Equilibrium, criteria thermal

Certain equilibrium states of thermodynamic systems are stable to small fluctuations others are not. For example, the equilibrium state of a simple gas is stable to all fluctuations, as are most of the equilibrium states we will be concerned with. It is possible, however, to carefully prepare a subcooled liquid, that is, a liquid below its normal solidiflcation temperature, that satisfies the equilibrium criteria. This is an tin-.stable equilibrium. state because the slightest disturbance, such as tapping on the. side of the containing ve.s.sel, will cause the liquid to freeze. One sometimes encounters mixtures that, by the chemical reaction equilibrium criterion (see Chapter 13). should react however, the chemical reaction rate is so small as to be immeasurable at the temperature of interest. Such a mixture can achieve a state of thermal equilibrium that is stable with respect to small fluctuations of temperature and pressure. If, however, there is a sufficiently large, but temporary, increase in temperature. so that die rate of the chemical reaction is appreciable for some period of time) and then the system... 

Along any pure-fluid, subcritical isotherm, the spinodal separates unstable states from metastable states. At the other end of an isotherm s metastable range, metastable states are separated from stable states by the points at which vapor-Uquid, phase-equilibrium criteria are satisfied. Those criteria were stated in 7.3.5 the two-phase situation must exhibit thermal equilibrium, mechanical equilibrium, and diffusional equilibrium. Since we are on an isotherm, the temperatures in the two phases must be the same, and the thermal equilibrium criterion is satisfied. 

A theoretical analysis of the stability of such colloidal crystals of spherical latex particles has been carried out by Marcel ja et al (28.). They employ the Lindemann criterion that a crystal will be stable if the rms thermal displacement of the particles about their equilibrium positions is a small fraction f of the lattice spacing R. Comparison with Monte Carlo simulations shows that f is about 0.1 for "hard crystals, and 0.08 for "soft crystals stabilized by long-ranged electrostatic forces. This latter criterion translates into a critical ratio... 

Equation (12-2) leads to the following criterion for spontaneity for a process occurring at constant temperature and pressure, but with the system in thermal and mechanical contact with the surroundings The Gibbs free energy decreases for a spontaneous (irreversible) process and remains constant for an equilibrium (reversible) process. 

Equilibrium in a multiphase system implies thermal, mechanical, and material equilibrium. Thermal equilibrium requires uniformity of temperature throughout the system, and mechanical equilibrium requires uniformity of pressure. To find the criterion for material equilibrium, we treat a two-phase system and consider a transfer of dn moles from phase p to phase a. First, we regard each phase as a separate system. Because material enters or leaves these phases, they are open systems and we must use Eq. (4) to write their change in internal energy ... 

The most important chemical thermodynamic property is the chemical potential of a substance, denoted /x.18 The chemical potential is the intensive property that is the criterion for equilibrium with respect to the transfer or transformation of matter. Each component in a soil has a chemical potential that determines the relative propensity of the component to be transferred from one phase to another, or to be transformed into an entirely different chemical compound in the soil. Just as thermal energy is transferred from regions of high temperature to regions of low temperature, so matter is transferred from phases or substances of high chemical potential to phases or substances of low chemical potential. Chemical potential is measured in units of joules per mole (J mol 1) or joules per kilogram (J kg 1). 

Such a quantity, denoted as 7 eff(co), and parameterized by the age of the system, has been defined, for real to, via an extension of both the Einstein relation and the Nyquist formula. It has been argued in Refs. 5 and 6 that the effective temperature defined in this way plays in out-of-equilibrium systems the same role as does the thermodynamic temperature in systems at equilibrium (namely, the effective temperature controls the direction of heat flow and acts as a criterion for thermalization). 

If the anisotropy is small enough that the differential term can be ignored, then the ratio of the surface energies is simply related to the cosine of the angle at which they meet. Note that one criterion for the application of Eq. 3 is that the facets are in local equilibrium. For this to be true, there must be no net growth or evaporation. If this is true, then thermal grooves at surface-grain boundary intersections should maintain a quasistatic profile and increase in width with the one quarter power of time [45]. 

The key then is to somehow calculate the probability with which a specific quantum state contributes to the average values. As far as thermal systems in thermodynamic equilibrium are concerned, this is the central problem addressed by statistical thermodynamics. W( therefore begin our discuasion of some core elements of statistical thermodynamics at the quantum level but will eventually turn to the classic limit, because the phenomena addressed by this book occur under conditions where a classic description turns out to be adequate. We shall see this at the end of this chapter in Section 2.5 where we introduce a quantitative criterion for the adequacy of such a classic description. 

In free disperse systems, in particular those with low concentration of dispersed phase, the nature of colloid stability and conditions under which the collapse occurs, are to a great extent dependent on thermal motion of dispersed particles, which may contribute to both stability and destabilization. For example, the necessary condition for sedimentation stability is sufficiently small particle size, so that the tendency of particles to distribute within the entire volume of disperse system due to the Brownian motion (an increase in entropy) would not be affected by gravity. As a quantitative criterion for the presence of noticeable amount of dispersed particles in equilibrium with a sediment one, for instance, may use the... 

To test whether a proposed state will involve one or more solid phases, we usually use the criterion (7.1.40) which states that the equilibrium situation is the one that minimizes the Gibbs energy at the specified T and P. To perform such a calculation we need a model for the solid-phase Gibbs energy, and those models, in turn, require experimental data for the solid phase. The solid-phase data most often used are thermal data, such as heat capacities and latent heats for phase transitions. 

Any thermal process requires another scale-dependent criterion (which is often neglected) that decides whether or not any (thermally) stable state is stable enough and in what scale, viewed dimensionally, still maintains its stability. When the approached stability is of a simple natural scale this problem is more elementary but when equilibrium exists in the face of more complicated couplings between the different competing influences (forces) then the state... 

We are more interested in chemical equilibrium, achieved after transfer of species between two or more phases or regions. The criteria for equilibrium here will directly allow the calculation of different concentrations of a given species in different phases. This calculation presumes the existence of thermal and mechanical equilibrium. If the region is subjected to an external force field, the criterion for equilibrium separation is affected by the external potential field. This and other related criteria will be indicated in Section 3.3.1 without extensive and formal derivations (for which the reader should refer to different thermodynamics texts and references). The development of such criteria will be preceded by a brief illustration of the variety of two-phase systems encountered in separation processes. Our emphasis will be on two immiscible phase systems. 

A system is in chemical equilibrium when there is no tendency for a species to change phases or chemically react. In Chapter 6, we will develop the analogous criterion to Equations (1.10) and (1.11) for chemical equilibrium. To be in thermodynamic equilibrium, a system must be in mechanical, thermal, and chemical equilibrium simultaneously, so that there is no net driving force for any type of change. ... 

It is a derived thermodynamic property, unlike the measured thermodynamic properties, temperature and pressure, that provide the criteria for thermal and mechanical equilibrium, respectively Although the chemical potential is an abstract concept, it is useful since it provides a simple criterion for chemical equilibria of each species i. 

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