# Linear Solvation–Energy Relationships (LSER) and Equation-of-State Thermodynamics: On the Extraction of Thermodynamic Information from the LSER Database

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{a}and σ

_{b}, reflecting the acidity and basicity characteristics, respectively, of the molecule. The weak dispersive interactions are reflected by the dispersion PSP, σ

_{d}, while the remaining Keesom-type and Debye-type polar interactions are, collectively, reflected by the polar PSP, σ

_{p}. The hydrogen-bonding PSPs are used to estimate a key quantity: the free energy change upon formation of the hydrogen bond, ΔG

^{hb}. Their equation-of-state characteristic permits also the estimation of the change in enthalpy, ΔH

^{hb}, and the entropy change, ΔS

^{hb}upon formation of the hydrogen bond.

_{x}, L, E, S, A, and B, corresponding to the McGowan’s characteristic volume V

_{x}, the gas–liquid partition coefficient L in n-hexadecane at 298 K, the excess molar refraction E, the dipolarity/polarizability S, the hydrogen bond acidity A, and hydrogen bond basicity B, respectively. These correlations are performed, in practical applications, through two basic LFER relationships that quantify solute transfer between two phases. The first relationship, Equation (1), quantifies solute transfer between two condensed phases [2,3,4,5,6,7,8,9,10,11,12]:

_{p}+ e

_{p}E + s

_{p}S + a

_{p}A + b

_{p}B + v

_{p}V

_{x}

^{S}) = c

_{k}+ e

_{k}E + s

_{k}S + a

_{k}A + b

_{k}B + l

_{k}L

^{S}is the gas-to-organic solvent partition coefficient.

^{S}= c

_{H}+ e

_{H}E + s

_{H}S + a

_{H}A + b

_{H}B + l

_{H}L

_{1}a

_{2}and B

_{1}b

_{2}. The key question is how this “solvation” information could be used for a valid estimation of the free energy change upon formation of these acid (1)–base (2) and base (1)–acid (2) hydrogen bonds. Similar questions apply to the estimation of hydrogen bonding change in enthalpy on the basis of Equation (3), which is consistent with the information obtained from Equation (2).

_{1}B

_{solvent}(1 − n

_{3}A

_{solvent})

_{2}A

_{solvent}(1 − n

_{4}B

_{solvent})

_{i}, of these equations are determined by fitting to the available experimental data for several solutes. All of these interesting correlations are useful in practice, but do not explain, at the fundamental or thermodynamic level, the observed linearity of Equations (1)–(3) and do not facilitate the above-mentioned extraction of thermodynamically meaningful information.

## 2. The Thermodynamic Framework

#### 2.1. Solvation Thermodynamics

_{1}, N

_{2},…, N

_{t}}, at temperature, T, and pressure, P, is given by the following defining equation in Ben-Naim’s mole/mole convention [48]:

_{i}is the chemical potential of component i, superscript IG denotes the ideal gas state, and Z is the compressibility factor,

_{m}

_{2}in Equation (8) is the molar volume of component 2 and ${\gamma}_{1/2}^{\infty}$ is the activity coefficient of solute 1 at infinite dilution in solvent 2. ${P}_{1}^{0}$ is the vapor pressure of the solute at temperature T, and ${\phi}_{1}^{0}$ its fugacity coefficient (typically set equal to 1 under ambient conditions). In order to proceed, we need an expression for ${\gamma}_{1/2}^{\infty}$ with explicit contributions from weak and strong intermolecular interactions, including hydrogen bonding ones, and this necessitates the adoption of an appropriate thermodynamic model. For this purpose, we will adopt here a statistical thermodynamic model, the basic elements of which are presented in the next sub-section.

#### 2.2. The Equation-of-State Model

_{sp}* and ε*, and one hydrogen bonding parameter, $\Delta {G}_{ij}^{hb}$, for each type of hydrogen bond, i-j, in which it may participate. Τhe specific hard core volume, v

_{sp}*, of the fluid provides with two key molecular parameters, the molar hard core volume, V* = M v

_{sp}*, and the number, r, of molecular segments of a constant hard core volume equal to 9.75 cm

^{3}/mol, or r = V*/9.75. Each segment interacts with its neighbors via segmental interaction energy, ε*. Thus, the molar interaction energy is given by E* = rε*, while the scaling temperature, T*, and pressure, P*, are defined by the central lattice–fluid equation: ε* = RT* = 9.75P*.

_{sp}*, as follows:

_{i}and a

_{i}, respectively, for each component i, as well as for the number of hydrogen bonds, N

_{ij}, between donors i and acceptors j in the system, or for the reduced ones, ν

_{ij}= N

_{ij}/rN. Each type of interaction i-j may be viewed as a quasi-chemical reaction of an acidic site (Acid

_{i}) and a basic site (Base

_{j}), of the form $Aci{d}_{i}+Bas{e}_{j}\hspace{1em}\rightleftarrows \hspace{1em}ABcomple{x}_{ij}$, and is characterized by the corresponding free energy change upon formation of bond i-j, $\Delta {G}_{ij}^{hb}=-RT\mathrm{ln}{K}_{ij}$, and the equilibrium constant, K

_{ij}. This free energy change may be split, in the classical manner, into enthalpic and entropic components, $\Delta {G}_{ij}^{hb}=\Delta {H}_{ij}^{hb}-T\Delta {S}_{ij}^{hb}$.

_{1}= N

_{1}/(N

_{1}+ N

_{2}) = N

_{1}/N = 1 − x

_{2}, and total number of segments, rN = r

_{1}N

_{1}+ r

_{2}N

_{2}, in which the molecules of component i (i = 1, 2) have d

_{i}donor sites and a

_{i}acceptor sites of type 1, the reduced number of free donor sites (non-hydrogen bonded) is given by:

#### 2.3. The Equation-of-State Model at Infinite Dilution

_{1}segments, the logarithm describing the probability of finding r

_{1}consecutive empty sites for its accommodation is given by the second term in the right-hand side of Equation (16). Thus, this second term is the cavitation term and reflects the difficulty of creating a cavity in the solvent volume in order to accommodate the solute molecule. The remaining terms are the charge terms of the solvation equation. ${\epsilon}_{ij}^{*}$ is the interaction energy for the contact of segments i and j. This refers to the non-specific or weak types of van der Waals (dispersion, and those arising from molecular polarizability and Keesom-type or Debye-type polarization) interaction. The contribution of strong specific (hydrogen bonding) interactions is accounted for by the last two terms in Equation (16).

_{11}in this equation is obtained from F

_{12}(cf. Equation (15)) by replacing subscript 2 with 1.

_{OW}).

_{i}s. As observed in Equation (16), the cavitation term (second term on the right-hand side of the equation) and the first charge term (third term in the equation) are “linearity” terms, that is, they are products of solute parameters with quantities (in parentheses) dependent exclusively on solvent properties. It is not clear, however, whether the two hydrogen bonding terms (last two terms in Equation (16)) are also “linearity” terms. This is examined in the next sub-section.

#### 2.4. On the Linearity of Hydrogen Bonding Contribution to Solvation Free Energy

_{2}′ in these equations is an exclusive property of the solvent (component 2). As can be observed, the hydrogen bonding contribution to solvation free energy depends only on the equilibrium constants K

_{ij}for the quasi-chemical reactions of hydrogen bonding between the proton donor (or acidic site), i, and the proton acceptor (or basic site), j.

_{12}= K

_{21}= K

_{22}= K. In this case, Equations (24) and (25) become identical, and the acid–base and base–acid contributions are equal, regardless of the validity or invalidity of linearity. Whether this central result conforms with the LSER model will be determined below.

_{ij}. It only gives the final form of the hydrogen bonding contribution in the form of the linearity sum (cf. Equation (2)), A

_{1}a

_{2}+ B

_{1}b

_{2}. If there is a thermodynamic basis to this linearity sum, Equations (24) and (25) should lead to it. The very form of the products in this sum indicates that the equilibrium constants, or the corresponding free energy changes upon hydrogen bond formation, should be expressed in terms of the acidity and basicity the LSER molecular descriptors, A

_{i}and B

_{j}.

_{i}and B

_{j}. We do not know anything about this expression a priori. Common practice in solving such problems in physics is to make plausible assumptions, starting from those with the greatest simplicity, and focusing on the consistency of their implementation. Whatever assumption is made, it should also apply to self-solvation of hydrogen-bonded compounds, like alkanols, water, etc.

_{1}a

_{2}. Thus, apart from a constant, the lnK term in Equation (28) should be of the form lnK = A

_{1}f(B

_{2},…), with the function f being an exclusive function of the solvent—component 2. Similarly, the acid (2)–base (1) interaction leads to the term B

_{1}b

_{2}or to the product B(1) × b(2). Thus, again, apart from a constant, the lnK term in Equation (28) should be of the form lnK = B

_{1}f(A

_{2},…), with the function f being an exclusive function of the solvent—component 2. However, upon self-solvation, A

_{1}= A

_{2}and B

_{1}= B

_{2}. All of these requirements are met by the following simple and plausible assumption:

_{ij}are reduced by a constant, which depends on the number of donor and/or acceptor sites of the hydrogen bond (see Supplementary Materials (SM)); it is therefore indicative of these aspects of hydrogen bonding. For solute–solvent systems with one donor and one acceptor each, the constant is equal to 0.62. For the self-solvation of such compounds, the constant becomes 0.31 = 0.62/2!, indicating that acid–base interaction is indistinguishable from base–acid interaction upon self-solvation.

_{ij}quantity, which is split into an LFER product (A

_{1}a

_{2}or B

_{1}b

_{2}) and a solvent-dependent constant, c

_{2}. It should be pointed out, again, that the hydrogen bonding contribution contains a constant solvent term. This is crucial to remember if we want to extract the hydrogen bonding information from the corresponding LFER terms (disregarding the LFER constant coefficient).

#### 2.5. The Essentials of the Partial Solvation Parameter (PSP) Approach

_{d}, reflects the weak intermolecular dispersive forces and is defined by the following equation:

_{x}, accounts for the majority of the contribution, and is weighted four times more heavily than the excess refractivity descriptor, E. The molar volume, V

_{m}, and the hard core molar volume, V*, are related through an LFHB-type equation (cf. Equation (9)), V

_{m}= V*/$\tilde{\rho}$. V* is closely correlated with the van der Waals volume of the molecule. If the LFHB scaling constant, ${v}_{sp}^{*}$, is available, then V* = M${v}_{sp}^{*}$. Alternatively, it may be estimated from V

_{x}through the equation: V*new = 11.357 + 99.492V

_{x}, which is a linear fit of LFHB scaling constants to V

_{x}, with R

^{2}= 0.9991, as shown in Figure 1. At 25 °C, this PSP is very close to the dispersive Hansen solubility parameter, δ

_{d}[24]. For non-polar compounds, this PSP is identical to the total Hildebrand solubility parameter, δ.

_{p}, reflects the weak and moderately strong polar interactions of the Keesom and the Debye types. If the molecule does not have a non-zero acidity or basicity LSER descriptor, this polar PSP is defined by the following equation:

_{p}, [24]. For polar and heterosolvated compounds, this PSP and the dispersion PSP are related to the total Hildebrand solubility parameter as follows:

_{dp}is also referred to as non-hydrogen-bonding PSP.

_{d}* + E

_{p}*. In practice, these are estimations, at first. If several data points are available with respect to density, they are used in combination with the equation of state, Equation (11), in order to refine the estimations.

_{hb}, which is defined as follows:

_{hb}= rν

_{hb}is the number of hydrogen bonds per mol. It is worth mentioning that σ

_{hb}contains information not only for the hydrogen bonding enthalpy, $\Delta {H}_{ii}^{hb}$, but also for the free energy, $\Delta {G}_{ii}^{hb}$, and the entropy, $\Delta {S}_{ii}^{hb}$, via the equilibrium constant, K

_{ii}($-RT\mathrm{ln}{K}_{ii}=\Delta {G}_{ii}^{hb}$). The equation-of-state approach, analogously to the plain Equation (23), includes information on the density of the compound, as well as on its molecular size, by means of the number of segments, r. The number of hydrogen bonds is then obtained using the following LFHB equation:

_{hb}over a broad range of external conditions. At 25 °C, this σ

_{hb}PSP is often close to the Hansen solubility parameter, δ

_{hb}[24].

_{hb}is available from external resources, Equation (42) can be used to determine σ

_{p}. When available, this route is preferred over that of Equations (37) and (38), since quite often the descriptor S is found to be given with relatively higher uncertainty [16].

_{Ha}and σ

_{Hb}is used to obtain the change in enthalpy upon formation of the hydrogen bond, as follows:

_{m,I}is the molar volume of compound i with the acidic site and V

_{m,j}is the molar volume of compound j with the basic site.

_{Ga}and σ

_{Gb}, or simply, σ

_{a}and σ

_{b}, is used to obtain the free energy change upon formation of the hydrogen bond, as follows:

_{H,i}and B

_{H,j}, via equations analogous to Equation (47). However, it should be made clear that the enthalpic descriptors are not independent, but are quantities derived from the corresponding free energy ones. The same holds true for PSPs. In essence, if $\Delta {G}_{ij}^{hb}$ is known over a range of temperatures, the corresponding derived quantity, $\Delta {H}_{ij}^{hb}$, could be obtained from an equation entirely analogous to Equation (19). Equivalently, one may obtain the entropy change from $\Delta {G}_{ij}^{hb}$, $\Delta {S}_{ij}^{hb}=-\left(\partial \Delta {G}_{ij}^{hb}/\partial T\right)$, and the change in enthalpy from the classical equation, $\Delta {H}_{ij}^{hb}=\Delta {G}_{ij}^{hb}+T\Delta {S}_{ij}^{hb}$. The reverse process may also be used if extensive data on enthalpic hydrogen-bonding PSPs are available.

## 3. Applications

_{or}, for which the enthalpy and entropy data are known [49]. As shown in Table S5, the discrepancies in the experimental data regarding the free energies of self-solvation of alkanols are almost always less than 1%.

^{S}, for the self-solvation free energy of alkanols, as given by the five products of the linearity Equation (2), are reported in Table 1. As shown, excluding cavitation contribution (lL), the main charge contribution to solvation free energy is hydrogen bonding. As observed in columns 4 and 5, the acid–base contribution, aA, is significantly different from the base–acid interaction, bB, for all alkanols. The difference (log) is 0.93 ± 0.06 for 1-alkanols and 0.65 for 2-alkanols. At present, there is no explanation for this difference. Thus, this hydrogen bonding information is not directly transferable at present.

_{ij}) for the former quantity, $\Delta {G}_{hb}^{S}$, which is part of the measurable overall solvation free energy. The same holds true for the corresponding enthalpies, although the difference in enthalpies is significantly reduced. As an example, in the case of self-solvation, we may start from the simple Equation (32) and examine the above differences.

_{ij}in Equation (30) or (32) were written without the constant term, as in the LSER model, this would imply than lnF

_{12}= −$\Delta {G}_{ij}^{hb}/2RT$(cf. Equation (31)) and the hydrogen bonding equilibrium constant, K

_{12}, were identical to the hydrogen bonding component of the solvation equilibrium constant, K

^{S}, as well as, of course, with F

_{12}. This would simplify things, and the differences described above would be zero. The correction constant to F

_{12}, however, implies that the two Ks are conceived differently by the two modeling approaches. Thus, the LSER quantity, −2.303RT(A

_{1}a

_{2}+ B

_{1}b

_{2}) = $\Delta {G}_{hb}^{S}$, cannot be considered identical to $\Delta {G}_{12}^{hb}$. The way hydrogen bonding contributes to the solvation free energy depends on the nature and multiplicity of the interacting sites, and this requires some corresponding correction to the plain sum of the LFER products, A

_{1}a

_{2}+ B

_{1}b

_{2}. Neither should F

_{12}be considered to be identical to K

_{12}. Similar concerns apply to all models based on divisions of intermolecular interactions.

_{hb}. Thus, the correlation of this parameter can be considered to be a direct test of the accuracy of the proposed $\Delta {H}_{12}^{hb}$ values. It seems that the hydrogen bonding parameters reported in Table 1, which have apparently been adopted by the LSER model [40], are not compatible with the corresponding solubility parameter data described in the literature [24,61].

_{a}and σ

_{b}are known from other sources, say, from the COSMO-RS model [35,36,37,38,39] or from molecular dynamics simulations, Equation (47) can be used to calculate the hydrogen bonding LSER molecular descriptors. This particular transfer, either from LSER to PSP or from PSP to LSER, is meaningful and useful when the same constant k is used in the equation. This constant may be obtained using Equation (31) if $\Delta {G}_{ij}^{hb}$ is known. In Table 4, the estimations of this constant are reported for alkanols based on the hydrogen bonding parameters, $\Delta {G}_{ij}^{hb}$, presented in the table. It can be seen that k is nearly constant. In fact, on the basis of analogous calculations performed for other solute–solvent systems, including aqueous ones, it seems that the values of k center around k = 33.9 or kR × 298.15 = 84,000 J/mol. The adoption of such a universal value for the constant k would greatly augment the predictive capacity of LSER and PSP, as well as other interconnected QSPR-type databases. However, the prerequisite for this remains the agreement on the values of $\Delta {G}_{12}^{hb}$ or $\Delta {H}_{12}^{hb}$ for several hydrogen-bonded compounds. In fact, the adoption of such a universal value for k would also require a rather minor change in the A and B LSER descriptors to A’ and B’, as reported in Table 4, in order to obtain the same solvation free energy as the product kRTA’B’.

_{d}, is mainly connected to the McGowan volume, V

_{x}, and, to a lesser extent, to the excess refractivity descriptor, E (cf. Equation (36)). Both V

_{x}and E are rather clearly defined, and practically speaking, Equation (36) is always considered to be valid. Since the total solubility parameter is very often known with good accuracy, Equation (42) permits the estimation of the polar PSP, σ

_{p}, or, equivalently, the LSER polarity descriptor, S, or the interaction energy, E

_{d}*. The polarity descriptor S is not as clearly defined as V

_{x}and E. Thus, the above transfer of information from σ

_{p}may be useful for verification or for a better estimation of S.

## 4. Discussion

_{1}B

_{2}.

_{2}

^{′}is an exclusive property of the solvent (component 2). The constant λ reflects the character of the hydrogen-bonding interaction. In one-donor–one-acceptor solute–solvent systems, λ = $2/\left(1+\sqrt{5}\right)$ and, upon self-solvation, λ = $1/\left(1+\sqrt{5}\right)$. In the case of two-donor–two-acceptor solute–solvent systems, λ = 2/4 and, upon self-solvation, λ = 1/4.

_{12}= F

_{21}and A

_{1}a

_{2}′ = B

_{1}b

_{2}′. What is even more interesting, though, is that both solvent coefficients, a

_{2}′ and b

_{2}′, are expressed explicitly by the plain relations, a

_{2}′ = kB

_{2}and b

_{2}′ = kA

_{2}.

_{1}a

_{2}is different from B

_{1}b

_{2}. Apparently, one or both of these products also contains the information of constant c

_{2}′. The constant λ does not show up when using the LSER approach, since it handles solute–solvent interactions exclusively as a one-donor–one-acceptor interaction. Thus, at present, the extraction of separate information on acidity and basicity contributions is not quite straightforward. If this were possible, this information could be transferred to the corresponding PSPs via Equations (45) and (46), and practically useful equation-of-state calculations could be performed.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## List of Symbols

Latin letters | |

a_{i} | Number of acceptor sites of type i |

a_{i} | LFER acidity coefficient for solvent i |

A_{i} | LSER acidity molecular descriptor of component i |

b_{j} | LFER basicity coefficient for solvent j |

B_{j} | LSER basicity molecular descriptor of component j |

c | LFER constant coefficient |

d_{i} | Number of donor sites of type i |

e | LFER solvent refractivity coefficient |

E | Excess refractivity LSER molecular descriptors |

F | Hydrogen bonding contribution term |

G | Free energy |

H | Enthalpy |

k | Proportionality constant |

K | Equilibrium constant |

l | LFER coefficient for gas-to-C16 partitioning |

L | LSER molecular descriptor for gas-to-C16 partitioning |

N | Mole number |

P | Pressure |

r | Number of molecular segments |

R | Gas constant |

s | LFER polarity coefficient |

S | Entropy |

T | Temperature |

v | specific volume |

V | Volume |

V_{x} | McGowan volume |

x | Mole fraction |

Z | Compressibility factor |

Greek Letters | |

γ | Activity coefficient |

δ | Solubility parameter |

ΔY | Change in quantity Y |

$\Delta {G}_{ij}^{hb}$ | Free energy change on hydrogen bond formation between donor i and acceptor j. |

ε* | Interaction energy |

ν | Fraction or reduced number of hydrogen bonds |

ξ | Correction factor to geometric-mean interaction energy |

$\tilde{\rho}$ | Reduced density |

σ | Partial solvation parameter |

φ | Fugacity coefficient |

ω | Molecular conformation parameter |

Superscripts | |

0 | Pure component |

∞ | Infinite dilution |

* | LFHB scaling property |

hb | Hydrogen bonding quantity |

IG | Ideal gas |

S | Solvation quantity |

Subscripts | |

1/2 | Property of solute (1) in solvent (2) |

0i | Fraction of free acceptor sites |

i0 | Fraction of free donor sites |

d | Dispersion quantity |

hb | Hydrogen bonding quantity |

i | Quantity pertaining to component i |

ij | Quantity pertaining to the interacting pair i, j |

m | Molar quantity |

p | Polar quantity |

sp | Specific |

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**Figure 1.**The hard core molar volumes of the LFHB model as a function of the corresponding McGowan volumes [16]. The straight line through the data obeys the equation V* = 11.357 + 99.49V

_{x}with R

^{2}= 0.9991; standard errors: 0.4963 (intercept), 0.0023 (slope).

**Figure 2.**Comparison of the experimental [49] solvation free energies of 1-alkanols with the corresponding LSER calculations as a function of carbon atoms in the molecule.

**Figure 3.**The LSER estimations of the solvation free energy of various solutes in 1-octanol, as a function of the corresponding experimental data [49].

**Table 1.**The five contributions (cf. Equation (2)) to the equilibrium constant (−logK

^{S}) for the self-solvation free energy of alkanols [16,60]: the hydrogen bonding contribution, $\Delta {G}_{hb}^{S}$, to the solvation free energy (in kJ/mol); the experimental [61] hydrogen bonding contribution, $\Delta {H}_{hb}^{S}$, to the self-solvation enthalpy (in kJ/mol); and the calculated hydrogen bonding contribution to the self-solvation entropy (in J/mol K) of alkanols.

Alkanol | eE | sS | aA | bB | lL | $\mathbf{\Delta}{\mathit{G}}_{\mathit{h}\mathit{b}}^{\mathit{S}}$ | $\mathbf{\Delta}{\mathit{H}}_{\mathit{h}\mathit{b}}^{\mathit{S}}$ | $\mathbf{\Delta}{\mathit{S}}_{\mathit{h}\mathit{b}}^{\mathit{S}}$ |
---|---|---|---|---|---|---|---|---|

METHANOL | 0.095 | 0.578 | 1.645 | 0.656 | 0.747 | −13.14 | −15.1 | −6.58 |

ETHANOL | 0.058 | 0.363 | 1.441 | 0.572 | 1.265 | −11.49 | −16.9 | −18.14 |

1-PROPANOL | 0.056 | 0.314 | 1.439 | 0.516 | 1.784 | −11.16 | −17.7 | −21.93 |

1-BUTANOL | 0.063 | 0.323 | 1.365 | 0.424 | 2.312 | −10.21 | −17.7 | −25.11 |

1-PENTANOL | 0.035 | 0.225 | 1.392 | 0.463 | 2.793 | −10.59 | −17.7 | −23.85 |

1-HEXANOL | 0.043 | 0.245 | 1.330 | 0.432 | 3.294 | −10.06 | −17.7 | −25.62 |

1-HEPTANOL | 0.045 | 0.234 | 1.323 | 0.388 | 3.837 | −9.77 | −17.7 | −26.61 |

1-OCTANOL | 0.043 | 0.236 | 1.283 | 0.364 | 4.349 | −9.40 | −17.7 | −27.84 |

1-DECANOL | 0.017 | 0.150 | 1.292 | 0.354 | 5.382 | −9.40 | −17.7 | −27.84 |

2-PROPANOL | 0.068 | 0.261 | 1.237 | 0.591 | 1.571 | −10.43 | −17.3 | −23.03 |

2-PENTANOL | 0.065 | 0.182 | 1.237 | 0.573 | 2.650 | −10.34 | −17.3 | −23.36 |

Solvent | ε*/ J mol ^{−1} | ε*_{s}/JK ^{1} mol^{−1} | v*_{sp0}/cm^{3} mol^{−1} | v*_{sp1}/× 10 ^{4} | $\mathbf{\Delta}{\mathit{H}}_{22}^{\mathit{h}\mathit{b}}$/ kJ mol ^{−1} | ΔS^{h}_{22}/JK ^{−1 }mol^{−1} | $\mathbf{\Delta}{\mathit{G}}_{22}^{\mathit{h}\mathit{b}}$/ kJ mol ^{−1} |
---|---|---|---|---|---|---|---|

METHANOL ^{1} | 4609 | −0.117 | 1.165 | 2.5 | −23.60 | −28.0 | −15.25 |

METHANOL | 4162 | 1.185 | 1.131 | 2 | −26.00 | −29.5 | −17.21 |

ETHANOL | 4134 | −0.107 | 1.128 | 0 | −24.05 | −27.5 | −15.85 |

1-PROPANOL | 4072 | 0.330 | 1.103 | −1 | −23.60 | −26.5 | −15.70 |

1-BUTANOL | 4092 | 0.610 | 1.097 | −1 | −23.40 | −26.5 | −15.50 |

1-PENTANOL | 4076 | 0.914 | 1.088 | −1 | −23.38 | −27.0 | −15.33 |

1-HEXANOL | 4058 | 1.090 | 1.079 | −1 | −23.08 | −26.5 | −15.18 |

1-HEPTANOL | 4117 | 0.730 | 1.081 | −1 | −23.10 | −27.0 | −15.05 |

1-OCTANOL | 4086 | 1.004 | 1.075 | −1 | −23.10 | −27.5 | −15.18 |

1-DECANOL | 4095 | 1.311 | 1.072 | −3 | −23.10 | −28.5 | −15.05 |

ISOPROPANOL | 3777 | 0.267 | 1.103 | −1 | −23.50 | −26.5 | −14.90 |

2-PENTANOL | 3784 | 0.949 | 1.072 | −2 | −23.50 | −26.5 | −14.75 |

^{1}Scaling constants for methanol accounting for δ

_{hb}[24].

Solvent | δ_{t}/MPa^{0.5} | δ_{hb}/MPa^{0.5} | ||||
---|---|---|---|---|---|---|

LSER calc | σ | Exp [61] | LSER calc | σ_{hb} | Exp [24] | |

METHANOL ^{1} | 27.8 | 30.2 | 29.4 | 17.9 | 22.8 | 22.3 |

METHANOL | 27.8 | 30.4 | 29.4 | 17.9 | 24.4 | 22.3 |

ETHANOL | 25.0 | 26.3 | 26.2 | 14.9 | 19.3 | 19.4 |

1-PROPANOL | 23.8 | 24.3 | 24.6 | 13.3 | 16.9 | 17.4 |

1-BUTANOL | 22.8 | 23.1 | 23.5 | 11.5 | 14.9 | 15.8 |

1-PENTANOL | 21.9 | 22.6 | 22.4 | 10.5 | 13.6 | 13.9 |

1-HEXANOL | 21.3 | 21.9 | 22.1 | 9.4 | 12.5 | 12.5 |

1-HEPTANOL | 21.0 | 21.7 | 21.8 | 8.6 | 11.7 | 11.7 |

1-OCTANOL | 20.5 | 21.0 | 21.0 | 8.6 | 11.0 | 11.9 |

1-DECANOL | 20.1 | 20.4 | 20.2 | 7.0 | 9.8 | 10.0 |

ISOPROPANOL | 22.8 | 23.6 | 23.8 | 12.6 | 16.4 | 16.4 |

2-PENTANOL | 21.1 | 21.8 | 21.8 | 10.2 | 13.6 | 13.3 |

^{1}With scaling constants for methanol best accounting for δ

_{hb}[24].

**Table 4.**The acidity and basicity LSER descriptors, A and B [16], the free energy change upon self-association, $\Delta {G}_{ij}^{hb}$, the corresponding kRT product, and the readjusted A’ and B’ descriptors to be used with a universal value of kRT = 84 kJ/mol.

Solvent | A | B | $-\mathbf{\Delta}{\mathit{G}}_{\mathit{i}\mathit{j}}^{\mathit{h}\mathit{b}}$/ kJ mol ^{−1} | −kRT/ kJ mol ^{−1} | A’ | B’ |
---|---|---|---|---|---|---|

METHANOL | 0.43 | 0.47 | 17.21 | 85.13 | 0.433 | 0.473 |

ETHANOL | 0.37 | 0.48 | 15.85 | 89.25 | 0.381 | 0.495 |

1-PROPANOL | 0.37 | 0.48 | 15.70 | 88.40 | 0.380 | 0.492 |

1-BUTANOL | 0.37 | 0.48 | 15.50 | 87.26 | 0.376 | 0.491 |

1-PENTANOL | 0.37 | 0.48 | 15.33 | 86.31 | 0.374 | 0.489 |

1-HEXANOL | 0.37 | 0.49 | 15.18 | 85.15 | 0.370 | 0.489 |

1-HEPTANOL | 0.37 | 0.48 | 15.05 | 84.73 | 0.369 | 0.485 |

1-OCTANOL | 0.37 | 0.49 | 14.90 | 83.89 | 0.365 | 0.485 |

1-NONANOL | 0.37 | 0.49 | 14.75 | 81.82 | 0.362 | 0.484 |

1-DECANOL | 0.36 | 0.49 | 14.60 | 82.29 | 0.361 | 0.482 |

ISOPROPANOL | 0.31 | 0.56 | 15.90 | 92.65 | 0.322 | 0.588 |

2-PENTANOL | 0.33 | 0.56 | 15.60 | 85.38 | 0.329 | 0.565 |

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**MDPI and ACS Style**

Panayiotou, C.; Zuburtikudis, I.; Abu Khalifeh, H.; Hatzimanikatis, V.
Linear Solvation–Energy Relationships (LSER) and Equation-of-State Thermodynamics: On the Extraction of Thermodynamic Information from the LSER Database. *Liquids* **2023**, *3*, 66-89.
https://doi.org/10.3390/liquids3010007

**AMA Style**

Panayiotou C, Zuburtikudis I, Abu Khalifeh H, Hatzimanikatis V.
Linear Solvation–Energy Relationships (LSER) and Equation-of-State Thermodynamics: On the Extraction of Thermodynamic Information from the LSER Database. *Liquids*. 2023; 3(1):66-89.
https://doi.org/10.3390/liquids3010007

**Chicago/Turabian Style**

Panayiotou, Costas, Ioannis Zuburtikudis, Hadil Abu Khalifeh, and Vassily Hatzimanikatis.
2023. "Linear Solvation–Energy Relationships (LSER) and Equation-of-State Thermodynamics: On the Extraction of Thermodynamic Information from the LSER Database" *Liquids* 3, no. 1: 66-89.
https://doi.org/10.3390/liquids3010007