The reaction rate for any species i is generally defined as the rate of change of quantity of i (commonly moles of i) per unit time per unit volume of the reactor. Typically, the reaction rate is a function of the concentration of the reacting species and temperature [2]. The mechanism of the reaction, or the exact pathway by which the species reacting transform into reaction products, depends on the nature of the species involved in the reaction. Clearly, the nature of the species also influences the reaction rate. The intrinsic rate of reaction depends only on these factors, and does not depend on the type of reactor used for conducting the reaction [4]. It is common to define the rate on the volumetric basis—that is, per unit volume of the reactor—when the reaction is a homogeneous reaction, where only a single phase is involved [2]. A large number of chemical reactions are homogeneous, taking place either in the gas or liquid phase. However, many other reactions are heterogeneous; that is, they involve two or more phases. Fluid (gas or liquid) phase reactions conducted using solid catalysts are common in the chemical industry. In such cases, the rate may be defined on the basis of the catalyst surface area (rate of change of moles per unit time per unit surface area) or catalysts mass (rate of change of moles per unit time per unit catalyst mass).
In all cases, quantification of the intrinsic rate of reaction involves expressing the rate as a function of concentrations (or pressures) of species involved in the reaction, and temperature2. Consider a simple reaction involving two species, A and B:
2. A catalyst does not participate in the reaction, and catalyst concentration does not appear explicitly in the rate expression. Its influence on the rate is incorporated in the rate constant in the rate expression.
A + B → R + S
The rate of the reaction can be written in terms of any one of the species involved in the reaction. It can be seen from this equation that the species reaction rates are interrelated through the stoichiometry of the reaction. Mathematically, this is as follows:
The negative sign associated with A and B signifies that these two species are reactants that are consumed in the reaction. The rate for these species is the rate of disappearance. Conversely, for R and S, the products of the reaction, the rate is the rate of generation of the species. Equation 9.2 is stating that the rate of disappearance of A is exactly the same as the rate of generation of R, and so on. It is common to use a power-law model to describe the dependence of the reaction rate on concentrations, as shown in equation 9.3:
Here, k is the rate constant, and α and β—the exponents of the concentrations of A and B—are the orders of the reaction with respect to A and B, respectively. The overall order of the reaction is the sum of the orders of the reactants, which is (α + β) for the previous expression [7]. The intrinsic rate constant and the reaction orders are independent of time and concentration of species. A first-order reaction implies that the rate is directly proportional to the concentration, and a second-order reaction means that the rate is proportional to the square of the concentration. A zero-order reaction does not exhibit any concentration dependence. While these are the most common reaction orders proposed for rate equations, other orders are possible, and it is not necessary for the order to be an integer [7].
The temperature dependence of the reaction rate is incorporated in the rate expression through the rate constant k. This dependence is typically expressed by the Arrhenius3 expression [6]:
3. Svante Arrhenius, 1903 Nobel Laureate in Chemistry, is one of the first scientists to work on carbon dioxide and greenhouse effect.
k = Ae-Ea/RT
In this expression, A is called the frequency factor, and Ea is the activation energy for the reaction. Both these parameters are determined experimentally [5].
It should be noted that the units of the rate constant depend on the reaction order. The rate is typically expressed on a volumetric basis, that is, in terms of mole per unit time per unit volume. If the concentrations are expressed in (mol/volume) units, then the units of rate constant are per unit time. On the other hand, if the reaction is zero-order, then the units of rate constant are the same as the units of the rate; that is, mol per unit time per unit volume.
The power-law model offers a useful and convenient expression to describe the rate-dependence on concentration. However, it should be realized that it is not necessarily an accurate and exact description of the changes occurring at the molecular level. Other complex expressions based on postulated reaction mechanisms can be derived and offer quantitative accuracy. However, these expressions will also involve more parameters that need to be determined experimentally with confidence to achieve the desired accuracy.
Determination of the intrinsic kinetics for a reaction involves experimental determination of reaction orders (with respect to species and overall) and the rate constant (including the frequency factor and activation energy). Determination of the intrinsic rate expression constitutes a fundamentally important topic in chemical engineering kinetics. Designing proper laboratory experiments and performing accurate data analysis are critically important for obtaining reliable estimates of rate parameters that, in turn, form the basis of reactor design.
Once the intrinsic kinetics is determined, it is coupled with the reactor behavior to obtain design equations for the reactor. The different reactor types mentioned in Chapter 3, “Making of a Chemical Engineer,” are described in more detail and quantitative expressions governing their behaviors are presented next.
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