On the Derivation of Thermodynamic Equilibrium Criteria

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.Fluid Phase Equilibria 136 1997 113

On the derivation of thermodynamic equilibrium criteria

Jose Luis de Medeiros a,), Marcio Jose Estillac de Mello Cardoso b,1 a

Escola de Qumica da UFRJ, Departamento de Engenharia Qumica, CT Bloco E, Cidade Uniersitaria 21949-900, Rio de Janeiro, RJ, Brazil

bInstiuto de Qumica da UFRJ, Departamento de Fsico-Qumica, CT Bloco A, Sala 408, Cidade Uniersitaria 21949-900,

Rio de Janeiro, RJ, Brazil

Abstract

It is well established that, besides the two thermodynamic formulations based on the fundamental equations . ..i.e., entropic formulation Ss S U, V, N ; Energetic Formulation Us U S, V, N , the equilibrium states of a

system can be represented by equivalent formulations derived by making Legendre transforms of either of thewfundamental equations H.B. Callen, Thermodynamics and an Introduction to Thermostatics, 2nd edn., Wiley,

xNY, 1985 . The main purpose of this article is to present in a new and consistent way, the derivation of the .equilibrium criteria for both thermodynamic potentials i.e., Legendre transforms of internal energy and

.Massieu functions i.e., Legendre transforms of entropy with the aid of concepts such as boundaries, reservoirs,

and certain auxiliary devices known as equilibrium or intermittent connections. q 1997 Elsevier Science B.V.

Keywords: Equilibrium criteria; Gibbs energy; Chemical potential

1. Introduction

The development of the equilibrium criteria of a general system in several different configurationsof interaction with its surroundings is accomplished with the application of evolutionary principles.

.Therefore, if one knows that some property will increase or decrease while the system moves towardequilibrium, under some kind of interaction with its surroundings, and that property will reach astationary value at the equilibrium condition, then it is possible to associate a variational principle to itw x1 . This principle, which must be expressed in the system coordinates, defines an equilibrium

criterium that is valid whenever the system is in a specific pattern of interactions.

)

Corresponding author. E-mail: [email protected]

e-mail: [email protected].

0378-3812r97r$17.00 q 1997 Elsevier Science B.V. All rights reserved. .PII S 0 3 7 8 - 3 8 1 2 9 7 0 0 1 3 3 - 7


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( )J.L. de Medeiros, M.J.E.d.M. Cardosor Fluid Phase Equilibria 136 1997 1132

w x.One can easily find in existing thermodynamic textbooks e.g., 1 5 the derivation of the differentthermodynamic potentials and Massieu functions from the internal energy and entropy functions. This

w xderivation can be elegantly performed by the application of the Legendre transform technique 1,4,6 .Nevertheless, the derivation of the correspondent equilibrium criteria is, in general, only alluded to

w xor developed in a less rigorous fashion. For instance, Callen 1 adopts a so-called virtual change inthe total internal energy for an isolated system in order to obtain the equilibrium criteria for the

energetic formulation.The main purpose of the present article is to derive the different equilibrium criteria of a genericsystem in terms of its state coordinates.

In Section 2, some basic concepts are reviewed and some new definitions are presented. Section 3develops the evolutionary equilibrium criteria for both the entropic and energetic formulations.

Section 4 deals with the application of the same approach to thermodynamic potentials considered asw x. Legendre transforms of the internal energy 1 and Massieu functions considered as Legendre

w x.transforms of the entropy 1 . Finally, Section 5 presents some final remarks and conclusions.

2. Basic definitions: boundaries, reservoirs and connections

With the aim of establishing a set of basic concepts necessary for the development of the presentw xarticle, some fundamental definitions are reviewed 1,2 and some new ones are also introduced. In

the following sections we shall need to use these concepts which will always have the same preciserole in all analyzed thermodynamic systems.

The first of these objects are the system boundaries, or walls. The boundaries exist to define aphysical portion of space occupied by the system and usually determine how it will interact with its

surroundings. Although boundaries are usually considered as perfectly impermeable walls i.e.,.aiming for the complete insulation of the system , they are more appropriately defined as restrictive

walls which allow the transfer of only a certain type of transferable entities, or extensive properties, .like energy as heat, volume, molecules of certain substance, etc. between the interior and the

.exterior of a system. In many instances, we may select movable boundaries for volume transfer , . diathermic boundaries for heat transfer , component permeable boundaries for transfer of heat and

.only of a certain kind of system components . Composed boundaries accept multiple kinds of transfersimultaneously.

Associated with each transferable entity, or extensive property, is a corresponding transferpotential, or intensive property. These concepts are conjugated. The transfer of a transferable entitybetween the interior and exterior sides of a system is accomplished by means of a specific drivingforce. The driving force arises if there is a finite difference, on the correspondent transfer potential,between both sides of the system boundary. Generally speaking, every kind of transfer entails a

.transfer of energy as mechanical, thermal or chemical energy . Volume, energy as heat and amountof a substance are examples of transferable entities, with, respectively, pressure, temperature andchemical potential being the correspondent transfer potentials.

In order to establish the transfer of an entity between any two systems, it is needed that bothsystems have boundaries appropriate to the transfer, and that both systems be linked by what we cancall a suitable connection. There are two kinds of distinguishable connections: the equilibriumconnection and the intermittent connection.


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( )J.L. de Medeiros, M.J.E.d.M. Cardosor Fluid Phase Equilibria 136 1997 113 3

Table 1

The basic characteristics of some reservoirs in both entropic and energetic representations

Letter symbol Exchanged entity Tranfer potential Constant extensive properties Reservoir equation

.a Heat 1r T V , N d S s 1r T dUa a a a

b Heat T V , N dU sT d Sb b b b

.c Volume Pr T U , N d S s Pr T dVc c c c

d Volume P S , N dU sy P dVd d d d

. .e Amount of kth component m r T U , V , N l/ k d S sy m r T dNk e e le l k ke .f Amount of kth component m S , V , N l/ k dU sm dN

k f f lf f k kf

The equilibrium connection acts instantaneously and equilibrates the connected systems withrespect to the driving force characteristic of that transfer. This means that the thermodynamicequilibrium is partially achieved, with respect to a specific potential, between two connected systems.

That is, the pertinent transfer potential is now uniform across both systems.In the intermittent case, the contact is partial and incomplete, allowing the flow of a given amount

of the transferable entity between the connected systems but without reaching the thermodynamicequilibrium. The intermittent connection should be employed with the aim of controlling a certain

.extensive property i.e., constantly maintaining the extensive property during a system state change .For example, we can keep the system in internal energy or entropy with a constant value by means ofan intermittent connection with a heat reservoir. This artifice is particularly useful in controlling the

.entropy of a system, which cannot be fixed by means of boundaries see Section 3 .Reservoirs are a special class of systems which are, in general, largely and uniformly single i.e.,

.non-composite systems. It is considered that all the processes that take place inside a reservoir arew x .quasi-static 1 . No irreversible processes like non-equilibrated chemical reactions are allowed to

happen inside a reservoir. A reservoir may be seen as a large reserve of a transferable entity stored ina condition, such that, it exhibits a definite value of the correspondent transfer potential, which iscalled the reservoir characteristic intensive parameter here. The nature and the size of the reservoirs

.assure that their characteristic intensive parameters e.g., temperature, pressure or chemical potentialhave constant and uniform values. The reservoirs are enclosed by boundaries suitable to the outwardor inward flow of the transferable entity that they store.

When connected to a system by means of an equilibrium connection, it must be assumed that the .system including all its subsystems had attained equilibrium with respect to the reservoir characteris-

tic intensive parameter. Therefore, since the reservoirs are considered very large in comparison to thesystem, the system itself acquires the reservoir characteristic intensive parameter as one of its stateintensive parameter.

In the most general case, the system can be considered as a composite one, in which each .subsystem is properly connected with a specific reservoir or a set of reservoirs . The system, single

or composite, plus all possible reservoirs connected to it constitute, what we shall call, a macrosys-tem.

Table 1 presents a list of different possible reservoirs including their main characteristics e.g.,


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.exchanged entities, transfer potentials, etc. . The reservoir operating equations are properly obtainedw xfrom either the energetic or entropic differential form of the fundamental equations 1 :

NC

dUs Td Sy PdVq m dN 1 . k kks1

NC1 P mk

d Ss dU q dVy dN 2 .

k

/ / /T T Tks1

.where the index k runs over the number of chemical components NC , and U, S and N are the totalk

internal energy, total entropy, and amount of substance of k species, respectively. We also havetemperature, T, pressure, P, and chemical potential of the k species, m .

k

.Some of the listed reservoirs Table 1 have a relatively easy physical interpretation. Nevertheless,other reservoirs are to be considered as purely theoretical artifacts. One can take as an example of thelast type of reservoirs, the one in which the only exchanged entity is the amount of the kth component .reservoir of type e or f of Table 1 .

It is not physically possible to accomplish a transfer of the amount of a substance without anequivalent transfer of the amount of energy as heat. Thus, for example, when the reservoir of type e is

.connected by means of an equilibrium connection to any system, there will be only a partialthermodynamic equilibrium between both of them. That is, we only consider that the chemical

.potential of the exchanged species k will be equal in both the system and in the reservoir of type e.No conclusion can be drawn about the equality of temperature.

It is important to note, however, that none of the listed reservoirs in Table 1 violate anythermodynamic law or postulate.

In the following sections, the system and the reservoirs will be represented by squares and theirboundaries by solid lines. This representation does not intend to suggest any kind of actual

geometrical proportions or spatial boundaries. The solid line represents an isolated system or.reservoir , that is, not any kind of transfer is allowed through its boundaries.

An appropriate connection between a system and a specific type of reservoir makes the transfer of

the reservoir characteristic transferable entity possible between them. For instance, a system may .perform volume expansion or contraction when it exchanges transfers volume with a volume

.reservoir, by means of a mobile or flexible wall the systemrreservoir connection .As already mentioned, a system and a reservoir might be connected in two distinct ways. A solid

.line , means that both the system and the reservoir are in thermodynamic equilibrium, with .respect to the reservoir transfer potential. A dashed line , on the other hand, represents an

.intermittent i.e., non-equilibrated contact between the system and the reservoir.

3. Equilibrium criteria for entropic and energetic formulations

In general, one starts the presentation of the different thermodynamic equilibrium criteria either .with the evolutionary principle for the system entropy entropic formulation , or with the evolutionary

. w xprinciple for the system internal energy energetic formulation 1 .Let us consider, for the entropic formulation, an isolated non-equilibrated multicomponent system

containing NS interacting subsystems. The isolated system, as a whole, does not exchange energy,


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volume or amount of substance with its surroundings. The system is then set free to internal transferof extensive properties, among its constitutive subsystems, letting the total system entropy increase.When the thermodynamic equilibrium is achieved, the system entropy attains its maximum possible

.value in accordance with the internal and external constraints placed upon the system. Mathemati-cally, this extremum principle can be expressed by the following equations:

NS

dUs 0 3 .

i

is1

NS

dV s 0 4 . iis1

NS

dN s 0 for ks 1, . . . ,NC 5 . . kiis1

NS

d S G 0 6 . iis1

.where the i and k indexes run over the number of subsystems NS and the number of chemical . .components NC , respectively. At the equilibrium, the inequality Eq. 6 turns into an equality.

In the application of the energetic formulation, we take into account a closed rigid-walled,non-equilibrated multicomponent system. The systems, as a whole, does not exchange volume oramount of substance with its surroundings. The system is then set free to internal transfers ofextensive properties, among its constitutive subsystems, letting the total system internal energydecrease. In this case, the system total entropy is fixed to its initial value by means of an intermittentconnection with a heat reservoir, in a way that it is allowed the draining from the system, of anycreated entropy. When the thermodynamic equilibrium is achieved, the system internal energy attains

.its minimum possible value in accordance with the internal and external constraints placed upon thesystem. Mathematically, this extremum principle can be expressed by the following equations:

NS

d S s 0 7 . iis1NS

dV s 0 8 . iis1

NS

dN s 0 for ks 1, . . . ,NC 9 . . kiis1

NS

dUF 0 10 . iis1

.where, as for the previous case, the i and k indexes run over the number of subsystems NS and the . .number of chemical components NC . At the equilibrium, the inequality Eq. 10 turns into an

equality.It must be stressed that both the entropic and the energetic formulations are completely equivalent.

The main difference between these two approaches is related with the fixed or controlled property. Inthe former formulation, the total internal energy of the system is simply fixed by the impermeability

of the system external boundaries to the transfer of any kind of energy thermal, mechanical or


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.chemical . In the latter case, however, the total system entropy cannot be fixed by means of anyappropriated wall or equilibrium connection. An intermittent connection with a cold heat bath must beemployed.

. ..It must be remarked, in passing, that for both entropic Eqs. 3 6 and energetic formulations . ..Eqs. 7 10 , the system of equations presented above are restricted to non-reacting systems. For

. .the reacting system case, Eq. 5 and Eq. 9 must be properly rewritten. This particularization,

however, is not essential for the argument developed in the present article.

4. Equilibrium criteria for thermodynamic potentials, Massieu functions

In this section, we present the derivation of the equilibrium criteria for different thermodynamic . potentials i.e., Legendre transforms of internal energy , and Massieu functions i.e., Legendre

.transforms of entropy .Table 2 presents the evolutionary principles for some examples of macrosystems system plus

.reservoirs configurations. Each line of this table corresponds to a specific macrosystem configurationwhich is identified by the line number. The first column shows a schematic representation of the

.macrosystem, which includes the system which can be either single or a composite one , the .reservoirs, the type of connections equilibrium or intermittent , and the type of enclosure for thewhole macrosystem. We adopt, as a representation for an isolated and a constant entropy macrosys-tem, the whole macrosystem being enveloped by a solid or a dashed line, respectively. The second

column presents the macrosystem evolution equations. For an isolated macrosystem solid line. .enclosure , one must apply the maximum entropy principle i.e., entropic formulation . For an

.isentropic macrosystem dashed line enclosure , on the other hand, one must apply the minimum .energy principle i.e., energetic formulation . The third and fourth columns describe the reservoir

equations and the correspondent system evolutionary principle with the appropriated fixed properties,respectively.

.The first set of 10 entries of Table 2 lines 1 to 10 correspond to the application of an extremum

principle for a macrosystem for which the reservoir equations are written in the same representation. .For example, for the macrosystem 1 , we have adopted the entropic formulation together with the

entropic representation of a heat reservoir. .The second set of 10 entries of Table 2 lines 11 to 20 describes a macrosystem and reservoir

.opposite representations. For instance, for the macrosystem 11 , we have adopted entropic formula-tion together with the energetic formulation of a heat reservoir.

.Finally, the last two entries of Table 2 lines 21 and 22 correspond to two different cases in whichwe have an intermittent connection between the system and a heat reservoir.

Let us now exemplify the derivation of some of the system evolution criteria from the givenmacrosystem evolution and reservoir equations.

.For the macrosystem 5 , we derive from Table 2:

d Sq d S q d S G 0 11 .a c .by substituting the two reservoir equations in Eq. 11 , one obtains

1 Pd Sq dU q dV G 0 12 .a c

T T


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Table 2

The evolutionary principles for some examples of macrosystem configurations


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.Table 2 continued


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.Table 2 continued


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.Table 2 continued

by taking into account the macrosystem constraints in internal energy and volume, as well as,considering the equilibrium connections of both reservoirs with the system, one finally gets,

U PVd Sy y G 0 13 . /T T

which is the system evolution equation. This evolutionary principle is associated with a specific set of .system properties T, PrT, N , which must be held constant during the system evolution towards the

equilibrium state. . .If Eq. 13 is multiplied by the negative of the system constant temperature yT , one derives,

d Uy TSq PV s dG F 0 14 . .w xwhere G stands for the Gibbs free energy, if one uses its definition equation 1,2 .

.Eq. 14 represents the system evolution equation which would be obtained by applying an . . .analogous derivation for the macrosystem 6 . Therefore, macrosystems 5 and 6 are called

conjugated macrosystems. Other pairs of macrosystems of Table 2 are also conjugated in this sense.


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. .The system evolution equations obtained for the macrosystems 11 to 20 are the same as the . .ones obtained for the first set of 10 entries of Table 2, except for the macrosystems 13 and 14 .

.For the macrosystem 13 , we have from Table 2:

d Sq d S G 0 15 .d

but, d S s 0, thus,d

d SG 0 16 .

which is the desired system evolution equation. But, we also obtain, from the internal energyconstraint, and the volume reservoir equation,

dUy PdV s 0 17 .d

by means of the macrosystem volume constraint, dVs ydV , and taking into account the equilibriumd

.connection between the system, and the volume reservoir, we can rewrite Eq. 17 as follows,

d Uq PV s dHs 0 18 . .

where H stands for the system enthalpy.Therefore, the set of the system properties which must be kept constant during the system evolution

are; P, H, N. . .Finally, we shall discuss the system evolution equations for the macrosystems 21 and 22 . First

of all, it should be mentioned that two last system evolution equations are the same as can be obtained . . .for the macrosystems 3 and 4 , respectively. For the macrosystem 21 , the intermittent connection

.is used to keep the system internal energy constant dUs 0 . .For the macrosystem 22 , we have from Table 2:

d Sq d S q d S G 0 19 .b d

but we have d S s 0 and also d Ss 0, being the last equality due to the intermittent connectiond

between the system and the heat reservoir b, thus,

d S Gs 0 20 .b

from the heat reservoir equation, one obtains,

dU Gs 0 21 .b

from the macrosystem internal energy constraint one obtains,

ydU s dUq dU Fs 0 22 .b d

by taking into account the macrosystem constraint in volume, and the equilibrium connection between .the system and the volume reservoir, we can obtain from Eq. 22 the system evolution equation,

d Uq PV s dHFs 0 23 . .

The set of the system properties which should be kept constant during the system evolution to theequilibrium are: P, S, N.


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5. Final remarks

In the present article, we have presented a new and thermodynamically consistent way of obtainingequilibrium criteria for both thermodynamic potentials and Massieu functions. The proposed approachhas been applied for various types of macrosystems configurations.

The technique developed in this article can be easily extended to more complex type of

macrosytem configurations, for example, in the case in which we must consider other types of work . w xbesides the mechanical volume work: gravitational work, surface work, etc. 6 .

List of symbolsa Helmholtz free energyH EnthalpyG Gibbs free energyN Amount of a substance of a chemical componentN Set of the amount of substance for the system

Chemical components

NC Number of chemical componentsNS Number of subsystemsP PressureS EntropyT TemperatureU Internal energyV Volume

Greek letters

m Chemical potential

Subscripts .a Heat reservoir entropic formulation .b Heat reservoir energetic formulation

.c Volume reservoir entropic formulation .d Volume Reservoir energetic Formulation

.e Amount of kth component entropic formulation .f Amount of kth component energetic formulation

i Subsysteml,k Chemical component

References

w x1 H.B. Callen, Thermodynamics and an Introduction to Thermostatics, 2nd edn., Wiley, NY, 1985.w x2 M.W. Zemansky, R.H. Dittman, Heat and Thermodynamics, An Intermediate Textbook, 6th edn., McGraw-Hill, NY,

1981.


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w x3 L.D. Landau, E.M. Lifshitz, Statistical Mechanics, 2nd edn., Pergamon Press, Oxford, 1969.w x4 L. Tisza, Generalized Thermodynamics, MIT Press, Cambridge, 1966.w x5 J. de Heer, Phenomenological Thermodynamics With Applications to Chemistry, Prentice-Hall, Englewood Cliffs, NJ,

1986.w x .6 R.A. Alberty, Legendre transforms in chemical thermodynamics, Chem. Rev. 94 1994 14571482.

On the Derivation of Thermodynamic Equilibrium Criteria

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