Enthalpy–Entropy Compensation (EEC) Effect ... - ACS Publications

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Enthalpy−Entropy Compensation (EEC) Effect: Decisive Role of Free Energy Animesh Pan,†,§ Tanmoy Kar,† Animesh K. Rakshit,*,‡ and Satya P. Moulik*,† †

Centre for Surface Science, Department of Chemistry and ‡Indian Society for Surface Science & Technology, Department of Chemistry, Jadavpur University, Kolkata 700032, India

ABSTRACT: The “enthalpy−entropy compensation” (EEC) effect has been a long-standing fascinating yet unresolved phenomenon in chemical thermodynamics. The reasons for the observation of EEC are not clear. Various views such as empirical, extrathermodynamic, error-related, solvation, and so forth as reasons for the H/S linear correlation are floating. Statistical reasons and a hidden Carnot’s cycle (involving microscopic “heating and cooling” machines) have also been proposed recently for the observation of EEC. In this work, we have attempted a different line of approach to understand and explain the phenomenon. In the EEC treatment, the enthalpy (ΔH) and entropy (ΔS) values of “similar processes” are considered keeping aside the role of the other important thermodynamic parameter, that is, the free energy (ΔG). Considering ΔG along with ΔH and ΔS, it is established that the conventional EEC plot is not appropriate and mathematically sound. Consideration of ΔG may account for correlations of different kinds, linear, nonlinear, and so forth. Reports of non- or anticompensation phenomenon also prevail in the literature. A realistic account of the role of ΔG along with ΔH and ΔS in the understanding of such EEC correlations using authentic literature data is presented and discussed herein. EEC has several facets. Planned studies on similar systems with a wide range of ΔG values are required for realistic evaluation of the EEC and antienthalpy entropy compensation manifestations.

1. INTRODUCTION Enthalpy (ΔH)−entropy (ΔS) compensation and enthalpy of activation (ΔH#)−entropy of activation (ΔS#) compensation for equilibrium and kinetic processes, respectively, have been much discussed for over nine decades.1 Many attempts were made to explain the phenomenon. They are empirical, extrathermodynamic, error-related, involvement of microscopic thermodynamic processes (statistical reasons), solvent effect, and so forth.2−8 Publications on this issue have enriched the literature, but the situation remains complex and unresolved. It is observed that for “similar processes” (or systems) at a constant temperature, the experimentally found sets of ΔH and ΔS values when plotted against each other, on many occasions, result in linear correlations. The slope of each line is different from the experimental temperature, and it is called the compensation temperature, Tcomp. The difference between Texpt and Tcomp is a measure of the degree of the compensation effect, which is divided into three categories, low, moderate, and large. The validity of the linear correlation, for similar processes and not for “nonsimilar processes”, is intriguing. Although a © 2016 American Chemical Society

physicochemical explanation of linear enthalpy−entropy compensation (EEC) for similar processes has been attempted, it has so far been impossible for nonsimilar processes. The opinions of Krug,1 Fisicaro et al.,9 Starikov,5,7 Sharp,10 and many others regarding the H/S relation are important. They have discussed the compensation temperature,1 role of the Gibbs free energy window,9,10 CP values,9 Carnot’s cycle in microphase transition (MPT),5,7 and so forth. Although they suggest the possibility of the EEC effect, they fail to explain its thermodynamic rationality or otherwise. Piguet and co-workers proposed that at the compensation temperature (Tcomp), ΔGcomp = ΔH − Tcomp ΔS, and this “linear EEC corresponds to a ‘fourth law’ of thermodynamics”.11,12 They used different interaction models to explain linear and nonlinear EEC observed in various molecular association processes. Ford derived a nonquantum justification claiming enthalpy/entropy Received: June 12, 2016 Revised: September 3, 2016 Published: September 8, 2016 10531

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The Journal of Physical Chemistry B (H/S) compensation (Tcomp > 0) when the minimum host− guest separation distances in the associated pairs remain reasonably constant for a series of intermolecular binding events.13,14 However, the linearity proposed in the above fourth law relation does not result from Ford’s model,12 and alternative physical justifications for parabolic15,16 or rectangular-hyperbolic17 correlations for H/S compensation are proposed. In terms of molecular statistical thermodynamics, Khakhel et al.18−20 have proposed three extrathermodynamic relationships for the “compensation phenomenon” “in different series of systems” introducing the “phase volume” term, Ω, and a parameter M whose physical meaning “is unclear”.20 The relations are of the types ΔH = TΔS + RTΔ ln Ω, ΔH = T(ΔS + RΔ ln Ω) − RTΔ ln M, and ΔH = T(ΔS + RΔ ln Ω) − RTΔ ln M1/2. Their slopes are the “compensation temperature”, and the intercepts are characteristics of the system states. These relations are supported by literature data on pyrene excimers, essentially in hydrocarbon (HC) solvents.21 The above interesting proposition requires support from results of different types of equilibrium systems for its generality. We have not yet come across any such attempt. In a recent publication,2 we revisited EEC (taking literature results, both of kinetics and equilibrium origin) to show probable conditions for the occurrence of EEC in a general way. We have spelled out the importance of ΔG in the EEC study, which is normally ignored.5 Cooper et al.6 have also proposed the requirement of “limited-free-energy windows” in EEC like others.9,10,22 Many compensation models have been proposed,22 although the approaches in those models are quite different from our approach. A detailed analytical treatment of this issue remains pending. In this article, we shall deal with the dependences between ΔH and ΔS inclusively in relation to ΔG. In addition to normally observed positive compensation (linear H/S correlation with a positive slope), there are examples of nonor anticompensation (linear H/S correlation with a negative slope) as well as no-compensation (correlations with very weak or nonacceptable slope).23−25,13 Besides these, nonlinearity in H/S plots has also been observed.12,15−17 Thus, the alleged “enigma” or “conundrum” terminology of EEC may not be totally impertinent. Herein, we make a detailed discussion on H/S correlation in relation to Gibbs free energy (G), which was not done in the past.

In comparison with eq 1a, the intercept (α) and the slope (β) of the plot should have the same dimensions as that of ΔG and T (=Texpt), respectively. For α = ΔG, β = T, the EEC equation becomes equivalent to eq 1a. To comply with this, the concerned similar processes should have unique and constant ΔG and T values. Processes with varied ΔG values at a constant T in a compensation plot (if found linear with a positive slope, eq 2a) should produce an intercept (α) that ought to be some function of all of the ΔG values of the concerned processes (maybe some sort of their averages or something different), and normally Texpt < β. Thus, the magnitude of the intercept is an uncertain part of the EEC plot that requires to be justified through experiment (or concept): this also equally applies to the other parameter, the slope, β. In practice, the ΔG values in the set of similar processes considered mostly fall within a narrow or moderate range to make the intercept (with their uncertainties) fairly close to their average. For a set with a large range of ΔG, the intercept may be largely different from the average, which is subsequently demonstrated. Thus, the associated ΔG values have a striking say on the nature of EEC, which was only scarcely considered in the past. We have shown earlier that the EEC plot of both similar and nonsimilar processes with very close ΔG values obeys eq 1a,2 a worthwhile demonstration of ΔH/ΔS correlation. Varied ΔG values tend to obey eq 2a and may even deviate from it and become nonlinear.28,29 Cooper et al.6 stated that the H/S linear correlation is the result of finite CP values, the quantum confinement effect, multiple weak interactions, and the limitedfree-energy window particularly in bimolecular interactions. In here, however, the involvement of ΔG in EEC will be demonstrated in reasonable detail using literature data of reliable sources to make a conclusion on the thermodynamic and mathematical basis of this interesting phenomenon. In the opinion of Starikov and Norden,30 EEC “has a def inite physical sense”. A correlation between ΔH and ΔS means that hidden or real physical factors are present in the system. A hidden Carnot-cycle model with successive microphase transitions has been proposed by them. They further state that the “hidden” factors “are not directly measureable with the experimental setup employed, but still possessed of clear physical meaning”. By integrating the Clausius−Gibbs equation (dU = T dS − P dV, where U, P, and V are the internal energy, pressure, and volume, respectively), keeping both T and P constant, and combining with the relation by definition (H = U + PV), they found the equation, H = aS + b (where “a” is the “so-called compensation temperature” and “b” has the dimensions of energy). They suggest that this equation represents “EEC for the particular isobaric−isothermal case”. But it is not clear from the fundamentals of the above thermodynamic derivation and the related statement on how the H/S correlation applies only to EEC, which is essentially constituted of “similar thermodynamic processes” and not of “nonsimilar processes”.2,27 In a previous report,2 we derived an equation from the fundamental thermodynamic relation, dS = dQ/T = dH/T (at constant pressure), which on integration produces, H = TS + C (a constant). At constant T, we get (H1 − H2) = T (S1 − S2) + (C1 − C2) or ΔH = z + T ΔS, where z = (C1 − C2). This is a general equation. Let us now examine the nature of these two equations (shown as eqs 3 and 4 below) along with eqs 1 and 2 (replacements of eq 1a and eq 2a for convenience)

2. BASIC INFORMATION From the premises of the first and second laws of thermodynamics, ΔG, ΔH, ΔS, and T are related by the equation ΔH = ΔG + T ΔS

(1a)

The experimentally determined ΔG values in a process can produce ΔH from its temperature coefficient (according to the van’t Hoff rationale), and then ΔS using eq 1a. It is also possible to compute ΔS from the temperature variation of ΔG. Directly determined ΔH by calorimetry may be used to get ΔS of the process (the results would expectedly be more accurate than that by the van’t Hoff method). Adopting these procedures in different similar processes at a constant T then produces different sets of ΔG, ΔH, and ΔS, from which the plot of ΔH against ΔS, that is, EEC, may sometimes appear as a linear relation (eq 2a).2,25−27 Thus ΔH = α + β ΔS

ΔH = ΔG + T ΔS (a well‐known thermodynamic relation)

(2a)

(1) 10532

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The Journal of Physical Chemistry B ΔH = α + β ΔS2,4

(2)

ΔH = b + aΔS 30

(3)

ΔH = z + T ΔS2

(4)

In eq 1, all of the terms are defined. At a known temperature, T, knowing the ΔG and ΔH of a process helps in the derivation of ΔS. Thus, T is the temperature at which the thermodynamic parameters are realized, and it is not the compensation temperature. Equations 2−4 have the same meaning, although eq 2 is empirically proposed, and eqs 3 and 4 are thermodynamically obtained, and may represent compensation processes with slopes and intercepts denoting the “compensation temperature” and the related “free-energy changes”. A linear plot between ΔH and ΔS with variable ΔG values (EEC phenomenon) is not in accordance with eq 1 but with eqs 2−4. However, it remains unclear how the thermodynamically derived eqs 3 and 4 can represent EEC that arises only for “similar processes”, and not for “dissimilar processes”. A hidden Carnot-cycle condition invoking MPTs and so forth is considered to explain the EEC effect,5 but the estimate and explanation of both the slope and intercept remain pending. The authors30 state that “a valid, nontrivial EEC is a necessary and suf f icient condition for the existence of a hidden thermodynamic cycle”. Then, the question of conditions for nontriviality of EEC obviously arises. However, the concept of MPTs for the interpretation of the observed EEC correlations is an advancement made in this field. But until convincing establishment of MPT in the thermodynamic domain, the scope of examining and analyzing the EEC effect in new light remains open. It should also be noted that the thermodynamic data are macrodata and to grasp that through MPT and the hidden Carnot-cycle mechanism is rather difficult, if not impossible. In this article, we address the problem in a rational perspective different from the conventional line/procedure. The importance of ΔG of the process on the EEC will be shown as a requirement to clarify some of its fundamental manifestations. So far, the observations and explanations of the phenomenon have remained essentially arbitrary and restricted to “limited-free-energy windows”. We show how narrow and broad free-energy windows can affect EEC presentations from an apparently linear to a curved geometry. Its mathematical (geometrical) relevance and consequence are also discussed. Combined plots of ΔG−ΔH−ΔS are presented here, which is a new and novel type of display for better revelation of systemspecific EEC phenomena.

Figure 1. ΔH vs ΔS plot of nonsimilar processes at nearly constant ΔG (7.90−8.21 kcal mol−1) at 298 K. Systems: blue square (Table 131 and Table 232); red full circle (Ben-Naim et al.33); pink triangle (Table 134).

similar and nonsimilar processes. It is displayed here as a representation of the unique feature of constancy of ΔG in the manifestation of enthalpy−entropy correlation of dissimilar chemical-equilibrium processes.2,27 To recognize the importance of the “free-energy window”, in Figure 2, and in the subsequent EEC plots, the locations of the free-energy values are shown on the ordinate (for better and comprehensive revelation of the effect in terms of ΔG). Figure 2A illustrates the compensation plots of different nonsimilar processes with varied ΔG values. Here, the associated ΔG values at ΔS = 0 are shown on the Y-axis. As the temperature T (298 K) is constant, parallel lines join the ΔGs with the (ΔH, ΔS) points (the apex points) in the two-dimensional space. The nature of correlation between ΔH and ΔS is clearly understood. It is nonsystematic and scattered. This is caused by the presence of nonsimilar processes with varied ΔG (−14.2 to −49.05 kcal mol−1). As expected, the linear or near-linear EEC effect is absent. The tested linear correlation is very poor (correlation coefficient = 0.2785), the slope, Tcomp, is 0.1046 K, and the intercept, α, is −0.874, whereas the average ΔG = −4.85 kcal mol−1: all parameter values are very unusual. The free-energy differences as well as the nonsimilarity of the processes considered produce the non-EEC phenomenon. In Figure 2B, an EEC plot at a constant temperature (298 K) for similar processes (thermodynamics of solvation of different amines) with varied ΔG values in the range of −4.3 to −3.04 kcal mol−1 is presented. The pattern of EEC is found to be nonlinear (although an apparent linear regression is tested, like that in many similar findings). This is an example of curved EEC for ΔG values in a small range (−4.3 to −3.04). Its correlation coefficient (0.9996) is much better than that of the linear regression (0.9803); Tcomp (262 K) < Texpt (298 K). The above results show that the use of ΔG in the EEC plot has a decisive role to better reveal the nature of the plotting process. The intercept, α = −5.199 < average ΔG = −3.67 kcal mol−1. In Figure 3A, results of Seldeen et al.31 on the binding of the TRE duplex to wild-type truncated constructs of bZIP of Jun are presented. Fair linearity (correlation coefficient = 0.9964) of the data is observed for a small range of ΔG (−6.60 to − 7.80 kcal mol−1): Tcomp = 291 K < Texpt = 298 K, and the intercept (α) = −8.91 < average ΔG = −7.2 kcal mol−1. Figure 3B

3. RESULTS Thermodynamic data treated herein are all collected from the literature with proper acknowledgment. The graphical illustrations are all newly constructed; no published figures and tables are reproduced. In Figure 1, we demonstrate an EEC plot of both similar and nonsimilar equilibrium processes at nearly constant ΔG values (∼8.0 kcal mol−1). The plot is excellently linear, with Tcomp (298.3 K; slope) equal to Texpt (298.15 K), obeying the requirements of eq 1. The requirement of constancy of ΔG in support of eq 1 was indicated in the literature.10 The ΔH versus TΔS plot in the inset is also shown for comparison. The slope of the line is 1.0008, which is very close to the expected value of unity. The intercepts in both cases have the magnitude −7.997 kcal mol−1. The constancy of ΔG cannot differentiate between 10533

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Figure 2. (A) Black, red (Jalal et al.35); blue, cyan (Graziano et al.36); pink, yellow (Seldeen et al.31); navy blue (Shi et al.37); wine red, orange (BenNaim and Marcus,33 Table 8); green (Ben-Naim and Marcus,33 Table 1). (B) Ben-Naim and Marcus,33 Table 1: black squareammonia, red circlemethyl amine, pink triangleethyl amine, blue down trianglepropyl amine, cyan diamondbutyl amine, dark green stardiethyl amine, orange pentagontriethyl amine.

in the range −3.81 to −10.88 kcal mol−1, the correlation coefficient = 0.9951, the intercept (α) = −1.35 > average ΔG = −7.39 kcal mol−1, and the slope Tcomp = 555.7 K ≫ Texpt = 298 K. The difference between Tcomp and Texpt is phenomenal. The EEC results of condensation of alkenes are illustrated in Figure 4B (ΔG range = −3.30 to −6.52 kcal mol−1). Again the plot is concave. Linear regression shows that the correlation coefficient = 0.9852, the intercept = −0.487 > average ΔG = −4.91 kcal mol−1, and the slope = 563.6 K ≫ Texpt = 298.15 K. Nonlinearity and a high Tcomp value are again found. The results of alkanols (layer C) and alkylbenzenes (layer D) are also nonlinear (C is convex, and D is concave). The parameter values are not presented here; they are listed in Table 1 with all other systems tested here. A systematic deviation from linear EEC with abnormal Tcomp values in the plots is noteworthy. Linear regression of the presented EEC results in the table shows that the intercept values of eq 2 are within the freeenergy ranges of the first four systems; those of the last four systems are below the range. Except for the first four systems, the Tcomp values of all of the systems appear exceptional. The Tcomp value of the first system is, on the other hand, lower than Texpt (an infrequent observation). Thus, the spread in the ΔG values has a definite say on the nature of the EEC correlation. Small sections of the plots in C and D may show linearity with small scatter. We may, therefore, state that inclusion of ΔG in the graphical EEC depiction can make the analysis simple and clear: a fairly large ΔG range can demonstrate nonlinearity, and it may also be manifested in a low range of ΔG (Figure 2B).

demonstrates the EEC course of binding of biological ligands to dsDNA oligos containing wild-type nucleotide variants (Seldeen et al.,31 Table 1), wherein 25 experimental points are treated with their corresponding ΔG values. Close data spreading is again observed for the ΔG values of −7.61 to −8.96 kcal mol−1. A linear regression produces a correlation coefficient = 0.9838, the intercept = −7.475 kcal mol−1 > average ΔG = −8.29 kcal mol−1, and Tcomp = 305 K > T expt = 298 K. Somewhat scattered points show weak EEC manifestation. In Figure 3C,D, two sets of results, C for Ca 2+ binding to mutant human lysozyme (Kuroki et al.38) and D for peptide binding to MHCII (Ferrante et al.,39 Table 1), are presented. In C, the ΔG range is −5.9 to −13 kcal mol−1, which for D is −17.6 to −43.3 kcal mol−1. The linear correlations of both are worse than the others. Thus, the spread in the ΔG values has a definite say on the EEC correlation. It seems that the greater the spread of ΔG values the worse the EEC correlations. Small sections of the plots in C and D may show linearity with small scatter. The former (with all of the scattered points, plot not shown) has a correlation coefficient = 0.9563, intercept (α) = −8.98 > average ΔG = −9.5 kcal mol−1, and Tcomp = 313 K > Texpt = 303 K, whereas the latter (see inset) has correlation coefficient = 0.9431, intercept (α) = −24.42 > average ΔG = −30.40 kcal mol−1, and Tcomp = 837.1 K ≫ Texpt = 310 K. Although some small sections of similar results may show fair linearity with better correlations, in reality, such correlations are not demonstrated. The importance of the spread of ΔG values in the formation of EEC is hence considered useful. It is ironical that such results are taken as examples in the literature in support of linear EEC.31,32 The nature of EEC correlations for reliable data of Ben-Naim and Marcus33 on the solvation thermodynamics of vapors of organic compounds condensing in their own liquids at 298 K is herein presented as a special demonstration. The systems considered are alkanes, alkenes, alkanols, and alkylbenzenes; their EEC results are displayed in Figure 4A−D. The courses are all nonlinear (three concave and one convex). Conventional linear regression results are shown for a rough comparative estimate. In the layer A of the figure, ΔG values for alkanes vary

4. DISCUSSION 4.1. EEC Features. The above presented and described ΔH/ΔS correlation diagrams in relation to ΔG have the following features: (1) If ΔGs of the treated processes (similar or dissimilar) at a constant experimental temperature are very close or constant, then the EEC correlation is linear with slope Tcomp equal to Texpt (Figure 1).2 (2) If ΔGs of the treated processes are unequal, then for dissimilar processes at a constant temperature, EEC 10534

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Figure 3. EEC plots for different types of binding interactions at 37 °C. (A) TRE duplex binding to truncated constructs of bZIP of Jun (Table 232). (B) binding of bZIP domains of Jun-Fos heterodimers to dsDNA oligos containing wild-type nucleotide variants (Seldeen et al.,31 Table 1). (C) Ca2+ binding to mutant human lysozyme (Kuroki et al.,38 Table 2). (D) Peptide binding to MHCII (Ferranate et al.,39 Table 1).

lowest ΔG and is arbitrarily extended to some length beyond the apex point (ΔH, ΔS), marked with a closed circle. The remaining five processes with five different values of the freeenergy changes (ΔG1, ΔG2, ΔG3, ΔG4, and ΔG5) should also produce five different lines parallel to line 1 with five different apex points, (ΔH1, ΔS1), (ΔH2, ΔS2), (ΔH3, ΔS3), and so forth, in the two-dimensional space, consecutively marked as points 1, 2, 3, 4, and 5. The joining of the apex points produces the linear EEC line with a slope, Tcomp, greater than Texpt in compliance with eq 2. According to the mathematical rule of straight lines, the apexes of the parallel lines with varied intercepts (the ΔGs) must deviate from line 1. The connected apexes should have a profile with greater inclination (than line 1) depending on the nature of the individual equilibrium processes considered in the EEC strategy. The observed profile may be linear or nonlinear (if linear it conforms to eq 2). For ΔG1 = ΔG2 = ΔG3 = ΔG4 = ΔG5 = ΔG, the parallel lines should merge with line 1 (Figure 5A), manifesting exact compensation (cf. Figure 1). In Figure 5B, the points are nonsystematic and only roughly linear (this may also be a practical option). In layer C, the course is nonlinear (concave, cf. Figure 4A,B,D; a convex course is also feasible cf. Figure 4C,

correlations with scattered points (Figure 2A) with poor or no real correlation occur.2 For similar processes, either weakly linear EEC correlations with scattered points (Figure 3C,D) or curved correlations with nonscattered and weakly scattered points (Figures 2B and 4C) may result. Their apparent linear correlations may show very large Tcomp values, with intercepts, ΔGs, significantly different from their averages. (3) The nonlinear (curved) EEC profiles originate for processes with wide ranges of ΔGs. In the narrow range, close data points make the correlation apparently linear (with incorrect analysis and conclusion). (4) A physicochemical meaning of the EEC is important and probably imperative. It is possible to offer some such meaning when similar type of systems are considered but certainly impossible for dissimilar type of systems even when ΔG values are very near to each other. For such systems, Tcom has no meaning. Let us now consider a geometrical perception of EEC at a constant temperature Texpt with reference to Figure 5A−D. In layer A, we represent an arbitrary EEC profile of six similar equilibrium processes. The straight line 1 is according to the 10535

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Figure 4. Solvation thermodynamics of nonionic solutes at 298.15 K. The solvation of vapors of (A) alkanes, (B) alkenes, (C) alkanols, and (D) alkylbenzenes condensing into their own liquids.33

Table 1. Presentation of Different EEC-Related Parameters system solvation of amines binding of TRE duplex to domains Jun-Fos binding of Jun-Fos hetero dimers to dsDNA oligos Ca2+ binding to mutant human lysozyme peptide binding to MHCII solvation of alkanes solvation of alkenes solvation of alkanols solvation of alkylbenzenes a

−ΔG (range) (kcal mol−1)

average (−ΔG) (kcal mol−1)

intercept (−ΔG) (kcal mol−1)

correlation coefficienta

Tcomp (Texpt) (K)

ref

3.04−4.3 6.6−7.8 7.61−9.0

3.67 7.2 8.29

5.19 8.9 7.48

0.9803 0.9964 0.9838

262 (298) 291 (290) 305 (298)

33 32 31

5.9−13.0 17.67−43.5 3.81−10.88 3.30−6.52 4.86−10.9 4.56−9.54

9.5 30.4 7.39 4.91 7.05 7.8

8.98 24.4 1.35 0.487 1.478 2.0

0.9563 0.9431 0.9951 0.9852 0.9686 0.9964

313 837 556 564 502 568

38 39 33 33 33 33

(303) (310) (298) (298) (298) (298)

Coefficients of linear regression.

C) are observed in practice (cf. Figures 1−4), and there may be other forms of EEC, which is hard to predict at present; they are expected to be case specific. We may add that as the intercepts and the line inclinations of the linear EEC plots so far remain nonexplainable, the EEC may also be called a “phantom effect”.5 In light of the usual data plotting procedure without cognizance of the related ΔG, the EEC profiles in a small

another practical option), guided by the thermodynamic nature of the involved processes. And the merging phenomenon of all of them with line 1 should occur following the arguments same as those proposed for layer A. Here lies the difference between eqs 1 and 2 (what we call the EEC effect). Thus, we see that the inequality between Tcomp and Texpt for a linear EEC is geometrically obvious, whereas the notion that EEC leads to linearity is not (it is conditional). The illustrated situations (A− 10536

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Figure 5. Probable ΔH−ΔS profiles of different types at a constant temperature with reference to associated ΔG values as intercepts at ΔS = 0 (scales arbitrary). A and B: Line 1, according to eq 1; Lines with points, according to eq 2; C: Line 1, according to eq 1; curve with points (nonlinear EEC), fits to the polynomial relation; D: Line 1, according to eq 1; Line with a negative slope, demonstration of AEEC (anti-EEC).

range of ΔG may look linear, and their slopes (Tcomp) are apparently revealed as a new finding. However, large (and some instances small) range of ΔG may make EEC nonlinear or curved. The relevance of ΔG in the EEC process has been discussed by Sharp,10 Liu and Guo,40 and us.2,27 From Figure 5A−C, it is obvious that Tcomp > Texpt . On limited occasions, Tcomp < Texpt may also be found. According to Benito-Perez,41 this can arise in kinetic EEC from measurement errors; a test of this proposition for equilibrium EEC is required. Of course, the rationale put forward in constructing Figure 5 does not support scope for lower inclination of the EEC plot than the slope of line 1; thus, Tcomp < Texpt is geometrically not permitted, and measurement errors would hardly lead to such a situation. In Figure 5, layer D presents a linear EEC with a negative slope, that is, AEEC in practice is not unlikely.13,23−25 In EEC with a positive slope, ΔH and ΔS are directly proportional; they are inversely proportional in AEEC to produce a negative slope. We have recently observed such a phenomenon in the formation of amphiphile micelles.27 It has been proposed that hydration has a say on the phenomenon.23,24 In micelle formation, hydrophobic hydration has been considered responsible for AEEC.27 It may be mentioned that the AEEC course may also be nonlinear, and the nature of the dealing processes should determine the AEEC pattern. This anti-EEC phenomenon warrants intimate and detailed exploration. 4.2. Overall Perspective. In this work, the EEC phenomenon is examined and explained on the basis of ΔG window. Although the effect of ΔG window on EEC has been mentioned in some earlier works, 6,7,9,10,22 no detailed discussion has come to our notice. Geometrical reasons for linear correlation are considered applicable. Careful data analysis has shown that the linearity is apparent, error

supported, and nonlinearity is not unexpected. The genesis of this idea (i.e., required stress on ΔG window in understanding EEC) is based on the consideration of all of the four thermodynamic parameters, ΔG, ΔH, ΔS, and T in the analysis. The neglect of ΔG makes EEC conspicuous; its inclusion helps in unambiguously realizing variations in EEC correlations. We have presented processes with Tcomp values closer to as well as much greater and smaller than Texpt (see Table 1). EEC at a constant ΔG producing Tcomp = Texpt for similar and nonsimilar processes (Figure 1) suggests that the inequalities between T comp and T expt arise from the presence of nonequal ΔGs of similar systems. The observed ΔH/ΔS correlations may be scattered, concave, and convex types (cf. Figures 3 and 4). These variations introduce nongenerality of the linear EEC process, and its manifestations, therefore, are conspicuous. The only generality it may show is linearity of the ΔH/ΔS plots with ΔGs covering a small range with their associated errors, bearing in mind that a curved profile can be a description of combinations of a number of linear courses of varied slopes. In other words, a curved profile seems to be more general and in that case the slopes are different at different points of the curve. Hence, the value of Tcomp becomes somewhat questionable. EEC results with large ranges of ΔGs are not much prevalent in the literature, and, therefore, apparent linear findings are mostly talked and discussed about. In Figure 5A−D, the possible deviations of different kinds from the constant ΔG course (line 1) are presented. In them, A and B stand for linear EEC (eq 2). Merging of the parallel ΔG rooted lines with line 1 means to do away with EEC and obey the thermodynamic equation (eq 1). The deviations in C and D from the expected EEC courses are nonsupportive of the mathematical requirements to generate eq 2. Starikov and 10537

DOI: 10.1021/acs.jpcb.6b05890 J. Phys. Chem. B 2016, 120, 10531−10539

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The Journal of Physical Chemistry B Norden5 have attempted to explain the EEC process considering a hidden Carnot’s cycle, wherein a MPT plays a crucial role in rationalizing EEC in relation to hydration and protein folding, functioning of molecular motors, and similar phenomena. However, generality of the Carnot cycle and associated MPT covering all EEC-exhibiting processes is yet to be analyzed for a convincing decision. At this point, let us discuss more on eq 2. For EEC42 with α ≠ 0, the process involves a cyclic reversible transition. The factor, α (≡ΔG), therefore, has a say on the EEC. When α = 0, the compensation between ΔH and TΔS becomes exact (slope = 1), otherwise it is inexact (slope > 1). An example in support can be the results of heating different pure solids with increasing temperature between 0 and 298 K,43 wherein exact compensation behavior with Tcomp = Tave is found. In the treatment, the expansions of the solids are taken to be negligible, making α or ΔG (the useful work) = 0 (in reality, the works are very small and practically emerge as a constant). The results agree with the illustration in Figure 1 for a constant ΔG ≈ 8.0 kcal mol−1. For variable ΔGs, EEC plots with linear correlations evidence Tcomp > Texpt. Therefore, ΔG has a controlling role in describing the nature of EEC. We may mention that a single, unidirectional, reversible phase transition can result5 under the condition ΔG = 0. In Figure 5 D, the EEC pattern is the reverse of what is shown in A−C. In there, the axes values are directly proportional (ΔH ∞ ΔS), in line with normal or expected findings, whereas in D, it is inversely proportional (ΔH ∞ 1/ ΔS), that is, a reverse finding. Such a phenomenon is called anti-EEC (AEEC).27 System-specific compensation behavior is further shown. This phenomenon is also reported in the literature;13,23−25,44 it is termed “noncompensation” for what we call “anticompensation”. According to Liu and Guo,40 “noncompensation behavior is very intriguing, but no theory has been proposed to explain it”; of course statistical mechanical theory of hydration is used for explanation. Graziano24 supports it by the reorganization of H-bonds by interaction of solutes in water. AEEC is recently discussed with respect to micelle formation in terms of hydrophobic hydration,27 that is, the formation of “ice-bergs” or “flickering clusters” surrounding the amphiphile hydrophobic tails. Ford13 from statistical mechanical analysis of bimolecular association of haloalkanes, ketones, alkanols, amines, and so forth in gas phase concludes the formation of different modes of compensation, either positive or negative ΔH−ΔS correlations; also there may be weak correlation in either way. Ryde45 has shown from theoretical calculations that ligand−receptor interaction may produce EEC in gas phase though that disappears in the presence of solvent. The molecular model of hydration predicts the importance of the attractive energy between solute and water resulting in anticompensation.23,24 For an intermolecular association process, Piguet also reports anticompensation arising from changing (decreasing) contact distances between the associated molecular pairs in solution.12 Clear evidence of AEEC for complexation of metal ions with amines in aqueous medium is also reported.44

2.

3. 4.

5.

6.

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and dissimilar) leading to an exact compensation (Tcomp = Texpt), otherwise the compensation becomes inexact and arbitrary. For similar systems, a narrow range of ΔG may also result in an apparent linear EEC plot with Tcomp ≠ Texpt. For a broad range, EEC may appear nonlinear (concave, convex, or a combination of both). In here Tcomp loses its importance. Nonsimilar processes at a constant temperature with varied ΔG produce randomly scattered points with no real correlation. So far EEC remains to be empirical although statistical mechanical treatment has made some advances in recent years. ΔH/ΔS correlation with variable ΔGs producing gradient Tcomp > Texpt is a geometrical consequence; its linearity is nonwarranted. Misconceptions of EEC are addressed. Consideration of a narrow free-energy window has been an oversimplified manipulative procedure to exemplify EEC. Analysis with proper cognizance of ΔG is required. A new and novel plotting procedure (ΔG−ΔH−ΔS correlation) has been used to make clear the visibility of nonlinearity in EEC; otherwise, we always eye for an apparent linear course, neglecting the inexactness in the pretext of measurement errors. Like EEC with a positive slope, EEC with a negative slope (anti-EEC or AEEC) is also found. Very weak EECs of both kinds are also possible to be designated as non-EEC. In addition to molecular assembly formation, hydration (solvation) and solute−solvent interaction are considered responsible for their occurrence. In AEEC, ΔH and ΔS are inversely correlated, whereas in EEC, the correlation is direct.

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (A.K.R.). *E-mail: [email protected]. Phone: +91-33-2414-6411. Fax: +91-33-2414-6266 (S.P.M.). Present Address §

Novozymes South Asia Pvt. Ltd., Whitefield, Bangalore 560066, India (A.P.).

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS A.P. thanks the Center for Surface Science (CSS), Department of Chemistry, Jadavpur University (JU), for granting Research Assistantship. T.K. thanks DSK (UGC), Govt. of India, for Post-Doctoral Research Fellowship at CSS (JU). A.K.R. thanks AICTE, New Delhi, for a former Emeritus Fellow position. S.P.M. appreciates the support from both Indian National Science Academy and Jadavpur University for Honorary Scientist position and Emeritus Professorship, respectively.

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5. CONCLUSIONS Pointwise conclusions are presented in what follows: 1. Like ΔH and ΔS, ΔG is an essential component of EEC. At constant ΔG, the thermodynamic equation (eq 1) is valid for all types of physical−chemical processes (similar

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DOI: 10.1021/acs.jpcb.6b05890 J. Phys. Chem. B 2016, 120, 10531−10539

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DOI: 10.1021/acs.jpcb.6b05890 J. Phys. Chem. B 2016, 120, 10531−10539