Substitution equilibria

Equation 33 is the familiar statistical relation between the equilibrium constants in a series of stepwise equilibria as derived by N. Bjerrum (II) for polyprotic acids and applied by J. Bjerrum (12) to complex ion equilibria. Substituting Equation 33 for K, into Equations 23 and 24 for 0f° and 0o° gives... 

Perdue, E.M, "Thermodynamics of Acid-Base Equilibria. Substituted Anilinium Ions, Pyridinium Ions and Thiophenol." Ph.D. Thesis, Georgia Inst. Tech., Atlanta, 1973. 

Among the different HBDs studied in the analysis of PT equilibria, substituted phenols are certainly the most versatile models for several structural reasons ... 

Structure of molecules ch4 What controls equilibria Substitution reactions at saturated C... 

You can see that, for every mole of Pb(OH)2 that dissolves, one mole of OH" is used up, and one mole of Pb(OH)f is formed. If we let s = moles of Pb(OH)2 that dissolve per liter, then s moles of Pb(OH)3 will be formed per liter, and there will remain (1.00 — s) moles of OH- per liter at equilibrium. Substitution of these equilibrium concentrations into the K, st expression gives... 

Pteridine in liquid ammonia at - 40° gives a mixture of the mono- and bisammonates (213 and 214) which consists of 60% 214 and 40% 213 at equilibrium.358 Higher temperatures more heavily favor the bis-adduct at equilibrium. Substituted pteridines behave similarly. These results are exactly analogous to the formation of mono- and bis-covalent hydrates by the pteridine cation in aqueous acid.359... 

When the acid dissociations are at equilibrium, substituting the expressions for the two acid dissociation constants of ATP yields... 

The presumption of reversibility implies that both of these processes are simultaneously at equilibrium. Substituting (5.4.72) into (5.4.71), we obtain... 

Recall from Chapter 13 that this activity term is referred to as Q rather than K because it refers to a metastable equilibrium. Substitution of equations (18.15) in their partial molar form gives... 

The intramolecular insertion of alkynes into a Pd-C bond has been observed and kinetically studied. The reaction involves a pre-equilibrium substitution of a phosphine ligand by the alkyne moiety, followed by rate determining insertion in a four-coordinate intermediate (Scheme 6.42). The longer the spacer chain (n) the more favorable ligand substitution K = 2.0(9) for = 2 versus K = 4.40(2) for n = 3) although the opposite is observed for the insertion step ( 2 = 7(3) for n = 2 versus k2 = 0.301(2) for n = 3). It seems that the short chain alkyne intermediate (n = 2) is strained enough to deviate from the usual perpendicular arrangement and adopts a conformation that places the alkyne closer to coplanarity and to the insertion transition state [133]. 

This is known as the Continuity equation. Equation (2.49) can be used to define the stress equilibrium. Substituting strain rates for strains, t for /a and using u=0.5, the viscous analogy of Hooke s Law can be written by summing the dilatational and deviatoric components (Eq. (2.73))... 

Phase rule. Generally in a liquid-liquid system we have three components. A, B, and C, and two phases in equilibrium. Substituting into the phase rule, Eq. (10.2-1), the number of degrees of freedom is 3. The variables are temperature, pressure, and four concentrations. Four concentrations occur because only two of the three mass fraction concentrations in a phase can be specified. The third must make the total mass fractions total to 1.0, +xg + Xc = 1.0. If pressure and temperature are set, which is the usual case, then, at equilibrium, setting one concentration in either phase fixes the system. 

Here we have used the fact that dp — 0 because the system is assumed to be in mechanical equilibrium. Substituting (16.8.4) into (16.8.1) we obtain... 

The most reliable estimates of the parameters are obtained from multiple measurements, usually a series of vapor-liquid equilibrium data (T, P, x and y). Because the number of data points exceeds the number of parameters to be estimated, the equilibrium equations are not exactly satisfied for all experimental measurements. Exact agreement between the model and experiment is not achieved due to random and systematic errors in the data and due to inadequacies of the model. The optimum parameters should, therefore, be found by satisfaction of some selected statistical criterion, as discussed in Chapter 6. However, regardless of statistical sophistication, there is no substitute for reliable experimental data. 

Substitution of Equations (2) and (3) into the equilibrium relations dictated by Equation (2-l)[Pg.99]

If the equilibrium ratios were known or specified. Equation (7-8) could be substituted in Equation (7-6) or Equation (7-9) in (7-7) to give implicit relations for a ... 

Hammen equation A correlation between the structure and reactivity in the side chain derivatives of aromatic compounds. Its derivation follows from many comparisons between rate constants for various reactions and the equilibrium constants for other reactions, or other functions of molecules which can be measured (e g. the i.r. carbonyl group stretching frequency). For example the dissociation constants of a series of para substituted (O2N —, MeO —, Cl —, etc.) benzoic acids correlate with the rate constant k for the alkaline hydrolysis of para substituted benzyl chlorides. If log Kq is plotted against log k, the data fall on a straight line. Similar results are obtained for meta substituted derivatives but not for orthosubstituted derivatives. 

By substituting relations (26) into equations (24), (25) we obtain the general solution of the equilibrium equations... 

If the surface is to be in mechanical equilibrium, the two work terms as given must be equal, and on equating them and substituting in the expressions for dx and dy, the final result obtained is... 

A simple, non-selective pulse starts the experiment. This rotates the equilibrium z magnetization onto the v axis. Note that neither the equilibrium state nor the effect of the pulse depend on the dynamics or the details of the spin Hamiltonian (chemical shifts and coupling constants). The equilibrium density matrix is proportional to F. After the pulse the density matrix is therefore given by and it will evolve as in equation (B2.4.27). If (B2.4.28) is substituted into (B2.4.30), the NMR signal as a fimction of time t, is given by (B2.4.32). In this equation there is a distinction between the sum of the operators weighted by the equilibrium populations, F, from the unweighted sum, 7. The detector sees each spin (but not each coherence ) equally well. 

The use of isotopic substitution to detennine stmctures relies on the assumption that different isotopomers have the same stmcture. This is not nearly as reliable for Van der Waals complexes as for chemically bound molecules. In particular, substituting D for H in a hydride complex can often change the amplitudes of bending vibrations substantially under such circumstances, the idea that the complex has a single stmcture is no longer appropriate and it is necessary to think instead of motion on the complete potential energy surface a well defined equilibrium stmcture may still exist, but knowledge of it does not constitute an adequate description of the complex. 

The logarithm of the equilibrium constant, K,. for the chemical equation shown in Figure 3-8a for a substituted benzoic acid can be related to the logarithm of the... 

The first term in this expansion, when substituted into the integral over the vibrational eoordinates, gives ifj(Re) , whieh has the form of the eleetronie transition dipole multiplied by the "overlap integral" between the initial and final vibrational wavefunetions. The if i(Rg) faetor was diseussed above it is the eleetronie El transition integral evaluated at the equilibrium geometry of the absorbing state. Symmetry ean often be used to determine whether this integral vanishes, as a result of whieh the El transition will be "forbidden". 

In Chapter 2 the Diels-Alder reaction between substituted 3-phenyl-l-(2-pyridyl)-2-propene-l-ones (3.8a-g) and cyclopentadiene (3.9) was described. It was demonstrated that Lewis-acid catalysis of this reaction can lead to impressive accelerations, particularly in aqueous media. In this chapter the effects of ligands attached to the catalyst are described. Ligand effects on the kinetics of the Diels-Alder reaction can be separated into influences on the equilibrium constant for binding of the dienoplule to the catalyst (K ) as well as influences on the rate constant for reaction of the complex with cyclopentadiene (kc-ad (Scheme 3.5). Also the influence of ligands on the endo-exo selectivity are examined. Finally, and perhaps most interestingly, studies aimed at enantioselective catalysis are presented, resulting in the first example of enantioselective Lewis-acid catalysis of an organic transformation in water. 

The relative basicities of aromatic hydrocarbons, as represented by the equilibrium constants for their protonation in mixtures of hydrogen fluoride and boron trifluoride, have been measured. The effects of substituents upon these basicities resemble their effects upon the rates of electrophilic substitutions a linear relationship exists between the logarithms of the relative basicities and the logarithms of the relative rate constants for various substitutions, such as chlorination and... 

The best-known equation of the type mentioned is, of course, Hammett s equation. It correlates, with considerable precision, rate and equilibrium constants for a large number of reactions occurring in the side chains of m- and p-substituted aromatic compounds, but fails badly for electrophilic substitution into the aromatic ring (except at wi-positions) and for certain reactions in side chains in which there is considerable mesomeric interaction between the side chain and the ring during the course of reaction. This failure arises because Hammett s original model reaction (the ionization of substituted benzoic acids) does not take account of the direct resonance interactions between a substituent and the site of reaction. This sort of interaction in the electrophilic substitutions of anisole is depicted in the following resonance structures, which show the transition state to be stabilized by direct resonance with the substituent ... 

Streitwieser pointed out that the eorrelation whieh exists between relative rates of reaetion in deuterodeprotonation, nitration, and ehlorination, and equilibrium eonstants for protonation in hydrofluorie aeid amongst polynuelear hydroearbons (ef. 6.2.3) constitutes a relationship of the Hammett type. The standard reaetion is here the protonation equilibrium (for whieh p is unity by definition). For eon-venience he seleeted the i-position of naphthalene, rather than a position in benzene as the referenee position (for whieh o is zero by definition), and by this means was able to evaluate /) -values for the substitutions mentioned, and cr -values for positions in a number of hydroearbons. The p -values (for protonation equilibria, i for deuterodeprotonation, 0-47 for nitration, 0-26 and for ehlorination, 0-64) are taken to indieate how elosely the transition states of these reaetions resemble a cr-eomplex. 

For this class of thiazoles most of the chemical and physicochemical studies are centered around the protomeric equilibrium and its consequences. The position of this equilibrium may be determined by spectroscopic and titrimetric methods, as seen in each section. A simple HMO (Hiickel Molecular Orbitals) treatment of 2-substituted compounds however, may, exemplify general trends. This treatment considers only protomeric forms 1 and 2 evidence for the presence of form 3 has never been found. The formation energy reported in Table 1 is the energy difference in f3 units. 

Physicochemical studies on aminothiazoles are mainly centered on two problems the position of imino-amino protomeric equilibrium and IsRR substitution effects on the thiazole nucleus. 

Charge diagrams suggest that the 2-amino-5-halothiazoles are less sensitive to nucleophilic attack on 5-position than their thiazole counterpart. Recent kinetic data on this reactivity however, show, that this expectation is not fulfilled (67) the ratio fc.. bron.c.-2-am.noih.azoie/ -biomoth.azoie O"" (reaction with sodium methoxide) emphasizes the very unusual amino activation to nucleophilic substitution. The reason of this activation could lie in the protomeric equilibrium, the reactive species being either under protomeric form 2 or 3 (General Introduction to Protomeric Thiazoles). The reactivity of halothiazoles should, however, be reinvestigated under the point of view of the mechanism (1690). 

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