Other Reaction Graphs

Several of the works discussed above include graph-theoretic calculations of, for example, the complexity of a graph. Unfortunately it is not clear in many cases what the implications are for the reaction network for differences in the complexity of various associated graphs, particularly when the differences are small. In some cases, the results seem counterintuitive in that the more complex graph is constructed from the physically more important reaction process. More study is needed of these issues. 

This material is based upon work supported by the National Science Foundation under Grant No. DMS-0426132. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. 

The author wishes to thank Ravi Datta, Hie Fishtik, Caitlin Callaghan, Kas Hemmes and Bill Martin for many useful discusses leading to this chapter, and Kim Ware for help in finding many of the references given here. 

Temkin, A. V. Zeigamik, and D. Bonchev, Chemical Reaction Networks, a Graph-Theoretical Approach, CRC, Boca Raton, 1996. 

Balaban, in Graph Theoretical Approaches to Chemical Reactivity, Ed. by D. Bonchev and O. Mekenyan, Kluwer, Dordrecht, 1994, pp. 137-180. 

Fisbtik, C. A. Callaghan, and R. Datta, J. Phys. Chem. B108 (2004) 5683. 

Stelling, N. Price, S. Klant, S. Schuster, and B. O. Palsson, TRENDS Biotech. 22 (2004) 400. See also the references therein. 

Introduction to Bond Graphs and their Applications, Pergamon press, Oxford, 1975. 

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Fig. 10. Schematic graph of a glycogen molecule after Meyer and Bernfeld (above). The monomeric unit is the glucose (below, left) which may for simplicity contracted to a graph (below, right). Group A can react only with group B or C, all other reactions are excluded104,112, U3>...

If Eq. (2.9) is appropriate, a graph of log A/Aq) vs. t should yield a straight line with a slope equal to —kc. However, based on this result alone, it is tenuous to conclude that Eq. (2.5) is the only possible interpretation of the data and that a straight-line graph indicates a first-order reaction. One can make these conclusions only if no other reaction mechanisms result in such a graphical relationship. 

Graphs of log (1 - A/Y0) vs. t are commonly used to test the validity of Eq. (2.10). However, Eq. (2.11), like Eq. (2.8), shows more complex behavior than simple graphical methods reveal. Thus, one should be cautious about making definitive statements concerning rate constants and particularly mechanisms, based solely on data according to integrated equations like those in Eqs. (2.9) and (2.10) unless other reaction mechanisms have been ruled out. 

Further it is emphasized that such directed reactions graphs are wide spread in chemistry conceivably even in other sciences as well. Sometimes the posets may be quite complex as with the briefly noted addition - reaction poset for hydrogenation of buckmisterfullerene - leading to a poset of >1014members. And sometimes the poset may be infinite, as with the noted alkane poset. Yet further the same types of progressive reaction posets may occur. For example the posets of ancestors or food webs in biology may often appear in the form of our reaction posets, and thence be susceptible to the same analyses. Thence there is much promise for our presently indicated ideas and techniques. 

A coherent organization of the welter of data presented in the graphs of this article is no easy task. However, the attempt must be made, knowing that future results will challenge some of the ideas set forth. The generally accepted behavior of the reaction classes is maintained hydrogenations are less structure-sensitive than are most other reactions. For any one... 

Some enzyme reactions can be studied colorimetrically when either the substrate or product can be converted chemically to a coloured product suitable for measurement in a u.v. or visible light spectrophotometer. In the case of alanine aminotransferase, the pyruvate formed in the reaction can be converted to pyruvate-2,4-dinitrophenylhydrazone by the addition of 2,4-dinitrophenylhydrazine (DNP). Addition of sodium hydroxide yields a product with an absorption maximum at 505 nm. Other examples of colorimetric procedures will be found in the last section. Colorimetric procedures are used for enzyme assays in the sampling mode, whereby samples of the reaction mixture are analysed at certain fixed times after starting the reaction. Graphs depicting the reaction rate must then be constructed by plotting amount of substrate transformed against time. 

Following the relation (5.8) this reaction graph may be decomposed in the following three ways chosen from many others... 

As can be seen for infinite recycle ratio where C = Cl, all reactions will occur at a constant C. The resulting expression is simply the basic material balance statement for a CSTR, divided here by the catalyst quantity of W. On the other side, for no recycle at all, the integrated expression reverts to the usual and well known expression of tubular reactors. The two small graphs at the bottom show that the results should be illustrated for the CSTR case differently than for tubular reactor results. In CSTRs, rates are measured directly and this must be plotted against the driving force of... 

These equations hold if an Ignition Curve test consists of measuring conversion (X) as the unique function of temperature (T). This is done by a series of short, steady-state experiments at various temperature levels. Since this is done in a tubular, isothermal reactor at very low concentration of pollutant, the first order kinetic applies. In this case, results should be listed as pairs of corresponding X and T values. (The first order approximation was not needed in the previous ethylene oxide example, because reaction rates were measured directly as the total function of temperature, whereas all other concentrations changed with the temperature.) The example is from Appendix A, in Berty (1997). In the Ignition Curve measurement a graph is made to plot the temperature needed for the conversion achieved. 

Simpson and Burt have studied the same reactions in the presence of various amounts of ethanol and have plotted graphs of phosphonate (81 R = Ph) and phenyl acetylene produced against moles of alcohol added. Acetylene in the product reached a maximum (around 60%) when two moles of ethanol were added and stayed fairly constant beyond this, which suggests that the attack-on-halogen contribution to the mechanism is approximately 60%. The rest of the reaction presumably follows some other mechanism and the authors suggest the addition-elimination route (79) in view of the isolation of the phosphonate (83) from the reaction of tri(isopropyl) phosphite with the bromoacetylene (84). 

Returning to the example sequence in Scheme 4.7 and using the minimum values of AE determined and setting the reaction yields also as minimum values, it is possible to evaluate the probabilities that each reaction will have an RME of at least 0.618 and also the probability that both reactions will achieve it simultaneously. Figures 4.10 and 4.11 illustrate the relevant regions in the graphs. We can conclude that the probabilities for the Petasis condensation and the coupling reaction are 77% and 94%, respectively. Since the reactions are independent of each other, that is, the individual probabilities are mutually exclusive, the combined probability that both reactions will have RME values of at least... 

Besada [12] described a spectrophotometric method for determination of penicillamine by reaction with nitrite and Co(II). Penicillamine is first treated with 1 M NaN02 (to convert the amino-group into a hydroxy-group), then with 0.1 M CoCl2, and finally the absorbance of the brownish-yellow complex obtained is measured at 250 nm. The process is carried out in 50% aqueous ethanol, and the pH is adjusted to 5.4— 6.5 for maximum absorbance. The calibration graph is linear over the concentration range of 0.25-2.5 mg per 50 mL, and the mean recovery (n = 3) of added drug is 99.7%. Cystine, cysteine, methionine, and other amino adds do not interfere. 

In order to test rate laws, a must be determined as a function of time using an appropriate experimental technique. If the reaction involves the loss of a volatile product as shown in Eq. (8.1), the extent of reaction can be followed by determining the mass loss either continuously or from sample weight at specific times. Other techniques are applicable to different types of reactions. After a has been determined at several reaction times, it is often instructive to make a graph of a versus time before the data are analyzed according to the rate laws. As will be shown later, one can often eliminate some rate laws from consideration because of the general shape of the a versus t curve. 

As an alternative to this traditional procedure, which involves, in effect, linear regression of equation 5.3-18 to obtain kf (or a corresponding linear graph), a nonlinear regression procedure can be combined with simultaneous numerical integration of equation 5.3-17a. Results of both these procedures are illustrated in Example 5-4. If the reaction is carried out at other temperatures, the Arrhenius equation can be applied to each rate constant to determine corresponding values of the Arrhenius parameters. 

Equation (4.49) describes the shape of the graph in Figure 4.6. Before we look at Equation (4.46) in any quantitative sense, we note that if RT In Q is smaller than AGf, then AGr is positive. The value of AGr only reaches zero when A Gf is exactly the same as RT In Q. In other words, there is no energy available for reaction when AGr = 0 we say the system has reached equilibrium . In fact, AGr = 0 is one of the best definitions of equilibrium. 

Also depicted on the graph in Figure 8.5 is the number of moles of magnesium sulphate produced. It should be apparent that the two concentration profiles (for reactant and product) are symmetrical, with one being the mirror image of the other. This symmetry is a by-product of the reaction stoichiometry, with 1 mol of sulphuric acid forming 1 mol of magnesium sulphate product. 

Figure 14 displays the product formation of H20, N2, C02, and CO. The concentration C(t) is represented by the actual number of product molecules formed at time t. Each point on the graphs (open circles) represents an average over a 250-fs interval. The number molecules in the simulation were sufficient to capture clear trends in the chemical composition of the species involved. It is not surprising to find that the rate of H20 formation is much faster than that of N2. Fewer reaction steps are required to produce a triatomic species like water, whereas the formation of N2 involves a much more complicated mechanism.108 Furthermore, the formation of water starts around 0.5 ps and seems to have reached a steady state at 10 ps, with oscillatory behavior of decomposition and formation clearly visible. The formation of N2, on the other hand, starts around 1.5 ps and is still progressing (as the slope of the graph is slightly positive) after 55 ps of simulation time, albeit slowly. 

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