Third-order integrated rate equation

For a third-order reaction, a plot of 1/(2[A,] ) versus time yields a straight line with positive slope kr (Fig. 1.6). 

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Figure 2 Kinetics of gas-phase propylene homometathesis at 0°C, catalyzed by (a) perrhenate/silica-alumina activated by SnMe4 (10 mg, 0.83 wt % Re) and (b) MeReOs on HMDS-capped silica-alumina (10 mg, 1.4 wt % Re). Solid lines are curve-fits to the first-order integrated rate equation. Solid squares first addition solid circles second addition open circles third addition of propylene (30 Torr) to the catalyst.

Find the integrated rate equation for a third-order reaction having the rate equation —dc/ ldt = kCf,. ... 

The mathematics involved for higher-order reactions become more difficult and such a treatment is beyond the scope of this book. For example, the integrated rate equations for the three types of third-order reactions are given in Table 3.2. 

Table 3.2. Integrated rate equations for the three types of third-order reactions...

A third order reaction can be the result of the reaction of a single reactant, two reactants or three reactants. If the two or three reactants are involved in the reaction they may have same or different initial concentrations. Depending upon the conditions the differential rate equation may be formulated and integrated to give the rate equation. In some cases, the rate expressions have been given as follows. 

We may develop similar expressions for second- and third-order reactions in which two or three molecules of A collide in the rate-limiting step. See, for example, Gordon, A. J. Ford, R. A. The Chemist s Companion lohn Wiley Sons New York, 1972 p. 135 for a listing of the differential and integrated forms of zero-, first-, and second-order rate equations. 

Plot of integrated form of rate equation for third-order reaction. 

The differential method of analysis (Problem 2.2) indicates that the reaction is third order. For a third-order reaction, the integral form of the rate equation is... 

The integral method is most appropriate when we have a good idea of the form of the rate equation, but need to test that form in a different context For example, suppose that previous analyses had shown that a certain reaction was second order in A and first order in B at temperatures 7i and T2. Now, a new set of data (say Ca versus time) becomes available at a third temperature T3. We need to confirm the existing rate equation, and obtain a value of the rate constant at the new temperature. The most straightforward way to do this would be to use the integral method, assuming that the rate equation — rA = kCfjCs remained valid. 

Thirdly one needs a drastic step to turn this integral equation into a differential equation. This is the Markov approximation , which comes in two varieties. The first variety consists in replacing ps(t — t) with ps(t). The error is of relative order rc/rm (where l/rm is the unperturbed rate of change due to S s) and of absolute order a2T2/rm. In this approximation one may as well omit the S s in the exponent of (4.13) and the result is the same as (3.19). The second variety takes the zeroth order variation of ps into account by setting ps(t — r) = e T sps(t). The result is the same as was obtained in (3.14) by means of the interaction representation, and the only requirement is arc <[Pg.444]

Expressions similar to equation (17) may easily be derived for various second-, third-, and higher-order reactions. These expressions are readily integrated for all second-order reactions and for many third- and higher-order reactions, yielding (in many cases) relations analogous to equation (18), which define useful concentration-time graphs. The dimensions of the rate constant k for an nth order reaction are (concentration) (time) ... 

In order to obtain crack-tip quantities such as the strain energy release rate g, the complex stress intensity factor K, and the mode-mixity xp, the following procedure may be adopted first, the strain energy release rate Q is directly computed via a contour integral evaluation - the J-integral method, or the VCCT second, the modulus of K, can be computed from Equation (10) and third, the crack surface displacements may be substituted in Equation (8) and with the knowledge of e, the parameter a is computed. Finally, the stress intensity factors may be expressed as ... 

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