Steady-state approximation substitution reactions

The steady-state approximation assumes that since I is very reactive, its concentration will be very low at any time during the reaction and it will not change appreciably. Therefore dl/dr = 0. Solving the above expression for the concentration of I and substitution into the rate law for the formation product gives... 

Note that these equations are again based on a pseudo-steady-state approximation such that the deactivation rate must be much slower than the diffusion or chemical reaction rates. These equations can be easily solved, as in Chapter 3, and the result substituted into the definition of the effectiveness factor, with the following results ... 

While Equation 1.35, in combination with Equation 1.32, can give the number-average degree of polymerization, it is important not to ignore the role of the transfer reactions. Even in the case where transfer to initiator and solvent is nonexistent (presumably by careful initiator choice and a solvent-free polymerization), transfer to monomer can never be avoided entirely. Another way to approach the problem is to consider the simplest definition of dp, that is, the total number of polymerized monomers units divided by one-half the number of chain ends. Here, it is worth considering the number of chain ends produced by each of the processes [17]. Neither propagation nor termination by combination produce any chain ends (n=0), while both initiation and termination by disproportionation produce one chain end ( =1), and transfer reactions actually create two chain ends (n=2). The steady-state approximation again allows the absolute number of each of these processes to be substituted by the overall rate of each ... 

One can make the quasi-steady state approximation (QSSA) for radicals (P and P). This assumes that radical reactions are fast compared with other reactions and so can be considered to be always at steady state thus the left-hand sides of Equations 16.65 and 16.66 may be set to zero. Solution of Equation 16.66 for P and substitution of Equation 16.65 into the result gives the concentration of live chains ... 

SN1 reactions, such as hydrolyses and substitution by anions of tertiary halogen-oalkanes, are good examples of reactions with a reversible first step. In these a carbonium ion is produced, which then reacts with water or anions. Chapter 6, Problem 6.5, illustrates the different rate expressions found after applying the steady state treatment, and after assuming a pre-equilibrium where the equilibrium lies very far to the left, i.e. where K is very small, and only a very, very small amount of R+ is present. It also looks at the conditions under which the amount of R+ present in a steady state could approximate to a pre-equilibrium. The discussion did not include the situation where K is not small. 

In this equation, 0.693 is a constant obtained during the derivation of the formula (log 0.5). If we substitute our hypothetical values as used above, we would obtain a t112 of approximately 14 minutes. This is an important value to know since the time required to reach a steady-state plateau, and maintain it, depends only on the half-life of the drug. In our case, therefore, it would take approximately 70 minutes (i.e., 5 half-lives) to reach approximately 97 percent of steady state. In first-order reactions t112 is independent of dose, since, under normal circumstances, i.e., therapeutic, the system is not saturated since dosage is in the subgram amount. 

The use of pH variation and isotope effects in transient kinetics can be illustrated with a recent study on dihydrofolate reductase. Analysis by steady-state methods had indicated an apparent p/fa of 8.5 that was assigned to an active site aspartate residue required to stabilize the protonated state of the substrate (59). In addition, it was shown that there was an isotope effect on substitution of NADPD (the deuterated analog) for NADPH at high pH but not at low pH, below the apparent p/fa This somewhat puzzling finding was explained by transient-state kinetic analysis. Hydride transfer, the chemical reaction converting enzyme-bound NADPH and dihydrofolate to NAD+ and tetrahydrofolate, was shown to occur at a rate of approximately 1000 sec at low pH. The rate of reaction decreased with increasing pH with a of 6.5, a value more in line with expectations for an active site aspartate residue. As shown in Fig. 14, there was a threefold reduction in the rate of the chemical reaction with NADPD relative to NADPH. Thus direct measurement of the chemical reaction revealed the full isotope effect. 

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