Pseudo first order reaction relaxation equations

Kinetic schemes involving sequential and coupled reactions, where the reactions are either first-order or pseudo-first order, lead to expressions for concentration changes with time that can be modeled as a sum of exponential functions where each of the exponential functions has a specific relaxation time. More complex equations have to be derived for bimolecular reactions where the concentrations of reactants are similar.19,20 However, the rate law is always related to the association and dissociation processes, and these processes cannot be uncoupled when measuring a relaxation process. 

If experimental conditions are carefully selected so that all individual steps are either first-order or pseudo-first-order processes, then, under transient or pre-steady-state conditions, the reaction time courses will take the form of a sum of exponentials (i.e., a linear combination of the individual rate equations for each relaxation), such that the observable, time-dependent changes in absorbance, AZ, are given by the relationship... 

Before analysing a model requiring linearization of the equations for a two step reaction we shall derive the relaxation equation for the condition that there is a pseudo first order initial binding step. Using the scheme ... 

For a sequence of first order reactions the relaxation times are clearly independent of reactant concentrations and the equations apply equally to the interpretation of large transients. The effects of changing the concentration, for instance of the ligand in the pseudo first order system, will be discussed later. Without such additional diagnostics, which are available in the case of concentration dependent systems, the four rate constants can only be estimated by numerical fitting procedures. If signals in terms of absolute concentrations for A, R and [AR] are available, the equilibrium constants can be evaluated and serve as a useful restriction for the numerical solutions. If the two relaxations are uncoupled, t, T2, then we can simplify from equations (6.2.20) ... 

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