Equilibrium development

When two or more phases, e.g. gas, liquid or solid, are in equilibrium, the principles of internal equilibrium developed in section A2.1.5.2 apply. If transfers between two phases a and p can take place, the appropriate potentials must be equal, even though densities and other properties can be quite different. 

In a closed container, equilibrium develops at any given temperature and pressure, w hile in an open condition (not enclosed) the liquid will continue to vaporize in air until the liquid is completely vaporized. In that situa-... 

Only in certain cases (a relatively high solubility of both components) is it possible to obtain equilibrium between a solid solution of known composition and an aqueous solution, because the solid solution is homogenized by a relatively fast recrystallization. In other instances, equilibrium develops between the surface of the particles of the solid solution and the aqueous solution. 

We have confirmed this, showing the H/D exchange in the protonation of the anion [( j -CH3C5H4)Mn(CO)jSi(D)Ph l-Np] (or SiH) with HCl (or DCl) which yields the same mixture of MnH-SiD and MnD-Sitf . Since this exchange takes place in spite of the fact that Mn-H and Si-H bonds are distinct and coupled in NMR, one may come to the conclusion that a slow equilibrium develops. 

In a more recent theory of swelling equilibrium developed by Flory in 1979 (19), the extent to which the swelling equihbrium deformation is non-affine is taken into account. The nonaffine behavior depends on the looseness with which the cross-links are embedded in the network and consequently is related to the network structure and the degree of swelling at equilibrium. The final expression is... 

In the calculation of the velocity profile of a hydrodynamic, fully developed, laminar flow we will presume the flow to be incompressible and all properties to be constant. The velocity profile of a fully developed, tubular flow is only dependent on the radius r, wx = wx(r) and wy = 0. Therefore the acceleration term gdw /dt in the Navier-Stokes equation (3.59) disappears body forces are not present, so fc = 0. An equilibrium develops between the pressure and friction forces. Balancing the forces on an annular fluid element, Fig. 3.32, gives... 

The term vaporization is used for any process of liquid becoming a gas. This includes evaporation and boiling. Evaporation takes place at a gas/liquid interface when temperature is less than the boiling point. Equilibrium develops between the gas and liquid phases when the rates of evaporation and condensation are equal. 

Steady states are among the phenomena that non-equilibrium thermodynamics studies. When one is not too far from equilibrium it can be shown that the steady states are stable. On the other hand, when far from equilibrium, certain systems can make transitions to states exhibiting "dissipative structures." The theory of non-equilibrium developed by I. Prigogine, is quite general and has been applied to a wide range of phenomena. It is the aim of this lecture to introduce this field with a few examples. 

Following the global equilibrium development of the preceding section one defines the thermodynamic potentials corresponding to the four alternative sets of the constrained-equilibrium states, for the alternative sets of the state parameters,... 

It is possible to show that the criterion for chemical equilibrium developed here is also applicable to systems subject to constraints other than constant temperature and pressure (Problem 8.4). In fact, Eq. 8.8-1, like the phase equilibrium criterion of Eq. 8.7-9, is of general applicability. Of course, the difficulty that arises in using either of these equations is translating their simple form into a useful prescription for equilibrium calculations by relating the partial molar Gibbs energies to quantities of more direct interest, such as temperature, pressure, and mole fractions. This problem will be the focus of much of the rest of this book. 

Shov. that the criterion for chemical equilibrium developed in the text,... 

Following this immediate shock the stratosphere cools. The increased CO2 in the stratosphere enhances the thermal emission. Because the stratospheric temperature increases with altitude, this has the effect that the cooling into space is larger than the absorption from layers below. This is in fact the fundamental reason for the COi-induced cooling in the stratosphere. After stratospheric cooling a new radiative equilibrium develops with the new doubled CO, concentration. This reduces the increased downward emission at the tropopause by about 0.2 W in - and the tropopause forcing is adjusted accordingly. 

In this chapter we apply the general criteria for equilibrium developed in Chap. 6 to systems in which chemical reactions may occur. In Sec. 8-1, we present a general discussion of chemical equilibrium in homogeneous and heterogeneous systems. The concept of a progress variable is introduced, and the conditions for chemical equilibrium are derived. The equilibrium constant is defined, and some of its properties are developed. A discussion of the Le Chatelier-Braun principle applied to chemical reactions is presented. In Sec. 8-2, the results of Sec. 8-1 are applied to chemical reactions in mixtures of real gases. 

At the same time, as the area being heated around the injection well enlarges, the losses of heat into the surrounding rocks also continue to increase. At a certain distance from the injection well, with the rate of heat input kept constant, the rate of advance of the burning front will ultimately slow down to almost zero. A dynamic equilibrium develops between heat input and heat losses. It is at that time that the efficiency of reservoir heating decreases. 

Figure 96.2 The chemical equilibrium of PVP-I in aqueous medium. In the solution, a chemical equilibrium develops. The complex-linked iodine is a reservoir for prolonged delivery.

The growth of drops in the flow can also occur due to condensation, but this can only happen in the absence of phase equilibrium. It was shown earlier that phase equilibrium develops much faster than dynamic equilibrium. Therefore, condensation-driven growth of drops in the separator is only possible if the DPC is located in the immediate vicinity of the separator, for instance, directly at the entrance. 

The following equation gives the overall equilibrium developing in this half-cell, involving the AgCI/Ag couple ... 

The study demonstrated that the larger the amount of initial water held by a closure, the more moisture is passed to the dry product. The product, however, will reach a maximum amount of moisture as an equilibrium develops between the closure and the dry product. 

A rarer case of a solid-state reaction that allows attainment of equilibrium develops within a metastable crystal as an exsolution event. An example of this occurs in the Si02—AI2O3—Zn0-Zr02 system. In this case, the equilibrium crystals formed are of smaller grain size than the original P-quartz crystals formed on the primary nuclei of tetragonal zirconia. The metastable... 

The development of a liquid-vapor pressure equilibrium is described in Section 15.4. An equilibrium between excess solute and a saturated solution is examined in Section 16.3. Both equilibria involve physical changes in which one of the two opposing rates remains constant until the other catches up with it. Here we study how a chemical equilibrium develops. This time both forward and reverse rates change as equilibrium is reached. 

Liquid-Vapor Equilibrium a liquid-vapor equilibrium develops in a closed system. 

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