Open access peer-reviewed chapter

Thermodynamic Stability Conditions as an Eigenvalues Fundamental Problem

By Francisco Nogueira Lima

Submitted: September 9th 2020Reviewed: December 25th 2020Published: January 13th 2021

DOI: 10.5772/intechopen.95777

Downloaded: 167

Abstract

Quadratic forms diagonalization methods can be used in addressing the stability of physical systems. Thermodynamic stability conditions appears as an eigenvalues fundamental problem, in particular when postulational approaches is taken. The second-order derivatives or appropriate relations between such derivatives of the energy, entropy or any considered thermodynamic potential, as Helmholtz, enthalpy and Gibbs, have interesting mathematical features that directly imply in the physical stability, obtained by use and as consequence of analytical techniques. Formal aspects on the thermal and mechanical stability become simple consequences, but no less formal, of the superposition of rigorously established physical laws, and appropriate applications of mathematical techniques.

Keywords

  • quadratic forms
  • Taylor’s series
  • themodynamic stability
  • eigenvalues
  • thermodynamic potentials

1. Introduction

In physics, there is a time-independent theory, namely, thermodynamics that is used to determine the macroscopic equilibrium of physical systems. In practice, to compute the equilibrium conditions and the physical properties of a system, a physicist must find a function that completely describes the system, being capable of capturing all involved properties. The existence of such a function arises as a postulate of the themodynamics, having an extremum to the equilibrium states [1]. The function is called entropy and has a maximum at final equilibrium state. On the other hand, the same understanding about the physical properties of the system can be extracted through another relevant physical function, namely, energy. This treatment of using energy function instead of entropy to investigate the physical properties is completely equivalent but now the energy has a minimum and its existence also occurs by postulational reason, as for entropy function. A broad discussion on themodynamic’s postulates can be found in Ref. [1].

In practical problems, it woud be impossible to computing the total energy of a system taking all time-dependent freedom degrees, such as atomic coordinates of the components of the system each with its translational, rotation energies, etc., among others time-dependent properties. The thermodynamics theory emerges from the fact that a great number of those freedom degrees are eliminated by considering statistical averages, and not macroscopically manifesting. Thus, as the physical principle of energy conservation keeps unaltered over decades, having been already rigorously tried and confirmed, a well-defined thermodynamic energy function appears somewhat intuitive. Indeed, the energy must be interpreted as a function capable of providing the macroscopic properties of the system. Besides, due to the complexity in measuring the energy of a system, it is relevant to assume some state whereby the energy is arbitrary defined as zero and measuring the energy in connection that state because only energy differences have any physical meaning [1, 2, 3].

There are equivalent approaches to investigate the thermodynamics properties of a system in terms of thermodynamic functions (or thermodynamic potentials) of Helmholtz, enthalpy and Gibbs instead of the energy or entropy. Such thermodynamic potentials are obtained by using Legendre transformations in order to change the original extensive variables, or part of them, in the function thermodynamic energy by the intensive variables. Besides, other thermodynamic functions (in addition to those already mentioned) can appear when making Legendre transformations in specific extensive parameters of the energy or in the extensive parameters of the entropy, such as grand canonical potential, and Massieu, Planck and Krammers functions. The function to be used must be defined by the practical characteristics of the problem and these last mentioned functions are less common in more elementary approaches of postulational thermodynamics [1, 4].

A solid understanding of postulational thermodynamic theory is necessary in order to investigate the thermal or mechanic stability of the most diverse systems. The increase in the thermal stability of DNA against thermal denaturation can be experimentally investigated using a methodology in which the differences or changes in the standard values of negativity and positivity of enthalpy and entropy, or even between them, are decisive for the study’s conclusions [5]. The formalism of free energy (or Helmholtz potential) can be used for practical determination of the level of stored energy accumulated in material during plastic processing applied as well as the stored energy for the simple stretching of austenitic steel [6]. There are an infinity of applications of thermodynamic theory in wich the stability of a system is intimately related to some physical feature of thermodynamic functions, and whose the convenience of the choice is determined by practical situation.

Interesting formalisms or analytical techniques that combine the superposition of the thermodynamic theory and mathematical methods appear as support for problems of applied physics aimed to investigate the stability conditions of a system, either through experimental or computational studies. In order to show of a physical point of view, as arises the thermal and mechanical stability of a system, let us invoke the known physical origin of the energy U, i. e., its existence is determined by a postulate and the same way we know that Uis a function of the extensive parameters, entropy S, volume Vand the mole numbers of the chemical components N1, N2, …, Nr. This physical consideration can be mathematically written as U=USVN1N2Nr. Similarly, entropy Sis a function of the extensive parameters, energy U, volume Vand the mole numbers of the chemical components N1, N2, …, Nr, and so S=SUVN1N2Nr[1].

In this chapter, we discuss in details the postulate of maximum entropy or minimum energy through which it is possible to see that the thermodynamic functions Sor U, or any potential/function derived them by Legendre transformations, have mathematical features that can be obtained of an eingenvalues fundamental problem, that is, the diagonalization of the hypersurfaces defined by U=USVN1N2Nror S=SUVN1N2Nrthat conveniently expanded in Taylor’s series provides the signs its second-order derivatives in an r+2-dimensional thermodynamic space. Besides, some relations between these derivatives by diagonalization of the quadratic form of U, Sor other thermodynamic function, naturally appear and as consequence relevant conclusions about the system stability. Quadratic forms appear in several physical problems, especially in quantum mechanics [7], and in thermodynamic theory this is not different. In particular, we precisely investigate the mathematical caracteristics of the hypersurface of energy and other thermodynamic functions for a system of single chemical component. In this case, it is possible to reduce the hypersurface USVN1N2Nr, in an r+2-dimensional thermodynamic space, to a three dimensional hypersurface where U=USVN(see that r=1). Analytical calculations of quadratic forms diagonalization are used to reveal the signs of the second-order derivatives of the three-dimensional thermodynamic functions. Accordingly, the stability conditions are obtained.

This chapter is organized as follows. In Section 2, we discuss the general procedures to diagonalize the thermodynamic energy as well as obtain Talyor’s series in an r+2-dimensional thermodynamic space. It is also presented the same way to entropy function. In Section 3, we diagonalize thermodynamic energy in a three-dimensional space, and derived Helmholtz, enthalpy, and Gibbs potentials as well as grand canonical potential. In addition, the signs of second-order derivatives of such thermodynamic functions are calculated. In Section 4, stability conditions are presented as consequences of the obtained signs in previous section. As it turns, we summarize our main findings and draw some perspectives in Section 5.

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2. The quadratic form of the energy hypersurface in an r+2-dimensional thermodynamic space

We already addressed in the introduction about the postulational existence of the thermodynamic energy function U=USVN1N2Nrthat is a function on extensive parameters entropy S, volume Vand the mole numbers of the chemical components N1, N2, …, Nr, where rrepresents the amount of chemical components in the system. Besides, Uis capable of describing all thermodynamic macroscopic properties of treated system. A formal discussion on extensive parameters can be found in Ref. [1]. However, understand them as those are dependent on the amount of matter or mass of the system.

Remembering the most general form of Taylor’s series for a function f=fx1x2xnof nvariables expanded around x10x20xn0[8]:

fx1x2xn= fx10x20xn0+ifxiΔxi+12!ij2fxixjΔxiΔxj+,E1

where Δxi=xixi0, and all partial derivatives are evaluated at x10x20xn0. Here xi0denotes the coordinates of some arbitrary stationary point around which the function is expanded, with zero index to differentiate it from all other points in the n-dimensional space.

Let us carefully expanding the energy USVN1N2Nrusing Taylor’s series given by Eq. (1) around S0V0N10N20Nr0point in r+2-dimensional space.

USVN1N2Nr=US0V0N10N20Nr0+USSS0+UVVV0+k=1rUNkNkNk0+12!2US2SS02+2UV2VV02+k=1r2UNk2NkNk02+ijij2UXiXjΔXiΔXj+,E2

where ΔXiXiXi0, with Xi=S,V,N1,N2,,Nrand Xi0=S0,V0,N10,N20,,Nr0. Notice that last term that explicitly appears in Eq. (2) in wich the simplified notation Xiis introduced represents all possible combinations of double partial derivatives obtained from the extensive variables of the energy. Besides, see that ijin the same term due to already computed previous terms to i=j.

By analogy with the one-variable differential calculus and due to the postulate of minimum energy (d2U>0, see Refs. [1, 2, 3]), taking a stationary point S0V0N10N20Nr0, we know that all first-order derivatives in Eq. (2) are null at this point

US=0,UV=0andUNk=0withk=1..r,E3

and therefore

USVN1N2Nr=US0V0N10N20Nr0+12!2US2SS02+2UV2VV02+k=1r2UNk2NkNk02+ijij2UXiXjΔXiΔXj+E4

Let us define in Eq. (4)ΔSSS0, ΔVVV0, ΔNkNkNk0, ΔUUSVN1N2NkUS0V0N10N20Nr0, and also U˜2!ΔU. Thus, it is possible rewriting Eq. (4) as follows.

U˜SVN1N2Nr=2US2ΔS2+2UV2ΔV2+k=1r2UNk2ΔNk2+ijij2UXiXjΔXiΔXj+E5

Notice that U˜in above expression must be interpreted the same way as the U, being only mathematically multiplied and suppressed by the constants 2!and US0V0N10N20Nr0, respectively. Physically, U˜also obeys minimum energy postulate and keep the dependence with the extensive parameters, U˜=U˜SVN1N2Nr. On the other words, U˜is the original energy function U, at less than a multiplicative constant, and additive. We should not forget that the expression given by Eq. (5) has more terms than those explicitly listed, with third-order, fourth-order derivatives and so on. However, if we take only terms until the second-order derivatives, it is possible to see that hypersurface defined by U˜is a complete quadratic form, in an r+2-dimensional thermodynamic space (see quadratic forms in Refs. [8, 9]). Then, some mathematical generalities can be extracted of the thermodynamic energy written as Eq. (6) below:

U˜= 2US2ΔS2+2UV2ΔV2+k=1r2UNk2ΔNk2+ijij2UXiXjΔXiΔXj.E6

The matricial form of the quadratic expression in Eq. (6) is given by

U˜=ΔSΔVΔN1ΔN2ΔNr2US2.2UV22UXiXj2UN122UXjXi.2UNr2ΔSΔVΔN1ΔN2ΔNr,E7

where the second-order derivatives above and below of main diagonal represent all combinations of double partial derivatives in relation to the extensive variables of the energy. Explicitly showing the terms of mixed partial derivatives in the matricial equation given by Eq. (7), we have

U˜=ΔSΔVΔN1ΔN2ΔNr2US22USV2USN12USNr2UVS2UV22UVN12UVNr2UN1S2UN1V2UN122UN1Nr2UNrS2UNrV2UNrN12UNr2ΔSΔVΔN1ΔN2ΔNr.E8

Resuming the previous discussion in which the extensive variables are compactly defined as Xi, we can also express the energy in Eq. (8) of a compact way

U˜=ΔXiTMΔXi,E9

where

M2US22USV2USN12USNr2UVS2UV22UVN12UVNr2UN1S2UN1V2UN122UN1Nr2UNrS2UNrV2UNrN12UNr2,E10

ΔXiis a column vector with ΔS, ΔV, ΔN1, …, ΔNrcomponents, and ΔXiTis the transpose of ΔXi. As Mis a symmetric matrix, a diagonalization procedure can be applied to simplify the investigation of mathematical features of U˜and its physical consequences. At first, the choice to expanding the thermodynamic energy in Taylor’s series up to the second-order is due to the appearance of a complete quadratic form with a known mathematics of many-variable calculus. Accordingly, the canonical form U˜=ΔXi'TDΔXi'obtained by diagonalization allows visualizing interesting physical features more easily. Notice that Dis the eigenvalues matrix of Mwith r+2-components, and the ΔXi'is the column eigenvector (with ΔS', ΔV', ΔN1', …, ΔNr'components) of the diagonal matrix Das well as ΔX'Tis the transpose. A review on quadratic forms diagonalization can be found in Ref. [9]. The canonical form to U˜can be expressed by Eq. (11)

U˜=ΔXiTDΔXi=ΔSΔVΔN1ΔN2ΔNrλS00000λV00000λN10000000000λNrΔSΔVΔN1ΔN2ΔNr=λSΔS2+λVΔV2+λN1ΔN12++λNrΔNr2.E11

See that in the canonical form of U˜given by Eq. (11) are eliminated the mixed partial derivatives of Eq. (6). Besides, the minimum energy postulate imposes to the function U˜SVN1N2..Nrin Eq. (11) the following mathematical condition:

U˜=λSΔS2+λVΔV2+λN1ΔN12++λNrΔNr2>0.E12

It is possible to see that this conditon occurs only when λS>0, λV>0, λN1>0, …, λNr>0for any sets of values of ΔS, ΔV, ΔN1, …, ΔNr. To obtain the λi(i=S,V,N1,,Nr) eigenvalues, it is necessary diagonalize M(see Eq. (8)) by solving the equation λiIMXi=0, where Iis an indentity matrix (see Ref. [9]) that provides the determinant below

2US2λ2USV2USN12USNr2UVS2UV2λ2UVN12UVNr2UN1S2UN1V2UN12λ2UN1Nr2UNrS2UNrV2UNrN12UNr2λ=0.E13

Observe that Eq. (13) implies an equation in λof r+2-degree. Besides, all λiare necessarily positive due to the minimum energy postulate.

So far, we have show some generalities about thermodynamic energy in an r+2-dimensional space. Notice that diagonalizing Mby solving Eq. (13) is not an easy task. For a system with great number of chemical components analytical solutions of Eq. (13) can become increasingly hard.

If we take the entropy of the system instead of energy, all above formalism remains valid by simple exchanging Uand Svariables in the equations. In addition, due to the maximum entropy postulate, all eigenvalues of second-order derivatives matrix (similar to Mby exchanging Uand S) must be negatives (λU<0, λV<0, λN1<0, …, λNr<0). Then, in this case we have Eq. (14) instead Eq. (12).

S˜=λSΔU2+λVΔV2+λN1ΔN12++λNrΔNr2<0.E14

In a two-dimensional thermodynamic space, a discussion on the eigenvalues of Mand the physical consequences of its positivity is presented in Ref. [10]. In this case, the conditions of thermal and mechanical stability are naturally demonstrated through the signs of the second-order derivatives of some thermodynamic function of two-variables. The two-dimensional problem arises when is considered a one-component system and, in particular, we can take the thermodynamic energy per mol, reducing the dependence of such energy function for only the variables entropy (s) and volume (v) per mol (u=usv).

The stability conditions of a thermodynamic system are intrinsically related to the signs of the second-order derivatives of the energy, being the exact calculating of the eigenvalues of Eq. (13) (of previously known signs) an important factor in order to understand the physical origin of the stability of the system. In next section, we present a discussion of eigenvalues of the energy in a three-dimensional thermodynamic space.

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3. Diagonalization of the energy in a three-dimensional thermodynamic space

Let us define the energy in a three-dimensional thermodynamic space. To do this, we consider a system with one chemical component and explicitly write the energy U=USVNin terms of the involved extensive parameters S,Vand N. Similarly of Eq. (3) and by analogy with one-variable calculus, we have US=UV=UN=0(at a stationary point (S0,V0,N0)) due to the minimum energy principle. Besides, in order to investigate the second-order derivatives of U(or U˜, there are no physical difference), a simple matricial quadratic form can be obtained by application of Eqs. (6), (7) and (8), as follows:

U˜=ΔSΔVΔN2US22USV2USN2UVS2UV22UVN2UNS2UNV2UN2ΔSΔVΔN=ΔXiTM3x3ΔXi,E15

where

M3x3=2US22USV2USN2UVS2UV22UVN2UNS2UNV2UN2E16

ΔXiis a column vector with ΔS, ΔV, ΔNcomponents, ΔXiTis the transpose of ΔXi, and M3x3is a symmetric matrix that provides three eigenvalues for U˜by diagonalization of M3x3. Thus, by using the canonical form of U˜combined with minimum energy principle, we know that all signs of the eigenvalues λ1, λ2, and λ3of M3x3are positive

U˜ =ΔXiTD3x3ΔXi=ΔSΔVΔNλ1000λ2000λ3ΔSΔVΔN=λ1ΔS2+λ2ΔV2+λ3ΔN2>0.E17

Note that D3x3in Eq. (17) is the eigenvalues matrix of M3x3given by Eq. (16). As in Eq. (13), here we need solve the eigenvalues equation λI3x3M3x3Xi=0that provides the following determinant

2US2λ2USV2USN2UVS2UV2λ2UVN2UNS2UNV2UN2λ=0.E18

The determinant given by Eq. (18) provides a third-degree equation in λ.

λ3+2US2+2UV2+2UN2λ2+2US22UV2+2US22UN2+2UV22UN2+2USN2UNS+2UVN2UNV+2USV2UVSλ+2US22UV22UN2+2USV2UVN2UNS+2UVS2UNV2USN2UNV2UVN2US22UNS2USN2UV22USV2UVS2UN2=0.E19

The above equation is commonly known as characteristic equation, and its solution necessarily imply in three positive roots due to the minimum energy postulate. After some algebraic manipulations [8, 11, 12] in order to solve Eq. (19) and considering λ1>0, λ2>0and λ3>0(three positive roots), we find the following relations

2US2>0,2UV2>0,2UN2>0asexpected fromonevariable calculus,E20
2US22UV22USV2UVS>0,E21
2US22UV22UN2+2USV2UVN2UNS+2UVS2UNV2USN2UNV2UVN2US22UNS2USN2UV22USV2UVS2UN2>0.E22

Observe that Eq. (22) is equivalent to the determinant of M3x3(see Eq. (16)), being positive to energy representaion, and so M3x3>0. Besides, considering that the product of the three roots x1x2x3=d/ain a general third-degree equation ax3+bx2+cx+d=0is a known expression of more elementary courses, Eq. (22) can be easily obtained due to the positivity of all eigenvalues of U˜(see that dis the last bracket term in Eq. (19), and a=1) in the condition of minimum introduced by the thermodynamic postulate. In addition, notice that first relation in Eq. (19) is the determinant of the upper left 1x1submatrix of M3x3, while Eq. (20) is the determinant of the upper left 2x2submatrix of M3x3.

In short, to obtaining in which conditions at equilibrium point (S0,V0,N0) U˜=U˜SVNhas a minimum in this three-dimensional thermodynamic space, the set of relations given by Eqs. (20)-(22) must occur, where the relations 2UV2>0and 2UN2>0in Eq. (20) were introduced for a more physical than mathematical reason during analytical solution of Eq. (19). A general approach about mathematical second derivative test for many variable functions can be found in Ref. [8].

We must solve Eq. (19) permuting Uand Sin an equivalent entropy representation. Besides, by imposing all negative values due to maximum entropy postulate, it is possible to obtain a set of relations as in Eqs. (20)-(22). Solving eigenvalues equation below

λ3+2SU2+2SV2+2SN2λ2+[2SU22SV2+2SU22SN2+2SV22SN2+2SUN2SNU+2SVN2SNV+2SUV2SVUλ+2SU22SV22SN2+2SSV2SVN2SNU+2SVU2SNV2SUN2SNV2SVN2SU22SNU2SUN2SV22SUV2SVU2SN2=0,E23

and imposing λ1<0, λ2<0and λ3<0(all negative eigenvalues due to maximum entropy postulate), we obtain

2SU2<0,2SV2<0,2SN2<0asexpected fromonevariable calculusE24
2SU22SV22SUV2SVU>0E25
2SU22SV22SN2+2SUV2SVN2SNU+2SVU2SNV2SUN2SNV2SVN2SU22SNU2SUN2SV22SUV2SVU2SN2<0.E26

As it happened for energy, here Eq. (24) is expected from one-variable calculus and its last two relations were introduced for a more physical than mathematical reason during analytical solution of Eq. (23). It is important to emphasize that although Eq. (25) keeps the same format and sign of Eq. (21), the sign in Eq. (26) for the entropy formalism is now negative. This should not cause any surprise and can be concluded even without explicitly calculate the three eigenvalues of characteristic equation due to the known expression to the product between the three roots, x1x2x3=d/ain a general third-degree equation ax3+bx2+cx+d=0. Then, as all eigenvalues are now negative, Eq. (26) is easy verified from characteristic equation (see Eq. (23) where dis the last bracket term, and a=1). The set of Eqs. (24)-(26) provides the mathematical conditions of maximum for entropy thermodynamic function S˜=S˜UVNat U0V0N0.

Some physical problems require the use of thermodynamic potentials of Helmholtz, enthalpy and Gibbs as well as the grand canonical potential instead of thermodynamic energy to be more easy solved. These thermodynamic functions are introduced in the next topic.

3.1 Second-order derivatives of other thermodynamic functions

By using Legendre transformations, it is possible to change the extensive variables, or part of them, in the thermodynamic energy function. In this subsection, we are considering the same energy of three extensive variables defined by U=USVNin which making appropriate Legendre transformations the intensive variables are introduced. A discussion on extensive and intensive thermodynamic variables can be found in Ref. [1]. Legendre’s transformation is, in short, a process of change of variables.

3.1.1 Helmholtz potential

In order to introduce Helmholtz potential that is an energy function that instead of being a function of S, Vand Nit is written in terms of T, Vand N, we need to make Legendre transformation (change Sby T) in extensive parameter S. This process of introducing intensive parameter Tis described below. Before let us write USVNas

dUSVN=USdS+UVdV+UNdN,E27

where the temperature can be defined by TUSwith Vand Nconstant, the pressure is defined by PUVwith Sand Nconstant, and the chemical potential is defined by μUNwith Sand Vconstant. With these definitions, we have to Eq. (27)

dU=TdSPdV+μdN.E28

Taking

dTS=TdS+SdTTdS=dTSSdT,E29

and substituting Eq. (29) into Eq. (28)

dU=dTSSdTPdV+μdNdUTS=SdTPdV+μdNdF=SdTPdV+μdN,E30

where

FUTSbeingFknownasHelmholtz potential.E31

See of the Eq. (30) that Fis a function of T, Vand N. Then F=FTVN, and the energy Fdefined as function of T, Vand Nhas modified its concavite in relation to the new introduced parameter by Legendre transformation in S, i. e., the second-order derivatives of Fon Tis negative now, keeping positive the signs of Fon Vand Nas in original energy (see Eq. (32) below).

2FT2<0,2FV2>0,2FN2>0.E32

It is a general fact that Legendre transformation change the sign of the second-order derivatives of the new introduced function in relation that intensive parameter. A demonstration of this consideration to molar Helmholtz potential f=fsvis shown in Re. [10], and a treatment on Legendre transformations can be found in Ref. [13]. Recently, the thermodynamic stability of chignolin protein was theoretically investigated by using of a computational methodology of decomposition of the Helmholtz energy profile that indicates that intramolecular interactions predominantly stabilized certain conformations of the protein [14]. Besides, in the same study the direct Helmholtz energy decomposition provides the predominant factor in the thermodynamic stability of proteins.

Following the same procedure used to derive the stability conditions of the energy and entropy functions, it is possible to obtain a complete set of relations that Helmholtz potential must obey. Mathematically Fis known as a saddle surface. This feature of Fstems from the imposition that some eigenvalue of the canonical form of F(similarly to the Eq. (17)) have opposite sign to the others. The saddle surface of Helmholtz of three variables has a maximum in relation to the temperature but a minimum in relation to the volume and mole number. The relations given by Eq. (32) are sufficient to conclude on the physical stability of a system, as demonstrated in Section 4, and the other expressions to the second-order derivatives of Fare not shown here. However, the curious reader can be computing all signs of the second-order derivatives to Helmoltz and to other thermodynamic functions that follow below, as already discussed to energy and entropy functions.

3.1.2 Enthalpy potential

The enthalpy potential is also mathematically a saddle surface. In this case, Legendre transformation is applied in the extensive parameter Vand introduced the intensive parameter P. Further, Hkeep unaltered with a minimum in relation to the entropy Sand Nbut becomes a maximum on P, and so H=HSPN. Remembering that dU=TdSPdV+μdN, then

dpV=PdV+VdPPdV=dPV+VdP,E33

and substituting Eq. (33) into Eq. (28), we have

dU=TdSdPV+VdP+μdNdU+PV=TdS+VdP+μdNdH=TdS+VdP+μdN,E34

where

HU+PVbeingHknownasenthalpy potential.E35

Due to Legendre transformations, it is possible to conclude that

2HS2>0,2HP2<0,2HN2>0,E36

and other inequalities can be obtained the same way as previously presented to energy and entropy functions,i. e., by diagonalization of HSPN.

3.1.3 Gibbs potential

It is possible to write a function obtained by double Legendre transformation in the extensive parameters Sand V, namely Gibbs potential. This is a function on introduced intensive variables Tand P. To do that, we combine Eqs. (29) and (33) into Eq. (28). Then,

dU=TdSPdV+μdNdU=dTSSdTdPV+VdP+μdNdUdTS+dPV=SdT+VdP+μdNdUTS+PV=SdT+VdP+μdNdG=SdT+VdP+μdN,E37

where

GUTS+PVbeingGknownasGibbs potential.E38

Legendre transformations provide the following relations, and G=GTPNas seen in Eq. (37).

2GT2<0,2GP2<0,2GN2>0.E39

Here the second-order derivatives in relation to Tand Pare negative now as well as the G=GTPNbecomes a surface of maximum in relation of these two parameters. See that energy keeps unaltered in relation to N, and Gibbs potential has a minimum in relation to mole number because Legendre transformations are applied only in Sand V, introducing Tand Prespectively. Besides, by diagonalization of quadratic form obtained by expanding of G, it is possible to compute other inequalities in additon those expressed by Eq. (39), as already discussed to the energy and entropy formalisms.

3.1.4 Grand canonical potential

A function of T, Vand μis known as grand canonical potential J. To obtaining J=JTVμlet us introduce the intensive parameter μof the extensive parameter Nas follows. Taking

dμN=Ndμ+μdNμdN=dμNNdμ,E40

and combining the above equation with Eq. (29) into (28), we have

dU=TdSPdV+μdNdU=dTSSdTPdV+dμNNdμdUdTSdμN=SdTPdVNdμdUTSμN=SdTPdVNdμdJ=SdTPdVNdμ,E41

where

J=UTSμN=FμNbeingJknownasgrand canonical potential.E42

Thus, by Legendre transformations in Sand N, Tand μintensive variables are introduced, respectively, the relations below are naturally obtained.

2JT2<0,2JV2>0,2Jμ2<0.E43

These relations indicate that Ghas now a maximum in relation to intensive parameters Tand μ, keeping a minimum on V. Legendre transformations applied in the entropy formalism are also useful to derive other thermodynamic functions that are not treated here. The appropriate choice of the thermodynamic function is relevant in practical problems. Besides, thermodynamic functions are convex functions of their extensive variables (positive signs of the second-order derivatives) and concave functions (negative signs of the second-order derivatives) of their intensive variables [1].

Novel geometric approaches aimed at obtaining thermodynamic relations in a systematic way for a number of thermodynamic potentials and formally derived the classical Gibbs stability condition has been recently investigated [15].

So far, we demonstrate the mathematical conditions that second-order derivatives of the thermodynamic functions must satisfied. In the next section, we use these conditions to directly obtain the mechanical and thermal stability of a general system.

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4. The stability conditions of a system

Let us start this section remembering some quantities of physical interest defined below [1, 2, 3]:

α1VVTE44
cVTNSTE45
cPTNSTE46
kT1VVPE47
kS1VVP,E48

where α(at pconstant) in Eq. (44) is the coefficient of thermal expansion, cVand cPin Eqs. (45) and (46) respectively, are the specific heats at Vor Pconstant, kT(Tconstant) in Eq. (47) is the isothermal compressibility and kS(Sconstant) in Eq. (48) is the adiabatic compressibility. All these quantities are relevant in physical applications and their exact values as well as their increase or decrease tendencies can say a lot about the stability of the physical system.

The thermal expansion is related to changes in dimensions of physical systems due to temperature variations. We can understand the behavior of materials on the macroscopic or microscopic scale when subjected to temperature changes by the abosolute values of αthat can be positive or negative.

Specific heats are useful to understand the thermal properties of physical systems in several length scales (macroscale and microscale). Besides, the specific heats are positive physical quantities associated to the thermal stability of the system, as will be mathematically demonstrated in this section.

The isothermal and adiabatic compressibilities are positive physical quantities, being related to the mechanical stability of the system. A deep comprehension of the physical origin of the mentioned quantities in terms of the signs of the second-order derivatives of thermodynamis functions, it is relevant to theoretical or experimental researchers.

In order to better investigate the physical consequences of the signs of the second-order derivatives of the energy, see the first relation in Eq. (20)

2US2>0.E49

Remembering the temperature definition T=US, we have by derivation of temperature Tside by side in relation to the Sentropy

TS=2US2>0.E50

Then, if we combine Eq. (50) and the definition of specific heat (at Vconstant) given by Eq. (45), it is possible to obtain

TNcV=2US2>0cV>0.E51

A positive specific heat (cV>0) is obtained due to the absolute temperature is positive. Besides, Nis a positive amount. The same physical conclusion can be obtained of the first relation in Eq. (32), 2FT2<0. As Fis a function of T, Vand N(F=FTVN), an infinitesimal of dFis given by

dF=FTdT+FVdV+FNdNE52

that compared with Eq. (30) provides

S=FT,E53
P=FV,E54

and

μ=FN.E55

If we take the derivation side by side of Eq. (53) in relation to Tconsidering Vand Nconstant

ST=2FT2.E56

It is possible to observe that the left side of Eq. (56) is relationed to the specific heat at Vconstant and the sign of the second-order derivatives can be checked by comparing with Eq. (32), and so

ST=2FT2<0,E57

and from definition of specific heat in Eq. (45)

NcVT<0NcvT>0cv>0.E58

Note that Eq. (58) represents the same result already obtained in Eq. (51), only taking different formalisms to thermodynamic function, and so analyzing distinct second-order derivatives. The specific heat must be interpreted as the necessary amount of heat to increase or decrease the temperature of the physical system. A negative specific heat would imply in an inexistent physical situation because we would have a system capable of receiving some quantity of heat (postive) and decreasing its temperature (negative dT). There is still another non-physical situation with negative specific heat in the hypothetical situation in which the system loses heat but increases its temperature.

We investigate now the signs of second-order derivatives of Gibbs potential. The relation given by first inequality in Eq. (39) provides an important conclusion to specific heat at Pconstant, with cP>0. To demonstrate that, let us take a differential element dGof Gibbs potential G=GTPN

dG=GTdT+GPdP+GNdN.E59

The above equation can be compared with Eq. (37), and we obtain

S=GT,E60
V=GP,E61

and

μ=GN.E62

Deriving Eq. (60) side by side in relation to Tat Pconstant, we have

ST=2GT2,E63

and from definition of specific heat at Pconstant in Eq. (46) and by comparing with the first inequality in Eq. (39)

NcPT=2GT2<0NcPT>0cP>0.E64

Notice that specific heat at Pconstant is also positive. The positivity of the specific heats previous shown is related to the thermal stability of the physical system. Then, it is possible to see that the thermal stability emerge as consequence of the signs of the second-order derivatives previously treated. Thus, appropriately computing the eigenvalues of the matricial energy or other thermodynamic function is essencial to finding the stability conditions.

Resuming Eq. (54) and by derivation of the left and right sides in relation to Vkeeping Tconstant

PV=2FV2.E65

Comparing Eq. (66) with the definition to isothermal compressibility in Eq. (47), we can obtain

PV=1VkT.E66

As the sign of the second-order derivative in Eq. (66) is positive, we have

1VkT=2FV21VkT=2FV2>0kT>0.E67

Notice that the sign of the second-order derivative of the appropriately chosen potential leads to a relevant relation for the sign of physical quantity of interest. Besides, in the definition given by Eq. (47) that increments of pressure in the system leads to decrease in volume due to the ever positive isothermal compressibility, and this is an intuitive conclusion. From Eq. (67) we mathematically demonstrated that isothermal compressibility is always positive due to specific features of the potentials. In particular, the positive value of kTappears from curvature of some chosen potential. The same way kS>0can be obtained from enthalpy potential through the the sign of the second relation (2HP2<0) in Eq. (36), and after some algebraic manipulations. A positive value of this physical quantity is associated with the mechanical stability of the physical system, as in kT.

It is relevant to clarify that αdoes not to have a positive defined sign that can be obtained from some function. The well-known case of the water shows that volume increases when temperature decreases below at 4oC, being negative αin this regime. Yet, thermodynamic books [1, 2, 3] show some relations between the physical quantities, as cp=cv+TVα2/NkT, cp/cv=kT/kSas well as cpcvand kTksobtained by reduction of thermodynamic derivatives and by using Maxwell’s relations. But this is not the purpose of this chapter.

It is worthy of emphasis that some stability condition can be deduced by the signs of the second-order derivatives of energy (or any thermodynamic function), as presented in this chapter. In a three-dimensional (or higher) thermodynamic space the complexity in obtain with success the stability conditions for some potential is associated to the matrix order of the second-order derivatives. Besides, to all cases one or several second-order relations must be manipulated to conclude about the thermal and mechanical stability of the system.

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5. Conclusions

In this chapter, we show the useful of specific linear algebra topics in addition with many-variable calculus that coupled to minimum energy postulate appear as in important insight to understand the stability of thermodynamic systems. We find the thermal and mechanical stability of physical systems are directly associated with the signs of the second-order derivatves of thermodynamic energy or other taken representation.

We present a general addressing to the energy representation in terms of matrial equations whereby the stability conditions arise of an eigenvalues fundamental problem. Besides, the minimum energy postulate provides the signs of the second-order derivatives. Accordingly, of a physical point of view the stabilility of a system occurs due to minimum energy postulate.

Formal caracteristics of postulational thermodynamic theory and, particularly, about the second-order derivatives of the thermodynamic functions are discussed with relevant consequences on the thermal and mechanical stability. The presented analytical formalism is an important support to conclude how the stability of a system arises, and can be useful in any field of the exact sciences. We hope that this methodology can be extended to higher-order matrices of energy as well as some of the obtained relations can be used in specific problems of applied physics.

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Acknowledgments

The author gratefully acknowledges the support provided by Brazilian agencies CAPES e CNPq. I would like to thank the following for their kind support: Instituto Federal do Piauí, Campus São Raimundo Nonato; and friend and colleague Israel A. C. Noletto for the private messages that contributed to the writing of this text.

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Francisco Nogueira Lima (January 13th 2021). Thermodynamic Stability Conditions as an Eigenvalues Fundamental Problem, Recent Developments in the Solution of Nonlinear Differential Equations, Bruno Carpentieri, IntechOpen, DOI: 10.5772/intechopen.95777. Available from:

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