The Gibbs stability theory

Stable equilibrium has minimum Gibbs free energy. The necessary (but not sufficient) condition is that the first derivative of the Gibbs energy is zero at the possible equilibrium states, and the second derivative of the Gibbs energy is positive 

Here x is a parameter that characterizes the state of the system (for example, the concentration of a gas). Stability in equilibrium plane requires that the second derivative of the entropy be negative 

The Gibbs stability theory provides necessary and sufficient conditions to investigate stability problems with well-defined boundary conditions in equilibrium state. Some examples of this are 

Fluctuations in thermodynamic properties determine the entropy change, which can be expanded in certain fluctuations. For an isolated system, the power series expansion of entropy in terms of fluctuation x is 

For a system with parts 1 and 2, consider a flow of energy dU from part 1 causing a small fluctuation in temperature 8T. Expansion of the total entropy of the parts (S) in terms of Ux and U2 yields 

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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. 

Dipole moments of solutes are also involved in the so-called reaction-field theory1112 which predicts generally the Gibbs energy of solvation, and from it the stability of con-formers as dependent on solvents. Besides the dipole the quadrupole moment is also taken into the calculations. For instance, conformational equilibria of cyclic halo ketones were predicted from the dipole moments of the two conformers with fair success13. However, the whole theory was criticised1415 that there is too much arbitrariness in the... 

Dobrowolski and co-workers reported calculations of the thermodynamic stability of the products derived from cycloaddition of vinylimine and ketene and of isocyanic acid and allene using MP2, density functional theory (DFT) (B3PW91), and Hartree-Fock (HF) methods with the 6-311+- -G basis set. Fourteen different structures were calculated for these cycloaddition reactions and the values of the free Gibbs energy showed that only 4- and 3-methylene-/3-lactams could be formed in considerable amount. 

One of the most important applications of the theory of PS is to biomolecules. There have been numerous studies on the effect of various solutes (which may be viewed as constituting a part of a solvent mixture) on the stability of proteins, conformational changes, aggregation processes, etc., (Arakawa and Timasheff 1985 Timasheff 1998 Shulgin and Ruckenstein 2005 Shimizu 2004). In all of these, the central quantity that is affected is the Gibbs energy of solvation of the biomolecule s. Formally, equation (8.26) or equivalently (8.28), applies to a biomolecule s in dilute solution in the solvent mixture A and B. However, in contrast to the case of simple, spherical solutes, the pair correlation functions gAS and gBS depend in this case on both the location and the relative orientation of the two species involved (figure 8.5). Therefore, we write equation (8.26) in an equivalent form as ... 

We note in passing that the Vrij theory, as modified by introducing the Gibbs elasticity, is an extension of Gibbs explanation of film stability (Section 10.7). 

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