Hydrophobic Hydration Processes. I: Dual-Structure Partition Function for Biphasic Aqueous Systems.

The thermodynamic properties of hydrophobic hydration processes have been analyzed and assessed. The thermodynamic binding functions result to be related to each other by the mathematical relationships of an ergodic algorithmic model (EAM). The active dilution d A of species A in solution is expressed as d A = 1/(Φ·x A) with thermal factor Φ = T -(C p,A/R) and (1/x A) = d id(A), where d id(A) = ideal dilution. Entropy function is set as S = f(d id(A),T). Thermal change of entropy (i.e., entropy intensity change) is represented by the equation (dS) d = C p dln T. Configuration change of entropy (i.e., entropy density change) is represented by the equation (dS)T = (-R dln x A)T = (R dln d id(A)) T . Because every logarithmic function in thermodynamic space corresponds to an exponential function in probability space, the sum functions ΔH dual = (ΔH mot + ΔH th) and ΔS dual = (ΔS mot + ΔS th) of the thermodynamic space give birth, in exponential probability space, to a dual-structure partition function { DS-PF }: exp(-ΔG dual/RT) = K dual = (K mot·ζth) = {(exp(-ΔH mot/RT))(exp(ΔS mot/R))}·{exp(-ΔH th/RT) exp(ΔS th/R)}. Every hydrophobic hydration process can be represented by { DS-PF } = { M-PF }·{ T-PF }, indicating biphasic systems. { M-PF } = f(T,d id(A)), concerning the solute, is monocentric and produces changes of entropy density, contributing to free energy -ΔG mot, whereas { T-PF } = g(T), concerning the solvent, produces changes of entropy intensity, not contributing to free energy. Entropy density and entropy intensity are equivalent and summed with each other (i.e., they are ergodic). From the dual-structure partition function { DS-PF }, the ergodic algorithmic model (EAM) can be developed. The model EAM consists of a set of mathematical relationships, generating parabolic convoluted binding functions R ln K dual = -ΔG dual/T = {f(1/T)*g(T)} and RT ln K dual = -ΔG dual = {f(T)*g(ln T)}. The first function in each convoluted couple f(1/T) or f(T) is generated by { M-PF }, whereas the second function, g(T) or g(ln T), respectively, is generated by { T-PF }. The mathematical properties of the thermodynamic functions of hydrophobic hydration processes, experimentally determined, correspond to the geometrical properties of parabolas, with constant curvature amplitude C ampl = 0.7071/ΔC p,hydr. The dual structure of the partition function conforms to the biphasic composition of every hydrophobic hydration solution, consisting of a diluted solution, with solvent in excess at constant potential.

Introduction

In 1980, Lumry[1] proposed to consider the thermodynamic functions enthalpy and entropy in biochemical reactions as divided into two parts, namely, thermal (or compensative) and motive (or work) functions, respectively. Moreover, Lumry regretted that very rarely the dual nature of water had been considered in biochemical equilibria. Lumry[2] observed that enthalpy, entropy, and volume data obtained for processes studied in aqueous solvent generally have been assumed to apply to a solute process, without consideration of the coupling between the process and the two-state equilibrium of water. It followed then that significant chemistry knowledge deduced from reactions in aqueous media may be wrong. Most free-energy information is likely to remain uncomplicated, but enthalpy, internal energy, entropy, and volume data are generally suspected, since have they been rarely analyzed as to take the two species of water into account. Another point touched by Lumry, again about biochemical equilibria in aqueous solvents, concerned the peculiar properties of the thermodynamic functions enthalpy and entropy. Lumry recommended to consider enthalpy and entropy as composed of two parts each, thermal and motive. Lumry,[2] however, stated that thermal and motive parts of enthalpy and entropy were not usually experimentally determinable. Lumry much appreciated also the property observed by Benzinger,[3] supported by Lee and Graziano,[4] that some of the components of the thermodynamic functions compensate each other and do not contribute to free energy.

We have been studying[5−8] for the last 10 years the thermodynamic properties of the hydrophobic hydration processes, including the reactions of protein denaturation and micelle formation, specifically considered by Lumry. In contrast, however, with the pessimistic opinion expressed by Lumry, we have succeeded to calculate numerically thermal and motive components of enthalpy and entropy in every one of the many hydrophobic processes examined, by taking advantage of constant hydrophobic heat capacity ΔC. By assuming that water is composed of cluster WI, cluster (iceberg) WII, and free molecules WIII, we have built up a molecular model for hydrophobic hydration processes that yields very significant self-consistent results, notwithstanding that the experimental data are referred to apparently different processes, from protonation of carboxylato anions and noble gas solubility in water to protein denaturation, micelle formation, and many others. Several points will be addressed:whereas in class B, the reaction, with opposite phase transition, iswith iceberg reduction. It is worth mentioning that the processes of iceberg formation in class A and iceberg reduction in class B, taking place in the solvent, with reduction or increment of solvent volume, respectively, produce changes in the thermodynamic properties of the solute.

Dual-structure partition function {} = {}·{} is indicative of the biphasic structure of every hydrophobic hydration system, with the motive function {} = f(T,did(A)) referred to the reacting solute and the thermal function {} = g(T) referred to the excess solvent, respectively.

Dual-structure partition function {} is valid for every hydrophobic hydration system.

Hydrophobic heat capacity ΔC is constant in each compound, independent of T.

Thermal free energy (−ΔGth/T) referred to {} is invariably zero.

The second law of thermodynamics is related to both entropy intensity and entropy density.

Every dual-structure partition function of probability space generates, in thermodynamic space, parabolic convoluted binding functions R ln Kdual = (−ΔGdual/T) = {f(1/T)*g(T)} and RT ln Kdual = (−ΔGdual) = {f(T)*g(ln T)}, constituting an ergodic algorithmic model (EAM). In consequence, constant hydrophobic heat capacity ΔC, which is inversely proportional to the geometrical constant curvature amplitude Campl (Campl = 0.7071/ΔC) of both parabolic binding functions, results to be an invariable property of every hydrophobic hydration process.

Condition of null thermal free energy (−ΔGth/T = 0) is an invariable property of the thermal partition function {}, which is referred to a large statistical molecule population (solvent) and can be treated by methods of statistical thermodynamics.

Motive partition function {} is monocentric, with linear van’t Hoff equation (−ΔGmot/T) = f(1/T) in thermodynamic space, as in any normal monophasic system with constant Kmot. The partition function {}, being referred to an ensemble composed by very few elements (moles, solute) ruled by binomial distribution, cannot be treated by methods of statistical thermodynamics.

Consistency or inconsistency of ergodic algorithmic model (EAM) with computer simulations of free-energy functions is discussed. Computer simulations concerning the dual-structure partition function {} requires necessarily the introduction of quasi-chemical approximations,[9] associating molecule statistical distributions of {} to mole binomial distribution of {}. The potential distribution theorem (PDT),[9] which is referred to a nonexistent monophasic system, is inconsistent with the ergodic algorithmic model (EAM).

The essential reaction of the hydrophobic effect is, in class A, the reaction of iceberg formation with phase transition

This manuscript is the first part of the three-part study of hydrophobic hydration processes:

Dual-structure partition function for biphasic aqueous systems.

Entropy density and entropy intensity: ergodic algorithmic model (EAM).

Validation of (EAM) model.

In Part II, the molecular interpretation of the entropy intensity changes and of the entropy density changes is discussed. The variability of entropy density is bound, at constant temperature T, to changes of ideal dilution did(A) = 1/xA multiplied by the thermal factor = f(τm2) = T–(. The thermal factor represents, at constant temperature, the thermal energy associated with each molecule. τm2 represents the squared mean sojourn time of each molecule. Sojourn time is the time spent by each molecule to run 1 length unit. The variability of entropy intensity is bound to changes of velocity of the molecules, produced by changes of temperature, through the same thermal factor, in systems, like as a pure liquid, whereby no concentration change is possible. Both processes, variation of entropy density and variation of entropy intensity, produce changes of energy dispersal, i.e., changes of entropy. The connections of EAM with computer simulations will be discussed. Computer simulations will be conditioned by the molar reactions taking place in every hydrophobic hydration process.

In Part III, the analysis is extended to a large population of compounds of very different size and very different molecular structure. The statistical analysis over a population of about 80 compounds with about 600 experimental points has confirmed that EAM is valid in every hydrophobic hydration process, leading to unitary values of entropy change and enthalpy change, with variability at the limit of the experimental error. Statistical validation states the general applicability of EAM and indicates that the same properties of hydrophobic hydration process will be found in every such process taking place in biological ambient and in every process of drug design.

Results and Discussion

Dilution, Thermal Energy Dispersal, and Entropy

For statistical thermodynamics, the number of possible locations (configurations) for a solute molecule in the whole solution volume is a measure of the state probability of that solute. The larger the number of possible locations, the larger the solvent volume. The state probability of a solute, therefore, is proportional to the volume in which the solute is dissolved, i.e., to solute dilution. The identification of solvent-to-solute ratio (dilution) as homologous with configuration multiplicity is an important connection to statistical thermodynamics. A basic statistical setting is the relationship between state multiplicity, state probability, and entropy density. If we suppose to subdivide the solvent into many submicroscopic cells, we can consider each cell as a possible location for a solute molecule. The larger the number of accessible cells, the more probable a molecule can find a cell for location. The cell multiplicity, therefore (expressed as the number of solvent molecules per solute unit), identifies the statistical state probability Ω, calculated by Boltzmann equationIn a solution, the number of available cells is directly proportional to dilution (expressed as solvent-to-solute ratio): consequently, dilution, which is a number expressing the dispersion of solute molecules, is also homologous with the number expressing statistical state probability Ω. If the concentration cA is expressed in molar fraction (solute-to-solvent ratio), its reciprocal, ideal calculated dilution did(A) = 1/xA, represents the solvent-to-solute ratio. Ideal dilution did(A), therefore, which is a parameter of matter dispersion, corresponds to an exponential probability factor in probability spacewhereby we show how ideal dilution did(A) is an exponential function homologous with statistical state probability Ω of statistical thermodynamics of eq .

The molecules, however, tend to disperse over all of the available accessible microcells, stirred by thermal energy in such a way that the molecules carry thermal energy to every available accessible microcell. This point has been stressed by Lambert[10] who has launched a campaign to inform students that using the simple numerical probability without any mention of energetic involvement might be misleading. Lambert speaks of energetic “enablement”. The energetic involvement recommended by Lambert is essential to obtain any entropy change from matter dispersion change.

Regarding this energetic involvement, we can recall an analogy. If we let a drop of purple solute fall into a colorless liquid solvent, the thermal energy will bring the solute molecules to disperse all over the whole volume of the solvent, thus obtaining a pale pink color. Suppose that energy identifies with a purple layer covering each molecule: the color is dispersed all over the solvent volume. Dispersion of energy is analogous to dispersion of color. With dispersion of energy measured by the thermodynamic function entropy density Sdens, we can suppose Sdens to be inversely proportional to the intensity of the solution color. For the moving solute molecules, every microcell of solvent is potentially a location. Each solute molecule, however, carries with it a portion of thermal energy, supposed to be purple, in such a way that dispersion of matter measured by ideal dilution did(A) = 1/xA becomes at the same time dispersion of “purple red” energy, i.e., entropy density. Without thermal agitation, the purple solute molecules would have been resting at the initial point, so the same resting would have occurred for energy, and as a consequence, we could not observe any entropy density change. The dispersion of matter did(A), therefore, is proportional to a change of entropy density in the thermodynamic space: we can write, at constant temperature, the entropy density differentialWe can introduce the energetic “enablement” recommended by Lambert by substituting the concentration xA by activity aA, where aA = Φ·xA. The Lambert’s thermal energy factor (THEF, with Φ = T–() representing the energetic involvement recommended by Lambert’s is a measure of the thermal energy, supposed to be “purple red”, associated with each molecule. THEF is a source of ergodicity of the chemical systems. The ideal dilution did(A) = 1/xA is transformed into the active dilution dA = 1/aA: the reciprocal factor (1/Φ), therefore, transforms the parameter of matter dispersion did(A) = 1/xA into the parameter of energy dispersion dA = 1/aA and hence into an entropy parameter. We can consider that the entropy Sdens can be expressed as the function of activity of A asThe active dilution dA can be setand the differential of entropy can be rewritten asBy developing eq , we obtainThe two terms of eq , representing entropy density and entropy intensity differential changes, respectively, are related to the changes of energy dispersion at molecular level: the first term indicates the change of energy dispersion in space, i.e., change of entropy density, whereas the second term corresponds to the change of energy dispersion in time, i.e., change of entropy intensity, respectively (see Part II, Section 2, for molecular interpretations of entropy density and entropy intensity).

Equation implies the formulation of the entropy function, bound to an extended second law of thermodynamics (see Appendix A). The reappraisal of the traditional formulation of the second law is necessary because the usual expression of the second law, with condition to entropy, dS = δrevQ/T ≥ 0 (being δrevQ/T = C dT/T = C dln T ≥ 0), is inadequate. It is, in fact, clearly referred to the only thermal entropy change, or entropy intensity change due to heat transfer, in conformity with Clausius’ definition of the second law. The existence of the configuration entropy changes or entropy density changes requires the extension of the validity of the second law. Entropy intensity and entropy density are equivalent (ergodic) and are summed up or subtracted to each other.

At constant temperature (dln T = 0), the entropy change reduces to eq , but we have to remind the reader that the reciprocal THEF (1/Φ ≠ 1) is implicitly active, even if constant for [∂(1/Φ)/∂T]T = 0. In such isothermal conditions, the variation of entropy density as a function of ideal dilution ln did(A) = ln(1/xA) can be measured by experimental determinations of variations of xA: every potentiometer or pH meter, in fact, becomes, at constant temperature, an effective entropy density meter. Entropy, therefore, is a function of ideal calculated dilution did(A), which is homologous, at constant temperature, with state probability.

We can now search for other functions homologous with state probability and dependent on dilution. To endeavor these functions, we recall the Boltzmann equation for statistical ensemblewhere Ω is a partition function referred to an ensemble of molecules as the function of kB = 1.3806 × 10–23 J K–1. The number Ω is an extremely large quantity calculated by statistical mechanics methods and not accessible by experiment. We calculate the Nth root of Ω (N is Avogadro number: NAv = 6.022 × 1023) and transform Ω into the partition function ZM referred to a population of moles as the function of R (R = 8.31451 J K–1 mol–1)where ZM is a molar partition function.aZM is, in principle, experimentally accessible, being on mole chemical scale. By differentiation, we obtainat constant T, whereby we put in evidence, in comparison to eq , the parallelism between entropy density, logarithm of partition function, and logarithm of ideal dilution.

At the same time, however, this is the point strongly supported by Lambert:[10] dilution is a measure of energy dispersion (i.e., energy dispersion (or energy dilution) means entropy concentration) because the molecules, moved by thermal energy, represented by the factor (1/Φ) (reciprocal THEF) in eq , tend to spread over every accessible solvent cell, thus changing dilution. This process goes on until it reaches the minimum concentration and consequently the maximum dilution compatible with the system conditions. In such a way, dilution from parameter of matter dispersion becomes a parameter of energy dispersion, i.e., a parameter of entropy density. Lambert regrets that statistical thermodynamic authors insist on the probabilistic aspect of multiplicity without any mention of energy involvement. We have shown above how the energy “enablement” of configuration entropy, as suggested by Lambert, can be explained by introducing the reciprocal of energetic factor THEF (1/Φ = T(), multiplying the ideal calculated dilution did(A). In contrast, pure thermal changes of entropy intensity can be observed whenever heat dispersion in time, due to changed velocity of the molecules, takes place even without any concentration change ((1/xA) = 1 in eq ), as for example in a solvent or in a pure nonreacting liquid. For the solvent or a pure liquid, in fact, it is constitutionally impossible to define a concentration change. In these systems, therefore, we can observe changes of entropy intensity only.

Configuration Change of Entropy: Entropy Density

Being on search for thermodynamic functions depending on dilution, which is homologous with configuration, we have analyzed the formation constant of the equilibrium A + B = ABBy considering that did(A) = 1/xA, did(B) = 1/xB, and did(AB) = 1/cAB, the formation constant can be rewritten as a ratio of ideal dilutionswith clear connection to the settings of statistical thermodynamics, whereby the configurations as state probability are homologous with ideal calculated dilutions. It is worth noting, in fact, that if the concentration is expressed in molar fraction, the ideal dilution did(A) = (1/xA) represents the molar (solvent-to-solute) ratio. The dilution is homologous with the statistical partition function Ω, which represents the molecular (solvent-to-solute) ratio. By calculating the logarithm R ln K at temperature T, we move from probability space (K) to thermodynamic spaceand by recalling that for component Aand so on for other terms, we can writeshowing that (R ln K), being formed by a sum of entropy density terms, is itself an entropic function. The character of total entropy density of R ln K = (−ΔG°/T) conforms to the statement of statistical thermodynamics that, according to the second law of thermodynamics, a system assumes the configuration of maximum entropy, at thermodynamic equilibrium. This state probability, in fact, maximizes the discrete Gibbs entropy.

In general terms, ln K is a specific value of a general equilibrium quotient ln QK. By considering that R  dln did(B) ∼ R  dln did(AB), the differential of equilibrium quotient QK can be expressed asA change of QK corresponds, therefore, at constant temperature, to a change of dilution and hence to a change of entropy density. The partition function in probability space is homologous with dilution. The logarithm of equilibrium quotient, the logarithm of the partition function, and the logarithm of equilibrium constant can be reported along the dilution axis in the diagram (Figure ), where we report the vector representation of Gibbs equation in thermodynamic space. In this diagram, we report the configuration (dilution) change of entropy (entropy density) on the abscissa axis and the thermal change of entropy (entropy intensity) on the ordinate axis.

Figure 1

Vector representation of Gibbs equation, in thermodynamic space. x axis: (configuration) entropy density, y axis: (thermal) entropy intensity, z axis (coplanar): projection of (−ΔG°/T) on z axis: −ΔΓØ/T = (−ΔG°/T) cos(π/4).

The next function analyzed has been the chemical potential (partial molar function)a function introduced to represent the dependence of free energy from concentration of each reactantwith xA the concentration of A, in molar fraction.b

For a given temperature, a molecule has a higher chemical potential in a high concentration sector and a lower chemical potential in a low concentration sector.after differentiationand with sign changed, we obtainThis equation shows how the differential d(−μ/T) also is a configuration change of entropy density and can be reported on the dilution axis of the diagram in Figure .

The Gibbs equation, referred to a monocentric partition function, is represented in Figure as a vector (bold type) compositionThe reaction is assumed to be exothermic (ΔØ < 0). We note that {−ΔØ/} (thermal entropy intensity) is by construction necessarily equal to ΔH (change of entropy density)Both are, in fact, legs of an isosceles right triangle. The equivalence demonstrates the ergodic property of the thermodynamic system. In fact, eq states that an enthalpy divided by temperature T (i.e., entropy intensity) is transformed by projection onto the dilution axis into a configuration change of entropy (i.e., entropy density). Thus, we find on the x axis a total change of entropy density vector ΔTOT = −ΔØ/, which represents the sum of the entropy vectors ΔH (entropy density equivalent to entropy intensity) and ΔØ (change of entropy density)This vector can also be represented along dilution axis as entropy density function ((dS) ≡ (dSdens)). Every configuration change of entropy or change of entropy density in reacting mole ensembles (REMEs)c can be reported, as dilution-equivalent on x axis (Table ). We want, however, to stress once again the point that the dilution differential R  dln dA is actually enabled to represent dispersion of energy, as a change of entropy density, only because the ideal calculated dilution did(A) is associated with the active dilution d(A) to the reciprocal thermal factor THEF (1/Φ = T() (cf. eq ).

Table 1

Dilution-Equivalent Entropy Density Functions, in REME Ensembles

dS	=–R dln xA	=R dln did(A)	(dS)T = R dln did(A)b
dS	=–R dln aA	=R dln dA	(dS)T = R dln did(A)b
dS	=R dln QK	=R dln dA	(dS)T = nR dln did(A)b,a
dS	=d(−μA/T)	=R dln dA	(dS)T = R dln did(A)b
dS	=d(−ΔG/T)	=R dln dA	(dS)T = nR dln did(A)b,a
dS	=R dln ZM	=R dln dA	(dS)T = nR dln did(A)b,a

n power of A in QK.

(dS) ≡ (dSdens).

One special point is worth noting, concerning the diagram in Figure and van’t Hoff function. By determining the equilibrium constant R ln K at different temperatures for any kind of reaction, we measure in any case changes of entropy density, which can be reported on dilution axis. Then, by calculating the derivative ∂(R ln K)/∂(1/T) for these configuration data (van’t Hoff equation), we “calculate” the molar enthalpy ΔHØ. In a general regular reaction, the derivative ∂(R ln K)/∂(1/T) is a constant ΔHØ. In hydrophobic reactions, we obtain the experimental enthalpy ΔHdual that varies following a straight line as in eq . In any case, ΔH/T is an entropy intensity: this means that, by applying van’t Hoff equation, we calculate entropy intensity changes from measurements of entropy density. If, in a different experiment on the same reaction, we employ a calorimeter to measure reaction enthalpy and obtain an experimental value of molar enthalpy ΔHØ or ΔHdual equal to that previously “calculated” from concentration (i.e., configuration) determinations (cf. eq ). This result is possible because of the ergodic properties of the thermodynamic systems (cf. Part II, Section 2[23]). The ergodic condition defines the equivalence between energy dispersion in time, or entropy intensity, due to temperature T and hence velocity of molecules, and energy dispersion in space, or entropy density, due to dilution d.

The correspondence of the enthalpy term (−ΔHØ/T) or entropy intensity term in Gibbs equation with a change of entropy density term labeled ΔSH can be explained. The entropy term ΔSØ can be calculated aswhere Tmax is the temperature at which (−ΔHØ/T) → 0, obtainable by extrapolation to (1/T) = 0 of van’t Hoff equation. According to the second law and Carnot cycle, the term (−ΔHØ/T) represents thermal entropy transferred to the environment in an irreversible process. Within Gibbs equation, however, the thermal entropy term (−ΔHØ/T) is transformed and measured as an equivalent configuration entropy density termThis entropy density term ΔSH is increasing when the temperature is decreasing, with the consequence that the equilibrium constant R ln K, which is an entropy density function, is increasing at low temperature. Hence the total entropy is increasing as well. Equation represents a further example of ergodic property, based on an equivalence between entropy intensity (thermal) and entropy density (configurational). The equivalencecorresponds to equivalence of entropy (cf. Part II, Section 2[23]).

The ergodic property of a system consists, experimentally, in the parallelism between the binding functions R ln Kdual = {f(1/T)*g(T)} and RT ln Kdual = {f(T)*g(ln T)} and the binding quotients dependent on dilution R ln QK = {f(1/did(A))*g(T)} and RT ln QK = {f(did(A))*g(ln T)}, respectively (thermal equivalent dilution (TED)) (cf. Part II, Section 3[23]).

Probability Space and Thermodynamic Space

The equilibrium constant K is related to free energy in thermodynamic space bythat by transformation into an exponential becomes a monocentric partition functionin probability space. The exponential probability factor is, therefore, represented by the equilibrium constant K, which has been shown, in analogy with QK, to be homologous with ideal calculated dilution did(A). This means that state probability too is homologous with dilution.

Altogether, eqs and 29 represent the connection between probability (or dilution) space and thermodynamic space. By analogy, we can move from Gibbs equation (eq ), in thermodynamic space, to an exponential expression in probability spaceThis equation tells us that the free-energy probability factor exp(−ΔGØ/RT) can be factorized into a product of two probability factors, namely, exp(−ΔHØ/RT), which depends on the reaction enthalpy of the thermodynamic space, and exp(ΔSØ/R), which depends on the reaction entropy of the thermodynamic space.

We have shown that, following the ideas of Lumry,[1,2] notwithstanding his pessimistic opinion on the experimental accessibility of thermal and motive components of the thermodynamic functions, we have calculated separated[7] thermal and motive entropy as well as thermal and motive enthalpy, respectively. We have found (cf. Part II, Section 3) that the observed enthalpy ΔHdual determined as derivative in ∂(1/T) of the binding function R ln Kdual = {f(1/T)*g(T)} can be represented in the thermodynamic space aswhere ΔC, hydrophobic heat capacity, is a constant independent of temperature T. On the other hand, we have found that the observed entropy ΔSdual determined as a derivative in ∂T of the binding function RT ln Kdual = {f(T)*g(ln T)} can be represented in the thermodynamic space bywhere the slope ΔC is numerically equal to the slope of the enthalpy function ΔHdual for the same compound. We have obtained the expressions in eqs and 32 by applying the extrapolation of ΔHdual to T = 0 and of ΔSdual to ln T = 0, respectively, by taking advantage of constantd ΔC. We have thus identified the intercepts ΔH0 and ΔS0 as the motive components ΔHmot and ΔSmot, respectively, foreseen but not calculated by Lumry. The constancy of heat capacity ΔC is necessary by both analytical geometry and chemistry constraints (cf. Part II, Section 3).[23] We can write, thereforeandThe accuracy of the extrapolation procedure, based on ΔC constant, has been confirmed by the successive self-consistent results (cf. Part III),[24] calculated for both motive and thermal components in every element of the many reactions examined.

Equation , whereby entropy ΔSdual, experimentally determined, is the result of a summation of entropy density (ΔSmot = ΔSdens) with entropy intensity (ΔSth = ΔSints), is a further proof of ergodic property of these systems. The ergodic hypothesis assumes the equivalence of configuration changes of entropy (entropy density), depending on space variables, with thermal changes of entropy, depending on time variables (entropy intensity) (i.e., the sum ΔSdual = ΔSdens + ΔSints is valid) (see Part II, Section 2). Each function in thermodynamic space yields an exponential function in probability space, as shown in Table . Therefore, from eqs –34 of the thermodynamic space, we can pass to the probability factors in probability (dilution) spaceandThe probability factors of eqs –36 can be grouped together in a unique product partition function at dual structureThe dual-structure partition function {} of eq , valid for every hydrophobic hydration reaction, results to be the product of two distinct partition functions: a motive partition function {} = f(T,did(A)) multiplied by a thermal partition function {} = g(T). {} is of concern of the reacting solute and can give origin to configuration changes of entropy, i.e., changes of entropy density, whereas {}, concerning the solvent, can produce thermal changes of entropy, i.e., changes of entropy intensity.

Table 2

Relationships between Thermodynamic Space and Probability Space

thermodynamic space	probability space
–ΔG°/T = R ln K →	exp(−ΔG°/RT) = K
–ΔGdual/T = −ΔHdual/T + ΔSdual →	exp(−ΔGdual/RT) = Kdual = exp(−ΔHdual/RT) exp(ΔSdual/R)
–ΔGdual/T = −ΔHmot/T – ΔHth/T + ΔSmot + ΔSth = (−ΔHmot/T + ΔSmot) + (−ΔHth/T + ΔSth) →	exp(−ΔGdual/RT) = Kdual = {exp(−ΔHmot/RT) exp(ΔSmot/R)}·{exp(−ΔHth/RT) exp(ΔSth/R)}
–ΔGth/T = −ΔHth/T + ΔSth = 0 →	exp(−ΔHth/RT) exp(ΔSth/R) = ζth = 1
(dS)T = R dln did →	exp(ΔST/R) = did
dS = kB dln Ω →	exp(S/kB) = Ωa

Nonreacting molecule ensemble (NoremE): S ≡ ΔSth.

The motive thermodynamic functions enthalpy (ΔHmot) and entropy (ΔSmot) are referred, in fact (see Part II, Section 4),[23] to a reacting mole ensemble (REME), where the difference ΔH between levels is on the mole scale a multiple of RT and ΔS is on the mole scale of the order of multiples of R (R = 8.314 J K–1 mol–1; capital M = mole). The mole ensemble is constituted by few elements (moles). The variability of S is, at constant temperature, of the order of R times the differential of logarithm of reciprocal concentration, expressing the molar solvent-to-solute ratio (see eq ).

The thermal functions enthalpy, ΔHth, and entropy, ΔSth, concerning the solvent, are referred to a nonreacting molecule ensemble (NoremE; small m = molecule), which is characterized by enthalpy levels very narrowly spaced, with interlevel separation of the order of magnitude kBT on the molecule scale, where kB is the Boltzmann constant (kB = 1.3806 × 10–23 J K–1). NoremE is composed of an extremely large number of elements (molecules). NoremE is independent of concentration or dilution so that only thermal changes of entropy intensity can be produced (cf. (dS) = Cp dln T in eq ).

We set a thermal probability factor referred to a NoremEThe thermal functions in the thermodynamic space show special properties. By introducing the explicit values of the differentials, we obtainand

If we calculate the integral of eq and then divide by Tup, we obtainewhere ΔSth is clearly identical to the result of the integration of eq . Therefore, we confirm thatconform to eq is a property of any thermal partition function {}. We obtain the relation for thermal free energywhich is invariably zero in accordance with eqs and 42. This result is in contrast with the formula widely reported in the literature (cf. eq 79 in Part. II, Section 8 and citations therein),[23] whereby the thermal free energy is erroneously stated to be different from zero.

Therefore, we can set the probability thermal factor aswhere (−ΔHth/T) ≈ −ΔC ln T and ΔSth = ΔC ln T

In conclusion, the division of the thermodynamic functions into two parts, motive and thermal, proposed by Lumry (and numerically calculated by us for all of the many compounds examined), is a consequence of the special dual-structure partition function {} (cf. eq ), valid for every hydrophobic hydration process.

The introduction of the dual-structure partition function {} = {}·{} has been inspired by the suggestion of Lumry who proposed that the experimental functions ΔHdual and ΔSdual in hydrophobic hydration were separated into two parts, respectively, ΔHdual = ΔHmot + ΔHtherm and ΔSdual = ΔSmot + ΔStherm. Because a sum of exponent means a product of exponentials, we proposed the exponential product of eq . The thermal exponential is subject to the condition exp(−ΔGtherm/T) = ζth = 1, with (−ΔGtherm) = 0. This condition has been considered as indicative of a system at constant potential. The solvent, in a very diluted solution, is in excess and it does not change its concentration i.e., it is at constant potential. Therefore, the thermal function is suited to represent the properties of the solvent. The dual structure of the partition function corresponding to the biphasic composition of the system is confirmed by the curved shape of the binding functions R ln Kdual = {f(1/T)*g(T)} and RT ln Kdual = {f(T)*g(ln T)} obtained by mathematical development of the dual-structure partition function, the binding functions result to be convoluted functions, whereby the primitive (f) function, either f(1/T) or f(T) of the motive component, which is linear, is modified by the secondary (g) function, either g(T) or g(ln T) of the thermal component. We can identify the thermal partition function {} = g(T) or g(ln T) as the partition function of the nonreacting solvent and the motive partition function {} = f(1/T,1/did(A)) or f(T,did(A)) as the partition function of the reacting solute.

Ergodic Algorithmic Model (EAM)

The ergodic algorithmic model (EAM), obtained by development of the dual-structure partition function {} of eq , is based on a complex network of mathematical relationships connecting the whole set of experimental thermodynamic data (see Part II, Table 3[23]).

Motive and Thermal Ensembles

We express the total statistical probability of the thermodynamic state of the system[7,8] in analogy with eq bywhere Kdual is the observed equilibrium constant, ΔHdual is the observed reaction enthalpy, and ΔSdual is the observed entropy change. The formation constant Kdual has been demonstrated to be homologous with a partition function of statistical thermodynamics (cf. Table ). At constant temperature, reciprocal concentration (i.e., dilution) in probability space is a measure of both formation constant and partition function.

The reciprocal concentration is related to the configuration changes of entropy density (cf. eq ). The exponential factor exp(−ΔGdual/RT) of the topological probability space, therefore, is homologous with the topological space of experimental reciprocal concentrations or dilutions.

An equilibrium constant is not the only probability parameter suitable to monitor the equilibrium conditions of the system: in the solubility of gases or liquids, the parameter is saturation concentration; in micelle formation, the parameter is critical micelle concentration; in protein denaturation, the parameter is the denaturation quotient Qden; and so on.

In Table , we can note the correspondence between product probability functions (block C) and observed probability functions (block D), thus showing how the functions of the ergodic algorithmic model (EAM) conform to the observed thermodynamic functions.

Table 3

Ergodic Algorithmic Model (EAM) from Probability Space to Thermodynamic Space

A	motive function	units	eq
probability space	exp(−ΔHmot/RT) exp(ΔSmot/R) = exp(−ΔGmot/RT) = Kmot		(46)
thermodyn. space	R ln Kmot = f(1/T) = (−ΔHmot/T) + (ΔSmot) = (−ΔGmot/T)	J K–1 mol–1	(47)
	RT ln Kmot = f(T) = −ΔHmot + TΔSmot = −ΔGmot	J mol–1	(48)

As shown in eq , the thermal probability factor (or thermal partition function {}) is multiplied by a motive probability factor (or motive partition function, {}). The latter function is referred to a reacting mole ensemble (REME) (see Part II, Section 4)[23] with equilibrium constant KmotThus, we obtain a total dual-structure product probability factor {}The total probability factor Kdual is the product of two partition functions: Kdual = Kmot·ζth with ζth = 1 (cf. eq ). The constant Kmot is the motive partition function {} of the solute, referred to a reacting ensemble (REME) with (−ΔGmot/RT) ≠ 0, whereas ζth is the thermal partition function {} of the solvent, referred to a nonreacting molecule ensemble (NoremE), with (−ΔGth/RT) = 0.

The curved binding functions (convoluted functions) obtained by developing {} (see block C in Table ) can be compared to the curved binding functions obtained by interpolation of the experimental data reported in a van’t Hoff plot (see block D in Table ). The tangents ΔHdual and ΔSdual of the experimental binding functions interpolating the experimental data are calculated as a sum of two terms (ergodic) as shown in eqs and 34, respectively. From the experimental data, we obtain the following expressions for enthalpy and entropy probability factors, respectivelywhich clearly conform to eq .

Binding Functions

By passing to the logarithms of the partition functions, we move from probability space, homologous with the reciprocal concentration (dilution) space, to thermodynamic space, where we measure energy, enthalpy, and entropy (see Appendix B). We thus obtain the expressions presented in block C of Table . We note that, in Table , the observed binding functionsandare expressed in J K–1 mol–1 (entropy scale) and J mol–1 (enthalpy scale), respectively, thus confirming that they are in the thermodynamic space.

The two functions in eq and in eq , respectively (see Appendix C), present diagrams with curvature, depending on the value of ΔC. The thermal factor can be either of class A when ΔC > 0 or of class B when ΔC < 0. If ΔC > 0, the binding functions (cf. Section ) are monotonic convex at constant curvature amplitude, whereas if ΔC < 0, the binding functions are monotonic concave at constant curvature amplitude. The curvature amplitude is a constant typical of each parabola and is inversely proportional to the constant heat capacity ΔC (Campl = 0.7071/ΔC) thus showing how ΔC is constant for mathematical conditions. On the other hand, because ΔC is bound to a phase transition of water from solvent WI to solute (WII) in all of the compounds of class A or from solute (WII) to solvent WI in all of the compounds of class B, respectively (see the definitions of the properties of class A and class B), ΔC results to be constant for chemical conditions also. Any phase transition from WI (solvent) to WII (iceberg) is characterized, in fact, by constant entropy change equal to an enthalpy change divided by temperature, which means that heat capacity could be labeled as entropic function ΔC = ΔH/T = ΔS. The passage of state of pure water to form an iceberg is analogous to evaporation, although with its own characteristics, with Δs = C = 75.36 J K–1 mol–1. The interpretation of Δs = C as a constant value of entropy change for a phase transition has the important implication that the existence of the phase transition WI (solvent) → WII (iceberg) (with nw = 1) explains the abnormal high value of heat capacity of liquid water. This high value is inconsistent with a simple redistribution of energies over degrees of freedom of a nonreacting water molecule, as for usual interpretation of heat capacity. The phase transition of water, in fact, should take place even when we determine heat capacity of pure liquid water by calorimetry, thus giving a prominent contribution to heat capacity.

By passing to the logarithms as in eqs and 69 and then differentiating, we obtain the relations collected in Table . In Table , we note that, by assuming ΔC constant and independent of temperature, we arrive at identical ΔC by both enthalpy route and entropy route. The same equality of ΔC from either enthalpy or entropy is obtained by treating the experimental data. We can conclude, therefore, that the experimental data conform to the ergodic algorithmic model (EAM) and contain in themselves ΔC constant, independent of temperature. We can thus confirm once again that the heat capacity ΔC is constant for both chemistry and analytical geometry constraints.

Table 4

Derivatives of Binding Functions

enthalpy	entropy
70	71
72	73
74	75
76	77
78

Characteristic Properties of Binding Functions

The relationships between experimental data, free energy, enthalpy, and entropy as expressed by the two binding functions R  ln Kdual = {f(1/T)*g(T)} = −ΔGdualØ/T and RT  ln Kdual = {f(T)*g(ln T)} = −ΔGdualØ are reported in Table 5 of Part II. Being ΔC ≠ 0, the functions R ln Kdual = {f(1000/T)*g(T)} and RT ln Kdual = {f(T)*g(ln T)} are curvilinear (Figure a,b) represented by second-degree polynomials (see Appendix C). It is worth mentioning that the convex curves in Figure a,b are referred to the experimental solubility data of helium in water. These kinds of processes concerning gases belong to class A of the hydrophobic hydration processes. If we calculate the derivatives ∂(R ln Kdual)/∂(1/T) and ∂(RT ln Kdual)/∂T and plot them against T and ln T, respectively, we obtain linear plots (Figure a,b). We remind that the indicated derivatives correspond to the tangents at any point of the curves in Figure a,b. The value of the tangent at any point in the diagram R ln Kdual = f(1/T) (cf. Figure a) corresponds to the value of −ΔHdual, whereas the value of the tangent at any point in the diagram RT ln Kdual = f(T) (cf. Figure b) corresponds to the value of ΔSdual. In these diagrams, TH is the temperature at which ΔHdual = 0 and corresponds to the minimum of the top diagram of Figure a. Analogously, TS is the temperature at which ΔSdual = 0 and corresponds to the minimum of the respective upper diagram of Figure b. According to the ergodic algorithmic model presenting curved binding functions, the apparent dual enthalpy (i.e., the experimental enthalpy) represented by the tangent of one binding function is composed of two terms, motive (i.e., entropy density) and thermal (i.e., entropy intensity) (cf. eq ).

Figure 2

Solubility of helium[11] (class A): (a) R ln Kdual = {f(1/T)*g(T)} = (−ΔGdual/T) = f(1000/T), to calculate enthalpy (see Figure a, below); (b) RT ln Kdual = {f(T)*g(ln T)} = (−ΔGdual) = f(T), to calculate entropy (see Figure b).

Figure 3

Water solubility of helium[11] (class A): (a) enthalpy plot, ΔHdual = 0 at TH; (b) entropy plot, ΔSdual = 0 at ln TS.

The motive enthalpy ΔHmot is independent of the temperature, whereas the thermal enthalpy ΔHth depends exclusively on the temperature with proportionality factor ΔC.At TH, the enthalpy is zero becausetherefore, by introducing eq into eq , we can calculateIn this equation, the extrapolation to T = 0 has been applied, and the extrapolation to ln T = 0 will be applied in eq : these procedures are legal because ΔC is constant.

On the other hand, with the same arguments already used for enthalpy and TH, we can calculate the motive entropyThe relationships presented so far are referred to solubility data of argon that is a reaction belonging to class A with ΔC > 0 and iceberg formation. The condition ΔC > 0 characterizes the binding functions as convex.

Analogous relationships are valid for reactions belonging to class B, with iceberg reduction (see Appendix C). The curves in the diagrams of class B present a maximum instead of a minimum, i.e., are concave. This is shown in an application of these relationships referred to processes of class B (Figure a,b). Talhout et al.[12] studied the binding affinity of a series of hydrophobically modified benzamidinium chloride inhibitors binding to trypsin, using isothermal calorimetry and molecular dynamic simulation techniques. The binding functions R ln Kdual and RT ln Kdual reported as the function of (1/T) and T, respectively, show curved plots, as expected for hydrophobic hydration processes (cf. Figure a,b) of class B. The affinity between ligand benzamidinium chloride and enzyme is due presumably to the formation of hydrophobic bonds as revealed by the negative values of the reaction heat capacity ΔC (ΔC < 0). The condition ΔC < 0 characterizes the functions as concave. The diagram (−ΔG/T) = f(1/T) with a maximum confirms that we are dealing with a hydrophobic hydration process of class B. The tangent of the curve isThis tangent is variable with temperatureAt the maximum, the tangent ΔHdual is nilWe define TH as the temperature at which ΔHdual = 0.

Figure 4

Hexabenzamidimium chloride binding to trypsin[12] (class B): (a) R ln Kdual = (−ΔGdual/T) = f(1000/T), with enthalpy as tangent; (b) RT ln Kdual = (−ΔGdual) = f(T), with entropy as tangent.

Analogously, the plot (−ΔGdual) = f(T) is a curve with a maximum at a different temperature. The tangent to this curve isIn this case also, the tangent varies with temperatureAt the maximum, ΔSdual is nilWe define TS as the temperature at which ΔSdual = 0. Talhout et al.[12] report the data ΔC, TS, and TH, concerning the binding curves of eight compounds (Table ). The thermodynamic data of every compound in the list conform to the ergodic algorithmic model (EAM).

Table 5

Characteristic Points TS and TH of Thermodynamic Diagrams of Substituted Benzamidinium Chlorides Binding to Pepsina

subst. R	ΔCp,hydr (J K–1 mol–1)	TS (K)	ln TS	TH (K)	ΔSmot (J K–1 mol–1)	ΔHmot (kJ–1 mol–1)	nw
H	–400	317.15	5.759375	250.15	2303.75	100.06	–5.31
Me	–420	319.15	5.765661	254.15	2421.578	106.74	–5.57
Et	–603	315.15	5.753049	277.95	3469.088	167.60	–8.00
n-Pr	–598	320.15	5.76879	277.05	3449.736	165.68	–7.94
i-Pr	–419	336.15	5.817557	281.55	2437.557	117.97	–5.56
n-Bu	–728	321.15	5.771908	285.15	4201.949	207.59	–9.66
n-Pent	–632	328.15	5.793471	282.75	3661.474	178.70	–8.39
n-Hex	–849	321.15	5.771908	285.15	4900.35	242.09	–11.27

Data of ΔC, TS, and TH from ref (12); data of ΔSmot, ΔHmot, and nw calculated by us.

Motive Functions Disaggregated as the Functions of ξw

The separation of the thermodynamic functions ΔHdual and ΔSdual into thermal and motive components as proposed by Lumry[1,2] and calculated by us by applying the ergodic algorithmic model (EAM)can be exploited to extract very useful pieces of information from the experimental binding functions. If we accept the scheme of the ergodic algorithmic model (EAM) that separates the motive and thermal functions, we haveAt temperature TH corresponding to the maximum in the diagram (−ΔGdual/T) = f(1/T), we haveand hence (Figure ) an exact compensation between thermal and motive enthalpyBy knowing ΔC from TED, we can calculate ΔHmot (Figure ). Analogously, for entropy in class B, we haveFrom ΔC, we can calculate for each compound, by applying TED (cf. eq below)where C = 75.36 J K mol–1 is the heat capacity of liquid water.

Figure 5

Calculation of ΔHmot from the data reported by Talhout et al.[12] for the ligand benzamidium chloride (class B).

Figure 6

Calculation of ΔSmot from the data reported by Talhout et al.[12] for the ligand benzamidium chloride (class B).

The motive functions ΔHmot and ΔSmot calculated for a homogeneous set of compounds can be disaggregated by plotting them against ξw = |nw| (Figure ) of each compound of the set. The enthalpy interpolating function (Figure A) for the set of ligands (substituted benzamidinium chlorides) studied by Talhout et al.[12] isthat is coincident with the mean value of class B ⟨Δhred⟩B = +23.7 ± 0.6 kJ mol–1 ξw–1 (see Table 4a in Part III).[24]

Figure 7

Disaggregation of motive functions in class B: (A) ΔHmot and (B) ΔSmot as functions of the pseudostoichiometric number ξw.

The entropy interpolating function (Figure B) in the same group of compounds isThe slope is compared to the mean value of unitary entropy of class B ⟨Δsred⟩B = +432 ± 4 J K–1 mol–1 ξw–1. The comparison of the unitary values calculated from the data of Talhout et al.[12] with the unitary values reported in Table 5 of Part III[24] demonstrates that the foundations of the model are sound. A constant value of ΔC, independent of temperature, is a necessary property to construct the linear functions ΔHdual = f(T) and ΔSdual = f(ln T) used by Talhout et al. Specifically, the numbers ξw = |n| calculated from the slopes of the linear thermal functions can be successfully employed to disaggregate the corresponding motive functions in a homologous series of compounds. All of the numerical values of the thermodynamic functions are self-consistent. The separation of the dual functions into thermal and motive components leads to coherent numerical results. These proofs represent further validations of the EAM model itself because the statistical analysis is indisputable (cf. Section 5 in Part III).[24]

Ergodic Properties

We can complete the description of the ergodic algorithmic model (EAM) by introducing the formal mathematical expression of ergodicity of hydrophobic hydration systems.

The analysis of the binding functions of the ergodic algorithmic model (EAM) assumes that the function entropy is a measure of “energy dispersal” in the system. The concept of “dispersion” can be clearly grasped by analogy with concentration xA of species A and its reciprocal ideal dilution did,A = 1/xA. did,A is a measure of the volume in which 1 mol is dispersed (see Appendix C). If we suppose to measure a quantity ε of energy density, the dispersion of energy is given by 1/ε. The thermodynamic function entropy S is proportional to such energy dispersal, 1/ε. The dispersal of energy can take place by two different mechanisms: (a) energy dispersion in time and (b) energy dispersion in space. The dispersion in time depends on the velocity of the molecules through the variable temperature T, whereas the dispersion in space depends on variable concentration xA or better on variable dilution did,A = 1/xA. The dispersion of energy in time increases with temperature T because the temperature is proportional to the squared mean velocity of the particles. In fact, if the molecule is running faster, then the energy carried by the molecule flows more rapidly through the cell, increasing the dispersion of energy over a longer path in time unit. The dispersion of energy in time is measured by the squared mean sojourn time, τm2. The sojourn time τ of ith particle is the time spent by one molecule to run the length unit (τ = 1/l). As for dispersion in space, if we imagine that each molecule is carrying an amount of energy, when we dilute the molecules, we dilute, at the same time, the energy associated to each molecule, thus increasing the dispersion of energy in space. We call entropy intensity the dispersion of energy in time as the function of temperature T, whereas we call entropy density the dispersion of energy in space as the function of dilution did(A). We obtain in such a way for entropy densityand for entropy intensityThe ergodic hypothesis assumes that the variability in time of entropy or entropy intensity in a thermodynamic system is equivalent to the variability in space of entropy or entropy density: entropy intensity and entropy density can be summed up (ergodicity) because both are parameters of energy dispersal. We have experimentally verified that in every hydrophobic hydration system, the equivalence can be experimentally found as thermal equivalent dilution (TED): variability of R ln K as the function of 1/T is parallel to the variability of R ln K as the function of 1/did,A.

The activity of a species A is set aswhere xA is the concentration of A in molar fraction and Φ = T– is the thermal factor. The thermal equivalent dilution (TED) principle becomeswhereby a change of entropy density, as obtained from the motive partition function {} = {f(1/T)*f(did(A))} is experimentally determined by measuring a parallel change of entropy intensity by means of the binding function R ln Kdual = {f(1/T)*g(T)} at constant did(A). The potential distribution theorem (PDT),[9] based on a monocentric partition function, without distinction of entropy density and entropy intensity, is inconsistent with the dual composition of biphasic hydrophobic hydration systems (Table ).

Table 6

Ergodic System

entropy density and entropy intensity
(dSdens)T = (−R dln xA)T → (entropy density)
(dSInt)xA = (Cp,A dln T)xA → (entropy intensity)
Ergodicity
(dSdens)T = (dSInts)xA
(−R dln xA)T = (R dln did,A)T
Thermal Equivalent Dilution (TED)
(R dln did,A)T = (Cp,A dln T)xA

On the other hand, the development of the theorem PDT called “quasi-chemical approximation” (improperly defined approximation) is valid because it considers the different distribution types of the two phases: molecule statistical distribution of the solvent (entropy intensity) and mole binomial distribution of the reacting solute (entropy density) (cf. Part II, Section 10.b).[23]

Hydrophobic Heat Capacity, ΔC

ΔC, Equilibrium Constant, and TED

The ergodic algorithmic model (EAM) is completed by the relationships between binding functions and activities of the reacting species. The TED method based on the assumption of the validity of the so-called “ergodic hypothesis” (see Part II, Section 2)[23] has appeared as a potent experimental tool to evaluate the number nw of moles of water WI involved in each reaction.[8] The number nw can be either positive or negative. The number nw will be positive in the processes of class A and negative in the processes of class B. The number nw, as determined by TED, was assumed at first as being simply proportional to the volume of the incoming solute and dependent on a generic concentration of water molecules W. During subsequent researches, however, it has been proved to correspond to the real number ξw = |nw| of water clusters WI involved in each reaction (cf. eqs and 99) and as such we can consider it. This has opened the way to determine the real number ξw, whereby the water units WII enter as reactant in every hydrophobic hydration process. The absolute value ξw = |nw| was adopted because in a group of reactions (class B) nw is negative and the introduction of the absolute value transfers any change of sign to the associated thermodynamic quantity, with meaningful thermodynamic and molecular implications.

The relationship between hydrophobic heat capacity C and number ξw = |nw| through TED can be found by recalling that eq can be written as a double derivative in ln TWe can set a dissociation constant, valid for class Awhere aA, aW, and aB indicate activities of the species. The changing of the equilibrium constant at different temperatures can be represented by a serieswhere R ln K0 = (−ΔG°mot/T).

The second moment of the distribution is represented in eq . Alternatively, the second moment can be written as the first derivative of the differential changes δ1 ln aA, δ1 ln aB, δ1 ln aW induced by the first moment (derivative)Because changes δ1 ln aA and δ1 ln aB compensate each other, their contribution is nulland then the hydrophobic heat capacity is expressed byThe TED principle for a species X with activity aX stateswhere C is the heat capacity and SX is the configuration change of entropy of X.

Therefore, we obtain for the factor (aW)ξ of water WII the following equalitywhere C = 75.36 J K–1 mol–1 is the isobaric heat capacity of liquid water.

Alternatively, by changing sign to ξw, we set an association constant, valid for class Band obtainwhich represents the curvature in the binding functions of class B that present a maximum. There is a relationship between change in virtual dilution and curvature analogous to that between ΔC and curvature. This means that the curvatures depend on the number ξw = |nw|. The determination of the pseudostoichiometric number nw can be achieved by the variation of virtual dilution ∂(−nw ln aW/∂ln T ≠ 0) and hence the variation of the equilibrium constant brought about by the Lambert thermal energy factor Φ of water molecules WII when the temperature is changing. The reaction coefficient ξw is qualified as “pseudo” because it is in general noninteger, indicating the ratio between volume of incoming moiety and volume of cluster WI. We can verify that the variation of the virtual dilution dW of water WII (dW = 1/aW) is the unique cause of the curvature (as convoluted function, cf. Figure ) in the plot (−ΔGdual/T) = f(1/T), as it is ΔC.

Figure 9

(A) Slope of ΔH = g(T) is identical to (B) the slope of ΔS = g(ln T) for lysozyme. Data from Pfeil and Privalov[20] measured at five different pHs (7–2); nw = ΔC /C > 0.

The coherence and self-consistency of the numerical results obtained by applying TED can be considered as the experimental proofs of the validity of the “ergodic theory or ergodic hypothesis”: we can now speak of “ergodic properties” of thermodynamic systems, dismissing the word “hypothesis”. The energy fluxes in intensity (thermal) and density (configurational) entropy are equal (cf. Figure in Part II).[23]

ΔC and Phase Transition in Water WI

The hydrophobic hydration processes are characterized by large values of heat capacity ΔC. The values of heat capacityare calculated in general by plotting the values of the observed enthalpy change ΔHdual against T and then by interpolating the points by a straight line of equationwhere ΔC is the hydrophobic heat capacity.[13] The same rule holds for both calorimetric and van’t Hoff enthalpy. There has been, however, some debate whether this equation is exactly valid or only approximately valid. The question, in other words, is whether the heat capacity ΔC is constant and independent of temperature. We have, therefore, controlled many and many times the linearity of the experimental data. We have experimentally studied,[14] by potentiometry, the protonation constants of about 40 carboxylic acids at different temperatures. In each case, we have found that the plot ΔHdual = ΔH0 + g(T) is invariably linear with constant slope ΔC. For a homogeneous group of about 10 carboxylic acids, the value of ΔC has been found to be equal. Therefore, in this group of acids, ΔC is constant by changing not only the temperature but also the acid. On the other hand, we have proved (see Part II, Section 3) that ΔC is necessarily constant for both mathematical and chemical constraints.

We have, then, analyzed the experimental data of solubility in water of more than 50 gases and liquids as determined in other laboratories.[15] Next, we passed to the study of the denaturation processes[16,17] and then to the study of micelle formation processes.[18] All of the experimental data of every hydrophobic hydration process give origin to linear plots of the function ΔHdual = ΔH0 + g(T). This behavior conforms to the ergodic algorithmic model (EAM), whereby ΔC is necessarily constant for analytical geometry constraint. The hydrophobic heat capacity is constant for chemical constraint also because ΔC = (ΔH/T) = ΔS represents the entropy change for state passage of waterA peculiarity of the data concerning each compound examined was that the experimental data of entropy ΔSdual when plotted against ln T presented linear plots of the function ΔS = g(ln T).[17] Moreover, the slope of the diagram ΔSdual = ΔS0 + g(ln T) (i.e., ΔC) was numerically equal to the slope found in the diagram ΔHdual = ΔH0 + g(T) for the same compound. It is possible, therefore, to set an equation similar to eq The equality of the coefficients in the two diagrams is possible only if ΔC is constant and independent of temperature, in accordance with the equal curvature amplitudes of both binding functions.

All of these findings representing the experimental evidence that ΔC is independent of the temperature actually conform to the ergodic algorithmic model (EAM).

Another significant characteristic of ΔC is that in certain processes, we find ΔC > 0 and in others, ΔC < 0. Even in the compounds with negative ΔC, we find the identity of the coefficients ΔC in the diagrams ΔHdual = ΔH0 + g(T) and ΔSdual = ΔS0 + g(ln T). We have defined the processes with positive heat capacity (ΔC > 0) as belonging to class A and those with negative slope (ΔC < 0) as belonging to class B. The hydrophobic heat capacity ΔC = ±ξwC (where C = 75.36 J K–1 mol–1 is the isobaric heat capacity of liquid water) depends on the stoichiometry of the reaction with phase transition between water components WI, WII. In class A, the reaction with phase transition iswhereas in class B, the reaction with opposite phase transition iswith iceberg reduction. It is worth noting that iceberg formation in class A and iceberg reduction in class B, respectively, produce changes of the thermodynamic properties of the solute. As any entropy change, in analogy with Trouton’s law, ΔC could be labeled as ΔH/T. For any phase-transition entropy, ΔC = ΔH/T is constant, independent of the temperature. In general, the phase transition takes place for each compound at a fixed temperature; in these cases, however, the condition holds at all temperatures of measurement.

The curvature amplitude of the binding functions is inversely proportional to the constant hydrophobic heat capacity ΔC, as confirmed by an analysis (see Section ) of the experimental data concerning the denaturation of proteins.

ΔC and Curvature Amplitude of Binding Functions

Coming specifically to protein denaturation, belonging to class A, we have shown[8] that in the plot (−ΔGdual/T) = {f(1/T)*g(T)}, the curves obtained at different pH and different temperatures[19] present the same curvature. In fact, the tangent of the function isThe family of tangents of a curve (i.e., the values of ΔHdual) calculated at different values of the abscissa are straight lines of variable slopes.

By calculating the derivative of ΔHdual with respect to the variable T, we obtain the heat capacityWe note that if the original function (−ΔGdual/T) = f(1/T) were a straight line, as in normal van’t Hoff plots, the derivative of eq would be a constant at any temperature and the derivative in eq would be zero. Therefore, if ΔHdual were constant at different temperatures, we should have ΔC = 0. On the other hand, if the function (−ΔGdual/T) = f(1/T) is a curve, then ∂(Hdual)/∂T ≠ 0, and hence we can conclude that the function is not a simple function f(1/T), rather it will come out to be a convoluted function (−ΔGdual/T) = {f(1/T)*g(T)}. If by deriving further eq , we obtainthen we conclude that ΔC is constant. Therefore, the curvature amplitude of van’t Hoff plot is constant as well. The reciprocal value of ΔC is a measure of the curvature amplitude of the function (−ΔGdual/T) = {f(1/T)*g(T)}. The experimental data for DMS derivative of chymotrypsinogen[19] are reported in Figure A. If the curvature of van’t Hoff plot is constant and independent of T, a vertical displacement downward, simply by changing pH, of the experimental values of log Qden gets curves. Labeled a–d in Figure A, without altering the curvature. From a single curve, therefore, one can obtain sections displaced downward to bring every section to values of log Qden around zero. The concentration quotient Qden = αden/(1 – αden), as it is well known, can be reliably measured, in fact, at values around 1 (i.e., log Qden = 0). The displacement produced by changing pH keeps the curvature constant because the constant curvature amplitude is inversely proportional to the constant ΔC (Campl = 0.7071/ΔC). The sections a–d, in fact, of the curve measured at different pHs result to be displaced downward to bring them at ordinate values around the line log Qden = 0. The experimental curves a–d obtained at different pHs can be brought again onto the same common curve e by a simple parallel upward displacement because they have constant curvature (Figure B). This type of curve conforms to the mathematical algorithm presented in Part II, Section 3 and to the properties of a geometrical parabola (see Appendix C) g(T), as shown by an analysis of the data obtained by Pfeil and Privalov.[20]

Figure 8

Constant curvature amplitude: (A) experimental curves for DMSCGN at different pH; (B) reconstituted cumulative curve, by parallel upward displacement of curves a–d onto e (ref (19)).

These authors have reported a list of values of enthalpy and entropy for native (HN and SN) and denatured (HD and SD) lysozyme.[21,22] The thermodynamic functions have been measured at six different values of pH (7–2). From the tabulated values, we have calculated the functions Hden = HD – HN and Sden = SDD – SN (we recall that Hden and Sden are the observed experimental thermodynamic functions analogous to Hdual and Sdual, respectively). The calculated values of Hden are reported in Table , and the calculated values of Sden are reported in Table . The observed values of enthalpy and entropy as functions of T and of ln T, respectively, are shown in Figure A,B.

Table 7

Values of Denaturation Enthalpy ΔHden at Different Temperatures and Different pHs, for Lysozyme (from Pfeil and Privalov[20])

T (K)	ΔHden (kJ), pH 7	ΔHden (kJ), pH 6	ΔHden (kJ), pH 5	ΔHden (kJ), pH 4	ΔHden (kJ), pH 3	ΔHden (kJ), pH 2
273.15	71.55	71.55	71.55	71.55	71.55	71.55
283.15	137.24	137.24	137.24		137.24	137.24
293.15	203.34	203.34	203.34	203.34
298.15	235.98	235.98	235.98		235.98	235.98
303.15	269.03	269.03	269.03	269.03	269.03	269.03
313.15	335.14	335.14	335.14	335.14	335.14	335.14
323.15	400.83	400.83	400.83	400.83	400.83	400.83
333.15	466.52	466.52	466.52	466.52	466.52	466.52
343.15	532.62	532.62	532.62	532.62	532.62	532.62
353.15	599.99	599.99	599.99	598.31	599.99	599.99
363.15	664.00	664.00	664.00	664.00	664.00	664.00
373.15	730.11	730.11	730.11	730.11	730.11	730.11

Table 8

Values of Denaturation Entropy Sden at Different Temperatures and Different pHs, for Lysozyme (from Pfeil and Privalov[20])

ln T	ΔSden (J K–1), pH 7	ΔSden (J K–1), pH 6	ΔSden (J K–1), pH 5	ΔSden (J K–1), pH 4	ΔSden (J K–1), pH 3	ΔSden (J K–1), pH 2
5.610	10.46		10.46	21.34	57.74	121.75
5.646	247.27		247.27	258.15	294.55	358.57
5.681	476.14	476.14	476.14		523.42	587.01
5.698	587.43	587.43	587.43	598.31		698.73
5.714	697.05	697.05	697.05	707.93	744.33	807.93
5.747	910.86	910.86	910.86	921.74	958.14	1022.15
5.778	1117.97	1117.97	1117.97	1128.84	1165.24	1229.26
5.809	1318.80	1318.80	1318.80	1329.68	1366.08	1429.67
5.838	1513.77	1513.77	1513.77	1524.65	1561.05	1624.65
5.867	1702.89	1702.89	1702.89	1713.77	1750.17	1813.76
5.895	1886.98	1886.98	1886.98	1897.86	1934.26	1997.86
5.922	2066.06	2066.06	2066.06	2076.94	2111.25	2176.

Even the data reported by Pfeil and Privalov,[20] therefore, confirm all of the properties typical of the ergodic algorithmic model (EAM) as found in any denaturation diagram. Both convoluted binding functions R ln Kdual= (−ΔGdual/T) = {f(1/T)*g(T)} and RT ln Kdual= (−ΔGdual) = {f(T)*g(ln T)} are convex, with the same constant curvature amplitude, as shown by the linear derivatives (tangents) ∂(R ln Kdual)/∂(1/T) and ∂(RT ln Kdual)/∂T reported in Figure A,B, respectively.

Conclusions

The analysis of the procedure followed by us in the study of hydrophobic hydration processes has made possible to set an ergodic algorithmic model (EAM) based on a dual-structure (motive/thermal) partition function, {}. From this dual-structure partition function {}, a homogeneous set of parabolic binding functions R ln Kdual = {f(1/T)*g(T)} and RT ln Kdual = {f(T)*g(ln T)} can be derived, suited to describe coherently all of the properties of this important series of reactions. These results have been obtained as a development of the suggestion put forward by Lumry[1,2] of considering the thermodynamic functions enthalpy and entropy as composed of two parts, thermal and motive, respectively. By examining the theoretical foundations of this proposal, we have concluded that the system of every hydrophobic hydration process is biphasic and that the dual-structure partition function {}represents the probability state of these processes. The probability state has been demonstrated to be homologous with dilution (i.e., reciprocal concentration) and as such it is amenable of experimental determination. On the other hand, by passing to the logarithm of the partition function, we move from probability space to thermodynamic space, whereby we can experimentally determine free energy, enthalpy, and entropy. The fundamental homology relationship between ideal dilution did, partition function ZM, formation quotient QK, and formation constant K (or other equivalent potential function) has permitted to show how the mathematical expression of each function, partition function, formation constant, formation quotient, or chemical potential can be considered, at constant temperature, as a configuration (dilution) change of entropy density (dS) = (R dln did(A)) and reported on the same abscissa axis in the geometrical plane where we can plot the vector representation of Gibbs equationThe vector representation is referred to the abscissa axis, where dilution and configuration change of entropy density (dS) = (R dln did(A)) are reported, and to the ordinate axis whereby thermal change of entropy intensity (−ΔHØ/T) is reported. The vector geometry shows how there is a necessary perfect equality between thermal entropy intensity vector −ΔØ/ and an entropy density component vector ΔH. The equality is necessary because both are legs of an isosceles right triangle. The equalityis the mathematical formulation of the ergodic property of the chemical systems. The ergodic condition requires that dispersion of energy in time is equivalent to dispersion of energy in space. In terms of molecular processes, this means that thermal change of entropy intensity, produced by temperature and, therefore, by molecular velocity, equals the change of entropy density produced by dilution (see Part II,[23] Section 2).

In any hydrophobic hydration process, the hydrophobic heat capacity, ΔC results to be a remarkable characteristic quantity. The hydrophobic heat capacity ΔC is a constant that behaves as a phase-transition entropy intensity change (ΔH/T), similar to the Trouton constant. The Trouton law states that the ratio ΔHevap/Teb = ΔSevap is constant for many liquids, independent of temperature. By accepting as legal,f being hydrophobic heat capacity ΔC actually constant, the extrapolation of ΔHdual to T = 0 and of ΔSdual to lnT = 0, we have been able to calculate the function ΔHmot separated from ΔHth as well as ΔSmot separated from ΔSth, respectively. From the separate functions in the thermodynamic space, we have gone back to the probability space. Thus, we have obtained a dual-structure product partition function {} valid for every hydrophobic hydration process.

The product partition function {} of eq is the product of a motive partition function multiplied by a thermal partition function.The thermal partition function {} is referred to the solvent, whereas the motive partition function {}is referred to the solute. The solvent is represented by a nonreacting molecule ensemble (NoRemE), whereas the solute is represented by a reacting mole ensemble (REME). The thermal functions ΔHth/T and ΔSth do not contribute to free energy because they compensate each other at any temperature, giving null thermal free energy (−ΔGth/T = −ΔHth/T + ΔSth = 0). On the other hand, {} is referred to the solute, yielding non-null motive free energy, (−ΔGmot/T ≠ 0). The introduction of the Lambert thermal energy factor (THEF) Φ = T–( associated with the concentration is the source of ergodicity of the thermodynamic systems, generating the thermal equivalent dilution (TED) principle.

From {}, an ergodic algorithmic model can be developed consisting of a set of parabolic binding functions R ln Kdual = {f(1/T)*g(T)} and RT ln Kdual = {f(T)*g(ln T)} (see Appendix C and Part II,[23] Section 2). The binding functions of any hydrophobic hydration process are curved parabolic functions. The geometrical properties of the binding functions are representative of the characteristics of the thermodynamic properties of each hydrophobic hydration process. In fact, the tangents to the binding functions correspond to the observed thermodynamic functions enthalpy ΔHdual and entropy ΔSdual, respectively: ΔHdual for each compound is a linear function of T with slope ΔC, whereas ΔSdual is a linear function of lnT with identical slope ΔC. This same coefficient ΔC is inversely proportional to the constant curvature amplitude of both parabolic binding functions in every compound examined and is, therefore, a constant independent of temperature T for each compound. Moreover, the hydrophobic hydration heat capacity ΔC results to be constant, independent of temperature, for chemical conditions, too. In fact, we have shown that ΔC = ±ξwC (where C = 75.36 J K–1 mol–1 is the isobaric heat capacity of liquid water), depends on the stoichiometry of the reaction (phase transition) between water components WI, WII, and WIII. The coefficient ±ξw is the power of the ligand WII in an association or dissociation constant, K = aA{(aW)±ξ·aB–1}, where aA, (aW)±ξ, and aB indicate activity of the species. The positive coefficient +ξw is referred to a reaction of iceberg formation (class A) with dissociation constant Kdiss, whereas the negative coefficient −ξw is referred to a reaction of iceberg reduction (class B) with association constant Kass. As a function of a pseudostoichiometric coefficient ±ξw, ΔC remains the same as far as the reaction is the same. For the same reason, the coefficient ΔC is equal for both enthalpy (ΔHdual = ΔHmot + g(T)) and entropy (ΔSdual = ΔSmot + g(ln T)) functions.

By a general survey of the literature concerning the hydrophobic hydration processes, we can conclude that the proposal of considering these systems as biphasic with the inherent adoption of the dual partition function represents a novelty and is promising of profitable results in this important field.

One more point of advancement of this paper is that, by applying the thermal factor Φ = f(τm2) = T–( to the concentration parameter xA, we have opened the way to computer calculations of the ergodic properties of chemical solutions.

Table B1

Development of Constant Km (Monocentered)

Km
Probability Space
probability →	Km = exp(−ΔG°/RT)	monocentered PF
Thermodynamic Space
thermodyn. function →	ln Km = (−ΔG°/RT)	property	(B.1)
idem →	R ln Km = −ΔG°/T = – ΔH°/T + ΔS	straight line	(B.2)
idem →	∂(R ln Km)/∂(1/T) = −ΔH°	slope −ΔH°	(B.3)
idem →	RT ln Km = −ΔG° = – ΔH° + TΔS°	straight line	(B.4)
idem →	∂(RT ln Km)/∂T = ΔS°	slope ΔS°	(B.5)

Table B2

Development of Dual Constant Kduala

Kdual
Probability Space
probability →	Kdual = exp(−ΔGdual/RT)	{DS-PF}	(B.6)
Thermodynamic Space
thermodyn. function →	ln Kdual = (−ΔGdual/RT)	property
idem →	R ln Kdual = −ΔGdual/T = a + bx + cx2 (x = 1/T)	curved, class A: convex (+) or class B: concave (−)	(B.7)
idem →	∂(R ln Kdual)/∂(1/T) = b + 2cx (convolution)	–ΔH°dual = −ΔHmot ± ΔCp,hydrTb (convolution)	(B.8)
idem →	RT ln Kdual = = a′ + b′x + c′x2 (x = T)	curved, class A: convex (+) or class B: concave (−)	(B.9)
idem →	∂(RT ln Kdual)/∂T = b′ + 2c′x (convolution)	=ΔSmot ± ΔCp,hydr ln Tb (convolution)	(B.10)

Degree symbol omitted.

ΔC = ±ξwC (with C = heat capacity of water).

Table B3

Development of Constant Kmotaa

Kmot
Probability Space
probability →	Kmot = exp(−ΔGmot/RT)	{M-PF}	(B.14)
Thermodynamic Space
thermodyn. function →	ln Kmot = −ΔGmot/RT	property	(B.15)
idem →	R ln Kmot = −ΔGmot/T = −ΔHmot/T + ΔSmot	straight line, slope: −ΔHmot	(B.16)
idem →	∂(R ln Kmot)/∂(1/T) = −ΔHmot	–ΔHmot	(B.17)
idem →	RT ln Kmot = −ΔGmot = −ΔHmot + T ΔSmot	straight line, slope: ΔSmot	(B.18)
idem →	∂(RT ln K)/∂T = ΔSmot	ΔSmot	(B.19)

Degree symbol omitted.

Table B4

Development of Constant ζthaa

ζth = 1
Probability Space
probability →	ζth = exp(−ΔGth/RT) = 1, −ΔGth = 0	{T-PF}	(B.20)
Thermodynamic Space
thermodyn. function →	ln ζth = −ΔHth/RT + ΔSth/R = 0	property	(B.21)
idem →	R ln ζth = −ΔGth/T = −ΔHth/T + ΔSth = 0	=0	(B.22)
idem →	∂(R ln ζth)/∂(1/T) = 0	=0	(B.23)
idem →	RT ln ζth = 0	=0	(B.24)
idem →	∂(RT ln ζth)/∂T = ΔSth	=0	(B.25)

Degree symbol omitted.