Page 64 - The engineering of chemical reactions

48 Reaction Rates, the Batch Reactor, and the Real World
true in any batch reactor where the volume is held constant, but not in a constant-pressure
reaction with gases if the number of moles changes with the reaction.
Batch reactors are usually operated at constant volume because it is easy to construct
a constant-volume closed container (as long as the pressure does not increase enough to
burst the vessel). However, in flow reactors the density frequently changes as the reaction
proceeds, even though the reactor volume is constant, and we need to be able to handle this
situation.
For reactions such as A -+ B and A + B + 2C with ideal gases the density clearly
does not change as the reaction proceeds if P and T remain constant. It is frequently also
a good approximation with reactions among gases with changing numbers of moles if the
reactants are diluted by an inert solvent. Constant density is also a good approximation
for most liquid solutions because the density of a liquid solution usually does not change
much as the reaction proceeds. As noted previously, the concentration of liquid water is
-55 moles/liter, and in almost any aqueous reaction the reactant will be diluted by many
moles of water per mole of reactants or products.
An important situation in which we must be concerned with variable density is with
nonideal gases or in which one of the reactants or products condenses or evaporates. For
example, the hydration of ethylene

CzH4 + Hz0 + C2HSOH
involves gases and liquids at typical temperatures and pressures. These systems can be very
complex to describe because any gases are usually very far from ideal and because they
involve both phase and reaction equilibrium considerations in addition to chemical reaction
rates. We will not consider these complicated situations until Chapter 12.
If the volume V of a batch reactor depends on conversion or time, then the derivations
of all of the previous equations are incorrect. We could find V(CA) and integrate the mass-
balance equation as before, but it is usually more convenient to use a different variable such
as the fractional conversion X. We finally write CA = NA / V and then substitute for NA (X)
and V(X) to find CA(X).
Consider the reaction

A + 3B, r=kC*
in a constant-pressure batch reactor. This is a situation with first-order kinetics but with
3 moles of product formed for every mole of reactant decomposed. We assume that we
start with NAP moles of pure A. If all species are ideal gases at constant pressure at
initial volume V,,, then at completion the volume of the reactor will be 3V,. When the
reaction has proceeded to a conversion X, the number of moles of A and B are given by
the relations

NA = NA,(~ - X)
NB = 3NA,X

c N = NA,(l + 2X)
and the volume V occupied by this number of moles will be

v = V,(l + 2X)