Equilibrium departures from

Figure 8. Hypothetical concentration profiles in gas and aqueous phases indicating gradient in reagent concentration due to flux of material into and within drop. Concentration scales of aqueous-phase (r < a) left ordinate and gas-phase (r > a) right ordinate are chosen so that the same coordinate on each scale represents the condition of phase equilibrium. Departure from the uniform profile at the "bulk" (r =oo) value represents the inability of mass transport to maintain the reagent concentration as the reagent is consumed by aqueous-phase reaction. (Reproduced with permission from Ref. 28. Copyright 1986 Lewis Publishers, Inc.)...

What is obtained in this way is the number of theoretical plates, which means that the liquid is assumed to be completely mixed and that the gas and liquid are in equilibrium. Departure from these conditions are accounted for by means of the Murphree efficiency. When the operating and equilibrium lines are straight, the overall tray efficiency, E, which is the ratio of the number of ideal to real plates, can be computed analytically from... 

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-... 

The plate buckling equations inherently cannot be derived from the equilibrium of a differential element. Instead, the buckling problem represents the departure from the equilibrium state when that state becomes unstable because the in-plane load is too high. The departure from the equilibrium state is accompanied by waves or buckles in the surface of the plate. That is, the plate cannot remain flat when the... 

The observation of the departure from cubic symmetry above Tm co-incident with the appearance of the central peak scattering serves to resolve the conflict between dynamic and lattice strain models. The departure from cubic symmetry may be attributed to a shift in the atomic equilibrium position associated with the soft-mode anharmonicity. In such a picture, the central peak then becomes the precusor to a Bragg reflection for the new structure. 

If the grains of sand are small, each step does not represent a very large departure from equilibrium between p and ptxt. This process is an example of a quasi-static process that is, one in which the process is never far from equilibrium during the expansion. 

In a discussion of these results, Bertrand et al. [596,1258] point out that S—T behaviour is not a specific feature of any restricted group of hydrates and is not determined by the nature of the residual phase, since it occurs in dehydrations which yield products that are amorphous or crystalline and anhydrous or lower hydrates. Reactions may be controlled by interface or diffusion processes. The magnitudes of S—T effects observed in different systems are not markedly different, which indicates that the controlling factor is relatively insensitive to the chemical properties of the reactant. From these observations, it is concluded that S—T behaviour is determined by heat and gas diffusion at the microdomain level, the highly localized departures from equilibrium are not, however, readily investigated experimentally. 

As noted before, short focal ratios imply severe difficulties in generating the aspherical departure from an ideal sphere. It is, however, directly related to the telescope length, hence to the structure and building sizes, which are major cost positions and performance issues in a telescope project -cost of large structures and buildings, improbable thermal equilibrium hence local turbulence, misalignments and flexures, etc. 

The dominant practice in Quantum chemistry is optimization. If the geometry optimization, for instance through analytic gradients, leads to symmetry-broken conformations, we publish and do not examine the departure from symmetry, the way it goes. This is a pity since symmetry breaking is a catastrophe (in the sense of Thom s theory) and the critical region deserves attention. There are trivial problems (the planar three-fold symmetry conformation of NH3 is a saddle point between the two pyramidal equilibrium conformations). Other processes appear as bifurcations for instance in the electron transfer... 

The concepts of invertibility and reversibility must be distinguished. Invertibility is the term proposed to be used for reactions that can be made to occur in both directions, regardless of the departure from thermodynamic equilibrium that is necessary to achieve this. Reversibility of a reaction means that it occurs with a minimum departure from the thermodynamic equilibrium state. 

A limit to mass transfer is attained if two phases come to equilibrium and the net transfer of material comes to a halt. For a process in practice, which must have a reasonable production rate, equilibrium must be avoided, as the rate of mass at any point is proportional to the compelling or driving force, which is the departure from equilibrium at that point. In order to evaluate driving forces, a knowledge of equilibria between phase is therefore fundamentally important. Several kinds of equilibria are important in mass transfer. 

The electrode processes that are reversible provide values for the equilibrium emfs of cells, which are related to the thermodynamic functions. The condition of reversibility is practically obtained by balancing cell emf against an external emf until only an unappreciable current passes through the cell, in order that the cell reactions proceed very slowly. It may, however, be pointed out that for many of the applications of electrometallurgy, it is clearly necessary to consider more rapid reaction rates. In that situation there is necessarily a departure from the equilibrium condition. Either the cell reactions occur spontaneously to produce electric energy, or an external source of electric energy is used to implement chemical reactions (electrolyses). 

This confirms Onsager s regression hypothesis, namely, that the flux following a fluctuation in an isolated system is the same as if that departure from equilibrium were induced by an externally applied force. 

A distillation calculation is to be performed on a multicomponent mixture. The vapor-liquid equilibrium for this mixture is likely to exhibit significant departures from ideality, but a complete set of binary interaction parameters is not available. What factors would you consider in assessing whether the missing interaction parameters are likely to have an important effect on the calculations ... 

The phenomenon of critical flow is well known for the case of single-phase compressible flow through nozzles or orifices. When the differential pressure over the restriction is increased beyond a certain critical value, the mass flow rate ceases to increase. At that point it has reached its maximum possible value, called the critical flow rate, and the flow is characterized by the attainment of the critical state of the fluid at the throat of the restriction. This state is readily calculable for an isen-tropic expansion from gas dynamics. Since a two-phase gas-liquid mixture is a compressible fluid, a similar phenomenon may be expected to occur for such flows. In fact, two-phase critical flows have been observed, but they are more complicated than single-phase flows because of the liquid flashing as the pressure decreases along the flow path. The phase change may cause the flow pattern transition, and departure from phase equilibrium can be anticipated when the expansion is rapid. Interest in critical two-phase flow arises from the importance of predicting dis-... 

Integration leads to 5.1.10. The form of this equation indicates that the reaction may be considered as first order in the departure from equilibrium, where the effective rate constant is the sum of the rate constants for the forward and reverse reactions. 

The soil equilibrium parameter, Uts, measures the departure from the Mn fractionation in the oxic native soil, which is caused by the saturation and incubation of the soil. This parameter can acquire values equal to or larger than one. Compared to the native state, an increase in the Uts indicates that some fraction(s) have been grossly enriched in Mn. Note the fact that a fraction that has lost some Mn will only reduce its own contribution to the Uts value, but its contribution will not be subtracted from that of the other fraction(s) since the product F, xUtfi can not acquire negative values. 

That is, the total increase in entropy (which is a measure of disorder ) comes from heat transferred across the system boundary (Sq). However, a flowing fluid is in a dynamic, or irreversible, state. Because entropy is proportional to the degree of departure from the most stable (equilibrium) conditions, this means that the further the system is from equilibrium, the greater the entropy, so for a dynamic (flow) system... 

We have noted at the end of Section 2 in II that in and near a reverse-biased p-n junction or diode, the departure from electron-hole equilibrium will invalidate (3) and (4) (3) is replaced by (25), with r+0 given by (24) with eFe for eF, and r0+ given by (23), with eFh (4) is replaced by an analogous equation. Thus while the first equality in each of the three flux equations (30)-(32) remains valid, the expression of J+ and / in terms of n0 needs to be modified. The appropriate equations are easily found to be... 

Equation (3.19) gives a first approximation to the temperature structure of an atmosphere in radiative equilibrium, and departures from greyness can also be treated approximately by defining a suitable mean absorption coefficient (see Chapter 5). The emergent monochromatic intensity at an angle 9 to the normal (relevant to some point on the solar disk) is also found by integrating the equation of transfer (3.11) ... 

Equilibrium (i.e. local steady-state) ionization leads in this regime to solar-corona-like conditions where col-lisional ionization is balanced by recombination and the degree of ionization is fixed by the temperature alone, the electron density cancelling out. However, here departures from equilibrium occur because the time taken to establish ionization equilibrium is not negligible with respect to the timescale of expansion. 

The existence of dark matter (either baryonic or non-baryonic) is inferred from its gravitational effects on galactic rotation curves, the velocity dispersions and hydrostatic equilibrium of hot (X-ray) gas in clusters and groups of galaxies, gravitational lensing and departures from the smooth Hubble flow described by Eq. (4.1). This dark matter resides at least partly in the halos of galaxies such as our... 

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