Equilibrium distance from

The equilibrium determines the extraction limit and is a component of a driving force for the process the greater the distance from equilibrium, the faster extraction will be. The rate of transfer of a component C between two phases A and B can be given by the following equation ... 

The relaxation time is thus the time at which the distance from equilibrium has been reduced to the fraction 1/e of its initial value. From equation 5.1.50, it is evident that... 

Fig. 4 Morse curves of the reactants and products at zero driving force (z is the elongation of the RX distance from equilibrium, P = V(,(2Jt p/ )Rx) with Vq representing the vibration frequency, p the reduced mass of the two atoms of the R—X bond, and the RX bond dissociation energy).

However at a critical distance from equilibrium, the system must choose between two possible pathways, represented by the bifurcation point Ac. The continuation of the initial pathway, indicated by a broken hne, indicates the region of instability. The concentration of the species A and the value of A assume quite different values, and the more so, the further from equilibrium. An important point is that the choice between the two branching directions is casual, with 50 50 probability of either. The critical point Ac has particular importance because beyond it, the system can assume an organized structure. Here the term self-organization is introduced as a consequence of the dissipative structures, dissipative in the sense that it results from an exchange of matter and energy between system and environment (we are considering open systems). 

Figure 5.21 Bifurcation far from equilibrium, (a) Primary bifurcation is the distance from equilibrium, at which the thermodynamic branching of minimal entropy production becomes unstable. The bifurcation point or critical point corresponds to the concentration (b) Complete diagram of bifurcations. As the non-linear reaction moves away from equilibrium, the number of possible states increases enormously. (Adapted, with permission, from Coveney and Highfield, 1990).

Fig. 2. Stability of the thermodynamic branch as a function of some parameter A that measures the chemical system s distance from equilibrium. In the linear range (i.e., for 0 sAssA ) the steady states belong to the thermodynamic branch (a) and are stable. Beyond this domain there may exist a threshold point Ac at which a new stable nonequilibrium branch of solutions (b) appears while the thermodynamic branch becomes unstable.

Isotope effects could be neglected. Indeed, the ratio of the apparent rate constants for the OH-D2 exchange ( oh) and for the OD-H2 exchange ( od) on the same sample was always close to unity. For all the hydroxyl groups on the different samples, 1.00 < AjohAod < 1.15. Figure 2 shows that the data are represented satisfactorily by a first-order equation at the distance from equilibrium. The standard deviation of the slope of the lines never exceeded 3%. The rate of exchange was immeasurably slow below 250° C. 

Figure 2. First-order plot at distance from equilibrium for the 3550 cm l band on the F49/70 sample...

The problem to be solved in this paragraph is to determine the rate of spread of the chromatogram under the following conditions. The gas and liquid phases flow in the annular space between two coaxial cylinders of radii ro and r2, the interface being a cylinder with the same axis and radius rx (0 r0 < r < r2). Both phases may be in motion with linear velocity a function of radial distance from the axis, r, and the solute diffuses in both phases with a diffusion coefficient which may also be a function of r. At equilibrium the concentration of solute in the liquid, c2, is a constant multiple of that in the gas, ci(c2 = acj) and at any instant the rate of transfer across the interface is proportional to the distance from equilibrium there, i.e. the value of (c2 - aci). The dispersion of the solute is due to three processes (i) the combined effect of diffusion and convection in the gas phase, (ii) the finite rate of transfer at the interface, (iii) the combined effect of diffusion and convection in the liquid phase. In what follows the equations will often be in sets of five, labelled (a),..., (e) the differential equations expression the three processes (i), (ii) (iii) above are always (b), (c) and (d), respectively equations (a) and (e) represent the condition that there is no flow over the boundaries at r = r0 and r = r2. 

The irreversibility inherent in the equations of evolution of the state variables of a macroscopic system, and the maintenance of a critical distance from equilibrium, are two essential ingredients for this behavior. The former confers the property of asymptotic stability, thanks to which certain modes of behavior can be reached and maintained against perturbations. And the latter allows the system to reveal the potentialities hidden in the nonlinearity of its kinetics, by undergoing a series of symmetry breaking transitions across bifurcation points. 

When the HSS solution of the chemical rate equations (la)—(lc) first becomes unstable as the distance from equilibrium is increased (by decreasing P, for example), the simplest oscillatory instability which can occur corresponds mathematically to a Hopf bifurcation. In Fig. 1 the line DCE is defined by such points of bifurcation, which separate regions of stability (I,IV) of the HSS from regions of instability (II,III). Along section a--a, for example, the HSS becomes unstable at point a. Beyond this bifurcation point, nearly sinusoidal bulk oscillations (QHO, Fig. 3a) increase continuously from zero amplitude, eventually becoming nonlin-... 

The rate of sorption in Eq. (9.24) is proportional to the distance from equilibrium. As noted earlier, in deriving Eq. (9.24), which is based on the generalized equilibrium theory of Fava and Eyring (1956), it is assumed that the reverse reaction and desorption rate are small enough to be neglected. Haque et al. (1968) satisfied this assumption by using large amounts of each of the clays (5-15 g) and low 2,4-D solution concentrations (1.3 mg l-1). 

Therefore the reversible process is an imaginary one. However, if a process proceeds with an infinitesimally small driving force in such a way that the system is never more than an infinitesimal distance from equilibrium, a condition which is virtually indistinguishable from equilibrium, then the process can be regarded as a reversible process. Thus a reversible process is infinitely slow. 

By nature, the BOVB method describes properly the dissociation process. As a test case, the dissociation curve of the FH molecule was calculated at the highest BOVB level (extended SD-BOVB), and compared with a reference full Cl dissociation curve calculated by Bauschlicher et al. [33] with the same basis set. The two curves, that were compared in Ref. 12, were found to be practically indistinguishable within an error margin of 0.8 kcal/mol, showing the ability of the BOVB method to describe the bonding interaction equally well at any interatomic distance from equilibrium all the way to infinite separation [12]. 

Figure 2.2. Thermodynamic branch. AXC indicates the critical distance from equilibrium state.

In some systems, the distance from equilibrium reaches a critical point, after which the states in the thermodynamic branch become metastable or unstable. This region is the nonlinear region where the linear phenomenological equations are not valid. We observe bifurcations and multiple solutions in this region. 

One of the conventional methods for establishing the existence of active transport is to analyze the effects of metabolic inhibitors. The second is to correlate the level or rate of metabolism with the extent of ion flow or the concentration ratio between the interior and exterior of cells. The third is to measure the current needed in a short-circuited system having similar solutions on each side of the membrane the measured flows contribute to the short-circuited current. Any net flows detected should be due to active transport, since the electrochemical gradients of all ions are zero (Ai// = 0, cD = c,). Experiments indicate that the level of sodium ions within the cells is low in comparison with potassium ions. The generalized force of chemical affinity shows the distance from equilibrium of the /th reaction... 

One of the best-known physical ordering phenomena is the Benard cells, which occur during the heating a fluid held between two parallel horizontal plates separated by a small distance. The lower plate is heated, and the temperature is controlled. The upper plate is kept at a constant temperature. When the temperature difference between the two plates reaches a certain critical value, the elevating effect of expansion predominates, and the fluid starts to move in a structured way the fluid is divided into horizontal cylindrical convection cells, in which the fluid rotates in a vertical plane. At the lower hot plate, the hot fluid rises later, it is cooled at the upper plate, and its density increases again this induces a movement downward, as seen in Figure 13.2. The Benard cells are one of the best-known physical examples of spontaneous structurization as a result of sufficient distance from equilibrium, which is the large temperature difference between the plates. The critical temperature difference ( A 7 )c can be determined from the... 

An approximate relationship between the degree of undersaturation of seawater with respect to calcite and the extent of dissolution can be established by comparing the saturation state at the various sediment marker levels with estimates of the amount of dissolution required to produce these levels. In Figure 9 the "distance from equilibrium (1 - 2) has been plotted against the estimated percent dissolution of the calcitic sediment fraction. Within the large uncertainties that exist in the amount of dissolution required to produce the FL and Rq levels, a linear relation between the degree of undersaturation and extent of dissolution can be established. The intercept of the linear plot with the FL and Rq levels indicates that approximately 15 percent more material has been lost than Berger s (12) minimum loss estimate of 50% and 10%, respectively. 

Figure 9. Relationship between the distance from equilibrium (1 — Q) of seawater with respect to calcite and the amount of dissolution which is estimated to have occurred. (A) Adelseck s (11) experimentally determined amount of dissolution (B) Berger s (12) minimum loss estimate for the amount of dissolution at the Ro level and foraminiferal lysocline.

The rate is proportional to the distance from equilibrium. The proportionality factor is the sum of the forward and reverse rate coefficients. 

The decrease of distance from equilibrium thus is a first-order process. 

This is the fractional distance from equilibrium, varying from x— 1 at start to x=0 at equilibrium and having the same value for A and P. In terms of x, the rate is... 

Equation 5.5 and the first-order plot also remain valid regardless of volume variations if the fractional distance from equilibrium is defined in terms of amounts N-t instead of concentrations Cj ... 

The relationship between the characteristic rate coefficient k and the time tx required to reduce the distance from equilibrium to the fraction x is analogous to that for irreversible first-order reactions (see eqn 3.23) ... 

The product concentration remained constant within experimental error from 47 to 62 hours. The value at 47 hours, 0.1328 M, can therefore be taken as the equilibrium concentration of lactone, CP . The corresponding values of x (fractional distance from equilibrium) are listed in the third column. A plot of Injt versus t is shown in Figure 5.4 (left diagram) and is seen to be nicely linear, indicating that the reaction is first order-first order (the reverse step actually is pseudo-first order as it involves H20 in large excess as a... 

In first order-first order reversible reactions, the rate of approach to equilibrium is proportional to the fractional distance from equilibrium, measured in terms of any quantity that is a linear function of the concentrations. The same rule holds true for any participant in reactions with first-order parallel steps. 

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