Chemical Processes - Isoconversional Kinetics of Thermally Stimulated Processes - page 207

From Eq. 4.71, one can derive the effective activation energy for nucleation of the
product phase as follows:

d ln wT ( ) 1 2 T

E R =− = + EZ + . (4.72)

D

− 1

( )

2

( )

3

d T

∆ T ∆ T

The temperature dependence of E that follows from Eq. 4.72 is similar to that found
for the solid-solid phase transition from a low-temperature phase II to a high-tem-
perature phase I on heating above the equilibrium temperature T 0 (Fig. 3.50). That
is, at temperatures just above T 0 (i.e., when ∆ T is close to zero), the effective energy
of the formation of the product phase may be quite large. However, as temperature
continues to rise, the E value would decrease toward the activation energy of diffu-
sion ( E D ) of the product molecules as they self-assemble into a crystalline lattice.
It should be stressed that the nucleation model of Jacobs and Tompkins applies
only to the physical aspect of the reaction 4.64. It describes the process of self-as-
sembly of randomly formed individual molecules of B into a nucleus of the phase B.
The model does not account for the chemical aspect of the reaction, i.e., the fact that
the formation of individual molecules of B requires breaking bonds in the reactant
A. Therefore, the application of this model is limited primarily to the induction pe-
riod of thermal decomposition. The presence of a distinct induction period in the in-
tegral kinetic curves measured under isothermal conditions (Fig. 1.5 and Sect. 1.1)
is an indication that the process rate is limited by slow nucleation. In agreement
with the predictions of Eq. 4.72, the nucleation activation energies derived from the
induction periods are sometimes estimated to be significantly larger [ 100 , 101 ] than
the activation energy of further stages of thermal decomposition.
The meaning of the equilibrium temperature T 0 as well as of the effective activa-
tion energies estimated beyond the induction period can be understood when con-
sidering the chemical aspect of thermal decomposition. Many of thermal decom-
positions are endothermic and reversible. Common examples include the thermal
decomposition of carbonates, oxides, hydrates, and sulfates. The general equation
of reversible decomposition can be written as:

k

1

ABC
s

⇔+ ,

(4.73)

s g

k

2

where k 1 and k 2 are the rate constants of the forward and reverse reaction. Because
the activities of pure solid phases are taken to be 1, the equilibrium constant of such
a process is:

(4.74)

K = 0, ,

C

where P 0,C is the equilibrium partial pressure of the gaseous product C. Since the
equilibrium constant depends on temperature in accord with the van't Hoff equa-
tion, Eq. 4.74 can be rewritten as: