Extent of reaction per unit volume

For Class II second-order rate expressions of the form of equation 3.1.10, the rate can be expressed in terms of the extent of reaction per unit volume as... 

In this case one obtains an expression for that cannot be manipulated to yield a simple algebraic form. However, if the concentration of one species is known as a function of time, the concentrations of all other species may be determined from the definition of the extent of reaction per unit volume that is,... 

As equation 3.3.48 indicates, the change in X is directly proportional to the extent of reaction per unit volume. Similarly, the change in X between times zero and infinity is given by... 

A somewhat more useful approach is to consider the relations that should exist between the extent of reaction per unit volume at time t and the extent at equilibrium. At equilibrium equation 5.1.19 becomes... 

The generalized physical property approach discussed in Section 3.3.3.2 may be used together with one of the differential or integral methods, which are appropriate for use with reversible reactions. In this case the extent of reaction per unit volume at time t is given in terms of equation 3.3.50 as... 

The analysis is very similar to that employed in proceeding from equation 5.1.12 to equation 5.1.16, but the physical situation is somewhat different. The reaction is first allowed to come to equilibrium with Ae and Be representing the equilibrium concentrations of species A and 5, and the equilibrium extent of reaction per unit volume in a constant volume system. Under these conditions the net rate of reaction is zero. 

Solving for the extent of reaction per unit volume that gives the maximum reaction rate... 

Finally, with Xj — XJ V defined to be the molar extent of reaction per unit volume. Eq. 8.3-6 can be written as... 

If one employs reactants in precisely stoichiometric proportions, the class II and class III rate expressions will reduce to the mathematical form of the class I rate function. Because the mathematical principles employed in deriving the relations between the extent of reaction per unit volume (or the concentrations of the various species) and time are similar to those used in Sections 3.1.1.1 and 3.1.1.2, we list only the most interesting results for third-order reactions. 

The most useful physical properties for use in kinetics studies are those that are an additive function of the contributions of the various constituents, the contribution of each species being a linear function of its concentration. Total pressure, absorbance, optical rotation, and the electrical conductivity of dilute solutions are all properties of this type. In this section we indicate how such physical property measurements may be easily related to the extent of reaction per unit volume. 

From the requirement that the reaction rate must become zero at equilibrium, equation (5.1.4) indicates that the equilibrium extent of reaction per unit volume ( ) is given by... 

Because these coupled reactions take place at constant volume (liquid phase), the evolution of the product distribution with increasing reactor space time can be determined using either three suitable species concentrations or three extents of reaction per unit volume. In Illustration 5.6 we employed an approach based on extents of reaction. To demonstrate an alternative approach in this solution, we have elected to utilize the concentrations of the three alkanolamines [(MEA), (DEA), and (TEA)] as the reaction progress variables. Material balance considerations dictate that the concentrations of the feed species are then given by... 

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