Page 47 - The engineering of chemical reactions

E l e m e n t a r y R e a c t i o n s 3 1

Eb are the energy barriers for forward and reverse reactions, A HR is the heat of the reaction
to be discussed later, and the horizontal scale is called the reaction coordinate, an ill-defined
distance that molecules must travel in converting between reactants and products. Polanyi
and Wigner first showed from statistical mechanics that the rates should be described by
expressions of the form as given in the boxed equation by a Boltzmann factor, exp( - E / R T),
which is the probability of crossing a potential energy barrier between reactant and product
molecules. In fact, it is very rare ever to find reaction-rate coefficients that are not described
with fair accuracy by expressions of this form.
This functional form of k(T) predicts a very strong dependence of reaction rates on
temperature, and this fact is central in describing the complexities of chemical reactions, as
we will see throughout this book.

q: Example 2-l How much does a reaction rate with an activation energy of 15,000 Cal/mole
“js vary when the temperature is increased from 300 to 3 10 K? From 300 to 400 K?

The ratio of the rate of this reaction at 3 10 K to that at 300 K,
klo e-EIRT~ exp[-15,000/(2 x 31O)l = 2 24
-=-
boo e-EIRT2 exp[-15,000/(2 x 300)] ’
(We use the approximation of R = 2 Cal/mole K). Between 300 and 400 K this
ratio is very large,
boo e-EIRT~ expli-15,000/Q x 40011 = 517
-=-=
boo e-EIRTz exp[-15,000/(2 x 300)]
This shows that for this activation energy an increase of temperature by 10 K
approximately doubles the rate and an increase of 100 K increases it by more than
a factor of 500.

This example shows why the temperature is so important in chemical reactions. For
many nonreacting situations a 10 K increase in T is insignificant, but for our example it
would decrease by a factor of two the size of the reactor required for a given conversion. A
decrease in the temperature by 100 K would change the rate so much that it would appear to
be zero, and an increase by this amount would make the rate so fast that the process would
be difficult or impossible to handle.
Let us consider finally the units of k. Basically, we choose units to make the rate (in
moles liter-t time-i) dimensionally correct. For r = kc:, k has units of liter”- 1 molei- n
time-l, which gives k(time-1) for IZ = 1 and k(liters/mole time) for II = 2. The units of k
for some common orders of reactions are shown in Table 2-l.

ELEMENTARY REACTIONS

We emphasize again that rate expressions are basically empirical representations of the
dependence of rates of reaction on concentrations and temperature, r(Cj , T). From the
preceding examples and from intuition one can guess the order of a reaction from its
stoichiometry. The forward rates appear to be proportional to the concentrations of reactant