  In this article, we show that if $X$ is an excellent surface with rational
singularities, the constant sheaf $\mathbb{Q}_{\ell}$ is a dualizing complex.
In coefficient $\mathbb{Z}_{\ell}$, we also prove that the obstruction for
$\mathbb{Z}_{\ell}$ to become a dualizing complex lying on the divisor class
groups at singular points. As applications, we study the perverse sheaves and
the weights of $\ell$-adic cohomology groups on such surfaces.
