Physical Processes - Isoconversional Kinetics of Thermally Stimulated Processes - page 67

fits the temperature dependence of the rate of vaporization or sublimation to the Ar-
rhenius equation, the resulting activation energy should provide a fair estimate for
the enthalpy of the process.
As fairly noticed by Price and Hawkins [ 19 ], the accommodation coefficient
in Eq. 3.5 should not be assumed to be unity when the mass loss measurement
is conducted in a flow of a purge gas at ambient pressure as typically is the case
of regular thermogravimetric analysis (TGA) runs. The actual measurements on
methylparaben by Chatterjee et al. [ 20 ] have produced an estimate of ʳ = 5.8 10 −5
that is too low to be meaningful. Some rational insights into the problem have been
provided by Pieterse and Focke [ 21 ], who suggested that in order to be applicable
to the conditions other than vacuum, the Langmuir equation needs to account for
diffusion of the vapor in surrounding gas. The equation derived by Pieterse and
Focke is as follows:

d
d

m
t

PD M

(3.9)

−=

zRT ,

where D is the diffusion coefficient of the vapor compound in the surrounding gas,
and z is the height of the pan occupied by the gas. Comparing Eq. 3.9 with Eq. 3.5
suggests that the value of the coefficient ʳ is:

= D
z

2 π M

RT

(3.10)

γ .

Equation 3.10 affords explaining the excessively small values of ʳ . The order of
magnitude of ʳ is determined primarily by the value of D whose typical order of
magnitude is about 10 −4 − 10 −5 m 2 s −1 . Substitution of the actual values D, z, T, and
M for vaporization of methylparaben yields ʳ = 4.8 10 −5 which is quite close to the
value experimentally found by Chatterjee et al. [ 20 ].
Following the same logic as above, we can use Eq. 3.9 to derive the activation
energy of vaporization or sublimation. The resulting expression is as follows:

dlnd d
d

( α

1

/ t )

=−

=+−

ER ∆ HE RT , (3.11)

D

−

T

where E D is the activation energy of diffusion. For diffusion of gases in gases, the
typical values of E D are quite small. Figure 3.2 demonstrates the Arrhenius plots for
the temperature dependence of the diffusion coefficient of several gases in helium
[ 22 ]. It is seen that the plots have nearly the same slopes. The E D values estimated
from these slopes fall in the range 5-6 kJ mol −1 . Considering that the RT term in
Eq. 3.11 has similar magnitude but its sign is opposite to E D , we can expect these
two terms to cancel each other at least partially. Therefore, we can conclude again
that the activation energy of vaporization or sublimation should generally provide a
reasonable estimate of the process enthalpy.