Equilibrium stable state

Results Equilibrium states Stable singular points... 

This true or idealized energy minimum is also sometimes referred to as the equilibrium state, stable or metastable, and discussions can become confused unless the distinction between the macroscopic, practical kind of equilibrium that we have defined and the more idealized, conceivable kind of equilibrium state is made clear. 

The most general term for a change in H is simply A//. This refers to any change in the enthalpy of any system between two equilibrium states (stable or metastable), not necessarily associated with a chemical reaction. A special case is the A// between the products and reactants of a chemical reaction, called A,.H, so this represents a subset of the more general term LH. A special kind of chemical reaction involves only pure compounds, whose thermodynamic parameters can be found in tables, and so a subset of all A,.// values can be called A,.//°, to indicate that all products and reactants are in their pure reference states. A special case of is the reaction in which a compound is formed from its elements, all in their pure reference states, and this is called fH°. Finally, we found that there are two conventions for defining the enthalpy of formation from the elements, one being the traditional or common sense method, where the compound and all its elements are at... 

Final equilibrium state. Stable, thick film. c > c.m.c. 

This is a one-dimensional system which may have stable and unstable equilibrium states corresponding to stable and saddle equilibrium states of the entire system (12.4.6) or (12.4.7). The evolution along Meq is either limited to one of the stable points, or it reaches a small neighborhood of the critical values of X. Recall, that we consider x as a governing parameter for the fast system and critical values of x are those ones which correspond to bifurcations of the fast system. In particular, at some x two equilibrium states (stable and saddle) of the fast system may coalesce into a saddle-node. This corresponds to a maximum (or a minimum) of x on Meq, so the value of x cannot further... 

A homogeneous metastable phase is always stable with respect to the fonnation of infinitesimal droplets, provided the surface tension a is positive. Between this extreme and the other thennodynamic equilibrium state, which is inhomogeneous and consists of two coexisting phases, a critical size droplet state exists, which is in unstable equilibrium. In the classical theory, one makes the capillarity approxunation the critical droplet is assumed homogeneous up to the boundary separating it from the metastable background and is assumed to be the same as the new phase in the bulk. Then the work of fonnation W R) of such a droplet of arbitrary radius R is the sum of the... 

The forces in a protein molecule are modeled by the gradient of the potential energy V(s, x) in dependence on a vector s encoding the amino acid sequence of the molecule and a vector x containing the Cartesian coordinates of all essential atoms of a molecule. In an equilibrium state x, the forces (s, x) vanish, so x is stationary and for stability reasons we must have a local minimizer. The most stable equilibrium state of a molecule is usually the... 

A plate buckles when the in-plane compressive load gets so large that the originally flat equilibrium state is no longer stable, and the plate deflects into a nonflat (wavy) configuration. The load at which the departure from the flat state takes place is called the buckling load. The flat equilibrium state has only in-plane forces and undergoes only ex-... 

So far we have discussed mainly stable configurations that have reached an equilibrium. What about the evolution of a system from an arbitrary initial state In particular, what do we need to know in order to be assured of reaching an equilibrium state that is described by the Boltzman distribution (equation 7.1) from an arbitrary initial state It turns out that it is not enough to know just the energies H ct) of the different states a. We also need to know the set of transition probabilities between ail pairs of states of the system. 

This is concerned with the fact that in the case of the relaxation time, roughly speaking only half of all Brownian particles should leave the initial potential minimum to reach the equilibrium state, while for the profile of the decay time case all particles should leave the initial minimum. Expression (5.120), of course, is true only in the case of the sufficiently large potential barrier, separating the stable states of the bistable system, when the inverse probability current from the second minimum to the initial one may be neglected (see Ref. 33). 

As with experimental work on polymer adsorption, experiments in the area of dispersion stability in the presence of polymers require detailed characterisation of the systems under study and the various controlling parameters (discussed above) to be varied in a systematic way. One should seek the answer to several questions. Is the system (thermodynamically) stable If not, what is the nature of the equilibrium state and what are the kinetics of flocculation If it is stable, under what critical conditions ( s, T, x> p etc.) can flocculation be induced ... 

Fig-1 Stable equilibrium state (left) given by molecules vibrating about their mean position in a free energy well in an unstressed solid. Metastable equilibrium state (right) given by molecules vibrating in an elevated free energy well in a stressed solid (above the stable minimum for an unstressed solid). 

Sometimes the calculation predicts that the fluid as initially constrained is supersaturated with respect to one or more minerals, and hence, is in a metastable equilibrium. If the supersaturated minerals are not suppressed, the model proceeds to calculate the equilibrium state, which it needs to find if it is to follow a reaction path. By allowing supersaturated minerals to precipitate, accounting for any minerals that dissolve as others precipitate, the model determines the stable mineral assemblage and corresponding fluid composition. The model output contains the calculated results for the supersaturated system as well as those for the system at equilibrium. 

The extra pieces of information describe the extent of the system - the amounts of fluid and minerals that are present. It is not necessary to know the system s extent to determine its equilibrium state, but in reaction modeling (see Chapter 13) we generally want to track the masses of solution and minerals in the system we also must know these masses to search for the system s stable phase assemblage (as described in Section 4.4). 

The calculation described to this point does not predict the assemblage of minerals that is stable in the current system. Instead, the assemblage is assumed implicitly by setting the basis B before the calculation begins. A solution to the governing equations constitutes the equilibrium state of the system if two conditions are met ... 

These multiple branches are not equivalent. From the contour lines in Figure 7.8, we can deduce which branch is stable, and which branch is not. The middle branch (C-C1) lies in a range where, for a given value of p, F increases with u. The derivative of F with respect to u is therefore positive which is just the criterion we found for an unstable equilibrium. Any fluctuation of u at constant p will drive the system away from the branch C-C. The opposite holds for the upper and lower branches A-C and A -C that lie in a range where F decreases when u increases. The derivative of F with respect to u is therefore negative and any concentration fluctuation around an equilibrium state along these branches dies out rapidly. The branches A-C and A -C are stable steady-states. 

First-order phase transition from a nucleonic matter to the strange quark state with a transition parameter A > 3/2 that occurs in superdense nuclear matter generally gives rise to a toothlike kink on the stable branch of the dependence of stellar mass on central pressure. Based on the extensive set of calculated realistic equations of state of superdense matter, we revealed a new stable branch of superdense configurations. The new branch emerges for some of our models with the transition parameter A > 3/2 and a small quark core (.Mcore 0.004 A- 0.03M ) on the M(PC) curve, with Mmax 0.08M and A 0.82M for different equations of state. Stable equilibrium layered... 

Of course it will be impossible, in a few pages, to give a complete description of phase diagram science, and only an outline of some noteworthy aspects will be presented. Notice that the phase diagrams presented first are equilibrium diagrams. These represent an important reference point and describe the final state (stable state) which can be reached in a reaction of the substances involved. Slow (or very... 

Notice, however, that in real situations, if for instance an alloy is subjected to a rapid change of temperature, it is possible that transformations occur which do not correspond to a sequence of true equilibrium states. It is possible that some transition will be missed or shifted to a different temperature or that a phase stable in a certain range of temperature (or pressure) will be retained in different conditions. In Table 2.4, as an aide-memoire, an indication is given of how the rate of temperature... 

Stable, metastable and unstable states a simple analogy. A simple mechanical model is shown in Fig. 2.37 a block on a stand may be in different equilibrium states. In A and C the centre of gravity (G) of the block is lower than... 

Figure 2.37. A simple mechanical system and its equilibrium states. Different positions of a block on a stand and the corresponding values of the gravitation potential energy are shown. Point G is the centre of gravity of the block. In A there is stable equilibrium, in C metastable, in B unstable.

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