Deviations from Equilibrium

We have repeatedly observed that the slowly converging variables in liquid-liquid calculations following the isothermal flash procedure are the mole fractions of the two solvent components in the conjugate liquid phases. In addition, we have found that the mole fractions of these components, as well as those of the other components, follow roughly linear relationships with certain measures of deviation from equilibrium, such as the differences in component activities (or fugacities) in the extract and the raffinate. 

With equilibrium concentrations the (small) deviation from equilibrium is given by... 

Tray efficiency 0 j is supposed to represent a measure of the deviation from equilibrium-stage mass transfer assuming backmixed trays. However, the estimate of tray efficiency requires accurate knowledge of the equihbrium vaporization constant. Any deviations between the actual equihbrium relation and that predicted by the database will be embodied in the tray efficiency estimate. It is a tender trap to accept tray efficiency as a true measure of the mass transfer hmitations when, in fact, it embodies the uncertainties in the database as well. 

AC)m is the mean eoneentration gradient, representing the deviation from equilibrium. Henee the rate is direetly related to eoeffieient K, whieh will generally inerease witli any inerease in mrbulenee sueh as inereased relative veloeity between the phases or agitation to the exposed surfaee area A and to the eoneentration differenee, whether it is a pressure or humidity differential or a solubility relationship. As a result ... 

We use primes to denote the quantities after collision, while unprimed symbols label those before the collision taking another time average of the equation, and using/(1)/(1) /a)/tt), which is valid for first order deviations from equilibrium, we obtain... 

A reaction at steady state is not in equilibrium. Nor is it a closed system, as it is continuously fed by fresh reactants, which keep the entropy lower than it would be at equilibrium. In this case the deviation from equilibrium is described by the rate of entropy increase, dS/dt, also referred to as entropy production. It can be shown that a reaction at steady state possesses a minimum rate of entropy production, and, when perturbed, it will return to this state, which is dictated by the rate at which reactants are fed to the system [R.A. van Santen and J.W. Niemantsverdriet, Chemical Kinetics and Catalysis (1995), Plenum, New York]. Hence, steady states settle for the smallest deviation from equilibrium possible under the given conditions. Steady state reactions in industry satisfy these conditions and are operated in a regime where linear non-equilibrium thermodynamics holds. Nonlinear non-equilibrium thermodynamics, however, represents a regime where explosions and uncontrolled oscillations may arise. Obviously, industry wants to avoid such situations ... 

Oscillations such as in Fig. 2.15 are quite regular and can be sustained for hours if the conditions are kept the same. Depending on the feed rate of the reactants, which determines how far the system deviates from equilibrium, the oscillations may become more complex, and develop into chaotic oscillations (see, for example, P.D. Cobden, J. Siera, and B.E. Nieuwenhuys, J. Vac. Sci. Technol. A10 (1992) 2487). 

Show that the rate can be expressed as a deviation from equilibrium as ... 

All of the discussions so far regarding distillation lines, residue curves and distillation boundaries have assumed equilibrium behavior. Real columns do not work at equilibrium, and stage efficiency must be accounted for. Each component will have its own stage efficiency, which means that each composition will deviate from equilibrium behavior differently. This means that if nonequilibrium behavior is taken into account, the shape of the distillation lines, residue curves and distillation... 

Finally, the surface tension is expanded about the equilibrium adsorption, T, and only the first term in the deviation from equilibrium is retained. 

Regular Perturbation Solution. To effect an analytical expression for the bubble-flow resistance, we consider fast sorption kinetics or equivalently, small deviations from equilibrium surfactant coverage making 0 large. Hence, a regular perturbation expansion is performed in 1/0 about the constant-tension case. The resulting equations for and rf are to zero and first order in 1/0 (21) ... 

For small deviations from equilibrium, we average r — rm 1 by the statistical segment distribution. For a Gaussian chain which is realized under... 

Small deviations from equilibrium are assumed, which implies small changes in the applied magnetic field as in the case of ac magnetic susceptibility experiments. 

Reaction kinetics enter into a geochemical model, as we noted in the previous chapter, whenever a reaction proceeds quickly enough to affect the distribution of mass, but not so quickly that it reaches the point of thermodynamic equilibrium. In Part I of this book, we considered two broad classes of reactions that in geochemistry commonly deviate from equilibrium. 

The first term on the right side of the solution represents the extent to which the silica concentration deviates from equilibrium. Since the term appears as a negative exponential function in time, its value decays to zero (as can be seen in Fig. 26.1) at a rate that depends on the surface area and rate constant. As t becomes large, the first term disappears, leaving only the equilibrium concentration meq. 

One should take careful note of the fact that in the nonadiabatic solvation, or frozen solvent" limit, it is the absence of solvation dynamics that is important. But is just this lack that is responsible for the deviation from equilibrium solvation, which instead assumes the dynamics are effective in always maintaining equilibrium. 

In the case of slight deviations from equilibrium, Tjon obtains the relation... 

Now system is disturbed so that it is no longer at equilibrium. The deviation from equilibrium, Ax may be considered as... 

Terrestrial BMOs have also been widely used for monitoring environmental contaminants. In particular, the lipid-like waxy cuticle layer of various types of plant leaves has been used to monitor residues of HOCs in the atmosphere. However, some of the problems associated with aquatic BMOs apply to terrestrial BMOs as well. For example, Bohme et al. (1999) found that the concentrations of HOCs with log KoaS < 9 (i.e., those compounds that should have attained equilibrium) varied by as much as 37-fold in plant species, after normalization of residue concentrations to levels in ryegrass (Lolium spp.). These authors suggested that differences in cuticular wax composition (quality) were responsible for this deviation from equilibrium partition theory. Other characteristics of plant leaves may affect the amount of kinetically-limited and particle-bound HOCs sampled by plant leaves but to a lesser extent (i.e., <4-fold), these include age, surface area, topography of the surface, and leaf orientation. 

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