Simple Isomerization Equilibrium

We start with the simplest case of a chemical reaction. Consider a system of N particles in an ideal-gas phase of volume V and temperature T. We assume that each particle has only two internal (say, electronic or vibrational) energy levels, which we denote by Ea and Eb, with degeneracies coa and cob, respectively. No other internal degrees of freedom are ascribed to the particles. 

q is the internal PF of a single particle. Thus, if we know the molecular parameters m, CO A, (Ob, Ea, Eb, Q can compute all the relevant thermodynamic quantities of our system. 

Until this point we have considered a system of one component. No chemical equilibrium was mentioned. Now we choose to view the same system from a different point of view. Each molecule of the system can be in one of the two states. We call a molecule occupying the state Aden A molecule, and a molecule occupying the state B di B molecule. We stress that this classification of molecules into two groups is purely theoretical. It does not depend on whether or not we can detect or isolate the A molecules or the B molecules. 

According to the general rule of ST, each term in the PF is proportional to the probability of the event characterized by that term. Here Q T, V, Na, Nb) is proportional to the probability of finding our (one-component) system in a state such that precisely Na molecules are in the state A and the rest, Nb molecules, in the state B. Thus 

With the help of this distribution function we can calculate three specific values of 

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Fig. 3. Variation of autocorrelation function with changes in the equilibrium constant in the fast reaction limit. A and B have different diffusion coefficients but the same optical (fluorescence) properties. This figure illustrates how, for the simple isomerization process, A B, a change in the diffusion coefficient is sufficient to indicate the progress of the reaction. This example is calculated for a two-dimensional (planar) system in the fast reaction limit (kf + k ) 4Dj /w2. Therefore, only a single diffusion process is...

No. The reaction catalyzed by phosphoglucose isomerase is a simple isomerization between an aldose and a ketose and involves the open-chain structures of both sugars. Since glucose 6-phosphate and fructose 6-phosphate are both reducing sugars, their Haworth ring structures are in equilibrium with their open-chain forms. This equilibration is very rapid and does not require an additional enzyme. Note that this isomerization reaction is of the same type as that catalyzed by triosephosphate isomerase. 

HYNES - You emphasized using ideas and techniques from near equilibrium kinetics to understand far from equilibrium phenomena. Have you considered now going back in the other direction For example, can you visualize a molecular system (probably more complex than say a simple isomerization) for which the double well plus Poisson noise far from equilibrium problem you described might apply ... 

Van der Zwan G and Hynes J T 1984 A simple dipole isomerization model for non-equilibrium solvation dynamics in reactions in polar solvents Chem. Phys. 90 21-35... 

Because of the severe demands placed on us for aceuraey if we tre to calculate an equilibrium constant, let us choose a simple reaction, the isomerization of but-2-eiic,... 

Case 1. Unimolecular Isomerization Consider a simple chemical reaction between species A and B at dynamic equilibrium ... 

The aim of the present study is to investigate the validity of the pairwise additivity of two-body and three-body potentials for He2Br2. These results are compared with ah initio calculations" and a simple model of the three-body potential is proposed to determine well depths and equilibrium structures for different isomeric configurations of the complex, as well as the minimum energy pathways through them. Additionally, variational methods are used to calculate the vibrational states of He2Br2. The wavefunctions of the lower states are analyzed in terms of probability distributions of the internal coordinates and the zero-point energy of the vdW cluster is evaluated. 

Understanding the relative rates of both productive heterocoupling and homodimerization reactions allows for the judicious selection of cross-partners that can participate in a highly selective CM reaction, even when equal stoichiometries of reactants are employed. There are five relevant equilibria and 10 rate constants in CM (Scheme 4 the rate constants for olefin E/Z isomerization have been excluded for simplicity). In the simple scenario where all the rates are similar, and the reaction can achieve equilibrium, the expected statistical cross-product yield is 50%. If, however, one olefin (e.g., R CH=CH2), as a consequence of either steric or electronic factors, reacts at a slower rate k- ) than the other reactions, such that k, k/ k-, and it is assumed that the productive cross-... 

The most active heterogeneous catalysts that are effective at room temperature are alkali metals supported on activated alumina. Simple C4-Cg alkenes, for example, were shown to yield equilibrium mixtures in short contact time over sodium on alumina.112116 Partial conversion of 1-butene at low temperature and in very short contact time (—60°C, 14 s) led to the stereoselective formation of m-2-butene.112 The changes in isomeric composition in the transformation of 4-methyl-l-pentene112 are as follows ... 

Carotenoids can be converted into mixtures of geometrical isomers under appropriate conditions, the most common being iodine catalyzed photoisomerization. This produces an equilibrium mixture of isomers, in general the all-trans isomers predominates. These isomers in an isomeric mixture cannot be measured separately by simple spectrophotometric determination. The usual method of subsequent measurement would be chromatographic separation, diode-array detection, and spectral analysis. In the absence of any definitive data on extinction coefficients for cfv-isomcrs, they are quantified against the all-trans isomer. Modem procedures involve the direct synthesis of c/.v-carotcnoids. 

When a mixture of cis- and frares-decalins 24 (61.5 38.5) are treated with the same solid superacid at 0°C, the thermodynamic equilibrium is rapidly achieved110 [Eq. (5.48)]. In the reaction, decalin 24 serves as the solvent, substrate, and the hydride donor. When the equilibrium is reached, the hydrocarbon can be separated from the catalyst by simple filtration. Perhydroindane 25 was also isomerized under... 

Let us note that in eqn. (59) the expressions f+ (c) and f (c) are the kinetic dependences that are written according to the law of mass action for the "natural brutto-reaction, i.e. for the reaction obtained by a simple addition of all cycle steps, and K fT) is the equilibrium constant for this reaction. However, as we mentioned above for the reaction of catalytic isomerization, the "natural brutto-equation should not necessarily have integer-valued coefficients. For the mechanism... 

To mimic c T-[(NH3)2Pt(H20)2]2+ with the more rapidly exchanging Pd11 and to prevent isomerization, it is necessary to employ the complex of ethylenediamine (en), [(en)Pd(H20)2]2+, that through chelation is necessarily cis. Upon titration with standard base an endpoint is reached after the addition of only one equivalent of base at pH 7.5, but the reversible titration curve is flattened on the pH axis and cannot be fitted with the equilibrium expression for a simple deprotonation. It was proposed that the mono-hydroxo complex dimerizes to abinuclear dihydroxo-bridged dimer [9], The two reactions and their equilibrium constant expressions follow. 

In this way, base catalyzes an equilibrium between the isomeric keto and enol forms of a carbonyl compound. For simple ketones and aldehydes, the keto form predominates. Therefore, a vinyl alcohol (an enol) is best described as an alternative isomeric form of a ketone or aldehyde. In Section 9-9F, we saw that an enol intermediate, formed by hydrolysis of an alkyne, quickly isomerizes to its keto form. 

Both the cis and trans geometrical isomers of chromic ferrioxamine B isomerize to equilibrium solutions with half-lives of several days at room temperature. This is considerably slower than that found for the simple tris hydroxamate complexes such as Cr(men)3 and is caused by the steric constraints of the ferrioxamine B ligand and its hexadentate chelation. 

N-Substituted 3-aminocrotonic esters (67) tend to adopt a planar or near-planar structure. As a consequence of their push-pull nature these molecules show an increased facility for Z-67-is-67 isomerization around the C=C double bond as well as restricted rotation around the C—N and C—COOR2 single bonds. The IR and XH-NMR spectra of simple 3-(alkylamino)crotonic esters have shown that these substances exist either in the liquid state or in solution as equilibrium mixtures of the Z and E configurations, respectively. The position of the equilibrium is solvent-dependent, and the energy difference between the isomers varies from ca 7.3 kJ mol-1 in non-polar solvents to ca 0.8 kJ mol"1 in dimethyl sulphoxide, the intramolecularly bonded Z-form 68 or 69 being the most stable111-113. 

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