Alternate Forms of the Equilibrium Constant

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 

The ratio of pressures in equation (9.10) is known as Kp and the ratio of fugacity coefficients is written as JQ. Hence, 

Finally, p, is related to x, the mole fraction in the gas phased by Dalton s law 

Alternate expressions can also be written for equilibrium constants in condensed phase reactions. For example, for the reaction 

The activity product aH-fioi-r is given the symbol Kw so that 

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

Alternative Forms of the Equilibrium Constant Dependence of A" on the direction of reaction and the balancing coefficients... 

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) ... 

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... 

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. 

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. 

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... 

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... 

This form of the fundamental equation, which applies at equilibrium, indicates that the natural variables for this system are T, nAx, and nAfi. Alternatively, P, nAx, and nAp could be chosen. Specification of the natural variables gives a complete description of the extensive state of the system at equilibrium, and so the criterion of spontaneous change and equilibrium is dG < 0 at constant 7( nAz, and nA/l or... 

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... 

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... 

Recycled solutions of 2 initiated ROMP as quickly as the recycled Ru(III) solutions, closer examination of which revealed NMR resonances identical to those of the alkene protons in 2 [25]. It was therefore suggested that a key step in the initiation process using Ru(III) was the in situ formation of a Ru(II)-alkene complex [27]. Current evidence supports the disproportionation of the Ru(III) species to form Ru(II) and Ru(IV) species, followed by formation of a Ru(II)-alkene complex [25]. The equilibrium constant for disproportionation is small, accounting for the poor initiation efficiency of the Ru(III) systems [30]. An alternative, the disproportionation of an equilibrium amount of Ru(III)-alkene complex to a Ru(II)-alkene complex and a Ru(IV) species, is unlikely since Ru(III)-alkene complexes are generally unstable. Formation of a ruthenium alkylidene, the requi-... 

PL shows a possible titration curve of a ligand that does not initially fit into the ATP-binding pocket. The ligand binds to the protein building an intermediate state. This intermediate state is in a conformational equilibrium with the complex. The titration ends with two separate peaks. This ensemble of titration curves is incompatible with a key/bolt mechanism described in (a), (c) The model P- -L-o-P +L< PL depicts possible titration curves of a weak binder to an alternative conformation of the kinase (e.g., the putative DFG-in/DFG-out). A conformational equilibrium already exists in the free form. The ligand binds only to one of the conformations. The titration starts with two peaks. One of them constantly decreases while the other signal broadens and then arises in a new sharp peak. 

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