Second-Order Rate Equations

Suppose a reaction has this second-order rate equation ... 

The conventional second-order rate equation is = A dCa< b. where is in moles per liter per second and Ca, Cb are in moles per liter. Because c = 1000 /A/a and Vc = 1000v Wa, where A a is Avogadro s number, we obtain... 

If cis the concentration of single-stranded DNA at time t, then the second-order rate equation for two complementary strands coming together is given by the rate of decrease in c ... 

This reaction cannot be elementary. We can hardly expect three nitric acid molecules to react at all three toluene sites (these are the ortho and para sites meta substitution is not favored) in a glorious, four-body collision. Thus, the fourth-order rate expression 01 = kab is implausible. Instead, the mechanism of the TNT reaction involves at least seven steps (two reactions leading to ortho- or /mra-nitrotoluene, three reactions leading to 2,4- or 2,6-dinitrotoluene, and two reactions leading to 2,4,6-trinitrotoluene). Each step would require only a two-body collision, could be elementary, and could be governed by a second-order rate equation. Chapter 2 shows how the component balance equations can be solved for multiple reactions so that an assumed mechanism can be tested experimentally. For the toluene nitration, even the set of seven series and parallel reactions may not constitute an adequate mechanism since an experimental study found the reaction to be 1.3 order in toluene and 1.2 order in nitric acid for an overall order of 2.5 rather than the expected value of 2. 

Newton has shown that no complications ensue from the reaction of the intermediate U(V) with oxygen, since the latter has no effect on the rate. A simple second-order rate equation applies, the disappearance of Pu(VI) being followed at 830 m/i, and the probable mechanism is... 

The complexity of the integrated form of the second-order rate equation makes it difficult to apply in many practical applications. Nevertheless, one can combine this equation with modem computer-based curve-fitting programs to yield good estimates of reaction rate constants. Under some laboratory conditions, the form of Equation (A1.25) can be simplified in useful ways (Gutfreund, 1995). For example, this equation can be simplified considerably if the concentration of one of the reactants is held constant, as we will see below. 

Let us look again at the association reaction described by Equation (A 1.22). If we set up the system so that there is a large excess of [/] relative to [E], there will be little change in [/] over the time course of El complex formation. For example, suppose that we set up an experiment in which E = InM (0.001 pM) and [/] = 1 pM. The maximum concentration of El that can be formed is limited by the lowest reactant concentration, in this case by [E. Hence, at infinite time, the concentration of free I will be [/] - [El] = 1.000 - 0.001 = 0.999 pM (Figure A1.5). This is such a small change from the starting concentration of free I that we can ignore it and treat [/] as a constant value in the second order rate equation. Thus... 

Reference has already been made (p. 82) to the fact that the reactions of some substrates, e.g. secondary halides, may follow a mixed first/second order rate equation. The question then arises whether such a reaction is proceeding via both SN2 and SN1 pathways simultaneously (their relative proportions depending on the solvent, etc.) or whether it is proceeding via some specific, in between mechanistic pathway. 

The reaction has been shown to follow a second order rate equation, rate = fc2[ROH][SOCI2], but clearly cannot proceed by the simple Sn2 mode for this would lead to inversion of configuration (p. 87) in the product, which is not observed. 

The term kcJKm describes the reaction of any enzyme and substrate at low substrate concentration. At low substrate concentration, the velocity of an enzyme-catalyzed reaction is proportional to the substrate concentration and the enzyme concentration. The proportionality constant is kcJKm and v = (/cc u/A ,)l l [E]x- If you re real astute, you ll have noticed that this is just a second-order rate equation and that the second-order rate constant is kcJKm. 

The kinetic measurements were performed by NMR monitoring of the reaction mixtures in the temperature range from —90 to —40°C. All of the reactions under study were approximated by the second-order rate equation. 

A two liter flask was filled with pure HI at 1.24 atm and 683°K, and the decomposition was followed by measuring the absorption of light by the iodine produced. Immediately after the last reading the flask was chilled and an analysis for iodine showed 1.17 g. Evaluate the constants of the second order rate equation. 

The decomposition of nitrogen dioxide, 2N02 = 2N0 + 02, has a second order rate equation. Data at different temperatures are tabulated. Find the Arrhenius parameters. 

This system of second order rate equations is solved numerically for M and several values of Pn with these values ... 

We obtain the second-order rate constant k2 as the slope of a graph drawn according to the integrated second-order rate equation. 

Figure 8.9 Kinetics of a second-order reaction the racemization of glucose in aqueous mineral acid at 17 °C (a) graph of concentration (as y ) against time (as V) (b) graph drawn according to the linear form of the integrated second-order rate equation, obtained by plotting 1 / A, (as V) against time (as V). The gradient of trace (b) equals the second-order rate constant k2, and has a value of 6.00 x 10-4 dm3mol 1s 1...

Although the kinetics is not completely known, the following second order rate equation is assumed... 

Theoretical studies are primarily concentrated on the treatment of flame blow-off phenomenon and the prediction of flame spreading rates. Dunskii [12] is apparently the first to put forward the phenomenological theory of flame stabilization. The theory is based on the characteristic residence and combustion times in adjoining elementary volumes of fresh mixture and combustion products in the recirculation zone. It was shown in [13] that the criteria of [1, 2, 5] reduce to Dunskii s criterion. Longwell et al. [14] suggested the theory of bluff-body stabilized flames assuming that the recirculation zone in the wake of the baffle is so intensely mixed that it becomes homogeneous. The combustion is described by a second-order rate equation for the reaction of fuel and air. 

Model 2. First note that the stoichiometry of Eq. 10 is symmetrical in A and B, so just interchange A and B in Model 1, put 2 = 0 we will get = k[BY, which is what we want. So the mechanism that will match the second-order rate equation is... 

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