Figure 6.
The proposed reaction pathway
between CH3I and Ag in Ag0-Aerogel.
In the shrinking core model, the reactions only happen on the reacting
surface, which separates the reacted layer and the unreacted core. Once
the Ag sites on one surface are fully consumed, the reactions proceed to
the next surface. As Figure 6 shown, during the adsorption,
CH3I first diffuses through the reacted layer and
reaches the reacting surface (0). On the reacting surface,
CH3I breaks reversibly into two free radicals (1),
CH3* and I*. Two
CH3* radicals bind with each other and
form C2H6 (2); I*radical binds with Ag and forms AgI (3); and
C2H6 diffuses out through the reacted
layer (4). In this process, to satisfy the observed 1.4 reaction order
by Tang et al.23, the reaction order of
CH3I is assumed to be n ; the orders of other
reactions are 1 to each component; and the reason will be discussed in
the following contents.
The formation/consumption rates of the components can be written as,
where k ’s are the reaction rate constant,D* is the diffusivity of
C2H6, and L is the thickness of
the reacted layer. The
stoichiometric coeffecient of C2H6 is
ignored in the equations since it does not impact the results.
Assuming pseudo-equilibrium, the steady state is established immediately
after the reactions proceed to the next surface,
CH3*, I* and
C2H6 may not accumulate on the reacting
surface and the rates, , , and
should be 0. Therefore, the and in Eq. 14 can be rewritten in term of
constants and (the detailed procedure can be found in the
Supplementary Materials ) which is,
where Φ is similar to the Thiele modulus, the ratio of the
reaction rate to the diffusion rate.34
As more than five parameters are included, determining the step-wise
would be impracticable. Considering two limiting cases, when , the
reaction rate would be proportional to; if , would be proportional to.
Since the reaction order would be between n/ 2 and n , the
integer that satisfies the observed 1.4 order is n = 2.
Additionally, Eq. 18 indicates that the temperature change may impact
the reaction rate. Generally, the relationship between k (orD ) and T can be written as Eq. 20,
which implies that the temperature change may impact the overall
reaction rate by changing the magnitude of .
Arrhenius Relationship and Eyring
Equation
The well-known Arrhenius relationship is commonly used in describing the
temperature dependence of reaction rate constant and diffusivity, which
is, 35
and the linear form is,
Where ks0 is the
pre-exponential factor (unit same as ks ),ΔE is the activation energy (kJ/mol) and R is the gas
constant (kJ/mol/K). Reported by Tang et al.23, the
CH3I adsorption on Ag0-Aerogel may be
a 1.4 order shrinking core process, where the calibratedks*((cm/s)∙(mol/cm3)1-n) can be
represented as . Therefore, the Arrhenius equation was applied to the
calibrated ks* and the plot
including the fitting results are shown in Figure 7.