How to determine the surface reaction rate and equilibrium constant for a reaction between a solid site species and gas species

Consider the following reaction mechanism [1](https://i.stack.imgur.com/NdbqE.jpg):


S1: $\ce{C-H (s) + H (g) <=> C^\star (s) + H2 (g)}$


S2: $\ce{C^\star (s) + H (g) => C-H (s)}$


S3: $\ce{C^\star (s) + C2H2 (g) => C-H (s) + H(g) + 2C(g)}$ Note that 2 C(g) in reality is not a gas but let's assumed it is for now.


S4: $\ce{C^\star (s) + C2H6 (g) <=> C-H (s) + C2H5(g)}$


The pre-exponential factor A in [cm3/(mol.s)][cm3/(mol.s)], n and activation energy E in cal/mol are respectively for each reaction:


S1: A = 4.2E+13, n=0, E=13,000


S2: A = 2.0E+13, n=0, E=0


S3: A = 8.0E+07, n=1.56, E=3,800


S4: A = 2.1E+11, n=0, E=4,400


$\ce{C-H (s)}$ are $\ce{C^\star (s)}$ are the surface concentration on a soot (nucleated carbon) particle. $\ce{C-H(s)}$ is hydrogen occupied carbon and $\ce{C^\star}$ is unoccupied carbon.


The volumetric reactions rates can be determine using the following equation since the gas species are dependent on the availability of site species [2] (I don't have any issues up to this stage):


r = kiCi$\frac{f\_sX\_s}{N\_A}$, i=1 to 4 ...... eq. 1


Where,


r = reaction rate, mol/m3.s


C = molar concentration of gaseous species in mol/m3


fs = ratio of number density of active surface sites on catalyst to that of soot.


Xs = number density of surface sites on soot particles 1/m2.


NA = Avogadro's number 1/mol.


Now this is where I am having problems:


According to [2] it was provided that the number density of Cs-H can be approximated to be 2.3 x 1015 cm-2 based on experimental data. Estimated by considering one site in a
rectangular area with one side being half-width of the benzene ring.


If this number was divided by Avogadro's number we get the site concentration which is in agreement with the site concentration found in other sources (for example [1](https://i.stack.imgur.com/NdbqE.jpg)) equivalent to 3.82 x 10-8 kmol/m2.


The software that I am using deals with the site species using the following equations:


R\*j,r = ApnrYjRj,r ...... eq. 2


Rj,r = Rkin,r(pn - $\frac{R\_{j,r}}{D\_{0,r}}$)N ...... eq. 3


Where,


R\*j,r = rate of particle species depletion kg/s.


Ap = particle surface area m2.


Yj = mass fraction of surface species $j$ in the particle.


nr = effectiveness factor (dimensionless).


Rj,r = rate of particle surface species reaction per unit area kg/m2.s.


Pn = bulk partial pressure of the gas phase species (Pa).


D0,r = diffusion rate coefficient for reaction $r$.


Rkin,r = kinetic rate of reaction $r$ (units vary).


N = apparent order of reaction $r$.


The software does not know the number density or site concentrations of any of the site species and deals with them as mass species. I can only enter mass fractions with values 0 to 1 depending on the fraction of each species present in the particle. However, I know that the particle at t = 0 s is 100 nm in diameter. Therefore, based on the initial diameter and the site concentration of 3.82 x 10-8 kmol/m2 for Cs-H, I can work out the mass fractions for each species at t = 0 s assuming Cs\* is zero initially (explanation why Cs\* is zero is at the end of this post and should not affect anything here). For this reason I defined 3 mass fractions (these represent Yj in eq. 2):


1. Cs-H (the mass species that is occupied by hydrogen) = 0.012 at t = 0 s.
2. Cs\* (the mass species that is unoccupied by hydrogen) = 0 at t= 0 s.
3. Cc (the mass species representing the particle core and does not participate in the reaction) = 0.988 at t = 0 s.


***So now for the main questions,***


***Q1 I am not sure how to use A (pre-exponential factor) with the units mentioned in paper [1](https://i.stack.imgur.com/NdbqE.jpg) to get Rj,r with units of kg/m2.s for each of the reactions S1 to S4.***


Update for Q1: I managed to find some sources mentioning that surface reactions like the above are considered unimolecular which is quite confusing if you have a pre-exponential factor with units of cm3 /mol.s instead of 1/s.
Additionally, I managed to find a paper with the same mechanism for Appel [4] which mentions the following: "The rate coefficient of H combination with surface radical, reaction S1, was lowered from 1 x 1014 cm3/mol.s to
2 x 1013 cm3/mol.s". Does this indicate that the volume here is the volume of the reactor and not the particle itself?


***Q2 Also, are there any issues in using the standard equilibrium constant equation when dealing with reverse of reactions S1 and S4?***


Surface reaction mechanism S1 to S4 shown in the attachment (S5 to S8 are only third body collision reactions and are not part of the surface reaction mechanism).


Here is why Cs\* is assumed zero at t=0s:


Simple answer to is very difficult to measure the number density or site concentration experimentally and it is usually calculated via a steady state assumption using the following [2]:


XCs = $\frac{k\_{1}[H] + k\_{-4}[C2H5]}{k\_{-1}[H2] + k\_{2}[H] + k\_{3}[C2H2] + k\_{4}[C2H6]}$ XCs-H


I will not be using this equation as there no way to incorporate it in the software. I need to allow the reaction rates for each reaction to dictate the mass fractions.


References


[1](https://i.stack.imgur.com/NdbqE.jpg) C. Kou, X. Yuan, Y. Luo, and X. Yuan, “Performance investigation of the solar thermal decomposition of methane reactor considering discrete and deposited carbon particles,” vol. 324, pp. 124401–124401, Sep. 2022, doi: <https://doi.org/10.1016/j.fuel.2022.124401>.


[2] M. Younessi-Sinaki and F. Hamdullahpur, “Numerical investigation on the number of active surface sites of carbon catalysts in the decomposition of methane,” vol. 60, no. 6, pp. 2228–2234, Feb. 2014, doi: <https://doi.org/10.1002/aic.14395>.


[3] ANSYS Fluent Theory Guide, <https://www.afs.enea.it/project/neptunius/docs/fluent/html/th/node136.htm>


[4] J. Appel, H. Bockhorn, and M. Frenklach, “Kinetic modeling of soot formation with detailed chemistry and physics: laminar premixed flames of C2 hydrocarbons,” Combustion and Flame, vol. 121, no. 1–2, pp. 122–136, Apr. 2000, doi: <https://doi.org/10.1016/s0010-2180(99)00135-2>.


[![Surface Reaction Mechanism](https://i.stack.imgur.com/NdbqE.jpg)](https://i.stack.imgur.com/NdbqE.jpg)


