Figure 8. Graphical representation for determining reaction order (n) of a reaction.
comes C0 — C = k0t, where C = concentration at time t, and C0 = initial concentration. According to this mathematical expression, a distinguishing feature for this type of reaction is a linear decrease in concentration as a function of time as illustrated in Figure 9. Typical reactions that have been represented by zero-order reactions include some of the autooxidation reactions. It is clear that zero-order reactions do not appear to occur as frequently in food systems as other reaction orders. In most cases, it is evident that the most common situation for this type of reaction is when the concentration of the reactants is so large that the system appears to be independent of concentration.
First-Order Reactions. A large number of reactions occurring in food systems appears to follow a first-order reaction. A mathematical expression for this behavior would be as follows: —dC/dt = C, where kx = first-order reaction rate constant. By integration, this equation becomes ln[C/C0] = kxt. Thus, according to this mathematical expression In (C) versus time will be a linear function where the slope corresponds to — k as shown in Figure 10. The term half-life (i1/2) of a reactant is commonly used and may be described as tV2 = In Hkx. These mathematical expressions clearly indicate that the half-life and the reaction rate for a true first-order reaction are independent of the initial concentration. Although in a number of systems this may be the case, often formulated products will not follow true first-order reaction kinetics, but rather a pseudo-first-order reaction. In fact, in formulated systems the presence of breakdown products may strongly influence the order of the reaction. However, for only a given value of initial concentration, the reaction may follow apparent first-order kinetics. To determine if a given reaction does indeed follow a pseudo-first-order kinetics, conditions for the kinetic study can be chosen to follow the technique of flooding. Through this approach, all but one of the concentrations are set sufficiently high that, compared with the one reagent present at lower concentration, the others are effectively constant during the time of the experiment. Since only one of the concentrations changes appreciably during the run, the effective kinetic order is reduced to the reac
Figure 9. Graphical representation for the determination of a Figure 10. Graphical representation for the determination of a zero-order rate constant (k0). first-order rate constant (kt).
tion order with respect to that one substance. If the order of the reaction is determined to be one, the reaction is said to follow a pseudo-first-order reaction. The degradation of ascorbic acid, for instance, has been primarily found to follow first-order kinetics in food systems. On the contrary, degradation of ascorbic acid in model systems has frequently been found to follow pseudo-first-order kinetics. It appears that the presence of breakdown products modifies the kinetics of deterioration of ascorbic acid, and, thus, its initial concentration will influence its rate of degradation. These factors, of course, serve to further complicate the prediction of nutrient retention.
Second-Order Reactions. Two types of second-order reaction kinetics are of importance. Type I, A + A P, where A is a reactant and P is a product, may be mathematically described as -dCJdt = k2 ■ C\. Type II reaction, A + B ->• P, where A and B are the reactants and P the product, may be mathematically described as —dCJdt — k2CACB, where CA = concentration of reactant species (A) at time (t), CB = concentration of reactant species (B) at time (t) and k2 = second-order reaction rate constant. For Type I, the integrated kinetic expression yields: [1/CA] — [1/C^] = k2t, which in terms of the half-life becomes i1/2 = Hk2 ■ CA(). For Type II, the integrated form yields: k2t = [l/(CAo - CBo)] • In [CBo • CA/CAo ■ C„], where CAo and CBo are the respective initial concentrations and CA and CB are the respective concentrations at time (t). It should be stressed, however, that Type II reactions do not have to necessarily follow a second-order reaction. For instance, for the particular case where component A is present in large amounts as compared with component B, the reaction may follow first-order kinetics with respect to B. A typical plot of second-order kinetics is presented in Figure 11.
Figure 11. Graphical representation for the determination of a second-order rate constant (k2) for a Type I reaction.
Many different types of reactions fall under the category of nonelementary reactions, which will only be mentioned by name. To characterize the kinetics of nonelementary reactions, one can assume a series of individual elementary reactions taking place. In these reactions, intermediates may not be observed or quantitated, either because they are present in very small amounts or because they are unstable. Such reactions would fall under three main categories: (1) consecutive or series reactions, (2) reversible or opposing reactions that attain a finite equilibrium, and (3) parallel or competitive reactions. The types of intermediates postulated may fall in any one of the following categories, namely, free radicals, ions and polar substances, molecules, and transition complexes (chain reactions and nonchain reactions). It should be mentioned that a classical example of a transition complex-chain reaction is the degradation of ^-carotene. It involves an autoxidation reaction involving three main periods: (1) induction (formation of free radicals), (2) propagation (free-radical chain reactions), and (3) termination (formation of nonradical products). A more in-depth analysis and mathematical approach to characterizing this reaction has been reported by several authors (86-89).
Another example of a transition complex reaction is that of a nonchain catalyzed type that may involve the interaction of a substrate with a catalyst to form a complex, followed by its decomposition to form a product. Upon decomposition, the catalyst is then regenerated and is capable of taking part in the reaction once again. An example of this type of transition complex reaction is that of enzyme-catalyzed reactions. It should be mentioned that most of the reactions occurring in biological systems are catalytic in nature. The basic principles for enzyme-catalyzed reactions have been presented by MichaelisMenten, who proposed the theory of complex formation. Although the general principle of chemical kinetics may apply to enzymatic reactions, the phenomenon of saturation with substrate is unique to enzymatic reactions. In fact, at low substrate concentrations the reaction velocity is proportional to the substrate concentration, and thus the reaction is first order with respect to the substrate. As the substrate concentration increases, the reaction progressively decreases, being no longer proportional to the concentration of the substrate and deviating from any firstorder kinetics. The reaction follows zero-order reaction kinetics, due to saturation with the substrate. For the particular case of enzyme kinetics, however, the cases of competitive and noncompetitive inhibition also add complications to the reaction and are thus not as easily defined by simple kinetics. A general description of enzyme-substrate kinetics can be found in most any classical biochemistry textbook (eg, 90, 91).
When considering reaction rates, it is clear that these values may be influenced by a large number of parameters, including temperature and pressure. In fact, equilibrium yields, chemical reaction rates, and product distribution may be drastically influenced by temperature. Since chem
ical reactions are accompanied by heat effects, if these are large enough to cause a significant change in temperature of the reaction mixture, these effects also need to be considered. This would be particularly important in reactor design. The effect of temperature for an elementary process may follow, in most cases, the Arrhenius equation: k = &0exP(-EJIT) where, k0 = frequency or collision factor, Ea = activation energy (cal/mol), R = gas constant (1.987 cal/mol • K), and T = absolute temperature (K). It is obvious that if the frequency factor and the activation energy could be evaluated from molecular properties of the reac-tants, it would be possible to estimate the values corresponding to the reaction rate. Unfortunately, our knowledge of kinetics is limited, particularly for complex systems, as would be the case of food systems or products.
It is, however, important to mention the collision theory as an approach to deal with kinetics. In Figure 12, the energy levels involved in a reaction are illustrated. According to the collision theory, upon the collision of reactive molecules, enough energy is generated to provide the necessary activation energy. Such a theory was used as the foundation for the determination of rate expressions based on the frequency of molecular collision required to generate a minimum energy.
Another theory, the activated-complex or transitionstate theory, has also been suggested. According to this approach, which still relies on reactions occurring due to collision between reactive molecules, an activated complex is formed from the reactants that eventually decomposes to generate products. The activated complex is in thermodynamic equilibrium with the reactants. Complex decomposition is, then, the limiting step. Regardless of the theories considered, they do not provide the means to rapidly and easily calculate activation energies from simple thermodynamic information. Thus in practical terms, one has to obtain basic kinetic information to be able to determine the effect of temperature as affecting reaction kinetics. Based on the Arrhenius equation, it is clear that if one plots the In k versus 1 IT, the slope would correspond to the activation energy divided by the gas constant. Moreover, this value by itself will not provide any idea on the reactivity of a given system, only information on temperature dependence of the reaction.
Although the Arrhenius equation is commonly used to describe temperature dependence of the reaction rate in most food systems, deviations may occur as reported by several authors, including Labuza and Riboh (92). In fact, a large number of factors may contribute to deviations.
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