Straight line reactions

In order to confirm the order of a reaction, the left hand side of the expression, when plotted against time, should give a straight line. Thus, if a plot of (1 /a-x) versus time is a straight line, reaction would be of second order. This is a simple and accurate method and is applied generally for determination... 

Initial Rate Assumption. The entire reaction progress curve, or at least a substantial portion of it, is typically required to accurately determine the rate constant for a first-order or second-order reaction. Nonetheless, one can frequently estimate the rate constant by measuring the velocity over a brief period (known as the initial rate phase) where only a small amount of reactant is consumed. This leads to a straight-line reaction progress curve see Fig. 6) which is drawn as a tangent to the initial reaction velocity. 

The techniques of monomolecular rate theory easily transform measured reaction data into a form where we can analyze apparent kinetics and the effects of intracrystalline diffusion by the use of selectivity data. Time dependency has been eliminated. Since selectivity is extremely reproducible and is independent of short-term aging effects, the number of experimental runs is reduced while data reliability is maintained. For catalyst evaluation at any temperature, it is necessary to determine the equilibrium composition and the straight-line reaction path. With this information any catalyst can be evaluated at this temperature with simply the additional information from a curved-line reaction path. The approach used in the application of monomolecular rate theory to the xylene isomerization selectivity kinetics is as follows. Reference is made to the composition diagram, Figure 1. 

Straight-Line Reaction Paths. For a three-component reversible monomolecular system only two straight-line reaction paths exist both can be observed experimentally. Normally, the slow straight-line reaction path is estimated as the tangent to any curved-line reaction path at the equilibrium composition. This path is subsequently determined more precisely in the laboratory. The locus of the second, or fast, straight-line reaction path is then calculated (1). 

The technique for determining straight-line reaction paths in this work differed from the usual experimental approach. Our approach also determined the straight-line reaction path by minimizing the deviation between the experimental data and those predicted. 

Figure 7. The two-dimensional ground state free energy surfaces for (a) DNOA and (b) DONA. The straight-line reaction paths are shown on the contour plots as dashed lines, and the minima are labeled according to the dominant VB states. Note that the Zp solvent coordinate is scaled by a factor of approximately four. (Reproduced from Ref.

Fig. 9. A typical three component system with equilibrium composition point E. The characteristic vectors are Xo, Xi, and X2. The translations of Xi and % to the end of Xo form the vector sums xi(0) = Xo + Xi and ij(0) = Xo + Xj. The translated vectors x l and x 2 represent straight line reaction paths along which the initial compositions x,(0) and aij(O) go to equilibrium. The extension of these vectors shown by x"i and x"2 in Fig. 9b also represent straight line reaction paths for two other initial compositions corresponding to choices of Xi and X2 in the direction opposite to the first choice.

Hence, composition points at the ends of the vectors , (<) and ax,(t) shift with time to the equilibrium point along the straight lines x and x 2 respectively the displaced characteristic vectors x l and x 2 are, therefore, straight line reaction paths for these compositions. All straight line reaction paths must be derived from characteristic vectors displaced along Xo since all such paths are expressible in the form of Eqs. (52) and (53). From the generalized form of Eqs. (48) and (49), we have... 

Fig. 10. The straight line reaction paths as coordinate axes for the characteristic species j 0, in the reaction simplex.

The characteristic roots are determined by transforming experimental compositions along appropriate reaction paths into the B system of coordinates. Equations (44) and (46) are used to compute the matrix from the matrix X determined from the straight line reaction paths and the equilibrium composition. Each observed composition (<) is transformed by the matrix X into 3(0 [Eq- (40)]- The decay of each 6, with time is given by the set of Eqs. (27) and the value of — X, can be determined from the slope of the straight line obtained from a graph of In bj vs time. 

We have seen that all w-component reversible monomolecular systems have n — 1 straight line reaction paths and n — 1 decay constants Xy. 

The location of the straight line reaction paths and the values of the lambda s depend upon the experimental conditions such as pressure, temperature, nature of catalyst, etc. They are, however, independent of the initial composition used. For a given experimental condition, the entire behavior of the reaction system for all initial compositions is specified when the straight line reaction paths and the decay constants X are known. Thus, all quantitative and qualitative information about the system is contained in the location of the straight line reaction paths and the values of the lambda s. Furthermore, all reaction paths, all time courses, and all rate constants are quickly and simply determined from them. In addition, the general behavior of such systems can be most easily visuahzed in terms of them they provide a panoramic view of the entire reaction system and provide the most useful and convenient formulation of the system available. 

A graphical method may be used to locate the straight line reaction paths in the reaction simplex for a three component system. Any convenient composition such as pure Ai is chosen as an initial composition and its reaction path determined. Sufficiently close to equilibrium the reaction path will be dominated by the B species with the smallest decay constant since the other B species will have decayed to a much greater extent by this time. Consequently, a linear extrapolation of the part of the reaction path near equilibrium back to the side of the reaction triangle gives a new... 

In principle, the graphical method used for the three component system can be used for a four component system since its reaction simplex is a tetrahedron it is not very convenient, however, to plot reaction paths in three dimensions and for systems with more components this is not available. Consequently, a method for representing a reaction path is needed that does not involve the reaction simplex directly. In the reaction simplex a reaction path is a single curve in an (n — l)-dimensional space. A reaction path also can be specified parametrically by n — 1 curves in two dimensional coordinate systems if the amounts of each of the various components j) is plotted in terms of another one of them, a,-, that is monotonic with time. A straight line reaction path in the n — l)-dimensional reaction simplex becomes n — 1 straight lines in this two dimensional graph. 

Thus, a single straight line reaction path in n — 1 dimensions becomes n — 1 straight lines in two dimensions with slopes and intercepts equal to (xmi/xji) and [a — xmi/xji)aj ] respectively. 

There is one additional precaution that must be taken for n-component systems after a straight line reaction path has been located, the B species that correspond to it must be removed from other initial compositions that are used to locate new straight line reaction paths. Otherwise, the same straight line path will be obtained all over again if this species happened... 

This composition is used as a new starting composition and its reaction path determined near equihbrium. Since we are very near the straight line reaction path, the twelve composition points given in Table I, forming the approximately straight hne portion near equihbrium, is fitted to a straight... 

Fig. 17. Reaction simplex for the hypothetical four component system given in text. The straight line reaction paths are shown.

Straight line reaction paths may occur that do not lie within the reaction simplex and cannot be observed in the laboratory. 

The positions of the displaced vectors x l and x 2 in composition space are shown in Fig. 25. The displaced vector x l does not lie in the plane of the reaction triangle and cannot be a straight line reaction path. This will be true for all choices of Xo(r) and Xi. For a displaced vector x, - to be in the plane of the reaction triangle, the sum of the elements of u(0) must be the same as the sum of the elements of the vector Xo(r) used to move X,. This can only occur if the sum of the elements of the vector x< is zero as is the case for characteristic vectors with X 5 0 but not for characteristic vectors with X = 0 since these vectors contribute to the mass of the system. 

The Determination of Straight Line Reaction Paths Lying Outside the Reaction Simplex... 

Let us use the reaction scheme (206) to illustrate the use of orthogonality relations in a subspace of the composition space and constraints to determine the missing displaced characteristic vector that lies outside the reaction simplex for systems with an infinity of equilibrium points. The value of the equilibrium composition for Ai A is Ui = 0.6000 and 02 = 0.4000. The logical initial compositions to use are mixtures of Ai and A2 these compositions will converge to the particular straight line reaction path within the reaction simplex shown in Fig. 26. The value of a fO) that we obtain is... 

The value of the equilibrium point for the reaction plane on which this initial composition lies is determined from the extrapolation of the measured part of the straight line reaction path x 2 to its intersection with the edge of the reaction simplex connecting as = 1 and 04 = 1 as shown in Fig. 26 the value is... 

The butene isomerization system discussed in Section III,B will be used for the illustration. The true straight line reaction paths are given by the heavy solid lines in Fig. 32. Let us imagine that the straight line reaction paths are rotated until the one corresponding to the slowest decaying characteristic species passes through the pure fraras-2-butene comer of the... 

Also let the straight line reaction paths be rotated so that the paths... 

For the straight line reaction path —----------------- for which the path... 

---