Concepts from Chemical Reaction Kinetics - Elements of Environmental Engineering: Thermodynamics and Kinetics - page 208

In this case, the rate of disappearance of A is given by

d [ ]

d t =

A

r =− k 1 [ A ]+ k 2 [ A ] ,

]
d t =

[

d C

k 1 [

] (5.60)

A ,

d [ D ]
d t = k 2 [

A ] .

Solutions for this case are also given in Table 5.1. First, both C and D increase
exponentially with a rate constant (k 1 + k 2 ) , and A decreases exponentially with a

rate constant (k 1 + k 2 ) . Secondly, the ratio of products, [ C ] / [ D ]= k 1 /k 2 at all times

is called the branching ratio .

5.4 ACTIVATION ENERGY
The rates of reactions encountered in nature are very sensitive to temperature. In
general, an increase in temperature of 10 ◦ causes an approximate doubling of the rate
constant. In the nineteenth century, the Swedish chemist Svänte Arrhenius proposed
an empirical equation based on a large number of experimental observations. This is
called the Arrhenius equation

A e − (E a /RT) ,

k = (5.61)

where A iscalledthe pre-exponentialfactor and E a the activationenergy .Theequation
is also written in an alternative form by combining the two terms

e − ( Δ G † /RT) , k ∞ (5.62)

G † is called the Gibbs activation energy . In this form k bears a strong
resemblance to the equilibrium constant K eq , which is a function of the Gibbs free
energy. The integrated form of the Arrhenius equation is
ln k 2
k 1

Δ

where

1
T 2 −

.

E a
R

1
T 1

=− (5.63)

In general, a plot of ln k versus 1 /T will give − E a /R as the slope and ln A as the

intercept.Theactivationenergyisinterpretedastheminimumenergythatthereactants
must possess in order to convert to products. In the gas phase, according to the kinetic
theory of collisions, reactions are said to occur if two molecules have enough energy
when they collide. Although a large number of collisions do occur per second, only
a fraction leads to a chemical reaction. The excess energy during those collisions
that lead to a reaction is equivalent to the activation energy, and hence the term
exp (

E a /RT) is interpreted as the fraction of collisions with large enough energy
to lead to reactions. The pre-exponential factor A is interpreted as the number of

−