Page 92 - The engineering of chemical reactions

76 Reaction Rates, the Batch Reactor, and the Real World

Assuming that we have an irreversible reaction with a single reactant and power-law
kinetics, r = kc:, the concentration in a constant-volume isothermal batch reactor is given
by integrating the expression
dCA
- = -kc;
dt
to obtain

& (Cin+I - (!;;+I) = -kt
forn + 1.
For second-order kinetics, the integrated rate expression is
- ’ - ‘=kt
-
C A CAo
so that a plot of l/CA versus t should give a straight line whose slope is -k. For first-order
kinetics, the appropriate plot is one of In CA versus t, and the slope of this plot is -k. [What
plot will give a straight line for half-order kinetics?]
One thus obtains a family of these isothermal lines from batch-reactor data for a given
CAM for different temperatures, as shown in the graphs of Figure 2-19 for IZ = 1, 2, and i.

Temperature dependence

We expect that the slopes k from this graph should depend on T as
k(T) = k, exp(-E/RT)
Taking the logarithm of both sides, we obtain

In k = In k, - $

so that a plot of In k versus l/T should give a straight line whose slope is -E/R, as shown
in Figure 2-20. If we extrapolate this Arrhenius plot to l/T = 0 (T = co), the value of
k is the preexponential k,. We frequently plot these data on a basis of log,,, for which the
slope is -E/2.303R.
Thus from this procedure we have the simplest method to analyze batch-reaction data
to obtain a rate expression r (CA, T) if the reaction is irreversible with a single reactant and
obeys power-law kinetics with the Arrhenius temperature dependence.

Figure 2-19 Plots of CA versus time, which give straight lines for an nth-order
irreversible reaction. The slopes of these lines give the rate coefficient k.