In thermodynamics, enthalpy–entropy compensation is a specific example of the compensation effect. The compensation effect refers to the behavior of a series of closely related chemical reactions (e.g., reactants in different solvents or reactants differing only in a single substituent), which exhibit a linear relationship between one of the following kinetic or thermodynamic parameters for describing the reactions:Liu, L.; Guo, Q. X. Isokinetic relationship, isoequilibrium relationship, and enthalpy-entropy compensation. Chem. Rev. 2001, 101, 673–695. Between the logarithm of the pre-exponential factors (or prefactors) and the activation energies where the series of closely related reactions are indicated by the index , are the preexponential factors, are the activation energies, is the gas constant, and , are constants. Between enthalpies and entropies of activation (enthalpy–entropy compensation) where are the enthalpies of activation and are the entropies of activation. Between the enthalpy and entropy changes of a series of similar reactions (enthalpy–entropy compensation) where are the enthalpy changes and are the entropy changes.
When the activation energy is varied in the first instance, we may observe a related change in pre-exponential factors. An increase in tends to compensate for an increase in , which is why we call this phenomenon a compensation effect. Similarly, for the second and third instances, in accordance with the Gibbs free energy equation, with which we derive the listed equations, scales proportionately with . The enthalpy and entropy compensate for each other because of their opposite algebraic signs in the Gibbs equation.
A correlation between enthalpy and entropy has been observed for a wide variety of reactions. The correlation is significant because, for linear free-energy relationships (LFERs) to hold, one of three conditions for the relationship between enthalpy and entropy for a series of reactions must be met, with the most common encountered scenario being that which describes enthalpy–entropy compensation. The empirical relations above were noticed by several investigators beginning in the 1920s, since which the compensatory effects they govern have been identified under different aliases.
Related terms
Many of the more popular terms used in discussing the compensation effect are specific to their field or phenomena. In these contexts, the unambiguous terms are preferred. The misapplication of and frequent crosstalk between fields on this matter has, however, often led to the use of inappropriate terms and a confusing picture. For the purposes of this entry different terms may refer to what may seem to be the same effect, but that either a term is being used as a shorthand (isokinetic and isoequilibrium relationships are different, yet are often grouped together synecdochically as isokinetic relationships for the sake of brevity) or is the correct term in context. This section should aid in resolving any uncertainties. (see Criticism section for more on the variety of terms)
compensation effect/rule : umbrella term for the observed linear relationship between: (i) the logarithm of the preexponential factors and the activation energies, (ii) enthalpies and entropies of activation, or (iii) between the enthalpy and entropy changes of a series of similar reactions.
enthalpy-entropy compensation : the linear relationship between either the enthalpies and entropies of activation or the enthalpy and entropy changes of a series of similar reactions.
isoequilibrium relation (IER), isoequilibrium effect : On a Van 't Hoff plot, there exists a common intersection point describing the thermodynamics of the reactions. At the isoequilibrium temperature , all the reactions in the series should have the same equilibrium constant ()
isokinetic relation (IKR), isokinetic effect : On an Arrhenius plot, there exists a common intersection point describing the kinetics of the reactions. At the isokinetic temperature , all the reactions in the series should have the same rate constant ()
isoequilibrium temperature : used for thermodynamic LFERs; refers to in the equations where it possesses dimensions of temperature
isokinetic temperature : used for kinetic LFERs; refers to in the equations where it possesses dimensions of temperature
kinetic compensation : an increase in the preexponential factors tends to compensate for the increase in activation energy:
Meyer–Neldel rule (MNR) : primarily used in materials science and condensed matter physics; the MNR is often stated as the plot of the logarithm of the preexponential factor against activation energy is linear:
where is the preexponential factor, is the activation energy, σ is the conductivity, and is the Boltzmann constant, and is temperature.Abtew, T. A.; Zhang, M.; Pan, Y.; Drabold, D. A. Electrical conductivity and Meyer-Neldel rule: The role of localized states in hydrogenated amorphous silicon. J. Non Cryst. Solids 2008, 354, 2909–13.
Mathematics
Enthalpy–entropy compensation as a requirement for LFERs
Linear free-energy relationships (LFERs) exist when the relative influence of changing substituents on one reactant is similar to the effect on another reactant, and include linear Hammett plots, Swain–Scott plots, and Brønsted plots. LFERs are not always found to hold, and to see when one can expect them to, we examine the relationship between the free-energy differences for the two reactions under comparison. The extent to which the free energy of the new reaction is changed, via a change in substituent, is proportional to the extent to which the reference reaction was changed by the same substitution. A ratio of the free-energy differences is the reaction quotient or constant .
The above equation may be rewritten as the difference () in free-energy changes ():
Substituting the Gibbs free-energy equation () into the equation above yields a form that makes clear the requirements for LFERs to hold.
One should expect LFERs to hold if one of three conditions are met: 's are coincidentally the same for both the new reaction under study and the reference reaction, and the 's are linearly proportional for the two reactions being compared. 's are coincidentally the same for both the new reaction under study and the reference reaction, and the 's are linearly proportional for the two reactions being compared. 's and 's are linearly related to each other for both the reference reaction and the new reaction.
The third condition describes the enthalpy–entropy effect and is the condition most commonly met.Anslyn, E. V.; Dougherty, D. A., Modern physical organic chemistry; University Science: Sausalito, CA, 2006; pp. 466–470.
Isokinetic and isoequilibrium temperature
For most reactions the activation enthalpy and activation entropy are unknown, but, if these parameters have been measured and a linear relationship is found to exist (meaning an LFER was found to hold), the following equation describes the relationship between and :
Inserting the Gibbs free-energy equation and combining like terms produces the following equation:
where is constant regardless of substituents and is different for each substituent.
In this form, has the dimension of temperature and is referred to as the isokinetic (or isoequilibrium) temperature.Exner, O. The enthalpy-entropy relationship. Prog. Phys. Org. Chem 1973, 10, 411–482.
Alternately, the isokinetic (or isoequilibrium) temperature may be reached by observing that, if a linear relationship is found, then the difference between the s for any closely related reactants will be related to the difference between 's for the same reactants:
Using the Gibbs free-energy equation,
In both forms, it is apparent that the difference in Gibbs free-energies of activations () will be zero when the temperature is at the isokinetic (or isoequilibrium) temperature and hence identical for all members of the reaction set at that temperature.
Beginning with the Arrhenius equation and assuming kinetic compensation (obeying ), the isokinetic temperature may also be given by
The reactions will have approximately the same value of their rate constant at an isokinetic temperature.
History
In a 1925 paper, F.H. Constable described the linear relationship observed for the reaction parameters of the catalytic dehydrogenation of primary alcohols with copper-chromium oxide.Constable, F. H. The mechanism of catalytic decomposition. Proceedings of the Royal Society of London Series A-Containing Papers of a Mathematical and Physical Character 1925, 108, 355–78.
Phenomenon explained
The foundations of the compensation effect are still not fully understood, though many theories have been brought forward. Compensation of Arrhenius processes in solid-state materials and devices can be explained quite generally from the statistical physics of aggregating fundamental excitations from the thermal bath to surmount a barrier whose activation energy is significantly larger than the characteristic energy of the excitations used (e.g., optical phonons). To rationalize the occurrences of enthalpy-entropy compensation in protein folding and enzymatic reactions, a Carnot-cycle model in which a micro-phase transition plays a crucial role was proposed.Starikov, E. B.; Norden, B. Entropy-enthalpy compensation as a fundamental concept and analysis tool for systematical experimental data. Chemical Physics Letters 2012, 538, 118–20. In drug receptor binding, it has been suggested that enthalpy-entropy compensation arises due to an intrinsic property of hydrogen bonds.Gilli, P.; Ferretti, V.; Gilli, G.; Borea, P. A. Enthalpy-Entropy Compensation in Drug-Receptor Binding. J. Phys. Chem. 1994, 98, 1515–18. A mechanical basis for solvent-induced enthalpy-entropy compensation has been put forward and tested at the dilute gas limit.Freed, K. F. Entropy-Enthalpy Compensation in Chemical Reactions and Adsorption: An Exactly Solvable Model. J Phys Chem B 2011, 115, 1689–1692. There is some evidence of enthalpy-entropy compensation in biochemical or metabolic networks, particularly in the context of intermediate-free coupled reactions or processes. However, a single general statistical mechanical explanation applicable to all compensated processes has not yet been developed.
Criticism
Kinetic relations have been observed in many systems and, since their conception, have gone by many terms, among which are the Meyer-Neldel effect or rule,Meyer, W.; Neldel, H. Concerning the relationship between the energy constant epsilon and the quantum constant alpha in the conduction-temperature formula in oxydising semi-conductors. Physikalische Zeitschrift 1937, 38, 1014–1019. the Barclay-Butler rule,Barclay, I. M.; Butler, J. A. V. The entropy of solution. Transactions of the Faraday Society 1938, 34, 1445–1454. the theta rule,Schwab, G. M. About the Mechanism of Contact Catalysis. Advances in Catalysis 1950, 2, 251–267. and the Smith-Topley effect.Manche, E. P.; Carroll, B. Compensation and the Smith-Topley Effects. Thermochimica Acta 1979, 31, 387–390. Generally, chemists will talk about the isokinetic relation (IKR), from the importance of the isokinetic (or isoequilibrium) temperature, condensed matter physicists and material scientists use the Meyer-Neldel rule, and biologists will use the compensation effect or rule.Yelon, A.; Movaghar, B.; Crandall, R. S. Multi-excitation entropy: its role in thermodynamics and kinetics. Reports on Progress in Physics 2006, 69, 1145–1194.
An interesting homework problem appears following Chapter 7: Structure-Reactivity Relationships in Kenneth Connors's textbook Chemical Kinetics: The Study of Reaction Rates: