Equilibrium general statements about

Le Chatelier s principle is a general statement about how any system in equilibrium—not just a chemical system—responds to change. 

Since the pioneering studies of Fischer, carbene complexes have become the subjects of very extensive investigations in several laboratories, and they have become known for a large number of the transition metals.100 A fairly general statement about their chemistry is that these compounds usually are neither prepared directly from free carbenes nor are they to be considered as sources of free carbenes, i.e. free carbenes are generally not in equilibrium with the carbene complexes. Instead, they are prepared by indi-... 

Notice that the subscript c has been left off K in these general statements. This reflects the fact that there are other equilibrium constants that these statements apply to, not just equilibrium constants involving concentrations. You will learn about other types of equilibrium constants in the next two chapters. For the rest of this chapter, though, you will continue to see the subscript c used. 

It is, of course, not easy to make statements about the relative contributions of phase boundary and diffusion potentials. Since the electrochemical behavior of membranes is generally reflected by the total membrane potential, we did not try to differentiate in this respect. The models described in my report may, however, approximate the selectivity of certain membrane systems in the equilibrium domain even when assuming the absence of diffusion potentials. 

Probably the only general statement which can be made about the experimental studies on zeolites is that the majority of published data is inapplicable directly to natural minerals. This is due either to the excessively high temperatures under which the experiments are performed, outside of the physical limits of zeolite stability, or to short time spans of observation which do not allow the silicates to come to equilibrium with the fluids of the experiments. Those studies designed to determine zeolite stability indicate that the most silica-poor alkali zeolite, analcite, is not stable above 180°C. More silica-rich species will be found below this temperature. However, the reasons for the crystallization of one or another of the silica-rich alkali zeolites are not yet elucidated. 

When discussing the general aspects of FTNMR, we have to remember that all principal statements about Fourier methods have been introduced for a strictly linear system (mechanical oscillator) in Chapter 1. In Chapter 2, on the other hand, we have seen that the nuclear spin system is not strictly linear (with Kramer-Kronig-relations between absorption mode and dispersion mode signal >). Moreover, the spin system has to be treated quantummechanically, e.g. by a density matrix formalism. Thus, the question arises what are the conditions under which the Fourier transform of the FID is actually equivalent to the result of a low-field slow-passage experiment Generally, these conditions are obeyed for systems which are at thermal equilibrium just before the initial pulse but are mostly violated for systems in a non-equilibrium state (Oberhauser effect, chemically induced dynamic nuclear polarization, double resonance experiments etc.). 

Polyphosphates do not precipitate metal ions from solution, but hold them in solution as soluble complexes, provided the ratio of cations to phosphates does not exceed some value related to a metastable solubility product. It should be obvious that a true solubility product does not exist for the same reasons a true solubility does not exist. If polyphosphates become overwhelmed, a precipitate will form and condensed phosphates function as orthophosphates in removing metal ions from solution. Rather than attempting to include an in-depth study of complexing agents and stability constants, a general statement will be made and readers are referred to published literature for a more comprehensive view. Perhaps the most perplexing issue about complexing, and equilibrium constants used to study and represent it, is the number of different constants used to represent a system. 

An alternative statement of this rule is that a molecule cannot be optically active if it is in equilibrium with a structure (or a conformation) that is achiral. Inherently chiral compounds have NO achievable achiral conformations. Because conformers differ only by rotations about single bonds, they are generally in equilibrium at room temperature. We can consider cyclohexane rings as though they were flat (the most symmetric conformation), and we should consider straight-chain compounds in their most symmetric conformations (often an eclipsed conformation). 

Equilibrium statistical mechanics is a first principle theory whose fundamental statements are general and independent of the details associated with individual systems. No such general theory exists for nonequilibrium systems and for this reason we often have to resort to ad hoc descriptions, often of phenomenological nature, as demonstrated by several examples in Chapters 1 and 8. Equilibrium statistical mechanics can however be extended to describe small deviations from equilibrium in a way that preserves its general nature. The result is Linear Response Theory, a statistical mechanical perturbative expansion about equilibrium. In a standard application we start with a system in thermal equilibrium and attempt to quantify its response to an applied (static- or time-dependent) perturbation. The latter is assumed small, allowing us to keep only linear terms in a perturbative expansion. This leads to a linear relationship between this perturbation and the resulting response. 

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