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Energy 89 (2015) 1029e1049

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Energy

journal homepage: www.elsevier .com/locate/energy

Thermodynamic chemical energy transfer mechanisms ofnon-equilibrium, quasi-equilibrium, and equilibrium chemicalreactions

Heui-Seol Roh*

Department of Mechanical Engineering, City College of New York, New York, NY 10031, USA

a r t i c l e i n f o

Article history:Received 22 January 2015Received in revised form20 May 2015Accepted 16 June 2015Available online 10 July 2015

Keywords:Thermodynamic chemical energy transfertheoryChemical non-equilibriumChemical quasi-equilibriumChemical equilibriumChemical and electrochemical reactionsFour control mechanisms

* Tel.: þ1 212 650 6759; fax: þ1 212 650 8013.E-mail address: hroh@ccny.cuny.edu.

http://dx.doi.org/10.1016/j.energy.2015.06.0490360-5442/© 2015 Elsevier Ltd. All rights reserved.

a b s t r a c t

Chemical energy transfer mechanisms at finite temperature are explored by a chemical energy transfertheory which is capable of investigating various chemical mechanisms of non-equilibrium, quasi-equi-librium, and equilibrium. Gibbs energy fluxes are obtained as a function of chemical potential, time, anddisplacement. Diffusion, convection, internal convection, and internal equilibrium chemical energy fluxesare demonstrated. The theory reveals that there are chemical energy flux gaps and broken discrete sym-metries at the activation chemical potential, time, anddisplacement. The statistical, thermodynamic theoryis the unification of diffusion and internal convection chemical reactions which reduces to the non-equilibrium generalization beyond the quasi-equilibrium theories of migration and diffusion processes.The relationship between kinetic theories of chemical and electrochemical reactions is also explored. Thetheory is applied to explore non-equilibrium chemical reactions as an illustration. Three variable separa-tion constants indicate particle number constants and play key roles in describing the distinct chemicalreactionmechanisms. The kinetics of chemical energy transfer accounts for the four controlmechanisms ofchemical reactions such as activation, concentration, transition, and film chemical reactions.

© 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Energy transfer is a central process in the evolution of the uni-verse, and one distinctive formof energy transfer is chemical energytransfer. Thermodynamics dealing with energy transfer and chem-ical energy transfer dealing with chemical reactions have variousapplication areas in science and engineering. The coupled mecha-nisms of thermodynamic and chemical reactions are definitelydemanded in diverse fields including phase transition, chemistry,electrochemistry, corrosion, thermal devices, microscopic organ-isms, physics, mechanical engineering, biology, and energy.

Extensive research has been conducted for developing a ther-modynamic theory to describe chemical reaction processes. Exist-ing theories of chemical kinetics include the Arrhenius equation,collision theory, potential energy surfaces, Gibbs flux formalism,Lagrangian formalism, and transition state theory [1e26]. Never-theless, chemical reaction mechanisms are so complicated thattheir entire understanding in terms of existing theories [27e29] is

not possible yet. The kinetics based on the conservation of mass orconcentration provides only the information about the rate ofchemical reactions and the rate constant. Their nominal forms alsodepend on the orders of chemical reactions [27e29]. The concen-tration equation for mass (or particle number) transfer [33] isuseful in investigating particle diffusion:

vC=vtþ va$VC ¼ D0V2Cþ C0i: (1)

Eq. (1) is closely related to chemical reactions when the internalgeneration Ci

0 is taken into account. However, the precise form ofthe internal convective energy generation is not exactly clarified sofar. The other existing theories for chemical reactions are estab-lished under the assumption of chemical equilibrium, so theycannot predict non-equilibrium chemical processes. Otherwise,they describe chemical reactions under some limited situations. Forexample, transition state theory assumes intermediate reactionstates which are described by a statistical formalism, and then theyare combined with the reaction equation, but their detailed dy-namic properties are not given explicitly [10e12,27]. Therefore, arobust theory of chemical reactions should provide essential dy-namic and static information about chemical reactions under a

Fig. 1. Chemical energy transfer in the hierarchy for energy transfer mechanisms.Electrochemical energy transfer is considered as a branch of chemical energy transfer.

H.-S. Roh / Energy 89 (2015) 1029e10491030

certain fundamental principle. We are here interested in a chemicalenergy transfer theory which are simultaneously capable of inte-grating thermodynamic, spatial, and temporal properties as well asdescribing non-equilibrium, quasi-equilibrium, and equilibriumprocesses.

An internal energy transfer theory [30] for thermodynamic non-equilibrium, quasi-equilibrium, and equilibrium has been pro-posed. The systematically unified transport theory is capable ofproducing the three energy transfer mechanisms of thermody-namics so that it estimates parameters in thermodynamic equi-librium and non-equilibrium states and clarifies the characteristicsof transport processes [30e36]. The kinetic theory is also applicableto understanding the chemical energy transfer mechanisms invarious chemical phenomena in nature.

We have furthermore proposed the statistical thermodynamictheory for electrochemical reactions [31,32] which is vital toexamine both diffusive and internal convective electrochemicalreactions. It is a transport theory for electrochemical reactionsleading to the integration of diffusion and internal convection, ageneralization beyond quasi-equilibrium theories for migrationand diffusion, and the unification of internal equilibrium andequilibrium. The Nernst equation is derived from the internalconvective and equilibrium processes, the time rate of electro-chemical reactions is obtained, and the spatial cross section isgained from the kinetic theory.

In these contexts, we propose a chemical energy transfer theoryfor non-equilibrium, quasi-equilibrium, and equilibrium chemicalenergy as a special case of the internal energy transfer theory [30].The theory is analogous to the statistical thermodynamic theory forelectrochemical reactions [31,32]. It is able to probe internalconvective chemical energy transfer in chemical reactions since it isa generalization beyond chemical energy transfer in quasi-equilibrium. It is thus a non-equilibrium extension beyond chem-ical equilibrium and quasi-equilibrium. It is based on the conser-vation of energy rather than the rate of chemical reactions based onthe conservation of mass which the existing theories [27e29]depend on. The chemical transfer theory provides the chemicalpotential and spatial dependence in addition to the temporaldependence of chemical reactions. The theory is likewise compe-tent to explore both dissociation and synthesis chemical reactionsregardless of the molecularity of a chemical reaction.

In the chemical energy transfer theory, the Gibbs energy fluxesare described as a function of chemical potential, time, anddisplacement simultaneously under a postulate that chemical po-tential, time, and displacement are independent and orthogonalvariables in extended phase spaces. This is the first rigorousmicroscopic kinetic theory for internal convective chemical energytransfer.

External convective energy transfer is formulated in addition tointernal convective energy transfer. Moreover, the thermodynamicbehavior of the equilibrium chemical process is clarified throughthe theory. A connection between a rate equation and a reactionquotient is also proved through the chemical energy transfer theorysince they are dependent on both concentration and temperature.

Fig. 1 sketches the hierarchy for energy transfer which consistsof heat, work, and chemical energy transfer [30e36]. Each energytransfer possesses three modes of conduction (or diffusion), con-vection, and radiation in addition to equilibrium. Aside from radi-ation, this paper concentrates on internal convection, conduction,internal equilibrium, and convection of chemical energy transfer toexplore chemical reactions. Moreover, the statistical thermody-namic theory of electrochemical reactions is reviewed as a branchof the chemical energy transfer.

The objective of this paper is hence to establish a chemical en-ergy transfer theory for chemical reactions and to apply the

chemical energy transfer theory to examine various chemical re-actions to clarify their features. The theory is utilized to examinechemical energy transfer mechanisms in equilibrium, quasi-equilibrium, and non-equilibrium. The chemical energy transfertheory can be applied to internal convective chemical processes at asolideliquid boundary as its demonstration. Chemical energy fluxprofiles are predicted as a function of chemical potential, time, anddistance. The four control regimes are also anticipated as a functionof chemical potential, time, and distance in analogy with electro-chemical reaction regimes: activation, concentration, transition,and film chemical reaction regimes.

2. Internal energy transfer theory

The internal energy transfer theory [30] has been proposed forthe transfer mechanisms of heat, work, and chemical energy. A totalsystem containing reactants and products is thermodynamicallyisolated from its surroundings, and has thermodynamic energytransfer due to thermal, mechanical, and chemical non-equilibriumbetween reactants and products. The local thermodynamic non-equilibrium is characterized as the intensive variables of temper-ature, pressure, and chemical potential [37].

2.1. Motivation of internal energy transfer

Internal energy transfer theory is motivated from the thermo-dynamic point of view in which internal energy transfer dependson intensive state variables as well as extensive state variables.Internal energy transfer in a quasi-equilibrium (quasi-static) pro-cess is conventionally described only by extensive state variables,but internal energy in a non-equilibrium process can be stated byincluding intensive state variables. Here, the dependence ofintensive state variables for internal energy transfer is discussed[30,31,33].

Euler's equation of thermodynamics for the internal energy is

U (T, S, P, V, m, N) ¼ TS � PV þ mN.

From Euler's equation, we have the first law of thermodynamics:

dU ¼ dQ � dW þ dG.

Rewriting the first law, we obtain

dU (T, S, P, V, m, N) ¼ (TdS � PdV þ mdN) þ (SdT � VdP þ Ndm) (2)

where

dQ (T, S) ¼ TdS þ SdT,

H.-S. Roh / Energy 89 (2015) 1029e1049 1031

dW (P, V) ¼ PdV þ VdP,

dG (m, N) ¼ mdN þ Ndm.

For a quasi-equilibrium process, the fundamental equation ofthermodynamics is given by

dU (S, V, N) ¼ TdS � PdV þ mdN,

where the GibbseDuhem equation is observed:

SdT � VdP þ Ndm ¼ 0.

On the other hand, for a non-equilibrium or equilibrium process,we have

dU (S, V, N) < TdS � PdV þ mdN.

Combining non-equilibrium, quasi-equilibrium, and equilibriumprocesses, we can rewrite

dU (S, V, N) � TdS � PdV þ mdN, (3)

which is the fundamental thermodynamic relation.Comparing Eq. (2) with Eq. (3), we can analyze that

SdT � VdP þ Ndm � 0. (4)

From the above equation, we have the conditions of distinct ther-modynamic processes:

For a quasi-equilibrium process, it reduces to the GibbseDuhemequation

SdT � VdP þ Ndm ¼ 0.

For a non-equilibrium or equilibrium process, it holds theinequality condition

SdT � VdP þ Ndm > 0.

Distinction between non-equilibrium and equilibrium depends onwhether temperature, pressure, and chemical potential are largerthan their corresponding activation temperature, activation pres-sure, and activation chemical potential.

In summary, we analyze that there are two contributions ininternal energy transfer in Eq. (2). The first contribution originatesfrom external variable changes, and the second contribution comesfrom the internal variable changes. The GibbseDuhem relationholds only for the quasi-equilibrium process.

Based on Eq. (2), we can specifically classify non-equilibrium,quasi-equilibrium, and equilibrium processes. We utilize the defi-nitions of the thermodynamic potentials:

Gibbs free energy, G ¼ U � TS þ PV ¼ mN,

Heat potential, Q ¼ U þ PV � mN ¼ TS,

Grand potential, W ¼ U � TS � mN ¼ PV.

We also employ the activation points of intensive variables fromequilibrium to non-equilibrium:

Activation temperature, Ta,Activation pressure, Pa,Activation chemical potential, ma,

Irreversible non-equilibrium processes take place if T > Ta, P > Pa,and m > ma while reversible equilibrium processes occur if T < Ta,P < Pa, and m < ma. Table 1 summarizes non-equilibrium, quasi-equilibrium, and equilibrium processes.

Irreversible non-equilibrium processes take place if T > Ta,P > Pa, and m > ma while reversible equilibrium processes occur ifT < Ta, P < Pa, and m < ma:

Internal energy transfer, dU (T, S, P, V, m, N) ¼(TdS � PdV þ mdN) þ (SdT � VdP þ Ndm),

Heat transfer, dQ (S, T) ¼ TdS þ SdT,

Work transfer, dW (V, P) ¼ PdV þ VdP,

Chemical energy transfer, dG (N, m) ¼ mdN þ Ndm,

Irreversible quasi-equilibrium processes take place if T > Ta,P > Pa, or m > ma while reversible quasi-equilibrium processes occurif T < Ta, P < Pa, or m < ma:

Internal energy process, dU (T, P, m) ¼ SdT � VdP þ Ndm ¼ 0,

Isothermal process, dQ (S) ¼ TdS if dT ¼ 0,

Isobaric process, dW (V) ¼ PdV if dP ¼ 0,

Constant chemical potential process, dG (N) ¼ mdN if dm ¼ 0.

Irreversible quasi-equilibrium processes take place if T > Ta,P > Pa, or m > ma while reversible quasi-equilibrium processes occurif T < Ta, P < Pa, or m < ma:

Internal energy process, dU (S, V, N) ¼ TdS � PdV þ mdN ¼ 0,

Isentropic process, dQ (T) ¼ SdT if dS ¼ 0,

Isochoric process, dW (P) ¼ VdP if dV ¼ 0,

Constant particle number process, dG (m) ¼ Ndm if dN ¼ 0.

In general, internal energy transfer indicated by Eq. (2) containscontributions from temperature change, pressure change, andchemical potential change due to the internal degrees of freedom inaddition to the external variable changes. In addition, we adopt thefact that internal energy transfer depends on space and time due tothe external degrees of freedom.

2.2. Internal energy transfer theory

We utilize a postulate that the internal energy flux (or internalenergy intensity) can be described as a function of temperature,pressure, chemical potential, time, and space:

u ¼ u (T, P, m, t, r).

It is assumed that the five variables are independent and orthog-onal. Furthermore, the expression of the internal energy flux can bewritten as a product of temperature, pressure, chemical potential,time, and space dependent terms:

u ¼ q(T)w(P)g(m)B(t)A(r).

The internal energy flux in thermodynamics under the postulateis described by a partial differential equation:

Table 1Non-equilibrium, quasi-equilibrium, and equilibrium processes of energy transfer.

Energy transfer Irreversible or reversible quasi-equilibrium Non-equilibrium or equilibrium

Chemical energy dG (m) ¼ Ndm if dN ¼ 0. dG (N) ¼ mdN if dm ¼ 0 dG (N, m) ¼ mdN þ NdmThermal energy

(heat)dQ (T) ¼ SdT if dS ¼ 0 dQ (S) ¼ TdS if dT ¼ 0 dQ (S, T) ¼ TdS þ SdT

Mechanical energy(work)

dW (P) ¼ VdP if dV ¼ 0, dW (V) ¼ PdV if dP ¼ 0 dW (V, P) ¼ PdV þ VdP

Internal energy dU (S, V, N) ¼ TdS � PdV þ mdN ¼ 0 dU (T, P, m) ¼ SdT � VdP þ Ndm ¼ 0 dU (T, S, P, V, m, N) ¼ (TdS � PdV þ mdN) þ (SdT � VdP þ Ndm)

H.-S. Roh / Energy 89 (2015) 1029e10491032

lV$u þ t Du/Dt ¼ 0 (5)

where u ¼ un is the total energy flux vector (or the power densityvector) with the unit vector n. The internal energy flux u is definedby internal energy per unit area and unit time. t is the total reactiontime which is defined by the mean reaction time t0 multiplied bythe particle number of the system, and l is the total mean free pathwhich is defined by themean free path l0 multiplied by the particlenumber of the system.

D/Dt represents the thermodynamic convective derivative:

D/Dt ¼ v/vt þ vT$VΤ¼ v/vt � [T0 v/vΤ þ (kBT0/ v0) v/vP þ kBT0 v/vm]/t (6)

where Τ is the temperature, P is the pressure, and m is the chemicalpotential. v0 is the equilibrium volume V0 per particle number, T0 isthe equilibrium temperature, and kB is the Boltzmann constant. InEq. (6), we introduce the thermodynamic velocity vector

vT ¼ �l/t (i þ j þ k)

and the thermodynamic derivative vector

VΤ ¼ (i T0v/vΤ þ j P0v/vP þ k m0v/vm)/l.

The intrinsic intensive property dimensions of temperature, pres-sure, and chemical potential play the roles of momentum di-mensions in a material space and their corresponding extensivedimensions of entropy, volume, and particle number do the roles ofspace dimensions or vice versa. They behave as independent di-mensions in a material space like the conventional space and mo-mentum dimensions.

Eq. (5) can be fully expressed as

lV$u � [T0 vu/vΤ þ (kBT0/v0) vu/vP þ kBT0 vu/vm] þ tvu/vt ¼ 0. (7)

The first term in the bracket of Eq. (7) represents the heat transfer,the second term the work transfer, and the third term the chemicalenergy transfer of the system in amicro-canonical ensemble. Eq. (7)may become the thermodynamic extension of the Liouville equa-tion since the thermodynamic intensive variables in Eq. (7) areincluded as phase spaces and the internal energy flux in Eq. (7) isproportional to the phase space distribution in the Liouville equa-tion [38,39].

We deal with the internal convective and external diffusioneffects in Eq. (7). Using themethod of separation of variables for theinternal energy flux

u (Τ, P, m, t, r) ¼ q(Τ)w(P)g(m)B(t)A(r), (8)

we derive five differential equations from (7):

vq/vΤ þ (a/T0) q(Τ) ¼ 0, (9)

vw/vP þ (zv0 /kBT0) w(P) ¼ 0, (10)

vg/vm þ (n/kBT0) g(m) ¼ 0, (11)

vB/vt þ (b /t) B(t) ¼ 0, (12)

V$А þ (k/l) А(r) ¼ 0. (13)

Eq. (9) represents the heat transfer equation, and Eq. (10) stands forthe work transfer equation. Eq. (11) shows the chemical energytransfer equation and leads to the Nernst equation. Eq. (12) and (13)reflect the temporal and spatial equations for energy transfer,respectively.

The five constants a, z, n, b, and k in the separation of variablesare introduced:

a: thermal particle number constant,z: mechanical particle number constant,n: chemical particle number constant,b: temporal particle number constant,k: spatial particle number constant.

The particle number constants in the above are constrained by:In nonrelativistic case,

a þ z þ n ¼ d,

d þ k � b ¼ 0.

In relativistic case,

a2 þ z2 þ n2 ¼ d2,

d2 þ k2 � b2 ¼ 0.

The thermal particle number constant in Eq. (9) is connected tothe entropy change by a ¼ ln U ¼ S/kB where U is the number ofmicrostates in the microcanonical ensembles. The equilibriumvolume per particle number in Eq. (10) is related to k by v0 ¼ V0/k. nin Eq. (11) represents the particle number change in specific par-ticle species. kr ¼ b/t ¼ 1/t0 in Eq. (12) stands for the rate constantin internal energy transfer and is the inverse of the mean reactiontime t0. k ¼ (b2 � d2)1/2 ¼ l/l0 in Eq. (13) is connected to the re-action cross section sc ¼ V/kl0 ¼ v0/l0.

As for the solutions of internal convection and diffusion energyfluxes (uu and ud) in the non-equilibrium process of nonzero T, P, m,t, and r, we find the following integrated solutions from (9)e(13),respectively:

q(ΤS) ¼ q0 exp (�aΤS/T0), (14)

w(PV) ¼ w0 exp (�zv0PV/kBT0), (15)

g(mN) ¼ g0 exp (�nmN/kBT0), (16)

B(tB) ¼ B0 exp (�btB/t), (17)

H.-S. Roh / Energy 89 (2015) 1029e1049 1033

А(rB) ¼ A0 exp (k$rB/l), (18)

wherek¼kn¼ (b2� d2)1/2nandn is theunit vectoralong thediffusiondirection of the energy flux. Eq. (14)e(16) reflect the Arrhenius typeequations.Eq. (15) represents theequationof state, Eq. (17)denotes thereaction frequency, and Eq. (18) indicates the steric factor.

To apply the boundary and initial conditions to Eq. (14)e(18), weutilize the excess temperature, pressure, chemical potential, time,and displacement measured from their relative equilibriums,respectively:

ΤS ¼ jT � T0j,

PV ¼ jP e P0j,

mN ¼ jm � m0j,

tB ¼ jt � t0j,

rB ¼ jr � r0jn.

At large TS, PV, mN, rB, or tB, the corresponding energy flux amplitudeapproaches to zero.

The process with nonzero a, z, and n represents the coupledtransfer with heat, work, and chemical energy transfer in a system,and the relevant thermodynamic potential change is the internalenergy change DU since the microcanonical ensemble system doesnot interact with environment. The system process takes placeunder the constant entropy, particle number, and volume. The in-ternal energy flux takes the form

u (ΤS, PV, mN, tB, rB) ¼ u00exp [�(akBΤS þ zv0PV þ nmN)/kBT0] exp (�btB/t) exp (k$rB/l)

¼ u00exp [(DQS � DWV þ DGN)/kBT0] exp (�btB/t) exp (k$rB/l)

¼ u00exp (DUB/kBT0) exp (�btB/t) exp (k$rB/l) (19)

where u00 ¼ q0w0g0B0A0.

The heat reaction quotient Zq ¼ exp(DQS/kBT0) for a heatensemble, the work reaction quotient Zw ¼ exp(�DWV/kBT0) for agrand canonical ensemble, and the Gibbs reaction quotientZg ¼ exp(DGN/kBT0) for a Gibbs ensemble. Hence, we analyze that

DQS ¼ (DQ � DQ0) ¼ �akBΤS ¼ �akBjT � T0j,

DWV ¼ (DW � DW0) ¼ zVPV ¼ zv0jP e P0j,

DGN ¼ (DG � DG0) ¼ �nmN ¼ �njm � m0j,

DUB ¼ (DU � DU0) ¼ DQS � DWV þ DGN.

3. Chemical energy transfer theory for chemical reactions

We are interested in chemical energy transfer which is regardedas a special case of the internal energy transfer theory22 with thedesignated particle number constants (nonzero n, b, and k;a ¼ z ¼ 0). A chemical reaction is a process that leads to thetransformation of one set of chemical substances to another. A totalsystem containing reactants and products is chemically isolatedfrom its surroundings and is in chemical non-equilibriumwhile thetwo phases of reactants and products are in thermal and me-chanical isolation. For the system in chemical non-equilibrium, thephysical mechanisms of energy transfer can be described byphysical parameters in local thermodynamic equilibrium.

The statistical thermodynamic theory for electrochemical re-actions [31,32] is generalized for a statistical thermodynamic theoryfor chemical reactions. Chemical reactions can be either spontaneousor non-spontaneous. The Gibbs potential in chemical energy transferis definedbyG¼SmiNi and theGibbs energy transfer per area per unittime between reactants and products can be defined by g ¼ (Smin0i -Smkn0k) where n0 is the particle density per area per unit time.

A chemical reaction involves a dissociation of a molecule intofragments or a chemical synthesis from twomolecules to a merger:

AB ———> B þ C,

B þ C ———> AB.

More general chemical reactions take the form

oO þ pP ———> rR þ sS.

The Gibbs energy change in a chemical reaction DG is defined by

DG ¼ G(products) � G (reactants) ¼ Gp � Gr. (20)

We adopt a postulate that the Gibbs energy flux can bedescribed as a function of chemical potential, time, and space:

g ¼ g (m, t, r).

Furthermore, the expression of the Gibbs energy flux can bewrittenas a product of chemical potential, time, and space dependentterms:

g ¼ g(m)B(t)A(r).

To describe chemical reactions, Eq. (7) can be expressed as

l V$g � kBT0 vg/vm þ t vg/vt ¼ 0 (21)

where g ¼ gn is the Gibbs energy flux vector with the unit vector n.The second term represents the internal convective Gibbs energytransfer in the chemically isolated system under constant temper-ature and constant pressure.

A hypothesis dealing with convective chemical reactions is thatchemical energy transfer is proportional to chemical energychange: g f Dm. A restriction is that the chemical energy transfersatisfies the Arrhenius type equation. The chemical potential playsthe role of an independent dimension such as space or time.

Eq. (21) is the extension of energy density conservation:

l V$g þ t vg/vt ¼ 0 (22)

where va ¼ l/t is the average particle speed and g ¼ gd þ gv is theGibbs energy flux of diffusion and convection. Eq. (22) is thetransform of the continuity equation

V$g þ emvn0/vt ¼ 0

where n0 is the particle number density, em is the Gibbs energychange per particle, and emn0 ¼ g/va is the Gibbs energy change perunit volume.

Fig. 3. Hierarchy in chemical reactions.

H.-S. Roh / Energy 89 (2015) 1029e10491034

The external Gibbs energy flux in the first term of Eq. (21) iscomprised of the convection energy flux gv and the diffusion energyflux gd:

g ¼ gd þ gv.

The internal Gibbs energy flux in the second term of Eq. (21) ismade of the internal convection energy flux gm and the internalequilibrium energy flux g0:

g ¼ gm þ g0.

Fig. 2 presents the chemical energy transfer theory dealing withthe four regions of non-equilibrium, irreversible quasi-equilibrium,reversible equilibrium and equilibrium. The theory offers governingequations for the internal equilibrium, internal convection, externalconvection, and conduction mechanisms. Fig. 3 exhibits relation-ships among these processes. Non-equilibrium reactions containinternal convection processes, and equilibrium reactions representinternal equilibrium processes. Quasi-equilibrium reactions ofexternal convection and diffusion include migration and concen-tration diffusion processes. Governing equations and solutions inthe four regions are described in the following sections.

Relationships among subjects in the chemical energy transfertheory of chemical reactions are shown in Fig. 4. The chemical energytransfer theory renders the reaction quotient, and thermodynamicsdefines the reaction quotient. In chemical equilibrium, the equilib-rium constant is defined and is estimated by statistical mechanics.Conventional transport phenomena near equilibrium are connectedto the equations of diffusion andmigration. The rate equation knownas the kinetics of chemical reactions is derived as a special case of thechemical energy transfer theory. Thermodynamics, statistical me-chanics, and kinetics viewpoints depend on energy, probability, andrate considerations, respectively.

4. Chemical energy transfer for non-equilibrium chemicalreactions

4.1. Non-equilibrium chemical reactions

The governing equation for non-equilibrium chemical reactionscan be expressed as

l V$gm � kBT0 vgm/vm þ t vgm/vt ¼ 0. (23)

The above equation is the extension of the continuity equation fordiffusion:

V$gd þ em vn0/vt ¼ 0

Fig. 2. Four chemical energy transfer regions of non-equilibrium, irreversible quasi-equilibrium, reversible quasi-equilibrium and equilibrium.

where n0 is the particle number density and em is the chemicalenergy per particle.

We deal with the diffusion and internal convective effects.When we use the method of separation of variables for the Gibbsenergy flux

gm(m, t, r) ¼ g(m) B(t) A(r),

we derive the following three differential equations from Eq. (23):

vg/vm þ (n/kBT0) g(m) ¼ 0, (24)

vB/vt þ (b /t) B(t) ¼ 0, (25)

V$A þ (k/l) A(r) ¼ 0. (26)

Eq. (24) denotes the chemical energy transfer. Eqs. (25) and (26)represent the temporal and spatial equations for chemical energytransfer, respectively.

The three constants n, b, and k in the separation of variables areintroduced, and their constraint is expressed as:

In nonrelativistic case,

n þ k � b ¼ 0.

In relativistic case,

n2 þ k2 � b2 ¼ 0.

The relativistic constraint is applied to classify different processeseven though nonrelativistic governing equations are addressed.Τhe separation constants indicate.

n: chemical particle number constant,b: temporal particle number constant,k: spatial particle number constant.

Fig. 4. Relationships among subjects in the chemical energy transfer theory ofchemical reactions.

H.-S. Roh / Energy 89 (2015) 1029e1049 1035

Eq. (25) represents the rate equation in chemical reactions.kr ¼ b/t ¼ 1/t0 in Eq. (25) stands for the rate constant in chemicalreactions and is the inverse of the mean reaction time t0.k ¼ (b2 � n2)1/2 ¼ l/l0 in Eq. (26) is connected to the reaction crosssection sc ¼ V/kl0, and n0 ¼ k/V is the particle number per unitvolume. We can make use of the particle number constants toclassify chemical energy transfer mechanisms.

With respect to the internal convection and conduction chem-ical energy flux (gm and gd) in the non-equilibrium process ofnonzero m, t, and r, we obtain the three integrated solutions from(24), (25), and (26), respectively:

g(mN) ¼ g0 exp (�nmN/kBT0), (27)

B(tB) ¼ B0 exp (�btB/t), (28)

A(rB) ¼ A0 exp (k$rB/l), (29)

where k ¼ kn ¼ (b2 � n2)1/2n and n is the unit vector in thediffusion direction of the Gibbs energy flux.

Eqs. (27)e(29) satisfy the initial and boundary conditions. Thechemical potential, time, and displacement variables are repre-sented as the excess quantities, respectively:

mN ¼ jm � m0j,

tB ¼ t � t0,

rB ¼ jr � r0jn. (30)

At large mN, rB, or tB, the corresponding chemical energy fluxamplitude converges to zero.

The chemical energy density is thus of the final form:

gm(mN, tB, rB) ¼ g00exp (�nmN/kBT0) exp (�btB/t) exp (k$rB/l) (31)

where g00 ¼ g0 B0 A0. The normalization condition for g0

0may yield

g00 ¼ n0 (ek þ Pv0) va ¼ (k/V)emva

where va ¼ (8kBT0/m0p)1/2 is the mean speed in thermalequilibrium.

Eq. (27) can be utilized to understand the Gibbs energy flux as afunction of chemical potential. It also has a form of the Arrheniustype equation and a connection to the Gibbs energy change.Therefore, the relationships among thermodynamics, statisticalmechanics, and chemical reactions can be explored. In terms of Eq.(27), the relative chemical energy transfer, actual chemical energytransfer, chemical energy reaction quotient, and chemical potentialare respectively defined by

DGN ¼ DG � DG0 ¼ �nmN ¼ �njm � m0j,

DG ¼ DG0 þ kBT0 ln Zg,

Zg ¼ exp (DGN/kBT0),

m ¼ m0 � (kBT0/n) ln Zg. (32)

These equations may be useful to explore the chemical energytransfer. In chemical reactions,

K ¼ exp (DGN/RT0)

where K is the reaction quotient and R is the universal gas constant.Therefore, a relation between the Gibbs reaction quotient and thereaction quotient is obtained:

Zg ¼ KNa

where Na is Avogadro's number.Once the process surpasses the activation chemical potential,

the zero or nonzero n process takes place. The process of n ¼ 0represents the chemical energy transfer process due to the diffu-sion mechanism after the process reaches the activation chemicalpotential. The zero n process is of irreversible diffusion while thenonzero n process is of irreversible internal convection.

Eq. (28) provides the time rate of a chemical reaction, and thereaction equation is given by Eq. (25). The temporal particle num-ber constant b gives the information of process speed. From Eq.(28), the behavior of time is obtained:

t ¼ t0 � (t /b) ln M ¼ t0 � t0 ln M

whereM¼ B/B0¼ gt/g0t with the exchange chemical energy flux g0t

is the temporal chemical (Gibbs) energy partition function. Sincethe conventional rate constant kr is related to the temporal particlenumber constant by kr¼ b/t¼ 1/t0, Eq. (28) is analogous to the ratelaw of the first order or the pseudo-first order chemical reactionunder the assumption that the quantity B is proportional to theconcentration:

vB/vt þ kr B(t) ¼ 0.

Eq. (29) leads to the spatial dependence of the convection en-ergy process. The spatial particle number constant k affords theinformation of the reaction cross section, the diffusion speed, andthe process stability in spaces. The relation between the reactioncross section sc and the spatial particle number constant k issc ¼ �V/kl0:

sc ¼ 1/n0l0.

From Eq. (29), the spatial coordinate is given by

r ¼ r0 � n(l/k) ln L ¼ (r0 � l0 ln L)n

where L¼ A/A0 ¼ gr/g0r with the exchange chemical energy flux g0r

is the spatial chemical energy (Gibbs) reaction quotient.

4.2. Special cases of chemical reactions

In the following, we address special cases of the chemical energytransfer theory (21): non-equilibrium, irreversible quasi-equilibrium, reversible quasi-equilibrium, and equilibrium energytransfer. The particle number constants of n, b, and k along with mN

and m0 especially play the major roles to describe the distinctiveprocesses. Table 2 classifies thermodynamic, spatial, and temporalenergy transfer mechanisms in equilibrium, reversible quasi-equilibrium, irreversible quasi-equilibrium, and non-equilibrium.Spontaneity conditions for chemical energy transfer processes areclosely connected to the values of the particle number constants asshown in Table 3. Table 4 displays the processes of reversible andirreversible quasi-equilibrium chemical energy transfer mecha-nisms. Table 5 summaries the thermodynamics laws of reversibleand irreversible chemical energy transfer mechanisms.

In non-equilibrium chemical reactions for a nonzero n, the Gibbsenergy change is not zero: nonzeroDGN. Two possible processes areanalyzed:

Unsteady and uniform chemical reactions with n2 � b2 ¼ 0(nonzero n, mN and b; k ¼ 0).

Non-uniform and steady chemical reactions with n2 þ k2 ¼ 0(nonzero n, mN, and k; b ¼ 0).

Table 2Thermodynamic, spatial, and temporal energy transfer mechanisms.

Energy transfer Thermodynamics Space Time

Chemical energy Equilibrium m ¼ 0 for njm � m0j r ¼ 0 for k$jr � r0jn t ¼ 0 for bjt � t0jReversible quasi-equilibrium n ¼ 0, m0 s 0 k ¼ 0, r0 s 0 b ¼ 0, t0 s 0

m0 ¼ 0, n s 0 r0 ¼ 0, k s 0 t0 ¼ 0, b s 0Irreversible quasi-equilibrium n ¼ 0, mN s 0 k ¼ 0, rB s 0 b ¼ 0, tB s 0

mN ¼ 0, n s 0 rB ¼ 0, k s 0 tB ¼ 0, b s 0Non-equilibrium nmN ¼ njm � m0j s 0 k$rB ¼ k$jr � r0jn s 0 btB ¼ bjt � t0j s 0

Internal energy Equilibrium T ¼ P ¼ m ¼ 0 for akBΤSþzv0PV þ nmN r ¼ 0 for k$jr � r0jn t ¼ 0 for bjt � t0jReversible quasi-equilibrium a ¼ z ¼ n ¼ 0 nonzero Τ0,P0,m0 k ¼ 0, r0 s 0 b ¼ 0, t0 s 0

Τ0 ¼ P0 ¼ m0 ¼ 0 r0 ¼ 0, k s 0 t0 ¼ 0, b s 0Irreversible Quasi-equilibrium a ¼ z ¼ n ¼ 0 nonzero ΤS, PV, mN k ¼ 0, rB s 0 b ¼ 0, tB s 0

ΤS ¼ PV ¼ mN ¼ 0 rB ¼ 0, k s 0 tB ¼ 0, b s 0Non-equilibrium akBΤS þ zv0PV þ nmN s 0 k$rB ¼ k$jr � r0jn s 0 btB ¼ bjt � t0j s 0

Table 3Spontaneity conditions for chemical energy transfer mechanisms.

Energy transfer Thermodynamics Space Time

Chemical energy Spontaneous n > 0 k > 0 b > 0Equilibrium n ¼ 0 k ¼ 0 b ¼ 0Non-spontaneous n < 0 k < 0 b < 0

Internal energy Spontaneous d > 0 k > 0 b > 0Equilibrium a ¼ z ¼ n ¼ d ¼ 0 k ¼ 0 b ¼ 0Non-spontaneous d < 0 k < 0 b < 0

H.-S. Roh / Energy 89 (2015) 1029e10491036

The governing equation for steady and uniform chemical energytransfer leads to

kBT0 vgm/vm � t vgm/vt ¼ 0. (33)

The chemical energy density takes the integrated solution:

gm(mN, tB) ¼ g00exp (�nmN/kBT0) exp (�btB/t)

where g00 ¼ g0B0 and b2 ¼ n2.

Non-uniform and steady chemical energy transfer is given bythe partial differential equation:

l V$gm � kBT0 vgm/vm ¼ 0. (34)

The chemical energy density is thus of the final form:

gm(mN, rB) ¼ g00exp (�nmN/kBT0) exp (k$rB/l)

where g00 ¼ g0A0 and k2 ¼ �n2.

5. Chemical energy transfer for irreversible quasi-equilibrium chemical reactions

The internal source Ci in Eq. (1) can be identified from Eq. (23).Eq. (23) can be written as

m[l V$C � kBT0 vC/vm þ t vC/vt]þ C[l V$m � kBT0 þ t vm/vt]¼ 0 (35)

Table 4Reversible and irreversible quasi-equilibrium chemical energy transfer mechanisms.

Energy transfer Process Variable Re(c

Chemical energy Chemical potential(Migration)

nm0

Particle number(Diffusion)

nm0

where m is the chemical potential and C is the particle numberdensity. In non-equilibrium, the differential Eq. (35) leads to twoequations for the intensive and the extensive variable under theassumption that m and C are orthogonal and independent:

l V$C � kBT0 vC/vm þ t vC/vt ¼ 0,

l V$m � kBT0 þ t vm/vt ¼ 0.

Rewriting the above equations, we find

V$jdd � (kBT0/t)vC/vm þ vC/vt ¼ 0,

V$jdm � (kBT0/t) þ vm/vt ¼ 0,

where Fick's first law jdd ¼ Cva ¼ �D0VC and the migration currentdensity jdm ¼ mva ¼ �DmVm are utilized under the assumption ofconstant temperature and constant pressure. The first is the con-centration diffusion equation for an isotropic process, and thesecond is the migration equation for an isotropic process:

D0V2C � vC/vt ¼ �(kBT0/t)vC/vm, (36)

DmV2m � vm/vt ¼ �kBT0/t, (37)

where D0 ¼ val0/3 is the concentration diffusion coefficient andDm ¼ val0/3 ¼ l0

2/3t0 is the migration coefficient. Ci ¼ (kBT0/t) vC/vmis obtained by comparing Eq. (1) with Eq. (36). This means chemicalinternal convection becomes the source in the concentration

versible quasi-equilibriumonvection)

Irreversible quasi-equilibrium(diffusion)

¼ 0s 0

n ¼ 0mN s 0

s 0¼ 0

n s 0mN ¼ 0

Table 5Thermodynamics laws of reversible and irreversible chemical energy transfer mechanisms.

Energy transfer Process Variable Reversible equilibriumand quasi-equilibrium

Irreversible non-equilibriumand quasi-equilibrium

Chemical energy Chemical potential Conservation DecreaseParticle number Conservation Increase

H.-S. Roh / Energy 89 (2015) 1029e1049 1037

diffusion equation. The condition for the particle number constantsin Eq. (36) leads to b2 ¼ n2 þ k2.

During quasi-equilibrium chemical reactions, two irreversibleprocesses are possible:

Migration with b2 ¼ n2 þ k2 (nonzero b and k; n ¼ 0),Concentration diffusion with b2 ¼ k2 (nonzero b and k; mN ¼ 0).Quasi-equilibrium transport theories for conduction chemical

reactions are considered as special cases of general chemical re-actions (1). Migration and diffusion equations are derived. Thegoverning equation for diffusion and migration is found to be

l V$gd þ t vgd/vt ¼ 0. (38)

The zero n process in Eq. (37) represents the chemical energymigration at constant particle number. The migration equationbecomes

DmV2m � vm/vt ¼ 0. (39)

The requirement for the particle number constants is b2 ¼ k2.The process of mN¼ jm�m0j ¼ 0 in Eq. (36) is the particle diffusion

at constant chemical potential. Eq. (36) leads to Fick's second law:

D0V2C � vC/vt ¼ 0. (40)

The constraint for the particle number constants is b2 ¼ k2.

6. Chemical energy transfer for equilibrium chemicalreactions

Chemical energy transfer for equilibrium is the reversible pro-cess of (nonzero n, b, and k; m ¼ t ¼ r ¼ 0). The equilibrium processtakes place at a nonzero m0¼ jm0p� m0

r jwhile the chemical potentialsbetween reactants and products are the same, mp¼ mr; m¼ 0 in (30).Note that m0 is a variable depending on the chemical potentialdifference between m0

p and m0r while m0 is the equilibrium chemical

potential. Similarly, the spatial and temporal equilibriums areattained at t ¼ r ¼ 0 in (30), but t0 and r0 are nonzero variables.

The equilibrium chemical potential, time, and displacementvariables are parameterized as the excess quantities, respectively:

m0 ¼ jm0p � m0r j ¼ jm � m0j,

t0 ¼ t0p � t0r ¼ t � t0,

r0 ¼ jr0p � r0r jn ¼ jr � r0jn.

Under the parameterization, the governing equation for equi-librium chemical reactions becomes

l V$g0 � kBT0 vg0/vm þ t vg0/vt ¼ 0. (41)

When the method of separation of variables for the Gibbs energyflux

g0(m, t, r) ¼ g0(m) B0(t) A0(r)

is used, three differential equations are obtained from (41):

vg0/vm � (n/kBT0) g0 (m) ¼ 0, (42)

vB0/vt � (b/t) B0 (t) ¼ 0, (43)

V$A0 � k A0 (r) ¼ 0. (44)

The three constants n, b, and k in the separation of variables arelinked to

n2 þ k2 � b2 ¼ 0.

Eqs. (42)e(44) then yield the following integrated solutionsrespectively:

g0 (m0) ¼ g0 exp (nm0/kBT0), (45)

B0 (t0) ¼ B0 exp (bt0/t), (46)

A0 (r0) ¼ A0 exp (�k$r0/l), (47)

where k ¼ kn ¼ (b2 � n2)1/2n and n is the unit vector in thediffusion direction of the Gibbs energy flux.

Eqs. (45)e(47) satisfy the initial and boundary conditions. Notethat Eqs. (27)e(29) lead to Eqs. (45)e(47) at T ¼ t ¼ r ¼ 0,respectively. The chemical energy density is thus of the final form:

g0(m0, t0, r0) ¼ g00exp (nm0/kBT0) exp (bt0/t) exp (�k$r0/l) (48)

where g0’ ¼ g0 B0 A0.The activation chemical potential ma, the activation time ta, and

the activation displacement ra are used in the above:

ma ¼ jm0p � m0r jmax ¼ jm � m0jmax,

ta ¼ jt0p � t0r jmax ¼ jt � t0jmax,

ra ¼ jr0p � r0r jmax ¼ jr � r0jmax. (49)

The integrated solutions of (45)e(47) satisfy g0(m0 ¼ 0) ¼ g0,B0(t0 ¼ 0) ¼ B0, and A0(r0 ¼ 0) ¼ A0 at the boundary and initialconditions of m0 ¼ 0, t0 ¼ 0, and r0 ¼ 0, which stand for chemical,temporal, and spatial equilibriums. Moreover, they satisfyg0(m0 ¼ ma) ¼ g1, B0(t0 ¼ ta) ¼ B1, and A0(r0 ¼ ra) ¼ A1 at the limitingconditions of m0 ¼ ma, t0 ¼ ta, and r0 ¼ ra. A variable with thesubscript a denotes the threshold value, and an amplitude variablewith the superscript 1 indicates its limiting value. Using the initialand limiting conditions, we can evaluate the particle numberconstants:

n ¼ (kBT0/ma) ln (g1/g0),

b ¼ (t/ta) ln (B1/B0),

k ¼ (l/ra) ln (A1/A0).

The Gibbs equilibrium constant has the relation

Zg0 ¼ exp (�DG0/kBT0),

H.-S. Roh / Energy 89 (2015) 1029e10491038

and the equilibrium constant K0 takes the form

K0 ¼ exp (�DG0/RT0)

¼ exp (nm0/RT0)

¼ exp (njm � m0j/RT0).

Rewriting the above equation, we get the chemical potential duringequilibrium chemical reactions:

m ¼ m0 þ (RT0/n) ln K0. (50)

7. Chemical energy transfer for reversible quasi-equilibriumchemical reactions

Chemical energy transfer for reversible quasi-equilibrium is theconvection process of (nonzero b and k; n ¼ t ¼ r ¼ 0 ormN ¼ t ¼ r ¼ 0). The spatial and temporal equilibriums are attainedat t ¼ r ¼ 0 in (30). The governing equation becomes

lV$gv þ t vgv/vt ¼ 0. (51)

The particle number constants have a relation:

k2 � b2 ¼ 0.

The equilibrium Gibbs energy flux thus has the final form:

gv (t, r) ¼ g00exp (bt0/t) exp (�k$r0/l)

where g00 ¼ B0 A0.

Eq. (51) leads to the chemical potential conservation (nonzero band k; mN ¼ t ¼ r ¼ 0) and particle number conservation (nonzero band k; n ¼ t ¼ r ¼ 0):

vm/vt þ V$(mva) ¼ 0,

vC/vt þ V$(Cva) ¼ 0. (52)

In the presence of convection and diffusion process, the con-centration diffusion-convection equation and the migration-convection equation are given by

V2C � (1/D0) vC/vt � (1/D0) va$VC ¼ 0,

V2m � (1/Dm) vm/vt � (1/Dm) va$Vm ¼ 0. (53)

8. Statistical thermodynamic theory for electrochemicalreactions

An electrochemical reaction is a branch of a chemical reaction,as emphasized in Fig. 1. The connection between chemical andelectrochemical reactions is considered here. The statistical ther-modynamic theory to describe electrochemical reactions has beenproposed [31,32]. A total system is in chemical non-equilibriumwhile it is in thermal and mechanical equilibrium. The currentdensity in thermodynamic non-equilibrium in continuous media ischaracterized as the intensive variable, electric potential.

Tomake sure that an electrochemical reaction is one of chemicalreactions, following relationships among thermodynamic proper-ties are described. The relation between the current density j andthe Gibbs energy transfer per unit area and unit time is given by

DG/tA ¼ zeFE/tA ¼ jE with the electric potential E. The relationbetween the Gibbs potential and the chemical potential is DG/N ¼ Dm where the other relations are Dm ¼ zeFE and N ¼ jV/zeFvwith the volume V and the velocity v. In the following, the currentdensity as a function of electric potential is demonstrated instead ofthe Gibbs energy flux as a function of chemical potential.

The partial differential equation for chemical reaction (23) leadsto a partial differential equation:

lV$gE � (RT0/F) vgE/vЕ þ t dgE/dt ¼ 0 (54)

where gE ¼ jEE ¼ gn is the Gibbs energy flux vector with the unitvector n. The Gibbs energy flux is defined as the Gibbs energychange per unit area and unit time.

Eq. (54) reduces to

E[l V$jE� (RT0/F) vjE/vЕþ t djE/dt]þ j[l V$E� RT0/Fþ t dE/dt]¼ 0.

Under the assumption that the current density and the electricpotential are orthogonal and independent, we obtain two equa-tions from the above equation:

l V$jE � (RT0/F) vjE/vЕ þ t djE/dt ¼ 0, (55)

V$jdm � RT0/Ft þ dE/dt ¼ 0, (56)

where jE is the internal convective current density and jdm is themigration current density. Eq. (55) is the governing equation for theelectric current density in electrochemical reactions, and Eq. (56) isthe migration equation.

Eq. (55) is the extension of current density conservation:

V$jdd þ (1/v) vjdd/vt ¼ 0 (57)

where va ¼ l/t is the average particle velocity. Note that jE leads tojdd in the absence of internal convection. Eq. (57) is the transform ofthe continuity equation

V$jdd þ q vne/vt ¼ 0

where ne is the number density and q is the electric change perparticle.

The second term in the left hand side of Eq. (55) represents theelectrochemical reaction which reflects the internal convectiveelectrochemical mechanism and is useful to demonstrate polari-zation mechanisms. Eq. (55) is analyzed under the assumption ofconstant temperature and constant pressure in which the system iselectrochemically isolated from its surroundings.

8.1. Non-equilibrium electrochemical reactions: the processes of(nonzero ze)

We have the differential equation for non-equilibrium electro-chemical reaction (55). Using the method of separation of variablesfor the current density

jE (E, t, r) ¼ j(Е) B(t) A(r),

we derive the following three differential equations from Eq. (55):

vj/vE þ (zeF/RT0) j(Е) ¼ 0, (58)

vB/vt þ (b /t) B(t) ¼ 0, (59)

V$A þ k A(r) ¼ 0. (60)

H.-S. Roh / Energy 89 (2015) 1029e1049 1039

The three constants ze, b, and k in the separation of variables areintroduced, and their relation is given by

ze2 þ k2 � b2 ¼ 0

where the constants denote:

ze: the electrochemical particle number constant,b: the temporal particle number constant,k: the spatial particle number constant.

ze ¼ nNa in Eq. (58) is the charge particle number constant and n isthe mole number of the charge particles. kr ¼ b/t ¼ 1/t0 in Eq. (59)stands for the rate constant in electrochemical reactions and is theinverse of the mean reaction time t0. k ¼ (b2 � ze2)1/2 ¼ l/l0 in Eq.(60) is connected to the reaction cross section sc ¼ V/kl0, andn0 ¼ k/V is the particle number per unit volume.

Then the three integrated solutions for (58), (59), and (60) areobtained respectively:

j(ЕP) ¼ j0 exp (�zeFEP/RT0), (61)

B(tB) ¼ B0 exp (�btB/t), (62)

A(rB) ¼ A0 exp (k$rB/l), (63)

wherek¼ kn¼ (b2� ze2)1/2n and n is the unit vector in the diffusiondirection of the current density. Eq. (61) has the form of theArrhenius type equation.

Eqs. (61)e(63) describe the electric potential dependence,temporal dependence, and spatial dependence of the currentdensity, respectively. When the initial and boundary conditions areapplied, the excess forms of the electric potential, time, anddisplacement variables relative to their relative equilibriums areutilized in Eqs. (61)e(63), respectively:

EP ¼ jEeE0j,tB ¼ t � t0,rB ¼ jr � r0jn. (64)

The current density is then found to be the form:

jE (EP, tB, rB) ¼ j00exp (�zeFEP/RT0) exp (�btB/t) exp (k$rB/l) (65)

where j00 ¼ j0B0A0.

In non-equilibrium electrochemical reactions for nonzero ze, theGibbs energy change is not zero: nonzeroDG. Two special processesare possible:

Unsteady and uniform electrochemical reactions withze2 � b2 ¼ 0 (nonzero ze, EP and b; k ¼ 0),

Non-uniform and steady electrochemical reactions withze2 þ k2 ¼ 0 (nonzero ze, EP, and k; b ¼ 0).

The governing equation for unsteady and uniform electro-chemical reactions is found to be

(RT0/F) vjE/vЕ � t vjE/vt ¼ 0. (66)

The current density then reduces to the form:

jE (EP, tB) ¼ j00exp (�zeFEP/RT0) exp (�btB/t)

where j00 ¼ j0 B0 and b2 ¼ ze2.

In this case of non-uniform and steady electrochemical re-actions, the governing equation becomes

l V$jE � (RT0/F) vjE/vЕ ¼ 0. (67)

The current density then takes the form:

jE (EP, rB) ¼ j00exp (�zeFEP/RT0) exp (k$rB/l)

where j00 ¼ j0A0 and k2 ¼ �ze2.

8.2. Irreversible quasi-equilibrium electrochemical reactions: theprocesses of (zero ze or zero EP)

During irreversible quasi-equilibrium electrochemical reactions,two processes are possible to occur:

Migration with b2 ¼ k2 (nonzero b and k; ze ¼ 0),Concentration diffusion with b2 ¼ ze2 þ k2 (nonzero b and k;

EP ¼ 0).In quasi-equilibrium process, the governing equation becomes

l V$gd þ t vgd/vt ¼ 0.

In the absence of the internal convection source (zero ze), thegoverning process is migration with constant particle number. Thehomogeneous migration equation for an isotropic process reducesto

DEV2E � vE/vt ¼ 0 (68)

where DE ¼ 1/m0s is the migration coefficient and s ¼ neq2t0/m isthe electric conductivity. The condition for the constants is b2 ¼ k2.

In the absence of the internal convection source (zeroEP ¼ jE � E0j), the governing process is the concentration diffusionwith constant electric potential. The homogeneous diffusionequation for an isotropic process becomes

D0V2 C � vC/vt ¼ 0. (69)

The relation for the constants is b2 ¼ k2.

8.3. Equilibrium electrochemical reactions: the process of (nonzeroze, b, and k; E ¼ 0)

The equilibrium process takes place at a nonzero E0 ¼ jE0p � E0r j;E¼ 0 in Eq. (64). Note that E0 is a variable depending on the electricpotential difference between E0p and E0r while E0 is the equilibriumpotential. Similarly, the spatial and temporal equilibriums areattained at t ¼ r ¼ 0 in (64), but t0 and r0 are nonzero variables.

The equilibrium variables are the relative electric potential,time, and displacement and can be parameterized as

E0 ¼ jE0p � E0r j ¼ jEeE0j,

t0 ¼ t0p � t0r ¼ t � t0,

r0 ¼ jr0p � r0r jn ¼ jr � r0jn.

Under the parameterization, the governing equation for equi-librium electrochemical reactions leads to

lV$j0 � (RT0/F) vj0/vЕ þ t vj0/vt ¼ 0. (70)

Using the method of separation of variables, we find governingequations from (70):

vj0/vЕ � (zeF/RT0) j0 (Е) ¼ 0, (71)

H.-S. Roh / Energy 89 (2015) 1029e10491040

vB0/vt � (b/t) B0 (t) ¼ 0, (72)

V$A0 � k A0 (r) ¼ 0, (73)

where the separation variable constants have the relation:

ze2 þ k2 � b2 ¼ 0.

Applying initial and boundary conditions, we obtain the equi-librium convection current densities from (71)e(73):

j(Е0) ¼ j0 exp (zeFE0/RT0), (74)

B(t0) ¼ B0 exp (bt0/t), (75)

A(r0) ¼ A0 exp (�k$r0/l). (76)

Eqs. (74)e(76) satisfy the initial and boundary conditions. Notethat Eqs. (61)e(63) lead to Eqs. (74)e(76) at E ¼ t ¼ r ¼ 0,respectively. The current density then reduces to the final form:

j0 (E0, t0, r0) ¼ j00exp (zeFE0/RT0) exp (bt0/t) exp (�k$r0/l) (77)

where j00 ¼ j0 B0 A0.

The activation potential Ea, activation time ta, and activationdisplacement ra are respectively defined by the maximumvalues inmagnitude:

Ea ¼ jE0p e E0r jmax ¼ jE e E0jmax,ta ¼ jt0p e t0r jmax ¼ jt e t0jmax,ra ¼ njr0p e r0r jmax ¼ njr e r0jmax. (78)

The integrated solutions of (74)e(76) satisfy j0(Е0 ¼ 0) ¼ j0,B0(t0 ¼ 0) ¼ B0, and A0(r0 ¼ 0) ¼ A0 at the initial conditions ofE0 ¼ 0, t0 ¼ 0, and r0 ¼ 0, which stand for electrochemical, tem-poral, and spatial equilibriums. Furthermore, they satisfyj0(Е0 ¼ Еa) ¼ j1, B0(t0 ¼ ta) ¼ B1, and A0(r0 ¼ ra) ¼ A1 at the limitingconditions of E0 ¼ Еa, t0 ¼ ta, and r0 ¼ ra. A variable with thesubscript a indicates the threshold value, and an amplitude variablewith the superscript 1 denotes its limiting value. Using the initialand limiting conditions, we can obtain the following relations:

ze ¼ (RT0/FEa) ln (j1/j0),

b ¼ (t/ta) ln (B1/B0),

k ¼ (l/ra) ln (A1/A0).

From (74), the electric potential becomes

E ¼ E0 þ (kBT0/zeq) ln Zg0

which produces the Nernst equation in an equilibrium electro-chemical reaction:

E ¼ E0 þ (RT0/nF) ln K0. (79)

8.4. Reversible quasi-equilibrium electrochemical reactions: theprocess of (nonzero b and k; ze ¼ E ¼ 0)

Reversible quasi-equilibriums are the convection electro-chemical reactions of (nonzero b and k; ze ¼ t ¼ r ¼ 0 orEP ¼ t ¼ r ¼ 0). The spatial and temporal equilibriums are attainedat t ¼ r ¼ 0 in (64), but t0 and r0 are nonzero variables. The gov-erning equation becomes

lV$jv þ t vjv/vt ¼ 0. (80)

The particle number constants have a relation:

k2 � b2 ¼ 0.

The current density is thus of the final form:

jv (t0, r0) ¼ j00exp (bt0/t) exp (k$r0/l)

where j00 ¼ B0 A0.

Eq. (80) leads to the particle number conservation (nonzero band k; ze ¼ t ¼ r ¼ 0):

vC/vt þ V$(Cva) ¼ 0.

This reduces to the conservation of convective electric charge or theconservation of convective mass (the continuity equation):

V$jv þ q vne/vt ¼ 0

where jv ¼ qneva. The conservation of electric potential is given by

vE/vt þ va$VE ¼ 0.

9. Chemical reactions and electrochemical reactions

The chemical energy transfer theory of chemical reactionsprovides accurate information about the chemical potential,displacement, and time dependence of chemical reactions. Inchemical reactions, the energy difference of reactants and productsis obtained in the chemical potential dependence, the reactioncross section is given in the spatial dependence, and the reactionrate is found in the temporal dependence. The Arrhenius equationis derived from Eq. (48). At the activation chemical potential ma,position ra, and time ta,

g1(m) ~ g0 (bta/t)(kra/l) exp (nma/kBT0).

After removing constants, the Arrhenius equation for the rateconstant can be given by

kr ¼ (bk/va) exp(�Q/RT0) ¼ kr0 exp (�Q/RT0)

where kr0 ¼ bk/va is the frequency factor which depends on thereaction frequency and steric factor,Q ¼ �nmaNA ¼ �nNAjm0p � m0

r jmax is the activation energy, andexp(�Q/RT0) reflects the fraction of collision with sufficient energyto produce a chemical reaction.

Fig. 5(a) and Fig. 5(b) respectively show the Gibbs energy fluxand the current density profiles during equilibrium processes. TheGibbs energy flux and the electric current density are respectively

g0(m0) ¼ g0 exp (nm0/kBT0),

j0(Е0) ¼ j0 exp (zeFE0/RT0).

The Gibbs energy flux at the global equilibrium chemical m0leads to the exchange chemical energy flux, and the current densityat the equilibrium electric potential E0 reduces to the exchangecurrent density: g0(m0) ¼ g0 and j0(Е0) ¼ j0. The limiting Gibbsenergy flux at the equilibrium process

g1(ma) ¼ g0 exp (nma/kBT0)

Fig. 5. Gibbs energy flux and current density. The activation chemical potential andelectric potential are measured from their global equilibriums of m0 and E0, respec-tively. (a) Gibbs energy flux profile as a function of chemical potential during anequilibrium process. The equilibrium process occurs in the regime of 0 < m0 < ma orm0 < m < (m0þma). (b) Current density profile as a function of electric potential during anequilibrium process. The equilibrium process occurs in the regime of 0 < E0 < Ea orE0 < E < (E0þEa).

Fig. 6. Gibbs energy flux and current density in the regime of 0 < r0 < ra orr0 < r < (r0þra). The activation displacement is measured from its global equilibriumr0. (a) Gibbs energy flux profile as a function of displacement during a convectionprocess. (b) Current density profile as a function of displacement during a convectionprocess.

H.-S. Roh / Energy 89 (2015) 1029e1049 1041

is attained at the activation chemical potential ma which is themaximum chemical potential of m0 ¼ jm0p � m0

r j ¼ jm � m0j. Theactivation chemical potential is associated with the work functiondifference between reactants and products. It is connected to theequilibrium constant Zg01 ¼ g1/g0:

ma ¼ (kBT0/n) ln Zg01.

ma is positive for a spontaneous reaction and negative for anonspontaneous reaction.

We determine that at the activation chemical potential ma, thereis the Gibbs energy flux gap between the limiting energy flux g1 atthe chemical valence band and the exchange energy flux g0 at thechemical conduction band. Before an external chemical potential isapplied, the system is in the equilibrium process with the exchangeGibbs energy flux g0. An applied external chemical potential isutilized tomake the equilibrium process proceed until the chemicalpotential reaches the activation chemical potential. At the activa-tion chemical potential, the equilibrium process reaches thelimiting Gibbs energy flux g1 as shown in Fig. 5(a). The chemicalenergy flux gap and the chemical energy gap between the equi-librium and non-equilibrium processes at the activation potentialma are dependent on the particle number constant n:

Gibbs energy flux gap: Dg ¼ g1 e g0,

Gibbs energy gap: DG ¼ nma.

Phase transitions caused by chemical energy transfer presentfirst order phase transitions since they are accompanied bydiscontinuous particle number gaps at activation chemical poten-tials. The order parameter for the phase transitions is the chemicalpotential change in regions of the internal equilibrium and internalconvection. Particle pairing (or dissociation) mechanisms appear atactivation chemical potentials.

Similarly, the Gibbs energy flux is drawn as a function ofdisplacement. Fig. 6(a) and (b) respectively represent the chemicalenergy flux and current density profiles in convection mechanismsuntil the reversible convection processes reach the initiation dis-placements of the irreversible quasi-equilibrium migration pro-cesses. The spatial chemical energy flux gap at the activationdisplacement is connected to the spatial particle number constantk. Once the process reaches the activation displacement, thenonzero k diffusion process occurs.

We recognize that there are four chemical energy transfer re-gions of internal thermodynamic space and external space asdepicted in Figs. 5 and 6. The chemical energy transfer regionsconsist of internal equilibrium, internal convection, external con-vection, and external diffusion. Internal convection is the irre-versible non-equilibrium process, and external convection is thereversible quasi-equilibrium process. Therefore, the chemical en-ergy transfer theory presents the Gibbs energy fluxes produced bythe internal equilibrium and internal convective chemical reactionsas well as external convective and external diffusion chemical re-actions. The four regions have categorical constraint conditionsillustrated in Tables 2, 4 and 5.

The chemical energy transfer theory predicts broken discretesymmetries at the activation points of the irreversible processestransferred from the reversible processes:

Chemical potential reversal symmetry breaking at the activationchemical potential ma,Space reversal symmetry breaking at the activation displace-ment ra,Time reversal symmetry breaking at the activation time ta.

Discrete symmetries are broken, and certain preferred di-rections are chosen during the irreversible chemical processes. Atthe activation chemical potential, the reversible equilibrium pro-cess is changed to the irreversible non-equilibrium process. The

H.-S. Roh / Energy 89 (2015) 1029e10491042

governing Eq. (41) for equilibrium is different from the governingEq. (23) for non-equilibrium. At the activation displacement, thereversible convection process is transformed to the irreversiblediffusion process. The governing Eq. (51) for convection leads to thegoverning Eq. (38) for diffusion. The particle number constants nand k change their front positive-negative signs as the indices ofenergy flux exponential functions at their corresponding activationvalues.

From Table 5, we analyze that the intensive and extensive var-iables of chemical potential and particle number are conservedproperties during the processes of internal equilibrium and con-vection. However, they are not conserved properties during theprocesses of internal convection and diffusion.We have known thatthe second law of thermodynamics is the entropy increase princi-ple. According to irreversible quasi-equilibrium Eqs. (36) and (37),we recognize that the particle number increase principle and thechemical potential decrease principle are analogous to the entropyincrease principle.

Fig. 7(a) shows the Gibbs energy flux profiles of internal equi-librium and external convection as a function of chemical potentialand space in chemical reactions. Along the two perpendicular axesof m and g, the reversible convective processes occur while the in-ternal equilibrium process takes place in the intermediate areabetween the two extreme convective processes. The two reversibleconvective processes contain unsteady, temporal contributions as afunction of space. We expect similar statements for the currentdensity profiles of internal equilibrium and external convection as afunction of electric potential and space in electrochemical re-actions, as shown in Fig. 7(b). Along the two perpendicular axes of Eand j, the reversible convective processes takes place while theinternal equilibrium process proceeds in the intermediate areabetween the two extreme convective processes.

We schematically obtain the energy flux profiles of internalconvective and external diffusive mechanisms as shown in Fig. 8.Fig. 8(a) represents the Gibbs energy fluxes of chemical reactions asa function of chemical potential and space. Fig. 8(b) reflects thecurrent densities of electrochemical reactions as a function of

Fig. 7. Energy flux profiles of internal equilibrium and external convection reactions.(a) Gibbs energy fluxes as a function of chemical potential and space in chemical re-actions. (b) Current densities as a function of electric potential and space in electro-chemical reactions.

electric potential and space. Along the two perpendicular axes of mand g or E and j, the irreversible diffusive processes such as themigration and diffusion process take place while the internalconvective process advances in the intermediate area between thetwo extreme conductive processes. The diffusive processes containunsteady, temporal contributions as a function of space while theinternal convective processes include both steady and unsteadycontributions as a function of space.

Fig. 9 schematically illustrates the four diffusion processes ofchemical reactions and electrochemical reactions in the absence ofinternal convective energy transfer. Eqs. (39) and (40) respectivelyreflect the migration and concentration diffusion processes inquasi-equilibrium chemical energy transfer. Eqs. (68) and (69)respectively depict the constant charge (migration) and constantelectric potential (diffusion) processes in quasi-equilibrium elec-trochemical energy transfer.

Gibbs energy transfer between the equilibrium and convectionprocesses takes place under the relation (98) among the particlenumber constants, and Gibbs energy transfer between the internalconvection and diffusion processes also arises under the require-ment (98). The equilibrium and convection processes governed byEq. (41) are reversible while the internal convection and diffusionprocesses governed by Eq. (23) are irreversible. The particle num-ber constant n represents the chemical energy transfer between thethermodynamic spaces and the conventional space and time.

The chemical energy fluxes in chemical energy transfer arecategorized into six modes:

g ¼ gdd þ gdm þ gv þ gm þ g0 þ gr

where gdd is the diffusion chemical energy flux due to the con-centration gradient VC,

gdm is the migration chemical energy flux due to the chemicalpotential gradient Vm,

Fig. 8. Energy flux profiles in internal convection and external diffusion energytransfer mechanisms. (a) Gibbs energy fluxes as a function of chemical potential andspace in chemical reactions. (b) Current densities as a function of electric potential andspace in electrochemical reactions.

Fig. 9. Energy diffusion processes in the absence of internal convection. (a) Chemicalreactions. (b) Electrochemical reactions.

H.-S. Roh / Energy 89 (2015) 1029e1049 1043

gv is the convective chemical energy flux,gm is the internal convective chemical energy flux due to thechemical potential change Dm,g0 is the internal equilibrium chemical energy flux due to theequilibrium change m0,and gr is the radiant chemical energy flux.

The internal convective chemical energy flux gm is generated in theinternal phase space of chemical potential while the diffusion en-ergy fluxes are created by the conductive diffusion and migrationprocesses. The two diffusive chemical energy fluxes of migrationand diffusion chemical energy are coupled more or less in phasetransitions. The diffusive chemical energy fluxes do not contributeto the chemical energy flux profile as a function of chemical po-tential even though they are closely related to the activationchemical potentials of chemical energy fluxes.

The Gibbs reaction quotient Zg can be defined by an extension ofthe Gibbs free energy G in analogy to the Gibbs partition functionZ0 ¼ �kBT0 ln G0 in statistical mechanics:

Zg ¼ exp (DGN/kBT0).

At equilibrium, an equilibrium Gibbs transfer function is alsodefined:

Zg0 ¼ exp (�DG0/kBT0).

Using the above two equations, we have the actual Gibbs energytransfer and the equilibrium Gibbs energy transfer:

DG ¼ DG0 þ kBT0 ln Zg ¼ kBT0 ln (Zg/Zg0),

DG0 ¼ G0p � G0

r ¼ �kBT0 ln Zg0 ¼ �kBT0 ln (Z0p/Z0r),

where G0p ¼ �kBT0lnZ0p and G0

r ¼ kBT0 lnZ0r are the equilibriumGibbs free energies for products and reactants, respectively, and Z0pand Z0r are the conventional Gibbs partition functions of productsand reactants, respectively. The Gibbs reaction quotient can beconnected to the reaction quotient in chemical reactions: Zg ¼ KNa.The conventional Gibbs partition function in statistical mechanicscan thus be related to the activity:

Z0p ¼ (a0p)Na

where a0p denotes the equilibrium activity of products.The Gibbs reaction quotient can also be expressed for sub-

systems with i species. In this case, we introduce the Gibbs reactionactivation zgi for a particle with species i. Then the reaction quo-tients for different statistical distributions reduce to:

MaxwelleBoltzmann distribution: Zg ¼ Pi zgiNa/Ni!,

BoseeEinstein distribution: Zg ¼ Pi zgiNa,

FermieDirac distribution: Zg ¼ Pi zgiNa,

where the reaction activation satisfies the required rule for indi-vidual particle distribution. If the de Broglie wavelength

lT ¼ (hf2/2pmkBT0)1/2

is small compared with the interparticle wave length, particles inthe system obey MaxwelleBoltzmann statistics. Otherwise, Bose-Einstein statistics or FermieDirac statistics is more appropriate.

Equilibrium chemical energy transfer and electrochemical en-ergy transfer are represented as two Eqs. (50) and (79), respectively.

m ¼ m0 � (RT0/n) ln K0,

E ¼ E0 � (RT0/zeF) ln K0.

In non-equilibrium, the two intensive variables play essential rolesin chemical reactions and electrochemical reactions.

m ¼ m0 � (kBT0/n) ln Zg,

E ¼ E0 � (kBT0/zeF) ln Zg.

The two equations lead to the same form

G ¼ G0 þ kBT0 ln Zg

since Zg ¼ KNa. The relative time and displacement are similarlyexpressed as

t ¼ t0 � (t/b) ln Zgt ,

r ¼ r0 � n(l/k) ln Zgr,

where Zgt ¼ gt/g0t and Zgr ¼ gr/g0r.

10. Chemical energy transfer mechanisms in non-equilibriumchemical reactions

The chemical energy transfer theory can be utilized to explorenon-equilibrium chemical energy transfer processes. We canreadily name several chemical reactions [27e29] belonging to thesecategories:

H2 þ I2 ——————> 2HI,

H2 þ Br2 ——————> 2HBr,

H2 þ Cl2 ——————> 2HCl,

2H2 þ O2 ——————> 2H2O,

2NO þ O2 ——————> 2NO2,

C2H6 ——————> C2H4 þ H2.

Non-equilibrium chemical energy transfer processes aredescribed at the interface of solid and liquid, as shown in Fig. 10(a).In the presence of an external chemical energy ge, the differential

Fig. 10. (a) Interface of solid and liquid. (b) Spatial cylindrical coordinates and thespatial particle number constant k. (c) Gibbs potentials at the solideliquid interface.

H.-S. Roh / Energy 89 (2015) 1029e10491044

equation for the chemical energy flux in chemical reactions takesthe following form from Eq. (24):

ge ¼ g þ (kBT0/n)dg/dm. (81)

In the cylindrical coordinates of r,f, and z, as shown in Fig.10(b),the chemical energy flux g and the chemical potential m depend on rand z since g and m are symmetric in the azimuthal angle f. TheGibbs potentials for a chemical reaction are depicted in Fig. 10(c). Qis the activation energy of a chemical reaction. The reactions arespontaneous if the Gibbs potential energy difference DG is negative.A chemical reaction is dependent on the applied Gibbs energy fluxor chemical potential which changes the Gibbs energies in reactioncoordinates.

Fig. 11 shows the Gibbs energy flux profiles as a function ofchemical potential during equilibrium and non-equilibrium pro-cesses in the presence of external chemical potential. The reversibleand irreversible Gibbs energy fluxes respectively take the forms:

g0(m0) ¼ g0 exp (nm0/kBT0),

g(mN) ¼ g1 [1 � exp (�nmN/kBT0)].

An externally applied chemical potential initiates to make anequilibrium process proceed until the chemical potential reachesthe activation chemical potential. Before the external chemicalpotential is applied, the system is in the equilibrium process withthe exchange Gibbs energy flux g0. At the activation chemical po-tential ma, the equilibrium process gains the limiting Gibbs energyflux g1 as shown in Fig. 11. The chemical energy flux gap betweenthe equilibrium and non-equilibrium processes depends on theparticle number constant n. Above the activation chemical poten-tial, a non-equilibrium process proceeds from the exchange Gibbsenergy flux g0 toward the limiting Gibbs energy flux g1. The non-equilibrium process is an irreversible process, and the limitingGibbs energy flux in equilibrium plays the role of an internal Gibbsenergy flux so that the non-equilibrium process eventually rises tothe limiting Gibbs energy flux as the excess chemical potentialincreases. The limiting Gibbs energy flux g1 is determined by theequilibrium process caused by the Gibbs energy difference

Fig. 11. Gibbs energy flux profiles as a function of chemical potential during theequilibrium process of m0 < m < (m0 þ ma) and the non-equilibrium process ofm > (m0 þ ma) in the presence of external chemical potential. The activation chemicalpotential is measured from its global equilibrium m0.

DG0 ¼ (G0p e G0

r ) between chemical reactants and products. Notethat the limiting Gibbs energy flux g1 is identical with the externalGibbs energy flux ge in Eq. (81).

We define the activation chemical potential ma, the activationtime ta, and the activation distance ra, respectively, as themaximum values in their magnitudes:

ma ¼ jm0p e m0r jmax ¼ jm e m0jmax,

ta ¼ jt0p e t0r jmax ¼ jt e t0jmax,

ra ¼ jr0p e r0r jmax ¼ jr e r0jmax.

The particle number constants are found to be

n ¼ (kBT0/ma) ln (g1/g0),

b ¼ (t/ta) ln (B1/B0),

k ¼ (l/ra) ln (A1/A0),

where g0 is the exchange Gibbs energy flux and g1 is the limitingGibbs energy flux.

The relativistic relation among the particle number constants isdetermined by

n2 þ k2 � b2 ¼ 0.

Whenwe apply the restriction to chemical reactions, we classify thefollowing three non-equilibrium regimes:

n2 ¼ b2 > 0 and k2 ¼ 0 for activation reactions,n2 ¼ b2 � k2 > 0 for concentration reactions,n2 ¼ b2 � k2 < 0 for transition reactions.

Moreover, the process conversion from non-equilibrium to equi-librium takes place in transition reactions:

n2 ¼ b2 � k2 > 0 for transition reactions,n2 ¼ b2 � k2 < 0 for film reactions.

Stable conditions for a process are given by

n2 > 0 for chemical stability,b2 > 0 for temporal stability,k2 > 0 for spatial stability.

In the chemical energy transfer processes at the interface ofsolid and liquid, Fig. 12(a) depicts the four control regimes formedby the pure chemical reaction processes: activation, concentration,transition, and film reactions. The four regimes are analogous to thefour regimes designed in the polarization processes of electro-chemical reactions, as shown in Fig. 12(b) [32]. There are threereaction barriers which depend on chemical potential, space, andtime. It implies that the three variables play the roles of the reactioncoordinates. The reaction barriers exist between the layers ofconcentration and transition reactions. Fig. 12(a) shows twoopposite chemical energy fluxes in concentration and transitionchemical energy transfer. The chemical energy flux by reactants islarger than the chemical energy flux by products in concentrationchemical energy transfer while the chemical energy flux by prod-ucts is larger than the reactant chemical energy flux in transitionchemical energy transfer. In concentration chemical energy trans-fer, the chemical energy flux by reactants governs while in filmchemical energy transfer, the chemical energy flux by products

Fig. 12. Four control regimes formed by reaction processes. (a) Chemical reactions. (b)Polarization in electrochemical reactions.

H.-S. Roh / Energy 89 (2015) 1029e1049 1045

governs. There are a maximum limiting energy flux in the con-vection mechanism, a maximum energy flux in the concentrationmechanism, two maximum and minimum limiting chemical fluxesin the transition mechanism, and a minimum limiting energy fluxin the film mechanism.

We here concentrate on the application of Eq. (81) to non-equilibrium chemical reactions. As Fig. 12 shows, the probablebehavior of chemical energy flux in the nonzeromodes of n in a non-equilibriumchemical reaction canbepredicted as a functionof excesschemical potential mN ¼ jmw e (m0 þ ma)j where ma is the activationchemical potential. In each chemical energy transfer regime of thefour distinct regimes, the particle number constant n plays theimportant role in characterizing the chemical energy flux. The zero nis relevant to the difussive chemical energy flux, and the nonzero n isconnected to the internal convective chemical energy flux.

In the presence of external chemical energy ge, Eq. (81) leads tothe differential equation in the four distinct regimes:

ge ¼ g þ kBT0 [(1/na)dgA/dmþ(1/nc)dgC/dm� (1/nt)dgT/dm � (1/nf)dgF/dm]. (82)

The second term in the right hand side of (82) includes the fourmechanisms of the chemical reactions. The four mechanisms areadditive in the chemical energy flux which is proportional tochemical potential difference. The total chemical energy flux is thusexpressed as the sum of the parallel chemical energy fluxes:

g ¼ gA þ gC þ gT þ gF. (83)

In a process dominated by activation chemical energy, the dif-ferential equation for the chemical energy flux leads to

geA ¼ gA þ (kBT0/na) dgA/dm (84)

where ge is the applied chemical energy flux to the solid wall.Solving Eq. (84), we find the chemical energy flux by convectionchemical energy, gA, given as a function of the excess chemicalpotential mA:

gA ¼ g1A [1 e exp (�namA/kBT0)] (85)

where g1A ¼ geA. The limiting Gibbs energy flux at the activationchemical potential of activation reaction maa is determined at theend of the equilibrium process:

g1A ¼ g0 exp (namaa/kBT0).

We may apply the analogous schemes to the four controlmechanisms and establish the total Gibbs energy flux in an equa-tion [30e36]. The total chemical energy flux can approximately begiven by the sum of chemical energy fluxes in the four chemicalreaction regimes:

g ¼ g1A [1 � exp (�namA/kBT0)] þ g1C [1 � exp (�ncm

C/kBT0)]þ [g1C exp (ntamT/kBT0) H(mtr � m) þ g1Fexp (�nt

bmF/kBT0)�H(mtr � m)] þ g1F exp (nfmF/kBT0) (86)

where H is the Heaviside step function. The transition chemicalpotential is expressed as

mtr ¼ [1/(ntb � nta)] [(ntb maf � nt

b mat) þ ln(g1C/g1F)].

The first term in the right hand side represents the Gibbs energyflux due to activation chemical energy, the second the Gibbs energyflux due to concentration, the third the Gibbs energy flux in tran-sition chemical energy, and the fourth the Gibbs energy flux in filmchemical energy. g1A, g1C, and g1F respectively indicate the limitingGibbs energy fluxes in activation, concentration, and film chemicalreactions.

We get the excess chemical potentials from the absolute acti-vation chemical potentials in the four mechanisms:

mA ¼ jmw e (m0 þ maa)j,mC ¼ jmw e (m0 þ mac)j,mT ¼ jmw e (m0 þ mat)j,mF ¼ jmw e (m0 þ maf)j, (87)

where m0 is the saturation chemical potential and mw is the wallchemical potential. The activation chemical potentials of maa, mac,mat, and maf relative to the global equilibrium chemical potential m0are positive quantities in chemical reactions.

In terms of (86), the chemical potential m in each chemical re-action mechanism can be summarized as

m ¼ m0 þ maa � (kBT0/na) ln ZgeA , m ¼ m0 þ mac � (kBT0/nc) ln ZgeC ,

m ¼ m0 þ mat þ (kBT0/nt) ln ZgeT , m ¼ m0 þ maf þ (kBT0/nf) ln ZgeF , (88)

where Zge ¼ 1 � gm/g1m in the activation or cencentration chemicalreaction, Zge¼ gm/g1m in the transition or film chemical reaction, andthe limiting chemical energy flux g1m depends on chemical poten-tial. Fig. 13 demonstrates schematic chemical energy fluxes as afunction of excess chemical potential. The control mechanisms arepresented, and the maximum limiting state of concentration re-action is regarded as an intermediate transition state.

Schematic diagrams for chemical energy transfermechanisms ofactivation and concentration chemical energy in chemical reactionsare sketched in Fig. 14. The activation chemical energy process is aslow, stable, and irreversible mode and the concentration chemicalenergy process is a fast, stable, and irreversible mode. The former isthe process with a single phase while the latter is the process withthe two phase components. They have different initiation chemicalpotentials depending on conductive chemical energy processes.

The chemical energy transfer theory thus predicts multi-stepreaction processes. For example, we can consider the hydro-geneiodine reaction. During an activation reaction, a bimolecularreaction takes place:

H2 þ I2 ——————> 2HI. (89)

During concentration and transition reaction, termolecular reaction(or three body association reaction) proceeds:

I2 ——————> 2I,H2 þ 2I ——————> 2HI. (90)

The reaction speed of the process (89) is slow, and the Gibbs energychange during the process is small. On the other hand, the reaction

Fig. 13. Schematic chemical energy fluxes as a function of excess chemical potential. (a) Nonspontaneous transition chemical reactions. (b) Spontaneous transition chemicalreactions.

H.-S. Roh / Energy 89 (2015) 1029e10491046

speed of (90) is fast, and the Gibbs energy change during the pro-cess is large. The process (90) is analogous to the transition in thetransition state theory [10e12], and the maximum limiting state ofconcentration reaction corresponds to the transition state. Hydro-genebromine reaction, hydrogenechlorine reaction, hydro-geneoxygen reaction, and decomposition of nitrogen pentoxidereaction are the analogous multi-step processes.

In the presence of the external chemical energy flux ge, thetemporal chemical energy transfer Eq. (25) yields

ge ¼ g þ (t/b) dg/dt (91)

where t is the mean reaction time. From Eq. (91), the temporalchemical energy flux has the integrated solution

g (tB) ¼ g1t [1 e exp (�btB /t)]. (92)

We combine (86) and (92) and then find the chemical energy flux inactivation chemical energy and concentration chemical energytransfer:

g(m,t) ¼ g1A [1 e exp (�namA/kBT0)] [1 e exp (�ba tA/t)]

þ g1C [1 e exp (�ncmC/kBT0)] [1 e exp (�bc tC/t)] (93)

where the temporal particle number constant ba in activation re-actions is less than the temporal particle number constant bc inconcentration reactions, and the diffusion process in activationreactions is slower than that in concentration reactions.

The excess times in the four chemical reaction mechanismsindicate

Fig. 14. Chemical energy transfer mechanisms. (a) As a function of chemical potential.(b) As a function of distance.

tA ¼ t e (t0 þ taa),tC ¼ t e (t0 þ tac),tT ¼ t e (t0 þ tat),tF ¼ t e (t0 þ taf), (94)

where t0 is the global equilibrium time. taa, tac, tat, and taf are theactivation times for the four chemical reaction mechanisms. Thetime in each chemical reaction mechanism is expressed as

t ¼ t0 þ taa � (t/ba) ln MeA,

t ¼ t0 þ tac � (t/bc) ln MeC,

t ¼ t0 þ tat þ (t/bt) ln MeT,

t ¼ t0 þ taf þ (t/bf) ln MeF, (95)

where Me ¼ 1 � gt/g1t in the activation or cencentration temporalchemical reaction, Me ¼ gt/g1t in the transition or film temporalchemical reaction, and the limiting chemical energy flux g1t de-pends on time. Since the temporal particle number constant bc inactivation reactions is less than bn in concentration reactions, thechemical process in activation reactions is slower than that inconcentration reactions.

Spatial behavior for the chemical energy flux is understood bychanging Eq. (26). In the presence of an external source we, it ismodified as

ge ¼ V$g þ (k/l) g (r). (96)

Eq. (96) has the integrated solution

g (rB) ¼ g1r [1 e exp ((k/l) n$rB)]¼ g1r [1 e exp (((n2 � b2)1/2/l) rB cos q)] (97)

where q is the angle between the vectors of n and z: ((n2 � b2)1/2/l)n$rB ¼ k zB cosq/l and tan q ¼ kr/kz.

We find the excess distances along the z axis in the fourchemical reaction mechanisms:

zA ¼ jz e (z0 þ zaa)j,zC ¼ jz e (z0 þ zac)j,zT ¼ jz e (z0 þ zat)j,zF ¼ jz e (z0 þ zaf)j, (98)

where z0 is the boundary position in global equilibrium. zaa, zac, zat,and zaf are the activation distances for the four chemical reactionmechanisms. The distance in each phase transition mechanism isindicated as

H.-S. Roh / Energy 89 (2015) 1029e1049 1047

z ¼ z0 þ zaa,z ¼ z0 þ zac þ (l/kc) ln LeC,z ¼ z0 þ zat � (l/kt) ln LeT,z ¼ z0 þ zaf � (l/kf) ln LeF, (99)

where Le ¼ 1 � gz/g1z in the cencentration spatial chemical reac-tion, Le ¼ gz/g1z in the transition or film spatial chemical reaction,and the limiting chemical energy flux g1z depends on distance. Thelayer in activation chemical reactions is uniform spatially,compared with the other layers.

Summarizing the above descriptions, we draw Fig. 15 whichdemonstrates the schematic diagrams of the chemical energyfluxes as a function of excess chemical potential, time, and distancein the four regimes of chemical reactions.

11. Discussions

Chemical energy is the ubiquitous form of available energywhich is the potential of a chemical substance to undergo atransformation through a chemical reaction. There are vital ad-vantages of using chemical energy in multi-disciplinary fields.Compared to other energy source, chemical energy sources areabundantly available. They are easily combustible, providinginstant energy in the form of heat, and their combustion efficiencyis relatively high. In the following, a few broad scopes of the pro-posed chemical theory are addressed.

Chemical energy transfer can be utilized along with otherthermodynamic energy transfer mechanisms. Thermodynamicenergy transfer consists of internal energy transfer, heat transfer,

Fig. 15. Schematic diagrams of the chemical energy flux in the four regimes ofchemical reactions. (a) As a function of excess chemical potential. (b) As a function oftime. (c) As a function of distance.

work transfer, and chemical energy transfer. In thermodynamicenergy transfer, the intensive variable changes such as tempera-ture, pressure, chemical potential, and electric potential changesare independent variables in describing the internal convectiveprocesses, and they are not dependent variables described by theexternal space and time coordinates. This is the reason why theexpression of the internal energy flux can bewritten as a product oftemperature, pressure, chemical potential, time, and spacedependent terms when the internal and external phase spaces aresimultaneously considered.

The justification of the postulate for the internal energy flux isbriefly discussed. There are seven fundamental quantities inphysics: distance, time, mass, temperature, electric current,amount of substance, and intensity of light. From these quantities,we can say temperature is an independent parameter. Furthermore,from the amount of substance and electric current, wemay analyzethat the particle number and the electric current become inde-pendent parameters respectively. When we combine the knowl-edge from thermodynamics with fundamental quantities, we canconfirm that the entropy is an independent parameter as the cor-responding extensive parameter of the temperature, the chemicalpotential is an independent parameter as the correspondingintensive parameter of the particle number, and the electric po-tential is an independent parameter as the corresponding intensiveparameter of the electric current. On the other hand, the internalenergy depends on the six parameters of T, S, P, V, m, and N as theinternal degrees of freedom. Since the parameters T, S, m, and N areindependent, we may assume that the parameters P and V are alsoindependent even though there is no explicit proof. As an implicitillustration, it may be pointed out that the PeV, TeV, PeT, orPeTeV property diagrams are frequently utilized to understandphase transition processes in thermodynamics.

Employing the product form of the internal energy flux u ¼ q(T)w(P)g(m)B(t)A(r) means the dependence of thermodynamic variablechanges to the internal energy for internal convective mechanisms[30e36]. In this formalism, the difference between internalconvective mechanisms and external diffusion mechanisms areclearly clarified. The two distinct mechanisms are separable ingeneralized internal and external coordinates. The internal convec-tivemechanisms of heat transfer, chemical energy transfer, andworktransfer are formulated based on empirical data in which each en-ergy transfer is a function of its corresponding intensive variable.Each energy flux is proportional to its intensive variable change:

qT f DT for heat transfer,wP f DP for work transfer,gm f Dm for chemical energy transfer.

The energy fluxes are independent of the external space andtime coordinates in these internal convective mechanisms. Theinternal degrees of freedom for temperature, pressure, and chem-ical potential as internal phase spaces do not depend on theexternal space and time coordinates. On the other hand, in theconductive quasi-equilibrium mechanisms, each energy flux isproportional to its intensive variable gradient:

qdT f VT for heat transfer (heat conduction),wdP f VP for work transfer (work diffusion),gdm f Vm for chemical energy transfer (migration).

These energy fluxes in conduction or diffusion mechanismsexplicitly depend on the external space and time coordinates.

As the explicit illustration of the chemical energy transfer the-ory proposed, the theory can be applied to several chemical re-actions. It predicts multiple step reaction processes and their

Table 6Comparison among energy transfer processes.

Energy transfer Heat transfer (solidification) Heat transfer (boiling) Work transfer Electrochemical reaction Chemical reaction

Conduction IsothermalIsentropic

IsothermalIsentropic

IsobaricIsochoric

DiffusionMigration

DiffusionMigration

Internal convection ConvectionNucleationTransientFilm

ConvectionNucleateTransitionFilm

ConvectionTurbulenceTransitionFilm

ActivationConcentrationResistanceFilm

ActivationConcentrationTransitionFilm

H.-S. Roh / Energy 89 (2015) 1029e10491048

corresponding process times. For example, we can consider thehydrogeneiodine reaction. During an activation reaction, a bimo-lecular reaction takes place, and the reaction consists of the mul-tiple processes. The reaction speed of the first process is slow, andthe Gibbs energy change during the process is small. On the otherhand, the reaction speed of the second process is fast, and the Gibbsenergy change during the process is large. Due to the lack of usefulmeasurement data for the purpose of the comparison with theo-retical calculations, only the sketches of theoretical predictions arepresented in the previous subsection. Therefore, more systematicmeasurements are demanded to evaluate the predictions.

Two phenomenological diffusion equations are derived from thechemical energy theory as special cases:

Migration equation,Concentration diffusion equation.

They are well defined equations and are classified as quasi-equilibrium processes. Their mechanisms depend on externalspace and time coordinates. The chemical energy theory predictsequilibrium and non-equilibrium processes in addition to them.These internal processes do not depend on the external space andtime coordinates, but do depend on internal degrees of freedom.

Table 6 explains the non- and quasi-equilibrium energy transferprocesses. The mechanisms of chemical reactions possess similarcharacteristics with polarization, solidification, boiling, and liq-uidevapor phase transition [30e36]. In the kinetics of chemicalreactions, the four control mechanisms of activation, concentration,transition, and film chemical energy are taken into account, and thecharacteristics of the chemical potential, temporal, and spatialdependence of chemical energy transfer are demonstrated. Thekinetics of electrochemical reactions (polarization) contains thefour control mechanisms of activation, concentration, resistance,and film polarization. Likewise, solidification, pool boiling, andliquidevapor phase transition predicts the four controlmechanisms.

In Eq. (21) or Eq. (23), we are interested in the chemical internalconvective effects under the assumption of constant temperatureand constant pressure. However, if the chemical reaction takesplace only under a constant pressure, the differential equation leadsto the enthalpy flux h:

l V$h � [T0 vh/vΤ þ kBT0 vh/vm] þ t vh/vt ¼ 0.

The nonzero a and N lead to the coupled transfer with heat andchemical energy transfer in a system and the enthalpy change DH isthe relevant thermodynamic potential change. Hence, the aboveequation offers the solution for the enthalpy flux (enthalpy per unitarea and unit time):

h (Τ, m, t, r) ¼ q(Τ)g(m)B(t)A(r) ¼ h0 exp (�DH/kBT0) exp (�bt/t) exp(�k$r/l)

where the temperature dependence reflects an Arrhenius typeequation.

The Higgs mechanism [40] in quantum field theory of physics isthe relativistic extension of the Ginzburg-Landau theory in non-relativistic thermodynamics. This implies spontaneous symmetrybreaking mechanisms can be described by the Ginzburg-Landautheory in thermodynamics. In the Ginzburg-Landau theory, thescalar fields are recognized as the order parameters during phasetransitions. The scalar fields of temperature, pressure, and chemicalpotential can be the order parameters which are independentvariables regardless of external space and time coordinates.Moreover, it may be feasible that thermodynamic energy can bequantized.

Chemical energy transfer can thus be applicable in multi-disciplinary areas: For instance, endothermic or exothermic chemi-cal reactions, biochemical reactions, polarization and electrodepo-sition, oxidation and reduction processes, connection withGinzburgeLandau theory, relativistic extension of thermodynamicenergy transfer, andquantization of thermodynamic energy transfer.

12. Conclusions

We propose the chemical energy transfer theory to clarify thecharacteristics of chemical energy transfer mechanisms of non-equilibrium, quasi-equilibrium, and equilibrium at finite tempera-ture. The statistical, thermodynamic theory is regarded as a specialcase of the statistical internal energy kinetics of heat, chemicalenergy, and chemical energy transfer. The chemical energy transfertheory provides the unified mechanisms of internal convective andexternal diffusive chemical reactions. It is the non-equilibriumgeneralization beyond the quasi-equilibrium chemical reactionsof the constant chemical potential and constant particle numberprocesses. The theory is capable of exploring both spontaneous andnonspontaneous chemical reactions or to both dissociation andsynthesis chemical reactions. It is applicable to any chemical re-actions regardless of their molecularity and order of reaction.

We apply the chemical energy transfer theory to non-equilibrium chemical reactions and discuss its logical predictions.The four control mechanisms, three limiting fluxes, and four acti-vation parameters are adopted as input parameters. The four re-gimes formed as a function of chemical potential, time, anddistance are suggested analogously to electrochemical reactions:activation, concentration, transition, and film chemical reactionregimes. The Gibbs energy fluxes are predicted as a function ofchemical potential, time, and displacement.

The thermodynamic chemical energy transfer theory is indis-pensable and promising in practical applications in science andengineering and is capable of being a pioneering fundamentalapproach for equilibrium, quasi-equilibrium, and non-equilibriumchemical reactions. The chemical energy transfer theory will havewidespread applications relevant to electrochemical reactions,biochemical reactions, and chemical reactions.

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