Reaction Kinetics
IV. DETERMINING KINETIC PARAMETERS A General Considerations
Compared to characterizing chemical reactions in chemical laboratory settings, characterizing reactions and changes in foods is often not easy. As pointed out
earlier, environmental factors such as pH, temperature, moisture content (and its closely related variable, water activity), oxidation-reduction potential, presence of constituents (such as microorganisms, enzymes, metal ions, and chelating or requestering substances), and state of the tissue (e.g., whole or mascerated, age, cultivar) all influence the rate of chemical reaction. Furthermore, the food itself is not homogeneous; i.e., there can be concentration gradients in whole tissue and temperature and pH profiles in the cross section. Due to this heterogeneity, it is essential to ultimately carry out reaction rate studies in the food itself. Furthermore, it is important to keep in mind that we are going to model the rate process, and the model will not necessarily reflect the actual mechanism of the reaction.
As we noted earlier, many reactions which are of interest in food process design and food stability are hydrolysis or oxidation-reduction reactions. The influence of water on reactions in foods will be covered more extensively in Chapter 5 on water activity. Here we will briefly introduce the subject. Hydrolysis and oxidation-reduction reactions are usually complex and composed of several reactions in series. Furthermore, hydrolysis reactions depend on Hþ and OH2, not on molecular water, H2O. Oxidation-reduction reactions may
involve free radicals and may involve molecular oxygen O2. For simplicity, the
rate of change for hydrolysis or oxidation and reduction can be written, respectively, as d C½ dt ¼ 2k C½ n H 2O ½ m ð18Þ and d C½ dt ¼ 2k C½ n½ Om ð19Þ
where n represents the order of the reaction for reactant C and [H2O]mand [O]m
represent the effective concentrations of water and oxidant (or reductant) of order m, respectively. This simplification should not be interpreted as the mechanism of the reaction.
Although reactions involving water are covered more extensively in Chapter 5 on water activity, a few observations will be made here. Water can influence the reaction rate in several ways. Since it is a reactant, it may become rate limiting as in the case of foods that have low water activity. It also serves as a solvent for distribution of reactants, catalysts, and products. In some cases, lack of water can limit redistribution of reaction products resulting in a reduced reaction rate or even reversing the reaction through the law of mass action. Oxidation- reduction reactions can be limited by low concentration of oxidant or reductant or increased by diffusion of molecular oxygen through the packaging material and into the food. Regardless of the simplicity of describing the kinetics
of oxidation-reduction reactions, they generally are very complex reactions mechanistically.
As a consequence of the potential influence of water and/or oxygen on reactions in food, it is commonplace to make simplifying assumptions. In foods with water activity greater than 0.9, it is generally assumed that water is neither a limiting reactant nor a limiting solvent, and therefore, hydrolysis reactions maybe assumed to be zero order with respect to water. For oxidative reactions, such simplifying assumptions are not generally made and the reactions are modeled as zero, first, or second order in oxygen. The general rule of thumb is to first assume a reaction is first-order only in the reactant of interest (colorant, flavor compound, vitamin, etc.) and zero order in all other constituents. If that model does not adequately represent the concentration of the desired reactant over time, then more complex models are assumed and statistically tested to determine one which best describes the disappearance of the reactant.
In the design of experiments to determine kinetic parameters, Hill (1977) suggested that several questions be addressed:
1. Is the reaction under study properly identified?
2. Are side reactions important? What is the stoichiometry of the reaction? 3. Are the conditions of the reaction properly specified?
4. Does the analytical method properly measure the extent of reaction? 5. Is the experiment properly planned to provide significant data, and is
the time period adequate to allow models to be distinguished? 6. Are the data reproducible?
7. Is there a good mechanism for quenching the reaction (i.e., stopping the reaction at precise times)?
Once the system is thoroughly understood, it is necessary to generate data on concentration over time. Table 2.1 provides a description of several reactor types, the type of data obtained, and the usual methods of treating the raw data to generate the kinetic parameters. For liquids or semisolids, all of these methods are applicable provided the material can be pumped or stirred. For solids, only the batch reactor is available.
Analysis of the data can be accomplished either by treating the data differentially, d C½ dt ¼ 2k C½ n ð20Þ or integrally, Z d C½ C ½ n ¼ Z 2k dt ð21Þ
B. Differential Methods
In the differential methods, the data generally are treated in the following manner: 1. Model approach
a. Choose a form of concentration dependence on time (i.e., zero, first, second order)
d C½
dt ¼ rate ¼ 2kF Cð Þ ð22Þ b. Determine reaction rate at various times (i.e.,dcdtversus t). This can be done either graphically from plots of C versus t or numerically using a polynomial to estimate the slope at various times.
c. Calculate F(C) at various times corresponding to the calculated d[C]/dt (i.e., C0, C1, C2, etc.).
d. Plot or regress reaction rate versus F(C). If the plot (or regression equation) is linear and goes through the origin [i.e., rate is zero when F(C) is zero], the assumed model is consistent with the data. If nonlinear or does not go through the origin (0, 0), then guess a new F(C) and try again.
e. Repeat items a through d until you are successful. 2. Unknown form of F(C). Then
d C½
dt ¼ rate rð Þ ¼ 2kF Cð Þ ¼ 2k C½
b ð23Þ
log r ¼ 2log k þ b log[C]. Plot or regress log r versus log C. The intercept at C ¼ 1 is2log k ( ¼ log 1/k) and the slope is b (the order of the reaction).
3. Initial rate measurement.
TABLE2.1 Reactor Types for Determining Kinetic Parameters
Reactor type
Data
procured Treatment of data Comments
Batch c vs t Differential Integral Limited to slow
reactions
Plug flow c vs Ø Differential Integral Control of flow
rate and plug flow is essential Continuous stirred
tank reactor
Rate vs c Material balance gives rate:
log-log plot of rate vs c gives order (n ) and k
Requires large conversions for good precision c ¼ concentration; t ¼ time; Ø ¼ holding time.
This method is generally restricted to very small conversions (i.e., [C]f/[C]i
$ 0.9; less than 10% conversion) and requires that the experiment be repeated at various initial concentrations. It is most useful for complex rate functions which would be difficult to integrate.
C. Integral Methods
Integral methods are obviously closely related to differential methods.
1. Graphical Procedure
This is a trial-and-error procedure in which one hypothesizes a reaction functional form.
d C½
dt ¼ 2kF Cð Þ ð24Þ Separating variables and integrating yields
Z ½Cf ½Ci d½C F½C¼ Z t 0 2 kdt ¼ 2kt ð25Þ
One then calculatesR½Cf
½Ciðd½C/F½CÞ from the data.
If plotting or regressingR½Cf
½Ciðd½C/F½CÞ versus time produces a linear
function going through the origin (0, 0) with slope2k, then the hypothesis on the reaction functional form was correct. If not, then try again.
Since many mathematical functions are linear over short ranges of variables, it is imperative that the experiment be carried out until at least 80 – 90% of the reactant has disappeared (i.e., 3 – 5 half-lives).
If the guessed functional form is incorrect, guessing for the next try can be improved by observing the curvature of the R½Cf
½Ciðd½C/F½CÞ versus time
plot. If it deviates in a concave manner, the assumed order is greater than the true order. If the deviation is convex, the assumed order is less than the true order.
2. Numerical Procedure
In this procedure, the rate constant is calculated between successive pairs of data points (concentration versus time). Thus, from n þ 1 measurements of c versus t, one can calculate n k’s. Regressing k on time using unweighted
linear least squares then produces k. For first-order, k ¼ P tiln C½ i 2 Pti P ln½Ci /n P ti 2 n h i 2Pt2 i ð26Þ
This method has limitations because all points are assigned equal weight and functions of the measured concentration must be used in equations defining residuals. These drawbacks can be somewhat overcome by using weighted linear least squares programs when the precision of measurement of concentration is a function of concentration.
D. Accuracy of Rate Constants
To be most useful, it is important to have accurate estimates of rate constants for use in design of processing and packaging systems for foods. There are two main variables affecting the accuracy of estimated rate constants: (1) analytical precision for measuring the reactant and (2) the percent change in the reactant species monitored. Benson (1960) produced Table 2.2 to show the influence of analytical errors and extent of reaction on percent error in the rate constant k.
Since food systems are chemically complex, it is not unusual to have 1 – 2% variability in the determination of concentration of chemical species. This variability results from lack of precision in the analytical method and, frequently, from sampling problems. Consequently, if you want to determine a rate constant with an error on the order of 5 – 10%, it is imperative to carry out the reaction at least through 30% reduction in reactant species and generally through at least one half-life. Unfortunately, there are many examples in the literature where rate constants and order of reaction are reported based on studies carried out under conditions where the results are not statistically valid.
TABLE2.2 Errors in Calculated Rate Constant Caused by Analytical Errors
Percent change in reactant species monitored
1 5 10 20 30 40 50 Analytical precision (%) Error in k (%) ^0.1 14 2.8 1.4 0.7 0.5 0.4 0.3 ^0.5 70 14 7 3.5 2.5 2 1.5 ^1.0 .100 28 14 7 5 4 3 ^2.0 .100 56 38 15 10 8 6 Source: Benson (1960).