Luis Guarracino
email: luisg@fcaglp.unlp.edu.ar

Damien Jougnot

A physically-based analytical model to describe effective excess charge for streaming potential generation in water saturated porous media

Derivation of a physically-based analytical model to determine the effective excess charge • Mechanistic explanation for the empirical dependence between effective excess charge and permeability • The new model reproduces experimental data for different media and a broad range of ionic concentrations This paper is published in

Introduction

The self-potential (SP) method is a passive geophysical method based on the measurements of the electrical field which is naturally generated in the subsurface. The SP method was first proposed by Robert Fox in 1830 [START_REF] Fox | On the electromagnetic properties of metalliferous veins in the mines of cornwall[END_REF] and it is considered as one of the oldest geophysical methods. Although SP data are relatively easy to measure, the extraction of useful information is a non-trivial task since the recorded signals are a superposition of different SP components. In natural porous media, signals are mainly generated by electrokinetic (water flux) and electrochemical (ionic fluxes or redox reactions) phenomena. For more details of this method and for an overview of all possible SP sources we refer to [START_REF] Revil | The Self-Potential Method: Theory and Applications in Environmental Geosciences[END_REF].

In the present study, we focus on the electrokinetic (EK) contribution to the SP, that is the part of the signal generated from the water flow in porous media (often referred to as streaming potential). The surface of the minerals that constitute porous media is generally electrically charged, creating an electrical double layer (EDL) containing an excess of charge that counterbalances the charge deficiency of the mineral surface (see [START_REF] Hunter | Zeta Potential in Colloid Science: Principles and Applications[END_REF][START_REF] Leroy | A triple-layer model of the surface electrochemical properties of clay minerals[END_REF]. Figure 1a shows how the EDL is composed: a Stern layer that contains only counterions coating the mineral with a very limited thickness, and a diffuse layer that contains both counterions and coions but with a net excess charge. The shear plane, that can be approximated as the limit between the Stern layer and diffuse layer (e.g. [START_REF] Leroy | A triple-layer model of the surface electrochemical properties of clay minerals[END_REF], separates the mobile and immobile part of the water molecules when subjected to a pressure gradient. This plane is characterized by an electrical potential called ζ-potential (see [START_REF] Hunter | Zeta Potential in Colloid Science: Principles and Applications[END_REF]. When the water flows through the pore it drags a fraction of the excess charge that give rise to a streaming current and an electrical potential field.

The EK phenomenon has been studied experimentally and theoretically for more than a century due its relevance in many practical applications in various fields (e.g., microfluidic, chemical engineering, fluid mechanics). In hydrogeophysics, its interest relies on the fact that SP data is sensitive to groundwater flow. Indeed, groundwater flow monitoring can otherwise be performed locally using rather intrusive methods (i.e. hydraulic head measurements between boreholes or in situ sensors, heat or chemical tracer evolution from a single well). However, given that SP only provides indirect measurements (electrical potential) of water flux, it is necessary to rely on a theoretical framework and petrophysical relationships that can predict the EK phenomenon at macroscopic scales.

Two main approaches to simulate streaming current generation in fully saturated porous media can be found in the literature. The classical approach relies on the use of the coupling coefficient, which is a rock-dependent property that relates the difference of hydraulic pressure to the difference in electrical potential. This property has first been described experimentally (e.g. Quincke, 1859; Dorn, 1880) and later quantified by [START_REF] Helmholtz | Studien über electrische grenzschichten[END_REF][START_REF] Von Smoluchowski | Contribution to the theory of electro-osmosis and related phenomena[END_REF]. The so-called Helmholtz-Smoluchowski coupling coefficient has been developed from a capillary-tube model and its final expression does not depend on the geometrical properties of the porous medium. Therefore, it has been used for any kind of medium under the assumption that the electrical conductivity of the mineral surface could be neglected. When it is not the case, alternative formula have been proposed by several researchers (e.g., [START_REF] Revil | Streaming potential in porous media: 2. theory and application to geothermal systems[END_REF][START_REF] Glover | Streaming potential coupling coefficient of quartz glass bead packs: Dependence on grain diameter, pore size, and pore throat radius[END_REF]. Various model using capillaries to predict the coupling coefficient under saturated and partially saturated conditions can be found in the literature (e.g. [START_REF] Rice | Electrokinetic flow in a narrow cylindrical capillary[END_REF][START_REF] Ishido | Experimental and theoretical basis of electrokinetic phenomena in rock-water systems and its applications to geophysics[END_REF][START_REF] Jackson | Multiphase electrokinetic coupling: Insights into the impact of fluid and charge distribution at the pore scale from a bundle of capillary tubes model[END_REF][START_REF] Jackson | On the validity of the thin and thick double-layer assumptions when calculating streaming currents in porous media[END_REF][START_REF] Thanh | A fractal model for streaming potential coefficient in porousmedia[END_REF]. The second approach to simulate streaming current generation is more recent and focuses on the excess charge that is dragged by the water flow. The first reference to this approach in the English literature can be found in [START_REF] Kormiltsev | Three-dimensional modeling of electric and magnetic fields induced by the fluid flow movement in porous media[END_REF] and more detailed theoretical frameworks can be found in Revil andLeroy (2004), Revil et al. (2007) and Revil (2016a,b). In this approach, the coupling parameter is the excess charge that is effectively dragged by water in the porous media. Then, the streaming current and streaming potential distribution can be generated by multiplying the effective excess charge by the water velocity. Note that both approaches describe the same physics; the main difference between them relies on the parameter (coupling coefficient or excess charge density) used to describe the electrokinetic coupling between fluid flow and streaming potential generation.

In this work, we focus on the excess charge approach in the framework proposed by Sill (1983) and modified by [START_REF] Kormiltsev | Three-dimensional modeling of electric and magnetic fields induced by the fluid flow movement in porous media[END_REF] and [START_REF] Revil | Electrokinetic coupling in unsaturated porous media[END_REF], where SP signals can be directly related to the water velocity:

∇ • (σ∇ϕ) = ∇ • Qv v D (1)
where, σ is the electrical conductivity (S m -1 ), ϕ the electrical potential (V) and Qv the excess charge density (C m -3 ) effectively dragged by the water flow at a given Darcy velocity v D (m s -1 ). From Eq. ( 1), it becomes clear that determining the effective excess charge is crucial to predict streaming potential signal in hydrosystems and to infer water flux from SP measurements. Experimental evidences has shown that the effective excess charge depends on the porous medium permeability [START_REF] Titov | Electrokinetic spontaneous polarization in porous media: petrophysics and numerical modelling[END_REF][START_REF] Jardani | Tomography of the darcy velocity from self-potential measurements[END_REF][START_REF] Bolève | Dyke leakage localization and hydraulic permeability estimation through self-potential and hydro-acoustic measurements: Self-potential 'abacus' diagram for hydraulic permeability estimation and uncertainty computation[END_REF]. The first relationship to estimate the effective excess charge from permeability can be found in [START_REF] Titov | Electrokinetic spontaneous polarization in porous media: petrophysics and numerical modelling[END_REF] (Fig. 2 of their paper). Some years later, [START_REF] Jardani | Tomography of the darcy velocity from self-potential measurements[END_REF] proposed a more precise empirical relationship that has been proven to be very useful as it decreases the number of variables to be estimated and provides good estimates of efective excess charge for different types of porous media. This relationship has been used in many studies involving electrokinetic phenomena, such as dam leakage (e.g. [START_REF] Bolève | Preferential fluid flow pathways in embankment dams imaged by self-potential tomography[END_REF][START_REF] Ikard | Characterization of focused seepage through an earthfill dam using geoelectrical methods[END_REF], surface-groundwater interaction (e.g. [START_REF] Linde | Self-potential investigations of a gravel bar in a restored river corridor[END_REF], seismoelectric studies (e.g. [START_REF] Mahardika | Waveform joint inversion of seismograms and electrograms for moment tensor characterization of fracking events[END_REF]Jougnot et al., 2013;[START_REF] Revil | The Seismoelectric Method: Theory and Application[END_REF][START_REF] Monachesi | An analytical study of seismoelectric signals produced by 1-d mesoscopic heterogeneities[END_REF], hydraulically-active fracture identification (e.g. [START_REF] Roubinet | Streaming potential modeling in fractured rock: Insights into the identification of hydraulically active fractures[END_REF] and permeability field characterization (e.g. [START_REF] Jardani | Stochastic joint inversion of temperature and self-potential data[END_REF][START_REF] Soueid Ahmed | Hydraulic conductivity field characterization from the joint inversion of hydraulic heads and self-potential data[END_REF]. However, Eq. ( 26) has been obtained regardless of pore water composition and other hydraulic parameters of porous media. Unfortunately, this relationship does not take into account the dependence of the EDL on the ionic concentration of the electrolyte (see the discussion in [START_REF] Jougnot | Monitoring of saline tracer movement with vertically distributed self-potential measurements at the HOBE agricultural test site, voulund, denmark[END_REF].

The most rigorous approach to construct an accurate description of any phenomena at macroscopic scales is to start with a pore-scale description and then apply an upscaling method to the microscopic equations [START_REF] Bernabé | 02 -physics of porous media: Fluid flow through porous media[END_REF]. In this study, the porous media is conceptualized as a bundle of capillary tubes with a fractal pore-size distribution. The porosity and permeability of the porous media are estimated using an equivalent medium theory. This geometrical description of the porous media and upscaling procedure have been successfully used to describe water flow in fractured media [START_REF] Guarracino | A fractal constitutive model for unsaturated flow in fractured hard rocks[END_REF][START_REF] Guarracino | A constitutive model for water flow in unsaturated fractured rocks[END_REF], the evolution of multiphase flow properties during mineral dissolution [START_REF] Guarracino | A fractal model to describe the evolution of multiphase flow properties during mineral dissolution[END_REF] and relations between hydraulic parameters [START_REF] Guarracino | Estimation of saturated hydraulic conductivity ks from the van genuchten shape parameter α[END_REF]. On the other hand, the effective excess charge in a single tube can be calculated from the radial distributions of excess charge and water velocity. Then the effective excess charge at the macroscopic scale is estimated using the flux-averaging technique proposed by [START_REF] Jougnot | Derivation of soil-specific streaming potential electrical parameters from hydrodynamic characteristics of partially saturated soils[END_REF]. In that paper, they show that by combining these hydraulic and EK properties, a closedform expression for the effective excess charge can be obtained in terms of permeability, porosity, tortuosity, ionic concentration, ζ-potential, and Debye length. The dependence of the developed model on ionic concentration, permeability, porosity and also grain size is tested using different set of experimental data. It is also shown that the proposed model allows to derive from physical concepts the empirical relation between effective excess charge and permeability found by [START_REF] Jardani | Tomography of the darcy velocity from self-potential measurements[END_REF].

Theoretical development

The proposed model is based on the macroscopic description of effective excess charge in porous media from the upscaling of pore-size flow and electrokinetic phenomena. The porous medium is conceptualized as an equivalent bundle of water-saturated capillary tubes with a fractal law distribution of pore sizes. First, we derive expressions for the most important macroscopic hydraulic parameters (porosity and permeability). Then, we obtain an approximate expression for the excess charge in a single tube and, from this result, we estimate the effective excess in the porous media in terms of the hydraulic parameters.

To derive both hydraulic and electrokinetic properties we consider as representative elementary volume (REV) a cylinder of radius R REV (m) and lenght L (m). The pores are assumed to be circular capillary tubes with radii varing from a minimum pore radius R min (m) to a maximum pore radius R max (m).

The cumulative size-distribution of pores whose radii are greater than or equal to R (m) is assumed to obey the following fractal law [START_REF] Tyler | Fractal processes in soil water retention[END_REF][START_REF] Yu | Permeabilities of unsaturated fractal porous media[END_REF][START_REF] Guarracino | A fractal model to describe the evolution of multiphase flow properties during mineral dissolution[END_REF]:

N (R) = R REV R D , ( 2
)
where D is the fractal dimension of pore and 0 < R min ≤ R ≤ R max < R REV . By using the Sierpinski carpet (a classical fractal object) [START_REF] Tyler | Fractal processes in soil water retention[END_REF] show that the fractal dimension of (2) ranges from 1 to 2, with the highest values associated with the finest textured soils. Capillary tube models with this type of fractal pore-size distributions have been successfully used to describe macroscopic process and hydraulic properties in different porous media (e.g. [START_REF] Yu | Permeabilities of unsaturated fractal porous media[END_REF][START_REF] Guarracino | A fractal model to describe the evolution of multiphase flow properties during mineral dissolution[END_REF][START_REF] Soldi | A simple hysteretic constitutive model for unsaturated flow[END_REF]. Note that the cumulative number of pores given by ( 2) decreases with the increase of pore radius R, then differentiating (2) with respect to -R we can obtain the number of pores whose radii are in the infinitesimal range R to R + dR:

dN = DR D REV R -D-1 dR.
(3)

Hydraulic properties

The porosity φ of the REV defined above can be straightforward computed from its definition:

φ = Volume of pores Volume of REV = Rmax R min V p (R)dN πR 2 REV L , (4)
where

V p (R) = πR 2 l (m 3
) is the volume of a single tube of radius R and length l (m). Substituting (3) into ( 4) and solving the definite integral we obtain

φ = τ D R 2-D REV (2 -D) R 2-D max -R 2-D min , (5)
where τ = l/L is the dimensionless hydraulic tortuosity of the capillary tubes [START_REF] Scheidegger | The physics of flow through porous media[END_REF]. Note that the model assumes a single value of τ for all capillary tubes, this value must be considered as a mean tortuosity value of all tube sizes. In order to obtain both the permeability and the excess charge of the REV we need to describe the water flow in a single tube. For a laminar flow rate, the velocity distribution inside the tube can be described by the Poiseuille model [START_REF] Bear | Dynamics of Fluids in Porous Media[END_REF]:

v(R, r) = ρ w g 4ητ R 2 -(R -r) 2 ∆h L , (6)
where r (m) is the distance from the pore wall (r = 0) to the center of the tube (r = R), ρ w the water density (kg/m 3 ), g the gravitational acceleration (m/s 2 ), η the dynamic viscosity (Pa s), and ∆h the pressure head drop across the REV (m). The average velocity v (m/s) in the capillary tube has the following expression:

v(R) = ρ w g 8ητ R 2 ∆h L . (7)
The total volumetric flow through the REV V Q (m 3 /s) is the sum of the volumetric flow rates of all individual tubes. According to ( 7) and (3) V Q can be computed as follows:

V Q = Rmax R min v(R)πR 2 dN = ρ w g 8ητ πDR D REV 4 -D R 4-D max -R 4-D min ∆h L . ( 8
)
On the basis of Darcy's law (macroscopic scale), V Q can also be expressed as

V Q = πR 2 REV ρ w g η k ∆h L , (9)
k being the intrinsic permeability of the porous media (m 2 ).

Then, combining ( 8) and ( 9) we obtain the following expression for permeability k in terms of the geometrical parameters of the porous media:

k = D 8τ (4 -D)R 2-D REV R 4-D max -R 4-D min . (10)
It is important to remark that a similar equations for φ and k have been recently derived by [START_REF] Soldi | A simple hysteretic constitutive model for unsaturated flow[END_REF] assuming constrictive capillary tubes. In the limit case of straight tubes both expressions for φ and k are identical (see equations ( 11) and ( 15) of their paper).

For most porous media it can be assumed that R min << R max [START_REF] Yu | Some fractal characters of porous media[END_REF]. Then, equations ( 5) and ( 10) can be reduced to

φ = τ D (2 -D)R 2-D REV R 2-D max , (11)
k = D 8τ (4 -D)R 2-D REV R 4-D max . (12)
Finally, combining ( 11) and ( 12) we obtain a simple relationship to estimate permeabilty from porosity k = γφ

4-D 2-D , (13)
where

γ = R 2 REV 8τ (4-D) 2-D τ D 4-D 2-D .
Note that for D = 1 the exponent of the porosity is 3 and ( 13) is equivalent to Kozeny's equation [START_REF] Kozeny | Über kapillare Leitung des Wassers im Boden:(Aufstieg, Versickerung und Anwendung auf die Bewässerung)[END_REF].

Electrokinetic properties

Since electrokinetic phenomenon is caused by the coupling of fluid flow and charge distribution at pore scale, the magnitude of this phenomenom will be mainly determined by the macroscopic hydraulic and electrical properties of the porous medium. Based on the previous description of hydraulic properties we will compute the effective excess charge density QREV v carried by the water flow in the REV (C/m 3 ). The effective excess charge density, also called dynamic excess charge depending on the authors and symbolized by Qv or Qeff v , has to be distinguished from the other excess charge densities contained in the pore space: Q v the total excess charge density (C/m 3 ), which includes all the charges from the Stern and diffuse layers of the EDL, and Qv the excess charge located only in the diffuse layer (C/m 3 ) (Fig. 1a) (see the discussion in [START_REF] Revil | Comment on "dependence of shear wave seismoelectrics on soil textures: a numerical study in the vadose zone" by f.i. zyserman, l.b. monachesi and l. jouniaux[END_REF]. It is important to remark that in this study we do not consider the charges that are located in the Stern layer as they are fixed on the pore wall and do not contribute to the streaming current.

We start this derivation by defining the excess charge distribution Qv in a capillary tube saturated by a binary symetric 1:1 electrolyte (e.g., NaCl) as follows

Qv (r) = N A e 0 C 0 e - e 0 ψ(r) k B T -e e 0 ψ(r) k B T , ( 14
)
where N A is the Avogadro's number (mol -1 ), e 0 the elementary charge (C), C 0 the ionic concentration far from the mineral surface (mol/m 3 ), ψ the local electrical potential in the pore water (V), k B the Boltzman constant (J/K), and T is the absolute temperature (K). For the thin double layer assumption (i.e., the thickness of the double layer is small compared to the pore size) the local electrical potential can be expressed [START_REF] Hunter | Zeta Potential in Colloid Science: Principles and Applications[END_REF]:

ψ(r) = ζe -r l D , (15)
l D = k B T 2N A C 0 e 2 0 , (16)
where ζ (V) is the ζ-potential on the shear plane (Fig. 1a), l D the Debye lenght (m), and the water dielectric permittivity (F/m).

As proposed by [START_REF] Jougnot | Derivation of soil-specific streaming potential electrical parameters from hydrodynamic characteristics of partially saturated soils[END_REF], the effective excess charge density QR v carried by the water flow in a single tube of radius R is defined by

QR v = 1 v(R)πR 2 A Q v (r)v(R, r)dA, (17)
being A the cross sectional area of the tube (m 2 ). Using a polar coordinate system with the pole located in the center of the tube, equation ( 17) becomes

QR v = 2 v(R)R 2 R 0 Q v (r)v(R, r)(R -r)dr. ( 18
)
In order to obtain a closed-form analytical expression of QR v we approximate the exponential terms of ( 14) by a four-term Taylor serie:

e ± e 0 ψ(r) k B T = 1 ± e 0 ψ(r) k B T + 1 2 e 0 ψ(r) k B T 2 ± 1 6 e 0 ψ(r) k B T 3 . ( 19
)
Substituting ( 19) in ( 14) and solving (18) we obtain

QR v = - 8N A e 2 0 C 0 ζ k B T (R/l D ) 4 6 -e -R l D R l D 3 + 3 R l D 2 + 6 R l D + 6 + 24N A e 2 0 C 0 ζ k B T (R/l D ) 3 2 -e -R l D R l D 2 + 2 R l D + 2 - 16N A e 2 0 C 0 ζ k B T (R/l D ) 2 1 -e -R l D R l D + 1 - 4N A e 4 0 C 0 ζ 3 3(k B T ) 3 (3R/l D ) 4 6 -e -3R l D 3R l D 3 + 3 3R l D 2 + 6 3R l D + 6 + 4N A e 4 0 C 0 ζ 3 (k B T ) 3 (3R/l D ) 3 2 -e -3R l D 3R l D 2 + 2 3R l D + 2 - 8N A e 4 0 C 0 ζ 3 3(k B T ) 3 (3R/l D ) 2 1 -e -3R l D 3R l D + 1 . ( 20
)
For the thin double layer assumption (i.e., the thickness of the double layer is small compared to the pore size) we consider l D << R and (20) can be reduced to

QR v = 8N A e 0 C 0 (R/l D ) 2 -2 e 0 ζ k B T - e 0 ζ 3k B T 3 . ( 21
)
Figure 2 (a) shows the effective excess charge QR v predicted by ( 20) and ( 21) for a ionic concentration C 0 = 1 mol/m 3 . Note that even though the number of terms of equation ( 21) is drastically reduced, both equations predict similar values of QR v . In order to test the general validity of (21)under the thin double layer assumption, we compare approximate values of QR v with exact values obtained by the numerical solution of (18) assuming pore-sizes R greater that 5 Debye lenghts. Figure 2 (b) presents the goodness of the fit for different values of ionic concentration C 0 . From the analysis of Figure 2, we conclude that equation ( 21) predicts fairly well the effective excess charge in capillary tubes for a wide range of radius and ionic concentration values.

The effective excess charge QREV v carried by the water flow in the REV is defined by where v D = ρwg η k ∆h L is the Darcy's velocity (m/s) (macroscopic scale). Substituting ( 21), ( 7) and ( 3) in ( 22) and assuming R min << R max yields

QREV v = 1 v D πR 2 REV Rmax R min QR v (R)v(R)πR 2 dN, (22)
QREV v = 8N A e 0 C 0 -2 e 0 ζ k B T - e 0 ζ 3k B T 3 4 -D 2 -D l D R max 2 . (23)
Finally, combining ( 11), ( 12) and ( 23) we obtain the following expression for QREV v :

QREV v = N A e 0 C 0 l 2 D -2 e 0 ζ k B T - e 0 ζ 3k B T 3 1 τ 2 φ k . ( 24
)
The above equation constitutes the main result of this paper. Note that (24) predicts the effective excess charge density in terms of both macroscopic hydraulic parameters (porosity, tortuosity and permeability) and electrokinetic parameters (ionic concentration, ζ-potential and Debye lenght). This equation gives insight into the role of macroscopic hydraulic parameters and it can be considered a starting point for designing non-invasive methods to monitoring groundwater flow using self-potential measurements. In order to study the role of porosity φ, permeability k, and ionic concentration C 0 on effective excess charge, we perform a parametric analysis of equation ( 24). The ionic concentration dependence of ζ-potential is assumed to obey the relation proposed by [START_REF] Pride | Electrokinetic dissipation induced by seismic waves[END_REF]:

ζ(C 0 ) = a + blog 10 (C 0 ), (25)
where a and b are fitting parameters. For this study we use the parameter values obtained by [START_REF] Jaafar | Measurement of streaming potential coupling coefficient in sandstones saturated with high salinity nacl brine[END_REF] on silicate-based materials for NaCl brine a=-6.43 mV and b=20.85 mV. Figure 3 summarizes the parametric analysis of log 10 QREV v for the following ranges of variability of ionic concentration C 0 , permeability k and porosity φ: 10 -1 mol/m 3 ≤ C 0 ≤ 10 3 mol/m 3 , 10 -20 m 2 ≤ k ≤ 10 -8 m 2 and 0.05 ≤ φ ≤ 0.5. Figure 3a shows the effect of porosity and permeability on QREV v for a fixed value of ionic concentration (C 0 = 100 mol/m 3 ). It can be observed that QREV v is strongly determined by permeability while porosity only produces a slightly increase of QREV v values. Figure 3b shows the effect of porosity and ionic concentration on QREV v . As shown in this panel, these parameters can change QREV v values in two orders of magnitude for a fixed value of permeability (k = 10 -14 m 2 ). An increase of QREV v can be observed when ionic concentration decreases and porosity increases. Finally, Figure 3c shows the role of permeability and ionic concentration on QREV v for a fixed value of porosity (φ = 0.2). It can be observed that the effect of permeability on QREV v is much more significant than ionic concentration. From this parametric analysis we can conclude that effective excess charge QREV v is highly sensitive to permeability values. However, porosity and ionic concentration can modify QREV v values in two orders of magnitude for a given value of permeability (see discussions in [START_REF] Jougnot | Derivation of soil-specific streaming potential electrical parameters from hydrodynamic characteristics of partially saturated soils[END_REF][START_REF] Jougnot | Monitoring of saline tracer movement with vertically distributed self-potential measurements at the HOBE agricultural test site, voulund, denmark[END_REF]. The increase of porosity or the decrease of ionic concentration produce an increase of QREV v values.

3 Comparison with the empirical relationship proposed by [START_REF] Jardani | Tomography of the darcy velocity from self-potential measurements[END_REF] The empirical relationship to estimate the effective excess charge QREV v from permeability k proposed by [START_REF] Jardani | Tomography of the darcy velocity from self-potential measurements[END_REF] reads as follows:

log 10 ( QREV v ) = A 1 + A 2 log 10 (k), (26)
where A 1 = -9.2349 and A 2 = -0.8219 are constant values obtained by fitting (26) to a large set of experimental data that includes various lithologies and ionic concentrations (Fig. 4). Note that this equation has been obtained regardless of pore water composition and other hydraulic parameters of porous media. The empirical relationship ( 26) can be derived from the proposed equation ( 24). Using ( 13) we obtain the following expression for porosity in term of permeability: φ = (k/γ) 2-D 4-D . Replacing this expression in ( 24) and taking the logarithm on both sides of the resulting equation, we obtain exactly (26) but with the following constants:

A 1 = log 10 N A e 0 C 0 γ 2-D 4-D -2 e 0 ζ k B T -e 0 ζ 3k B T 3 l D τ 2 , A 2 = -2 4-D . (27)
According to our model, the log 10 ( QREV v )-intercept A 1 mainly depends on chemical and interface parameters while the slope A 2 only depends on the fractal dimension of the pore size distribution (1 < D < 2). Note that the predicted range of the slope is -1 < A 2 < -0.666 and that the value obtained by [START_REF] Jardani | Tomography of the darcy velocity from self-potential measurements[END_REF] A 2 = -0.8219 corresponds to the fractal dimension D = 1.567.

Figure 4 shows the fit of ( 27) to an extensive set of QREV v data determined by several authors [START_REF] Ahmad | A laboratory study of streaming potentials[END_REF][START_REF] Cassagrande | Stabilization of soils by means of electroosmotic state-of-art[END_REF][START_REF] Friborg | Experimental and theoretical investigations into the streaming potential phenomenon with special reference to applications in glaciated terrain[END_REF][START_REF] Pengra | Determination of rock properties by low-frequency ac electrokinetics[END_REF][START_REF] Revil | Characterization of transport properties of argillaceous sediments. application to the callovo-oxfordian argillite[END_REF][START_REF] Revil | Electrokinetic coupling in unsaturated porous media[END_REF][START_REF] Revil | Petrophysical properties of saprolites from the oak ridge integrated field research challenge site, tennessee[END_REF][START_REF] Boleve | Streaming potentials of granular media: Influence of the dukhin and reynolds numbers[END_REF][START_REF] Jardani | Tomography of the darcy velocity from self-potential measurements[END_REF][START_REF] Glover | Streaming potential coupling coefficient of quartz glass bead packs: Dependence on grain diameter, pore size, and pore throat radius[END_REF][START_REF] Zhu | Formation velocity measurements using multipole seismoelectric lwd: Experimental studies[END_REF]Jougnot and Linde, 2013). The best fit of our model is obtained for fractal dimension D = 1.571. Note that this single value of D can fit a wide range of soil textures, then it can be considered a mean value of all pore size distributions.

It is worth mentioning that QREV v data displayed on Fig. 4 are obtained from electrokinetic coupling coefficient C EK values (V/Pa) using the following equation [START_REF] Revil | Constitutive equations for ionic transport in porous shales[END_REF][START_REF] Revil | Electrokinetic coupling in unsaturated porous media[END_REF]:

QREV v = - C EK ση k , ( 28
)
where σ (S/m) is the electrical conductivity, η (Pa s) the dynamic viscosity, and k (m 2 ) the permeability. The values of σ and k of each sample have been measured independently.

Permeability, k [m 2 ]

10 -20 10 -18 10 -16 10 -14 10 -12 10 -10 10 -8 10 -6

Effective excess charge,

[C m -3 ] 10 -4
10 -2 10 0 10 2 10 4 10 6 10 8

Clayed soils [START_REF] Cassagrande | Stabilization of soils by means of electroosmotic state-of-art[END_REF] Glacial tills [START_REF] Friborg | Experimental and theoretical investigations into the streaming potential phenomenon with special reference to applications in glaciated terrain[END_REF] Sand [START_REF] Ahmad | A laboratory study of streaming potentials[END_REF] Berea sandstone (Zhu & Toksoz, 2012) Saprolites [START_REF] Revil | Petrophysical properties of saprolites from the oak ridge integrated field research challenge site, tennessee[END_REF] Glass beads [START_REF] Boleve | Streaming potentials of granular media: Influence of the dukhin and reynolds numbers[END_REF][START_REF] Pengra | Determination of rock properties by low-frequency ac electrokinetics[END_REF] Limestones [START_REF] Revil | Electrokinetic coupling in unsaturated porous media[END_REF][START_REF] Pengra | Determination of rock properties by low-frequency ac electrokinetics[END_REF] Clayrocks [START_REF] Revil | Characterization of transport properties of argillaceous sediments. application to the callovo-oxfordian argillite[END_REF] Alluvium [START_REF] Jardani | Tomography of the darcy velocity from self-potential measurements[END_REF] Sandstone [START_REF] Pengra | Determination of rock properties by low-frequency ac electrokinetics[END_REF] Sand (Jougnot & Linde, 2013) Glass beads (Glover & Dery, 2010[START_REF] Jardani | Tomography of the darcy velocity from self-potential measurements[END_REF] This model with D = 1.571 If D = 1.527 allows us to retrieve the exact same trend as [START_REF] Jardani | Tomography of the darcy velocity from self-potential measurements[END_REF], we are left with a cloud of values that spread around the proposed model. To test our model with more accuracy, we focus on a selection of experimental data where all the parameters have been measured in the following section.

Application to laboratory data 4.1 Effect of salinity

In order to analyze the effect of salinity on QREV v we test the proposed model ( 24) with laboratory data obtained by [START_REF] Pengra | Determination of rock properties by low-frequency ac electrokinetics[END_REF]. These authors performed an exhaustive petrophysical characterization of a collection of rock and glass bead samples and they measured the electrokinetic coupling coefficient C EK for different NaCl brine concentrations. For this test QREV v values are obtained from C EK using (28). The only parameter of(24) which is not measured by [START_REF] Pengra | Determination of rock properties by low-frequency ac electrokinetics[END_REF] is the hydraulic tortuosity τ . Thus, we fit this parameter using a least square algorithm. The ionic concentration dependence of ζ-potential is assumed to obey (25).

Figure 5 shows the fit of ( 24) to experimental values of effective excess charge QREV v measured at different ionic concentrations C 0 for 3 sandstones samples of different permeabilities and 1 fused glass bead sample. The fitted values of tortuosity and measured values of porosity and permeability of each sample are listed in the figure caption. Experimental data show that QREV v decreases with the increase of ionic concentration, and this behaviour can be adequately described by the proposed model. The decrease of QREV v with the increase of C 0 was also predicted and discussed by [START_REF] Jougnot | Monitoring of saline tracer movement with vertically distributed self-potential measurements at the HOBE agricultural test site, voulund, denmark[END_REF].

Effect of porosity

To test the effect of porosity on QREV v estimates we fit the proposed model to experimental values of QREV v in terms of both k and k/φ obtained by [START_REF] Pengra | Determination of rock properties by low-frequency ac electrokinetics[END_REF] for C 0 = 0.2 mol/L in different porous media. Figure 6a shows the fit of (24) to experimental data in terms of k. The fitting parameter is the ratio φ/τ 2 and the RMS of the fit is 41.97 C/m 3 . Figure 6b shows the fit of the model to the same experimental values of QREV v but in terms of k/φ. In this case the fitting parameter is τ 2 and the RMS is reduced to 19.64 C/m 3 . Even though more experimental evaluations of the model are needed, this test shows that the inclusion of porosity improves the estimate of QREV v .

Effect of grain size

Among the available data in the SP literature, Glover and Déry (2010) studied the effect of grain size on the electrokinetic coupling coefficient C EK . These authors measured C EK on 12 packs of glass bead having different grain diameters d (d ∈ [1, 990] × 10 -6 m) at two pore water salinities: C 0 = 2 × 10 -4 and 2 × 10 -3 mol/L.

To test the performance of our model for different grain sizes, we used the measurements of Glover and Déry (2010) to predict the effective excess charge. For each grain size, we used the permeability and mercury porosity listed in their Table 1. For ionic concentrations C 0 = 2 × 10 -4 and 2 × 10 -3 2). Note that the glass bead diameter should not influence the interface properties of the quartz mineral, this is an artifact due to surface conductivity effects (see also [START_REF] Leroy | A double layer model of the gas bubble/water interface[END_REF][START_REF] Li | Influence of surface conductivity on the apparent zeta potential of calcite[END_REF].

In order to compare our model prediction with the measurements, we compute the coupling coefficient C EK using ( 28) and the electrical conductivity model proposed by [START_REF] Glover | Streaming potential coupling coefficient of quartz glass bead packs: Dependence on grain diameter, pore size, and pore throat radius[END_REF]:

σ = 1 F σ w + 4mF Σ S d , (29)
where σ w is the electrical conductivity of pore water (calculated from C 0 , see [START_REF] Sen | Influence of temperature on electrical conductivity on shaly sands[END_REF])); Σ S , the glass beads surface conductance; m, the cementation index; and F = φ -m , the formation factor of the bead samples. The only parameter which is not provided in the study is the hydraulic tortuosity τ .

Figure 7a shows the predicted effective excess charge for the two ionic concentrations as a function of the glass bead diameter and τ = 1.2 (this tortuosity has been chosen as it gives the best fit for all the data presented on Fig. 7b at once). We also plot in the figure the prediction of [START_REF] Jardani | Tomography of the darcy velocity from self-potential measurements[END_REF] empirical relationship which only depends on permeability k. Figure 7b illustrates the pretty good fit between the coupling coefficient measurements and predicted values of our model for both pore water salinities. Note that τ = 1.20 is the only fitting parameter of the model and this parameter allows to describe all measured data (different glass bead sizes and two ionic concentration values). It is worth mentioning that we could improve the fit between the proposed model and experimental data by adjusting τ for each glass bead diameter (especially for d 5 × 10 -6 m), but we prefer keeping the number of fitting parameters as small as possible.

Discussion and conclusion

The present study is focused on the estimate of the effective excess charge QREV not straightforward as a relationship between permeability and saturation needs to be derived for the fractal pore size distribution.

The first step of this study consists in the up-scaling of EK properties for a binary symetric 1:1 electrolyte from the EDL scale to the capillary tube or pore scale (i.e., from few nm to possibly several mm). The approximate expression (21) relates effective excess charge QR v to the radius of the tube R and chemical parameters (ionic concentration C 0 , ζ-potential and Debye length l D ). The accuracy of this equation has been tested to be correct as long as the electrical double layer from the walls of the capillary do not overlap (R > 5l D ), i.e. under the thin double layer assumption (Fig. 2). An extension of this model for EDL overlapping could be obtained by truncation of the diffuse layer (e.g. [START_REF] Gonçalvès | Introducing interacting diffuse layers in tlm calculations: A reappraisal of the influence of the pore size on the swelling pressure and the osmotic efficiency of compacted bentonites[END_REF].

The second step consists in the up-scaling of both hydraulic and EK properties from the pore scale (capillary tube) to the REV scale. The REV of porous media is conceptualized as a bundle of capillary tubes with a fractal pore-size distribution. Fractal models have been proven very useful to obtain different hydraulic properties of rocks and soils [START_REF] Tyler | Fractal processes in soil water retention[END_REF][START_REF] Yu | Permeabilities of unsaturated fractal porous media[END_REF][START_REF] Guarracino | Estimation of saturated hydraulic conductivity ks from the van genuchten shape parameter α[END_REF][START_REF] Guarracino | A fractal model to describe the evolution of multiphase flow properties during mineral dissolution[END_REF][START_REF] Soldi | A simple hysteretic constitutive model for unsaturated flow[END_REF]. In this study, the fractal distribution approach allows us to obtain simple expressions for porosity φ (11) permeability k (12) and effective excess charge QREV v (23) in terms of the fractal dimension D, the maximum pore radius R max and the REV radius R REV . Finally, combining these macroscopic properties we obtain the closed-form expression (24) to estimate QREV v from hydraulic (φ, k, τ ) and pore water chemistry (C 0 , l D , ζ) parameters.

The proposed model is derived from physical and geometrical concepts and provides a mechanistic framework for understanding the role of hydraulic parameters on QREV v . In particular, the model (2007) which relies on an increasing number of data sets with different lithologies and it is used by many researchers. It is important to remark that the link between effective excess charge and permeability is the subject of debates in the scientific community (e.g. [START_REF] Jouniaux | A review on electrokinetically induced seismo-electrics, electro-seismics, and seismo-magnetics for earth sciences[END_REF][START_REF] Zyserman | Dependence of shear wave seismoelectrics on soil textures: a numerical study in the vadose zone[END_REF][START_REF] Revil | Comment on "dependence of shear wave seismoelectrics on soil textures: a numerical study in the vadose zone" by f.i. zyserman, l.b. monachesi and l. jouniaux[END_REF] and our model provides a theoretical justification to this link. However, in this study we show that it would be better to link QREV v to the ratio of permeability and porosity (k/φ) instead of k (see Fig. 6). Another limitation of the relationship proposed by [START_REF] Jardani | Tomography of the darcy velocity from self-potential measurements[END_REF] is that it does not explicitly depend on the pore-water chemistry, i.e. on the ζ-potential, Debye length and ionic concentration. Indeed, the newly proposed model takes into account the ionic concentration and its ζ-potential dependence and is therefore able to reproduce laboratory data from [START_REF] Pengra | Determination of rock properties by low-frequency ac electrokinetics[END_REF] where all the parameters but one (tortuosity) have been estimated independently.

The proposed model shows that QREV v contains information on the lithology as predicted by Revil and Jardani (2013) (p. 64). Indeed, three petrophysical parameters are explicitly identified in the model: the permeability, k, the porosity, φ, and the hydraulic tortuosity, τ . It is worth to emphazise here that the relation between hydraulic and electrical tortuosity is not straightforward (e.g. [START_REF] Clennell | Tortuosity: a guide through the maze[END_REF]. In the present work, we test the simple model of [START_REF] Winsauer | Resistivity of brine-saturated sands in relation to pore geometry[END_REF] to determine the hydraulic tortuosity used in Eq. ( 24) based on parameters that can be measured electrically (see [START_REF] Clennell | Tortuosity: a guide through the maze[END_REF], for a discussion):

(τ e ) 2 = F φ,

where τ e is the hydraulic tortuosity determined electrically from the formation factor and the porosity. Table 1 show the comparison between the fitted tortuosity τ to the one predicted by (30). One can see that predicted and fitted tortuosities fall fairly close one to an other, which is a promising preliminary result. However, further work is needed to establish a better relation between these electrical and hydraulic properties, therefore to link the effective excess charge QREV v to electrical conductivity. The proposed model represents a major step forward in understanding the links between hydraulic and electrokinetic parameters in the framework of the excess charge approach. The model provides a theoretical basis for the empirical relationship with the medium permeability and describes the dependence of the excess charge on other petrophysical parameters. The simplicity of the model and its excellent fit to different set of experimental data opens-up new possibilities for a broader use of the SP method in hydrogeophysics studies based on the increasingly popular excess charge approach. The analytical development of a model for partially saturated porous media using the presented approach will be the natural next step to pursue this work.

acknowledgments

The data used to test the proposed model is listed in the references. This research is partially supported by Universidad Nacional de La Plata and Consejo Nacional de Investigaciones Científicas y Técnicas (Argentina). The authors strongly thank the editor and the three reviewer for their nice and really constructive comments.

Figure 1 :

 1 Figure 1: (a) Scheme of the electrical double layer for a given capillary of radius R. Distribution of the excess charge (b) and the pore vater velocity (c) from the shear plane (r = 0 m) to the center of the capillary (r = R = 10 -7 m). The excess charge is calculated using (14) for different ionic concentrations of NaCl and the Jaafar et al. (2009) model for the ζ-potential. Figure modified from Jougnot et al. (2015).

Figure 2 :

 2 Figure 2: (a) Analytical estimates of QR v using equations (20) and (21) for a ionic concentration C 0 = 1 mol/m 3 ; (b) numerical (points from Eq. 17) and analytical (lines from Eq. 21) estimates of QR v for different concentration values (R > 5l D ). The ζ-potential values are computed using (25).

Figure 3 :

 3 Figure 3: Parametric analysis of effective excess charge QREV v : (a) sensitivity to porosity and permeability for a fixed value of ionic concentration, (b) sensitivity to porosity and ionic concentration for a fixed value of permeability, (c) sensitivity to permeability and ionic concentration for a fixed value of porosity.

Figure 4 :

 4 Figure 4: Comparison between the effective excess charge QREV v as a function of the permeability k. Symbols represent experimental data from literature for different lithologies. Solid lines are the fit of the proposed model (with D = 1.571) and the Jardani et al. (2007) empirical relationship (26).

Figure 5 :

 5 Figure 5: Comparison between measured values of QREV v and predicted values using the proposed model (Eq. 24) for different ionic concentrations. Experimental data are obtained from Pengra et al. (1999). Hydraulic parameters of soil samples: a) φ = 0.223, k = 2.36 × 10 -12 m 2 , τ = 1.95, b) φ = 0.168, k = 9.09 × 10 -13 m 2 , τ = 1.83, c) φ = 0.067, k = 5.63 × 10 -15 m 2 , τ = 3.24, d) φ = 0.298, k = 5.07 × 10 -12 m 2 , τ = 1.90. Tortuosity values are obtained by fitting (24) to experimental data.

Figure 6 :

 6 Figure 6: Comparison between measured values of QREV v vs k/φ and predicted values using (24) for C 0 = 0.2 mol/L.

v

  in fully saturated porous media, a key parameter to understand and model the streaming potential generation. Based on physical and geometrical concepts we derive a closed-form analytical expression to estimate QREV v from electrokinetic and hydraulic parameters. The mathematical development involves up-scaling procedures at pore and REV scales, similar to the numerical up-scaling proposed by[START_REF] Jougnot | Derivation of soil-specific streaming potential electrical parameters from hydrodynamic characteristics of partially saturated soils[END_REF] for partially saturated soils. The extension of the present model to unsaturated conditions isGlass bead diameter, d [m]

Figure 7 :

 7 Figure 7: (a) Model prediction of QREV v for glass beads pack with different diameters at two different ionic concentrations: C 0 = 2×10 -4 and 2×10 -3 mol/L in blue and red, respectively. (b) Comparison between the measured and predicted coupling coefficients when using the proposed model with τ = 1.20. Note that the black dashed lines are the Jardani et al. (2007) empirical relationship prediction (Eq. 26) for the effective excess charge in (a) and the coupling coefficient in (b).

Table 1 :

 1 Comparison between fitted and predicted hydraulic tortuosities Material τ from fit τ e from Eq. (30) Glass beads packs from[START_REF] Glover | Streaming potential coupling coefficient of quartz glass bead packs: Dependence on grain diameter, pore size, and pore throat radius[END_REF]

	Mean value over 12 packs	1.20	1.24
	Samples from Pengra et al. (1999)
	Fontainebleau A	1.95	1.59
	Fontainebleau B	1.83	1.82
	Fontainebleau C	3.24	3.03
	Glass beads (200 µm)	1.90	1.64
	corroborates the empirical relationship (26) of Jardani et al.


Denis Michel
email: denis.michel@live.fr.

Test of the formal basis of Arrhenius law with heat capacities

Keywords: Kinetics, equilibrium, heat capacity, enthalpy

come

Test of the formal basis of Arrhenius law with heat capacities

The celebrated laws of Arrhenius and van't Hoff are the basements of teaching and research in physical chemistry. Various versions of the energetic restriction of reactions have been proposed before the currently used Arrhenius equation, which works pretty well but remains empirical [START_REF] Laidler | The development of the Arrhenius equation[END_REF]. A mathematical approach will be used here to ground these equations on rational bases. The probabilistic tool dedicated for this purpose is likely to be the geometric or exponential distribution, which proved sufficient to recover Boltzmann statistics [START_REF] Michel | Simply conceiving the Arrhenius law and absolute kinetic constants using the geometric distribution[END_REF][START_REF] Michel | A probabilistic rate theory connecting kinetics to thermodynamics[END_REF].

Exponential distribution of energy

Contrary to the historical equations of Arrhenius and van't Hoff, the new relations described here between energy, kinetic and equilibrium constants will be based on Boltzmann's laws of energy distribution. More precisely, the exponential distribution underlies both Boltzmann theory and the thermochemical constants. Indeed, to define the distribution of maximal randomness in mechanical systems, mathematics provides an unambiguous shortcut bypassing Hamiltonians and Lagrangians: this is necessarily the "memoryless" exponential distribution [START_REF] Michel | Simply conceiving the Arrhenius law and absolute kinetic constants using the geometric distribution[END_REF]. A probabilistic distribution relies upon a unique function called probability density function (PDF). That of the exponential law is simply

f (E) = 1 E e - E E (1)
where E is the mean value of the distribution. We can apply it to energy distribution by supposing that E is the number of energy units in a particle. This function monotonously decreasing towards high energies is illustrated in Fig. 1, which represents the densities of particles containing different amounts of energy. The low energy particles are the most numerous and give the dark lower levels, whereas the particles are less and less numerous when going up in the energy levels. Probabilities are obtained by integration of the PDF. The probability that a randomly picked particle has an energy equal to or higher than a certain threshold E ‡ is

P (E E ‡ ) = ∞ E=E ‡ f (E) dE = e - E ‡ E (2)
Note that the double dagger ( ‡), taken again here, is historical in the field of rate theories to symbolize the energy threshold. To be validated by statistical mechanics, this approach should also allow recovering Boltzmann's distribution. This requirement is fully satisfied since the probability that a particle has an energy level precisely of E ‡ , can indeed be obtained by integration of the same PDF, but between E ‡ and E ‡ + 1.

P (E = E ‡ ) = E ‡ +1 E=E ‡ f (E) dE =   1 -e - 1 E    e - E ‡ E = e - E ‡ E ∞ j=0 e - j E (3)
The latter form of Eq.( 3) is known as the partition function of Boltzmann, recovered here without need for the usual introductory treatments of statistical mechanics. Eq.( 2), that is less familiar in thermochemistry, quantifies the probability for a particle of exceeding a threshold energy. It could evoke the formula of Arrhenius

k = A e - E a k B T
but this is only apparent because as it is structured, the Arrhenius equation cannot be a probability [START_REF] Michel | A probabilistic rate theory connecting kinetics to thermodynamics[END_REF]. To use Eq.( 2) for constructing the energy restriction formula of rate constants, we must first agree on the nature of the average energy E . Identifying E as k B T allows to use the exponential law described above to recover in a simple manner different extensions of statistical mechanics, including energy densities and Maxwell velocity densities [START_REF] Michel | A probabilistic rate theory connecting kinetics to thermodynamics[END_REF]; but for an application to chemistry where the different molecules have different energies, the nature of E should be precised. Two formulas are possible depending on whether the interconvertible species belong to the same distribution or to two different distributions [START_REF] Michel | A probabilistic rate theory connecting kinetics to thermodynamics[END_REF].

Hypothesis of molecule type-specific energy distributions

Each type of molecule, with a specific name and defined by chemists on the basis of its covalent architecture, has its own specific energy which could be envisioned as its average energy. Within a homogeneous population of such a type of molecule, the energy of each individual molecule could take different values because of thermal fluctuations. For example, a double bond C=C could be more or less rotated or stretched. This view is illustrated in Fig. 1A, where E i and E j are mean energies corresponding to different values of E in Eq.( 2), apart of which individual molecules can take different energy levels. The probability that a given molecule has an energy higher than a threshold E ‡ is exactly described by Eq.( 2). As the two interconvertible species have their own average energy, this hypothesis will be called Model II. The extensions of this model have been thoroughly described in [START_REF] Michel | A probabilistic rate theory connecting kinetics to thermodynamics[END_REF].

Hypothesis of the general mean energy

In this second option illustrated in Fig. 1B, all molecules, regardless of their chemical classification, participate to a global energy distribution, characterized by a single average energy value. In this unique continuum of chemical energies, the covalent architecture used for the classification of molecules, has no more importance than secondary phenomena like bond twisting and stretching. As it is based on a unique energy distribution, this hypothesis will be called Model I. 2 Application of these assumptions to the Arrhenius formula

The two views of mean energies described above will be applied to the following generic form of rate constant

k = A (T ) e B (T ) (4a)
which is, like the Arrhenius equation, made of a preexponential and an exponential term, both depending on temperature. The preexponential factor A does not contribute to the reactional energy of the molecule but corresponds to a configurational recurrence frequency, that is to say the frequency at which reappears one of the molecular configurations favorable to the reaction, out of a total number Ω of possible configurations.

A (T ) = ΦT (4b) with Φ = k B /(Ω h),
where k B and h are the Boltzmann and Planck constants respectively. This preexponential factor will no longer be studied here because the different theories more or less explicitly agree on its nature, including the activated complex theory [START_REF] Michel | A probabilistic rate theory connecting kinetics to thermodynamics[END_REF]. By contrast, the meaning of the exponent of Arrhenius depends on the theories.

Model II

If the different types of reactants are assumed to have specific mean energies, the exponential term of Arrhenius would be the probability that a molecule of type j of mean energy E j , reaches a threshold E ‡ . This probability is predicted by the exponential distribution.

k ji /A ji = P (E E ‡ ) = e - E ‡ E j (T ) (5)
The dependence on temperature of this exponent is mediated by the mean energy E j (T ) only, because the threshold E ‡ is fixed and temperature-independent. When temperature increases, E j (T ) increases while E ‡ is unchanged, so that the reaction rate expectedly increases. In this respect, this theory is in perfect agreement with the traditional interpretation of the stimulatory role of temperature on reaction rates, through a shift of the whole Maxwell-Boltzmann distribution towards higher energies. As a consequence of this shift, the fractional particle population whose energy exceeds a certain threshold, increases. When defining the equilibrium constant as a ratio of such rate constants, the van't Hoff equation reads

K ji = A j A i e E ‡ ij    1 E i (T ) - 1 E j (T )    (6)
which also depends on temperature through the mean energies.

Model I

In the unique chemical energy continuum of overall mean value E u(T ) illustrated in Fig. 1B, the reactional energies E i (T ) and E j (T ), as well as the threshold energy E ‡ , are just particular energy levels. According to Eq.(3),

P (E = E i ) = e - E i E u    1 -e - 1 E u    (7a) P (E = E j ) = e - E j E u    1 -e - 1 E u    (7b) and P (E = E ‡ ) = e - E ‡ E u    1 -e - 1 E u    (7c)
In the usual conditions of chemistry, the concentration ratios of molecules with different energies are simply given by their relative probabilities [START_REF] Michel | A probabilistic rate theory connecting kinetics to thermodynamics[END_REF], so that the Arrhenius equation can now be defined as

k ji /A ji = P (E = E ‡ ) P (E = E i ) = e - E ‡ -E i E u (8a)
A ratio of probabilities, contrary to a probability, can range from 0 to infinity, like the Arrhenius equation. As the same holds for the reciprocal reaction, in equilibrium the concentration ratio between the interconvertible reactants, or equilibrium constant, gets rid of the energy threshold.

K ji = A j A i P (E = E ‡ ) P (E = E i ) P (E = E j ) P (E = E ‡ ) = A j A i P (E = E j ) P (E = E i ) = A j A i e E j -E i E u (8b)
In both hypotheses, the equilibrium distribution of the interconvertible molecules depends primarily on their energy difference, but under the hypothesis that the different molecules have all their own mean energy, this difference is further accentuated by the height of the energy barrier. The model of the generalized chemical continuum conforms to the principle of thermochemical equilibrium in that the energy threshold does not intervene in equilibrium. For this reason it appears simpler, but conversely its temperature dependence is more complex and this complexity is overlooked. Indeed, the exponent of rate constants depends twice on the temperature, contrary to what is suggested in the Arrhenius equation expressed as a straight line.

B i(T ) = - E ‡ -E i(T ) E u(T )
and that of the equilibrium constants depends three times on the temperature

B i(T ) -B j(T ) = E j(T ) -E i(T ) E u(T )
In this model, the access of a molecule to the energy threshold E ‡ is favored by an increase in temperature in two ways: (i) by increasing the general average populational energy, which increases the denominator; and (ii) by increasing the specific energy of the considered reactant molecule, which decreases the numerator. The first mechanism is general and applies to all molecules in the isothermal medium, whereas the second mechanism depends on each type of molecule.

Theoretical shapes of the Arrhenius plots in the two models

The energy of reactants, called enthalpy in thermochemistry, is rooted in statistical mechanics, but it results from the combination of many interfering mechanisms particularly complex at low temperature. For simplicity, let us imagine a theoretical reactant consisting of a single resonator of frequency ν (i.e. absorbing a single wavelength, contrary to the more advanced treatment of Debye), like an Einstein solid. In that respect, it may not be a coincidence if in his original article on heat capacities, Einstein took as a model the diamond [START_REF] Einstein | Die Plancksche Theorie der Strahlung und die Theorie der spezifischen Wärme[END_REF], which contains only one type of chemical bond (C-C). The discrete approach to thermal mean energy of Planck can be rapidly recovered starting from the number of ways to distribute E energy quanta in N particles

Ω = (N + E -1)! (N -1)! E! (9)
giving a single particle mean entropy of

S = 1 N ln Ω = 1 + E N ln 1 + E N - E N ln E N (10a)
The ratio E/N is equivalent to the mean number of energy quanta per particle U/hν [START_REF] Planck | On the law of distribution of energy in the normal spectrum[END_REF].

S = S k B = 1 + U hν ln 1 + U hν - U hν ln U hν (10b)
Introducing temperature through the fundamental entropy equation

dS dU = 1 T (11) yields k B hν ln 1 + hν U = 1 T (12a)
and

U = hν e hν/k B T -1 (12b)
where hν can also be defined as hν = hc/λ, where c is light velocity and λ is the spectroscopic wavelength of absorption. Hence, the number of energy quanta used in Eqs [START_REF] Laidler | The development of the Arrhenius equation[END_REF][START_REF] Michel | Simply conceiving the Arrhenius law and absolute kinetic constants using the geometric distribution[END_REF][START_REF] Michel | A probabilistic rate theory connecting kinetics to thermodynamics[END_REF], is

E = U hν = 1 e hν/k B T -1
As the heat capacity of a substance is its ability to increase its energy with temperature, it can be derived from Eq.(12b) as did [START_REF] Einstein | Die Plancksche Theorie der Strahlung und die Theorie der spezifischen Wärme[END_REF],

C = dU dT = (hν) 2 k B T 2 e hν/k B T e hν/k B T -1 2 (13)
The shape of this function is represented in Fig. 2A, which shows that increasing either the temperature or the wavelength, symmetrically increases heat capacity. The two models described above predict that the Arrhenius equation is not a straight line.

Model I

In the theory of the general mean energy, even without introducing the complexity of real heat capacities, the theoretical Arrhenius plot drawn for a single frequency is not linear. Taking k B T as the mean energy, Eq.(8a) becomes

ln(k/A) = - E ‡ k B T + hν k B T e hν k B T -1 (14)
represented in Fig. 2B.

Model II

If neglecting the logarithmic temperature dependence of the preexponential factor, replacing the energy in Eq.( 5) by that of Eq.(12b) gives,

ln k/A = - E ‡ hν   e hν k B T -1    (15)
whose Taylor series is

ln k/A = E ‡ hν   1 - ∞ j=0 hν k B T j /j!   (16a)
which can be limited to its first two terms when 1/T and ν are small enough to render the higher order terms negligible

ln k/A = - E ‡ k B T - E ‡ hν 2(k B T ) 2 (16b)
This second term can generate convex Arrhenius plots for high frequency reactants (Fig. 2C). Note that interestingly, the first term of Eq.( 16b) is close to the traditional slope when H is negligible compared to E ‡ . In the traditional interpretation, the Arrhenius slope is considered constant, which misleadingly suggests that the Arrhenius plot is a straight line. We see that both theories predict that Arrhenius plots are globally curved. In addition, these curvatures can be further complicated by irregularities of heat capacities, because these simplified plots hold for single resonators, whereas real molecules contain numerous juxtaposed resonators, absorbing at different wavelengths and participating in the global energy of the molecule in a cooperative manner. Hence, molecular energies cannot be determined in a bottom-up approach starting from known wavelengths, but they can be deduced by a reverse strategy from the measured heat capacities. 14)), concave at high reactant frequency. (C) Shape of Arrhenius plot with the hypothesis of a mean reactant energy (Eq.( 15)), convex at high reactant frequency. The energy thresholds are arbitrary and the temperature-dependence of the preexponential factor is not taken into account in these representations.

Experimental comparison of the approaches

In physics, theories are not chosen for their elegance but for their capacity to describe experimental observations. The chemical reaction selected for this test is the isomerization between cis-2-butene (Z) and trans-2-butene (E), because it has a series of advantages: it involves reciprocal reactions unimolecular in both directions. Very precise data are available in kinetics [START_REF] Kaiser | Study of the thermodynamics (Thermal and Cl Catalyzed) and kinetics of the cis and trans isomerizations of CF3CF=CHF, CF3CH=CHCF3, and CH3CH=CHCH3 in 100-950 torr of N2 diluent at 296-875 K: Effect of F and CF3 substitution on the isomerization process including the fluorine "cis effect[END_REF] and in equilibrium [START_REF] Kapteijn | Thermodynamics of the geometrical isomerization of 2-butene and 2-pentene[END_REF]. Moreover, the thermal isomerization of 2-butene is restricted by a quite high energy barrier (corresponding essentially to the energy needed to break the π bond), which should make it possible to clearly distinguish the two models described above, since the barrier plays a role in equilibrium in the former but not in the latter. Because of this barrier, the thermal conversion between the two isomers is not measurable at temperatures lower than 700 K. As a consequence, equilibrium data were obtained in presence of a catalyst [START_REF] Kapteijn | Thermodynamics of the geometrical isomerization of 2-butene and 2-pentene[END_REF]. Finally, the temperaturedependent heat capacities of these two isomers are also available, which will allow to approximate their energies.

Determination of absolute energies

The two forms of the exponential factor compared here require knowledge of the absolute energies of reactants.

The probabilistic form based on reactant-specific average energies needs this value for the denominator of the exponent (Eq.( 5)) while the form based on the energy continuum model needs it at the exponent numerator (Eq.(8a)). But absolute energies and enthalpies are notoriously difficult to determine.

What form of energy to use

The energy of the single resonator reactant of frequency ν used previously is theoretical and not available for determining the molecular energy of Eq.( 12b), but the principle of this type of energy will however be retained here, that is to say an energy equal to zero at T = 0 K, increasing with temperature and calculable as the integral of heat capacity, as theorized by Einstein [START_REF] Einstein | Die Plancksche Theorie der Strahlung und die Theorie der spezifischen Wärme[END_REF] (Eq.( 13)). This point is essential since absolute energies are unknown while heat capacities are often known, so we have first to explicit their temperature-dependence in the form of an integrable function. In the energy continuum model similar to the current treatment of thermochemistry, absolute enthalpies (H) will be used, but enthalpies can take negative values, which would lead to a division by zero at a certain temperature in Eq(8a), so only positive energies (written E) will be used for the Model II. Absolute enthalpies can in principle be obtained from heat capacities using the Kirchhoff approach

H(T ) = H(T 0 ) + T T0 C p (T ) dT ( 17
)
where T 0 is temperature of reference chosen for convenience. The first step is to determine a function satisfactorily adjusted to the heat capacities measured by the Thermodynamics Research Center [START_REF]Selected values of properties of chemical compounds[END_REF] and available in the NIST Chemistry WebBook. The 3 parameter exponential association

C p ≈ α β -e -γT (18)
is chosen here as a fitting function because it is easily integrable and valid for a wide temperature range over 300 K. The fitting parameters are, for the cis isomer α = 264.49, β = 1 and γ = 13.35 × 10 -4 (curve shown in Fig. 3A), and for the trans isomer α = 266.07, β = 0.9968 and γ = 12.97 × 10 -4 . The indefinite integrals F (T ) of these functions can then be used to calculate absolute enthalpies provided the enthalpy of formation is known at a given tempera-ture T 0 .

H(T ) = F (T ) -F (T 0 ) + H f (T 0 ) (19)
If the starting elements are taken as 0 J/mol, the values given in the NIST webBook are, for the cis isomer H f Z (298)= -7.7 kJ/mol and for the trans isomer is H f E (298)= -10.8 kJ/mol. [START_REF] Prosen | Heats of combustion, formation, and insomerization of ten C4 hydrocarbons[END_REF]. To link these values to the range of validity of the exponential fitting curve, the intermediate range is covered by a polynomial approximation (Appendix). The enthalpies found in this way are presented in Fig. 3A. For comparison, the values T C p (T ) are drawn in the same graph (Fig. 3B) to show that this direct approach overestimates the energies.

Based on the view of molecular energy as a positive number of energy quanta, like in Eq.(12b), and to avoid division by zero in the probabilistic approach, internal energies will be simply defined as

E(T ) = F (T ) -F (0) (20)
The models can now be compared numerically with the kinetic results.

5 Determination of energy threshold from the Arrhenius slope at T = 800 K

The rate equations of cis trans isomerization of 2-butene, determined experimentally [START_REF] Kaiser | Study of the thermodynamics (Thermal and Cl Catalyzed) and kinetics of the cis and trans isomerizations of CF3CF=CHF, CF3CH=CHCF3, and CH3CH=CHCH3 in 100-950 torr of N2 diluent at 296-875 K: Effect of F and CF3 substitution on the isomerization process including the fluorine "cis effect[END_REF] are

k Z→E = 9 × 10 13 × Exp - 31845 T s -1
and

k Z→E = 21 × 10 13 × Exp - 32725 T s -1
The slopes are expressed here in Kelvin units to eliminate a presupposed model. Written in this form, these equations are purely experimental and modelindependent, so they can be used as templates to calculate the parameters of the model-derived equations. Giving to the central experimental temperature 800 K the best degree of confidence, the parameters are adjusted such that at this temperature, the slope d(ln k)/d(1/T ) of the theoretical equations gives the experimental slope. The superposition of the resulting curves to the Arrhenius plots of [START_REF] Kaiser | Study of the thermodynamics (Thermal and Cl Catalyzed) and kinetics of the cis and trans isomerizations of CF3CF=CHF, CF3CH=CHCF3, and CH3CH=CHCH3 in 100-950 torr of N2 diluent at 296-875 K: Effect of F and CF3 substitution on the isomerization process including the fluorine "cis effect[END_REF], confirms the concavity of Model I (Fig. 4A) and convexity of Model II (Fig. 4B), both predicted theoretically (Fig. 2). Introducing energies in the Arrhenius equation should yield the same value for E ‡ by two different calculations starting from the two reciprocal reactions.

Model I

The two types of energies will be applied to this model.

Using absolute enthalpies

We find, for the cis isomer,

• A Z = 2.82 × 10 4 s -1
• E ‡ = 185.85 kJ/mol and for the trans isomer,

• A E = 25.00 × 10 4 s -1

• E ‡ = 190.83 kJ/mol
The preexponential factors defined as the intersects of the Arrhenius straight line with the vertical axis, are naturally no longer relevant in the two models using reactant energies. The values reported in [START_REF] Kaiser | Study of the thermodynamics (Thermal and Cl Catalyzed) and kinetics of the cis and trans isomerizations of CF3CF=CHF, CF3CH=CHCF3, and CH3CH=CHCH3 in 100-950 torr of N2 diluent at 296-875 K: Effect of F and CF3 substitution on the isomerization process including the fluorine "cis effect[END_REF] were obtained using the linear Arrhenius equation dependent on temperature only through RT and without taking into account the contribution of enthalpies. Since the experiments giving these results were conducted at temperatures around 800 K (1/T =0.00125) [START_REF] Kaiser | Study of the thermodynamics (Thermal and Cl Catalyzed) and kinetics of the cis and trans isomerizations of CF3CF=CHF, CF3CH=CHCF3, and CH3CH=CHCH3 in 100-950 torr of N2 diluent at 296-875 K: Effect of F and CF3 substitution on the isomerization process including the fluorine "cis effect[END_REF], the Arrhenius slopes obtained in these conditions must be compatible with the reactant enthalpies at this temperature, which are, using the Kirchoff approach described above, E Z (800) = 60 kJ/mol and E E (800) = 54 kJ/mol. The values of E ‡ found from the two reciprocal reactions, 186 and 191 kJ/mol, are not very distant, but the activation energies however, are no longer those currently reported. The previous values were E aZ = 264.6 kJ/mol and E aE = 271.9 kJ/mol. They are now E aZ = 125.2 kJ/mol and E aE = 136.3 kJ/mol at 800 K.

Using positive energies

The same treatment with positive energies gives very similar results

• A Z = 2.82 × 10 4 s -1

• E ‡ = 207.34 kJ/mol and for the trans isomer,

• A E = 27.00 × 10 4 s -1

• E ‡ = 214.28 kJ/mol

Model II

Using the hypothesis of reactant-specific energy distribution and positive energies, given that reactant energies are much higher than RT , the values of E ‡ will naturally be much higher than the previous ones. Moreover, since calculations from the reciprocal reactions use multiplications, the determination of E ‡ is expected to be much more sensitive to imprecision than the previous one based on subtractions. When incorporating the positive energies calculated above in the probabilistic rate constant of Eq.( 5) and adjusting the slope of the plot in Arrhenius coordinates at T = 800 K. One finds for the cis isomer,

• A Z = 2.43 × 10 6 s -1

• E ‡ = 1906.53 kJ/mol and for the trans isomer,

• A E = 6.25 × 10 6 s -1

• E ‡ = 1849.28 kJ/mol

The kinetic data thus give different parameter values depending on the model, but without being able to provide a discriminating criterion. The comparison of kinetics and equilibrium data will allow this discrimination.

6 Comparison of the data obtained from Arrhenius coordinates with the results obtained at equilibrium

The results of [START_REF] Kapteijn | Thermodynamics of the geometrical isomerization of 2-butene and 2-pentene[END_REF] obtained with catalysis are remarkably precise and compatible with previous, more partial measurements, as for instance [START_REF] Scott | Thermodynamic properties of cis-2-butene from 15 to 1500 K[END_REF]. Fig. 4C andD show the comparison between values of equilibrium constants K Z→E obtained in [START_REF] Kapteijn | Thermodynamics of the geometrical isomerization of 2-butene and 2-pentene[END_REF] and the ratios of kinetic constants k Z→E /k E→Z based on the values measured in [START_REF] Kaiser | Study of the thermodynamics (Thermal and Cl Catalyzed) and kinetics of the cis and trans isomerizations of CF3CF=CHF, CF3CH=CHCF3, and CH3CH=CHCH3 in 100-950 torr of N2 diluent at 296-875 K: Effect of F and CF3 substitution on the isomerization process including the fluorine "cis effect[END_REF] and interpreted as either (i) the linear Arrhenius equations reported in [START_REF] Kaiser | Study of the thermodynamics (Thermal and Cl Catalyzed) and kinetics of the cis and trans isomerizations of CF3CF=CHF, CF3CH=CHCF3, and CH3CH=CHCH3 in 100-950 torr of N2 diluent at 296-875 K: Effect of F and CF3 substitution on the isomerization process including the fluorine "cis effect[END_REF], (ii) the Arrhenius equations modified to include reactant energies (Model I) and (ii) the probabilistic equation where reactant energies are envisioned as mean values (Model II). All approaches confirm the predominance of the trans isomer at low temperatures and the accumulation of the cis isomer at higher temperatures, but none gives exactly the equilibrium results.

Obviously and even in case of uncertainties on the values of energies, the model II is disqualified since it predicts that cis-2-butene is almost absent below 300 K, contrary to the experimental measurements [START_REF] Scott | Thermodynamic properties of cis-2-butene from 15 to 1500 K[END_REF][START_REF] Kapteijn | Thermodynamics of the geometrical isomerization of 2-butene and 2-pentene[END_REF], in which this isoform is clearly present from 100 K. This difference is specific of the model II because it is due to the participation of the activation thresholds in equilibrium. As a matter of fact, using the kinetic model II, the presence of trans-2 butene in equilibrium can be shifted at lower temperatures by lowering E ‡ in the equations (not shown). 5). (C) Comparison of the kinetic results obtained in Arrhenius coordinates [START_REF] Kaiser | Study of the thermodynamics (Thermal and Cl Catalyzed) and kinetics of the cis and trans isomerizations of CF3CF=CHF, CF3CH=CHCF3, and CH3CH=CHCH3 in 100-950 torr of N2 diluent at 296-875 K: Effect of F and CF3 substitution on the isomerization process including the fluorine "cis effect[END_REF] with the equilibrium constant [START_REF] Kapteijn | Thermodynamics of the geometrical isomerization of 2-butene and 2-pentene[END_REF]. (D) Proportion of the trans isomer in equilibrium predicted by the different theories. The curves shown in (C) and (D) are (i) the equilibrium data of [START_REF] Kapteijn | Thermodynamics of the geometrical isomerization of 2-butene and 2-pentene[END_REF] (black line with dots) in which the cis isoform is always predominant at all temperatures tested; (ii) the ratio of Arrhenius equations of rate constants kZ→E/kE→Z of [START_REF] Kaiser | Study of the thermodynamics (Thermal and Cl Catalyzed) and kinetics of the cis and trans isomerizations of CF3CF=CHF, CF3CH=CHCF3, and CH3CH=CHCH3 in 100-950 torr of N2 diluent at 296-875 K: Effect of F and CF3 substitution on the isomerization process including the fluorine "cis effect[END_REF] (black dashed line, for which the switch between Z and E isomers occurs at 1000 K); the curve predicted in the Model I (blue plain line, for which the equivalence between Z and E isomers is obtained at 468 K); the curve predicted using the Model II (dotted line, equivalence between Z and E at 560 K).

Conclusions

The comparison between kinetic and equilibrium experimental data shows that the hypothesis that the Arrhenius exponential factor could be a probability, does not apply to a chemical system of homogenized temperature. This model can apply to other physical situations like two-temperature systems or the evaporation of a clump of hot matter [START_REF] Michel | Simply conceiving the Arrhenius law and absolute kinetic constants using the geometric distribution[END_REF], but not to thermochemistry. The fundamental principle that activation energies do not influence equilibrium ratios is verified, but it is very instructive to return to the theoretical bases to see what this principle implies. This reductionist approach sheds light (i) on the nature of enthalpies which are not average energies but fixed basal values which can be increased by thermal fluctuations, and (ii) on the nature of the Arrhenius equation, which is a ratio of exponential probabilities calculated within a single energy distribution. This universal energy continuum encompasses all types of molecules, irrespective of our classification of the chemical world into different molecular species. Covalent bonds, which are used to define molecules, have no more importance for defining molecular energy than other features such as non-covalent bonds, stretching, rotations and vibrations. This view of molecular energy is in fact consistent with its immaterial nature, homogenizing between colliding molecules without any consideration for their structure. In this global exponential distribution, all forms are metastable and the most energetic molecules are the rarest ones. The acceleration of reactions by temperature is mediated at two levels: the increase of the denominator of the exponent k B T , and the decrease of the numerator by reduction of the difference E a = E ‡ -H(T ). Curiously this latter role of temperature is generally ignored in the interpretation of the Arrhenius equation as a straight line. The usual explanation of the role of temperature on reaction rates is based on a shift towards the high energies of a Maxwell-Boltzmann distribution, which increases the fraction of molecules exceeding a fixed energy threshold. Strangely, this pedagogical interpretation completely holds for the hypothesis of reactant-specific mean energies disqualified here, but it is very partial under the hypothesis of the universal mean energy, since it does not take into account the enthalpy of the reactants. The introduction of reactant energy into the Arrhenius equation and the replacement of activation energies by energy thresholds, which are common to the two reciprocal reactions, would allow to anticipate the nonlinearity of Arrhenius plots. The simple knowledge of heat capacities allows to estimate this energy and to predict many kinetic and thermodynamic behaviors. (C) Adjustment between the measured heat capacities, and functions defined as the sum of elementary heat capacities.

Adjustment is good at high temperatures but not at low temperatures.

Figure 1 .

 1 Figure 1. Competing views of the energy restriction of rate constants. The quantities written in brackets are average values of exponential distributions, whereas the quantities written without brackets are particular energy levels belonging to these distributions. (A) A mean energy value is assigned to each type of molecule i or j. (B) The energies of the different types of molecules are not mean values but fixed values belonging to a common energy distribution with a general mean value E u .

Figure 2 .

 2 Figure 2. Minimalist modeling of heat capacity and Arrhenius plots for a theoretical reactant made of a single resonator. (A) Heat capacity symmetrically increases when increasing either temperature or λ. (B) Shape of Arrhenius plot obtained from the energy continuum hypothesis (Eq.(14)), concave at high reactant frequency. (C) Shape of Arrhenius plot with the hypothesis of a mean reactant energy (Eq.(15)), convex at high reactant frequency. The energy thresholds are arbitrary and the temperature-dependence of the preexponential factor is not taken into account in these representations.

Figure 3 .

 3 Figure 3. (A) Measured heat capacities and fitting function valid for high temperatures. (B) Absolute enthalpies (H, plain lines) and positive energies (E, dotted lines) of the 2-butene isomers, red for the cis isomer and blue for the trans isomer. For comparison, energies calculated as direct products T Cp(T ) are shown (diamonds corresponding to trans-2-butene).

Figure 4 .

 4 Figure 4. (A) Arrhenius plots of the reactions of 2-butene isomerization reported in [6] (dashed lines), superposed to those obtained by introducing reactant energies in the Arrhenius equation according to Eq.(8a). Cis to trans reaction in blue and trans to cis reaction in red. The theoretical parameters were obtained by adjusting the slopes at 800 K. (B) Corresponding results when reactant energies are intro-

Figure A1 .

 A1 Figure A1. Heat capacities of the cis and trans isomers of 2-butene. (A) Normalized Einstein heat capacities (Eq.(A.2))for the absorbed infrared wavelengths of cis-2-butene taken individually. (B) The complexity of real heat capacities is particularly evident at low temperatures, where the relative values obtained for the two isomers of 2-butene vary surprisingly. (C) Adjustment between the measured heat capacities, and functions defined as the sum of elementary heat capacities. Adjustment is good at high temperatures but not at low temperatures.

Acknowledgements. I thank very much Bill Kaiser and Freek Kapteijn for helpful discussions.

8 References

Appendices

A Assembly of single resonator heat capacities

Real molecular heat capacities differ from the theory of Einstein in that they do not derive from a single wavelength of absorption. A first attempt of construction of the global heat capacity of a molecule may consist in juxtaposing its internal resonators. The so-called Neumann-Kopp law stipulates that the heat capacity of a composite material is equal to the weighted average of the individual elemental heat capacities. Using this postulate, the measured heat capacities can be adjusted to a theoretical curve of the form

where C(λ j , T ) is the normalized shape of the Einstein capacity for the chosen wavelength λ j , ranging from 0 to 1 when either λ and/or T go from 0 to infinity. This equation is satisfactory for trans-2-butene, but is less successful for cis-2-butene (Fig. A1C). Although this isoform presents two additional absorption wavelengths compared to the cis counterpart (around 15 and 17 µm), the best result was in fact obtained using only one wavelength, of intermediate value 16 µm. This method however yields long and hardly tractable equations. In addition, it is moderately justified because a molecule cannot be approximated as a juxtaposition of resonators since they are non-independent and likely to cooperate. The different peaks of the infrared absorbance spectrum may be involved in different molecular phenomena including C-H, C-C and C=C bond stretching or rocking, which are not independent from each others. Hence, it may be preferable to use a more arbitrary but simpler formula such as the simple exponential function used in the main text and the polynomial function shown below, useful for intermediate temperatures.

B Polynomial fitting function for

heat capacities between 273 and 1500 K

For the cis isomer, α = 7.63, β = 31.35 × 10 -2 , γ = -1.63 × 10 -4 , δ = 4.08 × 10 -8 and = -4 × 10 -12 . For the trans isomer, α = -6.21, β = 34.13 × 10 -2 , γ = -1.86×10 -4 , δ = 4.91×10 -8 and = -5.07×10 -12 .