler to Bernal and Beyond. Rev. Mod. Phys. 2010, 82, 2633. (43) Finney, J. L. Bernal’s Road to Random Packing and the Structure of Liquids. Philos. Mag. 2013, 93, 3940−3969. (44) Pierotti, R.A. A. Scaled Particle Theory of Aqueous and Nonaqueous Solutions. Chem. Rev. 1976, 76, 717−726. (45) Reiss, H.; Frisch, H. L.; Lebowitz, J. L. Statistical Mechanics of Rigid Spheres. J. Chem. Phys. 1959, 31, 369−380. (46) Gharagheizi, F.; Mehrpooya, M. Prediction of Some Important Physical Properties of Sulfur Compounds Using Quantitative Structure−Properties Relationships. Mol. Diversity 2008, 12, 143. (47) Tang, K. E.; Bloomfield, V. A. Excluded Volume in Solvation: Sensitivity of Scaled-Particle Theory to Solvent Size and Density. Biophys. J. 2000, 79, 2222−2234. (48) Linstrom, P.J.; Mallard, W.G. NIST Chemistry WebBook, NIST Standard Reference Database Number 69 2018, DOI: 10.18434/ T4D303. (49) Sander, R. Compilation of Henry’s Law Constants (version 4.0) for Water as Solvent. Atmos. Chem. Phys. 2015, 15, 4399−4981. (50) Chickos, J. S.; Acree, W. E., Jr Enthalpies of Vaporization of Organic and Organometallic Compounds, 1880−2002. J. Phys. Chem. Ref. Data 2003, 32, 519−878. (51) Marenich, A. V.; Cramer, C. J.; Truhlar, D. G. Universal Solvation Model Based on the Generalized Born Approximation with Asymmetric Descreening. J. Chem. Theory Comput. 2009, 5, 2447− 2464. (52) Pollack, G. L.; Himm, J. F. Solubility of Xenon in Liquid nAlkanes: Temperature Dependence and Thermodynamic Functions. J. Chem. Phys. 1982, 77, 3221−3229. (53) Pollack, G. L.; Himm, J. F.; Enyeart, J. J. Solubility of Xenon in Liquid n-Alkanols: Thermodynamic Functions in Simple Polar Liquids. J. Chem. Phys. 1984, 81, 3239−3246. (54) Grimme, S.; Bannwarth, C.; Shushkov, P. A Robust and Accurate Tight-Binding Quantum Chemical Method for Structures, Vibrational Frequencies, and Noncovalent Interactions of Large Molecular Systems Parametrized for All spd-Block Elements (Z = 1− 86). J. Chem. Theory Comput. 2017, 13, 1989−2009. (55) Taylor, W. J. Average Length and Radius of Normal Paraffln Hydrocarbon Molecules. J. Chem. Phys. 1948, 16, 257−267. (56) Perdew, J. P.; Sun, J.; Garza, A. J.; Scuseria, G. E. Intensive Atomization Energy: Re-Thinking a Metric for Electronic Structure Theory Methods. Z. Phys. Chem. 2016, 230, 737−742. (57) Chai, J. D.; Head-Gordon, M. Long-Range Corrected Hybrid Density Functionals with Damped Atom−Atom Dispersion Corrections. Phys. Chem. Chem. Phys. 2008, 10, 6615−6620. (58) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Petersson, G. A.; Nakatsuji, H.; Li, X.; Caricato, M.; Marenich, A. V.; Bloino, J.; Janesko, B. G.; Gomperts, R.; Mennucci, B.; Hratchian, H. P.; Ortiz, J. V.; Izmaylov, A. F.; Sonnenberg, J. L.; Williams-Young, D.; Ding, F.; Lipparini, F.; Egidi, F.; Goings, J.; Peng, B.; Petrone, A.; Henderson, T.; Ranasinghe, D.; Zakrzewski, V. G.; Gao, J.; Rega, N.; Zheng, G.; Liang, W.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Vreven, T.; Throssell, K.; Montgomery, J. A., Jr.; Peralta, J. E.; Ogliaro, F.; Bearpark, M. J.; Heyd, J. J.; Brothers, E. N.; Kudin, K. N.; Staroverov, V. N.; Keith, T. A.; Kobayashi, R.; Normand, J.; Raghavachari, K.; Rendell, A. P.; Burant, J. C.; Iyengar, S. S.; Tomasi, J.; Cossi, M.; Millam, J. M.; Klene, M.; Adamo, C.; Cammi, R.; Ochterski, J. W.; Martin, R. L.; Morokuma, K.; Farkas, O.; Foresman, J. B.; Fox, D. J. Gaussian 16, Revision B.01; Gaussian, Inc.: Wallingford CT, 2016. (59) NIST Computational Chemistry Comparison and Benchmark Database; NIST Standard Reference Database Number 101, release 19, April 2018; Johnson, R. D., III, Ed.; http://cccbdb.nist.gov/ (retrieved December 17, 2018). (60) Ruiz-López, M. F.; Assfeld, X.; García, J. I.; Mayoral, J. A.; Salvatella, L. Solvent Effects on the Mechanism and Selectivities of Asymmetric Diels−Alder Reactions. J. Am. Chem. Soc. 1993, 115, 8780−8787.

(61) Bader, R. F.; Carroll, M. T.; Cheeseman, J. R.; Chang, C. Properties of Atoms in Molecules: Atomic Volumes. J. Am. Chem. Soc. 1987, 109, 7968−7979. (62) Mantina, M.; Chamberlin, A. C.; Valero, R.; Cramer, C. J.; Truhlar, D. G. Consistent van der Waals Radii for the Whole Main Group. J. Phys. Chem. A 2009, 113, 5806−5812. (63) Hildebrand, J. H. The Entropy of Vaporization as a Means of Distinguishing Normal Liquids. J. Am. Chem. Soc. 1915, 37, 970−978. (64) Everett, D. H. J. Chem. Soc. 1960, 2566. (65) Rabinovich, A.; Yasserman, A. A.; Nedostup, Y. I.; Veksler, L. S. Thermophysical Properties of Neon, Argon, Krypton, and Xenon; Hemisphere: Washington, D.C., 1988. (66) Streett, W. B.; Staveley, L. A. K. Experimental Study of the Equation of State of Liquid Argon. J. Chem. Phys. 1969, 50, 2302− 2307. (67) Letchworth-Weaver, K.; Arias, T. A. Joint Density Functional Theory of the Electrode-Electrolyte Interface: Application to Fixed Electrode Potentials, Interfacial Capacitances, and Potentials of Zero Charge. Phys. Rev. B: Condens. Matter Mater. Phys. 2012, 86, 075140. (68) Doncheski, M. A.; Robinett, R. W. Wave Packet Revivals and the Energy Eigenvalue Spectrum of the Quantum Pendulum. Ann. Phys. 2003, 308, 578−598. (69) Gilson, M. K.; Irikura, K. K. Symmetry Numbers for Rigid, Flexible, and Fluxional Molecules: Theory and Applications. J. Phys. Chem. B 2010, 114, 16304−16317. (70) Fernández-Ramos, A.; Ellingson, B. A.; Meana-Pañeda, R.; Marques, J. M.; Truhlar, D. G. Symmetry Numbers and Chemical Reaction Rates. Theor. Chem. Acc. 2007, 118, 813−826.

K


Solvation Entropy Made Simple - Journal of Chemical Theory and

Recommend Documents