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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, reactants differing only in a single substituent, etc.), which exhibit a linear relationship between one of the following thermodynamic parameters for describing the reactions:^[1]

(i) between the logarithm of the pre-exponential factors (or prefactors) and the activation energies

       lnA_i = α + E_a,i/Rβ

where the series of closely related reactions are indicated by the index i, A_i are the preexponential factors, E_a,i are the activation energies, R is the gas constant, and α and β are constants.

(ii) between enthalpies and entropies of activation (enthalpy-entropy compensation)

       ΔH^‡_i =  $α$  +  $β$ ΔS^‡_i

where H^‡_i are the enthalpies of activation and S^‡_i are the entropies of activation.

(iii) between the enthalpy and entropy changes of a series of similar reactions (enthalpy-entropy compensation)

       ΔH_i =  $α$  +  $β$ ΔS_i

where H_i are the enthalpy changes and S_i are the entropy changes.

When the activation energy is varied in the first instance, we may observe a related change in preexponential factors. An increase in A tends to compensate for an increase in E_a,i, which is why we call this phenomena 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, ΔH scales proportionately with ΔS. 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. Notably, one of three conditions must be met for linear free energy relationships (LFERs) to hold, the most common being that which describes enthalpy-entropy compensation. The empirical relations above were noticed by several investigators beginning in the 1920s, where they were known under several different aliases.

Related Terms

It is important to note that many of the more popular terms used in discussing the compensation effect are specific to their field or phenomena. In these contexts, the disambiguous terms are of course 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 to be painted. For the purposes of this entry it is thus important to note that different terms may be used to 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 β, all the reactions in the series should have the same equilibrium constant (K_i)

       ΔG_i(β) = α

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 β, all the reactions in the series should have the same rate constant (k_i)

       k_i(β) = exp(α)

isoequilibrium temperature : used for thermodynamic LFERs; refers to β in the equations where it possesses dimensions of temperature

isokinetic temperature (IKR) : used for kinetic LFERs; refers to β in the equations where it possesses dimensions of temperature

kinetic compensation :

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 with the equation:

       σ(T) = σ₀exp(-E_a/k_BT)

where lnσ₀ is the preexponential factor, E_a is the activation energy, σ is the conductivity, and k_B is Boltzmann's constant, and T is temperature.^[2]

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 Q.

       (ΔG^'₀ - ΔG^'_x) = Q(ΔG₀ - ΔG_x)

The above equation may be rewritten as the difference (δ) in free energy changes (ΔG):

       δΔG = QδΔG

Substituting the Gibbs free energy equation (ΔG = ΔH - TΔS) into the equation above yields a form that makes clear the requirements for LFERs to hold.

       (ΔH^' - TΔS^') = Q(ΔH - TΔS)

One should expect LFERs to hold if one of three conditions are met:

(1) δΔH 's are coincidentally the same for both the new reaction under study and the reference reaction, and the δΔS 's are linearly proportional for the two reactions being compared.

(2) δΔS 's are coincidentally the same for both the new reaction under study and the reference reaction, and the δΔH 's are linearly proportional for the two reactions being compared.

(3) δΔH 's and δΔS '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.^[3]

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 ΔH^‡_i and ΔS^‡_i:

       ΔH^‡ = βΔS^‡ + ΔH^‡₀

Inserting the Gibbs free energy equation and combining like terms produces the following equation:

       ΔG^‡ = ΔH^‡₀ - (T-β)ΔS^‡

where ΔH^‡₀ is constant regardless of substituents and ΔS^‡ is different for each substituent.

In this form, β has the dimension of temperature and is referred to as the isokinetic (or isoequilibrium) temperature.^[4]

Alternately, the isokinetic (or isoequilibrium) temperature may be reached by observing that, if a linear relationship is found, then the difference between the ΔH^‡'s for any closely related reactants will be related to the difference between ΔS^‡'s for the same reactants:

       δΔH^‡ = βδΔS^‡

Using the Gibbs free energy equation,

       δΔG^‡ = (1 - T/β)δΔS^‡

In both forms, it is apparent that the difference in Gibbs free energies of activations (δΔG^‡) 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.

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 ^[5]

Criticism

Kinetic relations have been observed in many systems and gone by many terms since their inception, among which are the Meyer-Neldel effect or rule^[6], the Barclay-Butler rule^[7], the theta rule^[8], and the Smith-Topley effect^[9]. 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 ^[10].

Athel Cornish-Bowden^[11] has argued enthalpy-entropy is probably a statistical artifact, which could result in observing a relationship between two variables which may at first seem to be trustworthy, but is false because the relationship being observed does not actually exist. Sometimes a graph that is plotted against temperature will be used to attain a value of enthalpy and entropy, when in fact this is not the proper way to measure these values. Assume a linear, Arrhenius plot of the natural log of the rate constant, ln(k), versus inverse temperature T⁻¹. Both the intercept and slope of this plot are used alongside transition-state theory to reveal a linear plot of ΔS^‡ versus ΔH^‡. This approach is fine for ΔH^‡. On the other hand, the ΔS‡ value is derived from an extrapolation that can be as high as twenty times the range of the measured data. One cannot hope for more than a broad idea of the true value of ΔS^‡.

Plots of ΔH^‡ vs. ΔS^‡ have been shown to exhibit an extremely strong linear relationship. Criticism of the enthalpy-entropy compensation arises from this strong correlation, which is “too good to be true.” It is believed these plots are a result of looking at the same variable in two different ways, and nothing experimentally significant is being shown.

Ideally, calorimetry experiments can yield the proper measurements of enthalpy and entropy.

References

^ Liu, L.; Guo, Q. X. Isokinetic relationship, isoequilibrium relationship, and enthalpy-entropy compensation. Chem. Rev. 2001, 101, 673-695.
^ 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, 2909-2913.
^ Anslyn, E. V.,; Dougherty, D. A., Modern physical organic chemistry; University Science: Sausalito, CA, 2006; pp. 466-470.
^ Exner, O. The enthalpy-entropy relationship. Prog.Phys.Org.Chem 1973, 10, 411-482.
^ 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, 355-378.
^ 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.
^ Barclay, I. M.; Butler, J. A. V. The entropy of solution. Transactions of the Faraday Society 1938, 34, 1445-1454.
^ Schwab, G. M. About the Mechanism of Contact Catalysis. Advances in Catalysis 1950, 2, 251-267.
^ Manche, E. P.; Carroll, B. Compensation and the Smith-Topley Effects. Thermochimica Acta 1979, 31, 387-390.
^ 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.
^ Cornish-Bowden, Athel (March 2002). "Enthalpy–entropy compensation: a phantom phenomenon". Journal of Biosciences. 27 (2): 121–126. doi:10.1007/BF02703768. PMID 11937682.{{cite journal}}: CS1 maint: date and year (link)

Category:Thermodynamics Category:Chemical thermodynamics

[1] Liu, L.; Guo, Q. X. Isokinetic relationship, isoequilibrium relationship, and enthalpy-entropy compensation. Chem. Rev. 2001, 101, 673-695.

[2] 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, 2909-2913.

[3] Anslyn, E. V.,; Dougherty, D. A., Modern physical organic chemistry; University Science: Sausalito, CA, 2006; pp. 466-470.

[4] Exner, O. The enthalpy-entropy relationship. Prog.Phys.Org.Chem 1973, 10, 411-482.

[5] 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, 355-378.

[6] 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.

[7] Barclay, I. M.; Butler, J. A. V. The entropy of solution. Transactions of the Faraday Society 1938, 34, 1445-1454.

[8] Schwab, G. M. About the Mechanism of Contact Catalysis. Advances in Catalysis 1950, 2, 251-267.

[9] Manche, E. P.; Carroll, B. Compensation and the Smith-Topley Effects. Thermochimica Acta 1979, 31, 387-390.

[10] 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.

[11] Cornish-Bowden, Athel (March 2002). "Enthalpy–entropy compensation: a phantom phenomenon". Journal of Biosciences. 27 (2): 121–126. doi:10.1007/BF02703768. PMID 11937682.{{cite journal}}: CS1 maint: date and year (link)

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