## 2 Reaction Mechanisms and Kinetics of Electrocatalytic Nitrate Reduction
### DFT Calculation as an Advanced Tool for eNO\({}_{3}\)R
#### 2.2.1 Insights into eNO\({}_{3}\)R: Descriptors and Rate Determining Steps (RDS)  
Descriptors are simplified parameters connected to fundamental surface properties that control both adsorption energies and activation energies for reactions. Moreover, a more suitable descriptor is the strength of surface-adsorbate bonds, which can be estimated from experimental data. This estimation facilitates the construction of a Sabatier-type 'volcano' plot, thus helping to identify optimal catalysts by correlating bond strength with catalytic activity [64]. For example, the binding energies of nitrate adsorbed (\(\Delta G_{\text{NO3}}\)) serve as the first descriptor in determining catalytic efficiency, as nitrate adsorption is recognized as the initial step involving a solution-mediated proton transfer without electron transfer, as previously described (This step has been previously identified as RDS) [65]. In addition to *NO\({}_{3}\), the formation of adsorbed *NO\({}_{2}\) and *NO represent additional key descriptors, with *NO\({}_{2}\) identified as the RDS and *NO as the potential-limiting step (PLS) in the process. Weak adsorption of NO\({}_{2}\) and NO can result in their desorption into the solution, thereby negatively impacting the overall efficiency of the reaction. Figure 5A presents the binding energies of *NO\({}_{3}\), *NO\({}_{2}\), and *NO on different transition metals. The weak binding of *NO\({}_{3}\) and *NO\({}_{2}\) on Au results in poor catalytic performance for nitrate reduction. Moreover, most transition metals show a high desorption barrier for *NO except for Ag and Au [66]. To confirm these results, [PERSON] et al. [66] used the volcano framework as illustrated in Figure 5B, based on the Sabatier principle to analyze the catalytic activity of transition metals in the eNO\({}_{3}\)R. They focus on the binding energies of nitrogen which correlate with \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Electrode Materials** & **Free Energy Diagram** & **Rate Determining Steps** & **Refs.** \\ \hline Cobalt manganese spinel nanoparticles (CoMn\({}_{2}\)O\({}_{4}\)) & & The desorption of *NH\({}_{3}\) & [62] \\ \hline Cobalt manganese spinel nanoparticles embedded in multichannel carbon fibers (CoMn\({}_{2}\)O\({}_{4}\)/NC) & & & \\ \hline Iron coordinated with nitrogen on an ordered mesoporous carbon framework (Meso-Fe-N-C) & & N\({}_{2}\) formation pathway & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._the binding energies of *NO adsorbates (\(\Delta\)G\({}_{\text{NO}}\)) and the binding energy of oxygen which correlates with *OH adsorbates (\(\Delta\)G\({}_{\text{OH}}\)). They also identify the PLS as the reaction with the most negative limiting potential, revealing that for metals like Cu, Ni, Rh, Pt, and Pd, the key steps are *NO \(\rightarrow\) *NOH or *NO \(\rightarrow\) H*NO which means that the protonation of *NO or *NO\({}_{3}\) is the slowest and most difficult step in the overall reaction pathway. Furthermore, Cu demonstrates superior catalytic activity despite a significant overpotential and is the most active and selective catalyst for eNO\({}_{3}\)R to ammonia among the other transition metals (It will be useful in the discussion in Section 4 for copper-based electrocataly). They also proposed strategies for enhancing nitrate reduction catalysis including (a) alloying with oxygen-affinitive elements to stabilize intermediates like *NOH and H*NO, though care must be taken to avoid *OH poisoning; (b) tethering ligands that geometrically interact with *NO/H*NO to stabilize the latter without significantly affecting *NO binding; (c) employing promoters to modify adsorption sites and binding energies; and (d) leveraging hydrogen bonding with intermediates such as *NHO/*NOH to achieve stronger stabilization and improved catalytic performance [66]. Another study by [PERSON] et al. [67] predicted the activity of nitrate reduction electrocatalysts under different applied potentials (\(-\)0.2, 0, 0.2, and 0.4 V).