For equilibrated system

Figure 2.32 shows a typical resultant biexponential decay of the hydrocarbon radical observed for equilibrating systems when the rate of decomposition of the peroxy radical becomes comparable with that of the addition reaction. The initial rapid decay is due to the establishment of the equilibrium (47, -47) and the rate coefficient which characterizes this... 

In order to delineate the effect of surfactant mass transfer on in situ behavior of oil ganglia, we carried out several oil displacement experiments using equilibrated and nonequilibrated oil/ micellar solution systems. For equilibrated systems, the oil displacement efficiency showed an excellent correlation with IFT and capillary number. However, for unequilibrated systems, the oil displacement efficiency depended on salinity. Below optimal salinity, the oil displacement efficiency almost remained the same for both equilibrated and nonequilibrated systems, whereas at and above optimal salinity the oil displacement efficiency was higher for nonequilibrated systems as compared to equilibrated systems. This was attributed to mass transfer rate effects in these systems. 

A comparison of equilibrated and nonequilibrated systems for oil displacement efficiency Figure 2 shows the IFT and the percent oil recovery as a function of initial TRS 10-80 concentration in 1% NaCl for equilibrated and nonequilibrated systems. It was observed that for the pre-equilibrated system, 94% oil was recovered at 0.05% TRS 10-80 concentration corresponding to minimum IFT at this concentration. However, for nonequilibrated systems, the maximum oil recovery shifted from 0.05% to 0.1% TRS 10-80 concentration. The maximum oil recovery for nonequilibrated systems was much lower than that observed for equilibrated systems (Figure 2). Since the amount of surfactant injected was the same for each run (0.125 gm), the maximum oil recovery was interpreted as a result of the capillary number vs. final oil saturation correlation (21). 

We propose that the interface is occupied with both water-soluble and oil-soluble species. For equilibrated systems, the surfactant species come from both sides of the interface and saturate the interface with surfactant molecules more quickly as... 

For equilibrated systems, there is an excellent correlation between the capillary number and oil recovery efficiency. However, in calculating capillary number for nonequilibrated systems, care should be exercised because the IFT measured in vitro may not be achieved in situ and, in certain cases, the interfacial viscosity and not interfacial tension, may be a predominant factor influencing the oil displacement efficiency. 

An example of the time effects in irreversible adsorption of a surfactant system is shown in Fig. XI-8 for barium dinonylnapthalene sulfonate (an oil additive) adsorbing on Ti02 (anatase). Adsorption was ineversible for aged systems, but much less so for those equilibrating for a short time. The adsorption of aqueous methylene blue (note Section XI-4) on TiOi (anatase) was also irreversible [128]. In these situations it seems necessary to postulate at least a two-stage sequence, such as... 

In many molecular dynamics simulations, equilibration is a separate step that precedes data collection. Equilibration is generally necessary lo avoid introducing artifacts during the healing step an d to en su re th at the trajectory is aciii ally sim u laiin g eq u i librium properties. The period required for equilibration depends on the property of Interest and the molecular system. It may take about 100 ps for the system to approach equilibrium, but some properties are fairly stable after 1 0-20 ps. Suggested tim es range from. 5 ps to nearly 100 ps for medium-si/ed proteins. 

Th e sim u lation or run tim e m eludes time for the system lo equilibrate at Ibe simulation temperature plus tbe time for data collection, while the trajectory evolves. Simulation timesdepend on the time scale of tbc property you are investigating. 

A sequence of successive configurations from a Monte Carlo simulation constitutes a trajectory in phase space with HyperChem, this trajectory may be saved and played back in the same way as a dynamics trajectory. With appropriate choices of setup parameters, the Monte Carlo method may achieve equilibration more rapidly than molecular dynamics. For some systems, then, Monte Carlo provides a more direct route to equilibrium structural and thermodynamic properties. However, these calculations can be quite long, depending upon the system studied. 

Extensions of this concept have utilized enamine hydrolysis (171, X = R N) and the quenching of the enolate anion (171, X = O ) e.g. ref. 353). a,(i-Un-saturated ketones are usually more stable than their p,y counterparts, but there are notable exceptions to this, and in such cases the deconjugated ketone may be isolated from the equilibrated system. For example, retro steroids (9, 10a) have a large proportion of A -3-ketone at equilibrium, and 17-ketones yield the more stable A -system on treatment with acid. ... 

Thermal equilibration between isolable, unsymmetrically substituted 3//-azepines has been noted on a few occasions,28-31 and for such systems it has been suggested31 that the [1,5]-H shifts observed in the tautomerism of the azepines always proceed in the direction of theC —N double bond rather than in the direction of the N —C double bond. 

Opposing reactions. Calculate half-times for equilibration in the triphenyl methyl system starting with all A or with the equilibrated mixture, for the conditions given in Table 3-3. Use algebraic equations, not the tabulated numerical values. Compare the latter with the t fi from the approximate solution given in Eq. (3-39). Compare the values of 4AT-15o and a, to assess whether Eq. (3-39) provides an adequate representation. 

Consider the interconversion of two chiral molecules to yield ultimately the racemic mixture. This is simply the situation of opposing first-order reactions of A and P, treated in Chapter 3, for the special case of an equilibrium constant of unity. Recall that for such an equilibrating system ke = kf + kr because of that, knc is one-half the experimental rate constant. 

Isolated unperturbed polyoxyethylene chains have been simulated on the 2nnd lattice [154], The literature contains RIS models for a large number of polyethers [124], and it is likely that most of these chains could be mapped onto the 2nnd lattice with little difficulty. It is also likely that the work on PP [156,158] can be extended to other vinyl polymers, such as poly(vinyl chloride). This capability should permit the construction and complete equilibration of amorphous poly(vinyl chloride) cells larger than those described to date. They may be large enough to address issues arising from the weak crystallization reported for these systems [174]. 

For equilibration processes, one must synthesize both oligomers and what are termed dimers, or disiloxanes. Our primary interest is in the utilization of these functional oligomers for the synthesis of both linear block or segmented copolymers, and also surface modified, oughened networks such as the epoxy and imide systems (3-27). The generalized structure of the oligomers of interest is shown in Scheme 1. 

In considering catalyzed olefin-cyclopropane interconversions, an important question arises concerning thermodynamic control and the tendency (or lack thereof) to attain a state of equilibrium for the system. Mango (74) has recently estimated the expected relative amounts of ethylene and cyclopropane for various reaction conditions and concluded that the reported results were contrary to thermodynamic expectation. In particular, the vigorous formation of ethylene from cyclopropane (16) at -78°C was stated to be especially unfavored. On the basis of various reported observations and considerations, Mango concluded that a reaction scheme such as that in Eq. (26) above (assuming no influence of catalyst) was not appropriate, because the proper relative amounts of cyclopropanes and olefins just do not occur. However, it can be argued that the role of the catalyst is in fact an important element in the equilibration scheme, for the proposed metal-carbene and [M ] species in Eq. (26) are neither equivalent nor freely interconverted under normal reaction conditions. Consequently, all the reaction pathways are not simultaneously accessible with ease, as seen in the published literature, and the expected equilibria do not really have an opportunity for attainment. In such a case, absence of thermodynamic control should not a priori deny the validity of Eq. (26). 

Both FEP and TI are carried out by systematically varying X from the initial state 0 to the final state 1. At each X point, equilibration of the system is performed, followed by data collection to determine the value of the ensemble for the equilibrated system. 

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