Entropy−Enthalpy Compensation in Chemical Reactions and

ARTICLE pubs.acs.org/JPCB

Entropy-Enthalpy Compensation in Chemical Reactions and Adsorption: An Exactly Solvable Model Karl F. Freed* James Franck Institute and Department of Chemistry, University of Chicago, Chicago, Illinois 60637, United States ABSTRACT: The free energies of reaction or activation formany systems respond in a common fashion to a perturbing parameter, such as the concentration of an “inert” additive. Arrhenius plots as a function of the perturbing parameter display a “’compensation temperature” at which the free energy appears to be independent of the perturber, an entropy-enthalpy compensation process. Thus, as the perturber’s concentration varies, Arrhenius plots of the rate constant or equilibrium constant exhibit a rotation about the fixed compensation temperature. While this (isokinetic/ isoequilibrium) component of the phenomenon of entropy-enthalpy compensation appears in a huge number of situations of relevance to chemistry, biology, and materials science, statistical mechanical descriptions have been almost completely lacking. We provide the general statistical mechanical basis for solvent induced isokinetic/isoequilibrium entropy-enthalpy compensation in chemical reactions and adsorption, understanding that can be used to control of rate processes and binding constants in diverse applications. The general behavior is illustrated with an analytical solution for the dilute gas limit.

I. INTRODUCTION The general phenomenon “enthalpy-entropy compensation” in essence refers to the experimental observation of a linear scaling between enthalpy (Δhi) and entropy (Δsi) for a set of related reactions (labeled by the index i) Δhi ¼ RþβΔsi where R and β are constants, with β called the “compensation” temperature, as well as a common isokinetic/isoequilibrium temperature for all van’t Hoff plots (see below). This compensation phenomenon has long been claimed to be observed in an extremely wide range of areas, such as thermal desorption kinetics,1 micellization,2-4 chromatography,5-9 Langmuir adsorption,1 water sorption,10 drug-receptor binding,11-17 melting,18,19 thermal transitions of peptides,20 protein and nucleic acid unfolding,21,22 solution extraction,23 ion hydration,24 conformational transitions,25 dielectric relaxation,26 formation of supramolecular27 and van der Waals complexes,28 capsules,29 DNA,30 ligand solvation,31 thermal death of bacteria, viruses, and yeasts,32 evolutionary adaptation of proteins,33 viscous flow of simple and polymeric liquids at high temperatures,34 conductivity of organic substances,35 electrical conductivity of single crystals,36 plasticization-antiplasticization transition,37,38 dispersion of quantum dots,39 dissociation of lipid complexes,40 thermal decomposition,41 thermal isomerization,42 oxidation43 and substitution reactions,44 proton45 and electron46,47 transfer reactions, depolymerization,48 photoisomerization,49 hydrolysis,50,51 cycloaddition,52-54 etc. Considerable controversy surrounds the subject of entropyenthalpy compensation, ranging, on one hand, from puzzlement and amazement at its occurrence and questions concerning its origins to, on the other hand, claims that the effect is either spurious or an artifact55 of a limited temperature range for the data or of a limited range for the free energies. Liu and Guo r 2011 American Chemical Society

provide a comprehensive review56 of the many sources of errors in assigning data as demonstrating entropy-enthalpy compensation, along with examples conforming to their stringent criteria. Gilli et al.57 suggest that observed entropy-enthalpy compensation in drug receptor binding probably arises because of a intrinsic property of hydrogen bonds. Nord
en and co-workers58 suggest various mathematical and thermodynamic mechanisms for entropy-enthalpy compensation, with particular application to DNA duplex stability. While enthalpy-entropy compensation has been ascribed in many diverse areas, a truly molecular statistical mechanical explanation of this phenomenon has been generally lacking. Our recent paper is the first attempt to fulfill this gap.59 Reference 59 develops a Flory-Huggins type theory for the influence of crowding on equilibrium self-assembly when an “inert” polymer additive is introduced into a solution of a selfassembling species M. The self-assembly process is idealized as a reversible mth order chemical reaction in which m identical monomers M1 directly associate into fully assembled entities Mm mM1 rfMm The calculated solubility ratio S/S0 of the associating species (where S0 denotes the solubility at infinite polymer dilution) exhibits an isoequilibrium behavior characteristic of many entropy-enthalpy compensation phenomena as follows: In particular, S/S0 displays an Arrhenius temperature dependence, and the Arrhenius curves for various additive concentrations jp intersect at a common point corresponding to the temperature TΘ at which the system behaves as if there were no molecular additives. At this temperature TΘ, the effect of the additive on the Received: November 4, 2010 Revised: January 4, 2011 Published: February 2, 2011 1689

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solubility becomes inverted from a decrease in solubility for T > TΘ to its enhancement for T < TΘ. As the additive concentration grows, the curves for ln(S/S0) rotate around 1/ TΘ in a counterclockwise direction. Alternatively, if jp is varied at a fixed T, then ln(S/S0) changes linearly with jp, and the slope of ln(S/S0) vs jp shifts progressively from positive to negative upon cooling. The tendency toward an increased stability of the assembled clusters, due to the modification of the entropy of assembly by repulsive polymer-M interaction, is progressively compensated by attractive interactions that alter the enthalpy of assembly. This finding represents a derivation for a special case of the isoequilibrium component of entropy-enthalpy compensation phenomenon and provides a prediction that has recently been confirmed by experimental studies of Minton and co-workers.60

II. THEORY The ubiquitous claims for the appearance of entropyenthalpy compensation in diverse phenomena suggests the existence of certain (or classes of) commonality. For example, many instances involve compensation in the linear response of equilibrium constants or rate constants to the presence of “inert” additives. The nominal equilibrium constant K for the reaction AþB a CþD

ð1Þ

is expressed in terms of the equilibrium constant K0 in the absence of the additive and the change ΔΔG in free energy of reaction due to the additive K ¼ ½C½D=½A½B ¼ K 0 expð-βΔΔGÞ

ð2Þ

where β = 1/kT. The perturbation ΔΔG often is found to vary linearly with the density jp of the additive as ΔΔG ¼ jp ða - bβÞ

ð3Þ

where a and b are constants, and consequently ln(K/K0) behaves as a function of T and of jp exactly as ln(S/S0) in our previous work. Within transition state theory, ΔΔG is the shift in activation free energy that often displays the same type of (isokinetic) dependence on temperature and additive concentration. Here we study in more generality this class of isoeqilibrium/ isokinetic entropy-enthalpy compensation processes that are characterized by eq 3 and are associated with the influence of inert additives and/or solvents on equilibrium constants and rate constants of chemical reactions. The goal here is in establishing how the phenomena emerge from a simple, but realistic and analytically tractable, limit (reactions in dilute gases) of a more general theory (for liquids) that may be applied to a class of chemical reactions to determine whether a common compensation temperature exists and that may also be extended and applied to more complex systems for comparison with experiments. Because both equilibrium constants and rate constants may be represented in terms of sums and/or differences of chemical potentials Δμ for the reacting species, the existence of generality in isoenergetic/isokinetic entropy-enthalpy compensation associated with the presence of additives boils down to an analysis of whether the coefficient of the term in the chemical potential that is linear in the concentration jp of the additive admits this type of compensation. Fortunately, Singer et al.61 have provided a

general theory for the shift in chemical potential of a molecule due to solvation by a fluid. The simplest limit for this system involves considering a reactant (or product) molecule in a dilute gas, a limit that renders the theory analytically tractable upon use of a square well potential between reactant and additive (i.e., solvent) and a model where all molecules have a single interaction site.68 The contribution to the reactant’s chemical potential linear in the solvent density is derived analytically in the following and, indeed, exhibits the generic isoenergetic/ isokinetic entropy-enthalpy compensation. The theory of Singer et al. appears to be exact when applied to a limiting dilute gas system of reactants and solvent that are described using a single interaction site for each. Hence, first introducing the restriction to single interaction sites and averaging their eq 2.8 over the reactants position produces the contribution Δμ to the excess chemical potential of the reactant due to the presence of the solvent as Z Z -βΔμ ¼ jp dr½cðrÞ-ð1=2Þh2 ðrÞþð1=2Þ

d3 k 2 c ðkÞχpp ðkÞ ð2πÞ3

ð4Þ where c(r) is the direct correlation function between the reactant and the solvent, h(r) is the reactant-solvent correlation function, c(k) is the Fourier transform of c(r), and χpp(k) is the Fourier transform of the solvent’s density-density correlation function.

III. DILUTE GAS LIMIT: ANALYTICAL RESULTS Taking the dilute gas limit and retaining only the contribution linear in the solvent density jp implies that both c(r) and h(r) reduce to the leading order f-function f(r) = exp[-βu(r)] - 1, with u(r) the reactant-solvent interaction potential, and χpp(k) reduces to jp in this order. These simplifications reduce eq 4 to the integrals Z Z d3 k 2 f ðkÞ -βΔμ ¼ jp dr½f ðrÞ-ð1=2Þf 2 ðrÞþðjp =2Þ ð2πÞ3 ð5Þ that are analytically tractable for a square well potential. The square well interaction between reactant and solvent is specified by the parameters ε, σ, and σγ, which, respectively, are the well depth, hard core reactant-solvent diameter, and outer diameter of the square well. The hard core contribution to the second virial coefficient is B0 = 2πσ3/3, and the quantity f = exp(βε) - 1, which becomes ∼βε in the high T approximation, enters into the contribution from the attractive portion of the potential. Each reactant has different values of ε, σ, and σγ, so labels are used when considering the contribution to the equilibrium constant or reaction rate, etc. Performing the integrals in eq 5, the excess chemical potential of the reactant to first order in the solvent density is a quadratic in f -βΔμ ¼ jp B0 f-2 þ 2ðγ3 -1Þf -½1þðγ3 -1Þf 2 -ð1=4πÞ½ð1þf Þ2

þγ3 f 2 þð3π1=2 =4Þf ð1þf Þg

ð6Þ

The treatment of a reacting system, e.g., A þ B = C þ D, merely requires considering sums a difference of the chemical potentials ΔΔG ¼ ΔμC þΔμD -ΔμA -ΔμB 1690

ð7Þ

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so that the equation for the compensation temperatures contains the three parameters (ε, σ, σγ) for each of the reactants. The compensation temperature is readily resolved numerically and can be solved analytically as the root of a quadratic equation when invoking the high-temperature limit of f ∼ βε. The high-temperature limit for ΔΔG is a quadratic in βε of the general form 2

βΔΔG ¼ jp ½aþbβþcðβÞ 

ð8Þ

The term in a involves a constant multiplied by sums and differences of hard core second virial coefficients. This contribution vanishes if the excluded volume is unchanged by the reaction, whereupon eq 8 displays the classic behavior of entropy-enthalpy compensation described in the Introduction, with the compensation temperature T0 = -c/b. Even when the excluded volume is not conserved in the reaction, the solution to the full equations may be analyzed to deduce general trends as to how, for instance, the compensation temperature for a equilibrium constant varies with relative values of the square well parameters, and the trends found can be used as initial guesses to guide theoretical treatments for more complicated systems and to guide and interpret molecular simulations. Applications to a series of chemical reactions will enable determining whether a common compensation temperature and, thus, full blown entropy-enthalpy compensation, exists for these systems. The description of the Langmuir isotherm follows from equating the chemical potentials for the adsorbed species with that in the bulk gas phase. The former is just the chemical potential for a two-dimensional gas. Application of eq 5 to a dilute two-dimensional gas yields -βΔμ2 ¼ jp πσ2 f-1þðγ2 -1Þf -ð1=2Þ½1þðγ2 -1Þf 2  þð1=2Þ½ð1þf Þ2 þγ2 f 2 -2f ð1þf Þg

ð9Þ

which may be combined with the contributions independent of jp to obtain the isotherm.

IV. DISCUSSION AND CONCLUSION The derivation of the isoenergetic/isokinetic component of entropy-enthalpy compensation for the dilute gas limit implies that the general theory can now be applied to series of chemical reactions to search for the existence of a common compensation temperature, as well as to reactants in a dense liquid and for systems of “inert” additives in solution (a ternary system of reactant, solvent, and additive) to consider the contribution of the reactant’s chemical potential that is linear in the concentration of the additive. The equations can be treated by standard liquid theory, e.g., the highly successful extended RISM theory,62 for application to more complicated systems involving reactions in solution. The full theory will aid in the control of rate processes and binding constants in diverse applications. ’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

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