Extent of reaction variable

A stoichiometric table for keeping track of the amounts or flow rates of all species during reaction may be constructed in various ways, but here we illustrate, by means of an example, the use of , the extent of reaction variable. We divide the species into components and noncomponents, as determined by a stoichiometric analysis (Section 5.2.1), and assume experimental information is available for the noncomponents (at least). 

The form of the dependence of / on will be determined by the order of the reaction with respect to A and B. Provided the reaction does indeed follow the overall stoichiometry in the above equation (in this particular case we have ru = vb = -1) then, once we know the initial concentrations of A and B, the concentration of B at any time can be determined uniquely if the concentration of A is known, i.e., these concentrations are not independent, and so the rate can be formally expressed as a function solely of one concentration or extent of reaction variable. 

As the reaction proceeds, the mole numbers and mole fractions of all species, and the total Gibbs energy of the reacting mixture, change. The number of moles of each reacting species in a closed system is not an independent variable (i.e., it cannot take on any value), but is related to the mole numbers of the other species and the initial mole numbers through the reaction stoichiometry. This is most easily taken into account using the molar extent of reaction variable of Chapters 2 and 8... 

Next, we write the mole fractions of both NO2 and N2O4 in terms of a single extent of reaction variable. This is most easily done using the mass balance table ... 

Write the material balances according to stoichiometry in terms of the extent of reaction variable, X. This gives us the following tabulation ... 

One important application of the variable-time integral method is the quantitative analysis of catalysts, which is based on the catalyst s ability to increase the rate of a reaction. As the initial concentration of catalyst is increased, the time needed to reach the desired extent of reaction decreases. For many catalytic systems the relationship between the elapsed time, Af, and the initial concentration of analyte is... 

The influence of reaction variables and catalyst is complex 19,62,83,84). It is difficult to formulate generalities from available data suffice it to note that much can be done to alter the extent of hydrogenolysis in compounds susceptible to this reaction. 

The design equations for a CSTR do not require that the reacting mixture has constant physical properties or that operating conditions such as temperature and pressure be the same for the inlet and outlet environments. It is required, however, that these variables be known. Pressure in a CSTR is usually determined or controlled independently of the extent of reaction. Temperatures can also be set arbitrarily in small, laboratory equipment because of excellent heat transfer at the small scale. It is sometimes possible to predetermine the temperature in industrial-scale reactors for example, if the heat of reaction is small or if the contents are boiling. This chapter considers the case where both Pout and Tout are known. Density and Q ut wiU not be known if they depend on composition. A steady-state material balance gives... 

Colby et al. [35] proposed an interesting experimental approach to measure the static exponents. They noticed that it is hard to accurately measure the chemical extent of reaction, p, and thus eliminated this variable (more precisely the distance from the gel point p — pc ) from the scaling relations. For example combining Eqs. 2-5 and 2-6 yields the following relation between the weight average molecular weight, Mw, and the characteristic radius, Rchar ... 

This equation is valid for all species Ah a fact that is a consequence of the law of definite proportions. The molar extent of reaction is a time-dependent extensive variable that is measured in moles. It is a useful measure of the progress of the reaction because it is not tied to any particular species A. Changes in the mole numbers of two species j and k can be related to one another by eliminating between two expressions that may be derived from equation 1.1.4. 

The variable / depends on the particular species chosen as a reference substance. In general, the initial mole numbers of the reactants do not constitute simple stoichiometric ratios, and the number of moles of product that may be formed is limited by the amount of one of the reactants present in the system. If the extent of reaction is not limited by thermodynamic equilibrium constraints, this limiting reagent is the one that determines the maximum possible value of the extent of reaction ( max). We should refer our fractional conversions to this stoichiometrically limiting reactant if / is to lie between zero and unity. Consequently, the treatment used in subsequent chapters will define fractional conversions in terms of the limiting reactant. 

The chemical composition of many systems can be expressed in terms of a single reaction progress variable. However, a chemical engineer must often consider systems that cannot be adequately described in terms of a single extent of reaction. This chapter is concerned with the development of the mathematical relationships that govern the behavior of such systems. It treats reversible reactions, parallel reactions, and series reactions, first in terms of the mathematical relations that govern the behavior of such systems and then in terms of the techniques that may be used to relate the kinetic parameters of the system to the phenomena observed in the laboratory. 

For reactor design purposes, the distinction between a single reaction and multiple reactions is made in terms of the number of extents of reaction necessary to describe the kinetic behavior of the system, the former requiring only one reaction progress variable. Because the presence of multiple reactions makes it impossible to characterize the product distribution in terms of a unique fraction conversion, we will find it most convenient to work in terms of species concentrations. Division of one rate expression by another will permit us to eliminate the time variable, thus obtaining expressions that are convenient for examining the effect of changes in process variables on the product distribution. 

We require a means to follow the progress of reaction, most commonly with respect to changing composition at fixed values of other parameters, such as T and catalytic activity. The method may involve intermittent removal of a sample for analysis or continuous monitoring of an appropriate variable measuring the extent of reaction, without removal of a sample. The rate itself may or may not be measured directly, depending on the type of reactor used. This may be a nonflow reactor, or a continuous-flow reactor, or one combining both of these characteristics. 

As in Example 3-3, cB is not independent of cA, but is related to it through equation 3.4-5, to which we add the extent of reaction to emphasize that there is only one composition variable ... 

Another stoichiometric variable that may be used is the extent of reaction, , defined by equation 2.3-6 for a simple system. For a complex system involving N species and represented by R chemical equations in the form... 

The procedure developed by Joris and Kalitventzeff (1987) aims to classify the variables and measurements involved in any type of plant model. The system of equations that represents plant operation involves state variables (temperature, pressure, partial molar flowrates of components, extents of reactions), measurements, and link variables (those that relate certain measurements to state variables). This system is made up of material and energy balances, liquid-vapor equilibrium relationships, pressure equality equations, link equations, etc. 

Furthermore, a variable classification strategy based on an output set assignment algorithm and the symbolic manipulation of process constraints is discussed. It manages any set of unmeasured variables and measurements, such as flowrates, compositions, temperatures, pure energy flows, specific enthalpies, and extents of reaction. Although it behaves successfully for any relationship between variables, it is well suited to nonlinear systems, which are the most common in process industries. 

Assume for the moment that there is a single irreversible reaction. Let be a progress variable describing the extent of reaction CKE 1. As Ej increases in infinitesimal increments d , the solution s analytical concentrations are perturbed and the activities of the species in solution change. Thus for each solute species sj, aj is a function of . 

E, progress variable for extent of reaction fraction reacted... 

With practice, though, we can usually eliminate unlikely reaction schemes, and in most cases devise a model that can simulate a reaction over a wide range of reaction variables to the extent that a model is almost certain to be the correct one. 

The success of this approach is illustrated by Fig. 2, which displays results for the reaction of Fig. 1. This plot depicts the linear increase in the amounts of primary products (including their derivatives) as a function of total product concentration, which is also linear in Y, the abscissa of Fig. 1. Experimental data treated in this manner thus indicate the selectivity to primary products formed by a reaction, regardless of the extent of secondary reactions, and meaningful interpretation of the effects of reaction variables becomes possible. 

The studies reported in this paper have focused on more complete elucidation of the nature of the interaction between the hydronium ion and active carbon. Both rate and extent of reaction have been studied as a function of several variables to obtain data which ultimately should contribute to a meaningful interpretation of pH effects on adsorption of organic solutes by active carbon. 

Table 1. List of the various functional groups in the star-molecule with a branched nucleus, the corresponding extents of reaction, and the auxiliary variables denoting the partner of reaction...

Functional group Extent of Reaction Auxiliary variables... 

The preceding section was based on the fundamental equation for G in terms of the extent of reaction, but in order to identify the D natural variables for a one-reaction system at equilibrium, we need to apply the condition for equilibrium ZvjjUi = 0 (equation 3.1-6) that is due to the reaction. That is done by using each independent equilibrium condition to eliminate one chemical potential from equation 2.5-5. This is more easily seen for a simple reaction ... 

Although free energies are related to equilibrium constants, it is concentrations that are either ultimately desired or experimentally determined. There are many ways to perform equilibrium calculations. We will approach such calculations through the extent of reaction, , defined in Eq. (5), because all concentrations can be expressed in terms of this single variable. In addition, use of the extent of the reaction allows us to perform equilibrium calculations in a very systematic manner. At equihbrium, the extent of reaction becomes the extent of reaction... 

---