Kinetic systems integrated rate expressions

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. 

Equations (5.1.5), (5.1.6), and (5.1.8) are alternative methods of characterizing the composition of the system as a function of time. However, for use in the analysis of kinetic data, they require an a priori knowledge of the ratio of k to. 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. 

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. 

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. 

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. 

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. 

Since kinetics for the system has already been provided in Chapter 1, the species concentration profiles achieved in the beaker may be computed using the standard batch reactor equation. This is done by numerically integrating the rate expressions with the new initial condition given by concentration 82. The batch profile resulting from this integration is shown in Figure 3.2, given by the curve a20. The profiles have been overlaid with the previous experiment for comparison. 

For a system of S chemical species and R reactions c is the S vector of concentrations, k the R vector of time independent parameters (rate coefficients), and f the vector of the R rate expression functions. If the overall reaction is isothermal and takes place in a well-mixed vessel, equation (1) comprises a detailed chemical kinetic model (DCKM) of the reaction. The integration of the model equations can present difficulties because the rate coefficients may vary from one another by many orders of magnitude, and the differential equations are stiff. Numerical methods for the solution of stiff equations are discussed by Kee et al. [1]. Efficient solvers for stiff sets of equations have been developed and are available in various software packages. Some of these are described in Chapter 5. Additional information can be found in Refs. [2,3]. 

The most realistic description of the kinetics of a unimolecular reaction is given by the RRKM method,3,16 which has been successfully used in the investigations of a wide variety of reaction systems. However, the general equation of the RRKM method is quite complex, and values of the rate constant at a given temperature and pressure are obtained by numerical evaluation of a complicated integral expression. This is an important limitation of the RRKM method because kinetic modeling studies require a simple expression, best in an analytical form, convenient for estimating the rate constant under any experimental conditions of pressure and temperature. 

The temperature-dependent physical constants in the mass balance (i.e., the kinetic rate constant and the equilibrium constant) are expressed in terms of nonequilibrium conversion x using the linear relation (3-42). The concept of local equilibrium allows one to rationalize the definition of temperature and calculate an equilibrium constant when the system is influenced strongly by kinetic changes. In this manner, the mass balance is written with nonequilibrium conversion of CO as the only dependent variable, and the problem can be solved by integrating only one ordinary differential equation for x as a function of reactor volume. 

A model that reproduces the homogeneous dynamics of a chemical reaction should, when combined with the appropriate diffusion coefficients, also correctly predict front velocities and front profiles as functions of concentrations. The ideal case is a system like the arsenous acid-iodate reaction described in section 6.2, where we have exact expressions for the velocity and concentration profile of the wave. However, one can use experiments on wave behavior to measure rate constants and test mechanisms even in cases where the complexity of the kinetics permits only numerical integration of the rate equations. 

Equation 56 indicates a first-order dependence of the rate of polymerization on the monomer concentration and a square-root dependence on the concentration of the initiator. These dependencies have been confirmed for the example of many polymerizing systems. It should be pointed out that deviations from equation 56 (such as chain-length-dependent rate coefficients or primary radical termination) are manifest in a change in the exponents associated with the initiator and monomer concentrations (386,387). The rate of polymerization will scale with a weaker than square-root dependence on [I] and a stronger than hnear dependence on [M]. Extreme dilution of monomer can also change the exponents of monomer and initiator concentration. Equation 56 is easily integrated to yield an expression which directly correlates the monomer conversion with the observed kinetic rate coefficient obs-... 

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