Stable equilibrium: definition

As soon as we had shown that is an available energy, from the definition of 39 and equation (13a), we could at once have inferred the relations (10)—(12) from the principle of dissipation of energy, for V must be a minimum in stable equilibrium. 

Recalling the classic definition of the triple point, say, for water as the intersection of the solid-vapour and the liquid-vapour curves, the analogy in Fig. 2.7 is the intersection of the II<->v. and I<->v. curves. Below the triple point only one of the solid phases (I) can exist in stable equilibrium with the vapour above the triple point only II... 

Note that the definition of stable equilibrium is based on small disturbances certain large disturbances may fail to decay. In Example 2.2.1, all small disturbances to X = -1 will decay, but a large disturbance that sends x to the right of X = 1 will not decay—in fact, the phase point will be repelled out to -l-oo. To emphasize this aspect of stability, we sometimes say that x = -1 is locally stable, but not globally stable. 

These definitions are motivated by the stable-equilibrium postulate, and their importance will become evident from the subsequent discussion. 

EquiUbrium between Solid, Liquid and Vapour. The Triple Point.—From the Phase Rule, F = n + 2 — r, it follows that when oaie component is present in three coexisting phases, the system is invariant. Such a system can exist in stable equilibrium only at one definite temperature and one definite pressure. This definite temperature and pressure at which three phases coexist in equilibrium, as an invariant system, is called a triple point. Although the commonest triple point in a one-component system is the triple point, solid, liquid, vapour (S—L—V), other triple points are also possible when, as in the case of ice, sulphur, and other substances, polymorphic forms occur. Whether or not all the triple points can be experimentally realised will, of course, depend on circumstances. We shall, in the first place, consider the triple point S—L—Y. 

The curve AC represents the composition of solutions which are in equilibrium, at different temperatures, with the solid component A and the curve BC, similarly, the composition of solutions in equilibrium with solid B. At the point C, where the two curves cut, both solid components can exist in equilibrium with a liquid solution of definite composition, corresponding with the point C. Point C gives the conditions of temperature and composition of the liquid phase under which the system —L can exist in stable equilibrium under constant... 

Example Spatial Oscillator.—A massive particle is restrained by any set of forces in a position of stable equilibrium (t.g. a light atom in a molecule otherwise consisting of heavy, and therefore relatively immovable atoms). The potential eneigy is then, for small displacement, a positive definite quadratic function of the displacement components. The axes of the co-ordinate system (x, y, z) can always be chosen to lie along the principal axes of the ellipsoid corresponding to this quadratic form. The Hamiltonian function is then... 

Although, as we have seen, each system has a minimum number of independent variables that must be specified or fixed for the system to be at stable equilibrium, systems frequently have more than this number of variables fixed, and they can then be said to be in metastable equilibrium states. In fact our definition of a metastable state can be restated as one that has more than the minimum number of constraints necessary to fix the equilibrium state. To illustrate what we mean by a metastable state, and the wide-ranging nature of the definition, we consider next three examples. 

We have just defined reversible processes in terms of the stable equilibrium surface, but completely analogous processes are also possible on metastable equilibrium surfaces, and the definition can be extended to include these. In fact, however, most discussions of reversible processes refer to stable equilibrium states and surfaces. 

What does it mean to say that our physical properties must be single-valued It means that our thermodynamic functions (such as equation (3.1)) can deal only with systems at equilibrium where, according to our definition, the properties of the system do not change with time. This equilibrium state may be stable or metastable, but generally speaking for systems more complex than a single component (e.g. a pure mineral) stable equilibrium states are almost always referred to." ... 

We must choose a system capable of having states other than the stable equilibrium state, preferably an infinite number of them, and the simplest possible one is probably the gas and piston arrangement previously used in Chapter 3 and repeated in Figure 5.2. The exterior wall is impermeable to energy and rigid, so the system is of constant U and V. The piston is movable and can be locked in any position. It is impermeable to the gas but it conducts heat so that the two sides are at the same temperature. According to our definition of S, the equilibrium position of the piston (that is, when the system has no additional constraints and the piston is free to move) is one of maximum entropy for the system, and any other position has lower entropy. If the two sides have equal amounts of the same gas, the curve illustrating this will be symmetrical. 

The surfaces in Figure 5.4 must not be confused with the stable equilibrium surface. Only the locus of points at maximum S or minimum U such as A are at stable equilibrium and this locus forms a contour on the stable equilibrium surface as shown in Figure 5.5. All the other points on the surfaces in Figure 5.4 are metastable according to our definitions and would appear at higher U and lower S values having the same total volume V, such as point B in Figure 5.5. 

According to the definition of Section LA, each solid-state polymorphic form constitutes a separate phase of the component. The phase rule can be used to predict the conditions under which each form can coexist, either along or in the presence of the liquid or vapor phases. One immediate deduction is that since no stable equilibrium can exist when four phases are simultaneously present, two polymorphic forms cannot be in equilibrium with each other and be in equilibrium with both their solid and vapor phases. When the two crystalline forms (denoted S, and Sj) are in equilibrium with each other, then the two triple points (Sj-Sj-V and S,-S2-L) become exceedingly important. 

Now we consider an arbitrary process p passing arbitrary states between two such states—initial a, and final ct/. Choosing the (stable equilibrium) reference state ao and a process p, from cto to oi, we can regard the process p, followed by process p as a combined process connecting ao and a/. Therefore it follows from the definition (1.31) for Sf (entropy in the state cry with ao as a reference state) and postulate S2 that... 

We now demonstrate nonuniqueness of the entropy 5 (1.31) of the state a which was defined relative to the reference (stable equilibrium) state ao. Choosing another (stable equilibrium) state, say a, as the reference state, then the same state a will have the (new) entropy S by (1.31). Denoting by po a process from oro to or and by p a process from a to a, we have by definition (1.31) and by combination of the processes (S2 in Sect. 1.2)... 

Definition of equilibrium is motivated similarly as in Sects. 1.2, 2.1, 2.2 and 3.8 [39, 52, 53, 56, 79, 98, 142, 143] (for non-linear models, see, e.g. [60, 71, 72]). For the regular linear fluid mixture model summarized at the end of previous Sect. 4.6, we define equilibrium by zero entropy production (4.301) as an equilibrium process going persistently through a unique equilibrium state, which is possible, as we shall see, if the body heat source is zero (4.303) and at zero rates of chemical reactions (4.302). By regularity conditions (see 1,2,3 at the end of Sect. 4.6), we exclude some unusual processes compatible with zero entropy production. We apply the regularity conditions on equilibrium states (moreover, regularity condition 3 follows for stable equilibrium states which will be discussed later in this Sect. 4.7). 

In thermodynamics, a metastable equilibrium state has at least three constraints. Two of these constraints apply to a stable equilibrium state, and the third prevents the system from achieving that state. On releasing the third constraint the system experiences a spontaneous process and achieves the stable equilibrium state. We have seen two examples so far, in Figures 4.1 and 4.6. These examples were chosen to follow from our definition of entropy, and show spontaneous processes having no overall energy change in the system. They show entropy acting as a thermodynamic potential. 

But we don t have to use Euler s theorem. We can simply expand our definition of G, which so far is restricted to closed (constant composition) systems. If we exclude chemical work, which means we deal only with systems at complete stable equilibrium, we know from Equation (4.65)... 

The definition of solubility permits the occurrence of a single solid phase which may be a pure anhydrous compound, a salt hydrate, a non-stoichiometric compound, or a solid mixture (or solid solution, or "mixed crystals"), and may be stable or metastable. As well, any number of solid phases consistent with the requirements of the phase rule may be present. Metastable solid phases are of widespread occurrence, and may appear as polymorphic (or allotropic) forms or crystal solvates whose rate of transition to more stable forms is very slow. Surface heterogeneity may also give rise to metastability, either when one solid precipitates on the surface of auiother, or if the size of the solid particles is sufficiently small that surface effects become important. In either case, the solid is not in stable equilibrium with the solution. See (21) for the modern formulation of the effect of particle size on solubility. The stability of a solid may also be affected by the atmosphere in which the system is equilibrated. 

Scientifically described for the first time in 1943 by Hoar and Schulman (2), the latter author coined the term microemulsion in 1959 to describe these optically isotropic transparent oil and water dispersions (3). Since this early work, many experimental and theoretical efforts have shown that these dispersions are actually solutions, namely thermodynamically stable equilibrium phases (4). Consequently, the most widely, but still not universally accepted definition of a microemulsion is that of a thermodynamically stable mixture of oil and water. Occasionally, the term microemulsion (5) or miniemulsion (6) is used to describe long-lived emulsions with ultra-small droplet sizes (30-100 nm). Sometimes, stable emulsions may be created by agitation of systems while passing through regions of the phase diagram where microemulsion phases form however, the final state is in the emulsion region (7, 8). In this present chapter, we use the most widely accepted definition of microemulsions, namely equilibrium phases of oil and water (9). 

The mathematical statement of the second law is associated with the definition of entropy S, dS = 8q /T. Entropy is a thermodynamic potential and a quantitative measure of irreversibility. For reversible processes, dS is an exact differential of the state function, and the result of the integration does not depend on the path of change or on how the change is carried out when both the initial and final states are at stable equilibrium. The entropy of a closed adiabatic system remains the same in a reversible process, and increases during an irreversible process. A system and its surrounding create an isolated composite system where the sum of the entropies of all reversible changes remains the same, and increases during irreversible processes. 

If the difference in concentration of the measured ion between the inner solution and the sample solution is too large (> 3 or 4 orders of magnitude), then deviations from the Nernst equation can be detected due to a diffusion potential inside the ion-selective membrane [68]. For more accurate measurements in such cases, the concentration of the inner fQling solution is adjusted to roughly match that of the sample solution. Most manufacturers use a 10 to 10" M inner solution. If the inner solutions is changed, one must remember to add a definite amount of chloride ion for a stable equilibrium Galvani potential at the Ag/AgCl shunt element. 

The most widely used reference electrode, due to its ease of preparation and constancy of potential, is the calomel electrode. A calomel half-cell is one in which mercury and calomel [mercury(I) chloride] are covered with potassium chloride solution of definite concentration this may be 0.1 M, 1M, or saturated. These electrodes are referred to as the decimolar, the molar and the saturated calomel electrode (S.C.E.) and have the potentials, relative to the standard hydrogen electrode at 25 °C, of 0.3358,0.2824 and 0.2444 volt. Of these electrodes the S.C.E. is most commonly used, largely because of the suppressive effect of saturated potassium chloride solution on liquid junction potentials. However, this electrode suffers from the drawback that its potential varies rapidly with alteration in temperature owing to changes in the solubility of potassium chloride, and restoration of a stable potential may be slow owing to the disturbance of the calomel-potassium chloride equilibrium. The potentials of the decimolar and molar electrodes are less affected by change in temperature and are to be preferred in cases where accurate values of electrode potentials are required. The electrode reaction is... 

The vertical spring and mass is an example of a stable system and by definition this means that an arbitrary small external force does not cause the mass to depart far from the position of equilibrium. Correspondingly, the mass vibrates at small distances from the position of equilibrium. Stability of this system directly follows from Equation (3.102) as long as the mechanical sensitivity has a finite value, and it holds for any position of the mass. First, suppose that at the initial moment a small impulse of force is applied, delta function, then small vibrations arise and the mass returns to its original position due to attenuation. If the external force is small and constant then the mass after small oscillations occupies a new position of equilibrium, which only differs slightly from the original one. In both cases the elastic force of the spring is directed toward the equilibrium and this provides stability. Later we will discuss this subject in some detail. 

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