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.