Theory of the Kinetics of Chemical Potentials in Heterogeneous Catalysis

heterogeneous catalysis. We have further shown our method by using the ammonia synthesis as an example. Our method is not only able to predict the range of optimal adsorption energies, in agreement with reported values, but it also is very simple without the need of extensive calculations of reaction barriers and detailed kinetic analyses.

We start with the chemical potential of an ideal gas. The dependence on its partial pressure p at a given temperature T can be expressed as where N i is the number of surface species or free sites, and q i is the partition function.
If one species occupies one site, we will have q 0 consists only of high-frequency vibrational modes of substrate atoms, and hence may be treated to be unity. For the localized adsorption of surface species, there is no translational and rotational modes, and q i (i>0) is constituted only by vibrational modes. 1,2,3 For such a system, the total partition function can be given by 1 and on taking logarithm and using Stirling's approximation we can obtain For surface species i, the chemical potential is Combining Eqs. (S3), (S5) and (S6), the chemical potential of species i can be obtained: Replacing N with surface coverage θ, Eq. (S7) can be further written as: where θ i and θ 0 are the surface coverage of species i and free sites. Defining i.e. the standard chemical potential of surface species i at temperature T, and replacing free site coverage θ 0 by θ * , we will obtain the form of Eq. (1) in the main text. The presence of free site coverage reflects the fact that in Langmuir adsorption model surface species need to accommodate at certain adsorption sites, while in gas or liquid phases there are no such counterparts that gaseous molecules or solutes need to be attached to. Note that the chemical potential of free sites is always equal to zero because its partition function is unity, and it will not appear in the equations below. Similar to gas phase molecules, the temperature dependence of the chemical potential of surface species can be given by: is quite different from that of gaseous molecules containing a large entropy term. In contrast, the entropies of simple adsorbed species (e.g. atoms or small molecule fragments) are usually very small, and often neglected in microkinetic treatment. In this work, we use

Elementary surface processes
In this section, we will employ chemical potentials as key kinetic parameters to express reaction rates and reversibility of several typical elementary surface processes; molecular adsorption, dissociative adsorption, association reaction and their reverse processes.

Molecular adsorption
For the adsorption of gas phase molecule A, A(g) + * A* the forward reaction rate + r can be written according to transition state theory (TST): 4 where ≠ q is the partition function of the transition state (TS) excluding the vibration between the molecule and the surface (superscript ≠ refers to TS), g q is the partition function of the molecule in the gas phase, and * θ is the surface coverage of free sites.
Combining with Eq. (S1), Eq. (S10) can be rearranged as: where o , ≠ μ is the standard chemical potential of the TS, equal to ≠ − q RT ln , and g μ is the chemical potential of the molecule in the gas phase.
Similarly, the reverse reaction rate − r is obtained: where Eq. (S8) has been used, and ad q and ad μ are the partition function and chemical potential of the molecule on the surface, respectively.
According to the De Donder relation, 5 the net reaction rate is: where z is the reversibility of the process. 6 From Eqs. (S11) and (S12), the reversibility is equal to: (S14)

Dissociative adsorption
With respect to the dissociative adsorption of gas molecule AB, AB(g) + 2* A* + B* the forward reaction rate can be given by: Also, the reverse reaction rate is: where A q and B q are the partition functions of A and B on the surface, respectively, and A μ and B μ are the chemical potentials of A and B on the surface, respectively.

For an surface association reaction between adsorbates A and B
A* + B* AB* + * the forward reaction rate can be written as (S18) and the reverse reaction rate is where AB q and AB μ are the partition function and chemical potential of AB on the surface, respectively. Hence, the reversibility is equal to According to the above derivation, we can see the reaction rate of a chemical process on the surfaces can be simply written into a production of three terms: (i) h T k B , close to 10 13 at 500 K; (ii) an exponential term of the difference between the chemical potentials of the TS and the reactants; and (iii) the surface coverage term of free sites n * θ , where n is the number of surface sites bounded with the reactants (Note that an adsorbed species A* counts one site). It is worth noting that the second term refers to the standard chemical potential o , ≠ μ of the TS with no correction of surface coverages, and the chemical potential μ of the reactant which includes the effect of surface coverage (see Eq. (S9)). As can be seen, the dependence of reaction rates on reactant surface coverages is folded into the coverage-dependent chemical potentials.

Quasi-equilibrium
In the simple kinetic model proposed in the main text, if (see Eq. (7)) has been used. This inequality leads to I R μ μ ≈ if realizing that a small exponent, say I R μ μ − being 0.1 eV, will give rise to about 10 of the exponential function at 500 K, indicating that adsorption reaches quasi-equilibrium at steady state.

Free site coverage on good catalysts
Rearranging Eq. (1)   In principle, one can solve Eq. (S24) to obtain the value of o I μ , and then substitute it to Eq. (S26), which can be further substituted in Eq. (S23) to acquire * θ . Here, we will make some assumption to estimate * θ . It is known that the slopes of BEP relations are normally between 0 and 1, varying with reaction types. As yet, the largest slope is found to be ~0.9 for the dissociation of N 2 , NO and CO, TSs being very final state like; the smallest slope is found to be ~0. 3