# Solvent and H/D Isotopic Substitution Effects on the Krichevskii Parameter of Solutes: A Novel Approach to Their Accurate Determination

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## Abstract

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## 1. Introduction

## 2. Fundamentals from Molecular Thermodynamics

#### 2.1. Molecular-Based Description of the Solvent Effect on the Solvation Behavior of a Solute

_{i}solute-labeled molecules undergo an alchemical mutation process (à la Kirkwood’s coupling-parameter charging) [27] in which the strength of their intermolecular potential parameters change from the original values of the solvent-solvent to those of the final solute-solute interactions. This step generates the desired non-ideal solution of the infinitely dilute $i-$solute in the $j-$solvent, and involves the following isothermal-isobaric Gibbs free energy change [28],

#### 2.2. Link between the Solvent Effect on the Solute’s Solvation and Its Krichevskii Parameter

## 3. Experimental Evidence of the Solvent H/D−Isotope Substitution Effects and Solvation Interpretation

#### 3.1. Identity of the Aqueous Solute Species and the Sources of Their Experimental Data

#### 3.2. Brute-Force Difference Approach to the Solvent $H/D-$Effect on the Krichevskii Parameter of a Solute

#### 3.3. Required Solvation Properties in the Molecular-Based Approach to the Solvent H/D−Effect on the Krichevskii Parameter

#### 3.4. Resulting Linear Representation for the Krichevskii Parameter ${\mathcal{A}}_{Kr}^{i,\alpha}=\Im \left({\Delta}_{h}{G}_{i,\alpha}^{\infty}\right)$

#### 3.5. Link between the Solvent H/D−Effect on the Krichevskii Parameter and Solute–Solvent Intermolecular Interaction Asymmetries

#### 3.6. Solvent H/D−Effect on the Krichevskii Parameter of the Emblematic Ideal Gas Solute

#### 3.7. Solvent H/D−Effect on the Krichevskii Parameter of the Emblematic Case of Lewis-Randall’s Quasi-Ideal Solutions

## 4. Discussion and Relevant Observations

## 5. Final Remarks and Outlook

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Symbols | |

${\mathcal{A}}_{Kr},{\mathcal{A}}_{Kr}^{i,j}$ | Krichevskii parameter of an $i-$solute in a $j-$solvent, i.e., $\underset{T,{\rho}_{j}^{o}\to critical}{\mathrm{lim}}{\left(\partial P/\partial {x}_{i}\right)}_{T\rho}^{\infty}$ |

${C}_{\alpha \beta}^{\otimes}\left(T,P\right)$ | direct correlation function integral, aka DCFI, for the $\alpha \beta -$interactions at the ⊗ conditions, either infinite dilution ∞ or pure component o |

$DCFI$ | direct correlation function integral |

${f}_{i}\left(T,P,{x}_{i}\right)$ | fugacity of the $i-$species |

${\mathcal{F}}_{\alpha}\left(T,P\right)$ | isobaric-isothermal function defined by Equation (6) |

${G}_{\alpha \beta}^{\otimes}\left(T,P\right)$ | Kirkwood-Buff integral for the $\alpha \beta -$interactions at the ⊗ conditions, either infinite dilution ∞ or pure component o |

${\mathscr{H}}_{i,j}^{IS}\left(T,P\right)$ | Henry’s law constant of an $i-$species in a $j-$solvent given by ${\mathscr{H}}_{i,j}^{IS}\left(TP\right)={f}_{i}^{o}\left(TP\right){\gamma}_{i}^{LR,\infty}\left(TP\right)$ |

$k$ | Boltzmann constant |

$KB$ | Kirkwood-Buff |

${\kappa}_{j}^{o}\left(T,P\right)$ | isothermal compressibility of the pure $j-$solvent |

${K}_{D}^{\infty}\left(T,P\right)$ | vapor-liquid solute distribution coefficient |

$M{W}_{\alpha}$ | molar weight of an $\alpha -$species |

${m}_{i}^{o}$ | reference molality of an $i-$solute |

${N}_{\alpha}$ | number of molecules of an $\alpha -$species |

$P$ | system pressure |

${\mathcal{P}}_{i\left(a\right)}^{\infty ,R}$ | generic isobaric-isothermal residual property of an infinitely dilute $i-$species in an $\alpha -$solvent |

${\hat{\mathcal{P}}}_{i}^{\infty}(T,P)$ | generic partial molar property of an infinitely dilute $i-$species |

${\mathcal{S}}_{i,j}^{\infty}\left(T,P\right)$ | structure making/breaking parameter of an infinitely dilute $i-$species in a $j-$solvent |

$SR,LR$ | short- and long-range contributions to the Kirkwood-Buff integral according to the Ornstein-Zernike equation |

$TCFI$ | total correlation function integral, aka Kirkwood-Buff integral |

$T,P$ | state conditions defined by the system temperature and pressure |

$T,\rho $ | state conditions defined by the system temperature and density |

$V\left(T,P,{x}_{i}\right)$ | system volume at the specified state conditions and composition |

${\widehat{\upsilon}}_{i}\left(T,P,{x}_{i}\right)$ | partial molar volume of the $i-$species |

${x}_{i}$ | liquid phase composition defined by the mole fraction of the $i-$species |

${z}_{j}^{o}\left(T,P\right)$ | compressibility factor $P/{\rho}_{j}^{o}kT$ for the pure $j-$solvent |

${\Delta}_{h}{G}_{i,j}^{\infty}\left(T,P\right)$ | standard solvation Gibbs free energy of the $i-$solute in the $j-$solvent |

$\Delta {g}_{tr}^{o}\left(T,P\right)$ | solvation Gibbs free energy of transfer of an infinitely dilute $i-$solute between two solvent environments |

$\Delta {G}_{i}^{*}{\left(T,P\right)}_{\sigma}$ | solvation Gibbs free energy according to Ben-Naim’s definition |

${\Delta}_{ij}^{\infty}\left(T,P\right)$ | linear combination of Kirkwood-Buff integrals related to the non-ideality of the dilute solution, i.e., ${\left({G}_{ii}^{\infty}+{G}_{jj}^{o}-2{G}_{ij}^{\infty}\right)}_{TP}$ |

${\widehat{\varphi}}_{i}\left(T,P,{x}_{i}\right)$ | partial molar fugacity coefficient of the $i-$species |

$\Im (\cdots )$ | a general function |

${\gamma}_{i}^{LR}\left(T,P,{x}_{i}\right)$ | Lewis-Randall’s activity coefficient of the $i-$species, i.e., ${\widehat{\varphi}}_{i}\left(T,P,{x}_{i}\right)/{\varphi}_{i}^{o}\left(T,P\right)$ |

${\mu}_{i}^{R}\left(T,P,{x}_{i}\right)$ | isobaric-isothermal residual chemical potential of the $i-$species at the specified state conditions and composition |

${\mu}_{i}^{r}\left(T,\rho ,{x}_{i}\right)$ | isochoric-isothermal residual chemical potential of the $i-$species at the specified state conditions and composition |

$\rho \left(T,P,{x}_{i}\right)$ | molar density of the system at the specified state conditions and composition |

Sub- and super-scripts | |

$c$ | critical condition for the pure $j-$solvent |

$o$ | pure component |

$\infty $ | infinite dilution |

$i$ | solute species |

$IS$ | ideal solution |

$j,k$ | solvent species |

$LR$ | Lewis-Randall |

$IG$ | ideal gas condition |

$IG\text{\_}i$ | special case of solute as an ideal gas $i-$species |

$LR-IS$ | special case of Lewis-Randall ideality when ${\left({G}_{ii}^{\infty}={G}_{ij}^{\infty}={G}_{jj}^{o}\right)}_{TP}$ |

$R$ | residual property at constant $\left(T,P,{x}_{i}\right)$ |

$r$ | residual property at constant $\left(T,\rho ,{x}_{i}\right)$ |

## Appendix A. Relation among Solvation Gibbs Free Energy Expressions

## Appendix B. Krichevskii Parameter of Solutes in Quasi-Ideal Solutions

## Appendix C. Relation between the Krichevskii Parameter ${\mathcal{A}}_{\mathit{K}\mathit{r}}$ and the Structure Making/Breaking Parameter ${\mathcal{S}}_{\mathit{i}\mathit{\alpha}}^{\mathbf{\infty}}$

## Appendix D. The Standard Hydration Gibbs Free Energy of Water Isotopomers and Their Link to the Corresponding Krichevskii Parameters

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**Figure 1.**Schematic of the isothermal-isobaric solvation cycle used to define the Gibbs free energy of transfer of a solute between two solvent environments, where we identify the four-stage mutation process, the type of residual properties involved, the resulting type of infinite dilution solution, and its corresponding Krichevskii parameter.

**Figure 2.**Isothermal-isochoric residual chemical potential of the infinite dilute $i-$solutes in light and heavy aqueous solutions as a function of the corresponding solute-solvent intermolecular interaction asymmetry ${\Delta}_{i\alpha}^{\infty}$ when $\alpha =\left({H}_{2}O,{D}_{2}O\right)$.

**Figure 3.**Isothermal-isobaric Gibbs free energy of transfer for an $i-$solute transfer from light to a heavy water as a function of the solvent-density weighted difference between the solute-${D}_{2}O$ and the solute-${H}_{2}O$ interaction asymmetry, ${\left({\rho}_{{D}_{2}O}^{o}{\Delta}_{i{D}_{2}O}^{\infty}-{\rho}_{{H}_{2}O}^{o}{\Delta}_{i{H}_{2}O}^{\infty}\right)}_{TP}$.

**Figure 4.**Isotopic substitution effect on the Krichevskii parameter of gases in water, $\Delta {\mathcal{A}}_{Kr}\equiv {\mathcal{A}}_{Kr}^{i,{D}_{2}O}-{\mathcal{A}}_{Kr}^{i,{H}_{2}O}$, as a function of the solvent-density weighted difference between the solute-${D}_{2}O$ and the solute-${H}_{2}O$ interaction asymmetry, ${\left({\rho}_{{D}_{2}O}^{o}{\Delta}_{i{D}_{2}O}^{\infty}-{\rho}_{{H}_{2}O}^{o}{\Delta}_{i{H}_{2}O}^{\infty}\right)}_{TP}$, according to the regressed data from Ref. [24] in comparison with the results from the molecular-based formalism as described by Equation (8).

**Table 1.**Krichevskii parameters of gases in light and heavy aqueous systems from regressed $E-$coefficient of Fernandez-Prini et al. [24] and corresponding standard Gibbs free energies of hydration.

Solute | ${\mathcal{A}}_{\mathit{K}\mathit{r}}^{\mathit{i}\mathbf{,}{\mathit{H}}_{\mathbf{2}}\mathit{O}}\mathbf{=}\mathbf{}\mathbf{0.5}\mathit{R}{\mathit{\rho}}_{\mathit{c}}^{\mathit{o}}{\mathit{E}}^{\mathit{i}\mathbf{,}{\mathit{H}}_{\mathbf{2}}\mathit{O}}$ ^{(a)} | ${\mathcal{A}}_{\mathit{K}\mathit{r}}^{\mathit{i}\mathbf{,}{\mathit{D}}_{\mathbf{2}}\mathit{O}}\mathbf{=}\mathbf{}\mathbf{0.5}\mathit{R}{\mathit{\rho}}_{\mathit{c}}^{\mathit{o}}{\mathit{E}}^{\mathit{i}\mathbf{,}{\mathit{D}}_{\mathbf{2}}\mathit{O}}$ ^{(a)} | ${\mathbf{\Delta}}_{\mathit{h}}{\mathit{G}}_{\mathit{i}}^{\mathit{\infty}}\mathbf{\left(}{\mathit{H}}_{\mathbf{2}}\mathit{O}\mathbf{\right)}$ ^{(b)} | ${\mathbf{\Delta}}_{\mathit{h}}{\mathit{G}}_{\mathit{i}}^{\mathit{\infty}}\mathbf{\left(}{\mathit{D}}_{\mathbf{2}}\mathit{O}\mathbf{\right)}$ ^{(b)} |
---|---|---|---|---|

$He$ | 1661 | 1673 | 19.418 | 19.482 |

$Ne$ | 1836 | 1780 | 19.038 | 19.132 |

$Ar$ | 1692 | 1656 | 16.242 | 16.321 |

$Kr$ | 1668 | 1642 | 14.773 | 14.987 |

$Xe$ | 1482 | 1487 | 13.404 | 13.655 |

$Rn$ | --- | --- | 11.590 | 11.432 |

${H}_{2}$ | 1675 | --- | 17.687 | 17.394 ^{(c)} |

${D}_{2}$ | --- | 1562 | 17.565 ^{(c)} | 17.222 ^{(c)} |

${N}_{2}$ | 1750 | --- | 18.149 | 18.251 |

${O}_{2}$ | 1688 | --- | 16.479 | 16.539 |

$C{H}_{4}$ | 1623 | 1617 | 16.231 | 16.388 |

${C}_{2}{H}_{6}$ | 1570 | --- | 15.535 | 15.671 |

${C}_{3}{H}_{8}$ | --- | --- | 16.108 | 16.265 |

$S{F}_{6}$ | 2103 | --- | 20.510 | 20.643 |

$CC{l}_{2}{F}_{2}$ | --- | --- | 14.383 | 14.512 |

$CCl{F}_{3}$ | --- | --- | 17.310 | 17.060 |

$C{F}_{4}$ | --- | --- | 20.960 | 21.111 |

**Table 2.**Isotopic substitution effect on the Krichevskii parameters of gases in light and heavy aqueous systems from regressed coefficient $E$ of Fernandez-Prini et al. [24].

Solute | ${\mathcal{A}}_{\mathit{K}\mathit{r}}^{\mathit{i}\mathbf{,}{\mathit{H}}_{\mathbf{2}}\mathit{O}}\mathbf{\left(}\mathit{a}\mathit{t}\mathit{m}\mathbf{\right)}$ ^{(a)} | $\mathit{R}\mathit{M}\mathit{S}\mathit{D}\mathbf{\left(}\mathit{a}\mathit{t}\mathit{m}\mathbf{\right)}$ ^{(b)} | ${\mathcal{A}}_{\mathit{K}\mathit{r}}^{\mathit{i}\mathbf{,}{\mathit{D}}_{\mathbf{2}}\mathit{O}}\mathbf{\left(}\mathit{a}\mathit{t}\mathit{m}\mathbf{\right)}$ ^{(c)} | $\mathit{R}\mathit{M}\mathit{S}\mathit{D}\mathbf{\left(}\mathit{a}\mathit{t}\mathit{m}\mathbf{\right)}$ ^{(d)} | $\mathit{H}\mathbf{/}\mathit{D}\mathbf{-}\mathit{E}\mathit{f}\mathit{f}\mathit{e}\mathit{c}\mathit{t}\mathbf{\left(}\mathit{a}\mathit{t}\mathit{m}\mathbf{\right)}$ |
---|---|---|---|---|---|

$He$ | 1661 | ±52 | 1673 | ±40 | 12 ± 92 |

$Ne$ | 1836 | ±108 | 1780 | ±32 | −56 ± 140 |

$Ar$ | 1692 | ±37 | 1656 | ±67 | −36 ± 104 |

$Kr$ | 1668 | ±52 | 1642 | ±11 | −26 ± 63 |

${H}_{2}$ | 1675 | ±76 | 1562 ^{(e)} | ±101 | −113 ± 177 |

$C{H}_{4}$ | 1623 | ±56 | 1617 | ±15 | −6 ± 71 |

Solute | $\mathbf{\Delta}{\mathit{g}}_{\mathit{t}\mathit{r}}^{\mathit{o}}\mathbf{\left(}\mathit{T}\mathbf{,}\mathit{P}\mathbf{\right)}$ ^{(a)} | ${\mathbf{\Delta}}_{\mathit{i}\mathbf{,}{\mathit{H}}_{\mathbf{2}}\mathit{O}}^{\mathbf{\infty}}\mathbf{\left(}\mathit{c}{\mathit{m}}^{\mathbf{3}}\mathbf{/}\mathit{m}\mathit{o}\mathit{l}\mathbf{\right)}$ ^{(b)} | ${\mathbf{\Delta}}_{\mathit{i}\mathbf{,}{\mathit{D}}_{\mathbf{2}}\mathit{O}}^{\mathit{\infty}}\mathbf{\left(}\mathit{c}{\mathit{m}}^{\mathbf{3}}\mathbf{/}\mathit{m}\mathit{o}\mathit{l}\mathbf{\right)}$ ^{(c)} |
---|---|---|---|

$He$ | −198.0 | 428.5 | 427.0 |

$Ne$ | −168.0 | 423.0 | 421.9 |

$Ar$ | −183.0 | 383.2 | 380.7 |

$Kr$ | −48.0 | 360.8 | 361.2 |

$Xe$ | −11.0 | 340.8 | 341.7 |

$Rn$ | −420.0 | 314.3 | 309.1 |

${H}_{2}$ | −555.0 | 403.3 | 396.4 |

${D}_{2}$ | −605.0 | 402.8 | 393.9 |

${N}_{2}$ | −160.0 | 410.0 | 408.9 |

${O}_{2}$ | −202.0 | 385.7 | 383.9 |

$C{H}_{4}$ | −105.0 | 382.0 | 383.4 |

${C}_{2}{H}_{6}$ | −126.0 | 371.9 | 371.2 |

${C}_{3}{H}_{8}$ | −105.0 | 380.3 | 379.9 |

$S{F}_{6}$ | −129.0 | 444.5 | 444.0 |

$CC{l}_{2}{F}_{2}$ | −133.0 | 355.1 | 354.2 |

$CCl{F}_{3}$ | −512.0 | 397.8 | 391.5 |

$C{F}_{4}$ | −111.0 | 451.1 | 450.8 |

Solute | $\mathbf{\Delta}{\mathcal{A}}_{\mathit{K}\mathit{r}}\mathbf{\left(}\mathit{a}\mathit{t}\mathit{m}\mathbf{\right)}$ ^{(a)} |
---|---|

$He$ | −25.50 |

$Ne$ | −24.14 |

$Ar$ | −22.42 |

$Rn$ | −26.97 |

${H}_{2}$ | −36.60 |

${D}_{2}$ | −38.26 |

${N}_{2}$ | −23.21 |

${O}_{2}$ | −23.15 |

$C{H}_{4}$ | −19.68 |

${C}_{2}{H}_{6}$ | −19.85 |

${C}_{3}{H}_{8}$ | −19.58 |

$S{F}_{6}$ | −23.96 |

$CC{l}_{2}{F}_{2}$ | −19.17 |

$CCl{F}_{3}$ | −34.80 |

$C{F}_{4}$ | −23.69 |

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**MDPI and ACS Style**

Chialvo, A.A.; Crisalle, O.D.
Solvent and H/D Isotopic Substitution Effects on the Krichevskii Parameter of Solutes: A Novel Approach to Their Accurate Determination. *Liquids* **2022**, *2*, 474-503.
https://doi.org/10.3390/liquids2040028

**AMA Style**

Chialvo AA, Crisalle OD.
Solvent and H/D Isotopic Substitution Effects on the Krichevskii Parameter of Solutes: A Novel Approach to Their Accurate Determination. *Liquids*. 2022; 2(4):474-503.
https://doi.org/10.3390/liquids2040028

**Chicago/Turabian Style**

Chialvo, Ariel A., and Oscar D. Crisalle.
2022. "Solvent and H/D Isotopic Substitution Effects on the Krichevskii Parameter of Solutes: A Novel Approach to Their Accurate Determination" *Liquids* 2, no. 4: 474-503.
https://doi.org/10.3390/liquids2040028