Equilibrium constant alternate forms

Alternative forms of the equilibrium constant can be obtained as we express the relationship between activities, and pressures or concentrations. For example, for a gas phase reaction, the standard state we almost always choose is the ideal gas at a pressure of 1 bar (or 105 Pa). Thus... 

Equations 5.1.5, 5.1.6, and 5.1.8 are alternative methods of characterizing the progress of the reaction in time. However, for use in the analysis of kinetic data, they require an a priori knowledge of the ratio of kx to k x. To determine the individual rate constants, one must either carry out initial rate studies on both the forward and reverse reactions or know the equilibrium constant for the reaction. In the latter connection it is useful to indicate some alternative forms in which the integrated rate expressions may be rewritten using the equilibrium constant, the equilibrium extent of reaction, or equilibrium species concentrations. 

A reaction dataset, such as the LLNL database, provides an alternative method for balancing reactions. Such a database contains reactions to form a number of aqueous species, minerals, and gases, together with the corresponding equilibrium constants. Reactions are written in terms of a generic basis set B, which probably does not correspond to set B, our choice of species to appear in the reaction. 

The two-state model was used to test whether characteristics of the low-temperature cryosolvent cause the equilibrium constant for complex formation, K(T), to fall precipitously as the temperature is lowered through T ij. In this case, the slow phase that appears below 250 K would correspond to un-complexed ZnCcP. This interpretation fails because within the transition range the fraction, f(T), is unaffected by a ten-fold reduction in the ratio, R = [Cc]/[CcP], whereas use of K(T) calculated from f(T) would predict a larger shift of f(T). Alternatively, the two-state model would apply if a low-temperature form of the complex were created by a change in ligation of either ZnP or FeP. 

SOLUBILITY PRODUCT. For ions that combine and form an insoluble phase, we use an alternative form of an equilibrium constant known as the solubility product. Consider the following reaction ... 

An alternate form of the rate constant predicted by transition state theory using a statistical mechanical approach for the equilibrium constant K is Eq. (R) ... 

In such a case, the condition that the polymer is formed by a mixed association of monomer units would be reflected in the numerical value of the association equilibrium constant. The value for unlike association differs from that for like association by a factor pp since the Keq is referred to Ma moles of monomer for like particle association and to Ma/p moles of each monomer for unlike particle association, while the measurements are referred to systems both containing Ma moles of monomer. Alternately it can be shown that this factor arises from a statistical entropy contribution. 

Suppose that an enzyme E has an alternative conformation E and that both take part in ionization equilibria. This happens with a-chymotrypsin on deprotonation of Ile-16, which is protonated in the active conformation. The equilibrium constants in scheme 3 are defined by K = [E ]/[E] K = [E H+]/[EH+] Ka = [H+][E]/[EH+] and K A = [H+][E ]/[E H+], The apparent equilibrium constant between E and E must consider all ionic forms and is defined by... 

Eq 18 is applicable to equilibrium constants in alternate form. For ideal gases the equilibrium constant Kp is expressed in terms of partial pressures (rather than fugacities) of products and reactants. Still another form of the equilibrium constant. JC. is exnressed in terms nf... 

Alternate forms of the equilibrium constant do vary with pressure. For example,... 

For such unstable carbocations, an alternative approach to pAR can be developed, by recognizing the relationship that exists between pATR and pAa implied in Equation (15) (p. 30). For carbocations with [3-hydrogen atoms, loss of a proton normally yields an alkene. Then, as discussed by Richard, pATR and pAa form two arms of a thermodynamic cycle, of which the third is the equilibrium constant for hydration of the alkene, pAH2o, as already indicated in Scheme 1. The relationship between these equilibrium constants is shown for the t-butyl cation in Scheme 4. In the scheme the equilibria are... 

When liquid and gas phases are both present in an equilibrium mixture of reacting species, Eq. (11.30), a criterion of vapor/liquid equilibrium, must be satisfied along with the equation of chemical-reaction equilibrium. There is considerable choice in the method of treatment of such cases. For example, consider a reaction of gas A and water B to form an aqueous solution C. The reaction may be assumed to occur entirely in the gas phase with simultaneous transfer of material between phases to maintain phase equilibrium. In this case, the equilibrium constant is evaluated from AG° data based on standard states for the species as gases, i.e., the ideal-gas states at 1 bar and the reaction temperature. On the other hand, the reaction may be assumed to occur in the liquid phase, in which case AG° is based on standard states for the species as liquids. Alternatively, the reaction may be written... 

A similar synthesis can be used to generate DBr from benzoyl bromide if it is desired to study DBr and HBr in addition or as an alternative to DCl and HCl. In fact, if an equimolar mixture of the benzoyl halides is used along with D2O, which is about 95 percent D, the product mix will contain all isotopic forms with reasonable intensity ratios for an infrared scan of aU four species in a single ceU. Such a mixture can also be used for the determination of equilibrium constant Kp for the H D exchange reaction... 

For consecutive reactions in which the desired product is formed in an intermediate step, excess reactant can be used to suppress additional series reactions by keeping the intermediate-species concentration low. A reactive distillation can achieve the same result by removing the desired intermediate from the reaction zone as it is formed. Similarly, if the equilibrium constant of a reversible reaction is small, high conversions of one reactant can be achieved by use of a large excess of the other reactant. Alternatively, by Le Chatelier s principle, the reaction can be driven to completion by removal of one or more of the products as they are formed. Typically, reactants can be kept much closer to stoichiometric proportions in a reactive distillation. 

Measurement of the in vitro efficacy of compounds as substrates is usually deduced by comparison of their k JK ratios where is the first-order rate constant for product formation and is the Michaelis equilibrium constant [38]. For those compounds which are classical, reversible inhibitors, K, the dissociation (or inhibition) equilibrium constant, and (kassoc) the rate constant for enzyme inhibition, are the most commonly reported kinetic values. These values may be measured while using either a high-molecular-weight natural substrate or a low-molecular-weight synthetic substrate. For alternate-substrate inhibitors, that is, compounds which form a stable complex (an acyl-enzyme ) that dissociates to enzyme and intact inhibitor or to enzyme and an altered form of the inhibitor, the usually reported value is K, the apparent K. For compounds which irreversibly inactivate the enzyme, the kinetics are usually measured under conditions such that the initial enzyme concentration [E] is much lower than the inhibitor concentration [I] which in turn is much lower than the Ky Under these conditions the commonly reported value is obs/[I]> the apparent... 

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