The Equilibrium Case

For a system in which the contact between the two phases is long and/or the rate of mass transfer is relatively high, the concentrations of different components in the two phases generally reach a state of equilibrium, i.e., a state where no further mass transfer is possible. More details regarding equilibrium states are usually covered in thermodynamics courses. Equilibrium relations relate the concentrations of the phases to each other such as in 

For narrow regions of concentration values, we can replace (6.30) by linear relations such 

In the equilibrium case we neither know, nor need to know the rates of mass transfer. The simple and systematic approach is to add equations (6.28) and (6.29) for both the cocurrent and the countercurrent cases and thereby use only one equation instead of two, coupled with the mass-balance equations which are the same for both flow cases. 

This is a single algebraic equation linking the two unknowns C/ and 7. The equilibrium relation (6.30) can be used in (6.32) to obtain 

Equation (6.33) can be solved for (7 and the value of (7 then follows from the equilibrium relation (6.30). 

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Our first result is now the average collision frequency obtained from the expression, (A3.1.10). by dividing it by the average number of particles per unit volume. Here it is convenient to consider the equilibrium case, and to use (A3.1.2) for f. Then we find that the average collision frequency, v, for the particles is... 

When a system is not in equilibrium, the mathematical description of fluctuations about some time-dependent ensemble average can become much more complicated than in the equilibrium case. However, starting with the pioneering work of Einstein on Brownian motion in 1905, considerable progress has been made in understanding time-dependent fluctuation phenomena in fluids. Modem treatments of this topic may be found in the texts by Keizer [21] and by van Kampen [22]. Nevertheless, the non-equilibrium theory is not yet at the same level of rigour or development as the equilibrium theory. Here we will discuss the theory of Brownian motion since it illustrates a number of important issues that appear in more general theories. 

Introduction.—Statistical physics deals with the relation between the macroscopic laws that describe the internal state of a system and the dynamics of the interactions of its microscopic constituents. The derivation of the nonequilibrium macroscopic laws, such as those of hydrodynamics, from the microscopic laws has not been developed as generally as in the equilibrium case (the derivation of thermodynamic relations by equilibrium statistical mechanics). The microscopic analysis of nonequilibrium phenomena, however, has achieved a considerable degree of success for the particular case of dilute gases. In this case, the kinetic theory, or transport theory, allows one to relate the transport of matter or of energy, for example (as in diffusion, or heat flow, respectively), to the mechanics of the molecules that make up the system. 

The failure to identify the necessary authigenic silicate phases in sufficient quantities in marine sediments has led oceanographers to consider different approaches. The current models for seawater composition emphasize the dominant role played by the balance between the various inputs and outputs from the ocean. Mass balance calculations have become more important than solubility relationships in explaining oceanic chemistry. The difference between the equilibrium and mass balance points of view is not just a matter of mathematical and chemical formalism. In the equilibrium case, one would expect a very constant composition of the ocean and its sediments over geological time. In the other case, historical variations in the rates of input and removal should be reflected by changes in ocean composition and may be preserved in the sedimentary record. Models that emphasize the role of kinetic and material balance considerations are called kinetic models of seawater. This reasoning was pulled together by Broecker (1971) in a paper called "A kinetic model for the chemical composition of sea water."... 

In the limiting law, all dominant contributions come from large distances precisely as in the equilibrium case we thus see immediately that the velocity field term may be neglected here because it falls off rapidly to zero as r - oo. 

In contrast with the equilibrium case, this equation still depends upon the charges of a and ft in a complicated manner. Although methods exist to treat the general case,8 we shall limit ourselves here to the case of a binary electrolyte, composed of species a and (a / ). The electroneutrality condition thus reads ... 

The most recent effort in this direction is the work of Cohen,8 who established a systematic generalization of the Boltzmann equation. This author obtained the explicit forms of the two-, three-, and four-particle collision terms. His approach is formally very similar to the cluster expansion of Mayer in the equilibrium case. 

Equation 17.75 is important as it illustrates, for the equilibrium case, a principle that applies also to the non-equilibrium cases more commonly encountered. The principle concerns the way in which the shape of the adsorption wave changes as it moves along the bed. If an isotherm is concave to the fluid concentration axis it is termed favourable, and points of high concentration in the adsorption wave move more rapidly than points of low concentration. Since it is physically impossible for points of high concentration to overtake points of low concentration, the effect is for the adsorption zone to become narrower as it moves along the bed. It is, therefore, termed self-sharpening. 

The behavior of a mixture is determined by a system of ordinary differential equations, while the required state, either equilibrium or stationary, is determined by a time-independent system of algebraic equations. Therefore, at first glance one would not expect any qualitative difference between the equilibrium and stationary states. Ya.B. shows that in the equilibrium case, even for an ideal system, a variational principle exists which guarantees uniqueness. Such a principle cannot be formulated for the case of an open system with influx of matter and/or energy. 

In this chapter we have presented multistage systems with special emphasis on absorption processes. We have studied multitray countercurrent absorption towers with equilibrium trays for both cases when the equilibrium relation is linear and when it is nonlinear. This study was accompanied by MATLAB codes that can solve either of the cases numerically. We have also introduced cases where the trays are not efficient enough to be treated as equilibrium stages. Using the rate of mass transfer RMT in this case, we have shown how the equilibrium case is the limit of the nonequilibrium cases when the rate of mass transfer becomes high. Both the linear and the nonlinear equilibrium relation were used to investigate the nonequi-librium case. We have developed MATLAB programs for the nonequilibrium cases as well. 

In the equilibrium case, the relevant kinetic equations without retardation terms have the solutions... 

When Ei differs sufficiently from E%, then with the change in pH two wave appear on polarographic curves. Furthermore, if n = n%, the change in the wave-heights is similar to that shown in Fig. 2. The difference between the thermodynamic and activation-controlled systems is that in the region in which the two waves are observed for the equilibrium case, described in Chapter 2.1, both waves are diffusion-controlled, whereas for the activation-controlled system discussed here, the more positive wave at i <0.15 ia possesses a kinetic character. 

Next, consider the situation in which both of the terms on the right-hand side of eqn. (74) are much larger in magnitude than their difference. Since the difference between the two is the particle current itself, we designate this as the small current limit. Although not quite the same as the equilibrium case of eqn. (75), nevertheless the relationship between the area densities will not deviate markedly from those predicted by the equilibrium expression given in eqn. (75). If, in this small current limit, we further restrict our consideration to barriers of equal height as viewed from the forward and reverse directions, then the area densities on the two sides of the barrier will be almost equal, viz. n(r) — n(f). 

We can have the equilibrium case of zero current, in which case we find from this expression that... 

PI] = 6 mM, based on the equilibrium constant value Keq = 7.5 x 105M. Assuming that all rate constants remain the same as for the physiological example, these concentrations yield Ai = 0.0333sec-1 and 2.2 = 0.0585 sec-1. In this case, setting pdS)) = 1, Equation (5.32) yields a = 1.31. The functions pvj+d) and fj(/) for the equilibrium case are plotted as dashed lines in Figure 5.7. 

Note that the relative variance of the timing probability distribution is considerably larger in the equilibrium case than in the physiological case. However, even in the physiological case, the system behavior is far from that of a perfect timer. In this near ideal case, r.v. 1 /2, which is the minimal value obtained by Equation (5.37) when 2,i 2,2. 

The quantity 11/ is a measure of the so-called disjoining action , introduced by Derjaguin in 1936 [12]. The disjoining pressure n [8] is determined by the long-range interaction forces between the surfaces of the film (normal to the both surfaces of tension there) and tends to zero when the film thickness is sufficiently large [5]. Eq. (3.15) proposes a more general definition of IT than that for the equilibrium case (Eq. (3.10))... 

The equilibrium case is readily obtained from the general, steady-state case just discussed. When the conversion of ES to products is slow compared with the reversible first step of Equation 29-13, the first step is essentially at equilibrium throughout. Mathematically, this occurs when ki is much smaller than Under these conditions, Equation 29-19 becomes... 

Let us consider a light excitation of electrons and holes (An = Ap) within a doped n-type semiconductor so that Am < no and A/ >p(). Then the Fermi level of electrons, fipn, remains unchanged with respect to the equilibrium case, whereas that of holes, Epp, is shifted considerably downwards, as illustrated in Fig. 1.17b. In many cases, however, the excitation of electron-hole pairs occurs locally near the sample surface because the penetration of light is small. Then the splitting of the quasi-Fermi levels is... 

Considering now the equilibrium case, the two partial currents given in Eq. (7.2) must be equal = / ). Substituting Eq. (7.3) into (7.2), one obtains after rearranging the equation... 

What this equation tells us is that a particular state of stress is nothing more than a linear combination (albeit perhaps a tedious one) of the entirety of components of the strain tensor. The tensor Cijn is known as the elastic modulus tensor or stiffness and for a linear elastic material provides nearly a complete description of the material properties related to deformation under mechanical loads. Eqn (2.52) is our first example of a constitutive equation and, as claimed earlier, provides an explicit statement of material response that allows for the emergence of material specificity in the equations of continuum dynamics as embodied in eqn (2.32). In particular, if we substitute the constitutive statement of eqn (2.52) into eqn (2.32) for the equilibrium case in which there are no accelerations, the resulting equilibrium equations for a linear elastic medium are given by... 

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