Concentration-Dependent Term of a Rate Equation

Before we can find the form of the concentration term in a rate expression, we must distinguish between different types of reactions. This distinction is based on the form and number of kinetic equations used to describe the progress of reaction. Also, since we are concerned with the concentration-dependent term of the rate equation, we hold the temperature of the system constant. 

First of all, when materials react to form products it is usually easy to decide after examining the stoichiometry, preferably at more than one temperature, whether we should consider a single reaction or a number of reactions to be occurring. 

When a single stoichiometric equation and single rate equation are chosen to represent the progress of the reaction, we have a single reaction. When more than one stoichiometric equation is chosen to represent the observed changes. 

reaction proceeds in parallel with respect to B, but in series with respect to A, R, and S. 

Consider a single reaction with stoichiometric equation 

---

Concentration-Dependent Term of a Rate Equation 17 which for a first-order reaction becomes simply... 

AG° tells us where a reaction equilibrium lies, in which direction it will proceed, and AG tells us how fast we will get there. The free energy of activation also has enthalpic and entropic components (8.7). We can describe the kinetics of organic reactions, just like any other, in terms of a rate equation. For example, the hydrolysis of iodomethane (8.8) shows the rate (Equation 8.9). Measured kinetics of this type tell us about the whole reaction process, whether one or many steps are involved. Another useful classification is the molecularity of processes, since this is applicable to individual steps in reaction mechanisms. Figure 8.8 shows typical unimolecular and bimolecular reaction steps. In the unimolecular process, a single molecule is involved in the RDS, and in many (but not all) cases, this will lead to the rate depending only on the concentration of substrate. In a bimolecular reaction, two molecules are involved in the RDS, and the rate depends on the concentration of each of them. 

In this equation it is the reaction rate constant, k, which is independent of concentration, that is affected by the temperature the concentration-dependent terms, J[c), usually remain unchanged at different temperatures. The relationship between the rate constant of a reaction and the absolute temperature can be described essentially by three equations. These are the Arrhenius equation, the collision theory equation, and the absolute reaction rate theory equation. This presentation will concern itself only with the first. 

The example reactions considered in this section all have the property that the number of reactions is less than or equal to the number of chemical species. Thus, they are examples of so-called simple chemistry (Fox, 2003) for which it is always possible to rewrite the transport equations in terms of the mixture fraction and a set of reaction-progress variables where each reaction-progress variablereaction-progress variable —> depends on only one reaction. For chemical mechanisms where the number of reactions is larger than the number of species, it is still possible to decompose the concentration vector into three subspaces (i) conserved-constant scalars (whose values are null everywhere), (ii) a mixture-fraction vector, and (iii) a reaction-progress vector. Nevertheless, most commercial CFD codes do not use such decompositions and, instead, solve directly for the mass fractions of the chemical species. We will thus look next at methods for treating detailed chemistry expressed in terms of a set of elementary reaction steps, a thermodynamic database for the species, and chemical rate expressions for each reaction step (Fox, 2003). 

In (5.297), the interpolation parameter is defined separately for each component. Note, however, that unlike the earlier examples, there is no guarantee that the interpolation parameters will be bounded between zero and one. For example, the equilibrium concentration of intermediate species may be negligible despite the fact that these species can be abundant in flows dominated by finite-rate chemistry. Thus, although (5.297) provides a convenient closure for the chemical source term, it is by no means guaranteed to produce accurate predictions A more reliable method for determining the conditional moments is the formulation of a transport equation that depends explicitly on turbulent transport and chemical reactions. We will look at this method for both homogeneous and inhomogeneous flows below. 

Composition affects the rate of reaction by fitting a simple expression to the data as shown in Figure 3-3. The concentration dependent term is found by guessing the rate equation, and seeing whether or not it fits the data. 

Equation (42) cannot be used if NO concentrations approach their equilibrium values, since the net production rate then depends on the concentration of NO, thereby bringing bivariate probability-density functions into equation (40). Also, if reactions involving nitrogen in fuel molecules are important, then much more involved considerations of chemical kinetics are needed. Processes of soot production similarly introduce complicated chemical kinetics. However, it may be possible to characterize these complex processes in terms of a small number of rate processes, with rates dependent on concentrations of major species and temperature, in such a way that a function w (Z) can be identified for soot production. Rates of soot-particle production in turbulent diffusion flames would then readily be calculable, but in regions where soot-particle growth or burnup is important as well, it would appear that at least a bivariate probability-density function should be considered in attempting to calculate the net rate of change of soot concentration. 

Glissmann and Schumacher ( ) interpreted their data in terms of a mixed mechanism including a direct bimolecular reaction 2O3 302- In the reinterpretation of these data, Benson and Axworthy (4) decided that there was no evidence for such a reaction. [A reappraisal of the more recent work of Ogg and Sutphen 9) similarly shows that their data do not require the introduction of such a direct bimolecular reaction.] For such a reaction to contribute, let us say 10% to the scheme proposed, k would have to be about 0.2 k (Equation 9) over the pressure and temperature range studied. The frequency factor of Reaction B would be expected to be about 10 to 10 times the frequency of collisions which would be about 2 X 10 liter/mole-sec. If = 0.2 ki is set at 100° C., must be between 18 and 21.5 kcal. per mole, depending on the steric factor used. The observed rate of decomposition of concentrated ozone (I) at low temperatures, where such a reaction has the best chance of being observed, verifies these frequency factors and activation energies as upper and lower limits, respectively, for such a proposed bimolecular path. 

Even though the absorption rate constant (kf) defines the rate of absorption, its accurate determination is largely dependent on the adequacy of the plasma concentration-time data associated with the absorption phase of the drug. When a drug is administered orally, as a conventional (immediate-release) dosage form, or injected intramuscularly as an aqueous parenteral solution, the absorption and disposition kinetics can often be analysed in terms of a one-compartment pharmacokinetic model with apparent first-order absorption. The plasma concentration-time curve is described by the equation... 

The catalytic mechanism in solution phase described by Equations (3.1) and (3.2) is usually described in terms of a reversible electron transfer for the Cat system (Equation (3.3)) followed by a reaction operating under conditions of pseudo first-order kinetics (Nicholson and Shain, 1964). Thus, the shape of cyclic voltammo-grams (CVs) depends on the parameter X = kc, t, where k is the rate constant for reaction (3.2) and c at is the concentration of catalyst. For low A, values, the catalytic reaction has no effect on the CV response and a profile equivalent to a singleelectron transfer process is approached. For high X values, s-shaped voltammetric curves are observed that can be described by (Bard and Faulkner, 2001) ... 

Thus the activation energy, in principle, depends on whether the rate equation is expressed in terms of concentrations or partial pressures. Also, the difference between and Ep depends on the temperature. In practice this difference is not significant. In this example, at a temperature of ITC,... 

Because the left hand side of Type (4) equations are simply derivatives of concentrations of reactants, the maximum tolerances of the dependent variable of the equation that is used to solve for the rate constants is simply the sum of squares of the maximum tolerances of the derivatives. Thus, even if terms on the right hand side are known, their errors are considered to be zero. In other words, the maximum tolerance associated with a dependent variable of an individual equation is always calculated in the same way, regardless of how many of the rate constants are known. This method of calculation effectively assumes that any data generated by the SIMULATOR module is considered to be exact. It should be noted that this assumption leads to a more restricted solution than that which would be obtained if maximum errors in all the known terms (that is, those with constant parameters) had been considered. 

The steady-state method in which differential equations are set up defining the time dependence, d[x]/dt, of concentrations of intermediates x involved in the process. In the steady state, d[xydt = 0 giving a rate equation containing potential-dependent terms. 

In Equation 58, the time-dependent terms between the braces contain the decay constant A. Therefore, the rate of change in Ra-226 concentration at any depth (dC/dt) depends on the decay rate constant. Thus, in the case of a first-order reaction (radioactive decay), the rate of change in concentration depends on the reaction rate constant, whereas it has been shown in the preceding section that for a zero-order reaction (oxygen consumption), the rate of change in concentration (dC/dt) is independent of its rate constant. 

The rate of transport of water is thus expressed by equation 8 in terms of an apparent diffusion coefficient, D, and the gradient of the overall concentration of water in the rubber, dC /dx. The theory predicts that the diffusion of water in rubber containing hydrophilic Impurities can be expressed in terms of a concentration dependent diffusion coefficient (equation 9). 

The terminology graphical rate equation derives from our attempt to relate rate behavior to the reaction s concentration dependences in plots constructed from in situ data. Reaction rate laws may be developed for complex organic reactions via detailed mechanistic studies, and indeed much of the research in our group has this aim in mind. In pharmaceutical process research and development, however, it is rare that detailed mechanistic understanding accompanies a new transformation early in the research timeline. Knowledge of the concentration dependences, or reaction driving forces, is required for efficient scale-up even in the absence of mechanistic information. We typically describe the reaction rate in terms of a simplified power law form, as shown in Equation 27.4 for the reaction of Scheme 27.1, even in cases where we do not have sufficient information to relate the kinetic orders to a mechanistic scheme. 

Given a specific reaction and type of reactor, it is usually possible to derive a theoretic concentration model which is able to anticipate the reaction s behaviour over time depending on the specific environmental conditions such as pressure, temperature, and reactant inflow rates. But although such a theoretical approach for a chemical process can be derived more or less easily, the relationships of the technical realization in terms of a chemical plant are far more complicated due to the fact that a theoretical model necessarily requires simplifications and assumptions. For example, a homogeneously distributed mixture inside a reactor is often assumed which is rarely the case in a real reactor. This typically leads to a stochastiflcation of theoretic relations, e.g. the differential equations may turn into stochastic differential equations (SDEs). This turns the concentration as calculated in (2.7) into a stochastic variable with associated distribution. For example, (2.7) can be turned into... 

The reaction rate is properly defined in terms of the time derivative of the extent of reaction [eqnation (3.0.1)]. One must define /c in a similar fashion to ensnre nniqneness. Definitions of k in terms of the various r, would lead to rate constants that would differ by the ratios of their stoichiometric coefficients. The units of the rate constant will vary depending on the overall order of the reaction. These units are those of a rate divided by an mth power of concentration (where m is the overall order of the rate law). Thns, from examination of equations (3.0.17) and (3.0.18),... 

---