Stability, Gibbs condition

However, a more realistic model for the phase transition between baryonic and quark phase inside the star is the Glendenning construction [16], which determines the range of baryon density where both phases coexist. The essential point of this procedure is that both the hadron and the quark phase are allowed to be separately charged, still preserving the total charge neutrality. This implies that neutron star matter can be treated as a two-component system, and therefore can be parametrized by two chemical potentials like electron and baryon chemical potentials [if. and iin. The pressure is the same in the two phases to ensure mechanical stability, while the chemical potentials of the different species are related to each other satisfying chemical and beta stability. The Gibbs condition for mechanical and chemical equilibrium at zero temperature between both phases reads... 

The Gibbs stability theory condition may be restrictive for nonequilibrium systems. For example, the differential form of Fourier s law together with the boundary conditions describe the evolution of heat conduction, and the stability theory at equilibrium refers to the asymptotic state reached after a sufficiently long time however, there exists no thermodynamic potential with a minimum at steady state. Therefore, a stability theory based on the entropy production is more general. 

Figure S.8 Schematic phase stabilities, Gibbs energies of phase formation, and equilibrium conditions of the terminating Al and the intermetallic phases AI9C02 and AlisCou.

For thermodynamic stability, the condition (a minimum and not a maximum in the Gibbs free energy change)... 

Let us consider the vapor phase of the composition Zg at the pressure Pg > Pj. Such a phase must be stable and obey the Gibbs condition for stability For any other composition z,... 

The inverse susceptibility in Eq. 11 is proportional to the reduced temperature r and is proportional to (1/Ts - 1/T) with Ts the spinodal temperature that becomes the critical temperature Tc in the case of the critical concentration. This means that the inverse susceptibility S Ho) is positive in the one-phase regime and becomes zero at the spinodal as well as at the critical temperature. This result is consistent with the Gibbs conditions of stability which according to [9]... 

That this should be so is a corollary of the Second Law of Thermodynamics which is concerned essentially with probabilities, and with the tendency for ordered systems to become disordered a measure of the degree of disorder of a system being provided by its entropy, S. In seeking their most stable condition, systems tend towards minimum energy (actually enthalpy, H) and maximum entropy (disorder or randomness), a measure of their relative stability must thus embrace a compromise between H and S, and is provided by the Gibb s free energy, G, which is defined by,... 

Figure 6.15 Schematic cell using a stabilized zirconia electrolyte to measure the Gibbs energy of formation of an oxide MO. The cell voltage, E, is measured under open-circuit conditions when no current flows.

It is sometimes convenient to fix the pressure and decrease the degrees of freedom by one in dealing with condensed phases such as substances with low vapour pressure. The Gibbs phase rule for a ternary system at isobaric conditions is Ph + F = C + 1=4, and there are four phases present in an invariant equilibrium, three in univariant equilibria and two in divariant phase fields. Finally, three dimensions are needed to describe the stability field for the single phases e.g. temperature and two compositional terms. It is most convenient to measure composition in terms of mole fractions also for ternary systems. The sum of the mole fractions is unity thus, in a ternary system A-B-C ... 

The conditions for mechanical instability can be derived from a set of criteria for the stability of equilibrium systems put forward by Gibbs [8], Considering instability with regard to temperature and pressure, the criteria for stability are... 

In general, the first derivative of the Gibbs energy is sufficient to determine the conditions of equilibrium. To examine the stability of a chemical equilibrium, such as the one described above, higher order derivatives of G are needed. We will see in the following that the Gibbs energy versus the potential variable must be upwards convex for a stable equilibrium. Unstable equilibria, on the other hand, are... 

Other stability conditions are obtained from the negativity of second derivatives with respect to V or N. (More generally, determinants of such second derivatives must also be negative in order to guarantee stability with respect to arbitrary combinations of energy, volume, and mass changes.) In summary, we can say that the Gibbs criterion of equilibrium for a closed system is equivalent to conditions of uniform intensive properties 7, P,... 

As mentioned in the Preface, our goal in Part III is not merely to re-generate the material of Parts I and II (as summarized in Section 8.9) in new mathematical dress. We re-derive (rather trivially) many earlier thermodynamic identities and stability conditions to illustrate the geometrical techniques, but our primary emphasis is on thermodynamic extensions (particularly, to saturation properties, critical phenomena, multicomponent Gibbs-Konowalow-type relationships, higher-derivative properties, and general reversible changes... 

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