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

: We establish a direct route for the accurate determination of the solvent effect on the Krichevskii parameter of a solute, based solely on the contrasting solvation behavior of the solute in the desired solvent relative to that of the reference solvent, i


Introduction
The Krichevskii parameter defines a finite-size quantity as the limiting critical value of the isothermal-isochoric rate of change of the system's pressure caused by the mutation (aka alchemical transformation) of an α−solvent particle into an i−solute species, in an otherwise pure solvent, i.e., lim [1]. Even though A Kr is a finite quantity, its magnitude and sign are the result of the underlying solute-solvent intermolecular interaction asymmetry [2][3][4], where the developing pressure perturbation propagates across the entire system given that the solvent's correlation length diverges at criticality [5,6].
The interest on the Krichevskii parameter has grown immensely since its inception [2], in part, because it has become a key quantity in the description and/or correlation of the thermodynamic behavior of dilute solutions, especially for non-electrolyte aqueous systems [7][8][9][10][11][12][13]. Its relevance has generated the urgency for experimental approaches to its determination, involving a variety of methodologies as discussed elsewhere [14][15][16]. Unfortunately, the accumulated tabulations of Krichevskii parameters, especially involving A i,α Kr = ℑ ∆ h G ∞ i,α representation in terms of the solute-solvent intermolecular interaction asymmetry, compare the resulting H/D−solvent effect on the Krichevskii parameters of selected aqueous gases, and consequently, interpret two emblematic cases of aqueous solutions involving either an ideal gas solute or an i−solute behaving as the solvating α−isotopic form of water. To complete the development, in Section 4 we provide a novel microstructural interpretation of the solvent effect according to a rigorous characterization of the critical solvation in terms of a finite unambiguous structure making/breaking parameter S ∞ iα (SR), and identify some relevant observations. Finally, we close the manuscript with some additional remarks and outlook.

Fundamentals from Molecular Thermodynamics
Before we attempt the determination of the solvent effect on the Krichevskii parameter of a solute, we need to identify some essential thermodynamic relations and corresponding molecular-based interpretations. For that purpose, in what follows, we provide the framework for the microscopic-to-macroscopic rigorous description of the solvation process of a solute and its concomitant solute transfer between two distinctive solvent environments at standard (T, P) state conditions.

Molecular-Based Description of the Solvent Effect on the Solvation Behavior of a Solute
Our goal here is to assess how the replacement of the j−solvent with a k−solvent affects the Krichevskii parameter of an i−solute, while avoiding the need for any experimental data of the solute behavior at the critical conditions of the two solvents. In Figure 1, we illustrate a thought experiment involving a four-step solvation-cycle path for a single i−solute species, first in an pure j−solvent, and then in a pure k−solvent at fixed (T, P) state conditions during the process of formation of the infinitely dilute systems. The process comprises N = N j solvent molecules in which N i << N j of them are distinguishable by their i−solute levels, with N i → 0 in the thermodynamic limit, i.e., lim N,V→∞ (N/V) TP = ρ o j (T, P) [25]. This initial system represents an ideal solution, from the viewpoint of the Lewis-Randall rule, because the residual properties of the (N − N i ) solvent-labeled and those of the N i solute-labeled molecules are all identical [1,26].
provide the framework for the microscopic-to-macroscopic rigorous description of the solvation process of a solute and its concomitant solute transfer between two distinctive solvent environments at standard T , P ( ) state conditions.

Molecular-Based Description of the Solvent Effect on the Solvation Behavior of a Solute
Our goal here is to assess how the replacement of the j -solvent with a k solvent affects the Krichevskii parameter of an i solute, while avoiding the need for any experimental data of the solute behavior at the critical conditions of the two solvents. In Figure 1, we illustrate a thought experiment involving a four-step solvation-cycle path for a single i -solute species, first in an pure j -solvent, and then in a pure k -solvent at   Then, the N 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], where the superscript R denotes an isobaric-isothermal residual property, while H IS i,j (T, P) and f o j (T, P) are the Henry's law constant of the i−solute in the pure j−solvent, and its corresponding fugacity at the (T, P) state conditions. Likewise, for the desired non-ideal solution of the infinitely dilute i−solute in the k−solvent, we have that Finally, we alchemically mutate the (N − N i ) j−solvent labeled particles into k−solvent along an isothermal-isobaric path described by the following Gibbs free energy change, so that, the corresponding change of Gibbs free energy ∆g o tr (T, P) involved in the transfer of the infinitely dilute i−solute from the j−solvent to the k−solvent can be extracted from For clarity sake, and to distinguish the behavior of the i−solute at infinite dilution, the subindices in the residual property P ∞,R i(a) describe the i−solute at infinite dilution in the α−solvent, i.e., α = j or α = k. Alternative solvation-cycle paths, associated with Ben-Naim solvation approach [29], are briefly described in Section SI-1 of the Supplementary Materials.

Link between the Solvent Effect on the Solute's Solvation and Its Krichevskii Parameter
Note that, while Equation (3) involves only the properties of the two pure solvents, Equations (1) and (2) represent the solvation of the infinitely dilute i−solute in either the j−solvent or the k−solvent. Consequently, the finite change of Gibbs free energy associated with these two processes can be written as follows [1], whereυ ∞ i and υ o α = 1/ρ o α denote the partial molar volumes of the i−solute at infinite dilution and the pure α−solvent, respectively. Moreover, since lim Kr describes the Krichevskii parameter of the i−solute in the α−solvent [2], we are able to express the integrand of Equation (5) in the reduced (ℜ) form as follows, where (· · · ) ℜ ≡ (· · · )/(· · · ) c represents a reduced quantity for the involved solute-solvent intermolecular asymmetry, and F α (T, P) defines a general and unique function of the state conditions of the α−solvent. The uniqueness of F α (T, P) comes from the fact that  [30] where γ ∞,LR,IG_i i (T, P) describes the Lewis-Randall activity coefficient of an ideal gas i−solute (IG_i) while z o j (T, P) = P/kTρ o j represents the compressibility factor of the pure j−solvent, and the superindex r identifies an isochoric-isothermal residual property. This equation provides the rigorous theoretical foundation to the empirical linear regression originally reported by Plyasunov and Shock [31], and embodies a fundamentally based route to the accurate estimation of the corresponding Krichevskii parameters [4], i.e., where A IG_i,α Kr = kT α,c ρ o α,c defines the Krichevskii parameter of the i−solute as an ideal gas in the α−solvent with α = (j, k) [32]. Equation (7) suggests two possible scenarios depending on the solvation properties available for the i−solute in the desired α−solvent: if we had the standard Gibbs free energy of solvation (or its associated Henry's constant) of an i−solute in both j− and k−solvents at standard conditions, we could determine the solvent effect on the Krichevskii parameter of an i−solute defined as the difference . Because Equation (8) involves only standard solvation Gibbs free energies at normal conditions, data typically available with significant accuracy, this expression represents a direct route for the evaluation of ∆A Kr , i.e., it provides an answer to the question (b) of the Introduction. Otherwise, if we had at our disposal the Gibbs free energy of transfer, ∆g o tr (T, P), rather than the standard Gibbs free energy ∆ h G ∞ i(k) (T, P), in addition to the Krichevskii parameter of the i−solute in the j−solvent, A i,j Kr , then we could proceed as follows. First, we need to link the two A i,α Kr expressions involving the solvents, α = (j, k), with the one describing the Gibbs free energy of transfer of the i−solute from the j−solvent environment to the k−solvent environment, Equation (4), at the typically measured standard (T, P) state conditions. For that purpose, we isolate ln H IS i,α (T, P) from Equation (7) so that, where we have identified the two distinctive residual chemical potentials, defined at either fixed (T, P) or its corresponding fixed (T, Moreover, because the process depicted in Figure 1 involves normal (standard) state conditions, the determination of the Krichevskii parameter A i,k Kr becomes straightforward. In fact, provided that the transfer free energy between the two solvents is available, it requires only the isobaric-isothermal and isochoric-isothermal residual chemical potentials of the pair of pure solvents involved, which are quantities accurately known for most solvents at normal conditions. In other words, Equation (11) provides an answer to question (a) in the Introduction, and leads to the essential foundations to expand the analysis in Section 3.4 of question (b) in the Introduction.

Experimental Evidence of the Solvent H/D−Isotope Substitution Effects and Solvation Interpretation
While the described molecular-based approach to the solvent effect on the Krichevskii parameter of an i−solute applies to any type of solvent, here we focus our attention on the special case of H/D−isotopic substituted aqueous solvents, i.e., water isotopomers. The rationale for this choice is twofold: (a) these near-critical aqueous environments are frequently found in electric power generation and pose significantly challenging to study experimentally [34][35][36], and (b) the small magnitude of the H/D−effect on the thermodynamic properties of the aqueous solvent makes the alluded brute-force subtraction approach unreliable as a result of the large uncertainties involved in the individual nearcritical quantities (vide infra). As a reference, and to be more precise, the typical uncertainties of the experimental Krichevskii parameters might reach ±50 MPa and even higher depending on the evaluation method [14,16] while the H/D−effect on the Krichevskii parameters might amount to a small fraction of the alluded uncertainty.
Typically, the hydration (solvation) behavior of a solute is analyzed in terms of standard thermodynamic quantities and their relation to the transfer process of a solute from the ideal gas phase to its standard state in solution. In fact, we have two alternative paths toward the determination of an i−solute standard state propertyP ∞ i (T, P) namely, through its standard state dissolution quantity ∆ solP where IG stands for and ideal gas phase, the superscripts (∞, o) describe environments at infinite dilution and pure component, respectively, while P o = 0.1 MPa [37]. Here, we are focusing onP ∞ i (T, P) =Ĝ ∞ i (T, P) ≡ µ ∞ i (T, P), the partial molar Gibbs free energy of the i−solute at infinite dilution under a diversity of manifestations, including the standard hydration (solvation) Gibbs free energies ∆ h G ∞ i (T, P o ) [38], Ben-Naim's solvation Gibbs free energy ∆G * i (T, P) σ [29], and the standard Gibbs free energy of solution ∆µ ∞ i (T, P) σ [39,40], whose meanings and their interrelations are provided in Appendix A.

Identity of the Aqueous Solute Species and the Sources of Their Experimental Data
The systems targeted here are aqueous solutions of gaseous solutes, where the aqueous environments are either light or heavy water, and the i−solute species include simple and noble gases, as well as some halogen-substituted light hydrocarbons. In particular, the list of i−solutes include He, Ne, Ar, Kr, Xe, Rn, H 2 , D 2 , N 2 , O 2 , CH 4 , C 2 H 6 , C 3 H 8 , SF 6 , CCl 2 F 2 , CClF 3 , and CF 4 . This selection is based on the availability of (either assumed or considered) reliable experimental data for either the Krichevskii parameters of the solutes in both light and heavy aqueous systems or their hydration and/or transfer Gibbs free energies. In fact, we invoked the study of the near-critical behavior of the Henry's law constant and vapor-liquid distribution coefficient of several solutes in light and heavy water by Fernandez-Prini et al. [24] complemented by available information on the Gibbs free energy of hydration of these gases in both aqueous environment as well as the their Gibbs free energy of transfer between light and heavy water [39,41,42].
In Table 1, we present the calculated Krichevskii parameters of the dissolved gases in light and heavy aqueous systems and the resulting standard Gibbs free energies of hydration from the regressions of Ref. [24] as well as from solubility measurements from Refs. [39,41,42] as explicitly indicated. In particular, in columns 2 and 3 of Table 1, we reveal the Krichevskii parameters of the i−solutes in light and heavy water as determined from the parameter E resulting from the regression of the solute distribution coefficients in Ref. [24]. Moreover, in columns 4 and 5, we display the corresponding data for the Gibbs free energies of hydration derived from the regression of the Henry's law constants in Ref. [24], and complemented with those calculated from solubility measurements in Refs. [39,41,42]. 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.

Brute-Force Difference Approach to the Solvent H/D−Effect on the Krichevskii Parameter of a Solute
The obvious first attempt to assess the solvent effect is the simple subtraction between the third and second columns of Table 1 as illustrated in Table 2 , where we also provide the quoted uncertainties from Ref. [24]. It becomes immediately evident that we cannot expect reliable results for the solvent (and particularly, for the H/D−isotopic substitution) effects from the corresponding values of the Krichevskii parameter of a solute because their subtraction will result in a magnification of the individual uncertainties [43]. 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].

RMSD(atm) (d) H/D−Effect(atm)
He Table 5 of Ref. [24]; (b) From Table 6 of Ref. [24]; (c) From E-coefficients in Table 9 of Ref. [24]; (d) From Table 10 of Ref. [24]; (e) After assuming that This contention is additionally supported by the analysis of the uncertainties associated with the determination of the Krichevskii parameters of an ideal gas solute in light and heavy water from the regression of their solute vapor-liquid distribution coefficients, systems for which we know the exact answer [44]. This scenario suggests the need for an alternative approach to assess directly the underlying isotopic effect and avoid the unreliable brute-force subtraction method.

Required Solvation Properties in the Molecular-Based Approach to the Solvent H/D−Effect on the Krichevskii Parameter
For the implementation of the approach proposed in Sections 2.1 and 2.2, we proceed with the calculation of the required hydration (solvation) properties as follows. Based on the data of Table 1, we can determine the Gibbs free energy of transfer ∆g o tr , Equation (4), in terms of the hydration Gibbs free energy of the i−solute in the two isotopic forms of the solvent, Alternatively, ∆g o tr can be determined according to Ben-Naim's scheme [45], i.e., where ∆ tr G * is given by the following difference of solvation quantities, after invoking the relations derived in Appendix A. Moreover, we calculate the underlying solute-solvent intermolecular interaction asymmetry for the i−solute, ∆ ∞ iα (T, P), according to the expression [46], , reads as follows, with the subscript (α) emphasizing that we are dealing with either α = j or α = k as the solvent. Then, from the solubility measurements, x sat i(α) , we estimate the activity coefficient γ LR,∞ i(α) according to [47], as a more accurate alternative to the conventional ln γ LR,∞ according to (A15) in Appendix A. The resulting values from Equations (12) and (15) are given in Table 3 below. Table 3. Gibbs free energy of transfer, ∆g o tr in J/mol, of gases in light and heavy aqueous systems from the correlations of the Henry's law constants in Ref. [24] and gas solubilities from Refs. [39,41,42].

Resulting Linear Representation for the Krichevskii Parameter
After recalling that A IG_i,α Kr = kT α,c ρ o α,c defines the Krichevskii parameter of the i−solute as an ideal gas in the α−solute as an ideal gas in the solvent with α = (j, k) [8,32], and considering the critical conditions of the light [48] and heavy water [49,50] (7) and (A4) as well as the corresponding residual chemical potentials µ o,R α (T, P) and µ o,r α (T, ρ o α ) of the pure α−solvent, we obtain the following linear representations for the Krichevskii parameter of an i−solute in light and heavy water, As we might have expected, the resulting linear hydration Gibbs free energy representations, Equations (19) and (20), exhibit slightly different slopes in their dependence on the relative (to that of the corresponding pure α−solvent) hydration free energies. These expressions highlight the size of the resulting H/D−isotopic substitution effect on the Krichevskii parameter of the solutes under investigation, i.e., such an effect is significantly smaller than the magnitude of the reported uncertainties of the individual Krichevskii pa- [24]. We have recently addressed the uncertainty issue according to a rigorous analysis of the behavior of the orthobaric-density dependence of the solute distribution coefficient of when the j−solvent was light water [23]. Moreover, we have illustrated how small experimental uncertainties of the solute distribution coefficient at high temperature can drastically affect the outcome of the regression, and consequently, the resulting effective Krichevskii parameter [44]. Indeed, by analyzing the behavior of j,c when the j−solvent was either light or heavy water, we found that there were no H/D−isotopic effects on the orthobaricdensity slope within the range of effective linearity as a consequence of the null solutesolvent interactions. However, as we replaced the ideal gas solute with a real gas (compare Figures 8 and 9 in Ref. [44]), the range of effective linearity of the T σ ln K ∞,He when the j−solvent was heavy water became narrower than that observed for the same solute in light water, i.e., a clear manifestation of the H/D−isotopic effect associated with the non-zero solute-solvent interactions. Obviously, this feature imposes a stronger constraint j,c effective linearity leading to the determination of the Krichevskii parameter. On the one hand, the closer T ⊕ is to T c , the better since it provides a more accurate representation of the asymptotic critical slope; on the other hand, the closeness of the chosen T ⊕ to T c is significantly constrained by the experimental challenges associated with highly compressible environments.

Link between the Solvent H/D−Effect on the Krichevskii Parameter and Solute-Solvent Intermolecular Interaction Asymmetries
Considering the nature of the aqueous systems under study, we can first invoke the following identity [30], where the superscript L α identifies the liquid phase of the α−solvent. Then, we introduce the accurate second-order composition representation for the partial molar excess free energy of an interacting solute, (see Appendix A of Ref. [30] and Appendix B of Ref. [46] for details while noting that the second-order expansion is unable to describe accurately the behavior of non-interacting solutes as discussed in Ref. [51]) to find a link between the magnitude of the Henry's law constant of an i−solute and a precisely-defined molecular measure of solute-solvent intermolecular interaction asymmetry, i.e., where we have invoked Equations (15) and (16) to describe the infinite dilution activity coefficient γ LR,∞,L α i (T, P) in Equation (22). Moreover, by introducing Equation (23) into Equation (7), we obtain a revealing linear dependence of the magnitude of the Krichevskii parameter of the i−solute and the solute-solvent intermolecular interaction asymmetry, ∆ ∞ iα (T, P), as follows, In other words, the Krichevskii parameter of an Therefore, according to Equations (7) and (24), the solvent effect on the Krichevskii parameter of an i−solute defined as the difference of Krichevskii parameters between the two solvents, This equation suggests that ∆A Kr can be interpreted as a A IG_i,α Kr −prorated difference of the solute-solvent intermolecular interaction asymmetry function an observation that we will analyze below. In fact, we should note that the isothermal-isochoric residual chemical potential of the gaseous solutes at infinite dilution in aqueous solutions, µ ∞,r i(α) (T, ρ o α ), exhibits a linear dependence with the corresponding solute-solvent intermolecular interaction asymmetry function ∆ ∞ iα (T, P) as illustrated in Figure 2 when the α−solvent is either light or heavy water at ambient conditions. We can also identify µ ∞,r IG_i(α) (T, ρ o α ) = 0 at a hypothetical value of the solutesolvent intermolecular interaction asymmetry, i.e., one that differs from the theoretical The reason for this difference resides in the fact that, as demonstrated in Ref. [51], the second-order expansion cannot provide an accurate description of the behavior of non-interacting solutes, i.e., it requires at least a six-order composition expansion. However, because the condition µ ∞,r IG_i(α) (T, ρ o α ) = 0 also means that In fact, the quadratic-composition dependent ln γ LR i (T, P, x i ) = ln γ ∞,LR i(j) (T, P)x 2 i that describes accurately the LR−nonideal behavior of these solutes [46] leads and identified as the red dot in Figure 3.
Note also that, while the second-order composition approximation for ln γ LR i (T, P, x i ) is not accurate for an IG_i solute as indicated above, we can still identify the IG_i solute in Figure 3, given that Equation (4)  in Ref. [51], the second-order expansion cannot provide an accurate description of the behavior of non-interacting solutes, i.e., it requires at least a six-order composition expansion. However, because the condition and identified as the red dot in Figure 3. Note also that, while the second-order composition approximation for According to the linear behaviors described above, we can alternatively express the solvent effect in terms of the solvation Gibbs free energies DG  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- According to the linear behaviors described above, we can alternatively express the solvent effect ∆A Kr in terms of the solvation Gibbs free energies ∆G * i,α (T, P) as follows, where ∆G * i,α (T, P) = kT ln H IS i,α /kTρ o α identifies the solvation Gibbs free energy of the i−solute in the α−solvent, while ∆G * i=α (T, P) = kT ln( f o α /kTρ o α ) TP denotes the Gibbs free energy of self-solvation of the α−solvent, i.e., µ o,r , for the α = (j, k) solvents [40,52]. Likewise, ∆A Kr can be equivalently written in terms of the isothermalisochoric residual chemical potential of the species, µ ⊕,r β (T, ρ o α ) = kT ln φ ⊕ β z o α , as follows,  (10) and (11) (28) which becomes the answer to question (a) in the Introduction as the desirable form of ∆A Kr . Equations (25)- (28) reveal that the solvent effect on the Krichevskii parameter of a solute results from a linear combination of the relative solvation Gibbs free energy of the i−solute in the pair of solvents. More specifically, the solvent effect on the magnitude of the Krichesvkii parameter for an i−solute becomes directly proportional to the difference of two similar quantities comprising two distinctive terms, i.e., (a) the Krichesvkii parameter for the ideal gas solute in the chosen solvents, and (b) the ratio between the solvation Gibbs free energy of the i−solute in the chosen solvents and their self-solvation counterparts. Given the relation between the solvation Gibbs free energy and the species residual free energies, Equation (27) for the α = (j, k) solvents. Moreover, because the residual quantities measure the contribution of the intermolecular interactions to the thermodynamic properties, the A IG_i,α Kr −prorated difference conveys the contribution of the difference of solute-solvent intermolecular interaction asymmetries to the solvent effect on the Krichevskii parameter. In fact, according to Equations (10) and (25) as well as the resulting linear behavior in Figures 2 and 3, the solvent effect on the Krichevskii parameter of these gaseous solutes becomes also effectively as illustrated in Figure 4.  , as a function of the solvent-density weighted difference between the solute- , according to the regressed data from Ref. [24] in comparison with the results from the molecular-based formalism as described by Equation (8).
In summary, Equations (25) Table 4 and Figure 4. In fact, the values reported in Table 4 are the outcome of the derived Equations (8), (27), and (28), which provided precisely the same answer, after invoking the properties from Table 1 and the required thermodynamic properties of the two solvents at ambient conditions from the corresponding equations of state. Among the three theoretically equivalent expressions for , Equations (8), (27) and (28), the first two do not require the reference Krichevskii parameter , and thus, for practical purposes, they might be the preferred routes.
, as a function of the solvent-density weighted difference between the solute- , according to the regressed data from Ref. [24] in comparison with the results from the molecular-based formalism as described by Equation (8). In summary, Equations (25)-(28) provide alternative direct routes to both (a) a simple test of consistency between the calculated (by any method) Krichevskii parameters of a common i−solute in the (j, k)−pair of solvents through the determination of the standard Gibbs free energy for the transfer of the given i−solute from the j−solvent to the k−solvent, and the standard thermodynamic properties of the two pure solvents, and (b) an accurate evaluation of the effect of the solvent on the Krichevskii parameter of the i−solute, i.e., Kr , when we replace the j−solvent with the k−solvent, avoiding the regression of near-critical properties of the solute in the corresponding solvents as illustrated in Table 4 and Figure 4. In fact, the ∆A Kr values reported in Table 4 are the outcome of the derived Equations (8), (27), and (28), which provided precisely the same answer, after invoking the properties from Table 1 and the required thermodynamic properties of the two solvents at ambient conditions from the corresponding equations of state. Among the three theoretically equivalent expressions for ∆A Kr , Equations (8), (27) and (28), the first two do not require the reference Krichevskii parameter A i,j Kr , and thus, for practical purposes, they might be the preferred routes.
Moreover, in Figure 4, we illustrate the comparison between the solvent H/D− isotopic substitution effect on the Krichevskii parameter as determined by the "brute-force" subtraction approach based on the regressed coefficient E of Fernandez-Prini et al. [24] and that from our proposed routes, including the reported uncertainties. This comparison provides clear evidence of the lack of reliability of the "brute-force" approach in the assessment of the H/D−isotopic substituted (or for that matter, any solvent) effect from the regression of the Henry law constant and vapor-liquid distribution coefficient of solutes, as we could have anticipated given the challenges behind the measurements at near-critical compared with those at standard ambient solvent environments.  (8), (27) and (28) provide identical outcomes at least up to the 10th decimal figure.

Solvent H/D−Effect on the Krichevskii Parameter of the Emblematic Ideal Gas Solute
It becomes instructive to analyze the effect of the change of solvent environment on the Krichevskii parameter for an ideal gas solute IG_i for which we know the answer beforehand, and consequently, to be able to check the validity of the approach. From Equation (4) (29) so that, after introducing (29) which is the expected answer, where we note that the critical conditions of the two solvents are obviously different.

Solvent H/D−Effect on the Krichevskii Parameter of the Emblematic Case of Lewis-Randall's Quasi-Ideal Solutions
The mixture of water isotopomers, e.g., D 2 O + H 2 O, falls into the category of quasiideal solutions within the Lewis-Randall reference [53]. In fact, in their study of highly dilute aqueous solutions of heavy water, Japas et al. [54] introduced the simplifying assumption of Raoult solution ideality to provide an estimation of the Krichevskii parameter of D 2 O(i) in H 2 O(j), and arrived to the following expression in terms of the saturation pressures of the two species, (for a detailed analysis of either Lewis-Randall or Raoult solution ideality of these systems see Appendix B) According to the description of either Hill et al. [49] or Herrig et al. [50] and Wagner and Pruss [48] equations of state for heavy and light water, respectively, we have from Equation (31)  and by the same quasi-ideal approximation for the infinitely dilute light water in heavywater solvent we can conclude, as discussed in details in Appendix B, that In fact, in Appendix B we have provided a molecular-based argument for the general scenario of a Lewis-Randall ideal solution to demonstrate rigorously that should the mixture of light and heavy water behave ideally, then , after assuming for the sake of argument that T c, Obviously, as highlighted by Jancsó and coll. [53,55], these isotopic mixtures exhibit small but non-negligible deviations from ideality, which are usually described by either the simplest symmetric excess Gibbs free energy representation [56]. In this context, we should note that in Table IV MPa is at odd. Unfortunately, the authors have not provided any information on the uncertainties for the two composition limiting derivatives involved, though we are aware of the fact that these quantities usually bear large uncertainties that would affect these outcomes [16]. In contrast, in a later report by Bazaev et al. [58], they determined that A D

Discussion and Relevant Observations
The proposed approach to the H/D−effect on the Krichevskii parameter leads naturally to the assessment of this parameter for binary systems comprising isotopomers. In fact, when the i−solute in solution with the original j−solvent is simply another j−species, i.e., the iconic special case of Lewis-Randall ideal solution, Equations (3) and (10) where the j− and k−species are simply two isotopomers. Therefore, depending on the solute-solvent intermolecular asymmetry and the fact that usually ϕ o k z o k ≪ 1 at normal conditions, Equation (34) translates into two potential scenarios, either Liquids 2022, 2 490 or, By noting that when the k−species is an ideal gas,φ ∞,IG_k [30], so that for the case of mixtures of light (j−species) and heavy water (k−species) we haveφ ∞ j (T, P) ⪅ ϕ o k (T, P), consequently, we expect A j,k Kr < 0 according to Equation (36).
Kr > 0 according to Equation (35). In other words, Equations (35) and (36) indicate that the sign of the Krichevskii parameter of an isotopic j−solute in an isotopic k−solvent will be decided by either Kr > 0 will be the outcome for the upper inequality, the middle equality and the lower inequality, respectively.
At this point, it is worth highlighting the subtle effect of the small perturbations of the solute-solvent intermolecular interaction asymmetry ∆ ∞ H 2 O D 2 O around zero, i.e., the Lewis-Randall (Raoult) solution ideality [59], on the non-ideality of these aqueous systems. On the one hand, the D 2 O + H 2 O system described as a slightly non-ideal solution characterized by the excess Gibbs free energy as suggested by Jancsó et al. [53], leads to after considering the definition γ LR,∞ i,j Consequently, as the solute-solvent interaction asymmetry ∆ ∞ ij (T, P) vanishes, we have that lim Therefore, the reference to the Lewis-Randall (Raoult) solution ideality for the description of the behavior of the represents the equivalent limiting behavior, as if the light and heavy water behaved identically, leading to R = 1, and consequently to = 0, which is obviously not consistent with the actual case for the water isotopomers. In fact, the symmetric nature of the βg E x H 2 O TP = x H 2 O x D 2 O ln γ LR,∞ description for the non-ideality of these mixtures is neither agreeing with the actual R(T, P) ̸ = 1 evidence, nor being compatible with the general Lewis-Randall (Raoult) behavior discussed in Appendix B which leads to On the other hand, because the two water isotopomers exhibit slightly different thermodynamic behaviors, e.g., we might expect a marginally asymmetric isothermal-isobaric composition representation for its excess Gibbs free energy. In other words, instead of the simple symmetric B4 form, we could have a more realistic Margules-type representation, i.e., and consequently, lim This analysis indicates that the relative magnitude of the two activity coefficients at infinite dilution, γ LR,∞ H 2 O,D 2 O and γ LR,∞ D 2 O,H 2 O , decides the sign of the corresponding Krichevskii parameters. In fact, after invoking Equations (35) and (36), and according to the equations of state for light [48] and heavy water [49,50] 15, then we can reach the following conclusions, and, < 0 counterpart. We should emphasize that the agreement/disagreement observed in these datasets must be taken with caution given not only the approximated nature of the quasi-ideality approach and its inherent inconsistency as discussed above, but also the magnitude of the H/D−effect relative to the size of the uncertainties in the experimentally measured coefficients associated with the calculation of the corresponding Krichevskii parameters. In fact, we should also note that the A D 2  isothermal compressibility of the pure solvent, it becomes obvious that the sign of A i,α Kr is defined by the sign of the diverging partial molar volume of the solute,υ ∞ i , a crucial feature for the appropriate interpretation of solvation phenomena in highly compressible environments [1,61]. In fact, the observed solvent H/D−effects on the Krichevskii parameter of a solute can be interpreted in terms of the solute's ability to perturb the solvent microstruc-ture, given that lim Kr , and the pressure perturbation (∂P/∂x i ) ∞ Tρ is directly linked to the structure making/breaking parameter for a general dissociative (ν = 1 for non-dissociative) i−solute [62,63], as follows where κ o α denotes the isothermal compressibility of the pure α−solvent. Consequently, we can immediately conclude that, The significance of Equation (44) becomes evident after noting that the sign of the pressure perturbation upon solute solvation (∂P/∂x i ) ∞ Tρ has been key in the characterization of the solvation behavior of solutes in near-critical solvents, so that according to Equation (43) an i−solute behaves as a structure-maker in an α−solvent environment when the system exhibits a (∂P/∂x i ) ∞ Tρ < 0 [62], and the solute is depicted as non-volatile [2] or attractive [64]. Conversely, an i−solute behaves as a structure-breaker in an α−solvent environment when the system responds with a (∂P/∂x i ) ∞ Tρ > 0 [62], and the solute is described as volatile [2] or weakly attractive and repulsive [64] in the jargon of supercritical fluid solutions [65,66].
More importantly, from the fundamental expression given by Equation (43), we can split S ∞ iα (T, P) into its solvation (i.e., short-range local density perturbation, SR) contribution while isolating its diverging (i.e., long-range or compressibility driven, LR) contribution associated with the propagation of the density perturbation as follows [8], In Equation (45) as the corresponding isobaricisothermal residual isothermal compressibility. Therefore, from Equations (43) and (45) we immediately find the desired explicit expression for the solvation finite contribution, whose divergent compressibility-driven contribution becomes, Moreover, as demonstrated in Appendix C, the solvation and compressibility-driven contributions to the structure making/breaking parameter S ∞ iα (T, P) are related as follows, (48) tells us that the long-range contribution to the structure parameter of any real solute in an α−solvent, S ∞ iα (LR), becomes proportional to its short-range counterpart S ∞ iα (SR) through the negative value of the structure parameter of the ideal gas i−solute in the real α−solvent environment at the prevailing state conditions, S ∞,IG_i iα (T, P). Consequently, from Equations (44) and (A41) of the Appendix C, we finally arrive to the following fundamental identity, so that, the solvent effect on the Krichevskii parameter becomes, The identity in Equation (49) emphasizes that the Krichevskii parameter of an i−solute ability to perturb the α−solvent environment is simply that of the corresponding ideal gas solute IG_i prorated by (−v) times the short range (finite) contribution to the structure making/breaking parameter at critical conditions. Any increase (decrease) in the i−solute ability to perturb the α−solvent environment as a structure-making effect, S ∞ iα > 0, will translate into a more (less) negative A i,α Kr . Otherwise, any increase (decrease) in the i−solute ability to perturb the α−solvent as a structure-breaking effect, S ∞ iα < 0, will translate into a more

Final Remarks and Outlook
In this work we have discussed the solvent effect on the Krichevskii parameter of an i−solute in a k−solvent, A i,k Kr , and addressed its accurate determination when we know either (a) not only the solvation behavior of the i−solute in a j−solvent but also, its Krichevskii parameter A i,j Kr , or (b) the solvation behavior of the i−solute in both solvents as well as the Gibbs free energy of transfer of the i−solute between the two solvents. For that purpose, we first proposed a general molecular thermodynamic approach based on a Gibbs free energy cycle at standard state conditions, and then, we applied it to the determination of the H/D−isotopic substitution effect on the Krichevskii parameter of gaseous solutes in light and heavy water.
Although theoretically equivalent, the choice among the resulting Equations (8), (27) and (29) would depend on the non-trivial condition of accuracy of the available data for the Krichevskii parameter A i,j Kr of the i−solute in the reference j−solvent. Consequently, it becomes more fruitful to assess directly the solvent effect as ∆A Kr , Equations (8) and (27), and after validating the accuracy of A i,j Kr , proceed with the evaluation of Kr . The proposed scheme, developed around a fundamentally based solvation formalism of dilute solutions, identifies the links between the standard solvation Gibbs free energy of the i−solute in the two participating solvent environments and the resulting Krichevskii parameters, thorough the linear relation between the latter and the standard solvation Gibbs free energy of the solute. Additionally, it provides a novel microstructural interpretation of the solvent effect on the Krichevskii parameter through the rigorous characterization of the critical solvation as described by a finite unambiguous structure making/breaking parameter S ∞ iα (SR) of the i−solute in the pair of α−solvent environments. The molecular thermodynamic foundations of the proposed approach, combined with the involvement of accurate standard solvation properties, provide a broader and encouraging outlook on the understanding, and consequent interpretation, of the solvent effect on the Krichevskii parameter of any solute in any solvent environment.

Appendix A. Relation among Solvation Gibbs Free Energy Expressions
The standard Gibbs free energy of hydration (solvation) ∆ h G ∞ i (T, P o ) of a solute species is the isothermal-isobaric change of free energy required to transfer the species from an ideal gas environment, at P o (either 1.0 atm [29,67], or 0.1 MPa [37,38]) and temperature T, into the solvent environment to form a hypothetical solution of unit molality, i.e., [29]  Note that, whenever the Henry's law constants are given at the saturation conditions T σ , then where we assume that H IS i,j (T, P o ) ∼ = H IS i,j (T, P σ ) given the negligibly small Poynting correction under these conditions. In fact, we can rewrite (A4) as follows, after invoking Equation (21), and recognizing that f o j (T, P o )/ f o j (T, P σ ) ∼ = 1 at ambient conditions. Moreover, from (A5) and its corresponding expression for an ideal gas solute, we find that, Then, from (A4)-(A6) we obtain, Alternatively, (A5)-(A7) provide the following relations, Finally, we consider the low-pressure gas solubility scenario as described by Wilhelm et al. [39,67] i.e., where the equivalence and validity of the approximation in the (A13) has been discussed elsewhere [47] so that, we can identify the following link [45], given that the negligible pressure difference (P σ − P o ). Moreover, from (A1) and (A14), we find