Reaction rates characteristic life time

Even though it is difficult to predict reaction rates in marine systems, the concepts of molecular diffusion and mechanisms of reaction underpin much of geochemical research at the air-water and sea floor-ocean boundaries. A basic knowledge of molecular diffusion and chemical kinetics is essential for understanding the processes that control these fluxes. This chapter explores the topics of molecular diffusion, reaction rate mechanisms and reaction rate catalysis. Catalysis is presented in a separate section because nearly all chemical reactions in nature with characteristic life times of more than a few minutes are catal5 ed. 

The characteristic life time of a reaction is a measure of the time required after initiation for it to reach completion. This period is frequently related to the rate constant for the reaction in a veiy clear and specific way. Solutions to some of the common zero-, first- and second-order rate equations are presented in Table 9.5. Examples of zero- and first-order reactions are discussed in this section application of the second-order equations to general catalytic processes will be presented in the section on catalysis. The last column of Table 9.5 lists the relations between r, the characteristic life time of the reactant with respect to the chemical reaction, and the rate constant for the reaction. The meaning of the characteristic life time depends upon the order and reversibility of the reaction. 

Whereas radioactive decay is never a reversible reaction, many first-order chemical reactions are reversible. In this case the characteristic life time is determined by the sum of the forward and reverse reaction rate constants (Table 9.5). The reason for this maybe understood by a simple thought experiment. Consider two reactions that have the same rate constant driving them to the right, but one is irreversible and one is reversible (e.g. k in first-order equation (a) of Table 9.5 and ki in first-order reversible equation (b) of the same table). The characteristic time to steady state tvill be shorter for the reversible reaction because the difference between the initial and final concentrations of the reactant has to be less if the reaction goes both ways. In the irreversible case all reactant will be consumed in the irreversible case the system tvill come to an equilibrium in which the reactant will be of some greater value. The difference in the characteristic life time between the two examples is determined by the magnitude of the reverse reaction rate constant, k. If k were zero the characteristic life times for the reversible and irreversible reactions would be the same. If k = k+ then the characteristic time for the reversible reaction is half that of the irreversible rate. 

Here, as in all first rate kinetics, a plot of the natural logarithm of the count rate vs. the time results in a straight line whose slope is proportional to the rate constant and whose intercept is In N. Another reaction rate characteristic, called the half-life, t. is the time required for the initial reactant concentration to be reduced to one half. For first order reactions, the t, independent of concentration, = 0.693/k. 

Rate Laws Experimental measurement of the rate leads to the rate law for the reaction, which expresses the rate in terms of the rate constant and the concentrations of the reactants. The dependence of rate on concentrations gives the order of a reaction. A reaction can be described as zero order if the rate does not depend on the concentration of the reactant, or first order if it depends on the reactant raised to the first power. Higher orders and fractional orders are also known. An important characteristic of reaction rates is the time required for the concentration of a reactant to decrease to half of its initial concentration, called the half-life. For first-order reactions, the half-hfe is independent of the initial concentration. 

Important classes of chemical reactions in the ground electronic state have equal parity for the in- and out-going channels, e.g., proton transfer and hydride transfer [47, 48], To achieve finite rates, such processes require accessible electronic states with correct parity that play the role of transition structures. These latter acquire here the quality of true molecular species which, due to quantum mechanical couplings with asymptotic channel systems, will be endowed with finite life times. The elementary interconversion step in a chemical reaction is not a nuclear rearrangement associated with a smooth change in electronic structure, it is aFranck-Condon electronic process with timescales in the (sub)femto-second range characteristic of femtochemistry [49],... 

The slope of this straight line is 16.91 x n V-1 at 25 °C. However, it is more common to use the inverse function E = Ei/2 + 2.303 x (RT/nF) log [(fi, - I) /I], with the slope 0.059/nV. Both functions are called the logarithmic analysis of DC polarogram. They both cross the potential axis at the half-wave potential, which corresponds to I = Ii/2. The main characteristic of fast and reversible electrode reactions is that the half-wave potential is independent of the drop life-time in DC polarography, or the rotation rate of the rotating disk electrode, or the radius of microelectrode. If this condition is satisfied, the slope of the logarithmic analysis indicates the number of electrons in the electrode reaction. 

For first-order reactions in closed vessels, the half-life is independent of the initial reactant concentration. Defining characteristic times for second- and third-order reactions is somewhat complicated in that concentration units appear in the reaction rate constant k. Integrated expressions are available in standard references (e.g., Capellos and Bielski, 1980 Laidler, 1987 Moore and Pearson, 1981). 

Reaction rates are often compared using a characteristic time. For example, the characteristic time might be defined as the time needed for the destruction of 50% of the reactants (a = p = 0.5). This time is called the reaction s half-life, ty,. When t = ty a= p = 0.5. The half-life for an unopposed reaction where 1 is found by setting m = 0.5 in Eq. (3.18). 

However, reactions between Ei and E2 can only be observed if the half-life of El is compatible with the time characteristics of the reaction under study. For instance, the time needed to achieve the equilibrium must be shorter than the lifetime of the involved radionuclide. Studies of the chemical properties of SHEs give rise not only to the concept of single-atom chemistry but also to one-atom-at-a-time chemistry. For that purpose, chemical processes with high reaction rates are required. 

To determine what conditions are required for mixing processes to affect reaction processes, we will use a number of concepts. Most important is the comparison of time constants of the various processes. The processes of interest are blending, mixing, mass transfer between phases, and chemical reaction. Some typical time constants are the blend time and reaction half-life. For simple exponential processes (first-order reactions), rates and characteristic times, such as reaction half-lifes, are related. The first-order rate equation is... 

Other blend time correlations were presented by Penney (1971), Khang and Levenspiel (1976), and Fasano and Penney (1991). Use of these correlation equations allows the estimation of blending times, which can be compared to molecular reaction times for all the reactions in the reactor. Even though local mixing time is the critical time for determining apparent reaction rate, blend time can be used in an approximate manner. If the characteristic molecular reaction time (e.g., the half-life) is much greater than the characteristic blend time... 

The development of a quantitative theory of a free-radical copolymerization implies the derivation of equations for the rate of the monomers depletion and the statistical characteristics of the chemical structure of macromolecules present in the reaction system at the given conversion p of monomers. Elaborating such a theory one should take into account a highly important peculiarity inherent to any free-radical copolymerization. This peculiarity is that the characteristic time of a macroradical life is appreciably less than the time of the process duration. Consequently, its products represent definitely... 

Important characteristics of zero-order reactions are that (1) a constant amount of drug is eliminated per unit time since the system is saturated (maximized) and (2) the half-life is not constant for zero-order reactions but depends on the concentration. The higher the concentration, the longer the half-life. Therefore, the term zero-order half-life has little practical significance since it can change and (3) zero-order kinetics is also known as nonlinear or dose-dependent. For example, if the body can metabolize ethanol at a rate of 10 ml per hour, then if one consumes 60 ml, it will take 3 hours to metabolize half of it (the half-life under these circumstances). However, if 80 ml is consumed the half-life will now become 4 hours. This is particularly significant regarding ethanol toxicity. 

The biodistribution of plasmid can be determined by measuring the rate of disappearance of radiolabeled DNA from the bloodstream and its accumulation in tissues or by the use of fluorescence microscopy to trace the leakage of dye-labeled plasmids from the vasculature. Pharmacokinetic analysis of in vivo disposition profiles of radiolabeled plasmid provides useful information on the overall distribution characteristics of systemically administered plasmids, with one critical limitation. The radiolabel represents both intact plasmid and its metabolites. The plasma half-life of plasmid is less than 10 min, and hence tissue distribution and pharmacokinetic parameters of plasmid calculated on the basis of total radioactivity are not valid at longer time points. Thus, polymerase chain reaction and Southern-blot analysis are required to establish the time at which the radiolabel is no longer an index of plasmid distribution. 

In pulse radiolysis experiments the radicals are produced homogeneously in the solution. The polarographic current, /, is determined by the concentration of radicals at the electrode surface, the rate at which they are oxidized or reduced and the rate at which they are replaced by other radicals diffusing to the surface from the bulk solution account also has to be taken of reactions of the radicals in the bulk solution. By measuring the current, at a fixed time after the pulse, as a function of the potential applied, one can obtain a polarogram which is characteristic of the redox behavior of the radical and so can be used to identify it. Information about the rate of electron transfer can be extracted from measurements of the time dependence of / at a fixed potential. For radicals which undergo self-reaction in the bulk solution, the appropriate relationship is given by Eq. 71, provided the time is shorter than the first half-life of the radical [140],... 

Most elementary reactions involve either one or two reactants. Elementaiy reactions involving three species are infrequent, because the likelihood of simultaneous three-body encounter is small. In closed, well-mixed chemical systems, the integration of rate equations is straightforward. Results of integration for some important rate laws are listed in Table 2.7, which gives the concentration of reactant A as a function of time. First-order reactions are particularly simple the rate constant k has units of s , and its reciprocal value (1/k) provides a measure of a characteristic time for reaction. It is common to speak in terms of the half-life ( 1/2) for reaction, the time required for 50% of the reactant to be consumed. When... 

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