Equilibrium constants microscopic

Self-consistency of postulated forward and reverse rate equations and their coefficients can be tested with the principles of thermodynamic consistency and so-called microscopic reversibility. The former invokes the fact that forward and reverse rates must be equal at equilibrium. The latter is for loops of parallel pathways and for catalytic cycles. Thermodynamic consistency allows the reverse rate equation to be constructed from the forward one if at least one of its reaction orders is known, and requires the ratio of the products of the forward and reverse rate coefficients to be equal to the thermodynamic equilibrium constant. Microscopic reversibility leads to several useful conclusions The products of the clockwise and counter-clockwise rate coefficients of a loop must be equal the product of the forward rate coefficients of a catalytic cycle must be equal that of the reverse rate coefficients multiplied with the equilibrium constant of the catalyzed reaction forward and reverse reaction must occur along the same pathway and the ratio of the products of forward and reverse rate coefficients must be the same along all parallel pathways from same reactants to same products. The latter two rules apply regardless of whether or not any of the reactions are catalytic. 

Lack of termination in a polymerization process has another important consequence. Propagation is represented by the reaction Pn+M -> Pn+1 and the principle of microscopic reversibility demands that the reverse reaction should also proceed, i.e., Pn+1 -> Pn+M. Since there is no termination, the system must eventually attain an equilibrium state in which the equilibrium concentration of the monomer is given by the equation Pn- -M Pn+1 Hence the equilibrium constant, and all other thermodynamic functions characterizing the system monomer-polymer, are determined by simple measurements of the equilibrium concentration of monomer at various temperatures. 

Without further refinement, the value of the equilibrium constant for reaction (7-73) seemingly would depend on [H+]. Clearly, that supposition violates a very fundamental precept of thermodynamics. With the help of microscopic reversibility, the dilemma is resolved. It specifies that the equilibrium constant is separately realized for each pathway ... 

Both these methods require equilibrium constants for the microscopic rate determining step, and a detailed mechanism for the reaction. The approaches can be illustrated by base and acid-catalyzed carbonyl hydration. For the base-catalyzed process, the most general mechanism is written as general base catalysis by hydroxide in the case of a relatively unreactive carbonyl compound, the proton transfer is probably complete at the transition state so that the reaction is in effect a simple addition of hydroxide. By MMT this is treated as a two-dimensional reaction proton transfer and C-0 bond formation, and requires two intrinsic barriers, for proton transfer and for C-0 bond formation. By NBT this is a three-dimensional reaction proton transfer, C-0 bond formation, and geometry change at carbon, and all three are taken as having no barrier. 

Because Kcff applies to a scheme that involves more that one equilibrium (see Eq. (1.7)), it is referred to as a macroscopic equilibrium constant, to distinguish it from the microscopic equilibrium constants KA and E, which describe the individual equilibria. 

In contrast to the reactions of the cycloamyloses with esters of carboxylic acids and organophosphorus compounds, the rate of an organic reaction may, in some cases, be modified simply by inclusion of the reactant within the cycloamylose cavity. Noncovalent catalysis may be attributed to either (1) a microsolvent effect derived from the relatively apolar properties of the microscopic cycloamylose cavity or (2) a conformational effect derived from the geometrical requirements of the inclusion process. Kinetically, noncovalent catalysis may be characterized in the same way as covalent catalysis that is, /c2 once again represents the rate of all productive processes that occur within the inclusion complex, and Kd represents the equilibrium constant for dissociation of the complex. 

Table 2 gives rate and equilibrium constants for the deprotonation of and nucleophilic addition of water to X-[6+]. These data are plotted as logarithmic rate-equilibrium correlations in Fig. 5, which shows (a) correlations of log ftp for deprotonation of X-[6+] and log Hoh for addition of water to X-[6+] with logXaik and log KR, respectively (b) correlations of log(/cH)soiv for specific-acid-catalyzed cleavage of X-[6]-OH (the microscopic reverse of nucleophilic addition of water to X-[6+]) and log( H)aik for protonation of X-[7] (the microscopic reverse of deprotonation of X-[6+]) with log Xafc and log XR, respectively. 

Fig. 5 Logarithmic plots of rate-equilibrium data for the formation and reaction of ring-substituted 1-phenylethyl carbocations X-[6+] in 50/50 (v/v) trifluoroethanol/water at 25°C (data from Table 2). Correlation of first-order rate constants hoh for the addition of water to X-[6+] (Y) and second-order rate constants ( h)so1v for the microscopic reverse specific-acid-catalyzed cleavage of X-[6]-OH to form X-[6+] ( ) with the equilibrium constants KR for nucleophilic addition of water to X-[6+]. Correlation of first-order rate constants kp for deprotonation of X-[6+] ( ) and second-order rate constants ( hW for the microscopic reverse protonation of X-[7] by hydronium ion ( ) with the equilibrium constants Xaik for deprotonation of X-[6+]. The points at which equal rate constants are observed for reaction in the forward and reverse directions (log ATeq = 0) are indicated by arrows.

We recognize the right-hand side of the equation as the equilibrium constant K. We give the term microscopic reversibility to the idea that the ratio of rate constants equals the equilibrium constant K. 

The principle of microscopic reversibility demonstrates how the ratio of rate constants (forward to back) for a reversible reaction equals the reaction s equilibrium constant K. 

SAQ 8,22 A simple first-order reaction has a forward rate constant of 120 s 1 while the rate constant for the back reaction is 0.1 s F Calculate the equilibrium constant K of this reversible reaction by invoking the principle of microscopic reversibility. 

One could go on with examples such as the use of a shirt rather than sand reduce the silt content of drinking water or the use of a net to separate fish from their native waters. Rather than that perhaps we should rely on the definition of a chemical equilibrium and its presence or absence. Chemical equilibria are dynamic with only the illusion of static state. Acetic acid dissociates in water to acetate-ion and hydrated hydrogen ion. At any instant, however, there is an acid molecule formed by recombination of acid anion and a proton cation while another acid molecule dissociates. The equilibrium constant is based on a dynamic process. Ordinary filtration is not an equilibrium process nor is it the case of crystals plucked from under a microscope into a waiting vial. 

The principle we have applied here is called microscopic reversibility or principle of detailed balancing. It shows that there is a link between kinetic rate constants and thermodynamic equilibrium constants. Obviously, equilibrium is not characterized by the cessation of processes at equilibrium the rates of forward and reverse microscopic processes are equal for every elementary reaction step. The microscopic reversibility (which is routinely used in homogeneous solution kinetics) applies also to heterogeneous reactions (adsorption, desorption dissolution, precipitation). 

In the application of the principle of microscopic reversibility we have to be careful. We cannot apply this concept to overall reactions. Even Eqs. (4.43) - (4.45) cannot be applied unless we know that other reaction steps (e.g., diffusional transport) are not rate controlling. In a given chemical system there are many elementary reactions going on simultaneously. Rate constants are path-dependent (which is not the case for equilibrium constants)and may be changed by catalysts. For equilibrium to be reached, all elementary processes must have equal forward and reverse rates... 

An accurate knowledge of the thermochemical properties of species, i.e., AHf(To), S Tq), and c T), is essential for the development of detailed chemical kinetic models. For example, the determination of heat release and removal rates by chemical reaction and the resulting changes in temperature in the mixture requires an accurate knowledge of AH and Cp for each species. In addition, reverse rates of elementary reactions are frequently determined by the application of the principle of microscopic reversibility, i.e., through the use of equilibrium constants, Clearly, to determine the knowledge of AH[ and S for all the species appearing in the reaction mechanism would be necessary. 

Table 1 Microscopic and Macroscopic Equilibrium Constants for the Curves in Figs. 2 and 3... 

The principle of detailed balance is a result of the microscopic reversibility of electron kinetics. A prerequisite for the establishment of thermal equihbrium requires that the forward and reverse rates are identical. For isothermal reactions, the equilibrium constant remains unchanged. The principle of detailed balance is of fundamental importance to estabhsh helpful relations between reaction and equilibrium constants because both are at the initial thermal equilibrium in addition, at the new equihbrium after the relaxation of the perturbation, the net forward and reverse reaction rates are zero. 

It follows that the equilibrium constant K is given by kf/kr. The reverse reaction is inverse second order in iodide, and inverse first-order in H+. This means that the transition state for the reverse reaction contains the elements of arsenious acid and triiodide ion less two iodides and one hydrogen ion, namely, H2As03I. This is the same as that for the forward reaction, except for the elements of one molecule of water, the solvent, the participation of which cannot be determined experimentally. The concept of a common transition state for the forward and reverse reactions is called the principle of microscopic reversibility. 

Acid dissociation constants do not tell us which protons dissociate in each step. Assignments for pyridoxal phosphate come from nuclear magnetic resonance spectroscopy [B. Szpoganicz and A. E. Martell, Thermodynamic and Microscopic Equilibrium Constants of Pyridoxal 5 -Phosphate, J. Am. Chem. Soc. 1984,106, 5513]. 

The two monoprotonated forms of pyridoxine are the tautomeric pair shown in Eq. 6-75 and whose concentrations are related by the tautomeric ratio, R = [neutral form]/[dipolar ion], a pH-independent equilibrium constant with a value of 0.204/0.796 = 0.26 at 25°C.75 Evaluation of microscopic constants for dissociation of protons from compounds containing non-identical groups depends upon measurement of the tautomeric ratio, or ratios if more than two binding sites are present. In the case of pyridoxine, a spectrophotometric method was used to estimate R. 

For pyridoxine p/C, and pfC2 were determined spectrophotometrically as 4.94 and 8.89. These values, together with that of R given above, were used to estimate the microscopic constants that are given in Eq. 6-74.75 Notice that the microscopic constants of Eq. 6-74 are not all independent if any three of the five equilibrium constants are known the other two can be calculated readily. In describing and measuring such equilibria it is desirable to select one pathway of dissociation, e.g., H2P —> HP(A) —> P, and to relate the species HP(B) to it via the pH-independent constant R. 

The concept of apparent values is very useful, and it appears in other phenomena, such as in pKa values (section L3). Quite often, a pKa value does not represent the microscopic ionization of a particular group but is a combination of this value and various equilibrium constants between different conformational states of the molecule. The result is an apparent pKa which may be handled titri-metrically as a simple pKa. This simple-minded approach must not be taken too far, and, when one is considering the effects of temperature, pH, etc., on an apparent Km, one must realize that the rate-constant components of this term are also affected. The same applies to kcat values. The literature contains examples in which breaks in the temperature dependence of kcat have been interpreted as indicative of conformational changes in the enzyme, when, in fact, they are due to a different temperature dependence of the individual rate constants in cal, e.g., k2 and k2 in equation 3.22. 

From Eqs. (38) and (39) together with the KH values in Table XXI the microscopic equilibrium constants K ai and Ka2 can be calculated. 

By the principle of microscopic reversibility all the reactions of the set 0tr must be severally at equilibrium if the set as a whole is at equilibrium. The constants x are the logarithms of the conventional equilibrium constants. 

The same approach can be applied not only to the bulk equilibrium constants (K) but also to the microscopic connection processes (given the symbol k). Recall that the macroscopic equilibrium constant is simply the sum of all the microscopic equilibrium constants. For example, if an acid (H2A) has two non-equivalent ionisable protons there are two distinct but equivalent ways to remove a proton to produce HA- and hence there are two microscopic equilibrium constants kx and k2) for this deprotonation process. Thus the macroscopic acid dissociation constant, K = k1+k2. Don t get confused between microscopic equilibrium constants and rate constants, both of which have the symbol k. So, in terms of... 

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