The Langmuir adsorption isotherm of the over-potentially deposited hydrogen (OPD H) for the cathodic $H_2$ evolution reaction (HER) at the $poly-Au|0.5M;H_2SO_4$ aqueous electrolyte interface has been studied using cyclic voltammetric and ac impedance techniques. The behavior of the phase shift $(0^{circ}leq{-phi}leq90^{circ})$ for the optimum intermediate frequency corresponds well to that of the fractional surface coverage $(1geq{ heta}geq0)$ at the interface. The phase-shift profile $({-phi};vs.;E)$ for the optimum intermediate frequency, i.e., the phase-shift method, can be used as a new method to estimate the Langmuir adsorption isotherm $( heta;vs.;E)$ of the OPD H for the cathodic HER at the interface. At the poly-$Au|0.5M;H_2SO_4$ electrolyte interface, the equilibrium constant (K) and standard free energy $({Delta}G_{ads})$ of the OPD H are $2.3 imes10^{-6};and;32.2;kJ;mol^{-1}$, respectively.