^{*}

Among all statements of Second Law, the existence and uniqueness of stable equilibrium, for each given value of energy content and composition of constituents of any system, have been adopted to define thermodynamic entropy by means of the impossibility of Perpetual Motion Machine of the Second Kind (PMM2) which is a consequence of the Second Law. Equality of temperature, chemical potential and pressure in many-particle systems are proved to be necessary conditions for the stable equilibrium. The proofs assume the stable equilibrium and derive, by means of the Highest-Entropy Principle, equality of temperature, chemical potential and pressure as a consequence. A first novelty of the present research is to demonstrate that equality is also a sufficient condition, in addition to necessity, for stable equilibrium implying that stable equilibrium is a condition also necessary, in addition to sufficiency, for equality of temperature potential and pressure addressed to as generalized potential. The second novelty is that the proof of sufficiency of equality, or necessity of stable equilibrium, is achieved by means of a generalization of entropy property, derived from a generalized definition of exergy, both being state and additive properties accounting for heat, mass and work interactions of the system underpinning the definition of Highest-Generalized-Entropy Principle adopted in the proof.

The literature reports that Second Law statement can be enunciated in terms of existence and uniqueness of stable equilibrium for a given value of energy content, compatible with a given composition of constituents and compatible with a given set of parameters of any system A. This statement implies that each subsystem of a whole system has to be individually in stable equilibrium and that the composite of all subsystems mutually interacting with each other has to be in stable equilibrium as well. Stable equilibrium is proved to be a sufficient condition for equality of temperature, equality of potential and equality of pressure, or thermodynamic potentials, in many-particle systems interacting with an external reservoir R by heat, mass and work mutual exchange. Considering the inverse logical inference, the sufficiency of stable equilibrium for equality of thermodynamic potentials is equivalent to the necessity of equality of temperature, potential and pressure, inside the system and between system and reservoir, for stable equilibrium of the composite system-reservoir AR. The proof of sufficiency of stable equilibrium, or equality of thermodynamic potentials, is achieved by Gaggioli through the Lowest-Energy Principle [

The Second Law implies the definition of thermodynamic entropy property for a composite resulting from a system A and a reservoir R mutually interacting to determine the state of stable equilibrium of the composite AR [

The present analysis focuses on many-particle systems A constituted by particles interacting each other through inter-particle potential energy, here addressed to as potential, and inter-particle kinetic energy, namely temperature, determined by particles relative position and velocity respectively and constituting the system configuration at any state. The thermodynamic state of A can be global equilibrium or global non-equilibrium [

The reservoir R consists of an auxiliary system behaving at constant temperature

The method adopted is based on the canonical expression of entropy property

where

The set of necessary conditions for stable equilibrium consisting of equality of temperature

This definition of thermodynamic entropy is underpinned by equal temperature as the sole necessary condition for the stable equilibrium of AR and for this reason is based only on the constant temperature

Finally, the proof of definition of entropy property in which

The canonical definition of physical exergy property is based on the amount of heat and work interactions occurring until the system reaches the mutual stable equilibrium with the reservoir. In particular, the (thermal) exergy between two thermodynamic states

and corresponds to the maximum net useful work, obtained by means of a weight process resulting from the difference of generalized available energy between the (variable) temperature T of system A and the (constant) temperature

Physical exergy canonical definition reported in the literature [

where the term

According to the procedure reported in the literature [

(thermal) generalized available energy

The above equation of entropy is associated to, and is determined by, heat interaction of system A with the reservoir R at temperature

The expression of thermal entropy property depends solely on temperature

and entropy depending on temperature, potential and pressure as well being the inverse equation of the former:

However, the Highest-(Thermal)-Entropy Principle couldn’t be able to prove the necessity of equal potential and equal pressure since the canonical definition of (thermal) entropy does not account for the equal potential

It is noteworthy that the necessity of equal temperatures within the composite AR implies that the assumption of stable equilibrium determines solely equal temperature and not necessarily equal potential and equal pressure which, therefore, do not ensure stable equilibrium itself as, instead, is assumed. Hence, equality of potential and equality of pressure have to become necessary conditions to be satisfied jointly with equality of temperature. Once equal temperature, potential and pressure within AR are assumed as a set of necessary conditions, then this set of conditions has to be proved to be also sufficient. Therefore, these equality conditions have to be necessary and sufficient with the consequence that stable equilibrium has to be sufficient and necessary condition as well. Thus, stable equilibrium is proved to be a necessary condition if it is derived from equality of temperature, equality of potential and equality of pressure, here addressed to as equality of generalized potential, within AR composite. This bi-univocal logical inference represents the objective of this study and will be proved in following sections.

From differential standpoint, if reference is made to the canonical definition of entropy property

exact differential according to Pfaff theorem and Schwarz relation. In fact, considering a process characterized by both heat interaction and work interaction, the Gibbs equation for an ideal gas is

Stable equilibrium is proved to be a sufficient condition for equality of potential between two interacting systems, in addition to equality of temperature and pressure. The proof reported in the literature assumes stable equilibrium, for a given energy content, to derive equality of potential using the Highest-(Thermal)-Entropy Principle where entropy depends solely on

A system A with n chemical constituents is considered interacting with a chemical reservoir R assuming restricted stable equilibrium in the composite system-reservoir undergoing processes at equal temperature and pressure but different chemical potential so that mass can flow from system to the reservoir characterized by variable amount of constituents. The maximum net useful work withdrawn from A interacting with R, undergoing a process from initial state 0 to final state 1, until the system-reservoir composite AR reaches the chemical stable equilibrium state, corresponds to the chemical exergy

Kotas [

composite of system and reservoir undergoes a “mass interaction” determining a “useful work”. Mass interaction is characteristic of chemical energy transfer and it is moved by the difference of chemical potential between the system and the chemical reservoir. For this reason the expression of chemical exergy reported by Moran and

Sciubba [

tion [

A definition of chemical entropy can be derived from chemical energy and chemical exergy according to the method previously adopted and the general definition of Equation (1). To do so, if the concept of (chemical) generalized available energy is again considered, the formulation of chemical exergy should be translated into

from both chemical energy and (chemical) generalized available energy which depend on mass interaction:

properties, chemical entropy is an additive property as well and considering that the generalized reservoir allows a correlation between chemical generalized available energy and chemical exergy, then chemical entropy can be formulated as:

Chemical entropy is associated to and is determined by mass interaction between A and R at constant potential

Chemical entropy definition can be proved by means of PMM2. One could imagine a machine with a two- phase fluid operating at constant temperature between two different and constant pressures. The impossibility of PMM2 does not allow to withdrawn work interaction without net changes in the environment here constituted by the reservoir interacting with the system by mass transport. The proof, already provided by Gyftopoulos and Beretta to define the thermodynamic entropy, is general and no specific mention is made to its physical meaning, or to specific assumptions relating to the characteristics of the system, the number and type of particles, the type of potential as well as to the thermodynamic state. This is the rationale behind the generality of the theorems and proofs which, therefore, may be considered still valid also for chemical entropy here defined. This results from the fact that chemical entropy is an inherent property of any system, in any state. Therefore, the chemical entropy can be adopted to state the Highest-Chemical-Entropy Principle which can be assumed as complementary to the Lowest-Chemical-Energy Principle.

Necessity of equality of chemical potential for stable equilibrium can be proved by the Highest-Chemical- Entropy Principle which can also be adopted for the proof of sufficiency to infer chemical stable equilibrium from potential equality. This bi-directional logical implication can be worded as the necessity and sufficiency of chemical potential equality between the system and chemical reservoir, given the equality of temperature and pressure of the system-reservoir composite (restricted stable equilibrium).

Also in this case, from differential standpoint, the chemical entropy as for Equation (3) depends on the potential of reservoir

entropy

system, Gibbs equation is

theorem, it follows that

In case of adiabatic process of an open system undergoing mass interaction, Gibbs equation assumes the form

more since

for which again Schwarz relation is not valid anymore since

Assuming that

Schwarz condition being

Stable equilibrium is proved to be a sufficient condition also for equality of pressure within the AR composite in addition to equal temperature and equal potential [

The weight process represents the experimental measure of the maximum net useful work interaction

(thermal) reservoir

the maximum net useful heat

the mechanical reservoir

fore, mechanical exergy property accounts for the maximum net useful heat

system A releasing the minimum non-useful work

and considering again the meaning of generalized reservoir, then:

This relation expresses the mechanical exergy corresponding to the amount of generalized mechanical availa-

ble energy of system A converted into heat interaction

with respect to the constant temperature

the minimum amount of work interaction

process where the heat interaction is withdrawn at T_{R} from the thermal reservoir to be converted into

In fact, work interaction along the isothermal expansion process, could not be considered useful because it has to be entirely converted into heat by means of an inverse Joule cycle releasing heat (non-useful) to the thermal reservoir and non-useful work to the mechanical reservoir at a lower and constant pressure.

Thus, the maximum net useful heat

rises up to higher temperature the heat input withdrawn from the thermal reservoir constant temperature T_{R} while releasing, along the isothermal process at constant T_{R}, the minimum amount of non-useful work interaction expressed as follows [

The term

servoir, converted into high temperature heat

The definition of mechanical exergy, formulated by Equations (4a) and (4b), constitutes the basis to derive the expression of mechanical entropy using the same procedure adopted for thermal entropy and chemical entropy:

chanical generalized available energy and mechanical exergy:

The former Equation (5), substituted in the latter Equation (6), implies the expression of mechanical entropy [

This expression is a consequence of the Second Law and the stable equilibrium state in a system-reservoir composite AR.

The proof of definition of entropy provided by Gyftopoulos and Beretta [

The definition of mechanical entropy can be used to state the Highest-Mechanical-Entropy Principle applicable to those processes causing changes in volume of the system. As regard pressure equality between system and mechanical reservoir, this condition can be proved, using the Highest-Mechanical-Entropy Principle, to be a necessary condition of mutual stable equilibrium between system and reservoir that needs to be complied, in addition to equality of temperature and equality of potential, to ensure the equilibrium state of the composite system-reservoir extending to pressure the stable equilibrium restricted to temperature and potential. On the other hand, the same Highest-Mechanical-Entropy Principle can be adopted to prove the sufficiency of pressure equality or, in different terms, to prove stable equilibrium from pressure equality within system-reservoir composite. This procedure is reported in next sections.

As far as the Pfaff theorem and Schwarz relation are concerned,

tiplied by the integrating factor

from which

Although readers could consider the following clarification as obvious, it is worthy further underline that the concept of mechanical entropy here defined is correlated to the thermodynamic work only and has nothing to do with mechanical work. To better clarify, mechanical work is intended as the transfer work done by an external system behaving as an incompressible fluid, such as a liquid interacting with an hydraulic turbine whereas the thermodynamic work consists of the expansion work done by the internal system behaving as a compressible fluid, such as a vapor (steam) or a gas interacting with the expander of a gas turbine or with a cylinder-piston device.

The definition of exergy is characterized by the property of additivity because it is defined with respect to an internal subsystem of the whole system [

Generalized exergy can be regarded as the sum of generalized physical exergy, resulting from the sum of thermal exergy and mechanical exergy [

Likewise, the additivity of the entropy property is proved considering the additivity of energy and generalized exergy from which it is defined. On the basis of the additivity of the entropy property, the generalized entropy results as the sum of entropy contributions, each derived from the corresponding exergy contribution, related to the generalized potential constituted by the set of temperature, chemical potential and pressure

Generalized entropy is derivable from generalized exergy for any state of the system, equilibrium or non- equilibrium with respect to generalized reservoir having auxiliary function only. For this reason, as the method expressed in Equation (1) is adopted, it depends on the properties of both system, namely system’s generalized potential

Going back to Pfaff theorem demonstrated for each one of entropy components,

Although entropy property depends on temperature, potential and pressure (or generalized potential) by virtue of the Stable-Equilibrium-State Principle, its contributions are related to and dependent on each one of the specific interaction constituting the transfer of each specific energy, namely thermal energy, chemical energy and pressure energy, constituting the internal energy of whole system.

The generalized statement of Highest-Entropy Principle is based on the definition of generalized entropy property. Hence, it is possible to state that the Highest-Generalized-Entropy Principle results from the contribution of thermal entropy, chemical entropy and mechanical entropy as well as the property of additivity. Then the Highest-Generalized-Entropy Principle is consistent with the Highest-Thermal-Entropy Principle, Highest-Chemical- Entropy Principle and Highest-Mechanical-Entropy Principle proved for each entropy property related to heat, mass and work interaction between system and reservoir. Therefore, making reference to the statement formulated by Gyftopoulos and Beretta [

On the other hand, if all states with same value of generalized entropy are considered, and assuming the Stable-Equilibrium-State Principle as the relationship between energy and entropy as reported in previous Section 3, one can infer the Lowest-Generalized-Energy Principle implying that among all states of a system characterized by given value of entropy, number of constituents and parameters, the generalized energy of the unique stable equilibrium state is lower than the generalized energy of any other state with the same value of generalized entropy, number of constituents and parameters.

The present section describes the proof that generalized potential equality is a condition necessary and sufficient for stable equilibrium (or that stable equilibrium is a condition sufficient and necessary for generalized potential equality). This proof is included in a paper already published by the author [

According to the proof theory, deriving a thesis from a hypothesis implies the logical proof of hypothesis sufficiency and, vice versa, deriving the hypothesis from the thesis implies the logical proof of hypothesis necessity. The opposite is valid if thesis is replaced by hypothesis or, consistently, if hypothesis is replaced by thesis. The proof that equality of temperatures, potentials and pressures within the whole composite system-reservoir are necessary condition of stable equilibrium, hence “Equilibrium => Equality”, is described by Gyftopoulos and Beretta [

Thus, Gaggioli adopts the Lowest-Energy Principle and Gyftopoulos and Beretta adopt the Highest-Entropy Principle, and since entropy depends on the difference between energy and generalized available energy which express the First Law and Second Law respectively, then entropy remains the suitable quantity to attain the proofs of both sufficiency and necessity. Therefore entropy property should account for equality of potential and equality of pressure, in addition to equality of temperature, in order to constitute the procedure for such a proof. First and Second principles are mutually correlated to each other:

The proof of the necessity of stable equilibrium (or the sufficiency of generalized potential equality) consists of deriving stable equilibrium from equality and may be established through the “reductio ad absurdum” assuming that temperature, potential and pressure of system and reservoir are equal while the system-reservoir composite is not in stable equilibrium. Indeed, this condition is admitted by the sufficiency of stable equilibrium as the only

condition which does not “necessarily” imply that stable equilibrium is a consequence of equality of temperature, potential and pressure in the system-reservoir composite so that the equality may be compatible with non-equi- librium. This equality of generalized potential would thus be able to move the system into a non-equilibrium state without undergoing any net change of the environment, or, would be able to generate a weight process by means of a PMM2 which is impossible according to the Second Law statement based on stable equilibrium as assumed. The consequence is that equality must imply stable equilibrium, that is, equality must be a sufficient condition for stable equilibrium (or stable equilibrium must be a necessary condition for equality). The proof of this sufficiency can be based on the Highest-Generalized-Entropy Principle where generalized entropy depends on thermal, chemical and mechanical contributions associated to heat, mass and work interactions between system and reservoir

Finally, on the basis of the bi-univocal logical inference between stable equilibrium and generalized potential equality, and also considering the validity of Stable-Equilibrium-State Principle which correlates stable equilibrium with all properties characterizing the system in each thermodynamic state [

The result of the present study is twofold. Firstly, it consists of a proposal to encompass the Second Law statement by a bi-univocal logical inference between stable equilibrium, assumed as a hypothesis, and generalized potential equality derived as the thesis of a theorem of necessity and sufficiency of stable equilibrium for generalized potential equality in many-particle systems. The second result is that the proof is achieved by means of the definition of Highest-Generalized-Entropy Principle in which the generalized entropy is derived from generalized exergy property.

A consequence emerging from the generalized definition of entropy is that chemical entropy is an intrinsic and independent property of any system in any state and represents additional contribution, with respect to thermal entropy, determined by potential of the system itself characterized by its atomic and molecular configuration, regardless of the content of thermal entropy. Moreover, chemical entropy characterizes mass interactions of a system with a reservoir even in case no heat interactions are occurring while mass is entering or leaving the system. On the other hand, work interaction, representing the mechanical exergy, can also be correlated to the chemical exergy associated to an amount of mass interaction released to the system. As far as chemical entropy is concerned, future developments may be envisaged for atomic and molecular systems undergoing non-equili- brium phenomena determining specific geometrical configurations within the systems. In particular, Entropy Generation Minimization (EGM) methodology [

Future developments may concern further discussions about generalized entropy, characterizing heat, mass and work interactions depending on temperature, potential and pressure here restricted to many-particle systems. An extension of this generalized paradigm to definitions, properties and theorems relating to few-particle systems and non-equilibrium states will be based on, and account for, recent treatments of thermodynamic entropy property [

PierfrancescoPalazzo, (2016) A Generalized Statement of Highest-Entropy Principle for Stable Equilibrium and Non-Equilibrium in Many-Particle Systems. Journal of Modern Physics,07,344-357. doi: 10.4236/jmp.2016.73035