The Langmuir adsorption isotherms of the over-potentially deposited hydrogen (OPD H) fur the cathodic $H_2$ evolution reaction (HER) at the poly-Au and $Rh|0.5M;H_2SO_4$ aqueous electrolyte interfaces have been studied using cyclic voltammetric and ac impedance techniques. The behavior of the phase shift $(0^{circ}{leq}{-phi}{leq}90^{circ})$ for the optimum intermediate frequency corresponds well to that of the fractional surface coverage $(1{geq}{ heta}{geq}0)$ at the interfaces. The phase-shift profile $({-phi};vs.;E)$ for the optimum intermediate frequency, i.e., the phase-shift method, can be used as a new electrochemical method to determine the Langmuir adsorption isotherm $({ heta};vs.;E)$ of the OPD H for the cathodic HER at the interfaces. At the poly-Au|0.5M $H_2SO_4$ aqueous electrolyte interface, the equilibrium constant (K) and the standard free energy $({Delta}G_{ads})$ of the OPD H are $2.3 imes10^{-6}$ and 32.2kJ/mol, respectively. At the poly-Rh|0.5M $H_2SO_4$ aqueous electrolyte interface, K and ${Delta}G_{ads}$ of the OPD H are $4.1 imes10^4;or;1.2 imes10^{-2}$ and 19.3 or 11.0kJ/mol depending on E, respectively. In contrast to the poly-Au electrode interface, the two different Langmuir adsorption isotherms of the OPD H are observed at the poly-Rh electrode interface. The two different Langmuir adsorption isotherms of the OPD H correspond to the two different adsorption sites of the OPD H on the poly-Rh electrode surface.