Pseudo-order rate equations

Equation 5.22 is therefore referred to as a pseudo-order rate equation in fact it is pseudo-first-order. 

If one were to mix fixed concentrations E and / at time zero and then measure the concentration of El complex as a function of time after mixing, the data would appear to be described by the pseudo-first-order rate equation ... 

In the simplest case a 1 1 complex is formed between the host in solution and the guest immobilized on the surface. The response of the SPR sensor, R, is proportional to the concentration of the complex formed, and thus pseudo-first-order rate equations can be used to analyze the data.73 If no host is initially bound the function for R... 

The optical rotation of the mixture approaches zero (a racemic mixture) over time, with apparent first-order kinetics. This observation was supported by the semi-log plot [ln(a°D/ aD) vs time], which is linear (Figure 1). It has been shown that this racemization process does in fact follow a true pseudo-first-order rate equation, the details of which have been described by Eliel.t30 Therefore, these processes can be described by the first-order rate constant associated with them, which reflects precisely the intrinsic rate of racemization. Comparison of the half-lives for racemization under conditions of varying amino acid side chain, base, and solvent is the basis for this new general method. 

Figure 7.5. Time courses for reactions as in Figure 7.4 wherein IR peak areas at 1740 cm 1 for various time points were fitted to a pseudo-iirst-order rate equation. Rate constants were obtained by the best fit as in Figure 7.2.

Equation 1.5 is the familiar form of first-order rate equation and indicates that the rate of absorption is proportional to drag concentration. K, is a pseudo-rate constant and is dependent on the factors D, A, k and h. 

If the majority of reactions occur in the solid at 77°K and it is assumed that the energy flows into and out of the radical-molecule cage with a frequency, v, similar to that of the lattice vibrations (v = 10 sec ) then the extent of reaction at any time, t, is given by the first-order rate equation (pseudo-unimolecular)... 

By considering that in the course of the runs the oxygen bubbling in the dispersion is not stopped, its concentration in the liquid phase does not change during the occurrence of substrate degradation and, moreover, it is always in excess. Then the 0qx term of Equation (7) does not depend on time, that is, it is a constant. By putting k = k"-0ox> Equation (7) turns a pseudo-first-order rate equation ... 

As the amount of the catalyst increases (Fig. 4A) the conversion of toluene does not follow a nearly linearly increase as expected for a first order rate equation of hydrocarbon depletion (the low hydrocarbon conversion indicates pseudo-differential conditions with respect to hydrocarbon axial profile), because the concentration of oxygen also affects the reaction rate. This is confirmed in Fig. 4B showing the dependence of the conversion of toluene on the Oa/toluene ratio in the feed. The rate of toluene depletion thus is not limited by the rate of adsorption of the hydrocarbon, but probably by the consecutive steps of oxidation. 

Here kp should more appropriately be called an apparent rate constant or overall rate constant since both undissociated (ion pair) and dissociated (free ions) species usually exist and their propagation rate constants are different (discussed later). Integration of this pseudo-first-order rate equation gives the time dependence of the monomer concentration as... 

A simplified two-step fitting procedure has found application in many publications (21, 23). For the sake of completeness, this approach shall also be discussed here. It requires the following substitutions and results in a pseudo-first order rate equation. 

Table 5.6 provides information taken from the kinetic reaction profile for Br" in Figure 5.7b. Use this information to determine a value for the pseudo-order rate constant in Equation 5.22. 

Given the form of an experimental rate equation which has a particular order, determine the units in which the experimental rate constant, or pseudo-order rate constant, would typically be expressed. (Questions 4.2, 5.2, 5.3 and Exercise 5.2)... 

The variable-time method, as employed for first- or pseudo-first-order reactions, also uses the integral form of the first-order rate equation (Eqn. 18.15). Solving for [A]o yields... 

B] hardly changes as reactant A disappears. This assumption is so common in kinetic investigations that it has a name the pseudo-first-order assumption. We write pseudo-first-order rate equations as... 

The set of coupled equations (5.2) can be readily solved using matrix methods. The advantage of this approach is that it can easily be generalized to treat any number of coupled first-order (or pseudo-first-order) rate equations. For simplicity consider a two-step process and define a column vector constructed from the concentrations... 

---