Equilibration at a Temperature

Naturally, the study of non-equilibrium properties involves different criteria although the equilibrium state and evolution towards the equilibrium state may be important. 

In deciding the convergence of these averages, the RMS deviation of a value from its average (i.e., Dx) may be a very useful indicator. 

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Fig. 4.5.4 Heat stability of Periphylla luciferases A, B, C and L in 20 mM Tris-HC1 buffer (pH 7.8) containing 1 M NaCl and 0.05% BSA (solid lines) or 0.01% LCC (dotted lines). The buffer (1 ml) containing a luciferase sample was placed in a glass test tube that had been pre-equilibrated at a temperature in a water-bath. After 2 min, the test tube was briefly cooled in cold water, and then luciferase activity in 10 jl1 of the solution was measured in 3 ml of the pH 7.8 buffer containing 0.3 i,M coelenterazine at 24° C. From Shimomura et al., 2001.

Another thermal history consists of thermal equilibration at a temperature above the glass-transition range followed by rapid quench to a temperature within the glass-transition range. Isothermal annealing is then followed until equilibriinn is reached. A relaxation function is defined as ... 

It is also recommended that the initial load be applied after the polymer equilibrates at a temperature below its Tg for a certain period of time (usually 10-15 minutes). Sircar and Chartoff (1994) demonstrated poor reproducibihty of coefficient of thermal expansion with a number of elastomeric samples when the load was apphed at room temperature. 

The simplest method that keeps the temperature of a system constant during an MD simulation is to rescale the velocities at each time step by a factor of (To/T) -, where T is the current instantaneous temperature [defined in Eq. (24)] and Tq is the desired temperamre. This method is commonly used in the equilibration phase of many MD simulations and has also been suggested as a means of performing constant temperature molecular dynamics [22]. A further refinement of the velocity-rescaling approach was proposed by Berendsen et al. [24], who used velocity rescaling to couple the system to a heat bath at a temperature Tq. Since heat coupling has a characteristic relaxation time, each velocity V is scaled by a factor X, defined as... 

Fig. 10.2.4 Influence of temperature on the peak light intensity and total light of Luminodesmus photoprotein, when 25 pi of a solution of the photoprotein was injected into 2ml of lOmM Tris-HCl buffer, pH 8.3, containing 0.1 mM ATP and 1 mM MgCl2, that had been equilibrated at various temperatures.

This test was prepared and is limited to type 1 (low-density) polyethylenes. Specimens are annealed in water or steam at 212°F (100°C) for 1 h and then equilibrated at room temperature for 5-24 h. After conditioning the specimens are nicked according to directions given. The specimens are bent into a U shape in a brass channel and inserted into a test tube that is then filled with fresh reagent (Igepal). The tube is stoppered with an aluminum-covered cork and placed in a constant temperature bath at 122°F (50°C). 

The reference electrode-solid electrolyte interface must also be non-polarizable, so that rapid equilibration is established for the electrocatalytic charge-transfer reaction. Thus it is generally advisable to sinter the counter and reference electrodes at a temperature which is lower than that used for the catalyst film. Porous Pt and Ag films exposed to ambient air have been employed in most previous NEMCA studies.1,19... 

It is important to emphasize that this lattice database is highly idealized compared to real databases. Unlike the lattice database, real databases cannot be treated as thermodynamic ensembles of protein-ligand complexes equilibrated at room temperature [33,34]. Two of the more straightforward reasons are mentioned here. First, real databases are inherently biased toward strong binders (K < 10 pM), because weak binders are difficult to crystallize and of lesser interest. Second, as mentioned above, real databases are not composed of a representative selection of proteins and ligands, and their compositions are biased toward peptide and peptidomimetic inhibitors and certain protein superfamilies. In contrast, because only one protein and four ligand types are used, the lattice database should have representative ligand compositions. 

As in the 1,2-dichloroethane case too, transient EMF and SHG responses to KSCN were observed for the nitrobenzene membranes without ionic sites. This suggests that here too not only SCN but also K ions are transferred into the nitrobenzene phase. Salt extraction into the bulk of the organic phase, in analogy to similar observations previously reported for neutral ionophore-incorporated liquid membranes without ionic sites [55], was indeed independently confirmed by atomic absorption spectrometry. Figure 15 shows the concentration of K in nitrobenzene equilibrated at room temperature with a 10 M aqueous solution of KSCN as a function of equilibration time. The presence of the ion exchanger TDDMA-SCN efficiently suppresses KSCN extraction into the organic phase but in its absence a substantial amount of KSCN enters the nitrobenzene phase. The trends of the EMF and the SHG responses are therefore very similar in spite of the different polarities of the plasticizers. 

PPII helix-forming propensities have been measured by Kelly et al. (2001) and A. L. Rucker, M. N. Campbell, and T. P. Creamer (unpublished results). In the simulations the peptide backbone was constrained to be in the PPII conformation, defined as (0,VO = ( — 75 25°, +145 25°), using constraint potentials described previously (Yun and Hermans, 1991 Creamer and Rose, 1994). The AMBER/ OPLS potential (Jorgensen and Tirado-Rives, 1988 Jorgensen and Severance, 1990) was employed at a temperature of 298° K, with solvent treated as a dielectric continuum of s = 78. After an initial equilibration period of 1 x 104 cycles, simulations were run for 2 x 106 cycles. Each cycle consisted of a number of attempted rotations about dihedrals equal to the total number of rotatable bonds in the peptide. Conformations were saved for analysis every 100 cycles. Solvent-accessible surface areas were calculated using the method of Richmond (1984) and a probe of 1.40 A radius. 

Combining solids that have previously been equilibrated at different relative humidities results in a system that is thermodynamically unstable, since there will be a tendency for moisture to distribute in the system so that a single relative humidity is attained in the headspace. As shown in Fig. 7, moisture will desorb into the headspace from the component initially equilibrated at a higher relative humidity and sorb to the component initially equilibrated at a lower relative humidity. This process will continue until both solids have equilibrated at the final relative humidity. The final relative humidity can be predicted a priori by the sorption-desorption moisture transfer (SDMT) model [95] if one has moisture uptake isotherms for each of the solid components, their initial moisture contents and dry weights, headspace volume, and temperature. Final moisture contents for each solid can then easily be estimated from the isotherms for the respective solids. 

Situations that depart from thermodynamic equilibrium in general do so in two ways the relative concentrations of different species that can interconvert are not equilibrated at a given position in space, and the various chemical potentials are spatially nonuniform. In this section we shall consider the first type of nonequilibrium by itself, and examine how the rates of the various possible reactions depend on the various concentrations and the lattice temperature. 

A photoextrusion of a nitrogen molecule from a partially saturated tetrazolo[l,5- ]pyridine derivatives has been described by Quast et al. <1998EJ0317> (Scheme 11). The starting bicyclic compound 39 when irradiated at low temperature (at -60 °C) afforded annulated iminoaziridine 40 as a mixture of (E)- and (Z)-isomers. These two geometric isomers equilibrated at higher temperature (20 °C). Upon heating of the mixture of ( )-40 and (Z)-40, a thermal cycloreversion took place with methyl isocyanide elimination to afford the dihydropyrrole 41. 

Many other examples of contrasting behaviour have been discovered. For example all-cis-cyclodecapentaene (VII) photochemically equilibrate at low temperatures with trans 9, 10 dihydronapthalene by a conrotatory six electron electrocyclic reaction but it is converted thermally into cis-9, 10 dihydronaphthalene by disrotatory closure. 

The possible preparation of InAs by crystallization from the melt depends also on the liquidus shape (especially in the In-rich region). A summary of previous liquidus measurements was reported by De Winter and Pollack (1986) who employed a source dissolution method based on the equilibration, at a fixed temperature, of a known quantity of high-purity indium with single crystals of InAs, the weight loss of which was determined. The experiments were carried out under a flux of hydrogen purified via permeation through palladium. 

As has been discussed, ordinary formamides have a barrier of about 21 kcal/ mol, which is a little less than that required for the isolation of atropisomers at room temperature. This means that, at a temperature slightly lower than ambient, it may be possible to obtain stable rotamers. This possibility was first realized by Gutowsky, Jonas, and Siddall (40). They used a uranyl nitrate complex of N-benzyl-N-methylformamide (4) crystallized from dichloromethane. When the crystals were washed with ice water to strip off the uranyl nitrate, a mixture of E and Z forms (Z/E = 1.6) was obtained. Since the equilibrium mixture gives a Z/E value of 0.8, it was possible to perform a kinetic study of equilibration... 

Other studies have provided additional data on the relative stabilities of the lithium aldolates 14 and 15 derived from the condensation of dilithium enediolates 13 (Rj = alkyl, aryl) with representative aldehydes (eq. [ 10]) (16). Kinetic aldol ratios were also obtained for comparison in this and related studies (16,17). As summarized in Table 4, the diastereomeric aldol chelates 14a and ISa, derived from the enolate of phenylacetic acid 13 (R = Ph), reach equilibrium after 3 days at 25° C (entries A-D). The percentage of threo diastere-omer 15 increases with the increasing steric bulk of the aldehyde ligand R3 as expected. It is noteworthy that the diastereomeric aldol chelates 14a and 15a (Rj = CH3, C2HS, i-C3H7) do not equilibrate at room temperature over the 3 day period (16). In a related study directed at delineating the stereochemical control elements of the Reformatsky reaction, Kurtev examined the equilibration of both... 

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