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

^{1}

^{2}

^{*}

## Abstract

**:**

## 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

## References

- Chialvo, A.A. Solvation Phenomena in Dilute Solutions: Formal, Experimental Evidence, and Modeling Implications. In Fluctuation Theory of Solutions: Applications in Chemistry, Chemical Engineering and Biophysics; Matteoli, E., O’Connell, J.P., Smith, P.E., Eds.; CRC Press: Boca Raton, FL, USA, 2013; pp. 191–224. [Google Scholar]
- Sengers, J.M.L. Solubility Near the Solvent’s Critical Point. J. Supercrit. Fluids
**1991**, 4, 215–222. [Google Scholar] [CrossRef] - Chialvo, A.A.; Cummings, P.T. Solute-induced Effects on the Structure and the Thermodynamics of Infinitely Dilute Mixtures. AlChE J.
**1994**, 40, 1558–1573. [Google Scholar] [CrossRef] - Chialvo, A.A. On the Krichevskii Parameter of Solutes in Dilute Solutions: Formal Links between its Magnitude, the Solute-solvent Intermolecular Asymmetry, and the Precise Description of Solution Thermodynamics. Fluid Phase Equilibria
**2020**, 513, 112546. [Google Scholar] [CrossRef] - Fisher, M.E. Correlation Functions and the Critical Region of Simple Fluids. J. Math. Phys.
**1964**, 5, 944–962. [Google Scholar] [CrossRef] - Munster, A. Critical Fluctuations. In Fluctuation Phenomena in Solids; Burgess, R.E., Ed.; Academic Press: New York, NY, USA, 1965; pp. 180–264. [Google Scholar]
- Japas, M.L.; Sengers, J.M.H.L. Gas Solubility and Henry’s Law Near the Solvent’s Critical Point. AlChE J.
**1989**, 35, 705–713. [Google Scholar] [CrossRef] - Chialvo, A.A.; Cummings, P.T. Comments on “Near Critical Phase Behavior of Dilute Mixtures”. Mol. Phys.
**1995**, 84, 41–48. [Google Scholar] [CrossRef] - Akinfiev, N.N.; Diamond, L.W. Thermodynamic description of aqueous nonelectrolytes at infinite dilution over a wide range of state parameters. Geochim. Cosmochim. Acta
**2003**, 67, 613–629. [Google Scholar] [CrossRef] - Plyasunov, A.V.; Shock, E.L. Prediction of the vapor-liquid distribution constants for volatile nonelectrolytes in water up to its critical temperature. Geochim. Cosmochim. Acta
**2003**, 67, 4981–5009. [Google Scholar] [CrossRef] - Anisimov, M.A.; Sengers, J.V.; Sengers, J.M.H.L. Chapter 2—Near-critical behavior of aqueous systems. In Aqueous Systems at Elevated Temperatures and Pressures; Palmer, D.A., Fernández-Prini, R., Harvey, A.H., Eds.; Academic Press: London, UK, 2004; pp. 29–71. [Google Scholar]
- Orakova, S.M.; Rasulov, S.M.; Abdulagatov, I.M. Experimental study of the isomorphism behavior of weakly (CVX) and strongly (CPX, KTX) singular properties of 0.082 n-hexane+0.918 water mixtures near the upper critical point. J. Mol. Liq.
**2013**, 187, 7–19. [Google Scholar] [CrossRef] - Chialvo, A.A.; Crisalle, O.D. On density-based modeling of dilute non-electrolyte solutions involving wide ranges of state conditions and intermolecular asymmetries: Formal results, fundamental constraints, and the rationale for its molecular thermodynamic foundations. Fluid Phase Equilibria
**2021**, 535, 112969. [Google Scholar] [CrossRef] - Harvey, A.H.; Crovetto, R.; Sengers, J.M.H.L. Limiting vs. Apparent Critical Behavior of Henry’s Constant and K Factors. AlChE J.
**1990**, 36, 1901–1904. [Google Scholar] [CrossRef] - Abdulagatov, A.I.; Stepanov, G.V.; Abdulagatov, I.M. The critical properties of binary mixtures containing carbon dioxide: Krichevskii parameter and related thermodynamic properties. High Temp.
**2007**, 45, 408–424. [Google Scholar] [CrossRef] - Plyasunov, A.V. Values of the Krichevskii Parameter, AKr, of Aqueous Nonelectrolytes Evaluated from Relevant Experimental Data. J. Phys. Chem. Ref. Data
**2012**, 41, 033104. [Google Scholar] [CrossRef] - Abdulagatov, A.I.; Abdulagatov, I.M.; Stepanov, G.V. Crossover equation of state and microstructural properties of infinitely dilute solutions near the critical point of a pure solvent. J. Struct. Chem.
**2001**, 42, 412–422. [Google Scholar] [CrossRef] - Blanco, S.T.; Gil, L.; Garcia-Gimenez, P.; Artal, M.; Otin, S.; Velasco, I. Critical Properties and High-Pressure Volumetric Behavior of the Carbon Dioxide plus Propane System at T = 308.15 K. Krichevskii Function and Related Thermodynamic Properties. J. Phys. Chem. B
**2009**, 113, 7243–7256. [Google Scholar] [CrossRef] - Gil, L.; Martinez-Lopez, J.F.; Artal, M.; Blanco, S.T.; Embid, J.M.; Fernandez, J.; Otin, S.; Velasco, I. Volumetric Behavior of the {CO
_{2}(1) + C_{2}H_{6}(2)} System in the Subcritical (T = 293.15 K), Critical, and Supercritical (T = 308.15 K) Regions. J. Phys. Chem. B**2010**, 114, 5447–5469. [Google Scholar] [CrossRef] [PubMed] - Rivas, C.; Blanco, S.T.; Fernandez, J.; Artal, M.; Velasco, I. Influence of methane and carbon monoxide in the volumetric behaviour of the anthropogenic CO
_{2}: Experimental data and modelling in the critical region. Int. J. Greenh. Gas Control.**2013**, 18, 264–276. [Google Scholar] [CrossRef] - Plyasunov, A.V. Empirical evaluation of the Krichevskii parameter for aqueous solutes. J. Mol. Liq.
**2017**, 239, 92–95. [Google Scholar] [CrossRef] - Wilhelm, E. Solubilities, Fugacities and All That in Solution Chemistry. J. Solut. Chem.
**2015**, 44, 1004–1061. [Google Scholar] [CrossRef] - Chialvo, A.A.; Crisalle, O.D. On the Linear Orthobaric-density Representation of Near-critical Solvation Quantities: What Can We Conclude about the Accuracy of this Paradigm? Fluid Phase Equilibria
**2020**, 514, 112535. [Google Scholar] [CrossRef] - Fernández-Prini, R.; Alvarez, J.L.; Harvey, A.H. Henry’s constants and vapor-liquid distribution constants for gaseous solutes in H
_{2}O and D_{2}O at high temperatures. J. Phys. Chem. Ref. Data**2003**, 32, 903–916. [Google Scholar] [CrossRef] [Green Version] - Hansen, J.P.; McDonald, I.R. Theory of Simple Liquids, 3rd ed.; Academic Press: Amsterdam, The Netherlands, 2006. [Google Scholar]
- Haile, J.M. On the Use of Computer Simulation to Determine the Excess Free Energy in Fluid Mixtures. Fluid Phase Equilibria
**1986**, 26, 103–127. [Google Scholar] [CrossRef] - Kirkwood, J.G. Statistical mechanics of liquid solutions. Chem. Rev.
**1936**, 19, 275–307. [Google Scholar] [CrossRef] - Chialvo, A.A.; Cummings, P.T.; Simonson, J.M.; Mesmer, R.E. Solvation in High-Temperature Electrolyte Solutions. II. Some Formal Results. J. Chem. Phys.
**1999**, 110, 1075–1086. [Google Scholar] [CrossRef] [Green Version] - Ben-Naim, A. Solvation Thermodynamics; Plenum Press: New York, NY, USA, 1987. [Google Scholar]
- Chialvo, A.A.; Crisalle, O.D. On the behavior of the osmotic second virial coefficients of gases in aqueous solutions: Rigorous results, accurate approximations, and experimental evidence. J. Chem. Phys.
**2019**, 150, 124503. [Google Scholar] [CrossRef] [PubMed] - Plyasunov, A.V.; Shock, E.L. Estimation of the Krichevskii parameter for aqueous nonelectrolytes. J. Supercrit. Fluids
**2001**, 20, 91–103. [Google Scholar] [CrossRef] - Chialvo, A.A.; Cummings, P.T.; Kalyuzhnyi, Y.V. Solvation effect on kinetic rate constant of reactions in supercritical solvents. AlChE J.
**1998**, 44, 667–680. [Google Scholar] [CrossRef] - Abbott, M.M.; Nass, K.K. Equations of State and Classical Solution Thermodynamics—Survey of the Connections. ACS Symp. Ser.
**1986**, 300, 2–40. [Google Scholar] - Conrad, J.K.; Tremaine, P.R. A study of the deuterium isotope effect on zinc(II) hydrolysis and solubility under hydrothermal conditions using density functional theory. Chem. Eng. Sci.
**2022**, 254, 117596. [Google Scholar] [CrossRef] - Plumridge, J.; Arcis, H.; Tremaine, P.R. Limiting Conductivities of Univalent Cations and the Chloride Ion in H
_{2}O and D_{2}O Under Hydrothermal Conditions. J. Solut. Chem.**2015**, 44, 1062–1089. [Google Scholar] [CrossRef] - Trevani, L.N.; Balodis, E.C.; Tremaine, P.R. Apparent and standard partial molar volumes of NaCl, NaOH, and HCl in water and heavy water at T = 523 K and 573 K at p = 14 MPa. J. Phys. Chem. B
**2007**, 111, 2015–2024. [Google Scholar] [CrossRef] [PubMed] - Dohnal, V.; Fenclova, D.; Vrbka, P. Temperature dependences of limiting activity coefficients, Henry’s law constants, and derivative infinite dilution properties of lower (C-1-C-5) 1-alkanols in water. Critical compilation, correlation, and recommended data. J. Phys. Chem. Ref. Data
**2006**, 35, 1621–1651. [Google Scholar] [CrossRef] - Majer, V.; Sedlbauer, J.; Wood, R.H. Chapter 4—Calculation of standard thermodynamic properties of aqueous electrolytes and nonelectrolytes. In Aqueous Systems at Elevated Temperatures and Pressures; Palmer, D.A., Fernández-Prini, R., Harvey, A.H., Eds.; Academic Press: London, UK, 2004; pp. 99–147. [Google Scholar]
- Wilhelm, E.; Battino, R.; Wilcock, R.J. Low-Pressure Solubility of Gases in Liquid Water. Chem. Rev.
**1977**, 77, 219–262. [Google Scholar] [CrossRef] - Moine, E.; Privat, R.; Sirjean, B.; Jaubert, J.N. Jaubert, Estimation of Solvation Quantities from Experimental Thermodynamic Data: Development of the Comprehensive CompSol Databank for Pure and Mixed Solutes. J. Phys. Chem. Ref. Data
**2017**, 46, 033102. [Google Scholar] [CrossRef] - Scharlin, P.; Battino, R. Solubility of 13 Nonpolar Gases in Deuterium-Oxide at 15-Degrees-C-45-Degrees-C and 101.325-Kpa—Thermodynamics of Transfer of Nonpolar Gases from H
_{2}O to D_{2}O. J. Solution Chem.**1992**, 21, 67–91. [Google Scholar] [CrossRef] - Scharlin, P.; Battino, R. Solubility of CCl
_{2}F_{2},CClF_{3},CF_{4}, and c-C_{4}F_{8}in H_{2}O and D_{2}O at 298 K to 318 K and 101.325 KPa. Thermodynamics of Transfer of Gases from H_{2}O to D_{2}O. Fluid Phase Equilibria**1994**, 95, 137–147. [Google Scholar] [CrossRef] - Taylor, J.R. An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements; University Science Books: Sausalito, CA, USA, 2022. [Google Scholar]
- Chialvo, A.A.; Crisalle, O.D. Linear Orthobaric-density Approach to the Krichevskii Parameter of a Solute from the Vapor-Liquid Distribution Coefficient: What can we learn about its accuracy from systems whose precise behavior is known? Fluid Phase Equilibria
**2023**, 565, 113651. [Google Scholar] [CrossRef] - Ben-Naim, A.; Marcus, Y. Solvation Thermodynamics Of Nonionic Solutes. J. Chem. Phys.
**1984**, 81, 2016–2027. [Google Scholar] [CrossRef] - Chialvo, A.A. Gas solubility in dilute solutions: A novel molecular thermodynamic perspective. J. Chem. Phys.
**2018**, 148, 174502. [Google Scholar] [CrossRef] - Chialvo, A.A.; Crisalle, O.D. Solvation behavior of solutes in dilute solutions novel formal results, rules of thumb, and potential modeling pitfalls. Fluid Phase Equilibria
**2019**, 496, 17–30. [Google Scholar] [CrossRef] - Wagner, W.; Pruss, A. The IAPWS formulation 1995 for the thermodynamic properties of ordinary water substance for general and scientific use. J. Phys. Chem. Ref. Data
**2002**, 31, 387–535. [Google Scholar] [CrossRef] - Hill, P.G.; Macmillan, R.D.C.; Lee, V. A Fundamental Equation of State of Heavy Water. J. Phys. Chem. Ref. Data
**1982**, 11, 1–14. [Google Scholar] [CrossRef] - Herrig, S.; Thol, M.; Harvey, A.H.; Lemmon, E.W. A Reference Equation of State for Heavy Water. J. Phys. Chem. Ref. Data
**2018**, 47, 043102. [Google Scholar] [CrossRef] [Green Version] - Chialvo, A.A. On the Solvation Thermodynamics Involving Species with Large Intermolecular Asymmetries: A Rigorous Molecular-Based Approach to Simple Systems with Unconventionally Complex Behaviors. J. Phys. Chem. B
**2020**, 124, 7879–7896. [Google Scholar] [CrossRef] [PubMed] - Ben-Naim, A. Molecular Theory of Solutions; Oxford University Press: Oxford, UK, 2006. [Google Scholar]
- Jancsó, G.; Rebelo, L.P.N.; Van Hook, W.A. Non-ideality in Isotopic Mixtures. J. Chem. Soc.
**1994**, 23, 257–264. [Google Scholar] [CrossRef] - Japas, M.L.; Prini, R.F.; Horita, J.; Wesolowski, D.J. Fractionation of Isotopic-species between Coexistening Liquid and Vapor: Complete Range, including the Asymptotic Critical Behavior. J. Phys. Chem.
**1995**, 99, 5171–5175. [Google Scholar] [CrossRef] - Jancsó, G. Are isotopic mixtures ideal? Pure Appl. Chem.
**2004**, 76, 11–17. [Google Scholar] [CrossRef] [Green Version] - Jancso, G.; Rebelo, L.P.N.; Van Hook, W.A. Isotope Effects in Solution Thermodynamics: Excess Properties in Solutions of Isotopomers. Chem. Rev.
**1993**, 93, 2645–2666. [Google Scholar] [CrossRef] - Abdulkadirova, K.S.; Wyczalkowska, A.K.; Anisimov, M.A.; Sengers, J.V. Thermodynamic properties of mixtures of H
_{2}O and D_{2}O in the critical region. J. Chem. Phys.**2002**, 116, 4597–4610. [Google Scholar] [CrossRef] - Bazaev, A.R.; Abdulagatov, I.M.; Magee, J.W.; Bazaev, E.A.; Ramazanova, A.E. PVTx measurements for H2O+D2O mixtures in the near-critical and supercritical regions. J. Supercrit. Fluids
**2003**, 26, 115–128. [Google Scholar] [CrossRef] - Mazo, R.M. Statistical Mechanical Theory of Solutions. J. Chem. Phys.
**1958**, 29, 1122–1128. [Google Scholar] [CrossRef] - Chialvo, A.A. Alternative Approach to Modeling Excess Gibbs Free Energy in Terms of Kirkwood-Buff Integrals. In Advances in Thermodynamics; Matteoli, E., Mansoori, G.A., Eds.; Taylor & Francis: New York, NY, USA, 1990; pp. 131–173. [Google Scholar]
- Sengers, J.M.H.L. Critical Behavior of Fluids: Concepts and Applications. In Supercritical Fluids; Kiran, E., Sengers, J.M.H.L., Eds.; Fundamentals for Applications, Kluwer Academic Publishers: Dordrecht, The Netherlands, 1994; pp. 3–38. [Google Scholar]
- Chialvo, A.A. On the Solute-Induced Structure-Making/Breaking Effect: Rigorous Links among Microscopic Behavior, Solvation Properties, and Solution Non-Ideality. J. Phys. Chem. B
**2019**, 123, 2930–2947. [Google Scholar] [CrossRef] [PubMed] - Chialvo, A.A.; Crisalle, O.D. Can Jones-Dole’s B-coefficient Be a Consistent Structure Making/Breaking Marker?. Rigorous molecular-based analysis and critical assessment of its marker uniqueness. J. Phys. Chem. B
**2021**, 125, 12028–12041. [Google Scholar] [CrossRef] [PubMed] - Debenedetti, P.G.; Mohamed, R.S. Attractive, Weakly Attractive and Repulsive Near-Critical Systems. J. Chem. Phys.
**1989**, 90, 4528–4536. [Google Scholar] [CrossRef] - Petsche, I.B.; Debenedetti, P.G. On the Influence of Solute-Solvent Asymmetry upon the Behavior of Dilute Supercritical Mixtures. J. Phys. Chem.
**1991**, 95, 386–399. [Google Scholar] [CrossRef] - Chialvo, A.A. Solute-Solute and Solute-Solvent Correlations in Dilute Near-Critical Ternary Mixtures: Mixed Solute and Entrainer Effects. J. Phys. Chem.
**1993**, 97, 2740–2744. [Google Scholar] [CrossRef] - Wilhelm, E.; Battino, R. Thermodynamic Functions of Solubilities of Gases in Liquids at 25 degrees C. Chem. Rev.
**1973**, 73, 1–9. [Google Scholar] [CrossRef] - Crovetto, R.; Fernandez-Prini, R.; Japas, M.L. Solubility of Inert Gases and Methane in H
_{2}O and D_{2}O in the Temperature Range of 300 to 600K. J. Chem. Phys.**1982**, 76, 1077. [Google Scholar] [CrossRef] - Alvarez, J.; Fernandez-Prini, R. A semiempirical Procedure to Describe the Thermodynamics of Dissolution of Non-Polar Gases in Water. Fluid Phase Equilibria
**1991**, 66, 309–326. [Google Scholar] [CrossRef] - Van Ness, H.C.; Abbott, M.M. Classical Thermodynamics of Nonelectrolyte Solutions; McGraw Hill: New York, NY, USA, 1982. [Google Scholar]
- Kirkwood, J.G.; Buff, F.P. The Statistical Mechanical Theory of Solution. I. J. Chem. Phys.
**1951**, 19, 774–777. [Google Scholar] [CrossRef]

**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 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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