Enthalpy–Entropy Compensation (EEC) Effect: A Revisit - The Journal

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Enthalpy−Entropy Compensation (EEC) Effect: A Revisit Animesh Pan,† Tapas Biswas,†,§ Animesh K. Rakshit,*,‡ and Satya P. Moulik*,† †

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

‡

ABSTRACT: A short account of the developments and perspectives of IKR (iso-kinetic relation) and EEC (enthalpy (H) − entropy (S) compensation) has been presented. The IKR and EEC are known to be extra thermodynamic or empirical correlations though linear H−S correlation can be thermodynamically deduced. Attempt has also been made to explain the phenomena in terms of statistical thermodynamics. In this study, we have briefly revisited the fundamentals of both IKR and EEC from kinetic and thermodynamic grounds. A detailed revisit of the EEC phenomenon on varied kinetic and equilibrium processes has been also presented. Possible correlations among the free energy (ΔG), enthalpy (ΔH), and entropy (ΔS) changes of different similar and nonsimilar chemical processes under varied conditions have been discussed with possible future projections.

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INTRODUCTION The well-known thermodynamic relation (1)

ΔH = ΔG + T ΔS

applies to all chemical equilibrium processes. Interestingly, it has also been empirically found that for some closely related processes the enthalpy and entropy changes are often linearly related by the following relation

ΔH = α + β ΔS

Figure 1. Plots of thermodynamic parameters in arbitrary scale: (A) Compensation plot. (B) Iso-kinetic plot with a crossing point (i) and no crossing (ii).

(2)

where α and β are constants, and such phenomenon is known as “enthalpy−entropy compensation” (EEC). It has been further observed that for a series of similar chemical kinetic processes the following relations are also valid E Rb

ln A = a +

It should be noted here that there are examples where both EEC and IKR occur or only one of them appears.2−4 (cf. Figure 1B). Two excellent reviews5,6 were published in 1990s dealing with isokinetic relationship and their effect in undertaking the mechanistic and structural investigation. The heat−bath theory, which was said to be responsible for the IKR was discussed, too. The presence of IKR or EEC or both has been observed in various fields such as micellization,7−10 microemulsion,11 Langmuir monolayer,12 solution thermodynamics,13 food chemistry,14 and others. Various names of these effects like Meyer−Neldel effect,15 θ-rule,16 and so on are also used in literature. It is known that out of the three thermodynamic parameters ΔG, ΔH, and ΔS in eq 1 only ΔG and ΔH are independently determinable, but ΔS cannot be so determined. Thus, for a series of systems, similar or otherwise, which have the same ΔG, the changes in the ΔH will be compensated for by the TΔS quantity. Such “compensation of entropy and

(3)

ΔH # = c + dΔS #

(4)

where a, b, c, and d are constants, E is the activation energy, R is the gas constant, and A is the pre exponential factor of the wellknown Arrhenius equation in chemical kinetics k (rate constant) = A exp(−E /RT ) #

(5)

#

The terms ΔH and ΔS are the process enthalpy and entropy of activation, respectively. Equations 3 and 4 are known as iso-kinetic relations (IKRs), which was first observed and proposed by Constable.1 In general, the IKR and the EEC are used interchangeably, although it is known that some differences exist between the two. In the case of EEC, the systems follow eq 2 (Figure 1A), whereas for IKR to be true ΔG# versus T plots should produce a series of lines intersecting at a particular temperature called the iso-kinetic temperature (Figure 1B (i)). © 2015 American Chemical Society

Received: October 10, 2015 Revised: November 25, 2015 Published: December 7, 2015 15876

DOI: 10.1021/acs.jpcb.5b09925 J. Phys. Chem. B 2015, 119, 15876−15884

Article

The Journal of Physical Chemistry B

ΔS of a set of similar equilibrium processes are plotted. Both are linear. In the insets of all of the Figures ΔH versus TΔS plots are shown as complementary demonstrations. The compensation effect is an apparent phenomenon found in many processes; in some processes it may be absent. From eq 1 it is obvious that for various similar as well as dissimilar processes when their ΔG values have the same value at a constant experimental temperature then the linear plot between ΔH versus ΔS will have the slope equal to the experimental temperature. This slope, called the compensation temperature, is obviously equal to the experimental temperature; however, it should be noted that, in general, the “compensation temperature” (Tcomp) (spoken of only for similar type of systems and not for dissimilar types) may be different from the experimental temperature (Texpt). It is also to be noted that the term β in eq 2 is equivalent to Tcomp. Starikov et al.29 have stated that the EEC is well known, “although the physical roots of it are still not completely understood”. They stated that “it may be rationalized in terms of hidden but physically real factors implying a Carnot-cycle model in which a microphase transition (MPT) plays a crucial role.” Furthermore it has been shown that EEC in a general format is not inimical to conventional thermodynamics. The statistical thermodynamics has been used in an attempt to explain it. Thus, EEC is not an obvious thermodynamical phenomenon but is observed only under specific mathematically defined equilibrium conditions. It has also been stated30 that entropy−enthalpy compensation is a natural consequence of finite ΔCp values and is in general the result of quantum confinement effects, multiple weak interactions, and limited free-energy windows. In a 2013 publication, Starikov31 opined that with EEC there is “still a lot of difficultiesin particular, those with conclusively elucidating the entropy notion’s exact meaningits physical−chemical sense”. The above statement invokes that the statistical thermodynamical explanation of EEC still has some limitations. The compensation between ΔH and ΔS is easy to rationalize when the free energy changes of the similar systems are same or almost same. The problem arises in explaining the linear ΔH−ΔS plot when the associated ΔG is not constant. It has been shown32,33 that free energies of ionization of substituted phenol in gas phase is fairly linearly related to the free energies of solution of the substituted phenols; that is, ΔG (ionization, gas phase) is proportional to ΔG (ionization in solution phase). Following Leffler34,35 one can write

enthalpy within the (fixed) Gibbs energy change” is ubiquitous.17 The reality of EEC and/or IKR has been questioned by many researchers. Most often it has been suggested that they are artifacts that arise due to statistical reasons, that is, of mathematical origin, and no chemical reason needs be associated with it;18 however, it has been stated that “the pattern is real, very common and a consequence of the properties of liquid water as a solvent regardless of the solutes and the solute processes studied”.19 Furthermore, it has been claimed that “kinetic compensation and iso-kinetic relations are real phenomena, whose underlying origins are understood”.20 It has been proposed21 that for small values of activation energy, which cannot cross the reaction threshold limit, there is no IKR. High ΔH# and ΔS# are responsible for IKR. Statistical “Ftest” should be used to provide probability support for the presence of IKR. Low probability means absence of IKR. Barrie22 has also conceded to this. In general though, the experimental errors are expected to be relatively more in kinetic studies rather than in equilibrium studies. In a recent paper Perez-Benito23 from statistical analysis has shown that kinetic enthalpy−entropy compensation with T comp < Texpt is experimental-error-based; those with Tcomp > Texpt are “real and meaningful”. In the following, we shall essentially present and discuss some details of EEC. The empirical “Linear Free Energy Relation” (LFER), for example, Hammett equation,24,25 and the empirical EEC are related to each other. The LFER has been accepted as an axiom and is better understood, and there have been attempts to explain EEC on the basis of LFER. Hammett was the first to state, “The idea that there is some sort of relationship between the rate of a reaction and the equilibrium constant is one of the most persistently held and at the same time most emphatically denied concepts in chemical theory.”26 He showed that for a reaction between a series of methyl esters with NMe3, the rates of the reactions were directly related to the ionization constants of the corresponding carboxylic acids in water. Thus, a plot of logarithm of the reaction rate constant (log k) with logarithm of the ionization constant (log K) produced a reasonably good straight line. Because the log k is related to the activation freeenergy change and log K is related to the free-energy change for the ionization of the carboxylic acids in water, the above straight line relation is termed the “linear free energy relation” (LFER).27 An implicit meaning of the linear Hammett plot is that ΔH and ΔS are linearly related in a series, as shown in Figure 2. In Figure 2A, the ΔH# and ΔS# of a similar set of kinetic processes are plotted, whereas in Figure 2B the ΔH and

Δ(ΔG)(solution) = β Δ(ΔG)(ionization)

(6)

where β is a constant. Now Δ(ΔG1)(solution) = Δ(ΔH )(solution) − T1Δ(ΔS)(solution)

(7)

Δ(ΔG2)(solution) = Δ(ΔH )(solution) − T2Δ(ΔS)(solution)

(8)

Therefore Figure 2. (A) Sen Gupta et al.28 on kinetic results of oxidation of aldoses by V2+ ion; free-energy range 89.8 to 95.9 kJ mol−1. (B) Jalal et al.12on adsorption of spreading of carboxylic acids at air/water interface; free-energy range −1.34 to +0.471 kcal mol−1.

Δ(ΔS)(solution) = (Δ(ΔG1)(solution) − Δ(ΔG2) (solution))/(T2 − T1) 15877

(9)

DOI: 10.1021/acs.jpcb.5b09925 J. Phys. Chem. B 2015, 119, 15876−15884

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The Journal of Physical Chemistry B

that “for homologous series of molecules with repeated interactions studied in vacuum, EEC is a rule. However, if water molecules are added, the relation is blurred and it can be predicted that for a real binding reaction in water solution, both enthalpy−entropy compensation and anticompensation can be observed.”

Δ(ΔS)(solution) = [(β1 − β2)/(T2 − T1)]Δ(ΔG) (ionization)

Δ(ΔS)(solution) = δ Δ(ΔG)(ionization)

(10) (11)

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Now if δ ≠ 0, then Δ(ΔG)(solution) = β1Δ(ΔS)(solution)/δ

OBJECTIVE The EEC has been a matter of concern for a long time, although a real solution or understanding of the process still remains unknown. It favorably applies to similar systems and restrictedly to nonsimilar systems. The range of ΔG changes associated with the systems treated has a positive say on the EEC process. It is neither thermodynamic and nor nonthermodynamic (therefore, extra thermodynamic). How the correlation deviates from the thermodynamic rule (eq 1) can be measured from T comp − T expt values. Effectively, four thermodynamic quantities, ΔG, ΔH, ΔS, and T, describe a process, so there may arise several correlative combinations. In EEC we consider only ΔH, ΔS, and the T, keeping ΔG in the background. The EEC at constant ΔG but different temperatures, the EEC with different ΔG and different temperatures, and likes could be looked into also for a broad understanding of the thermodynamic perspective. There is also a requirement of direct determined ΔH in a calorimeter and its use in the EEC for a better and more revealing knowledge of the process. These constitute the basic premise of the present study. Chodera and Mobley45 have recently discussed this aspect. We have comprehensively addressed these points to understand the EEC and its related perspectives. We are revisiting the EEC phenomenon from the nonstressed angles and concept to make the issue fundamentally more clear and acceptable to the concerned minds. Thermodynamically derived or measured parameters showing extra-thermodynamic or empirical correlations should not qualify for an axiomatic acceptance, although LFER has been so; a clear and critical justification of the phenomenon on fundamental or experimental ground is warranted. We may herein show that a linear relation between ΔH and ΔS can be found from fundamental consideration, that is, dS = dQ/T = dH/T at constant pressure. On integration, we get, H = TS + λ (a constant). For constant T with λ as a function of the system, we get the relation, H1 − H2 = T (S1 − S2) + (λ1 − λ2) or ΔH = TΔS + Z (constant having the same unit as H and TS, that is, kJ mol−1). Comparing this relation with eqs 1 and 2, we get for (1) Z = ΔG and for (2) β = T and Z = α. Consequently, α is related to the free-energy changes of the chemical processes considered for EEC demonstration. A similar type of derivation has also been previously shown by Starikov.46 Experimentally, an excellent H−S relation has been also observed for different types of pure solids.47

(12)

Therefore, we can write β1Δ(ΔS)(solution)/δ + T1Δ(ΔS)(solution) = Δ(ΔH )(solution)

(13)

Δ(ΔH )(solution) = [(β1/δ) + T1]Δ(ΔS)(solution) = [constant]Δ(ΔS)(solution)

(14)

This indicates that a linear relationship between the enthalpy and the entropy of solvation exists. From the LFER, Leffler34,35 did show that such a linear relationship between ΔH‡ and ΔS‡ does exist. Therefore, the LFER and the compensation phenomenon are interrelated. It may be mentioned here that Linert et al.36 have also deduced such relationship. They actually showed that different iso-parametric relations are deducible from IKR. Both support and criticism of this view are found in literature37 because there are systems where compensation phenomenon exists in the absence of LFER or they exist separately. It has been argued that the physical origin of LFER is not obvious and hence such a correlation is not proper.2,38,39 Hepler40,41 has proposed that both the enthalpy and the entropy can be divided into external and internal components. In other words, Δ(ΔG) = Δ(ΔH ) − T Δ(ΔS) = Δ(ΔHint) + Δ(ΔHext) − T Δ(ΔSint) − T Δ(ΔSext)

(15)

where the internal components arise from the substituent effect in the gas phase and the external components come from the substituent effect in the solution phase. Hepler considered that Δ(ΔSint) = 0 and Δ(ΔHext) = β Δ(ΔSext)

(16)

Therefore, Δ(ΔG) = Δ(ΔHint), if β is approximately equal to Texpt. Therefore, Tcomp = Texpt if ΔG for all of the systems in the series are the same or almost same. Such exact compensation can be explained within the realm of the well-known thermodynamic laws previously mentioned; however, there are examples where linear relationship between ΔH and ΔS is difficult to rationalize by the laws of thermodynamics when the spread in ΔG is large and still linearity in ΔH − ΔS is observed. This may justify the proposition of “extra thermodynamic compensation”. In this case the slope of the line dΔH/dΔS is called the compensation temperature (Tcomp), which suggests some inherent peculiarity of the system “that cannot be a priori deduced from the laws of statistical thermodynamics”.42 According to Krug,43 “The compensation temperature is the temperature at which enthalpy variations precisely cancel entropy variations such that the rate or equilibrium constants are completely invariant”. It has been recently shown by Ryde44

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EEC AND OTHER POSSIBLE THERMODYNAMIC CORRELATIONS In this work we are revisiting the EEC from macroscopic thermodynamic view points. The data used for the evaluation have been taken from published works; no new measurements have been herein performed as the literature is amply rich with authentic results. We have acquisitioned data on both kinetic and equilibrium processes with proper acknowledgments and treated them in different way according to our objectives. We may mention that the systems considered for illustration and discussion fall in the categories of weak (Tcomp moderately 15878

DOI: 10.1021/acs.jpcb.5b09925 J. Phys. Chem. B 2015, 119, 15876−15884

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The Journal of Physical Chemistry B differs from Texpt) to strong (Tcomp strongly differs from Texpt) EEC manifestations. A comprehensive discussion on the thermodynamic relevance of the EEC in relation to the possible additional correlations among the thermodynamic parameters has been presented to make the EEC perspective more clear. Both kinetic and equilibrium processes of different origins are considered. On the basis of the overall transition states of the treated processes the activation parameters (ΔG#, ΔH#, and ΔS#) have been also similarly treated and correlated. For a simplified presentation, only the ΔG, ΔH, and ΔS are mentioned in the text. The related kinetic results are shown in the illustrations and the Table. The kinetic and equilibrium processes have shown a parallelism between them supporting that the kinetic compensation is a real phenomenon.20,22 Ryde44 has shown from theoretical studies that EEC arises for the binding of two molecules (ligand and receptor) in the gas phase; in the solution phase the binding is more complicated by the solvent effect. All physical−chemical processes are guided by the thermodynamic rules. Equation 1 represents such a rule. In general, ΔG can be experimentally determined whose temperature dependence can yield ΔS and their use in the eq 1 and then produces ΔH. Alternatively, ΔH can be as well found from the temperature dependence of ΔG, and consequently ΔS follows from the equation. It is also true that ΔH can be directly determined (in a calorimeter), which may be a temperature-dependent thermodynamic parameter. It should be noted that the previously mentioned indirect procedure of calculating ΔH and ΔS and using them to find the EEC effect has been criticized,37,45,48,49 although plenty of use of it is found in literature.12,19 Physicochemical processes can be of two general categories: “similar” and “nonsimilar”. To suit to our purpose, in this discussion we shall mainly consider similar processes. Nonsimilar processes will also be briefly discussed toward the end. From Figure 3A−D it is obvious that the experimental temperatures and the slopes of the lines, that is, the compensation temperatures, are quite different, and the differences are small (Figure 3A) to very large (Figure 3D). The associated free-energy differences are 76.8 to 80.0 kJ/mol (i.e., only 3.2 kJ/mol) in Figure 3A, whereas it is pretty large in Figure 3D (11.9−45.78 kJ/mol). Narrow free-energy range brings Texpt and Tcomp closer. Hence, we summarize that without constant intercepts (i.e., constant free-energy changes), EEC plots at a given experimental temperature yield slopes (Tcomp) different from Texpt. The wider the spread of the ΔG, the greater the deviation of the Tcomp from the Texpt. PerezBenito23 has shown that this corresponds to real kinetic compensation effect. Thus, EEC is perhaps an “undefined type of nonthermodynamic quantity” if at all it can be considered as a recognizable phenomenon. For a true thermodynamic correlation, that is, Texpt ≈ Tcomp, the free-energy spread (in Figure 3A) is required to be very narrow. The deviations of the slopes from unity (1.0) in the inset of the Figures are the manifestations of the inequalities between Texpt and Tcomp. It is to be noted that the compensation effect shown in Figure 3A has to be apparent and not real.50 Because the examples of reactions presented in Figure 3A−C are all biological and nonelementary in nature, they may be questioned; nevertheless, elementary reactions are rarely prevalent in nature. The said shortcomings are also associated with nonelementary laboratory experiments. The observed compensation, therefore, has to be on an overall thermodynamic basis and not on each step of multistep transition state basis. This consideration/concept

Figure 3. (A) Johnston et al.50 on kinetic results of fish myofibriliar ATPase activity; free-energy range 76.8 to 80.0 kJ mol−1. (B) Nikolic et al.51 on reaction of benzoic acid with diazodiphenylmethane in different solvents; free-energy range 70.9 to 78.6 kJ mol−1. (C) Kuroki et al.52 on Ca2+ binding to mutant human lysozymes and organic chelators at 5.5 pH; free-energy range −5.8 to −9.6 kcal mol−1. (D) Ben-Naim et al.53 results on the solvation of vapors condensing into their liquids; free-energy range −2.82 to −10.9 kcal mol−1.

also applies to the presently reported kinetic processes. In this connection, we also present below two other important combinations of the three related thermodynamic parameters ΔG, ΔH, and ΔS (and the corresponding kinetic quantities). (1) Correlation between ΔH and ΔS at constant ΔG but varied T. (2) Correlation between ΔH and ΔS when both ΔG and T vary. Such plots are presented in Figure 4A−D. In the main diagrams of A and B again good straight lines are obtained; the slopes of the courses stand closely to the average temperature, that is, Tave (kinetics) and Tave (equilibrium), respectively. The kinetic process (shear viscosity of a microemulsion at many shear rates) shows very close Tcomp and Tave, where the free energy of activation has a very small spread between 15.2 and 15.3 kJ mol −1 . For the equilibrium process the matching between Tcomp and Tave is excellent because the spread in ΔG is very small, only −17.2 to −17.7 kJ mol−1. Normally, processes with not a very large temperature range are studied in practice and hence they produce straight lines with Tave as the slope. These are thermodynamic compensation and follow eq 1. Over a wide range of temperature what type of graphical profiles and Tave would result requires to be tested. (Poor linear correlation is expected.) The insets in the Figure routinely show the dependences of the ΔH on TΔS. Slopes 0.997 in panel A and 0.998 in panel B with excellent correlations are found as expected from thermodynamics. The examples and illustrations of different types previously presented constitute similar processes. According to our knowledge, nonsimilar processes are not so far considered (tested) in literature for their EEC performance. We have examined several such systems for EEC correlation. Three kinetic processes (Hassan et al.58 on oxidation of carboxymethyl cellulose, Izatt et al.56 on ion inclusion in macrocycles, and Neyhart et al.59 on oxidation of sugar) are considered at 15879

DOI: 10.1021/acs.jpcb.5b09925 J. Phys. Chem. B 2015, 119, 15876−15884

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The Journal of Physical Chemistry B

Figure 5. (A) Hassan et al.58 on kinetic results for oxidation of carboxymethyl cellulose (free-energy values are 80.35, 80.84 kJ mol−1), Izatt et al.56 on ion inclusion in macrocycles (free-energy values are 79.9, 82.1 kJ mol−1), and Neyhart et al.59 on oxidation of sugar (freeenergy value is 82.1 kJ mol−1). (B) Rosen et al.60 on equilibrium adsorption parameter for N-dodecylpyridinium halides (free-energy values are −31.3, −30.7, −317, −30.5, −31.7, −30.1, −31.3, −29.5 kJ mol−1), Ben-Naim et al.53 on solvation vapors condensing into their liquids (free-energy values are −30.6, −31.5, −31.5, −31.0 kJ mol−1), and Chatterjee et al.61 on micellization parameters of different surfactants (free-energy values are −31.9, −32.8 kJ mol−1). (C) Results from dissimilar systems like Rosen et al.60 (free-energy values are −32.4, −31.7, −31.7, −30.1, −31.3 kJ mol−1), Chatterjee et al.61 (free-energy values are −35.1, −41.8, −33.2 kJ mol−1), Ben-Naim et al.53 (free-energy values are −22.7, −39.9, −50.4, −34.4 kJ mol−1), and Rekharsky et al.55 (free-energy values are −5.6, −12.4, −4.2, −3.0, −9.0, −3.0, −20.5, −25.7, −7.8 kJ mol−1).

Figure 4. (A) Acharya et al.54 on viscosity; free-energy range values are 15.2 to 15.3 kJ mol−1. (B) Rekharsky et al.55 on organic molecules complexes with cyclodextrins (CDs); free-energy range −17.2 to −17.7 kJ mol−1. (C) Izatt et al.56 on complexation of cations with macrocycle; free-energy range 26.8 to 100 kJ mol−1. (D) Rekharsky et al.55 on organic molecules complexes with CDs; free-energy range −1.0 to −30.6 kJ mol−1. The profiles of the other type (previously mentioned class 2) are found to be random and noncorrelative (panels C and D) and do not follow the compensation rule. Although a nonlinear compensation phenomenon has also been suggested57 the present illustrations do not fall in that class.

nearly equivalent average free-energy values of 79.4 kJ mol−1. The corresponding enthalpy and entropy values are plotted in Figure 5A. Three equilibrium processes (Rosen et al.60 on adsorption of dodecylpyridinium halides at the air/water interface, Ben-Naim et al.53 on solvation of vapors condensing on their liquids, Chatterjee et al.61 on micellization of surfactants in water) are also considered at nearly equivalent ΔG values of average −31.1 kJ mol −1. The corresponding ΔH and ΔS plots are presented in Figure 5B. At a constant ΔG# or ΔG, for nonsimilar systems, we expect (also stated in the Background section) ΔH# versus ΔS# or ΔH versus ΔS plots to be linear with slopes equal to Texpt. The expectation is fulfilled: Tcomp (kinetics) and Tcomp (equilibrium) found are 303 and 296, respectively; the corresponding Texpt are 298 and 298, respectively. The agreements are close. They are rarely documented in literature. An important conclusion can therefore be made. For a process (similar or nonsimilar), as long as the ΔG# or ΔG is close (i.e., constant) plots ΔH# versus ΔS# and ΔH versus ΔS are linear with slopes equal to the experimental temperature. When ΔG# or ΔG values are not close (or constant) and the temperature is not constant, then both of the above dependences are found to be noncorrelative (or random Figure 4C,D). They are also depicted in Figure 5 C. The results show that for nonsimilar processes, EEC is not found. We may add that herein presented kinetic and equilibrium processes that evidence EEC show Tcomp > Texpt, and in line with Perez-Benito23 proposition (cited above) they are all “real and meaningful” and not experimental-error-related. In general, a large number of EEC citations in literature also fall in this line. We may here add that the conclusion found from the results of Figure 5B matches that of Lambert and Leff47 where at very small (nearly zero or constant) work the plot between the entropy and enthalpy of many nonsimilar solids

produced a nice linear correlation with a compensation. Like Figure 5, the plotted results also obeyed the thermodynamic eq 1. We now make a critical assessment of the eqs 1 and 2 and their thermodynamic implications. For a ready reference the equations are herein rewritten. ΔH = ΔG + T ΔS

(1)

ΔH = α + β ΔS

(2)

Combining them we get ΔG = α + (β − T )ΔS

(17)

Equation 1 tells that at a constant ΔG the plot between ΔH and ΔS is linear to yield a slope T = Texpt. Both similar and nonsimilar processes, obeying the above condition, are guided by the equation as expected. Equation 2 represents similar systems (with varied ΔG) that describe a linear plot between ΔH and ΔS producing an intercept, α (a constant), and a slope β (i.e., Tcomp ≠ Texpt). The combined relation 17 shows that α = ΔG when β = T (or Texpt), which can be also understood by comparing eqs 1 and 2. Now, the significance of the parameters α and β is the point of concern. They are the free-energy change and the temperature under a hypothetical condition so far remaining as an unproven fact. Thus, we may state that at β the concerned similar processes are expected to produce constant or nearly constant ΔG(α) values. The values of the above two parameters found from the herein reported and 15880

DOI: 10.1021/acs.jpcb.5b09925 J. Phys. Chem. B 2015, 119, 15876−15884

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The Journal of Physical Chemistry B Table 1. Essential Findings of Different Similar Systems Considered for EEC in This Study authors

ΔG/ΔG# range (kJ mol−1/kcal mol−1)

α(intercept) (kJ mol−1/kcal mol−1)

β(slope) (K)

*Sen Gupta et al. (Figure 2) Jalal et al.12 (Figure 2)a Johnston et al.50 (Figure 3) *Nikolic et al.51 (Figure 3) Kuroki et al.52 (Figure 3)a Ben-Naim et al.53 (Figure 3)a *Acharya et al.54 (Figure 4)b *Rekharsky et al.55 (Figure 4)b *Hassan et al.58 (Figure 5) *Izatt et al.56 (Figure 5) *Neyhart et al.59 (Figure 5) Rosen et al.60 (Figure 5) Ben-Naim et al.53 (Figure 5) Chatterjee et al.61 (Figure 5)

89.8 to 95.9 −1.34 to +0.471 76.8 to 80.0 70.9 to 78.6 −5.8 to −9.6 −2.82 to −10.9 15.2 to 15.3 −17.2 to −17.7 80.4, 80.8 79.9, 82.1 82.1 −31.3, −30.7, −31.7, −30.5, −31.7, −30.1, −31.3, −29.5 −30.6, −31.6, −31.5, −30.9 −31.9, −32.8

93.4 −1.63 78.2 82.3 −8.49 −1.46 15.3 −17.6 81.4

389.4 (Texpt = 303) 320.6 (Texpt = 298) 300.6 (Texpt = 291) 399.3 (Texpt = 303) 337.22 (Texpt = 303) 547.82 (Texpt = 298.15) 295 (Tave = 297) 298.4 (Tave = 298.8) 303 (Texpt = 298)

−31.1

296 (Texpt = 298)

28

a ΔG values are in kcal mol−1. *Asterisked results are on kinetics bIt may seen that the ΔG# values of Acharya et al.54 and Rekhersky et al.55 are very close; the corresponding Tcomp and Texpt are also close within experimental error. The results follow the thermodynamic eq 1.

thermodynamic parameters herein dealt with. In the ITC or the calorimetry in general the heats absorbed or evolved from the processes other than what we want to study are also simultaneously measured. There the enthalpy is an integral value whereas the enthalpy normally found from the van’t Hoff rationale is a differential value.66,67 Solvation, ionization, orientation, aggregation, deaggregation, steric phenomenon, and so on make their contributions, and the measured enthalpy in a calorimeter is an algebraic sum of the contributions of all the involved processes. The derived thermodynamic parameters from this enthalpy ought to be apparent (but not what is intended). Physicochemical processes in solvent media have such a multiprocess effects on the measured ΔH. In this stage, we, refrain from presenting such results and their discussion here. A detailed presentation of EEC with the ITC data will be taken up in a separate study. In this connection, it may be added that multiple entropy values in both kinetic21,22 and equilibrium40,41 processes have been also proposed. The above stated multiprocess effects on the measured enthalpy in a calorimeter may have an apparent similarity with the aforesaid multiple entropy values but the multi-entropy concept is related to the specific physicochemical processes in question, whereas in the calorimetric measurements the involved heat is related to all possible processes that may arise in the reaction environment. Thus, the two propositions have a basic difference.

illustrated systems with their associated experimental ranges of ΔG are presented in Table 1. It is found that Texpt < Tcomp (β), and α may either fall within the range of the process free energies or beyond. Therefore, if similar physical−chemical processes are performed at Tcomp or β and the corresponding free energies are all similar or nearly similar to α then the EEC is a rational concept. We hope that such attempts will be made in the future to justify EEC.

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EFFECT OF DIRECT DETERMINED ΔH ON EEC There has been a pertinent question about the interdependency of ΔH and ΔS on ΔG. In practice, ΔG in the chemical equilibrium processes is determined from the equilibrium constant (K). From the temperature dependence of K, the corresponding entropy and hence enthalpy terms are estimated. Of the latter two parameters, ΔH can be also directly determined by calorimetry; direct determination of the ΔS is difficult. We may, therefore, treat the results taking the direct determined values of ΔG and ΔH and the derived values of ΔS as described above to test the EEC. Recently, isothermal titration calorimetry (ITC) applied to the surfactant selfaggregation or micellization and other equilibrium forming interaction or binding processes provided such results enriching the literature.8,61−65 With reference to micellization it may be stated that from the same ITC experimental runs at different temperatures both ΔGM (free energy of micellization), and ΔHM (enthalpy of micellization) values can be directly found. These values could be used in eq 1 to get the corresponding ΔSM (the entropy of micellization) values that are derived. The values are considered more accurate than the indirect route of getting ΔHM and ΔSM using the van’t Hoff rationale; however, from the temperature dependence of ΔGM we can also find out the van’t Hoff enthalpy of micellization ΔHM (van’t Hoff) as well as ΔSM (van’t Hoff) and compare with the calorimetric results. This has shown significant differences for ionic surfactants and small differences for non ionic surfactant systems. A preliminary trial of calorimetric results has produced some features of EEC in micelle-forming amphiphilic systems that differ from the features produce by the van’t Hoff rationale. Although direct determination of ΔHM by calorimetry is more accurate than van’t Hoff procedure (results not shown), a caution may be worth mentioning here on the use of the calorimetric enthalpy values in the overall calculation of the

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COMPREHENSION AND CONCLUSIONS EEC has been a matter of discussion, research, and criticism for over half a century, but the issue is yet to get a pragmatic and acceptable solution. As spelled out by many, the phenomenon may have the following major origins: (1) It arises from the uncertainties and errors in the measurements of the parameters ΔH and ΔS and is an extra-thermodynamic relation.42 (2) It comes into effect by the influence of the hidden Carnot cycles in the concerned processes.29,31 (3) It arises as a consequence of the solute−solute and solute−solvent interactions in relation to solvent structure.7,19,68 (4) Its presence in the gas-phase interaction has been noted.44 (5) It is a natural consequence of finite ΔCp values, quantum confinement effects, multiple weak interactions, and limited free-energy windows.30 Basically, two fundamental thermodynamic quantities enthalpy and entropy changes of both kinetic and equilibrium origins are used in the 15881

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then with respect to ΔH, ΔG= constant. Linear ΔH − ΔS compensation follows immediately from the relationship ΔG = ΔH − TΔS.” This supports our stress on the importance of ΔG in the EEC conundrum. A range of ΔG (not very close or exact) should always have an element of mathematical invalidity (which we have stressed upon) to appear as a nonexact compensation (Texpt ≠ Tcomp) imagined with an extrathermodynamic or empirical concept. It is to be noted that H−S linear relation can be thermodynamically framed, as has been done here as also shown in literature.46 We look forward to further development in this interesting and important area.

analyses, which are essentially obtained from the basic relation, [ΔG or ΔG#] = [ΔH or ΔH#] − [TΔS or TΔS#]. The first term is found from experiments, and the other two are indirectly obtained from the temperature dependence of the first (although the second for an equilibrium process could also be found directly from calorimetry, the third is not a directly determined quantity). Thus, a pertinent question arises whether a linear correlation between ΔH and ΔS is permissible without the cognizance of ΔG. The possible combinations of the parameters (ΔG#, ΔH#, and ΔS# or ΔG, ΔH, and ΔS) at constant and varied temperatures with available literature data can be worth consideration and discussion, which has been the subject matter of this study. We find that the associated ΔG can decide the existence of the EEC. We also believe that the points in the ΔH − ΔS linear plot must be rational and least random. If they are, EEC may be thought of but not otherwise. If the associated ΔG values fall in a wide range, then a well visible EEC occurs (category I); if the range is narrow, then the EEC is weak (category II), and if the free energy changes are very small or nearly invariant, then the EEC is of category III making Tcomp = Texpt as per the thermodynamic requirement (eq 1). The above observations very often apply to similar processes. Nonsimilar processes, we find, comply with the category III. Normally, in our analysis ΔH and ΔS are considered, and the results are rationalized in different ways (Figure 4A−D), which has so far remained undone. In testing EEC, the enthalpy values used are indirectly determined quantities; direct determined enthalpy by calorimetry could be an improved way of testing EEC, which has been only restrictedly attempted, but the direct determination of ΔH of an equilibrium process by calorimetry has an advantage as well as a disadvantage in that it measures the integral heat (heat produced or absorbed from all possible processes in addition to that intended in the system), whereas the van’t Hoff process (procedure) is a differential one that aims only to the intended process.61,62,66 Sorting out the other nonintended heats from the calorimetry results ought to be a complex and difficult proposition, although it is required for a better analysis of the EEC. In this connection, we drop a few lines on the explanation recently proposed by Starikov29,31 from the view points of statistical thermodynamics. The EEC has been attributed to the contributions of hidden Carnot or Kinetic Cycles at the microscopic level in the concerned systems. The EEC has been attributed to the onset of thermodynamic equilibrium between the system and its surrounding.69 He stressed that a proper “isolated system” is never available in the conventional physicochemical conditions and there is always energy transfer with the surroundings. Heating and cooling phenomena are the consequences of “artificial refrigerator” and “artificial heat pump”. The proposition has novelty, but the existence of the proposed “refrigerator” and “heat pump” requires support from concept and reasoning from rational internal conditions. Starikov et al. have discussed this in a series of publications.70−72 Constant ΔG for processes similar or nonsimilar may manifest equivalence between the experimental and the compensation temperature; variable free-energy changes of different similar or related processes mostly at a constant temperature produce the so-called EEC, which has led researchers to search for an explanation. From the thermodynamic point of view we believe that eq 1 has a say in this. In the summary of his work, based on statistical tests of several systems, Sharp42 stated, “if the range of ΔG’s measured in a series of experiments is much smaller than the range of ΔH’s,

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AUTHOR INFORMATION

Corresponding Authors

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

T.B.: Sagardighi Thermal Power School, S.g T.P.P, P.O.Monigram, Murshidabad, Pin-742237, India. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS A.P. thanks the Centre for Surface Science, Department of Chemistry, for granting Research Assistantship. T.B. also thanks UGC, Govt. of India for a Junior Research Fellowship. 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 an Honorary Scientist position and Emeritus Professorship, respectively.

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