The equilibrium assumption

Now the equilibrium assumption is applied to Scheme XVI. From the above considerations, Cc = KcaPb, which, combined with Eq. (3-131), gives... 

For Scheme XIV, and for each of the following sets of rate constants, calculate the exact relative concentration cb/ca as a function of time. Also, for each set, calculate the approximate values of cb/ca using both the equilibrium assumption and the steady-state approximation. 

Let us examine the equilibrium assumption of transition state theory. Consider a reversible elementary reaction at equilibrium. Because the initial and final states are at equilibrium, assuredly the transition state is in equilibrium with each of these. (It follows that for a reaction at equilibrium, transition state theory is exact insofar as the equilibrium assumption is concerned.)... 

Now suppose that, from this equilibrium situation, the final state is instantaneously removed. The production of transition state species by the product state will cease. However, the production of transition state species by the reactant state is unaffected by this suppression of the final state, and, according to the third postulate of the theory, the rate of reaction is a function of the transition state concentration formed from the reactant state. This is the usual argument for the equilibrium assumption. Despite its apparent artificiality, the equilibrium assumption is generally considered to be fairly sound, with the possible exception of its application to very fast reactions. ... 

Here Z represents the reaction products. M is the transition state the double dagger symbol will always signify a quantity or structure relating to the transition state. Scheme I incorporates the equilibrium assumption by writing the conversion of the initial state into the transition state as an equilibrium. This assumption then allows us to apply statistical mechanics to the rate problem making use of Eq. (5-32), we have... 

A first-order rate constant has the dimension time, but all other rate constants include a concentration unit. It follows that a change of concentration scale results in a change in the magnitude of such a rate constant. From the equilibrium assumption of transition state theory we developed these equations in Chapter 5 ... 

The extension to rates draws on the equilibrium assumption of transition state theory to yield the analogous result, with rate constants replacing the equilibrium constants of Eq. (6-96). Kresge has generalized this argument, the result being... 

It is also possible to derive a rate equation for a reaction sequence which does not differ essentially from that which led to (16) without introducing the equilibrium assumption (12)36. For convenience the mechanism is now rewritten as... 

Note how the partition function for the transition state vanishes as a result of the equilibrium assumption and that the equilibrium constant is determined, as it should be, by the initial and final states only. This result will prove to be useful when we consider more complex reactions. If several steps are in equilibrium, and we express the overall rate in terms of partition functions, many terms cancel. However, if there is no equilibrium, we can use the above approach to estimate the rate, provided we have sufficient knowledge of the energy levels in the activated complex to determine the relevant partition functions. 

Successful lead optimization can drive the affinity of inhibitors for their target enzymes so high that the equilibrium assumptions used to derive the equations for calculating enzyme-inhibitor K, values no longer hold. 

Equation 14.11 explicitly contains the equilibrium assumption of transition state theory, i.e. that the transition state and the excited molecules are in chemical equilibrium. 

Using the equilibrium assumption of transition state theory the ratio of concentra-... 

G. E. Briggs and J. B. S. Haldane, Biochem J., 19, 338 (1925), on the other hand, employed a steady-state assumption in place of the equilibrium assumption... 

Moreover, the number of activated complexes is caleulaled by the equilibrium assumption. 

Most studies of contaminant transport in slurry walls have relied on the equilibrium assumption, either for convenience or because of the long residence times associated with diffusion-dominated transport. As an alternative, Rabideau and Khandelwal (1998a) have proposed the simple two-compartment mass transfer model to describe nonequilibrium sorption in soil/bentonite systems ... 

The reasonable agreement between the Kd values from our study (2045 mL/g) and Cantrell s work (2600 mL/g) suggests that both studies similarly accounted for the effects of calcium competition in the experimental design. Also, the ability of a single Kd to describe data from several column durations (10 to 60 days) indicates that the equilibrium assumption is reasonable. However, the limited number of applicable studies suggests that more research is needed to confirm these conclusions. 

The label thermal can apply to any of the ET types just described and implies that the initial state is at thermal equilibrium (i.e., all the molecular and medium modes are at thermal equilibrium with the given electronic state). Ultrafast PIET processes may be sufficiently fast (ultrafast) that the equilibrium assumption is no longer valid [11]. 

This is precisely the behaviour predicted by the Kira mechanism, provided that the formation of the silene-ROH complex is reversible and the proton transfer steps are rate-limiting. The complete mechanism is shown in Scheme 4, while equation 27 gives the predicted expression for the pseudo-first-order rate constant for decay of the silene, derived assuming the steady-state approximation for the silene-alcohol complex. Equation 27 reduces to the quadratic expression in [ROH] of equation 28 when k c (A h + A h [ROII ), i.e. under the conditions of the equilibrium assumption for the complex. In practice, it is difficult to distinguish between the two situations given by equations 27 and 28. The experimentally determined second- and third-order rate constants roh and k2ROH are defined in equations 29 and 30, respectively, in terms of the mechanism of Scheme 4 and using the... 

The equilibrium assumption is justified for many reactions with short rchem and long tr. Rate data for a large number of first- and second-order aqu(x>us... 

Distillation columns are made to separate at least two different components. There are several different types of columns. The assumption of equilibrium between the liquid and vapour that leave each tray is still in common use to model tray distillation. The work of Krishna and Wesselingh shows, however, that non-equilibrium models give results that are very different from those obtained with the equilibrium assumption. 

When using a conserved scalar approach with the equilibrium assumption, it is necessary to carry out computations of (1) equilibrium composition of gas from... 

Hence, substrate and enzyme concentrations can be determined in the same manner as for the steady-state case for enzyme reactions in which ki is small and the equilibrium assumption holds. Enzyme concentrations are determined under conditions in which the substrate concentration is large, while substrate concentrations are determined when [S] [Pg.892]

Curtiss, C. F., The Equilibrium Assumption in the Theory of Absolute Reaction Rates/ University of Wisconsin, Report CM-476, June 1948. 

Instantaneous equillibrium between sorbed and soluble phases is Inaccurate desorption is slower than adsorption. "Macropore" flow has been defined as rapid and deep Infiltration of rainfall at the onset of a storm as the rain follows preferential flowpaths In the soil. Hence, not all the aldicarb In the surface zones will "see" the water as It flows by, and desorb according to the equilibrium assumption. It Is difficult to determine which, If any, of these possibilities explain the malntainance of high aldicarb concentrations near the soil surface. Nonetheless, the high assigned partition coefficient Is the way In which the model artificially duplicates this behavior. 

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