# Ergodic Algorithmic Model (EAM), with Water as Implicit Solvent, in Chemical, Biochemical, and Biological Processes

^{*}

## Abstract

**:**

_{eq}= {f(1/T)* g(T)} and RTlnK

_{eq}= {f(T)* g(lnT)}, where the primary linear functions f(1/T) and f(T) are modified and transformed into parabolic curves by the secondary functions g(T) and g(lnT), respectively. Convoluted functions are consistent with biphasic dual-structure partition function,

**{DS**-

**PF**

**} = {M**-

**PF**

**} ∙ {T**-

**PF**

**} ∙ {ζ**

_{w}}**,**composed by (

**{M**-

**PF**

**}**(Density Entropy),

**{T**-

**PF**

**}**) (Intensity Entropy), and

**{ζ**(implicit solvent). In the present paper, after recalling the essential aspects of the model, we outline the importance of considering the solvent as “implicit” in chemical and biochemical reactions. Moreover, we compare the information obtained by computer simulations using the models till now proposed with “explicit” solvent, showing the mess of information lost without considering the experimental approach of the EAM model.

_{w}}## 1. Introduction

_{dual}, or other potential function RlnK

_{eq}, measured at different temperatures, as the function of (1/T), obtaining the convoluted parabolic binding function α) RlnK

_{dual}= {f(1/T)* g(T)}. If we plot the values of RTlnK

_{dual}as the function of T we obtain the convoluted parabolic binding function β) RTlnK

_{dual}= {f(T)* g(lnT)}. The binding functions are convoluted functions, where the primary linear functions f(1/T)and f(T)are modified and transformed into parabolic curves by the secondary functions g(T) and g(lnT), respectively. The constant factor R (gas constant) is introduced as a normalisation constant, to guarantee that we are referring to an Avogadro number of particles. Convoluted functions are consistent with biphasic dual-structure partition function,

**{DS**-

**PF**

**} = {M**-

**PF**

**} ∙ {T**-

**PF**

**} ∙ {ζ**composed by (

_{w}},**{M**-

**PF**

**}**(Density Entropy),

**{T**-

**PF**

**}**) (Intensity Entropy), and

**{ζ**(Implicit Solvent). In contrast, the functions obtained by simulation methods do not consider the free energy as a thermodynamic potential function (−ΔG/RT) = (−ΔH/RT) + (ΔS/R) composed by two terms (Intensity Entropy (−ΔH/RT) and Density Entropy (ΔS/R)), specific for each hydrophobic hydration process. Simulations, referred to a conventional mono-phasic potential function or pseudo-free energy function, as proposed by other researchers, ignore the existence of the different structures of Intensity Entropy (−ΔH/RT) and Density Entropy (ΔS/R). The formal functions used in simulations are inconsistent with the dual biphasic structure of every experimental hydrophobic hydration system, with partition function composed by the product of a partition function

_{w}}**{M**-

**PF**

**} ∙ {T**-

**PF**

**}**multiplied by the partition function of the implicit solvent (

**{ζ**). The list of thermodynamic information elements provided by the Ergodic Algorithmic Model (EAM) and calculated by us for about fifty compounds was analysed by statistical analysis methodologies and successfully validated [8].

_{w}}#### Thermal Equivalent Dilution (TED): Ergodicity

_{w}C

_{p}

_{,w}dlnT = Rdlnd

_{id,A}

_{id,A}= 1/x

_{A}is ideal dilution; x

_{A}is the molecular fraction of species A; C

_{p}

_{,w}= 75.36 J K

^{−1}mol

^{−1}is the molar heat capacity of liquid water and R = 8.314 J·mol

^{−1}K

^{−1}is the gas constant. The expression in Equation (1) can be considered as the ergodicity parameter of a system.

_{A}/R) and y = exp(−ΔH

_{A}/RT). By referring one equation to a given process, we choose one hyperbole of the family, corresponding to a point P on the auxiliary axis z of the diagram. Then, by choosing (Figure 1b) the value of ordinate y (exp(−ΔH

_{A}/RT)) and the value of abscissa x (exp(ΔS

_{A}/R)) of point A (or of point B, or C), we choose a point on that hyperbola, thus reaching all the information elements available. Instead, this point A (or B or C) cannot be chosen by computer simulation. This is the essential information element that we lose by employing computer simulations: we cannot fix a point referring to the specific system on the chosen hyperbola. On the auxiliary coplanar axis z, we read the scale of values of point P, exp(−ΔΓ/RT) = (exp(−ΔG/RT) cos 45°, with x

_{P}= y

_{P}. By computer simulation, we might perhaps choose the correct hyperbola P, but we cannot fix any point A for a specific reaction, losing all the essential information elements carried by the coordinates of A, −ΔH

_{A}/RT, Intensity Entropy and ΔS

_{A}/R, Density Entropy, respectively.

## 2. Free Energy

#### 2.1. Thermodynamic Free Energy

_{A}represents Density Entropy and the ordinate y

_{A}represents Intensity Entropy. The hyperbolas of the exponential diagram are transformed into a set of parallel lines, orthogonal to the diagonal auxiliary axis (−ΔΓ/T) = (−ΔG/RT)

**∙**(cos 45°). The coordinates of point A can be determined by experimental determination of RlnK

_{mot}= (−ΔG/T) = (−ΔH/T) + (ΔS/T), measured at different temperatures T, and analysed following the Ergodic Algorithmic Model (EAM). In contrast, computer simulations [10] might even guess (Figure 1a) the correct vector P (−ΔΓ/T), marked in blue in Figure 1a, but ignore the vectors (

**ΔS**

_{A}) + (

**−**

**ΔH**

_{A}

**/T**), marked in red in Figure 2b, thus losing all the information elements carried by these vectors. Computer simulations, in search for a potential function RlnK = f(x) to represent as the function of a variable x, calculate the function

_{(P)}= (−ΔG(P)°/RT) = (−

**ΔΓ**

**/RT**)/(cos 45°)

_{(P)}/T) = (ΔS

_{(P)}) on the diagonal, thus cancelling any difference, between the enthalpy parameter (or Intensity Entropy) and the entropy parameter (or Density Entropy). The computer-simulated function, not considering the difference between the two terms, Intensity Entropy and Density Entropy, cancels all the information elements contained there, and characterises each specific reaction, A, B, or C, at points A, B, or C, respectively. By applying the Ergodic Algorithmic Model (EAM) to a set of potential parameters (e.g., RlnK

_{dual}or RTnK

_{dual}), referring to one specific hydrophobic hydration process, experimentally determined at different temperatures, one can calculate the binding functions

_{dual}= {f(1/T)·g(T)}

_{dual}= {f(T)·g(lnT)}

_{mot}= (−ΔH

_{mot}/T) + (ΔS

_{mot})

_{mot}= (−ΔH

_{0}) + (TΔS

_{mot})

_{w}, the pseudo-stoichiometric number of water molecules W

_{I}. The pseudo-stoichiometric number ±ξ

_{w}is calculated from the curvature of the binding function, that is a concave function, with ΔC

_{p}

_{,}

_{hydr}= ξ

_{w}C

_{p}

_{,w}>0 for Class A, or a convex function, with ΔC

_{p}

_{,hydr}= −ξ

_{w}C

_{p}

_{,w}< 0 for Class B. ξ

_{w}= |n

_{w}| is the number of water clusters W

_{I}involved in each process, and C

_{p}

_{,w}= 75.36 J·K

^{−1}mol

^{−1}is the molar heat capacity of liquid water.

_{simul}= f(−ΔG/T), without specifying coordinates of any point A, B, or C, etc. The assignment of these equations means specifying, in the hyperbolic diagrams, the coordinates of each point, A, B, or C, each point referring to one specific different hydrophobic reaction, with specific properties and specific interrelations between properties for each family of compounds.

_{simul}is not suited to the study of these systems because it refers to a model of a monophasic system, not conforming to the biphasic composition [7] with implicit solvent of diluted aqueous systems and of biological solutions. The inadequacy of the monophasic model for simulations leads to the loss of essential information elements and to a pseudo-free energy potential function.

#### 2.2. Pseudo-Free Energy Function

_{simul}= f(−ΔG/T). They optimize the function exp(−ΔG/RT), but forget that thermodynamic free energy should be written and calculated as

**·**exp(ΔS/R)

_{w}= 1, referring to water as an implicit solvent at constant thermodynamic potential μ

_{s}.

- (i)
- (ii)
- (iii)
- (iv)

**·**exp(ΔS/R)

**+**(ΔS/R)

_{s}(implicit solvent) and a solution with ergodic properties. These systems are represented in probability space by a dual-structure partition function.

## 3. Explicit Solvent and Implicit Solvent

**{DS**-

**PF**

**} = {M**-

**PF**

**} · {T**-

**PF**

**}**·

**{ζ**

_{w}

**}**of the aqueous systems as proposed by EAM model [9].

**{ζ**

_{w}

**}**represents the partition function of implicit solvent, at constant thermodynamic potential μ

_{s}. These words explain the difference between pseudo-free energy calculations and the Ergodic Algorithmic Model (EAM). Pseudo-free energy calculations are applied to “explicit” solvent molecular systems, with homogenous composition whereas EAM requires an “implicit” solvent model for a dual system with biphasic composition. The dual system is composed by solvent in excess (“implicit” solvent) and diluted solute, as proved by statistical analysis of errors, extended to a large population of hydrophobic hydration processes [8]. Specifically, we assume [9] that every aqueous solution, and particularly biological solutions, has a biphasic composition, constituted by a solvent at constant potential (i.e., “implicit” solvent) and a diluted solute. None of the systems studied by perturbation theory [10,11] and by free energy perturbation (FEP) calculation [15] with determination of gradients and integration thereof, i.e., the partition function

**{ζ**

_{w}

**},**give any contribution to the free energy of the whole system, conforming to the characteristics of the “implicit” solvent. The assumption of the “explicit” solvent for these systems would mean introducing a specific variable partition function for the explicit solvent. This hypothetical function should be suited to calculate a variable free energy contribution by the solvent, i.e., by the component at constant potential, whose contribution to free energy is constantly zero. In other words, we should introduce a partition function for the explicit solvent with the properties of implicit solvent, which would mean accepting the implicit solvent as the correct choice.

## 4. Free Energy: Intensity Entropy and Density Entropy Components

_{mot}= (−ΔH

_{mot}/T) + (ΔS

_{0})

_{mot}= (−ΔH

_{0}) + (ΔS

_{mot})T

**A**→

**B**) = F

_{B}− F

_{A}= k

_{B}T ln < exp[−(E

_{B}− E

_{A})/k

_{B}T] >

_{B}is Boltzmann’s constant. The triangular brackets < and > denote an average over a simulation run for state A. In practice, one runs a normal simulation for state A, but each time a new configuration is accepted, the energy for state B is also computed. The difference between states A and B may be in the atom types involved, in which case the ΔF obtained is for “mutating” one molecule onto another, or it may be a difference of geometry, in which case one obtains an enthalpy map along one or more reaction coordinates. In practice, the averages in Equation (14) are obtained using the so-called λ-dynamics: E(λ) = (1−λ) · E

_{A}+ λ · E

_{B}, where λ changes smoothly from 0 to 1 (i.e., from state A to state B) in order to improve the sampling of states [25].

## 5. Procedure According to Ergodic Algorithmic Model (EAM)

**{DS**-

**PF**

**}**=

**{M**-

**PF**

**}**·

**{T**-

**PF**

_{th}

**}**· {ζ

_{w}}, whereby the Motive Partition Function

**{M**-

**PF**

**}**, referring to Density Entropy, is multiplied by a thermal partition function

**{T**-

**PF**

**}**, referring to Intensity Entropy, and by {ζ

_{w}}, at constant potential (implicit solvent). In mathematical format, we consider a partition function

_{dual}/RT) = {(exp(−ΔH

_{mot}/RT) (exp(ΔS

_{mot}/R)} = K

_{dual}= K

_{mot}·ζ

_{w}

_{w}= 1. According to EAM, the integration of differential functions cannot be applied to free energy calculation because the chemical reacting component, represented by

**{DS**-

**PF**

**},**is ruled by binomial distribution, with the partition function composed by a limited sum of finite arithmetic elements. EAM, however, considers an infinitesimal statistical distribution of molecules within sublevels h

_{j}

_{,i}of each macrolevel H

_{i}(see Figure 3).

_{i}to H

_{i}

_{+1}of the transformed molecule. This level-to-level passage can be a continuous process suitable for integration. The whole reaction also involves (Figure 4) a change in the multiplicity of states and this transformation of moles can be expressed by finite numbers, suitable for mathematical finite transformation corresponding to the coefficients of chemical equation. We can consider the contributions by integration terms as producing Intensity Entropy whereas the contributions by multiplicity mathematical terms as producing Density Entropy, respectively. The question now is: how can we exploit the computer programs on the market to calculate the Ergodic Algorithmic Model?

**A**→

**B**) = H

_{B}− H

_{A}= k

_{B}T ln < exp(<(−H

_{B}+ H

_{A})/k

_{B}T) >

- (a)
- Calculation of convoluted binding functions from experimental determinations; then, by applying EAM we derive Rln K
_{mot}. - (b)
- Calculation of simulation functions according to HEP.

_{0}to level H

_{1}with a gradual change in molecular structure from one to another shape, with a change in wave function. This passage consists of a change in Intensity Entropy: the whole mole passage of Intensity Entropy corresponds to the experimental determination of −ΔH/T in the not Hoff function or similarly, in EAM to the slope of RlnK

_{mot}= −ΔH/T + ΔS. The multiplicity within each macrolevel corresponds to Density Entropy change and is measured in experimental thermodynamic space by the ΔS component and in EAM by the slope of RTlnK

_{mot}= −ΔH + T ΔS. There is correspondence between the two components of the equation −ΔG/T = −ΔH/T + ΔS and the terms calculated from the experimental data by applying RlnK

_{mot}in EAM.

## 6. Thermodynamic Functions and Information Elements

_{dual}= {f(1/T)·g(T)} and by the primary function f(T) of the binding function RTln

_{dual}= {f(T)·g(lnT)}. The Ergodic Algorithmic Model (EAM) consists of the calculation and processing of the convoluted binding functions, to obtain the motive functions RlnK

_{mot}and RTlnK

_{mot}(Table 2). The two motive functions contain the correct information elements concerning the leading hydration process of the reaction, together with the thermodynamic properties of the chemical process associated with hydration. In carboxylic acids, for example, the reaction associated with hydration is protonation of the base.

_{dual}= {f(1/T)·g(T)} and RTln

_{dual}= {f(T)·g(lnT)} have a parabolic shape. The sign of these binding functions is sensible to the sign of ΔC

_{p}

_{,hydr}(Figure 5).

## 7. Dual-Structure Partition Function. Molecular Structures

_{simul}= f(1/T) is monocentric and with a Boltzmann distribution with “explicit solvent”, whereas EAM assumes that hydrophobic systems are represented by a dual-structure partition function

**{**

**DS-PF**

**}**=

**{**

**M-PF**

**}**·

**{**

**T-PF**

**}**· {

**ζ**

_{w}

**}**

**{M**-

**PF**}, referring to the Density Entropy component ((−ΔS)), is multiplied by a thermal partition function

**{T**-

**PF**

**}**, referring to the Intensity Entropy component (−ΔH/T), as finite mathematical components. The partition function

**{ ζ**

_{w}

**}**refers to the solvent at constant thermodynamic potential μ

_{s}.

_{dual}= {f(1/T)·g(T)} and RTlnK

_{dua}

_{l}= {f(T) · g(lnT)} are obtained. From the secondary function g(T) of the convoluted binding function, RlnK

_{dual}= {f(1/T)·g(T)},

- (a)
- (Reaction A {–ξ
_{w}W_{I}(solvent) → ξ_{w}W_{II}(iceberg, solute)}). - (b)
- Convexity (ΔC
_{p}_{,hydr}= ξ_{w}C_{p}_{,w}> 0). - (c)
- Iceberg size and water stoichiometry: ξ
_{w}.

_{dual}= {f(T) g(lnT)}.

- (a)
- (reaction: B {–ξ
_{w}W_{II}(iceberg, solute)} → ξ_{w}W_{I}(solvent). - (b)
- Concavity (ΔC
_{p}_{,hydr}= ξ_{w}C_{p}_{,w}< 0). - (c)
- Iceberg size and water stoichiometry: ξ
_{w}.

_{0}

^{(}

^{ξ}

^{w}

^{ = 0)}= –86.4J K

^{−1}mol

^{−1}, (see reference [5]) with the Trouton constant ΔH

_{eb}/T

_{eb}= ΔS

_{evap}= +86.9 ± 1.4 J K

^{−1}mol

^{−1}, we can infer that the passage of gas molecules from the free state (Figure 8) to the trapping into the solvent implies a change in entropy (as change in kinetic energy) that is exactly the opposite of that for the passage from liquid state to vapor state (i.e., configuration entropy change for the condensation of gas is equal to the opposite of evaporation, –ΔS

_{evap}).

## 8. Structures of Thermodynamic Functions

- (1)
- The simulated free energy functions RlnK
_{simul}, as calculated by any simulation functions, either FEP, or TI, MV, MC, etc., and applied to hydrophobic hydration processes are inadequate to represent the properties of biphasic systems with implicit solvent. Many important thermodynamic functions are ignored by these calculations and all the important information elements carried by these functions are completely lost. - (2)
- The hydrophobic hydration systems have a biphasic composition, constituted by a “non-reacting” molecule ensemble, at constant potential formed by the solvent and by a “reacting” mole ensemble formed by diluted solutes. The distribution of molecules over the sublevels h
_{i}_{,j}of each macrolevel H_{i}follows a Boltzmann distribution, whereas the distribution of moles of reactants over macrolevels H_{i}follows a binomial distribution. The Ergodic Algorithmic Model (EAM) is adequate to the biphasic composition of aqueous solutions and suited to evaluate correctly the many essential information elements.

_{app}= {f(1/T)·g(T)} and (b) RTlnK

_{app}= {f(T)·g(lnT)} as developed from a dual-structure partition function

**{DS**-

**PF**

**}**=

**{M**-

**PF**

**} · {T**-

**PF**

**},**where the thermal partition function

**{T**-

**PF**

**}**is the thermal partition function of implicit solvent.

## 9. Conclusions

_{s}, to calculate the thermodynamic properties of hydrophobic hydration systems, is highly recommended, instead of computer simulations with explicit solvent, the evaluation of only binding affinity not being enough either from a speculative or practical point of view. Every hydrophobic hydration process necessarily has the properties of an ergodic model, statistically validated. For a correct thermodynamic analysis of any hydrophobic hydration process in chemical, biochemical, and biological systems, these properties must be searched for.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Probability Space |

Dual-Structure Partition Function (for Biphasic Systems) |

{DS-PF} = {T-PF} · {M-PF}{ζ_{w}}} = K_{mot}·1 |

{M-PF} = exp(ΔS_{mo}_{t}/R), {T-PF} = exp(−ΔH_{mot}/RT), exp(−ΔG_{mot}/RT) = K_{mot} |

exp(−ΔH_{mot}/RT) exp(ΔS_{mot}/R) = exp(−ΔG_{mot}/RT) = K_{dual} |

exp(S_{dens}/R) = a_{A}^{−1} = x_{A}^{−1} Φ^{−1} = x_{A}^{−1}· T^{Cp,}^{A/R} |

exp(S_{int}/C_{p,}_{A}) = T |

K_{dual} = K_{mot} · ζ_{th} |

ζ_{th} = 1 (Implicit Solvent, at constant Thermodynamic Potential μ_{s}) |

{M-PF} = K_{mot} → (Solute, Density Entropy parameter) |

{T-PF} = ζ_{th} = 1 → (Solute, Intensity Entropy parameter) |

a_{A} = x_{A} · Φ (a_{A} = activity of A, Φ = T^{−Cp,}^{A/R}: Lambert thermal factor) |

Ergodicity → (Dilution → Temperature) |

(1/x_{A}) = d_{id}_{(A)} → dilution thermal factor T^{Cp,}^{A/R} = 1/Φ |

Thermodynamic Space |

(A) Binding Function RlnK_{dual} = (−ΔG_{dual}/T) = {f(1/T)·g(T)} |

1.Curved convoluted function ({f(1/T)_{*}g(T)}) |

Rln K_{dual} = {(−ΔH_{dual}/T) + (ΔS_{dual})} = {(−ΔH_{mot}/T) + (−ΔH_{th}/T)} + {(ΔS_{mot}) + (ΔS_{th})} |

∂(RlnK_{dual})/∂(1/T) = −ΔH_{dual} = −ΔH_{mot} − ΔH_{th} = −ΔH_{mot} − ΔC_{p.hydr}T (J mole^{−1}) |

2. Linear function (f(1/T)) |

RlnK_{mot} = RlnK_{dual} ={(−ΔH_{dual}/T) + (ΔH_{th}/T)} + {(ΔS_{dual}) − (ΔS_{th})} = −ΔH_{mot}/T + ΔS_{mot} |

∂(RlnK_{mot})/∂(1/T) = −ΔH_{mot} (J mole^{−1}) |

(B) Binding Function RTlnK_{dual} = (−ΔG_{dual}) = {f(T)·g(lnT)} |

1.Curved convoluted function ({f(T)·g(lnT)}) |

RTln K_{dual} = {−ΔH_{dual} + T ΔS_{dual}} = {−ΔH_{mot} − ΔH_{th}} + T {ΔS_{mot} + ΔS_{th}} |

∂(RTlnK_{dual})/∂T = ΔS_{dual} = ΔS_{mot} + ΔS_{th} = ΔS_{mot} + ΔC_{p,hydr}lnT (J K^{−1} mole^{−1}) |

2.linear function (f(T)) |

RTlnK_{mot} = {−ΔH_{dual} + ΔH_{th}} + T {ΔS_{dual} − ΔS_{th}} = −ΔH_{mot} + T ΔS_{mot} |

∂(RTlnK_{mot})/∂T = ΔS_{mot} |

(C) Activity: a_{A} = f(x_{A},T) |

dS_{Dens} = −Rdln a_{A} = (−Rdln x_{A})_{T} + (C_{p}_{,A} dlnT)_{x} |

(D) Ergodicity |

Density Entropy (ΔS/R) and Intensity Entropy (ΔH/RT) |

(dS_{Dens})_{T} = (−Rdln x_{A})_{T} = (Rdlnd_{id}_{,A})_{T} → (Changing Density Entropy) |

(dS_{Int})_{x}_{A} = (C_{p}_{,A} dlnT)x_{A} → (Changing Intensity Entropy) |

TED (Thermal Equivalent Dilution) |

(dS_{Den})_{T} = (dS_{Int})x_{A} |

n_{w} (Rdln d_{id}_{,A})_{T} = n_{w} (C_{p,}_{A} dlnT) x_{A} |

(E) Hydrophobic Heat Capacity |

ΔC_{p,hydr} = ±ξ_{w} C_{p,w}; C_{p,w} = 75.36 J K^{−}^{1}mol^{−1} (molar heat capacity for liquid water) |

## Appendix B

**Table A2.**Validation of the Ergodic Algorithmic Model (EAM). For analysis of unitary thermodynamic functions (*), see reference [8].

(a) Analysis within Classes | ||

Class A:iceberg formation | Unit | Relative error |

<Δh_{for}>_{A}= −22.7 ± 0.07 | kJ⋅mol^{−1} ⋅ξ_{w}^{−1} | ±3.1% |

<Δs_{for}>_{A} = −445 ± 3 | J⋅K^{−1}⋅mol^{−1}⋅ξ_{w}^{−1} | ±0.7% |

Class B:iceberg reduction | Unit | Relative error |

<Δh_{red}>_{B} = +23.7 ± 0.6 | kJ⋅mol^{−1} ⋅ξ_{w}^{−1} | ±2.51% |

<Δs_{red}>_{B} = +432 ± 4 | J⋅K^{−1}⋅mol^{−1}⋅ξ_{w}^{−1} | ±0.9% |

(b) Comparison among Classes | ||

Enthalpy | Entropy | |

<Δh_{for}>_{A} = −22.7 ± 0.7 kJ mol^{−1} ξ_{w}^{−1} | <Δs_{for}>_{A} = −445 ± 3J K^{−}^{1} mol^{−}^{1} ξ_{w}^{−1} | |

<Δh_{red}>_{B} = +23.7 ± 0.6 Kj mol^{−1} ξ_{w}^{−1} | <Δs_{red}> _{B} = +432 ± 4 J K^{−}^{1} mol^{−}^{1} ξ_{w}^{−1} | |

mean abs.value <Δh>_{A,B} = 22.95 ± 0.75 | mean abs.value <Δs>_{A,B} = 438.5 ± 6.5 | |

Student’s ratio: 0.75/0.92 = 0.815 | Student’s ratio: 6.5/5 = 1.3 |

**Hypothesis**: absolute values in Class A and B are equal

**: hypothesis accepted (mean in Class A = mean in Class B with sign reversed).**

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**Figure 1.**Hyperbolic partition functions in probability space. (

**a**) exp(−ΔΓ/T) = exp(−ΔG·(cos 45°)/RT); (

**b**) exp(−ΔG

_{A}/RT) = exp(−ΔH

_{A}/RT) exp(ΔS

_{A}/R).

**Figure 2.**Vector representation of free energy in thermodynamic space (

**a**) RlnK = (

**−**Δ

**Γ**

**/T**)

**/**cos 45° = (−ΔG/T); (

**b**) RlnK =

**−**Δ

**G**/

**T**=

**−**Δ

**H**/

**T**+ Δ

**S**.

**Figure 3.**The distribution of molecules over sublevels h

_{i}

_{,j}of each macrolevel H

_{i}follows a statistical distribution: passage of molecules from population h

_{0}

_{,j}to population h

_{i}

_{,j}can be calculated by integration (HEP).

**Figure 4.**The distribution of moles over each macrolevel H

_{i}follows a binomial mathematical distribution, not calculated by integration. Now, we have two methods for calculating partition functions.

**Figure 5.**(

**a**) The convexity of the parabolic function depends on the positive coefficient (ΔC

_{p}

_{,hydr}> 0); (

**b**) The concavity of the parabolic function depends on the negative coefficient (ΔC

_{p}

_{,hydr}< 0).

**Figure 8.**Change in kinetic thermal energy (entropy change) for gas trapping in a cage and for passage of vapor to liquid (condensation).

∂(RlnK_{mot})/∂(1/T) = −ΔH°_{mot} (J mol^{−1}) |

∂(RTlnK_{mot})/∂T = ΔS°_{mot} (J K^{−1} mol^{−1}) |

ΔC_{p}_{,hydr} = ±ξ_{w} C_{p}_{,w} |

with C_{p}_{,w} = 75.36 J·K^{−1}mol^{−1} (molar heat capacity for liquid water) |

ΔC_{p}_{,hydr} = +ξ_{w} C_{p}_{,w} (Class A, convex binding function) |

ΔC_{p}_{,hydr} = −ξ_{w} C_{p}_{,w} (Class B, concave binding function) |

ENTHALPY | ENTROPY |
---|---|

RlnK_{dual} = −ΔG°/T = −ΔH_{dual}/T + S_{dual.}= −{ΔH _{mot}+ΔC_{p}_{,hydr,}T}/T + ΔS’where ΔC _{p}_{,hydr} = 2202(ΔH_{dual})/∂TΔC _{p}_{,hydr} = isobaric heat capacity(*) | RT lnK_{dual} = −ΔG° = −ΔH_{dual} + T ΔS_{dual}= −ΔH’+T{ΔS _{mot} + ΔC_{p}_{,hydr} lnT}where ΔC _{p}_{,hyd} = ∂(ΔS_{dual})∂lnTΔC _{p}_{,hydr} = isobaric heat capacity(*) |

Motive Function from primary function f(1/T) in RlnK _{app} = {f((1/T)_{*}g(T)} | Motive Function from primary function f(T) in RTlnK _{app} = {f((T)_{*}g(lnT)} |

R lnK_{mot} = −(ΔH_{mot}/T) + ΔS | RT lnK_{mot} = −ΔH_{mot} +T ΔS° |

_{p}

_{,hydr}numerically equal for enthalpy and entropy: the same constant curvature amplitude (mathematical constraint), the same molecular event, i.e., the same passage of state of water (chemical constraint).

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**MDPI and ACS Style**

Fisicaro, E.; Compari, C.; Braibanti, A.
Ergodic Algorithmic Model (*EAM*), with Water as Implicit Solvent, in Chemical, Biochemical, and Biological Processes. *Thermo* **2021**, *1*, 361-375.
https://doi.org/10.3390/thermo1030022

**AMA Style**

Fisicaro E, Compari C, Braibanti A.
Ergodic Algorithmic Model (*EAM*), with Water as Implicit Solvent, in Chemical, Biochemical, and Biological Processes. *Thermo*. 2021; 1(3):361-375.
https://doi.org/10.3390/thermo1030022

**Chicago/Turabian Style**

Fisicaro, Emilia, Carlotta Compari, and Antonio Braibanti.
2021. "Ergodic Algorithmic Model (*EAM*), with Water as Implicit Solvent, in Chemical, Biochemical, and Biological Processes" *Thermo* 1, no. 3: 361-375.
https://doi.org/10.3390/thermo1030022