Multiple equilibrium

FIGURE 3 Multiple equilibria computed for present-day climate (a) and for the climate of the last glacial maximum (c). For mid-Holocene conditions, only one solution is obtained (b). A summary of the results of Claussen (1997) (a), Claussen and Gayler (1997) (b), and Kubatzki and Claussen (1998) (c). 

Interestingly, the stability of the atmosphere-vegetation system seems to change with time experiments with mid-Holocene vegetation yield only one solution, tlie green Sahara (Claussen and Gayler, 1997), while two solutions exist for the Last Glacial Maximum (LGM) (Kubatzki and Claussen, 1998). 

So far, no other regions on earth in which multiple eciuilibria could evolve on a large scale have been identified Levis et al. 

Why do we find multiple solutions in the subtropics, but none at high latitudes—and why for the present-day and LGM climates, but not for mid-Holocene climate Claussen et al. (1998) analyze large-scale atmospheric pattern in present-day, mid-Holocene, and LGM climates. They find that velocity potential 

A more ecological interpretation of multiple equilibria is given by Brovkin et al. (1998). They develop a conceptual model of vegetation-precipitation interaction in the western Sahara which is applied to interpret the results of comprehensive models. The conceptual model finds three solutions for present-day and LGM climate one of these, however, is unstable to infinitesimally small perturbations. The humid solution is shown to be less probable than the arid solution, and this explains the existence of the Sahara desert as it is today. For mid-Holocene climate, only one solution is obtained. Application of the conceptual model to biospheric feedbacks at high latitudes (Levis et al, 1999) yields only one solution for the present-day conditions. 

Consider the general case, in which a macromolecule P has n identical binding sites, and reacts with a ligand A,. one molecule of which can become attached to each of the sites. Attachment of one molecule of A to P has an association constant 

In the last reaction, all the sites have become occupied. The total concentration of bound A is given by 

If we make use of the equilibrium expressions such as (11.10) through (11.12), we see that 

The simplest situation is when all of the sites are the same and there is no cooperativity, i.e., binding at any site is independent of the occupation of other sites. We would then expect (11.18) to reduce to the Langmuir form..The reason it does so may be seen as follows. 

At first sight, we might think that we should set all the K s equal i.e., 

Another frequently used technique involves the detection of some physical or chemical change in either biomacromolecule or the ligand upon binding, such as differential spectrometry or changes in biological activities. The fractional saturation, 0, which expresses the fraction of sites in biomacromolecule occupied by ligand, A can be written as 

The primary tritium and deuterium isotope effects were found to be 0.91 and 0.62 ppm, respectively, again suggesting either a very strong hydrogen bond or a tautomeric equihbrium. 

In the soHd state, a similar large deuterium isotope effect on the C-1 CS is observed. No temperature variation was found, but as the mole fraction was close 

Isotope effects on chemical shifts can be used to reveal tautomeric equilibria in symmetric as well as asymmetric tautomeric systems. The isotope effects may also provide information about hydrogen-bond potentials. A number of different isotope effects, short-range as well as log-range, may be used. Deuterium isotope effects are by far the most useful, both from an effect point of view and also because deuterium is easy to introduce in most situations. Heavy isotopes are not used extensively. and are excellent as observing nuclei for secondary isotope 

Primary isotope effects have the advantage that the chemical shifts of the tautomers usually can be estimated well based on the enormous amount of data available. The drawback is that only a single number is obtained. They are therefore often used in conjunction with other data. 

Theoretical calculations are becoming more and more reliable, and it can be foreseen that these will play a much larger role in the future. 

UV/VIS spectroscopy (just referred to as UV in the following text) has a great advantage of being a very sensitive technique. This means that, in concentration terms, it will supplement NMR and IR results that are usually obtained at a higher concentration. However, the low concentration used in this technique requires 

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So far, as in Equation (3.33), the hydrolyses of ATP and other high-energy phosphates have been portrayed as simple processes. The situation in a real biological system is far more complex, owing to the operation of several ionic equilibria. First, ATP, ADP, and the other species in Table 3.3 can exist in several different ionization states that must be accounted for in any quantitative analysis. Second, phosphate compounds bind a variety of divalent and monovalent cations with substantial affinity, and the various metal complexes must also be considered in such analyses. Consideration of these special cases makes the quantitative analysis far more realistic. The importance of these multiple equilibria in group transfer reactions is illustrated for the hydrolysis of ATP, but the principles and methods presented are general and can be applied to any similar hydrolysis reaction. 

A property of K that you will find very useful in this and succeeding chapters is expressed by the rale of multiple equilibria, which states... 

The product expression, at the right, is clearly the equilibrium constant for the overall reaction, as it should be according to the rule of multiple equilibria. 

Because Equation (1) + Equation (2) = Equation (3), we have, according to the rule of multiple equilibria (Chapter 12),... 

Strategy Apply the rule of multiple equilibria to find K. Then work with the expression for K to find the molar solubility of AgCL... 

Multiple equilibria, rule of A rule stating that if Equation 1 + Equation 2 = Equation 3, then K3 = K1X K2,370,439 Multiple proportions, 27... 

An important class of cycles with non-linear behavior is represented by situations when coupling occurs between cycles of different elements. The behavior of coupled systems of this type has been studied in detail by Prigogine (1967) and others. In these systems, multiple equilibria are sometimes possible and oscillatory behavior can occur. There have been suggestions that atmospheric systems of chemical species, coupled by chemical reactions, could exhibit multiple equilibria under realistic ranges of concentration (Fox et ai, 1982 White, 1984). However, no such situations have been confirmed by measurements. 

Now we are ready to organize the data and the unknowns and do the calculations. The spectator ion is easiest to deal with [K- -] = 2 [S03 -] = 2 (0.075 M) = 0.15 M There are multiple equilibria affecting ion concentrations, so we must work with more than one concentration table, starting with the dominant equilibrium. Set up a concentration table to determine concentrations of the ions generated by this reaction ... 

In principle, the calculation of concentrations of species of a complexation equilibrium is no different from any other calculation involving equilibrium constant expressions. In practice, we have to consider multiple equilibria whenever a complex is present. This is because each ligand associates with the complex in a separate process with its own equilibrium expression. For instance, the silver-ammonia equilibrium is composed of two steps ... 

R.Tauler and E. Casassas, Application of principal component analysis to the study of multiple equilibria systems — Study of Copper(ll) salicylate monoethanolamine, diethanolamine and triethanolamine systems. Anal. Chim. Acta, 223 (1989) 257-268. 

Measurement of pH-dependent equilibria can also be used to identify coordination isomerization reactions in addition to stepwise dissociation, such as in the case of the iron(III) complex of exochelin MN (59). Here, a combination of spectrophotometric and potentiometric titration characterized multiple equilibria involving second-sphere protonation, coordination isomerization, and stepwise dechelation, and is illustrated in Fig. 8. 

Words that can be used as topics in essays 5% rale buffer common ion effect equilibrium expression equivalence point Henderson-Hasselbalch equation heterogeneous equilibria homogeneous equilibria indicator ion product, P Ka Kb Kc Keq KP Ksp Kw law of mass action Le Chatelier s principle limiting reactant method of successive approximation net ionic equation percent dissociation pH P Ka P Kb pOH reaction quotient, Q reciprocal rule rule of multiple equilibria solubility spectator ions strong acid strong base van t Hoff equation weak acid weak base... 

This problem involves the concept of multiple equilibria. The dissociation constants given in the example are related to the following reactions ... 

For multiple equilibria dissociation constants (such as polyprotic acids), K for the overall reaction is the product of the equilibrium constants for the individual reactions. Therefore,... 

The relaxation approach has played an important role in our understanding of the mechanisms of complex formation in solution (Chap. 4) 39,i4o -pjjg qj computer programs has now eased the study of multiple equilibria. For example, four separate relaxation effects with t s ranging from 100 xs to 35 ms are observed in a temperature-jump study of the reactions of Ni with flavin adenine dinucleotide (fad) (Eqn. (8.121)). The complex relaxation... 

Steinhardt, J. Reynolds, J. A. "Multiple Equilibria in Proteins" Academic Press New York, 1969 234 pp. 

Steinhardt, J., and Reynolds, J. A., Multiple Equilibria in Proteins. Academic Press, New York, 1970. 

The dinuclear Sr complex of the ditopic ligand 17 increases the rate of basic ethanolysis ofthe malonate derivative 19 by a surprising 5700-fold, but increases by only 9.5-fold the rate of cleavage of 14 [28]. It is remarkable that such a huge rate-enhancement occurs under extremely dilute conditions, namely 15 pM 19 and 30 pM 17-Sr2. A slightly lower rate enhancement is observed in the presence of 17-Ba2. It seems likely that under the dilute conditions of the catalytic experiments several crown-complexed metal species occur simultaneously (Scheme 5.4). Given the plethora of species involved in such a complicated system of multiple equilibria, quantitative kinetic treatment is out of reach. Nevertheless, a comparison with the reactivity of model compounds, particularly that of the malonate derivative 20, provides insight into the composition of the reactive intermediate (Table 5.8),... 

The choice between driving on the right-hand or left-hand side of the road is one with multiple equilibria, since countries... 

Multiple equilibria pose a formidable problem for the rational-expectations argument. A convention equilibrium like rules of the road cannot emerge by rational expectations, if the situation offers no clue to what others will do. (I shall have more to say about clues later.) Multiple equilibria with different winners and losers, as in the second version of the erosion story, are even less hospitable to rational expectations. In this circumstance many things might happen. The situation might remain indefinitely out of equilibrium. The realization of one equilibrium rather than another could happen by accident. One set of individuals might be sufficiently powerful to impose the equilibrium that favored them over other people. What can be ruled out is the realization of one equilibrium by tacit coordination and rational anticipation. 

The greatest obstacle to planning arises from the lack of reliable theories of society. What we have is a toolbox of mechanisms, not a set of laws. We cannot predict how rational people behave under conditions of uncertainty or multiple equilibria, nor whether their behavior will be governed by rationality or by social norms. Even more fundamentally, we cannot predict how preferences and norms might themselves come to change as a result of policy reforms. We cannot predict technical change. As... 

Such reactions can occur on oxides of Fe, Al, and Ti. Carboxylic acids and phenols are common reactive moieties of the sorbates /. Often the resultant bound species is not really known. Given this additional sorption mechanism, the situation becomes more complicated, and multiple equilibria must be modeled (Stone et al. 1993) to account for all species ... 

These assumptions are identical to those in the ideal theory of multiple equilibria used to analyze low molecular weight ligand binding to proteins 4-47 49). Figure 7 shows how these assumptions can be considered in more refined treatments of adsorption from solution 45). 

Fig. 11, Scheme for the analysis of protein adsorption data using the methods of multiple equilibria of proteins... 

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