Kinetics integrated rate expressions

Our initial experimental results indicated that the kinetic model— first order in liquid phase CO concentration—was the leading candidate. We designed an experimental program specifically for this reaction model. The integrated rate expression (see Appendix for nomenclature) can be written as ... 

First-order kinetics. Show that the first-order integrated rate expression can be written as... 

Equations 5.1.5, 5.1.6, and 5.1.8 are alternative methods of characterizing the progress of the reaction in time. However, for use in the analysis of kinetic data, they require an a priori knowledge of the ratio of kx to k x. To determine the individual rate constants, one must either carry out initial rate studies on both the forward and reverse reactions or know the equilibrium constant for the reaction. In the latter connection it is useful to indicate some alternative forms in which the integrated rate expressions may be rewritten using the equilibrium constant, the equilibrium extent of reaction, or equilibrium species concentrations. 

For second-order kinetics, the integrated rate expression is... 

One problem for kineticists is that only a relatively slight increase in complexity of the kinetic scheme results in differential equations which cannot be integrated in a straightforward manner to give a manageable analytical expression. When this happens the differential equations have to be solved by either numerical integration or computer simulation. This is a mathematical limitation of the use of integrated rate expressions which is not apparent in the kinetic scheme. Typical schemes which are mathematically complex are... 

Kinetic theory tells us that the extent of completeness of a reaction is expressed by the integrated rate expression ... 

Direct observation of intermediates (or lack thereof) provides credence to any mechanistic assignment. Integrated rate expressions for the intermediates will generally be less convoluted than the products since they are further upstream in the kinetic cascade. However, it is often difficult to observe independent spectroscopic signatures for each of the four PCET states. This is partly a consequence of the inherent coupling between electronic states and protonic states in PCET systems. In addition, PCET systems have not incorporated design elements for independent spectroscopic signatures of the proton and the electron. 

While, in principle, due allowance for these effects can be incorporated into any quantitative kinetic analysis, in practice the integration is made more complicated or the rate expressions become intractable. The incorporation of additional, and sometimes imperfectly defined, parameters does not always represent a meaningful refinement of the approach. 

In equation (13.11), the first term corresponds to the catalysed part of the reaction and the remaining terms, which make a relatively small contribution, apply to the uncatalysed part. Kinetic data at constant acidity were in good agreement with the integrated form of the calculated rate expression. The rate coefficients k2,k, k, and the ratio k. jk were evaluated. Almost linear plots of log 2 versus log [ ] were obtained at four temperatures with slopes close to —1.8. This result suggests that the dominant activated complex is that formed by loss of two ions, viz. 

Exponential rate expressions are also useful in deriving kinetic equations because they can be substituted into differential equations, which can then be integrated. For example, from Scheme 2 the differential equation describing the rate of appearance of unchanged drug in urine may be written ... 

It is obvious that to quantify the rate expression, the magnitude of the rate constant k needs to be determined. Proper assignment of the reaction order and accurate determination of the rate constant is important when reaction mechanisms are to be deduced from the kinetic data. The integrated form of the reaction equation is easier to use in handling kinetic data. The integrated kinetic relationships commonly used for zero-, first-, and second-order reactions are summarized in Table 4. [The reader is advised that basic kinetic... 

Assume that the reaction follows first-order kinetics. The integral form of the reaction rate expression is given by equation 3.1.8. 

A reaction rate constant can be calculated from the integrated form of a kinetic expression if one has data on the state of the system at two or more different times. This statement assumes that sufficient measurements have been made to establish the functional form of the reaction rate expression. Once the equation for the reaction rate constant has been determined, standard techniques for error analysis may be used to evaluate the expected error in the reaction rate constant. 

It is also possible to use integral methods to determine the concentration dependence of the reaction rate expression and the kinetic parameters involved. In using such approaches one again requires a knowledge of the equilibrium constant for use with one of the integrated forms developed in Section 5.1.1. 

Equation 8.3.4 may also be used in the analysis of kinetic data taken in laboratory scale stirred tank reactors. One may directly determine the reaction rate from a knowledge of the reactor volume, flow rate through the reactor, and stream compositions. The fact that one may determine the rate directly and without integration makes stirred tank reactors particularly attractive for use in studies of reactions with complex rate expressions (e.g., enzymatic or heterogeneous catalytic reactions) or of systems in which multiple reactions take place. 

This problem may be solved by linear regression using equations 3.4-11 (n = 1) and 3.4-9 (with n = 2), which correspond to the relationships developed for first-order and second-order kinetics, respectively. However, here we illustrate the use of nonlinear regression applied directly to the differential equation 3.4-8 so as to avoid use of particular linearized integrated forms. The method employs user-defined functions within the E-Z Solve software. The rate constants estimated for the first-order and second-order cases are 0.0441 and 0.0504 (in appropriate units), respectively (file ex3-8.msp shows how this is done in E-Z Solve). As indicated in Figure 3.9, there is little difference between the experimental data and the predictions from either the first- or second-order rate expression. This lack of sensitivity to reaction order is common when fA < 0.5 (here, /A = 0.28). 

For a more detailed analysis of measured transport restrictions and reaction kinetics, a more complex reactor simulation tool developed at Haldor Topsoe was used. The model used for sulphuric acid catalyst assumes plug flow and integrates differential mass and heat balances through the reactor length [16], The bulk effectiveness factor for the catalyst pellets is determined by solution of differential equations for catalytic reaction coupled with mass and heat transport through the porous catalyst pellet and with a film model for external transport restrictions. The model was used both for optimization of particle size and development of intrinsic rate expressions. Even more complex models including radial profiles or dynamic terms may also be used when appropriate. 

The integral reactor can have significant temperature variations from point to point, especially with gas-solid systems, even with cooling at the walls. This could well make kinetic measurements from such a reactor completely worthless when searching for rate expressions. The basket reactor is best in this respect. 

For kinetic expressions which are just a little more complicated than first oder kinetics, the rate cannot be integrated by hand. Very complex kinetic expressions are easy to treat by numerical methods, but analytical treatment is only possible for extremely simple kinetic expressions. 

Fortunately, the kinetics of the wall loss, measured from the decay of the reactive species in the absence of added reactant, are generally observed to be first order, so that corrections for these processes can be readily incorporated into the kinetic analyses. When these wall losses are significant, the integrated form of the rate expression (T) for reaction (17) of A + B becomes... 

The rate expressions Rj — Rj(T,ck,6m x) typically contain functional dependencies on reaction conditions (temperature, gas-phase and surface concentrations of reactants and products) as well as on adaptive parameters x (i.e., selected pre-exponential factors k0j, activation energies Ej, inhibition constants K, effective storage capacities i//ec and adsorption capacities T03 1 and Q). Such rate parameters are estimated by multiresponse non-linear regression according to the integral method of kinetic analysis based on classical least-squares principles (Froment and Bischoff, 1979). The objective function to be minimized in the weighted least squares method is... 

Write a simulation program to integrate the kinetic rate expressions for Go and Oco in time, starting with initial coverages 0q1 and 0co- What is the maximum value of T0R(C02) (in s-1) At what elapsed time does this occur What are the coverages Go and Oco at the maximum T0R(C02) ... 

Only a relatively few kinetic schemes for complex reactions have differential rate expressions which can be integrated straightforwardly. Examples are the following. 

This sort of analysis is very important in the formulation of the steady state approximation, developed to deal with kinetic schemes which are too complex mathematically to give simple explicit solutions by integration. Here the differential rate expression can be integrated. The differential and integrated rate equations are given in equations (3.61)—(3.66). 

A more quantitative analysis of the batch reactor is obtained by means of mathematical modeling. The mathematical model of the ideal batch reactor consists of mass and energy balances, which provide a set of ordinary differential equations that, in most cases, have to be solved numerically. Analytical integration is, however, still possible in isothermal systems and with reference to simple reaction schemes and rate expressions, so that some general assessments of the reactor behavior can be formulated when basic kinetic schemes are considered. This is the case of the discussion in the coming Sect. 2.3.1, whereas nonisothermal operations and energy balances are addressed in Sect. 2.3.2. 

The first, called the integral method of data analysis, consists of hypothesizing rate expressions and then testing the data to see if the hypothesized rate expression fits the experimental data. These types of graphing approaches are well covered in most textbooks on kinetics or reactor design. 

In kinetic studies it is not necessary to use rates for determination of the rate parameters. In eqs 2-4 r represents the rate expression for the reaction under consideration, whatever its mathematical form. This can be inserted in eq 2, and subsequently integrated after separation of variables, leading to eq 5. The result may be an implicit expression, containing the rate parameters, describing the relationship between space-time and conversion. It will be shown later that this relationship can also be used for parameter determination. 

In general the interpretation of the data is somewhat more complicated than for the differential method. Especially for an unknown complicated kinetic functions, the derivation of the correct reaction rate expression RA from experimental results using Equations 5.41 and 5.33 is more cumbersome than fitting Equations 5.30 and 5.33. This is especially true for complex reaction networks, as in the isomerization and cracking reactions of crude oil fractions, where the integral method is very laborious with which to derive individual rate constants. 

A second kind of rate law, the integrated rate law, will also be important in our study of kinetics. The integrated rate law expresses how the concentrations depend on time. As we will see, a given differential rate law is always related to a certain type of integrated rate law, and vice versa. That is, if we determine the differential rate law for a given reaction, we automatically know the form of the integrated rate law for the reaction. This means that once we determine either type of rate law for a reaction, we also know the other one. 

This chapter is devoted to numerical integration, and more specifically to the integration of rate expressions encountered in chemical kinetics. For simple cases, integration yields closed-form rate equations, while more complex reaction mechanisms can often be solved only by numerical means. Here we first use some simple reactions to develop and calibrate general numerical integration schemes that are readily applicable to a spreadsheet. We then illustrate several non-trivial applications, including catalytic reactions and the Lotka oscillator. 

The kinetic data available for a particular reaction are examined to determine if they fit a simple kinetic expression. For example, for a first-order reaction, a plot of log [A] versus t yields a straight line with a slope of —A /2.303. For second-order reactions, a plot of 1/[A] versus t is linear with a slope of k. Figure 3.7 shows such plots. Alternatively, the value of k can be calculated from the integrated expression over a sufficient time range. If the value of k remains constant, the data are consistent with that rate expression. 

Kinetic rate expressions are well known to exhibit hard-to-fit analytical forms. Moreover, most of them cannot be integrated to present a usable analytical form. We must therefore collect and fit data that reports instantaneous rates rather than cumulative concentrations. The use of kinetics to study reaction mechanisms is greatly hampered by these constraints. The only solution that can be envisioned is to acquire massive amounts of reliable, error-free, data. To achieve this we must clean up the raw experimental data by the skillful application of powerful methods of error correction. Only then is there the prospect that the data will reveal the underlying reaction mechanism. In the following chapters we present the necessary experimental methods for acquiring vast amounts of rate data and outline the early stages of the development of error correction techniques designed to deal with raw and noisy kinetic rate data. 

---