Equilibrating systems

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. ... 

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. 

When JV /V -carbonyldiimidazole (CDI) is reacted with a primary amine in a 1 1 molar ratio, the product is an imidazole-A-carboxamide. However, these compounds dissociate in solution into isocyanates and imidazole even at room temperature,[1] forming a rapidly equilibrating system. Because of this equilibrium, primary imidazole-A-carboxamides can also be prepared from isocyanates and imidazole. 

UU is the Hamiltonian difference (the perturbation) the angle brackets represent a canonical ensemble average performed on an equilibrated system designated by the subscript. Usually the kinetic component of the Hamiltonian is not included in the free energy calculation, and A- //, n can be replaced by the potential energy difference AU = Ui - U0. 

Note that the total amount of metals exceeded that of light non-metals so that Earth was very reducing and water could not have existed in an overall equilibrated system. 

Isotope effects, magnetic, magnetic field effects and, on the products of organic reactions, 20, 1 Isotope effects, on nmr spectra of equilibrating systems, 23,63 Isotope effects, steric, experiments on the nature of, 10,1... 

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. 

Note that and are the mole fractions of L and H in the equilibrated system. 

The reason for the different behavior of 0 and 0 is quite simple. Suppose we start with an equilibrated system and follow the BI, 0 (X). If, at some point, we freeze-in the equilibrium L H, then at that point X = X (hence also... 

Note that we have switched from the notation a and b for the two sites to L and H. This has been done in order to emphasize the equivalence of the system b of this section with the system discussed in Section 3.5. There, L and H originated from freezing-in an equilibrated mixture H. Here, on the other hand, the PF in Eq. (4.4.9) pertains to a mixture of L and H, with = Mg=M, not necessarily originating from freezing-in an equilibrated system. 

This is also equivalent to a mixture of two different binding systems. However, here we stress the case of a mixture that is obtained from an equilibrated system. It is only in such a case that one might misinterpret spurious cooperativity as genuine see Section 4.8 for an experimental example. 

The calculation of IV (succinic) was carried out twice, for / (C = C ) = 1.3 A and / (Cju - Cjj) = 1.54 A. If the correlation were determined by the electrostatic interaction alone, we should have weakened the interaction upon increasing the proton-proton distance. In Table 4.4, we see that by increasing the C - C distance (keeping all other parameters fixed) we actually increase the value of IV. This is clearly due to the relatively large effect of the indirect positive correlation in the equilibrated system. Unfortunately, the relative contribution of the direct and indirect correlations cannot be determined from the experimental data. The computed values of IV = kgT In y(l, 1) are shown in brackets in Table 4.4. It is seen that a large negative contribution to IV(1,1) is due to the indirect cooperativity. 

It is clear from both Figs. 8.20 and 8.21 that CTP acts as an inhibitor while ATP acts as an activator. It is now established that both CTP and ATP bind to the regulatory binding site, which differs from the catalytic sites. The remarkable similarity between the equilibrium Bis on the one hand and the kinetic data on the other hand confirms the assertion made at the beginning of this chapter, that allosteric effects can be studied in equilibrated systems. 

First we show that if we start with an equilibrated system and freeze-in the conversion between the species a, then, at that point, the slope of the curve 0 (X) is larger than the slope of the curve 0 ( ),... 

We have seen how a comparison of the equilibrium constant estimated from kinetic data for the forward and reverse directions (i.e. K = kf/k ) with that obtained by measurements on the equilibrated system, may be used to provide strong support (or otherwise) for a particular reaction scheme (see also Chap. 8 Pd(II)). The kinetic approach may be useful also for providing information on thermodynamic data not otherwise easily available. 

Type III. Equilibrated systems with a relatively uniform oxygen isotope composition in all lithologies. These systems require a large water/rock ratio, temperatures between 500 and 800°C, and life times around 5 x 10 y. 

Provided we consider an equilibrated system, so that h 2 ti = t, then from Eq. (76), we have... 

Vanadium is present as V " in stoichiometric VPP however, the latter can host V " or V species as defects, without undergoing substantial structural changes (5,6). Therefore, the role of the different V species in the catalytic behavior of VPP in n-butane oxidation has been the object of debate for many years (7-9). Moreover, the catalyst may contain crystalline and amorphous vanadium phosphates other than (VO)2P207 (10) for instance, outer surface layers of V phosphates may develop in the reaction environment, and play active roles in the catalytic cycle. This is particularly true in the case of the fresh catalyst, while the equilibrated system (that one which has been kept under reaction conditions for 100 hours at least) contains only minor amounts of compounds with V species other than V. ... 

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