Proposal for Generalized Exergy and Entropy Properties Based on Stable Equilibrium of Composite System-Reservoir

doi:10.4236/jmp.2013.47A2008

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Journal of Modern Physics, 2013, 4, 52-58

http://dx.doi.org/10.4236/jmp.2013.47A2008 Published Online July 2013 (http://www.scirp.org/journal/jmp)

Proposal for Generalized Exergy and Entropy Properties

Based on Stable Equilibrium of Composite

System-Reservoir

Pierfrancesco Palazzo

Technip, Rome, Italy

Email: [email protected]

Received April 20, 2013; revised May 25, 2013; accepted June 27, 2013

which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT

The present theoretical study represents a proposal aimed at investigating about the possibility of generalizing the ca-

nonical entropy-exergy relationship and the reservoir concept. The method adopted assumes the equality of pressure

and chemical potential as necessary conditions of mutual stable equilibrium between a system and a reservoir in addi-

tion to the equality of temperature that constitutes the basis for defining entropy as deriving from energy and exergy

concepts. An attempt is made to define mechanical and chemical entropy as an additional and additive component of

generalized entropy formulated from generalized exergy property. The implications in exergy method and the possible

engineering applications of this appro ach are outlined as future developments among the domains involved.

Keywords: Adiabatic Availability; Available Energy; Generalized Exergy; Generalized Entropy; Reservoir; Stable

Equilibrium; Exergy Method

1. Introduction 2. Theoretical Model and Method

The formulation of the Seco nd Law is based on the exis-

tence and uniqueness of the state of stable equilibrium of

a system interacting with a reservoir “among all the

states of a system that have a given value of the energy

and are compatible with a given set of values of the

amounts of constituents and of the parameters” [1]. This

general statement requires compliance with the necessary

conditions of equ ality of (absolute) temperature, pressure

and chemical potential between system and reservoir.

The Second Law implies the definition of entropy prop-

erty in relation to temperature. The entrop y of system

The following assumptions are posited: 1) the analysis is

focused on “simple systems” according to the terminal-

ogy and definitions adopted by Gyftopoulos and Beretta

[1]; 2) the external reference system behaves as a reser-

voir as defined in the literature [1]; 3) the system may be

large or small, even at molecular level, and may experi-

ence states of equilibrium and nonequilibrium [1]; 4)

physical and chemical exergy are accounted for; 5) ki-

netic energy and the potential energy of the system as a

whole are neglected. These assumptions are posited in

order to constitute the requirements complying with the

fundamental framework to which the addressed literature

refers.

can also be derived as the difference of energy and

exergy associated with the generalized available

energy

with respect to a reservoir at constant

temperature R

The method adopted assumes the equality of pressure

and chemical potential of the composite system-reservoir

T [1]. The Second Law, expressed as in-

dicated above, has to account for pressure and chemical

potential in add ition to temperature, hence the purpose of

this present study is to conceive a generalized formula-

tion of exergy, and consequently of entropy, based on

equality of pressure and chemical p oten tial, in additio n to

temperature, between a system

and a reservoir

at constant temperature

R as additional necessary conditions for mutual stable

equilibrium, other than equality of temperature. The for-

mulation of the entropy property is derived from exergy,

related to a reservoir at constant temperature, constant

pressure and constant chemical potential. This set of ge-

neralized potentials equality constitutes a necessary and

sufficient condition for the stable equilibrium of a com-

posite system-reservoir consistent with the Second Law

T, constant pressure

P and

constant chemical potential



P. PALAZZO 53

statement.

3. Thermal Entropy Derived from Thermal

Exergy

The behaviour of the exergy property is characterized by

additivity because it is defined considering an external

reference system or an internal part of the system itself

that behaves as a reservoir [1,2]. Exergy is derived from

the generalized available energy that is a consequence of

the concept of adiabatic availability when the system

interacts with a reservoir [1]; the definitions of available

energy and exergy are based on the mutual stable equi-

librium of a system

with a reservoir at constant

temperature R



MAX

W

and account for work interaction by

means of a weight process; this implies that, as defined in

these terms, (thermal) exergy expresses the maximum net

useful work connecting two states 0 and 1

obtained by means of a weight process resulting from the

available energy between variable temperatures Tof

system

and constant temperature

R of reservoir

as expressed by the following equations adopting the

symbology used by Gyftopoulos and Beretta [1]:









MAX

PVV







MAX

10 1

101 0

TAR R

EX W

UUT SS





 (1)

The definition of entropy is derived from the differ-

ence between energy and available energy [1] and is here

defined as thermal entropy by virtue of the consid-

erations discussed above:



10 10

SS EE

T

 





1 0

R R







(2)

These equations, reported in the addressed literature,

constitute the outset of the present study underpinned by

the consideration that equality of temperature is a neces-

sary condition that is not sufficient to prove that a system

is in stable equilibrium with the reservoir. Indeed, even

though two interacting systems are in thermal stable

equilibrium owing to equality of temperatures, the two

systems may experience nonequ ilibrium states due to the

non-null difference between pressures or chemical poten-

tials. The equality of total potential and pressure between

system and reservoir should therefore constitute the set

of additional necessary conditions to ensure the stable

equilibrium so that the equality of the complete set of

“generalized potentials” constitutes a necessary and suf-

ficient condition for mutual stable equilibrium in com-

pliance with the Second Law as worded by Gyftopoulos

and Beretta [1]. As a consequence of the procedure

adopted for its definition by the said Authors [1], entropy

is an additive property that can be generalized to include

the contribution of additional components fulfilling to the

conditions of equality of pressure and chemical potential

that guarantee the mutual stable equilibrium of the com-

posite system-reservoir.

An intuitive rationale of the pro cedure here adopted to

define entropy may be explained considering that internal

energy is characterized by a “hybridization” of ordered

and disordered (due to distribution of molecule’ position

and velocity of system’s particles) energy status. Entropy

may be regarded as the measure of the amount of disor-

dered energy – released to the reservoir – resulting from

the difference of hybrid energy – ordered and disordered

– and available energy – ordered energy – transferred, as

useful interaction, to th e external system.

4. Mechanical Entropy Derived from

Mechanical Exergy

The formulation of thermal exergy in Equation (1) ex-

presses the weight process as the result (and the measure)

of the maximum net useful work that can be extracted

from a system interacting with a thermal reservoir. The

weight process can also be regarded as the minimum net

useful work delivered to the system and hence can be as-

sociated with the maximum net useful heat according to

the concepts of equivalence and interconvertibility [3-5].

In this case, the weight process is calculated as the result

of the interaction of the system with a mechanical reser-

voir at constant pressure

P. The mechanical reservoir

is here assumed to possess the same properties as a ther-

mal reservoir [1,2,6] that is characterized and defined

without explicit reference to any specific form of gener-

alized potential.

The equality of pressu re between system and reservo ir

is assumed to be an additional necessary condition of

mutual stable equilibrium. This condition should there-

fore be complied with equality of temperature to ensure

that the status of composite system-reservoir is one of

stable equilibrium.

If the concept of generalized available energy



[1]

is now referred to, then the formulation of mechanical

exergy should be translated into the following expression

where the superscript “M” stands for “Mechanical reser-

voir” since the composite system-reservoir undergoes a

“work interaction” and the physical meaning becomes

the “maximum” net useful heat of the system:



MAX

101 0

MAR RR

EXQ

 (3)

The definition of mechanical exergy is the basis for

deriving the expression of mechanical entropy using the

procedure adopted for thermal exergy and thermal en-

tropy:



1010 10

MM RR

SS EE





 (4)





P. PALAZZO

This expression of mechanical entropy can be demon-

strated considering that the procedure conceived by Gy-

ftopoulos and Beretta proving the formulation of (ther-

mal) entropy [1], does not impose any restriction with re-

spect to the form of energy and generalized available

energy that constitute the expression of entropy. There-

fore the same procedure can be considered valid regard-

less of the physical nature of the properties involved. Th e

minimum amount of weight process corresponds to the

maxim um amount of heat interact i on:



10 1

MIN

AR RR











1 0

ln ln

VVV









(5)

which expresses the (minimum) amount of work interac-

tion absorbed from mechanical reservoir at constant pres-

sure

P and constant

V (total volume coincides with

specific volume in the particular case of a reservoir). Equ-

ation (5), substituted in the former relation, expresses the

mechanical entropy:



1 0

ln lnRVV

NET

WG

SS (6)

5. Chemical Entropy Derived from Chemical

Exergy

The chemical potential generated by interaction among

the molecules of an open system constitutes a component

of internal energy and is defined as a form of “disordered

energy”, as thermal energy is [7].

The maximum work of a reversible (internally and ex-

ternally) chemical reaction is expressed by the Gibbs

function so that REV . The Van’t Hoff equilib-

rium open system at constant temperature and constant

pressure is a suitable device for reproducing a typical

chemical reaction where A, B are reactants and C, D are

products [7] :

KRT





AB CD

nA nBnCnD  (7)

is adopted to express maximum work tak ing into account

the internal mechanism of chemical reaction by means of

the expression depending on the equilibrium constant

of the reaction.

The Gibbs relation, obtained from mass and internal

energy balance, is as follows:

dddd

Vn Sn

UTSPVn



 



 



 



 



iVSn













(8)

where

iVSn











-ith





represents the chemical potential

of the constituent.

Chemical exergy is defined by Kotas [7] as “the ma-

ximum work obtainable from a substance when it is

brought from the environmental state to the dead state by

means of processes involving interaction only with the

environment”. The chemical reservoir can be character-

ized according to the definition proposed by Gyftopoulos

and Beretta [1] as a “reservoir with variable amounts of

constituents”. The whole chemical and physical process

is modelled by a schema subdivided in two typical steps

[7]: the initial molecular system form’s rearrangement

(molecular structure) and the final molecular system’s di-

mensional change (geometry-kinematics or pressure and

temperature). The two representative processes are: che-

mical reaction open system and physical operation open

system; the two processes in series provide an expression

of maximum net useful work withdrawn from the system

expressed by Equa t i o n [ 7]:

MAX 1

10 0

CAR RP

EXWRT P







(9a)

Mass interaction is the characteristics of the chemical

energy transfer and is moved by the difference of chemi-

cal potential between the system and the reservoir.

In the more general case of a mixture of chemical

constituents:

MAX

10 ln

CAR

EXWRTx x







 

MAX

101 0

CAR RR

EXW 





(9b)

The equality of total potentials is accounted for as an

additional necessary condition of mutual stable equilib-

rium between the system and the reservoir other than the

equality of temperature [1]. This implies a definition of

chemical entropy derived from chemical exergy and che-

mical energy in line with the methodology previously

adopted.

If the concept of generalized available energy is now

again considered, the formulation of chemical exergy

should be translated into the following expression:

(10)

where the superscript “C” stands for “Chemical reser-

voir” since the composite of system and reservoir under-

goes a “mass interaction”.

Now that chemical exergy is defined, and considering

that entropy is an additive property, the expression used

for entropy associated with heat interaction can be ex-

tended to chemical potential depending on mass interac-

tion:



101010

CRR

SS EE







 

(11)





This constitutes the expression of the chemical entrop y

derived from chemical exergy based on the equality of

P. PALAZZO 55

chemical potential which constitutes a necessary condi-

tion for mutual stable equilibrium between the system

and the chemical reservoir.

Since entropy is an inherent property of all systems

[8-11], chemical entropy would be characterized by the

chemical potential of all atoms and sub-molecules that

constitute all compounds and determine the supra-mo-

lecular architecture and configurations of all molecular

systems. The dimensions and shapes of molecular struc-

tures play, in this perspective, a fundamental role in de-

termining the minimum entropy level that ensures the

stability of matter and its capability to react with other

co-reactants, as well as to undergo endogenous or ex-

ogenous processes. Typical is organic chemistry and bio-

chemistry in which same chemical formula are incapable

of describing materials and compounds that display dif-

ferent physical and chemical characteristics and proper-

ties. The vibration degree of freedom is an additional

aspect correlated to molecular structure complexity.

6. Generalized Exergy and Entropy

The definition of a thermo-chemical-mechanical reser-

voir characterized by constant temperature, pressure and

chemical potential implies the additivity [1,2] of the com-

ponents constitu ting the generalized exergy:

EX EX

EX EX (12)

The internal energy balance of the composite system-

reservoir, adopting the symbology in [1], provides the

amount of weight process due to therm a l , mechanical and

chemical contributions:







TEM ,

SYSTEM RESERVOIR

SYSTEM , SYS

SYSTEM ,

GARAR

EX WQM



 

 



 





,RQ

(13)

where:

UQ ,RC

is the minimum heat interaction

with the thermal reservoir;

UM is the mini-

mum mass interaction with the chemical reservoir and

,RW

UW

is the minimum work interaction with the

mechanical reservoir.

The isothermal process r ealizes energy conversion and

at the same time an entropy conversion from thermal en-

tropy to mechanical entropy that occurs due to simulta-

neous heat interaction and work interaction. Both con-

versions are accounted for in the physical exergy expres-

sion:



SYSTEM RESERVO

TOTAL

P V

TOTAL,

TR C

EX UU

UUTS

TS S



 

 

 

(14)

where the term

R represents the contribu-

tion to entropy conversion only occurring inside the res-

ervoir and the terms

TS

TR CR

PVTSS



 

GTMC

SSSS 

ddd0STVPn

RR R

represent the contribution transferred from the system to

the reservoir. It is noteworthy that entropy conversion is

inherent in energy conversion and that entropy conver-

sion requires the additional term that contributes to ex-

ergy balance expressed in the above formulation which

therefore considers the effect of both energy and entropy

conversion processes. Considering that entropy is an ad-

ditive property constituted by components deduced from

the corresponding generalized exergy’s components, the

generalized entropy may be defined as the sum of ther-

mal, mechanical and chemical terms:

7. Remarks upon the Gibbs-Duhem Relation

The Gibbs-Duhem relation [1] constitutes a condition

among all intensive properties temperature—pressure and

chemical potential—that define the state of a heteroge-

neous system. If the system is homogeneous and consti-

tuted by one con stituent only, th ere are no phase changes

or chemical reaction mechanisms inside the system im-

plying the system itself is at chemical equilibrium and

the Gibbs-Duhem relation is:





 (15)

Chemical potential



is defined as the component of

internal energy ge nerated by the interactions of in ter-par-

ticle positions and relative distance, except for kinetic

potential (due to inter-particle relative velocity). Consid-

ering these assumptions, the system model characteristics

can be assimilated to those ado pted in the Kinetic Theory

of Gas [12] which, in p articular, considers molecules un-

dergoing elastic repulsive interaction forces (Lennard-

Jones) on collision with other molecules and with the

wall of the container but otherwise exert no attractive in-

teraction forces (Van der Waals) on each other or on the

container wall [13]. The container walls represent a geo-

metrical volume constraint con dition imposed on the sys-

tem.

In the special case of the system’s undergoing an iso-

thermal process, the pressure changes are due to the

change of volume that determines the frequency of parti-

cle collisions [12]; if the system is characterized as as-

sumed, then the differential of chemical potential among

all atoms or molecules is due to the temperature that is

the only inter-particle kin etic energy transformed into in-

ter-particle potential energ y due to repulsive collisio n in-

teractions (attractive interactions are negligible by as-

sumption). Considering that no chemical reactions occur

inside the system as assumed, then d0



, as reported

by Kotas [6] for systems with a fixed chemical composi-

P. PALAZZO

tion, and the Gibbs-Duhem relations becomes:

ddST VP

d0T

d0

(16)

where and V are not null both being inherent pro-

perties of any system [8]. If the system undergoes an

isothermal reversible process, then the temperature re-

mains constant by definition, so that ; on the

other side, is not null in the same isothermal proc-

ess and therefore an inconsistency appears. The same in-

consistency is displayed if an isobaric process is ac-

counted for using the Gibbs-Duhem relation where pres-

sure remains constant, so that , and temperature

does not. This inconsistency involves the intensive prop-

erties temperature and pressure which determine the

thermo-mechanical conversions involved in the concept

of available energy and exergy. The alternative ap proach

set forth in the present study resolves this inco nsistency.

The chemical potential expressed by the Gibbs func-

tion reveals that entropy property is variable in the iso-

thermal process and consequently the chemical potential

is not constant, which is in contradiction with the as-

sumption set forth.

The chemical potential that appears in assumed system

model is due exclusively to the repulsive interactions on

collisions and depends solely on molecules’ velocity, so

that temperature constitutes the first contribution to pres-

sure. The second contribution is due to the (specific) vo-

lume determining the frequency of collisions between

molecules and the external system which, on the other

side, does not determine the chemical potential due to at-

tractive interactions among the molecules which does not

exist in the model assumed.

If d



 and and considering attractive in-

teractions as negligible, repulsive interaction potential is

equal to kinetic poten tial (transformation during collision

only). The Gibbs rel at i on:

d0T

dVQ WddUTSP



  (17)

can be reformulated in different terms adopting thermal

entropy and mechanical entropy as defined in Equation

(6):

 

dQ Wdd

UT SS



  (18)

and transformed, by using the state equation PV RT

 



T S

valid in this special case, and consistent with the Kinetic

Theory of Gases [12], in the following form:



UTS

TSS





 

0U

TOTAL 0

SSS



  (19)

This, associated with the temperature, takes into ac-

count either heat or work interactions contributing to va-

riations in internal energy.

It is noteworthy that, in the case of work interaction,

the work put into the system, which is considered posi-

tive, corresponds to a decrease of mechanical entropy as

per Equation (6). This is the opposite of heat interaction

that is positive if thermal entropy increases. In other

terms, heat inpu t causes thermal entropy to increas e, work

input causes mechanical entropy to decrease, therefore

work depends on pressure in the opposite manner that

heat depends on temperature. If the system releases work,

it increases mechanical entropy because mechanical en-

ergy, distributed among the particles that constitute the

system, progressively becomes similar to thermal energy

i.e. energy distributed by the velocity of the same parti-

cles. Increasing volume means that pressure is progres-

sively determined by particles’ kinetic energy (tempera-

ture) with respect to the contribution of the frequency of

collisions among particles and with the external system

surface determined by the volume. Pressure is thus pro-

gressively the more like temperature as volume in-

creases.

The balance of entropy, along the isothermal expan-

sion reversible process d, is deduced from the

equation



 T

S where is the ther-

mal entropy input due to heat flowing into the system

and

S

TOTAL

S

TOTAL 0

SSS

is the mechanical entropy output from the

system which is the opposite of the mechanical entropy

input flow inherent in the expansion work output from

the system itself. The physical meaning is that conver-

sion from a kinetic form into a geometric form, required

to convert heat into work, necessitates an increase in me-

chanical entropy.

Due to the fact that mechanical entropy must enter the

system because of work output, then the mechanical en-

tropy direction is inverted compared to the thermal en-

tropy direction in the system’s entropy balance. Total en-

tropy is consequently constant if either thermal and me-

chanical entropy enter into the system. The rationale of

this statement is that these two entrop y components have

an opposite origin and elide each other.

If the term is expressed by means of



, the total entropy, resulting from

the addition of thermal and mechanical components of

entropy, implies that thermal entropy would be constant

in an isothermal reversible process that requires heat in-

teraction by means of thermal entropy exchange.

On the other side, pressure does not derive from in-

ter-particle chemical potential and is just the mechanical

effect produced by the temperature itself (apparent po-

tential).

The Gibbs relation, expressed in terms of the Equation

(19), resolves the apparent inconsistency highlighted in

the Gibbs-Duhem relation. In fact, this can be reformu-

lated as follows:

P. PALAZZO 57



dS n





T M

TS n

UTS

TS TS



 (20)

The Euler relation is obtained from the Gibbs relation

by integration at constant temperature and constant

chemical potential [1], so that:



TOTAL

UTS

TSSn TS



 

S nP



  (21)

If compared with the classical expression of the Gibbs

relation UT V



 PV, the term corre-

sponds to the term

TS

. Moreover, in the case of an

isothermal process (and absence of chemical reactions so

that chemical potential is constant) it requires that total

entropy is constant also implying that thermal entropy

variation is equal to mechanical entropy variation.

It is noteworthy that, in the case of an ideal system as

assumed, internal energy U is associated with the kinetic

energy of the molecules, and thus to temperature only;

however, internal energy components depend on the

terms



 and which both depend on volume

as well. This dependence ensures that real systems in

which thermodynamic cond itions are affected by interac-

tions among molecules that determine the potential en-

ergy, are characterized by volume. In differential terms,

the Euler relations are:

PV

d d

n n





TOTAL TOTAL

dd d

dddd

TM TM

TMTM

TSSTn n

TS SSST

TSTSST ST





 



d0VP n





 

(22)

On combining the Gibbs relation and Euler relation

expressed in the terms set forth, the Gibbs-Duhem rela-

tion ddST







dd0

STn



assumes the form:

TOTAL

STS









TPV





(23)

where: constitutes the inter-particle kinetic potential

component of internal energy resulting in the

macroscopic work interaction transferred by means of a

weight process;



constitutes the inter-particle chemi-

cal potential component of internal energy resulting in

the macroscopic work interaction transferred by

means of a weight process. The kinetic and chemical

constitute the two fundamental potentials at microscopic

inter-particle level interacting at macroscopic level that

constitute the hierarch ical geometric and kinematic struc-

ture.

The dualism of kinetic potential and chemical potential

constitutes the inh erent structure of potentials even in the

special case of an ideal system for which inter-particle

potential energy in null. In this case, in fact, potential

energy still exist in the form of repulsive reaction poten-

tial energy that is due to kinetic energy transformed on

collision only, without macroscopic effects on the entire

system.

This different form of the Gibbs-Duhem relation re-

solves the apparent inconsistency in the special case of

the isothermal ideal process. In fact, d0n





d0T

d0T

because

the system model is ideal and remains the only

condition to be satisfied since no longer appears in

the Gibbs-Duhem relatio n as expressed in Equ ation (23).

The rationale of these statements can also be found in the

behaviour of elements, molecules and atoms, constituting

the system as a whole. In fact, in the isothermal process,

the temperature and subsequent intermolecular repulsive

interactions on each collision are constant and the varia-

tions of kinetic potential and chemical potential (due to

intermolecular repulsive interactions) are therefore null:



and consequently d0





0T

. In the case of the

isobaric process, temperature and pressure are variable or

kinetic potential and chemical po tential (due to repulsive

interactions at each inter-particle collision) both change

along the isobaric process: d and consequently





. Even in the case of an ideal system, there is du-

alism and symmetry of kinetic energy and potential en-

ergy among the molecules so that and d0d0T



.

Pressure is the mechanical effect of the contribution

related to kinetic interaction and related potential and

chemical interaction and related potential. In this per-

spective, pressure can viewed as the outcome of the

temperature and chemical potential of a complex mul-

ti-particle system, converted into work interaction with

the external reservoir and with the external weight proc-

ess.

Finally, notwithstanding the restrictions assumed for

the model adopted, the behaviour of the system is coher-

ent with expectations in terms of phenomena and ten-

dency of the properties in the general case of real sys-

tems and processes where each particle experiences at-

tractive interaction with all others, and does not contra-

dict the fundamentals reported in the literature.

8. Conclusion

The three additive components of exergy discussed in

this study constitute the components of generalized ex-

ergy that depends on temperature, pressure and chemical

potential and, at microscopic level, on the kinetic energy

and potential energy generated by interactions among the

molecules of the system. Moreover, three components of

entropy property have been inferred from the corre-

sponding exergy components. In particular, chemical

exergy and entropy are correlated to the molecular struc-

ture of matter due to the composite of molecules geome-

P. PALAZZO

try and chemical bonds characteristics. The aim of seek-

ing a property related to molecular or supra-molecular

architecture is to obtain a method able to pred ict a-priori

stability as well as capability in self-assembling proc-

esses and the related intermediate phases of chemical

compounds that are not available in the environment and

which could undergo a building process by means of na-

no-sciences technologies. Such a method would make it

possible to design materials characterized by properties

that could be evaluated prior to being realized and to

confirm predictions by means of experiments and labo-

ratory tests. Finally, the method of Entropy Generation

Minimization (EGM) [14-16] associated with the Ex-

tended Exergy Accounting (EEA) [17,18] could be fur-

ther generalized to provide an overarching paradigm for

analysing the whole process.

REFERENCES

[1] E. Gyftopoulos and G. P. Beretta, “Thermodynamics:

Foundations and Applications,” Dover Publications, New

York, 2005.

[2] R. A. Gaggioli, International Journal of Applied Thermo-

dynamics, Vol. 1, 1998, pp. 1-8.

[3] W. R. Dunbar, N. Lior and R. A. Gaggioli, Journal of

Energy Resources Technology, Vol. 114, 1992.

[4] R. A. Gaggioli, D. H. Richar dso n and A. J. B owman, Jour-

nal of Energy Resources Technology, Vol. 124, 2002, pp.

105-109. doi:10.1115/1.1448336

[5] R. A. Gaggioli and D. M. Paulus Jr. “Available Energy—

Part II: Gibbs Extended,” Transaction of the ASME 2002,

2002.

[6] R. A. Gaggioli, “The Dead State,” Proceedings of 25th

ECOS, Perugia, 2012.

[7] T. J. Kotas, “The Exergy Method of Thermal Plant Ana-

lysis,” Reprint Edition, Krieger Publishing Co mpany, Krie-

ger, 1995.

[8] E. P. Gyftopoulos, International Journal of Thermody-

namics, Vol. 9, 2006, pp. 107-115.

[9] G. P. Beretta, International Journal of Thermodynamics,

2008.

[10] E. Zanchini, International Journal of Thermodynamics,

2010.

[11] E. Zanchini and G. P. Beretta, International Journal of

Thermodynamics, Vol. 13, 2010, pp. 67-76.

[12] H. A. Jakobsen, “Chemical Reactor Modeling,” Springer-

Verlag, Berlin, Heidelberg, 2008.

doi:10.1007/978-3-540-68622-4_2

[13] B. H. Bransden and C. J. Joachain, “Physics of Atoms

and Molecules,” 2nd Edition, Prentice Hall, Upper Saddle

River, 2003.

[14] T. L. Hill, “Statistical Mechanics. Principles and Selected

Applications,” Dover, New York, 1987.

[15] E. Sciubba, Entropy, Vol. 12, 2010, pp. 1885-1866.

doi:10.3390/e12081855

[16] E. Sciubba, International Journal of Thermodynamics, Vol.

14, 2011, pp. 11-20.

[17] M. Belli and E. Sciubba, International Journal of Exergy,

Vol. 5, 2006.

[18] E. Sciubba, Energy, Vol. 28, 2003, pp. 1315-1334.

doi:10.1016/S0360-5442(03)00111-7