## 1 Introduction: Thermodynamics of Hydrogen Evolution Reaction
In theory, the region of underpotential discharge of \(\rm H^{+}\) is confined to the range of potentials between \(-2.1\) V and 0 V.  
In our previous study [15] based on quantitative coulometric and micro-gravimetric analysis, it was demonstrated that in the case of the Pt electrode in the solution of 0.5 M \(\rm H_{2}SO_{4}\), the reversible electrochemical discharge of \(\rm H_{2}O\) molecules proceeds according to the following reactions (Figure 1):  
\[\rm 2H_{2}O+Q_{\rm dd}\rightleftarrows\rm(H_{3}O^{+})_{ad}+OH^{-},E^{0}=0.26 \leavevmode\
obreak\ \leavevmode\
obreak\ V\leavevmode\
obreak\ (SHE) \tag{6}\]\(Q_{ad}\) represents the adsorption charge, which is drawn to the electrode surface to compensate the charge of adsorbed H\({}_{3}\)O\({}^{+}\) or H\({}_{2}\)\({}^{+}\) ions. This charge is approximately equivalent to one electron per one H\({}_{3}\)O\({}^{+}\) or H\({}_{2}\)\({}^{+}\) ion, and it does not participate in the overall faradaic HER, as this term cancels upon summing of Equations (6)-(8). At \(E=0\) V (SHE), steps (6) and (7) become fast, and the overall process, which in the case of aqueous solutions, is written as:  
\[2\text{H}_{2}\text{O}+2e^{-}\rightleftarrows\text{H}_{2}+2\text{OH}^{-} \tag{9}\]  
corresponds to the general HER/HOR Equation (1). Equation (6) represents the process of H\({}_{2}\)O dissociation and adsorption of the H\({}_{3}\)O\({}^{+}\) ion on the electrode surface and is reflected by the first reversible Langmuir-type adsorption peak in voltammograms of the Pt electrode in sulphuric acid solution [15]. Reaction (7) is the reduction of (H\({}_{3}\)O\({}^{+}\))\({}_{ad}\) leading to the formation of the adsorbed (H\({}_{2}\)\({}^{+}\))\({}_{ad}\) ion and corresponds to the second reversible Langmuir-type adsorption peak, whereas Equation (8) represents the actual hydrogen evolution step on the Pt electrode, which determines the value of equilibrium potential \(E^{0}_{\text{2H}^{+}/\text{H}_{2}}=0\) V (SHE) and the magnitude of exchange current i\({}_{0}\). As one can see from Equations (6)-(9), the role of the intermediate in the reversible H\({}_{2}\) evolution process is played by the adsorbed hydrogen molecular ion (H\({}_{2}^{+}\))\({}_{ad}\). The existence of such an ion has been corroborated in [16], as well as by a very high energy cost required for H\({}^{+}\) emission [17]. The binding energy between the H\({}^{+}\) ion and H atom within H\({}_{2}^{+}\) is 255.7 kJ mol\({}^{-1}\)[18], which is tantamount to the energy of adsorption indicated for the H\({}_{\text{ad}}\) atom and is sufficient for Reaction (7), which, in fact, is the reduction of the first H\({}^{+}\) ion, to proceed. Thus, there is no need to invoke H\({}_{ad}\) atoms to explain the mechanism of reversible discharge of H\({}_{3}\)O\({}^{+}\) ions. The energy of adsorption in the case of the (H\({}_{2}^{+}\))\({}_{ad}\) intermediate is just 11.6 kJ mol\({}^{-1}\), as can be derived from the value of the standard potential of Equation (7) according to \(-\Delta G_{\text{ads}}=nFE^{0}\), whereas the \(\Delta G_{\text{ads}}\) of (H\({}_{3}\)O\({}^{+}\))\({}_{ad}\) (Equation (6)) is 25.1 kJ mol\({}^{-1}\)[15]. Such adsorbed intermediates are characteristic of Pt group metals. The maximum coverage of the Pt electrode surface by either (H\({}_{3}\)O\({}^{+}\))\({}_{ad}\) or (H\({}_{2}\)\({}^{+}\))\({}_{ad}\) is just about 1/3 of the monolayer [15]. At \(E^{0}=0\) V (SHE), the equilibrium surface concentration of (H\({}_{2}^{+}\))\({}_{\text{ad}}\) should be proportional to the bulk concentration of H\({}_{3}\)O\({}^{+}\) and hydrogen pressure \(p_{\text{H}_{2}}\), because otherwise, the existence of a reversible hydrogen electrode with \(E=0.000-0.059\text{pH}-0.0295\lg p_{\text{H}_{2}}\) would not be possible [10]. Figure 1: Mechanism of the reversible stepwise water reduction/hydrogen oxidation via the formation of ionic pairs at different potentials on the Pt electrode [15]. This mechanism accounts quantitatively for the charge and mass changes on the electrode’s surface. SHE, standard hydrogen electrode.