It is a general notion in interfacial electrochemistry that the stability of adsorbate phases that only contain hydrogen atoms should be independent of the pH value of the electrolyte on the scale of the reversible hydrogen electrode, whereas the stability of adsorbate phases that do not contain any hydrogen should be independent of the pH value on the scale of the standard hydrogen electrode. In this Perspective, it will be argued on the basis of a grand-canonical approach that such a Nernstian behavior can only be reproduced if the free energy of the adsorbate phase is independent of the electrochemical control parameters. In general, this should not be true, so that the Nernstian behavior should be the exception rather than the rule. Still, structural and chemical factors will be discussed that might lead to a Nernstian behavior. This requires an analysis of the electrochemical electrolyte/electrode interface on the atomistic level. At the same time, this analysis also provides a guideline for the validity of grand-canonical simulations using the concept of the computational hydrogen electrode in which the dependence of the energy of adsorbate phases on pH and electrode potential is neglected.