- Email: [email protected]
On the derivation of thermodynamic equilibrium criteria
Fluid Phase Equilibria 136 Ž1997. 1–13
On the derivation of thermodynamic equilibrium criteria Jose´ Luis de Medeiros
a, )
, Marcio Jose´ Estillac de Mello Cardoso ´
b,1
a
Escola de Quımica da UFRJ, Departamento de Engenharia Quımica, CT Bloco E, Cidade UniÕersitaria ´ ´ ´ 21949-900, Rio de Janeiro, RJ, Brazil b Instiuto de Quımica da UFRJ, Departamento de Fısico-Quımica, CT Bloco A, Sala 408, Cidade UniÕersitaria ´ ´ ´ ´ 21949-900, Rio de Janeiro, RJ, Brazil
Abstract It is well established that, besides the two thermodynamic formulations based on the fundamental equations Ži.e., entropic formulation S s SŽU, V, N .; Energetic Formulation U s UŽ S, V, N .., the equilibrium states of a system can be represented by equivalent formulations derived by making Legendre transforms of either of the fundamental equations wH.B. Callen, Thermodynamics and an Introduction to Thermostatics, 2nd edn., Wiley, NY, 1985x. 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 Ži.e., Legendre transforms of internal energy. and Massieu functions Ži.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 configurations of interaction with its surroundings is accomplished with the application of evolutionary principles. Therefore, if one knows that some property will increase Ž or decrease. while the system moves toward equilibrium, under some kind of interaction with its surroundings, and that property will reach a stationary value at the equilibrium condition, then it is possible to associate a variational principle to it w1x. 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.
) 1
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 Ž 9 7 . 0 0 1 3 3 - 7
2
J.L. de Medeiros, M.J.E.d.M. Cardosor Fluid Phase Equilibria 136 (1997) 1–13
One can easily find in existing thermodynamic textbooks Ž e.g., w1–5x. the derivation of the different thermodynamic potentials and Massieu functions from the internal energy and entropy functions. This derivation can be elegantly performed by the application of the Legendre transform technique w1,4,6x. Nevertheless, the derivation of the correspondent equilibrium criteria is, in general, only alluded to or developed in a less rigorous fashion. For instance, Callen w1x adopts a so-called virtual change in the 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 generic system in terms of its state coordinates. In Section 2, some basic concepts are reviewed and some new definitions are presented. Section 3 develops the evolutionary equilibrium criteria for both the entropic and energetic formulations. Section 4 deals with the application of the same approach to thermodynamic potentials Ž considered as Legendre transforms of the internal energy w1x. and Massieu functions Ž considered as Legendre transforms of the entropy w1x.. 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 present article, some fundamental definitions are reviewed w1,2x and some new ones are also introduced. In the following sections we shall need to use these concepts which will always have the same precise role in all analyzed thermodynamic systems. The first of these objects are the system boundaries, or walls. The boundaries exist to define a physical 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 Ž 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, Žlike 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 Ž for volume transfer. , diathermic boundaries Ž for heat transfer. , component permeable boundaries Ž for transfer of heat and only of a certain kind of system components. . Composed boundaries accept multiple kinds of transfer simultaneously. Associated with each transferable entity, or extensive property, is a corresponding transfer potential, or intensive property. These concepts are conjugated. The transfer of a transferable entity between the interior and exterior sides of a system is accomplished by means of a specific driving force. 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 Ž as mechanical, thermal or chemical energy. . Volume, energy as heat and amount of a substance are examples of transferable entities, with, respectively, pressure, temperature and chemical potential being the correspondent transfer potentials. In order to establish the transfer of an entity between any two systems, it is needed that both systems have boundaries appropriate to the transfer, and that both systems be linked by what we can call a suitable connection. There are two kinds of distinguishable connections: the equilibrium connection and the intermittent connection.
J.L. de Medeiros, M.J.E.d.M. Cardosor Fluid Phase Equilibria 136 (1997) 1–13
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 b c d e f
Heat Heat Volume Volume Amount of k th component Amount of k th component
1r T T Pr T P mk r T mk
Va , Na V b , Nb Uc , Nc Sd , Nd Ue , Ve , Nle Žl / k . Sf , Vf , Nlf Žl / k .
d Sa s Ž1r T .dUa dUb sT d S b d Sc s Ž Pr T . dVc dUd sy P dVd d Sl syŽ m k r T . d Nke dUf s m k d Nkf
The equilibrium connection acts instantaneously and equilibrates the connected systems with respect to the driving force characteristic of that transfer. This means that the thermodynamic equilibrium 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 thermodynamic equilibrium. The intermittent connection should be employed with the aim of ‘controlling’ a certain extensive property Ži.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 of an 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 Ž see Section 3. . Reservoirs are a special class of systems which are, in general, largely and uniformly single Ž i.e., non-composite. systems. It is considered that all the processes that take place inside a reservoir are quasi-static w1x. No irreversible processes Ž 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 in a condition, such that, it exhibits a definite value of the correspondent transfer potential, which is called the reservoir characteristic intensive parameter here. The nature and the size of the reservoirs assure that their characteristic intensive parameters Ž e.g., temperature, pressure or chemical potential. have constant and uniform values. The reservoirs are enclosed by boundaries suitable to the outward or 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 Žincluding all its subsystems. had attained equilibrium with respect to the reservoir characteristic intensive parameter. Therefore, since the reservoirs are considered very large in comparison to the system, the system itself acquires the reservoir characteristic intensive parameter as one of its state intensive 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 Ž or a set of reservoirs. . The system, single or composite, plus all possible reservoirs connected to it constitute, what we shall call, a macrosystem. Table 1 presents a list of different possible reservoirs including their main characteristics Ž e.g.,
J.L. de Medeiros, M.J.E.d.M. Cardosor Fluid Phase Equilibria 136 (1997) 1–13
4
exchanged entities, transfer potentials, etc... The reservoir operating equations are properly obtained from either the energetic or entropic differential form of the fundamental equations w1x: NC
dU s T d S y PdV q
Ý m k d Nk
Ž1.
ks1
dSs
1
ž / T
dU q
P
ž / T
NC
dV y
Ý ks1
mk
ž T /dN
k
Ž2.
where the index k runs over the number of chemical components Ž NC. , and U, S and Nk are the total internal energy, total entropy, and amount of substance of k species, respectively. We also have temperature, T, pressure, P, and chemical potential of the k species, m k . Some of the listed reservoirs ŽTable 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 the last type of reservoirs, the one in which the only exchanged entity is the amount of the k th component Žreservoir of type e or f of Table 1.. It is not physically possible to accomplish a transfer of the amount of a substance without an equivalent transfer of the amount of energy as heat. Thus, for example, when the reservoir of type e is connected Žby means of an equilibrium connection. to any system, there will be only a partial thermodynamic equilibrium between both of them. That is, we only consider that the chemical potential of the exchanged species Ž 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 any thermodynamic law or postulate. In the following sections, the system and the reservoirs will be represented by squares and their boundaries 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 Žor 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 Žtransfers. volume with a volume reservoir, by means of a mobile or flexible wall Ž the systemrreservoir connection.. As already mentioned, a system and a reservoir might be connected in two distinct ways. A solid ., means that both the system and the reservoir are in thermodynamic equilibrium, with line Ž respect to the reservoir transfer potential. A dashed line Ž — . , on the other hand, represents an intermittent Ži.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 Ž entropic formulation. , or with the evolutionary principle for the system internal energy Ž energetic formulation. w1x. 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,
J.L. de Medeiros, M.J.E.d.M. Cardosor Fluid Phase Equilibria 136 (1997) 1–13
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volume or amount of substance with its surroundings. The system is then set free to internal transfer of 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 Ž and external. constraints placed upon the system. Mathematically, this extremum principle can be expressed by the following equations: NS
Ý dUi s 0
Ž3.
is1 NS
Ý dVi s 0
Ž4.
is1 NS
Ý d Nki s 0 Žfor k s 1, . . . ,NC.
Ž5.
is1 NS
Ý d Si G 0
Ž6.
is1
where the i and k indexes run over the number of subsystems ŽNS. and the number of chemical components ŽNC. , respectively. At the equilibrium, the inequality Eq. Ž 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 or amount of substance with its surroundings. The system is then set free to internal transfers of extensive properties, among its constitutive subsystems, letting the total system internal energy decrease. In this case, the system total entropy is fixed to its initial value by means of an intermittent connection with a heat reservoir, in a way that it is allowed the draining from the system, of any created entropy. When the thermodynamic equilibrium is achieved, the system internal energy attains its minimum possible value in accordance with the internal Ž and external. constraints placed upon the system. Mathematically, this extremum principle can be expressed by the following equations: NS
Ý d Si s 0
Ž7.
is1 NS
Ý dVi s 0
Ž8.
is1 NS
Ý d Nki s 0 Žfor k s 1, . . . ,NC.
Ž9.
is1 NS
Ý dUi F 0
Ž 10.
is1
where, as for the previous case, the i and k indexes run over the number of subsystems ŽNS. and the number of chemical components Ž NC. . At the equilibrium, the inequality Eq. Ž 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. In the 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 Žthermal, mechanical or
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J.L. de Medeiros, M.J.E.d.M. Cardosor Fluid Phase Equilibria 136 (1997) 1–13
chemical. . In the latter case, however, the total system entropy cannot be fixed by means of any appropriated wall or equilibrium connection. An intermittent connection with a cold heat bath must be employed. It must be remarked, in passing, that for both entropic Ž Eqs. Ž 3. – Ž 6.. and energetic formulations ŽEqs. Ž 7. – Ž 10.. , the system of equations presented above are restricted to non-reacting systems. For the reacting system case, Eq. Ž 5. and Eq. Ž 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 Ži.e., Legendre transforms of internal energy. , and Massieu functions Ž i.e., Legendre transforms of entropy.. Table 2 presents the evolutionary principles for some examples of macrosystems Ž system plus reservoirs. configurations. Each line of this table corresponds to a specific macrosystem configuration which is identified by the line number. The first column shows a schematic representation of the macrosystem, which includes the system Žwhich can be either single or a composite one. , the reservoirs, the type of connections Ž equilibrium or intermittent. , and the type of enclosure for the whole macrosystem. We adopt, as a representation for an isolated and a constant entropy macrosystem, 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 Ž solid line enclosure., one must apply the maximum entropy principle Ž i.e., entropic formulation. . For an isentropic macrosystem Ždashed line enclosure. , on the other hand, one must apply the minimum energy principle Ži.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 Ž 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 Ž 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 Ž lines 11 to 20. describes a macrosystem and reservoir opposite representations. For instance, for the macrosystem Ž 11. , we have adopted entropic formulation together with the energetic formulation of a heat reservoir. Finally, the last two entries of Table 2 Ž lines 21 and 22. correspond to two different cases in which we 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 given macrosystem evolution and reservoir equations. For the macrosystem Ž5., we derive from Table 2: d S q d Sa q d Sc G 0 by substituting the two reservoir equations in Eq. Ž11. , one obtains 1 P d S q dUa q dVc G 0 T T
Ž 11.
Ž 12.
J.L. de Medeiros, M.J.E.d.M. Cardosor Fluid Phase Equilibria 136 (1997) 1–13
Table 2 The evolutionary principles for some examples of macrosystem configurations
7
8
Table 2 Žcontinued.
J.L. de Medeiros, M.J.E.d.M. Cardosor Fluid Phase Equilibria 136 (1997) 1–13
J.L. de Medeiros, M.J.E.d.M. Cardosor Fluid Phase Equilibria 136 (1997) 1–13
Table 2 Žcontinued.
9
J.L. de Medeiros, M.J.E.d.M. Cardosor Fluid Phase Equilibria 136 (1997) 1–13
10
Table 2 Žcontinued.
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,
ž
d Sy
U
PV y
T
T
/
G0
Ž 13.
which is the system evolution equation. This evolutionary principle is associated with a specific set of system properties Ž T, PrT, N ., which must be held constant during the system evolution towards the equilibrium state. If Eq. Ž13. is multiplied by the negative of the system constant temperature ŽyT ., one derives, d Ž U y TS q PV . s dG F 0
Ž 14.
where G stands for the Gibbs free energy, if one uses its definition equation w1,2x. Eq. Ž14. represents the system evolution equation which would be obtained by applying an analogous derivation for the macrosystem Ž6. . Therefore, macrosystems Ž5. and Ž 6. are called conjugated macrosystems. Other pairs of macrosystems of Table 2 are also conjugated in this sense.
J.L. de Medeiros, M.J.E.d.M. Cardosor Fluid Phase Equilibria 136 (1997) 1–13
11
The system evolution equations obtained for the macrosystems Ž 11. to Ž 20. are the same as the ones obtained for the first set of 10 entries of Table 2, except for the macrosystems Ž 13. and Ž 14. . For the macrosystem Ž13., we have from Table 2: d S q d Sd G 0
Ž 15.
but, d Sd s 0, thus, dSG0
Ž 16.
which is the desired system evolution equation. But, we also obtain, from the internal energy constraint, and the volume reservoir equation, dU y PdVd s 0
Ž 17.
by means of the macrosystem volume constraint, dV s ydVd , and taking into account the equilibrium connection between the system, and the volume reservoir, we can rewrite Eq. Ž17. as follows, d Ž U q PV . s d H s 0
Ž 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 Ž 21. and Ž 22. . First of all, it should be mentioned that two last system evolution equations are the same as can be obtained for the macrosystems Ž3. and Ž4., respectively. For the macrosystem Ž 21. , the intermittent connection is used to keep the system internal energy constant Ž dU s 0.. For the macrosystem Ž22., we have from Table 2: d S q d S b q d Sd G 0
Ž 19.
but we have d Sd s 0 and also d S s 0, being the last equality due to the intermittent connection between the system and the heat reservoir b, thus, d S b Gs 0
Ž 20.
from the heat reservoir equation, one obtains, dUb Gs 0
Ž 21.
from the macrosystem internal energy constraint one obtains, ydUb s dU q dUd Fs 0
Ž 22.
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. Ž22. the system evolution equation, d Ž U q PV . s d H Fs 0
Ž 23.
The set of the system properties which should be kept constant during the system evolution to the equilibrium are: P, S, N.
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J.L. de Medeiros, M.J.E.d.M. Cardosor Fluid Phase Equilibria 136 (1997) 1–13
5. Final remarks In the present article, we have presented a new and thermodynamically consistent way of obtaining equilibrium criteria for both thermodynamic potentials and Massieu functions. The proposed approach has 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 besides the mechanical Žvolume. work: gravitational work, surface work, etc. w6x. List of symbols a H G N N
Helmholtz free energy Enthalpy Gibbs free energy Amount of a substance of a chemical component Set of the amount of substance for the system
Chemical components NC NS P S T U V
Number of chemical components Number of subsystems Pressure Entropy Temperature Internal energy Volume
Greek letters m
Chemical potential
Subscripts a b c d e f i l,k
Heat reservoir Ž entropic formulation. Heat reservoir Ž energetic formulation. Volume reservoir Ž entropic formulation. Volume Reservoir Ženergetic Formulation. Amount of k th component Ž entropic formulation. Amount of k th component Ž energetic formulation. Subsystem Chemical component
References w1x H.B. Callen, Thermodynamics and an Introduction to Thermostatics, 2nd edn., Wiley, NY, 1985. w2x M.W. Zemansky, R.H. Dittman, Heat and Thermodynamics, An Intermediate Textbook, 6th edn., McGraw-Hill, NY, 1981.
J.L. de Medeiros, M.J.E.d.M. Cardosor Fluid Phase Equilibria 136 (1997) 1–13
13
w3x L.D. Landau, E.M. Lifshitz, Statistical Mechanics, 2nd edn., Pergamon Press, Oxford, 1969. w4x L. Tisza, Generalized Thermodynamics, MIT Press, Cambridge, 1966. w5x J. de Heer, Phenomenological Thermodynamics With Applications to Chemistry, Prentice-Hall, Englewood Cliffs, NJ, 1986. w6x R.A. Alberty, Legendre transforms in chemical thermodynamics, Chem. Rev. 94 Ž1994. 1457–1482.
On the derivation of thermodynamic equilibrium criteria Jose´ Luis de Medeiros
a, )
, Marcio Jose´ Estillac de Mello Cardoso ´
b,1
a
Escola de Quımica da UFRJ, Departamento de Engenharia Quımica, CT Bloco E, Cidade UniÕersitaria ´ ´ ´ 21949-900, Rio de Janeiro, RJ, Brazil b Instiuto de Quımica da UFRJ, Departamento de Fısico-Quımica, CT Bloco A, Sala 408, Cidade UniÕersitaria ´ ´ ´ ´ 21949-900, Rio de Janeiro, RJ, Brazil
Abstract It is well established that, besides the two thermodynamic formulations based on the fundamental equations Ži.e., entropic formulation S s SŽU, V, N .; Energetic Formulation U s UŽ S, V, N .., the equilibrium states of a system can be represented by equivalent formulations derived by making Legendre transforms of either of the fundamental equations wH.B. Callen, Thermodynamics and an Introduction to Thermostatics, 2nd edn., Wiley, NY, 1985x. 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 Ži.e., Legendre transforms of internal energy. and Massieu functions Ži.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 configurations of interaction with its surroundings is accomplished with the application of evolutionary principles. Therefore, if one knows that some property will increase Ž or decrease. while the system moves toward equilibrium, under some kind of interaction with its surroundings, and that property will reach a stationary value at the equilibrium condition, then it is possible to associate a variational principle to it w1x. 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.
) 1
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 Ž 9 7 . 0 0 1 3 3 - 7
2
J.L. de Medeiros, M.J.E.d.M. Cardosor Fluid Phase Equilibria 136 (1997) 1–13
One can easily find in existing thermodynamic textbooks Ž e.g., w1–5x. the derivation of the different thermodynamic potentials and Massieu functions from the internal energy and entropy functions. This derivation can be elegantly performed by the application of the Legendre transform technique w1,4,6x. Nevertheless, the derivation of the correspondent equilibrium criteria is, in general, only alluded to or developed in a less rigorous fashion. For instance, Callen w1x adopts a so-called virtual change in the 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 generic system in terms of its state coordinates. In Section 2, some basic concepts are reviewed and some new definitions are presented. Section 3 develops the evolutionary equilibrium criteria for both the entropic and energetic formulations. Section 4 deals with the application of the same approach to thermodynamic potentials Ž considered as Legendre transforms of the internal energy w1x. and Massieu functions Ž considered as Legendre transforms of the entropy w1x.. 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 present article, some fundamental definitions are reviewed w1,2x and some new ones are also introduced. In the following sections we shall need to use these concepts which will always have the same precise role in all analyzed thermodynamic systems. The first of these objects are the system boundaries, or walls. The boundaries exist to define a physical 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 Ž 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, Žlike 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 Ž for volume transfer. , diathermic boundaries Ž for heat transfer. , component permeable boundaries Ž for transfer of heat and only of a certain kind of system components. . Composed boundaries accept multiple kinds of transfer simultaneously. Associated with each transferable entity, or extensive property, is a corresponding transfer potential, or intensive property. These concepts are conjugated. The transfer of a transferable entity between the interior and exterior sides of a system is accomplished by means of a specific driving force. 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 Ž as mechanical, thermal or chemical energy. . Volume, energy as heat and amount of a substance are examples of transferable entities, with, respectively, pressure, temperature and chemical potential being the correspondent transfer potentials. In order to establish the transfer of an entity between any two systems, it is needed that both systems have boundaries appropriate to the transfer, and that both systems be linked by what we can call a suitable connection. There are two kinds of distinguishable connections: the equilibrium connection and the intermittent connection.
J.L. de Medeiros, M.J.E.d.M. Cardosor Fluid Phase Equilibria 136 (1997) 1–13
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 b c d e f
Heat Heat Volume Volume Amount of k th component Amount of k th component
1r T T Pr T P mk r T mk
Va , Na V b , Nb Uc , Nc Sd , Nd Ue , Ve , Nle Žl / k . Sf , Vf , Nlf Žl / k .
d Sa s Ž1r T .dUa dUb sT d S b d Sc s Ž Pr T . dVc dUd sy P dVd d Sl syŽ m k r T . d Nke dUf s m k d Nkf
The equilibrium connection acts instantaneously and equilibrates the connected systems with respect to the driving force characteristic of that transfer. This means that the thermodynamic equilibrium 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 thermodynamic equilibrium. The intermittent connection should be employed with the aim of ‘controlling’ a certain extensive property Ži.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 of an 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 Ž see Section 3. . Reservoirs are a special class of systems which are, in general, largely and uniformly single Ž i.e., non-composite. systems. It is considered that all the processes that take place inside a reservoir are quasi-static w1x. No irreversible processes Ž 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 in a condition, such that, it exhibits a definite value of the correspondent transfer potential, which is called the reservoir characteristic intensive parameter here. The nature and the size of the reservoirs assure that their characteristic intensive parameters Ž e.g., temperature, pressure or chemical potential. have constant and uniform values. The reservoirs are enclosed by boundaries suitable to the outward or 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 Žincluding all its subsystems. had attained equilibrium with respect to the reservoir characteristic intensive parameter. Therefore, since the reservoirs are considered very large in comparison to the system, the system itself acquires the reservoir characteristic intensive parameter as one of its state intensive 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 Ž or a set of reservoirs. . The system, single or composite, plus all possible reservoirs connected to it constitute, what we shall call, a macrosystem. Table 1 presents a list of different possible reservoirs including their main characteristics Ž e.g.,
J.L. de Medeiros, M.J.E.d.M. Cardosor Fluid Phase Equilibria 136 (1997) 1–13
4
exchanged entities, transfer potentials, etc... The reservoir operating equations are properly obtained from either the energetic or entropic differential form of the fundamental equations w1x: NC
dU s T d S y PdV q
Ý m k d Nk
Ž1.
ks1
dSs
1
ž / T
dU q
P
ž / T
NC
dV y
Ý ks1
mk
ž T /dN
k
Ž2.
where the index k runs over the number of chemical components Ž NC. , and U, S and Nk are the total internal energy, total entropy, and amount of substance of k species, respectively. We also have temperature, T, pressure, P, and chemical potential of the k species, m k . Some of the listed reservoirs ŽTable 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 the last type of reservoirs, the one in which the only exchanged entity is the amount of the k th component Žreservoir of type e or f of Table 1.. It is not physically possible to accomplish a transfer of the amount of a substance without an equivalent transfer of the amount of energy as heat. Thus, for example, when the reservoir of type e is connected Žby means of an equilibrium connection. to any system, there will be only a partial thermodynamic equilibrium between both of them. That is, we only consider that the chemical potential of the exchanged species Ž 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 any thermodynamic law or postulate. In the following sections, the system and the reservoirs will be represented by squares and their boundaries 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 Žor 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 Žtransfers. volume with a volume reservoir, by means of a mobile or flexible wall Ž the systemrreservoir connection.. As already mentioned, a system and a reservoir might be connected in two distinct ways. A solid ., means that both the system and the reservoir are in thermodynamic equilibrium, with line Ž respect to the reservoir transfer potential. A dashed line Ž — . , on the other hand, represents an intermittent Ži.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 Ž entropic formulation. , or with the evolutionary principle for the system internal energy Ž energetic formulation. w1x. 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,
J.L. de Medeiros, M.J.E.d.M. Cardosor Fluid Phase Equilibria 136 (1997) 1–13
5
volume or amount of substance with its surroundings. The system is then set free to internal transfer of 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 Ž and external. constraints placed upon the system. Mathematically, this extremum principle can be expressed by the following equations: NS
Ý dUi s 0
Ž3.
is1 NS
Ý dVi s 0
Ž4.
is1 NS
Ý d Nki s 0 Žfor k s 1, . . . ,NC.
Ž5.
is1 NS
Ý d Si G 0
Ž6.
is1
where the i and k indexes run over the number of subsystems ŽNS. and the number of chemical components ŽNC. , respectively. At the equilibrium, the inequality Eq. Ž 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 or amount of substance with its surroundings. The system is then set free to internal transfers of extensive properties, among its constitutive subsystems, letting the total system internal energy decrease. In this case, the system total entropy is fixed to its initial value by means of an intermittent connection with a heat reservoir, in a way that it is allowed the draining from the system, of any created entropy. When the thermodynamic equilibrium is achieved, the system internal energy attains its minimum possible value in accordance with the internal Ž and external. constraints placed upon the system. Mathematically, this extremum principle can be expressed by the following equations: NS
Ý d Si s 0
Ž7.
is1 NS
Ý dVi s 0
Ž8.
is1 NS
Ý d Nki s 0 Žfor k s 1, . . . ,NC.
Ž9.
is1 NS
Ý dUi F 0
Ž 10.
is1
where, as for the previous case, the i and k indexes run over the number of subsystems ŽNS. and the number of chemical components Ž NC. . At the equilibrium, the inequality Eq. Ž 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. In the 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 Žthermal, mechanical or
6
J.L. de Medeiros, M.J.E.d.M. Cardosor Fluid Phase Equilibria 136 (1997) 1–13
chemical. . In the latter case, however, the total system entropy cannot be fixed by means of any appropriated wall or equilibrium connection. An intermittent connection with a cold heat bath must be employed. It must be remarked, in passing, that for both entropic Ž Eqs. Ž 3. – Ž 6.. and energetic formulations ŽEqs. Ž 7. – Ž 10.. , the system of equations presented above are restricted to non-reacting systems. For the reacting system case, Eq. Ž 5. and Eq. Ž 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 Ži.e., Legendre transforms of internal energy. , and Massieu functions Ž i.e., Legendre transforms of entropy.. Table 2 presents the evolutionary principles for some examples of macrosystems Ž system plus reservoirs. configurations. Each line of this table corresponds to a specific macrosystem configuration which is identified by the line number. The first column shows a schematic representation of the macrosystem, which includes the system Žwhich can be either single or a composite one. , the reservoirs, the type of connections Ž equilibrium or intermittent. , and the type of enclosure for the whole macrosystem. We adopt, as a representation for an isolated and a constant entropy macrosystem, 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 Ž solid line enclosure., one must apply the maximum entropy principle Ž i.e., entropic formulation. . For an isentropic macrosystem Ždashed line enclosure. , on the other hand, one must apply the minimum energy principle Ži.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 Ž 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 Ž 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 Ž lines 11 to 20. describes a macrosystem and reservoir opposite representations. For instance, for the macrosystem Ž 11. , we have adopted entropic formulation together with the energetic formulation of a heat reservoir. Finally, the last two entries of Table 2 Ž lines 21 and 22. correspond to two different cases in which we 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 given macrosystem evolution and reservoir equations. For the macrosystem Ž5., we derive from Table 2: d S q d Sa q d Sc G 0 by substituting the two reservoir equations in Eq. Ž11. , one obtains 1 P d S q dUa q dVc G 0 T T
Ž 11.
Ž 12.
J.L. de Medeiros, M.J.E.d.M. Cardosor Fluid Phase Equilibria 136 (1997) 1–13
Table 2 The evolutionary principles for some examples of macrosystem configurations
7
8
Table 2 Žcontinued.
J.L. de Medeiros, M.J.E.d.M. Cardosor Fluid Phase Equilibria 136 (1997) 1–13
J.L. de Medeiros, M.J.E.d.M. Cardosor Fluid Phase Equilibria 136 (1997) 1–13
Table 2 Žcontinued.
9
J.L. de Medeiros, M.J.E.d.M. Cardosor Fluid Phase Equilibria 136 (1997) 1–13
10
Table 2 Žcontinued.
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,
ž
d Sy
U
PV y
T
T
/
G0
Ž 13.
which is the system evolution equation. This evolutionary principle is associated with a specific set of system properties Ž T, PrT, N ., which must be held constant during the system evolution towards the equilibrium state. If Eq. Ž13. is multiplied by the negative of the system constant temperature ŽyT ., one derives, d Ž U y TS q PV . s dG F 0
Ž 14.
where G stands for the Gibbs free energy, if one uses its definition equation w1,2x. Eq. Ž14. represents the system evolution equation which would be obtained by applying an analogous derivation for the macrosystem Ž6. . Therefore, macrosystems Ž5. and Ž 6. are called conjugated macrosystems. Other pairs of macrosystems of Table 2 are also conjugated in this sense.
J.L. de Medeiros, M.J.E.d.M. Cardosor Fluid Phase Equilibria 136 (1997) 1–13
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The system evolution equations obtained for the macrosystems Ž 11. to Ž 20. are the same as the ones obtained for the first set of 10 entries of Table 2, except for the macrosystems Ž 13. and Ž 14. . For the macrosystem Ž13., we have from Table 2: d S q d Sd G 0
Ž 15.
but, d Sd s 0, thus, dSG0
Ž 16.
which is the desired system evolution equation. But, we also obtain, from the internal energy constraint, and the volume reservoir equation, dU y PdVd s 0
Ž 17.
by means of the macrosystem volume constraint, dV s ydVd , and taking into account the equilibrium connection between the system, and the volume reservoir, we can rewrite Eq. Ž17. as follows, d Ž U q PV . s d H s 0
Ž 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 Ž 21. and Ž 22. . First of all, it should be mentioned that two last system evolution equations are the same as can be obtained for the macrosystems Ž3. and Ž4., respectively. For the macrosystem Ž 21. , the intermittent connection is used to keep the system internal energy constant Ž dU s 0.. For the macrosystem Ž22., we have from Table 2: d S q d S b q d Sd G 0
Ž 19.
but we have d Sd s 0 and also d S s 0, being the last equality due to the intermittent connection between the system and the heat reservoir b, thus, d S b Gs 0
Ž 20.
from the heat reservoir equation, one obtains, dUb Gs 0
Ž 21.
from the macrosystem internal energy constraint one obtains, ydUb s dU q dUd Fs 0
Ž 22.
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. Ž22. the system evolution equation, d Ž U q PV . s d H Fs 0
Ž 23.
The set of the system properties which should be kept constant during the system evolution to the equilibrium are: P, S, N.
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J.L. de Medeiros, M.J.E.d.M. Cardosor Fluid Phase Equilibria 136 (1997) 1–13
5. Final remarks In the present article, we have presented a new and thermodynamically consistent way of obtaining equilibrium criteria for both thermodynamic potentials and Massieu functions. The proposed approach has 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 besides the mechanical Žvolume. work: gravitational work, surface work, etc. w6x. List of symbols a H G N N
Helmholtz free energy Enthalpy Gibbs free energy Amount of a substance of a chemical component Set of the amount of substance for the system
Chemical components NC NS P S T U V
Number of chemical components Number of subsystems Pressure Entropy Temperature Internal energy Volume
Greek letters m
Chemical potential
Subscripts a b c d e f i l,k
Heat reservoir Ž entropic formulation. Heat reservoir Ž energetic formulation. Volume reservoir Ž entropic formulation. Volume Reservoir Ženergetic Formulation. Amount of k th component Ž entropic formulation. Amount of k th component Ž energetic formulation. Subsystem Chemical component
References w1x H.B. Callen, Thermodynamics and an Introduction to Thermostatics, 2nd edn., Wiley, NY, 1985. w2x M.W. Zemansky, R.H. Dittman, Heat and Thermodynamics, An Intermediate Textbook, 6th edn., McGraw-Hill, NY, 1981.
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w3x L.D. Landau, E.M. Lifshitz, Statistical Mechanics, 2nd edn., Pergamon Press, Oxford, 1969. w4x L. Tisza, Generalized Thermodynamics, MIT Press, Cambridge, 1966. w5x J. de Heer, Phenomenological Thermodynamics With Applications to Chemistry, Prentice-Hall, Englewood Cliffs, NJ, 1986. w6x R.A. Alberty, Legendre transforms in chemical thermodynamics, Chem. Rev. 94 Ž1994. 1457–1482.