We study the vicinal surface of the restricted solid-on-solid model coupled with the Langmuir adsorbates which we regard as two-dimensional lattice gas without lateral interaction. The effect of the vapor pressure of the adsorbates in the environmental phase is taken into consideration through the chemical potential. We calculate the surface free energy $f$, the adsorption coverage $��$, the step tension $��$, and the step stiffness $\tilde��$ by the transfer matrix method combined with the density-matrix algorithm. Detailed step-density-dependence of $f$ and $��$ is obtained. We draw the roughening transition curve in the plane of the temperature and the chemical potential of adsorbates. We find the multi-reentrant roughening transition accompanying the inverse roughening phenomena. We also find quasi-reentrant behavior in the step tension.

7 pages, 12 figures (png format), RevTeX 3.1, submitted to Phys. Rev. B